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RHEOLOGY FOR POLYMER MELT PROCESSING
RHEOLOGY SERIES
Advisory Editor: K. Waiters FRS, Professor of Applied Mathematics, University of Wales, Aberystwyth, U.K.
Vol. 1 Numerical Simulation of Non-Newtonian Flow (M.J. Crochet, A.R. Davies and K. Waiters)
Vol. 2 Rheology of Materials and Engineering Structures (Z. Sobotka)
Vol. 3 An Introduction to Rheology (H.A. Barnes, J.F. Hutton and K. Waiters)
Vol. 4 Rheological Phenomena in Focus (D.V. Boger and K. Waiters)
Vol. 5 Rheology for Polymer Melt Processing (Edited by J-M. Piau and J-F. Agassant)
RHEOLOGY FOR POLYMER MELT PROCESSING
E d i t e d by
J-M. Piau Laboratoire de Rheologie, Domaine Universitaire, Grenoble, France
and
J-F. Agassant CEMEF Ecole des Mines, Valbonne Cedex, France
1996
E l s e v i e r
A m s t e r d a m - L a u s a n n e - N e w Y o r k - O x f o r d - S h a n n o n - T o k y o
ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands
L i b r a r y oF Congress Cata loging- in -Publ icat ion Data
Rheology fo r polymer melt process|ng / [ed i ted by] J.-M. Piau and d. -F. Agassant.
p. cm. - - (Rheo]ogy ser ies ; vo l . 5) Inc]udes b i b l i o g r a p h i c a l re ferences and index. ISBN 0-444-82236-4 (a l k . paper) 1. Polymers--Rheology. 2. Po]ymer mel t ing. I . Piau. J.-M.
I I . Agassant, J . -F . I I I . Ser les: Rheology ser ies ; 5. TP1150.R49 1996 668.9--dc20 96-31377
CIP
ISBN: 0 444 82236 4
�9 1996 ELSEVIER SCIENCE B.V. All rights reserved.
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Printed in The Netherlands.
P R E F A C E
This book presents the main results obtained by different laboratories involved in
the research group "Rheology for polymer melts processing" and belonging to various
French universities, schools of engineering and to the CNRS (Centre National de la
Recherche Scientifique - France). This research group was created in 1987 and is supported
by the CNRS, the French Ministry for Research and Technology and three major French
companies (ELF-ATOCHEM, MICHELIN and RHONE-POULENC). The group has
comprised up to 15 research laboratories with different skills (chemistry, physics, material
sciences, mechanics, mathematics) but with a common challenge : to make progress in
understanding the relationships between macromolecule species, their rheology and their
processing. The first problem was to find a language common to all the experts in these
different fields in order to promote effective cooperations. This was achieved through
regular technical meetings (a minimum of two meetings every year) as well as open
meetings (in Paris, Sophia Antipolis, ViUard de Lans, Biarritz, Le Mans, Strasbourg, and
Paris again for the final meeting) where recognized international scientists were invited.
Some crucial issues of polymer science have been addressed ; correlation of
viscoelastic macroscopic bulk property measurements and models, slip at the wall,
extrusion defects, correlation between numerical flow simulations and experiments.
Significant research results have been obtained, and have been or will be published in
International Journals or congress proceedings. In addition, every participant has benefited
in his own research activities from being a member of the "Rheology for polymer
processing" research group and is now better prepared for developing further cooperation
as well as for addressing new problems.
The selection of results provided in the present book is unique in the sense that it
allows one to grasp the key issues in polymer rheology and processing at once, through a
series of detailed state of the art contributions. Generally these issues can only be found in
different books. Each paper was reviewed by experts and by the book editors and some
coordination was established in order to achieve a readable and easy access style.
vi
Nevertheless, each author remains responsible for his own contribution. Papers have been
gathered in sections covering successively : "Molecular dynamics", "Constitutive equations
and numerical modelling", "Simple and complex flows". However, each paper can be read
independently.
This book can be considered as an introduction to the main topics in polymer
processing. It is therefore intended to be useful for graduate students as well as for
scientists in academic or industrial research laboratories �9 polymer suppliers now have the
opportunity through the development of new catalysts and new polymer blends to propose
speciality polymers which are well adapted to a specific type of production but this entails
better control of the relationships between macromolecular structure, rheology and
processing. On the other hand, machine or tool makers want to define the best tools and
processing conditions for obtaining maximum throughput free of defects, at a controlled
final temperature, with minimum trial and error. Computer-assisted design is starting to be
commonly used but this calls for strong numerical algorithms as well as realistic
constitutive equations. Finally, polymer converters are often small companies but with a
high level of innovation. They also need appropriate numerical software in order to choose
the right polymer and machine to achieve the best product properties.
We hope that this book will help to enhance the interest of the scientific and
technical community for the fascinating field of rheology and polymer sciences. We are
very proud to be surrounded by he enthusiastic colleagues of the "Rheology for polymer
processing" research group and we thank them for their assistance in the preparation of this
book.
J. M. Piau, J. F. Agassant
~ 1 7 6
VII
C O N T E N T S
P R E F A C E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
I. MOLECULAR DYNAMICS
I. 1 The reptation model : tests through diffusion measurements in linear polymer melts L. l.,6ger, H. Hervet, P. Auroy, E. Boucher, G. Massey I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 The rep ta t ion mode l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Diffusion measurements in po lymer systems ............................... 6 Interpretation and comparison with rheometrical data ..................... 11 C o n c l u s i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2 Polybutadiene : NMR and Temporary elasticity J.P. Cohen Addad I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 T e m p o r a r y ne twork s t ructures .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Segmental motions : dynamic screening effect ............................. 28 Mol ten high polymers �9 semi-local dynamics ............................... 33 C o n c l u s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.3 Chain relaxation processes of uniaxially stretched polymer chains : an infrared dichroism study J.F. Tassin, L. Bokobza, C. Hayes, L. Monnerie I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Theoret ical background in Infrared dichroism .............................. 38 E x p e r i m e n t a l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Theore t ica l basis of interpretat ion ............................................ 41 Results and discussion on isotopically labeled chains ..................... 44 Results and discussion on isotopically labelled 6-arm stars .............. 49 Results and discussion on binary blends of long and short chains ....... 55 C o n c l u s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
1.4 Chain conformation in elongational and shear flow as seen by SANS R. Muller, C. Picot I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 M e t h o d o l o g y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 E l o n g a t i o n a l f low . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 S h e a r f low . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Conc lus ions and perspec t ives ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
~ 1 7 6
V I I I
1.5 Molecular rheology and linear viscoelasticity G. Marrin, J.P. Montfort I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Linear viscoelastic behaviour of linear and flexible chains - basics and
p h e n o m e n o l o g y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 The case of entangled monodisperse linear species : pure reptafion ...... 105 Entangled model -branched po lymers ........................................ 114 Entangled polydisperse linear chains : double reptation ................... 119 Effects of non entangled chains .............................................. 129 P rob lems still pending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
II. CONSTITUTIVE EQUATIONS AND NUMERICAL MODELLING
II. 1 Experimental validation of non linear network models C. Carrot, J. Guillet, P. Revenu, A. Arsac I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Theore t i ca l aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Exper imen ta l aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Experimental validation of the Wagner model ............................... 167 Experimental validation of the Phan Thien-Tanner model ................... 176 C o n c l u s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
II.2 Mathematical analysis of differential models for viscoelastic fluids J. Baranger, C. Guillop6, J.C. Saut In t roduct ion. The models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Maxwell type models : loss of evolution and change of type ............. 201 S teady f lows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 U n s t e a d y f lows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 S tab i l i ty i ssues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Numerical analysis of viscoelast ic f lows .................................... 225 C o n c l u s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
II.3 Computation of 2D viscoelastic flows for a differential constitutive equation Y. Demay I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Computat ion of a purely viscous flow ....................................... 240 Finite elements method for viscoelastic flows .............................. 244 A p p l i c a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 C o n c l u s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
III. SIMPLE AND COMPLEX FLOWS
III. 1 Validity of the stress optical law and application of birefringence to polymer complex flows R. Muller, B. Vergnes I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 General relationships and usefulness of birefringence measurements ... 257 Validi ty of the stress optical law .............................................. 264 Application to complex flow studies ......................................... 277 C o n c l u s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
III .2 Comparison between experimental data and numerical models J. Guillet, C. Carrot, B.S. Kim, J.F. Agassant, B. Vergnes, C. B6raudo, J.R. Clermont, M. Normandin, Y. B6raux I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 M a t e r i a l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 C o n s t i t u t i v e equa t ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 F low g e o m e t r i e s and exper iments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 N u m e r i c a l m o d e l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 Compar ison between numerical results and experiments .................. 317 C o n c l u s i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
111.3 Slip at the wall L. IAger, H. Hervet, G. Massey I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 Local determinat ion of the velocity at the wall .............................. 338 M o l e c u l a r mode l s and discuss ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 C o n c l u s i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
III .4 Slip and friction of polymer melt flows N. E1 Kissi, J.M. Piau I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 M e a n s u s e d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 F low in h igh surface energy dies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 F low in dies with low surface energy ....................................... 372 D i s c u s s i o n and conc lus ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
III.5 Stability phenomena during polymer melt extrusion N. E1 Kissi, J.M. Piau I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 Exper imen ta l facilities and flow curves ..................................... 391 Visua l i za t ion of ups t r eam flow .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Observa t ion of stable flow - sharkskin ...................................... 402 Observation of unstable flow for slightly to moderately entangled
p o l y m e r s - mel t f racture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 Highly en tangled polymers - flow with slip ................................ 413 C o n c l u s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
S U B J E C T I N D E X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
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Rheology for Polymer Melt Processing J-M. Piau and J-F. Agassant (editors) 1996 Elsevier Science B.V.
The reptation model: tests th rough d i f fus ion m e a s u r e m e n t s in l inear polymer
mel ts
L. LEGER, H. HERVET, P. AUROY, E. BOUCHER, G. MASSEY, Laboratoire de Physique de la Mati~re Condens6e, URA CNRS 792, Coll~ge de France, 11 Place Marcelin-Berthelot, 75231 PARIS Cedex 05, FRANCE
1. INTRODUCTION
The wide range of applications of polymer materials relies in part on their unique viscoelastic behaviour. In the melt state, a polymer sample can be strongly deformed (deformation ratio of several hundred of percent) without breakage, and then recover its original shape almost completely if the stress is relaxed after a short enough time. This memory is completely lost after long periods of time, and the same material flows like any ordinary viscous liquid. A number of methods have been developed to investigate this behaviour [ 1 ] and to try to relate the time after which the system behaves like a simple liquid to its molecular characteristics. The Brownian motion, in polymer melts or in sufficiently concentrated solutions of linear flexible macromolecules, also presents unique features, the most well known being a self diffusion coefficient scaling as Mw -2, with Mw the weight average molecular weight of the polymer chains [2], for large molecular weights. Viscoelasticity (response to an external mechanical excitation) and diffusion (response to thermal fluctuations) must have their origin in the same dynamical process at the molecular level, and any model proposed to explain one aspect must also consistently account for the other.
Early in the development of this area it was suggested that the unique dynamics of dense long polymer chains was due to molecular "entanglements" [3], first introduced as a rather loose concept which expresses the strong restrictions that uncrossable, unidimensional objects must exert on the motion of their neighbours. The reptation model, proposed by P.G. de Gennes in 1971 [4], gives a framework to describe the motion of entangled chains, and leads to a universal description of the linear viscoelasticity (description developed by S. Edwards and M. Doi [5, 6]) of long flexible polymer systems. One interesting feature of this model is that it provides laws for the variation of the key physical quantities of the linear
viscoelasticity as for example the zero shear viscosity, 1"10, or the terminal relaxation time, Tt, with the molecular parameters of the system such as the polymerisation index of the chains, N, (related to the molecular weight through M = Nm, with m the monomer molecular weight) and the polymer volume fraction if one is dealing with a solution.
These laws are in very good qualitative agreement with the large number of available experimental data on the dynamic behaviour of linear flexible entangled polymers, but quantitative departures still remain between the experiments and the predictions, as for example the fact that the exponent of the power law which characterises the variation of the zero shear viscosity with the molecular weight is observed to be 3.3 or 3.4 [3, 6] rather than the predicted value 3. These deviations have lead to a long controversy on the validity of the reptation model, and have stimulated a series of experimental and theoretical investigations to try to understand the limitations of this model and to propose the necessary modifications to obtain a better description of the dynamic properties of liquid polymers [7 to 22].
In the present paper, after a rapid presentation of the reptation model in its simplest version, in order to pinpoint the underlying hypothesis, we discuss the interest of complementary self diffusion and viscoelastic measurements, and present the currently available methods for measuring diffusion in entangled polymer systems. Then, results obtained on polydimethylsiloxane (PDMS), a model liquid polymer well above its glass temperaturd at room temperature will be described, and the consequences on the limits of the entangled regime as seen from diffusion measurements, compared to what is observed in rheometry, will be discussed.
2. THE REPTATION MODEL:
2.1. One linear chain among fixed obstacles: The Brownian diffusion of one long polymer chain trapped among fixed obstacles has
been described quantitatively by P.G. de Gennes in 1971 [4]. As schematically presented in Fig. l a, the chain cannot cross the obstacles. At any time, its shape depends on the actual monomers-obstacles interactions: the chain is confined by the obstacles. The only way for the chain to change its configuration is to find a path among the obstacles through small fluctuations of its contour, folding or unfolding locally. The leading motion takes place at the
�9 �9 �9 �9 �9 �9 �9 �9 �9
�9 �9 �9 �9 �9 �9 �9 �9 �9 " " 0 ~ " ~ " �9 " " ~ ' ~ ' L ' - I'- " �9
�9 .
�9 . . . . . - .
a_ _b
t \
Figure 1" Schematic representation of one chain among obstacles, a) The chain is constrained by the obstacles, b) By local fluctuations, the chain changes its conformation. The probability of forming a loop (dashed line) is very small (strong entropy loss), and the role of the extemities is dominant, c) The chain reptates like a snake in the virtual tube (thin line) envelope of all the topological constraints exerted on it by the obstacles. The tube is progressively redefined from the extremities, as schematically presented by the two situations at time t and t', with t'>t.
extremities of the chain, as a consequence of the low probability for the chain of forming a long loop at any other central part (figure tb), the formation of a large loop implying a large entropy loss (the two sides of the loop have to follow the same path among the obstacles). time. As a result, the extremities can engage between new obstacles and progressively pull the chain in a new environment (figure l c). This is the reptation process. As a consequence, the central part of the chain remains trapped by the same topological constraints over a long period of A way of taking these constraints into account in an averaged manner was proposed by S. Edwards [23]: one can assume that the chain is trapped in a virtual tube, or envelope made of all the obstacles which directly surround it (thin line on figure l c). The chain can move freely along the curvilinear axis of the tube, without encountering obstacles, while it cannot escape out of the tube laterally. At any time, by fluctuations of its local kinks, the chain leaves some parts of the tube, and creates new parts. The detailed statistical description of the process [4] leads to definite predictions for the molecular weight dependence of the diffusion coefficient, that can be reconstructed using simple arguments that we present now.
The chain is assumed to be an ideal Gaussian chain made of N monomers of size a (that is indeed the case in a polymer melt [24]) and its average end to end vector has a length R0 = Nl/2a. For a free ideal chain, the dynamics has been modelled by Rouse [25], assuming that the chain is similar to a collection of springs, connected by beads with the size of a monomer, where all the friction is concentrated. This leads to a longest relaxation time of the chain proportional to N 2, and to a translational diffusion coefficient of the centre of mass inversely proportional to N. In the presence of the obstacles, the local fluctuations of the chain contour which involve distances smaller than the diameter of the tube, d, are not affected by the tube. The chain can thus be considered as a necklace of beads of size d, the average distance between the obstacles. The corresponding average number of monomers inside one bead is Nd, with d = Ndl/2a. The portion of chain inside a bead obeys Rouse-like dynamics, because the monomers inside the bead are not affected by the tube. The longest relaxation time of one bead is
"eR(d) = ZlNd 2 , (1)
and the corresponding diffusion coefficient is
D d -- ~,D1 (2) Nd
where ~1 and D1 are the monomer characteristic time and diffusion coefficient respectively.
"el, D! and "eR, DR are related by general diffusion laws"
"elDl -- a 2,
"eR (d)Dd = d 2. (3)
The mobility of the whole chain, free to move along the curvilinear axis of the tube is N/Nd smaller than the mobility of one bead, as the friction on the full necklace is the friction on one bead times the number of beads. The diffusion coefficient of the chain along the tube is
D t ~ ~DI (4) N
The reptation time, TR, or the time it takes for the chain to renew its configuration is related to Dt through a relation analogous to eq.3: DtTR = Lt 2, with Lt = d(N/Nd) the total length of the tube. Thus
( % ) 3 NfifN = %1 �9 TR "t:d (d) d d (5)
TR is the longest relaxation time of the chain constrained by the obstacles. It is much
longer than the longest Rouse time of the free chain, z R (N) --- xlN 2. It should be noticed that the measurable self diffusion coefficient Ds of the chain is not Dr: the tube is contorted, with a Gaussian configuration, and when the chain travels a distance Lt along the tube it only travels
a distance R0 in a given direction of the real space, so that DsT R -- Ro 2 = Na 2, and
D s -- D 1 N d
N 2" (6)
The diffusion is much slower (by a factor N) than for a Rouse-type free chain.
2.2. Polymer melts: It is tempting to apply the ideas developed in section 2.1. to describe the dynamics of
long linear polymer molecules in the melt state. The average radius of the chain is R0 -- N 1/2a [24]. The volume spanned by one chain, R03, is much larger, for long chains, than the volume
effectively filled by the monomers of that chain, l) -- Na 3. In the melt state where the monomers are closely packed, the chains are thus interpenetrated. Since they cannot cross each other, they are strongly constrained, in a way somewhat similar to the situation of one chain among the fixed obstacles of section 2.1. One can again define a tube, envelope of all the topological constraints exerted on one chain by its surrounding neighbours. But one is now faced with two major problems. 1) All the chains in the system move, and the obstacles are not permanent. In order to apply reptation ideas to a polymer melt, it is necessary to assume or to establish that the evolution of the tube due to the motions of all the surrounding chains, is slower than the reptation. 2) The tube diameter is no longer an external parameter, it represents the average distance perpendicular to the local chain direction that one monomer can travel, due to the local chain flexibility, before being blocked by the surrounding chains. A complete determination would require a description of the actual monomer-monomer interactions and is still out of reach, despite strong efforts to do so [19, 26 to 28], and at present it is not fully understood what factors determine the tube diameter in polymer melts. It has been introduced as a phenomenological parameter, through an average number of monomers necessary to get an entanglement, Ne, with d -- Nel/2a. This is in fact the definition of an entanglement: it is not a knot between two chains, but an ensemble of constraints which the chains collectivelly exert on each other, so that the diffusive motions of the monomers are no longer isotropic, when the explored distance is large enough.
Then, both the longest relaxation time TR and the self diffusion coefficient can be estimated through eq. 5 and 6 respectively, replacing Nd by Ne. One gets:
N~N (7) TR ='1:1 e'
D s = D ! Ne / / / 2 . (8)
If chains with a polymerisation index smaller than Ne are used, they are no longer efficiently
constrained by their neighbours, and the reptation picture no longer holds. A Rouse-like dynamics should be recovered, with a self diffusion coefficient proportional to 1/N.
When reptation is used to develop a description of the linear viscoelasticity of polymer melts [5, 6], the same underlying hypothesis ismade, and the same phenomenological parameter Ne appears. Basically, to describe the relaxation after a step strain, for example, each chain is assumed to first reorganise inside its deformed tube, with a Rouse-like dynamics, and then to slowly return to isotropy, relaxing the deformed tube by reptation (see the paper by Montfort et al in this book). Along these lines, the plateau relaxation modulus, the steady state compliance and the zero shear viscosity should be respectively:
G N 0 = kT (9) Nea3 '
6 J 0 =
5GN ~ (10) ~2
no = ~ G N O T R �9 (11)
Of course, all these relations are expected to be valid only if the reptation description holds, i.e. if the motion of the tube due to the dynamics of all the surrounding chains is much slower than the reptation of one chain.
Eq. I 1 provides an easy way of checking the validity of the reptation model: the zero shear viscosity should depend on the polymerisation index of the chains like N 3. Experimentally, the observed exponent is larger, 3.3 to 3.5, except perhaps when extremely large molecular weights are used [29]. The reason for the discrepancy has been debated for many years and remains not totally elucidated. It is indeed a puzzling question: the dependences of the storage modulus versus frequency, for various molecular weights have been observed to agree well with the Doi-Edwards predictions [30], and the molecular weight dependence of the self diffusion coefficient has often been observed to agree well with the reptation prediction [2, 31 to 35]. Moreover,. more local tests of the reptation dynamics, as for example determinations of the monomer concentration profiles in a macroscopic diffusion experiment starting with a step-like labelled chains profile, through neutrons techniques, appear to agree very well with the reptation picture [34, 36, 37].
The question of the detailed limits of validity of the reptation model thus remains a pending question. What appears puzzling is the fact that, on one hand, the reptation model and the Doi - Edwards' description of the linear viscoelasticity work so well both qualitatively and quantitatively for some experiments, while, on the other hand, they seem unable to account for all the existing data. This may suggest that the reptation model does not contain the whole story of linear polymer dynamics, and that one needs to learn more on other possibly competing processes.
The obviously weak points in the use of reptation ideas to describe the dynamics of linear polymer melts are the two hypotheses mentioned above. Their validity is not easy to check experimentally and it is also not easy to understand how to relate Ne to the molecular properties of the polymer. In fact, it is not simple to determine Ne in a reliable manner: prefactors not taken into account in the simple arguments leading to eq. 7 to 11 may exist and the question is not a trivial one. Severalroutes are possible: one relies on measurements of GN 0. A strong effort has been made recently by L.J. Fetters and coworkers [38] to relate the value of the critical molecular weight between entanglements, as deduced from the plateau modulus, to the molecular structure of the polymer. The correlations they obtain are remarkable and should allow one to predict how one can imagine to define a given chemistry in order to reach a given rheology. Such correlations rely on measurements performed on high molecular weight polymers, well above Ne, i.e. highly entangled. Another possible route
consists in following the well-admitted idea that a way of estimating Ne is to locate the crossover between entangled and non-entangled regimes by looking for the appearance of a plateau modulus when the molecular weight is progressively increased, or for a change of slope in a log-log plot of the zero shear viscosity versus the molecular weight. It is not clear however that this is the best way to do so: decreasing the molecular weight accelerates the dynamics of all the chains in the media, and it may well be that the crossover region could not be described, even qualitatively by reptation ideas. When decreasing the molecular weight one can eventually enter into a regime in which the chains are still entangled i.e. dynamically constrainted by each other, but in which the reptation hypothesis is no longer valid, due to the onset of collective motions of the chains. The reptation picture is a one chain picture, the topological constraints exerted on one chain by its neighbours being taken into account in an average way, through the tube notion. Collective effects are not neglected in this framework, and if they become the dominant dynamical process, the reptation model no longer holds.
To try to characterise experimentally the importance of the collective effects on the dynamics of one chain, diffusion measurements reveal to be a unique tool. Mixtures of long and short chains can be used, along with labelling techniques which permits one to follow the diffusion of either the short or the long chains. Thus, diffusion measurements allow one to decouple the question of the dynamics of the surrounding chains and the question of the cross over between entangled and non-entangled behaviour. When few short chains (index of polymerisation N) are immersed into much larger chains (index of polymerisation P), one can expect that for large enough P, the motions of the surrounding chains will be frozen down on the time scale of the motion of the test chains. These labelling techniques should enable one to characterise the crossover region between entangled and disentangled behaviour, by measuring the diffusion coefficient as a function of N, in situations where the motions of the environment remain slow (P>>N). Then, the importance on the dynamics of the collective motions can be characterised, varying the ratio P/N, and comparing, at fixed N, the data for N = P and N << P. Similar experimental tests are not easy to obtain from rheometrical measurements, even if such experiments have indeed been conducted [15, 18], because rheometry is sensitive to all the chains present in the sample, and one loses the selectivity associated with the labelling. The use of diffusion measurements to help understand the dynamical behaviour of linear polymers has been recognised for many years [ 10,11,12,31 to 35,39]. Systematic experiments have already been performed, varying the relative molecular weights of the labelled and unlabelled chains, but, when approaching the non-entangled regime, one has to be carreful to avoid spurious effects associated with a possible additional molecular weight dependence in the dynamics, associated with a variation of the free volume, and of the local friction coefficient (hidden in the prefactor Dl of equation 8) with the molecular weight.
We present here the results of such a systematic investigation on the dependence of the self-diffusion coefficient of flexible polymer chains as a function of P and N, conducted on polydimethylsiloxane (PDMS). This model polymer is well above its glass temperature at room temperature (To_ = -120~ so that one can expect that spurious effects associated with the variation of the flee volume and of the local monomer-monomer friction coefficient with the molecular weights of the chains are minimised.
3. DIFFUSION M E A S U R E M E N T S IN P O L Y M E R SYSTEMS:
3.1. Materials and techniques: The technique we have choosen to measure the diffusion coefficient of polymer chains
is based on fluorescence recovery after photobleaching, using interference fringes. The principles of the technique and the details of the experimental set up have already been described [40]. This technique requires that the chains under investigation are labelled with a photobleachable fluorescent probe. A strong advantage of the fluorescence recovery after photobleaching technique, compared to other optical techniques of self-diffusion
measurements, such as, for example, the Forced Rayleigh light scattering technique [ 10, 32], is the irreversibility of the bleaching reaction. This is particularly important in polymer melts with high molecular weights, where extremly slow diffusion is expected. We have used
narrow molecular weight fractions of c~-co OH terminated PDMS (obtained by conventional fractionated precipitation techniques from Rh6ne-Poulenc or Petrarch commercial products or by anionic polymerisation when one wanted to get chains with only one terminal OH). The labelling is obtained by first modifying the OH extremities of the chains by reaction with an aminosilane, CH3-O-Si-(CH2)n-NH2, with n = 3 or 4, in excess in toluene at 100~ for 24 hours. After several cycles of precipitation dissolution, the polymer is left to react with an excess of chloro-7-nitrobenzo-2-oxa-1,3 diazole (C1-NBD), in toluene, at room temperature during 16 hours, in the dark. After separation of the polymer from the unreacted NBD, one obtains PDMS chains end labelled with NBD, with a yield which can reach 90% . Then, if
one starts with an cz-co OH terminated PDMS, most of the labelled chains bear two fluorescent probes, one at each extremity. Care has to be taken to conduct all the steps of the labelling in anhydrous conditions to avoid backbiting by the unsubstituted amine. The labelled polymer has a maximun absorption wavelength of 468 nm, and a maximum emission wavelength arround 510 nm. The analysed samples are all made of mixtures of labelled and unlabelled polymer, the characteristics of the couples used are reported in table 1. The labelled chain concentration has always been kept smaller than the first overlap concentration for the correponding molecular weight, so that these chains are independent of each other. In fact, typical label concentrations were always smaller than l ppm. We have checked that under such conditions the measured diffusion coefficient was independent of the concentration of labelled chains, whatever the molecular weights of the labelled and unlabelled chains. We have also checked that the label was not affecting the measured values of the diffusion coefficient, comparing our results for N = P with self-diffusion measurements performed on the same samples by pulsed field gradient NMR [41 ]. This
Table I (Couples of labelled and unlabelled chains used in this study).
average molecular weight of labeled
PDMS chains (g/mole)
3700 3700 6600 6600 14000 14000 22500 99500 41500 41500 68O0O 68000 106000 106000 320700 320700
polydispersity N index
50 1.1 50 1.1 89 1.1 89 1.1 189 1.1 189 1.1 304 1.13 304 1.13 561 1.07 561 1.07 919 1.11 919 1.11 1432 1.03 1432 1.03 4334 1.18 4334 1.18
average molecular weight of
unlabeled PDMS chains (g/mole)
962000 3700
962000 6400
962000 14000
787000 24400
962000 41500
962000 68000
962000 106000 787000 320700
13000 50
13000 86
13000 189
10635 330
13000 561
13000 919
13000 1432
10635 4334
polydispersity index
1.27 1.1
1.27 1.1
1.27 1.1
1.22 1.09 1.27 1.07 1.27 1.11 1.27 1.03 1.22 1.18
technique does not require labeling, since it is provided by the local Larmor frequency of the spins during the field impulsion. As a consequence, pulse field gradient NMR cannot give selective informations on the N chains immersed in P chains: all the spins rotate under the effect of the magnetic field, with a frequency which only depends on the actual field value they feel, whatever the chain to which they pertain. In order to measure the diffusion coefficient of the chains, one first shines on the sample a brief and intense pulse of light
(typical duration % = 100 ms and typical power 150 mW), with a fringe pattern. The fringes are obtained by recombining, in the sample, two beams coming from the same laser. Under the action of this intense pulse of light, a fraction of the fluorescent probes located in the bright fringes are photobleached. Using the same fringe pattern, but strongly attenuated in order to avoid further photobleaching, the fluorescence of the sample is excited and monitored. The position of the interference fringes in the sample can be modulated at a frequency F, by modulating the position of one of the mirrors used to recombine the two beams with respect to the other. If the modulation of the position of the fringes is symmetrical with respect to the bleaching fringes position and has an amplitude equal to one half of the fringe spacing, the fluorescence intensity is modulated, at the frequency 2F. The amplitude of this 2F component is proportional to the amplitude of the spatial modulation of the concentration in fluorescent probes resulting from the photobleaching. When this concentration modulation relaxes to zero due to the diffusion of the fluorescent chains, the 2F component of the fluorescence intensity relaxes to zero exponentially [40]. In Fig. 2a, such a typical relaxation is reported, along with its best fit to a single exponential decay. In Fig. 2b, the corresponding inverse of the relaxation times are reported as a function of Q2, with
Q = 27z/i, with i the interfringe spacing. The linear dependence is characteristic of a diffusion process, and allows the extraction of the diffusion coefficient D (slope of the straight line). The typical relative uncertainty in the diffusion coefficients is 3%, for one set of experiments, and of 10% when one takes into account the reproducibility from sample to sample.
~2oo
~ 800
400
o 0
0 : :3
1 a_
V l , w -
, o V , , , , 0 20 40 60
t(s) Q2 (cm-2)
Ib i i ' i i
0.4
~ 0.2
0 4 10 6 8 10 6 1.2 10 7'
Figure 2: a) typical experimental decay of the 2F component of the signal (full line) and the corresponding fitted exponential curve (dashed line), b) plot of the inverse of the relaxation time of the exponential decay versus the square of the wave vector, the linear fit indicates that the process is diffusive and the slope of the line is equal to the diffusion coefficient.
When the labelled and the unlabelled chains have the same length, as the influence of the label on the dynamics appears to be negligible, (identity of the diffusion coefficients measured by the photobleaching technique and by pulse field gradient NMR), the experiment gives access to the self-diffusion coefficient of the chains. When different molecular weights are used for the labelled and unlabelled species, what is measured is a mutual diffusion coefficient. As we shall compare the values of the diffusion coefficients obtained in these two situations, in all the following we shall omit to specify the subscript self or mutual, and speak in terms of diffusion coefficient, D, making clear in each particular situation which sample is under consideration.
3 . 2 . R e s u l t s -
E v
n
1 0 - 1 1 -
10 -~3 .-
10 -is
10 ~
' ' . . . . . . I ' ' . . . . . ' 1 ' ' ' ' ' ' " -
o o N=P
�9 N<<F ~ x n c o re f51 121 O
[] o i
DO " N , -
Ne
o 13
o -
N -2 ' ~ o
o
o
n n i n n I , , , , , , , , I , , , , , , , , I ,
10 z 10 3 10 4
N
Figure 3: Diffusion coefficient versus the polymerisation index of the labelled chains, N, in the case N<<P (filled circles, fixed matrix) and N=P (open squares) at T = 23~ When the matrix is frozen, the power law D - N -2 p0, typical of the reptation process, is observed down to very low values of N, leading to an evaluation of the minimum number of monomers to create an entanglement for PDMS: Ne = 100. For comparison, data from reference 51, with P - N and T = 60~ are reported as open circles.
10
In Fig. 3, all the diffusion coefficients obtained at a temperature of 23~ are reported as a function of the polymerisation index of the labelled chains N, in log scales, for two situations: a) the labelled chains have the same polymerisation index as the surrounding chains (N = P, open squares) and b) P >> N (P = 1.2 104 , filled circles). Several remarkable points appear on this figure"
1) For N > 1000, all the data are well represented by a power law D = N -2 p0, compatible with the picture of labelled chains moving by a reptation process, in a frozen environment. This is quite similar to what has already been established for polystyrene [ 11, 39], and we have not developed strong efforts to get a larger number of high molecular weights labelled samples to fully re-establish this point on PDMS, preferring to concentrate on the far less well-known range of the low molecular weights.
2) For 100 < P < 1000, the measured diffusion coefficients for N = P no longer follow the N -2 reptation prediction. In the same range of N values, D remains proportional to N -2 if P >> N, i.e. if the motion of the chains surrounding the test chain are frozen down during the diffusion time of the test chain. The comparison of the data obtained with N = P and with N << P clearly puts into evidence the acceleration of the dynamics associated with the matrix chains, similarly to what has yet been observed with other polymers [ 11, 12, 42 to 44] or in solutions [ 10]. This acceleration, by a factor close to three, can be attributed to the constraint release mechanism [7, 8, 13], the effects of fluctuations of the test chain inside its tube [9] being a priori the same in the two situations P = N and P >> N.
3) It is interesting to notice that, when P >> N, the data follow the law D = N -2 even for quite small values of N for which it is usually admitted that the chains are no longer entangled. For PDMS, the critical molecular weight between entanglements deduced from the cross over of the zero shear viscosity is in the range 20 000 to 30 000 [45, 46], and that deduced from the plateau modulus is 13 000 [46], i.e. to a Ne value of 175, larger than the smallest value for which we observe that the N -2 law starts to be obeyed. If we take as a criteria for entangled behaviour a diffusive motion in a frozen environment obeying the N -2 dependence (reptation behaviour), we can estimate that the average number of monomers between entanglements is Ne = 100. But 10 entanglements per chain are needed for the appearance of the reptation behaviour when P = N, i.e. when collective effects are present.
For comparison, we have also reported in Fig. 3 the data obtained by Appel and Fleischer [51] with PDMS at 60~ using the pulsed field gradient NMR technique to measure the self-diffusion coefficient. Except for the slightly higher values due to the increased temperature, it is remarkable that exactly the same trends are observed for the two sets of data corresponding to N = P.
4) For N < 100, a departure from the N -2 law is observed even in the case P >> N, and indicates a cross over towards another dynamical regime. The two data points we have in this low molecular weight domain seem compatible with the expected N -I Rouse-like behaviour. However, we have not been able to fully establish this molecular weight dependence, due to the difficulty of synthesising controlled samples in this very small molecular weight range (for low molecular weights, the sensitivity of the fractionation by precipitation drops down, and these samples have to be anionically synthesised, but even doing so, for very small molecular weights, the initiation time of the reaction becomes dominant and does not allow one to get reasonably monodisperse samples).
5) An important question is to decide how far one can believe that a self-diffusion coefficient varying like N -2 is characteristic of reptation. It has been argued that additional molecular weight dependences could exist and compensate for departures from the N -2 law [48 to 52]. Such an effect can come from the local monomer-monomer friction coefficient which appears as a prefactor in equation 8, hidden in the diffusion coefficient DI. Several processes can combine and lead to a local friction which is molecular weight dependent, and which decreases when the polymer molecular weight is decreased. This is, for example, the
case of the glass transition temperature which is expected to depend strongly on molecular weight for Mw smaller than 10 000, a fact which should affect the local free volume and the local friction and thus give an acceleration of all the dynamics at small molecular weights [48,49]. In order to estimate the importance of such local friction effects in the range of molecular weights investigated, we have conducted diffusion measurements of a small probe, CH3-CH2-CH2-NBD, (twenty times smaller than the smaller molecular weights investigated) mixed in PDMS melts, as a function of the PDMS molecular weight. The results are reported in Fig. 4: there is no drastic variation of the local friction when the molecular weight of the PDMS melt is decreased down to 4200 g/mole, which means that our data can be compared with simple reptation predictions, and that the points made in 3) are indeed meaningful.
4 1 0 .6
Od
E 0 - 6
v 2 1 0 a
0 10 ~
I I I I l i l l i I I I I I I I I I I I I I I I I
{ {{ t i t ! i i l t l I I I I I I I I J I t i t t t t t
10 2 10 3 10 4
N
Figure 4: Self diffusion coefficient of the small probe, CH3-CH2-CH2-NBD, versus the PDMS polymerization index (now only unlabelled PDMS is used). Given the low accuracy of the experiment at such high values of the diffusion coefficient (relative uncertainty larger than 10%, contrary to the data on figure 3, which are all slower by more than one decade and thus far easier to measure accurately), it seems that there is no significant change of the friction factor in the range of molecular weights investigated.
4. I N T E R P R E T A T I O N AND COMPARISON WITH R H E O M E T R I C A L DATA:
Both the questions of the transition from Rouse to reptation dynamics and of what fixes the average distance between entanglements in polymer liquids has been the subject of a number of recent theoretical and experimental investigations.
From the theoretical point of view, the first refinement of the reptation approach has been to introduce the collective dynamics of the chains in terms of the constraint release process[7, 8, 13]. Due to the motions of the surrounding chains, some constraints which constitute the tube may disappear during one reptation time, and thus give more freedom to the test chain. Quantitative attempts have been made to take into account these additional
12
degrees of freedom, in regimes where the constraint release process is only a weak correction to reptation, i.e. when the motions of all the chains in the system can be described by reptation. Then, two independent dynamical processes have to be considered: reptation inside the tube, and modification of the tube due to constraints release. To release one constraint, an extremity of one surrounding chain has to leave the vicinity, within one tube diameter, of the test chain. The probability of such an event is 1/TR(P), (the surrounding chains move by reptation) and the tube can be considered as a Rouse chain made of N/Ne subunits, characterised by a subunit jump frequency 1/TR(P). The corresponding constraint release time is
IN) 2 Tc~ = TR ~ee " (12)
The characteristic time of motion is Teff, with, for N = P:
T2ff= lZR , Tren = ~ R 1+ (13)
(the inverse of the times add for independent processes). Constraint release clearly appears to be a weak correction as soon as N is much larger than Ne, and the molecular weight dependences contained in eq. 12 and 13 lead to a weak acceleration of the diffusion for N larger than Ne, while pure reptation is recovered for N >> Ne, in good agreement with experiments. However, for N close to Ne the above description certainly no longer holds: all the chains in the system move by both reptation and constraint release, and their motions are accelerated compared to pure reptation, due to the additionnal degrees of freedom of tube renewal. The constraint release process can no longer be treated as a small perturbation to reptation. Attempts have been made to describe the motion in a self consistent way [ 10], but are not really satisfactory: they lead to an unphysical divergence of the characteristic time of the motions of the chains and are unable to correctly describe the crossover towards the non- entangled dynamical regime, because they do not introduce the Rouse dynamics as another competting process. This question has been addressed recently by D. Pearson et al [49], with an extensive investigation of both the self-diffusion and the zero shear viscosity as a function of molecular weight in polyethylene samples of low polydispersity. The technique used to measure the self-diffusion coefficient, pulse field gradient NMR does not allow for measurements at fixed matrix, and the data of reference 49 have to be compared with our data for P = N. Trends very similar to what we have observed in PDMS are clearly seen in polyethylene, i.e. a regime at large molecular weights well described by simple reptation arguments (Ds --- N -2) and an acceleration of the diffusion at lower molecular weights, associated with additional degrees of freedom such as tube renewal or possibly fluctuations of the test chains inside their tube. The important point made by Pearson et al is that the crossover between the Rouse-like regime
and the entangled one is most clearly evidenced if one reports the product riDs, which is independent of the local friction, as a function of the molecular weight: in the Rouse dynamical regime this product is expected to be independent of molecular weight, while it should increase linearly with Mw, in the pure reptation regime. In order to perform the same kind of analysis of our data, we have measured the zero shear viscosity of our PDMS sample. These measurements have been kindly performed for us by R. Muller from I.C.S. Strasbourg, and are reported, along with low molecular weights data from reference 45 in Fig. 5.
In Fig. 6, we have reported the product riD, for PDMS, as a function of the polymer molecular weight. One has to notice that for the viscosity measurements one is always in a
13
situation with N = P, and thus the filled symbols cannot be interpreted as corresponding to chains moving in a frozen environment. In a way very similar to what has been obtained in polyethylene, we observe a low molecular weight regime which could be Rouse-like, and a transition towards an entangled regime for both sets of data.
1 0 4 ....
10 3 _-..-
10 2 ...-
r~O
1 1 0 _--- =
13_ v
10 0 _-.-
1 0 -1 .._
10-2 L
10 ~
I I I I I I I 1 ~ I I I I I I I I i
f I I I I I IIIII
I I I I I I I I -
Q
. ,
I I I I I I I I I I I I lllli
1 0 2 10 3 10 4
N
Figure 5: Zero shear viscosity as a function of the polymerisation index of the chains for PDMS at T = 27~ The data up to N = 400 are from reference 45, while the larger molecular weights have been measured at I.C.S. Strasbourg by R. Muller and give an exponent 3.37 for the viscosity - molecular weight law.
One can try to locate a critical polymerisation index above which the data are no longer compatible with a Rouse-like dynamics, Ne' = 500, lager than the Ne = 100 value determined from the diffusion measurements in a frozen matrix. This is an illustration of the fact that the two processes, Rouse motion and entangled motion are in competition: the slowest process is the one which is indeed observed.When the matrix chains are mobile, the entangled dynamics becomes more rapid than pure reptation, and the Rouse motion can dominate the dynamics for larger molecular weights than when the matrix chains are immobile. In fact, in the crossover region, the chains are still entangled (this appears on the diffusion data with N << P), but they do not obey the simple reptation nor the simple-Rouse-like dynamics. This fact may be of practical importance in situations where the entanglements
14
directly affect a local physical property, such as the strenghthening of polymer interfaces or the determination of the friction between a polymer melt and a solid wall as discussed in the chapter on flow with slippage. It is clear however that as long as one experimentally determines the critical molecular weight between entanglements by the location of a crossover in the dependence of a physical property versus the molecular weight, there is no reason
100
10
-
(N<<P) " �9 1]D/TIDRou~ e F "
[] rlD/rlDRous e (N=P)
11) r/)
0 r r
E!
[] E!
n /
N y
e
0 . ' 1 , ~ ~ , ~ , l l l t i . . I , , , ~ 1 ~ , , , ~ . , ,
101 10 z 103 104 N
Figure 6" Product of riD versus the index of polymerisation of the PDMS, for N<<P (full circles) and for N=P (open squares). From this plot, a critical index of polymerisation for a departure from Rouse - like dynamics Ne' = 500 can be estimated.
for getting the same result from determinations based on different physical properties, the width of a crossover being highly dependent on the way the considered property is sensitive to the various competing processes. It would be quite interesting to build a complete description of the dynamics in this crossover region. Attempts in that direction have appeared very recently, based on a different way of estimating the bulk zero shear viscosity of a polymer liquid, due to de Gennes [50]: it relies on the fact that when two chains are entangled at a given point, in order to let one of them move at a given velocity V, the other one has to relax the constraint, and thus to travel along its own tube a distance comparable to its tube length, during the time the first chain only moves a tube diameter. The velocity of the second chain along its own tube is thus N/Ne larger than the relative velocity between the two chains. Then. estimating the total number of independent entanglements per chains, one can compute the total dissipation and obtain the bulk viscosity. Assuming N/Ne independent entanglements per chain allows to recover the reptation prediction for the viscosity. The interest of this approach is that first, it can be applied to more complicated situations than the zero shear
15
viscosity [50], and second that it can be used with different hypothesis on the number of entanglements per chain, or on the relative dynamics of the two chains. It has been applied to the problem of the cooperative motion of polymer melts [27] and to the complicated question of the dynamics of polymer melts of chains with structures more complicated than linear [52]. This is certainly a fruitful way of analysing the complicated dynamics of entangled polymer systems and has to be developed.
5. CONCLUSIONS
After a review of some remaining open questions raised by the reptation model, we have shown systematic self diffusion measurements, obtained by an optical technique which allows one to benefit from the selectivity associated with the labelling of a few chains among the others. This should help to clarify the question of the transition between entangled and non- entangled dynamical regimes, by getting rid of the collective motions of the chains. We have shown that the dynamics of linear chains could be described by simple reptation arguments, assumingthat each chain essentially sees a frozen environment during its reptation time, as soon as the number of entanglements per chain is larger than 10. Below this value, the dynamics is strongly accelerated mainly through the constraint release mechanism: a simple reptation behaviour is recovered when the matrix is frozen down, indicating that the fluctuations of the chains inside their tube are not dominant in this acceleration of the motion. This simple reptation behaviour is observed (with the frozen matrix) down to molecular weights which are usually considered to be below the critical molecular weight between entanglements, i.e. down to 7000 g/mole for PDMS (Ne = 100). We think that this is an indication that, in what is usually considered as a dynamical crossover region, the chains are entangled but do not behave as if they were due to competition between different dynamical processes. In the molecular weight range where the number of entanglements per chain is
smaller than 10, the dynamics as characterised through the dependence of the product riDs versus the number of entanglements per chain N/Ne appears to be quite universal: the same behaviour is observed in PDMS, in polyethylene, and also in polystyrene / benzene solutions [10]. Such a dynamical behaviour has certainly to be incorporated when one wants to model the properties of polydisperse melts of linear chains. It is also important in branched systems in which the reptation is expected to become very slow and thus susceptible to become more easily hidden by constraint release. In any case, a better characterisation of the crossover between entangled and non-entangled behaviour, on many different polymers, should help to understand what, on the molecular level, governs the value of the critical molecular weight between entanglements.
REFERENCES:
I. J. D. Ferry, Viscoelasticity of Polymers, Third Edition, John Wiley and Sons eds., New York (1980). 2. P. G. de Gennes, and L. L6ger, Ann. Rev. Phys. Chem., 33 (1982) 49 3. W. W. Graessley, Adv. Polym. Sci., 16 (1974) 1. 4. P.G. de Gennes, J. Chem. Phys., 55 (1971) 572. 5. M.Doi, and S. F. Edwards, J. Chem. Soc. Faraday Trans. II, 74 (1978) 1789. 6. M. Doi and S. F. Edwards. Theory of Polymer dynamics, Oxford University Press, (1986). 7. M. Daoud and P. G. de Gennes, J. Polym. Sci., Polym. Phys. Ed., 17 (1979) 1971. 8. J., Klein, Macromolecules, 11 (1978) 852. 9. M. Doi, J. Polym. Sci., Polym. Phys. Ed., 21 (1983) 667. 10. M.F. Marmonier and L. L6ger, Phys. Rev. Lett., 55 (1985) 1078.
16
1 1. P.F. Green and E.J. Kramer, Phys. Rev. Lett., 55 (1985) 2145" Macromolecules, 19 (1986) 11 O8. 12. B. Smith, Phys. Rev. Lett., 52,(1984) 45. 13. W. W. Graessley, Adv. Polym. Sci., 47 (1982) 67. 14. M.J. Struglinski and W.W. Graessley, Macromolecules, 18,(1985) 2630. 15. M. Rubinstein, E. Helfand, D.S. Pearson, Macromolecules, 20 (1987) 822. 16. M. Doi, W.W. Graessley, E.Helfand and D.Pearson, Macromolecules, 20 (1987) 1900. 17. H. Watanabe, T. Sakamoto and T. Kotaka, Macromolecules, 18 (1985) 1436. 18. J.P. Montfort, G. Marin and P. Monge, Macromolecules, 19 (1986) 393 and 1979. 19. W. Hess, Macromolecules, 19 (1986) 1395. 20. T.A., Kawasalis and J. Noolandi, Phys. Rev. Lett., 59 (1987) 2674. 21. J.L. Viovy, J. Polym. Sci., Polym. Phys. Ed., 23 (1985) 2423. 22. J.D. des Cloizeaux, Macromolecules, 23 (1990) 4678. 23. S.F. Edwards, Proc. phys. Scoc. Lond., 92 (1987) 9. 24. P.J. Flory, J. Phys. Chem., 17 (1949) 303" and in Statistic of Chain Molecules, New York: Interscience. 25. P.E. Rouse, J. Chem. Phys., 75 (1953) 1996. 26. W. Hess, Macromolecules, 20 (1987) 2587, and Macromolecules, 21 (1988) 2620. 27. R.B. Bird, C.F. Curtiss, R.C., Amstrong and O. Hassager, Kinetics Theory, 2 nd Ed.; Dynamics of Polymeric Liquids, Vol. 2, Wiley Interscience: New York, (1987). 28. M.F. Herman and Ping Tong, Macromolecules, 26 (1993) 3733. 29. R.H. Colby, L.J. Fetters and W.W. Graessley, Macromolecules, 20 (1987) 2226. 30. D. Pearson, Rubber Chemistry and technology, 60 (1987) 439 31. J. Klein and B.J. Briscoe, Proc. R. Soc., Lond. A, 365 (1979) 53. 32. L. L6ger, H. Hervet and F. Rondelez, Macromolecules, 14 (1981) 1732. 33. C.R. Bartel, B. Crist, L.J. Fetters and W.W. Graessley, Macromolecules, 19 (1986) 785. 34. C.R. Bartel, B. Crist and W.W. Graessley, Macromolecules, 17 (1984) 2702. 35. M. Tirrell, Rubber, Chemistry and Technology, 57 (1984)523. 36. G. Reiter and U. Steiner, J. Phys. II, 1 (1991) 659. 37. R.J. Composto, E.J. Kramer and D.M. White, Polymer, 31 (1990) 2320. 38. L.J. Fetters, D.J. Lohse, D. Richter, T.A. Witten, A. Zirkel, Macromolecules, 27, (1994) 4639. 39. P. F. Green, P. J. Mills, C. J. Palmstrom, J. W. Mayer, E. J. Kramer, Phys. Rev. Lett., 53, (1984) 2145. 40. J. Davoust, P. Devaux and L. lager, The EMBO Journal, 1 (1982) 1233. 41. B. Deloche, private communication. 42. J. Klein, Macromolecules, 14 (1981) 460. 43. S.F. Tead and E.J. Kramer, Macromolecules, 21 (1988) 1513. 44. M. Antonietti, J. Coutandin and H. Sillescu, Macromolecules, 19 (1986) 793. 45. R.R. Rahalker, J. Lamb, G. Harrion, A.J. Barlow, W. Hawthorn, J.A. Semlyens, A.M. North and R.A. Pethrick, Proc. R. Soc. Lond., A394, (1984) 207. 46. C.L. Lee, K.E. Polmanteer and E.G. King, J. Polym. Sci. A2, 8 (1970) 1909. 47. G.C. Berry and T. Fox, Adv. Polym. Sci., 5 (1968) 261. 48. D.S. Pearson, G. ver Strate, E. von Meerwall and F. Schilling, Macromolecules, 20 (1987) 1133. 49. D.S. Pearson, L.J. Fetters, W.W. Graessley G. ver Strate and E. yon Meerwall, Macromolecules, 27 (1994) 711. 50. P.G. de Gennes. MRS Bulletin, (1991) 20 51. M. Appel and G. Fleischer. Macromolecules, 26 (1993) 5520 52. F. Brochard-Wyart, A. Ajdari. L. Leibler, M. Rubinstein and J.L. Viovy, Macromolecules. 27 (1994) 803
Rheology for Polymer Melt Processing J-M. Piau and J-F. Agassant (editors) �9 1996 Elsevier Science B.V. All rights reserved. 17
P o l y b u t a d i e n e : N M R a n d T e m p o r a r y e l a s t i c i t y
J.P. Cohen Addad
Laboratoire de Spectromdtrie Physique associd au CNRS, Universitd Joseph Fourier Grenoble I, B.P. 87, 38402 St Martin d'H~res Cedex, France
1. INTRODUCTION
The purpose of this study was to give an insight into molecular properties which underlie the linear viscoelastic behaviour of molten polymers. Properties were probed from proton magnetic dipoles attached to polymeric chains or to small molecules in concentrated polymeric solutions.
1.1. H i e r a r c h y of r a n d o m m o t i o n s It is well-known that the specific character of properties of polymeric
systems, observed above the glass transition temperature, lies in the existence of a broad relaxation spectr~lm associated with a wide variety of internal random motions which occur along any linear macromolecule. The broadness of the relaxation spectrum originates from the linkage of monomeric units which induces a collective dynamic behaviour of skeletal bonds. As a consequence of the linear structure of macromolecules, monomeric units are involved in a hierarchy of random motions ; this hierarchy corresponds to a single chain property which can be detected from viscoelastic properties when measurements are performed on dilute solutions of polymer chains : a quantitative description has been given to the collective dynamic behavior of monomeric units which belong to one chain [1]. It shows that the time interval required to observe the full rotation of one Gaussian chain, in a dilute solution, is proportional to N 1-5 , where N is the total number of skeletal bonds; this time interval is proportional to N 2, in a melt, for short chains (N< 102), where hydrodynamic interactions can be neglected [2].
1.2. A s y m m e t r y and t ime sca le of ro ta t ions It will be shown tha t chain dynamic properties are conveniently
investigated within the frequency range going from about 1 to 1010 Hz, by using several NMR methods suitably adapted for different frequency domains. It is worth emphasizing that no phenomenological par_ameters will be introduced to analyse observed NMR properties. The magnetic relaxation of protons, attached to polymer molecules, is mainly sensitive to the result of the space average of tensorial magnetic interactions established between nuclear spins ; this average is induced by internal fluctuations which occur within one chain. These fluctuations arise from two kinds of basic random motions. The first kind corresponds to rotations of monomeric units about neighbouring units along one
18
chain ; the second kind is associated with transitions of rotational isomers through the finite number of states of discrete conformations of any monomeric unit. Tensorial magnetic interactions are sensitive to these internal local rotations. More precisely, the nuclear magnetic relaxation is governed by the degree of isotropy of random rotations of skeletal bonds with respect to the reference axis defined by the direction of the strong steady magnetic field which is applied to the spin-system to partly order nuclear dipoles ; it is also governed by the rate of random change of orientations of skeletal bonds. These are the two key points which must be taken into consideration to analyse NMR observations. The main feature about the hierarchy of random motions within one chain concerns the degree of anisotropy of rotations of monomeric units which is associated with the time scale of observation ; the shorter the time interval of measurement, the higher the degree of anisotropy of detected rotational motions of skeletal bonds. Consequently, from the NMR point of view, the hierarchy of random motions which occur along one chain, implies that the result of the space average of tensorial magnetic interactions depends on the time scale of observation. Fast and non-isotropic rotations are detected on a short time scale (_- 10-9 s) while a long time interval is required to observe isotropic rotations of skeletal bonds which involve the diffusion of one chain as a whole. Considering protons a t tached to one chain, for example, it may be stated, from the fundamenta l analysis which is given to NMR properties, tha t magnet ic interactions are averaged to zero, provided the time interval required to observe the full rotation of one chain is shorter than 10-5s. [3]. A typical value of this time interval is 10 -6 s at 217~ for undiluted polystyrene (N = 102) [4] . For longer chains (N > 103), in dilute solution in a viscous solvent, the time interval required to observe the full rotation of one chain is much longer than 10:5 s. and then, only a partial space average is detected from NMR.
1.3. C h a i n coup l i ng j u n c t i o n s In the case of long macromolecules in a melt, it is well-known tha t a
collective behaviour of chains results from topological interactions between segments; this collective behaviour is taken into consideration by assuming the existence of characterist ic uncrossable segments along each polymer. The number of skeletal bonds which determines the mean length of these segments varies from two to three hundred; it depends on the chemical nature of the polymer. As a consequence of the existence of uncrossable chain segments, the hierarchy of random motions of monomeric units, along one chain, becomes more marked; it gives rise to two dispersions in the relaxation spectrum of conformational fluctuations of one chain [5]. The presence of a well-defined separation between the two sets of correlation times implies that orientational correlations of skeletal bonds, within short segments, are disconnected from long range correlations of displacements of segments in one chain. It is like considering that macromolecules are coupled to one another by junctions that can be pictured as temporarily fixed points, also called entanglements; these vanish when coupling junctions dissociate according to a complex relaxation process which has been described as a reptational motion [6]. The time interval required for the vanishing of temporarily fixed points is usually much longer than 10 -5 s. The duration of the presence of these chain junctions is such that it allows the definition of a temporary network structure. Correspondingly, the relaxation of
19
the magnetisation of protons attached to polymer chains is expected to exhibit a dual behaviour. On the one hand, the magnetic relaxation process results from the fast random isomerisation of monomeric units; on the other hand, the irreversible dynamics of the magnetisation is sensitive to the topological hindrance created by the presence of entanglements. This dual behaviour parallels viscoelastic properties of polymeric systems : the degree of constraint is closely related to elasticity while the segmental mobility can be associated with viscosity. The separation between the two dispersions of relaxation times of one chain is also revealed from the plateau domain which characterises the time evolution of the relaxation modulus of any molten polymer; this plateau is associated with the modulus of temporary elasticity [4].
1.4. Transverse or longi tudinal magnet i c re laxat ions Any polymeric chain observed above the glass transition temperature is
known to exhibit a fractal nature corresponding to the self-similarity of its spatial properties considered as a function of the number of skeletal bonds in the chain (with a fractal exponent equal to 2 or 5/3) ; the fractal nature results from large amplitude conformational fluctuations which govern also the dual spin- system response. Nuclear magnetic properties observed in polymeric systems show a marked axial symmetry due to the presence of a strong magnetic field (1 Tesla) applied to the spin-system to induce a macroscopic magnetisation which results from a paramagnetism effect. In high polymer melts, the magnetisation of nuclear spins presents a relaxation process, along the direction of the steady magnetic field, analogous to that of the longitudinal magnetisation observed in ordinary liquids whereas the transverse component exhibits a relaxation behaviour analogous to that of nuclear spins embedded in a solid. It is worth noting that the treatment of the response of the spin-system requires the use of the framework of quantum mechanics. The spin-system is represented by a density matrix operator which applies both in the spin space and in the three- dimensional space where nuclei move randomly. The quantum analysis which underlies all NMR observations will not be evoked here. It will be kept in mind that the relaxation of the transverse magnetisation of protons attached to chains will be closely related to the dispersion of quantum phases of wave functions which are associated with nuclear spins ; the relaxation process of the magnetisation of protons which occurs along the direction of the steady magnetic field win be related mainly to a quasi-resonant exchange of energy between the spin-system and the thermal reservoir of energy of molecules which carry nuclear spins. In a short way, it may be considered that, for the sake of simplicity, quantum degrees of freedom will not be evoked throughout this Chapter while macromolecular degrees of freedom associated with translation and rotations will be taken into consideration in order to investigate properties of polymeric system.
The principle of the NMR approach to semi-local properties of polymeric melts is considered in Section 2 ; it is shown how the existence of a temporary network structure is detected from the relaxation of the t ransverse magnetisation of protons attached to chains. The observation of segmental motions from the longitudinal relaxation of proton magnetisation is described in Section 3 ; it is also shown how local motions in concentrated polymeric solutions can be probed from the diffusion process of small molecules. Section 4 is devoted to the analysis of the effect of entanglement relaxation on NMR properties.
20
2. T E M P O R A R Y NETWORK STRUCTURES
In this Section, the attention is focused on properties of the transverse re laxat ion of protons at tached to polymer molecules; it is sensitive to the presence of temporary network structures in molten polymers. Any high polymer melt is pictured as an ensemble of chain segments with temporarily fixed ends.
2.1. Anisotropy and tensorial interactions of spins One of these segments is now considered (Fig.l). It is supposed that
CH:
B < r >
Figure 1. Schemat ic representa t ion of one chain segment, also called submolecule, in a melt ; proton groups are drawn for illustration, only.
it is observed at a t empera ture higher than tha t of the glass transition; consequently, it undergoes conformational fluctuations. However, the random rotation in space of monomeric units which form this chain segment is hindered by the fLxing of its ends. The probability of orientation of skeletal bonds, considered at any time, is not isotropic; it is higher along the direction of the end- to-end vector than along the direction perpendicular to this vector. The non- isotropic character of random rotations of monomeric units is reflected by the space average of tensorial magnetic interactions of nuclear spins attached to the chain segment: these interactions are not averaged to zero. The fixing of segment ends gives rise to residual tensorial interactions of spins which accompany the reduction of entropy associated with conformational fluctuations. The strength of averaged magnetic interactions depends on the physical state of the chain segment, determined by its end-to-end vector which will be called r, hereafter, and the number of skeletal bonds that it contains; this number is called n. Let
21
r HD( p, n )deno te the residual spin Hamil tonian in the case of dipole-dipole
interactions; HD(p, n) is expressed as r
HD(p,n) =0.3{2pz 2 - py2 _ px 2 } A ( a / a ( n ) ) 2 ~ , (1)
where the steady magnetic field direction is along the Z-axis; a is the mean skeletal bond length and c(n) 2 is the mean square end-to-end distance between
segment ends [7]. The normalized p vector is defined as p = r/c(n). Orientational correlations of three skeletal bonds, located along one segment, are involved in the definition of the A parameter ; this is called the second order stiffness of the
i chain [7]. The H D spin-spin interaction reads
i HD = Z Ap,p,Bp,p, , (2)
p,p'
the sum is extended to all p and p' nuclear spins located on the chain segment; Ap,p, spin operators are supposed to commute with the Zeeman energy and B p,p' numerical factors result from transformations of spherical harmonics from monomeric unit frames to the reference frame ascribed to the r elongation vector of the chain segment. Details about these numerical factors are not given here; these are of order unity.
Equat ion (1) applies to weakly s t re tched segments; the s tat is t ical framework of description appears through the expression of cffn) 2. For Gaussian segments :
c(n) 2 = nCoo a 2 , (3)
where Coo is called the characteristic ratio of the chain; it describes the first order stiffness pa ramete r which is defined from orientat ional correlations of two skeletal bonds located on one segment. Equation (1) expresses actually an effect of the reduction of spin-spin interactions, induced by conformational fluctuations which occur within one segment with fixed ends. The effect of reduction is given by a factor which is called f and such that
r HD( p, n) = f , (4)
with
f = (a/c~)2 , (5)
22
the factor f is roughly equal to 1](nC~). The estimate of f can be compared with the time scale of experimental relaxation curves which are found to last over a few milliseconds, for protons ; for n = 2 xl0 2, C~ = 5, f is equal to 10-3. This corresponds to a transverse relaxation rate of 10 2 rad.s -1, in agreement with the observed time scale of relaxation.
2.2. Semi . loca l NMR probe Considering one chain segment characterized by a normalized p stretching
vector, and a number n of skeletal bonds, the transverse magnetic relaxation function of nuclear spins, attached to this segment, is generally expressed as a product
Mx(t, p, n) = M: (t, p, n) ~r(t). (6)
The contribution to the relaxation process induced by the presence of residual interactions of spins is given by
M~ (t, p, n) = 'Tr ( M: (t) Mx )/ q'r ( Mx 2 ) , (7)
where
r r M x(t) = e x p ( i ~ t ) Mx exp(-i H D t ) . (8)
Mx is the x-component of the spin operator. The contribution to the relaxation process induced by conformational fluctuations of one chain segment is represented by the @r(t) function. The time evolution of this function is
determined by the spin-Samiltonian �9 ~ - H D ( P , n). The dynamics of random
motions of monomeric units is detected from the @r(t) function while deviations
from isotropic rotations of these units are revealed from the M~ (t, p, n) function.
The @r(t) function accounts also for dipole-dipole interactions which exist between nuclear spins located on different chain segments.
Taking Eq.1 and 8 into consideration, it is clearly seen that the transverse magnetisat ion is sensitive to properties delocalised over one chain segment; consequently, it probes semi-local properties over about 50A.
2.3. Polymeric system The transverse magnetic relaxation observed over a whole polymeric
system, above the glass transition, is heterogeneous. It is the sum of relaxation processes assigned to each chain segment which participates in the formation of
23
the temporary network structure. Let M~ denote the total relaxation function; it is determined by the average carried out over the distribution of all end-to-end vectors and the distribution of chain segment lengths [8]. The statistical structure of the polymeric system is represented by the probability distribution function of the normalized p vector; it is called G(p)while the probability distribution function of the number of monomeric units in segments is called Q(n); G(p) will be called structure function, hereafter. The relaxation function is expressed as
M~ = Cr(t) ~: Q(n)f G(p) M: (t, p, n) dp (9)
r The existence of the HD( p, n)spin-H~miltonian gives rise to a solid-like
relaxation process of the transverse magnetisation while the Cr(t) function has a liquid-like character. These two types of behaviours can be discriminated from each other in the following way. A solid-like echo can be formed at a time ~ after
the beginning of the transverse relaxation process; let ~ (t, p, n ) denote the function which represents this echo which is also known as the pseudo-solid spin- echo. Then, the spin-echo which is observed, is described by the product
0
0.8
0.6
0.4
0.2
0
$ ~
�9 �9
�9
�9
@ ee~
§ % -~& +.4-.'.
' i f ' - F . .
. . . . ; . . . . . . 1 - - ' - - " - ' - - - I . I . _ _ r ~ t , , .
2 4 6 8 10
t ( m s )
Figure 2. Pseudo-solid spin-echoes recorded from a polybutadiene solution (0.88 w/w) in toluene at 254 K ; relaxation function (+) ; echo origins 0.8 (0), 1.6 (O), 2.4 ( . ) ms.
24
r Ex (z; t, p, n ) = E x (z; t, p, n ) Cr(t) �9 0o)
The pulse sequence which is applied to the spin system to observe the pseudo- solid spin-echo has been described elsewhere [7] ; it is determined by
9 0 ~ - z / 2 - 1 8 0 ~ - ~ / 2 - 9 0 ~ - ( t - r 1 8 0 ~ - ( t - z ) / 2 .
Characteristic pseudo-solid spin-echoes recorded from polybutadiene are shown in Fig.2 ; the polymer concentration in deuterated toluene was 0.88 (w/w). The fraction of monomeric units in the cis conformation was 0.96 (M = 110,000 g/tool.).
Considering Eq. 6, the M~ (t, p, n) and Or(t) functions are dis t inguished from
each other in the following way. The time derivative of Ex (t, p, n ), at t = z, is expressed as
aE• (z; t ~ , p, n ) _ dM:(t---z) at dt Or(Z) + M: (z, p, n) d(~r(t---'r (11) - - dt '
0 _. I " - i
-o.2 ; =
- 0 . 4 L- -
I " -o.6 I I 1 __. �9
-O.8
-1
i - 1 4
u
== g. § ~
- I . 4
T - - - - T I �9
�9 9
: i
-I
. . . . . . . . .
~" 7 �9 '- ~ " ~ 7
i 7 - ? t
7
7 I
7
i
i 7
7 7 ]
t
I 7
0 0.5 I 1.5 2 2.5
~ ( m s )
Figure 3. Logarithmic derivative of the r function drawn as a t ime function from polybutadiene at several temperatures �9 T = 252 (ll), 264 (A), 274 (O), 295 (~), 314 (~) and 334 K (O).
25
and, the time derivative of Mx(t, p, n), at t = r is written as
dMx(t--~, p, n) dM:(t=z) dt -- dt ~r(~) + Mr (% P, n) d(~r(t=r " (12)
Let p' and p denote these two time derivatives, respectively. The addition and the subtract ion of equations (11) and (12) lead to the determination of time
r t derivatives of M x( , p, n) and @r(t) functions, at t=~"
dCr(t=z) dt ~ / O r ( C ) = (p+p')/2Mx(z) (13)
The time derivative of the logarithm of the Cr(t) function is illustrated in Fig. 3, in the case of molten polybutadiene; the molecular weight was Mn = 77000, the vinyl content was 8%. Temperatures of observation were 252, 264, 274, 295, 314 and 334 K ; the r function is experimentally found to be an exponential time function [9].
2.4. Property of t ime scal ing The most str iking feature about NMR investigations into dynamical
properties of high polymer systems lies in the observation of a gel like behaviour of the t ransverse magnetic relaxation. This is i l lustrated by comparing the proton transverse relaxation and pseudo-solid echoes observed on high molecular weight polybutadiene chains, in a melt, at room temperature, with the relaxation curve and echoes observed on a vulcanized polybutadiene. These two systems are undist inguished from each other by observing the properties of the proton t ransverse magnet isa t ion [7]. This result shows clearly that there exists a separation between the two dispersions of relaxation times of chain fluctuations which occur in a melt. During this separation, it may be considered tha t any given chain in a melt behaves like in a permanent gel; it is divided into submolecules and each submolecule has temporarily fixed ends. Let Ne denote the mean number of skeletal bonds which determine the length of one submolecule; Ne is defined by analogy with the description of the property of elasticity of permanent gels
Ne = ppRT / G M m , (14)
pp is the pure polymer density while G~ is the modulus of temporary elasticity and Mm is the mean molar weight of one skeletal bond. It is considered that the mean number Ne is also involved in NMR properties. However, the relevant experimental quanti ty which is directly associated with the characterisation of
structures, in melts, is usually G ~ rather than Ne. In this work, the transverse
26
magnetic relaxation analysed by using directly G~. The entangled polymeric is
system is described from Eq. 9 where Q(n) =1.
NMR investigations into properties of the temporary network structure formed in polybutadiene were made by varying both the modulus of temporary elasticity and the polymer concentration r
i) The modulus of temporary elasticity is predicted to vary upon addition of a good solvent according to the expression
o ~_a (15) GN (r = G N ,
with a varying from 2 to 2.2, for different polymers [4].
ii) Variations of G were also induced by changing the concentration Xl,2 of monomeric units in the vinyl-1,2 conformation.
iii) Considering Eq. 1, changes of the temporary network structure are expected to be reflected by variations of the time scale of the t ransverse relaxation function. Correspondingly, relaxation curves must obey a property of superposition, by applying an appropriate factor to the time scale of observation.
Relaxation curves recorded from polybutadiene solutions have been shown to satisfy such a property, provided measurements are performed at a temperature higher than a temperature threshold which will be called Ts, hereafter ; Ts is higher than the Tg glass transition temperature of the polymeric system (Ts- Tg = 80K). Above this temperature threshold, the mathematical structure of the relaxation function is kept invariant except for the time scale. Because of the property of superposition, normalized relaxation curves can be characterized by chosing a given amplitude A and by measuring the time interval corresponding to this amplitude
1: A = (A G A )-1( ~ / a )2 Mx-1 (A), (16)
where Mx-I(A) is the inverse function of Mx(t). For polybutadiene in concentrated solutions, it has been shown that
0.45 I:A = (1 + G(Xl~2)~2. 2 )(T-Tg(~,Xl,2) - 50).8.10 -3 , (17)
the time interval is given in milliseconds and G ~ is expressed in Pascals and divided by 106. The above equation applies to different chain microstructures obtained by varying Xl,2. The glass transition temperature Tg(~,xl,2) depends on both the polymer concentration and the concentration xl,2 of vinyl monomeric units. The linear dependence of ~A upon the T-Tg temperature interval is illustrated in Fig. 4, for several chain microstructures [10].
27
&
0
50 100 150 2gO T-Tg
Figure 4. The charac te r i s t i c t ime ZA, measu red from severa l chain microstructures of polybutadiene, is represented for A = 0.6 ; vinyl contents are xz,2 = 0.22 (Q), 0.40 (A), 0.58 (A) and 0.82 0).
The presence of a temporary network structure manifests itself through the first
factor in the right hand side of Eq. 17. It comes from the M~(t) component of the relaxation function. The second factor results from fast non-isotropic motions of monomeric units which give rise to the Cr(t) function defined in equations (6) and
(9). It must be noticed that the reference temperature Ti = Tg(r +50 which appears in NMR properties, is close to the reference temperature Ti introduced from viscoelastic measurements [11]. The second factor reflects a free volume effect ; it depends necessarily on an expansion coefficient which is about equal to 10-3K.
It is seen that NMR properties, specific to polymeric systems, are not due to the absence of motions but to the non-isotropic character of these motions. It is worth emphasizing that the search for a characterisation of dynamical properties leads actually to the observation of non-isotropic rotations of monomeric units which result from the presence of entanglements. Asymmetry properties are observed instead of dynamic ones. Equation (17) is general and applies also to polyisobutylene solutions or to polyisoprene by using different values of the parameters.
28
A sharp analysis of relaxation functions is obtained by using
equations (11) and (12). They yield an exponential time function for ~r(t), associated with fast and non-isotropic motions whereas Mxr(t) is found to depend on deviations from isotropic rotations, only. This is an illustration of the dual behaviour of the spin-spin response.
3. S E G M E N T A L MOTIONS : D Y N A M I C S C R E E N I N G E F F E C T
In this Section, the attention is focused on the longitudinal relaxation of the magnetisation of protons attached to polymer molecules. Local motions are probed from this relaxation mechanism and from measurements of the friction coefficient of host small molecules moving randomly through the polymeric system. Segmental motions are expected to be independent of chain molecular weight variations �9 long range fluctuations are screened when dynamic properties are observed in a short space scale.
3.1. Frequency-temperature scaling The longitudinal relaxation also called spin-lattice relaxation is
induced by a quasi-resonant exchange which occurs between the Zeeman energy of the spin system and the thermal energy of molecules which carry nuclear magnetic dipoles. The condition of quasi-resonant exchange of energy is used to calibrate the time scale of relaxation of molecular
processes; the precession frequency coo of nuclear spins, also called Larmor frequency, plays the role of a reference. The maximum of the spin-lattice relaxation rate occurs when the Larmor frequency is approximately equal to the inverse of the correlation time which characterizes molecular motions [3]. The exact calculation of relaxation rates necessitates an accurate description of random molecular motions which are nowadays well described in the case of small molecules in an ordinary liquid. In the case of polymeric systems, the calculation of relaxation rates comes against the description of segmental motions; the difficulty arises from the necessity to take collective rotational isomerizations of monomeric units into consideration. Several correlation times must be introduced to account for the collective random monomeric rotations. As a consequence of the existence of a spectrum of relaxation times instead of a single correlation time, the spin-lattice relaxation rate is usually found to depend on the Larmor frequency on both sides around the maximum observed by varying the temperature of the polymeric system. A few models of segmental motions which involve several skeletal bonds have been proposed until now [12, 13]. In this Chapter, experimental results are reported only to give a short illustration of the analysis which has been already proposed elsewhere [14]. Measuring the T1-1 spin-lattice relaxation rate of protons or of 13C nuclei attached to
polymer chains, the function defined by the ration co0/T1 is considered. The
analysis is based on the assumption that the co0/T1 ratio obeys a property of
homogeneity with respect to the variable determined by the product of the co 0 Larmor frequency and one characteristic correlation time. Such a property implies that the spectrum of correlation times associated with random segmental motions represents multiples of one characteristic time whatever its exact definition : let zlC(T,f) denote this correlation time. The ratio 0~0/T1
29
co0/T 1 = '6' (cool: 1 c), (18)
where Xl c may depend on both the polymer concentrat ion ~ and the temperature of the polymeric system. Equation (18) shows that a frequency- temperature or a frequency-concentrat ion scaling can be applied to the arralysis of experimental results.
It is considered that several monomeric units are involved in segmental motions which are observed from the spin-lattice relaxation rate. More precisely, it is assumed that collective rotational isomerizat ions of a small number of succesive monomeric units are converted into a translational random motion of a short segment. An effective friction coefficient of one
monomeric unit can be associated with this translational motion : let ~f(T,~) denote this friction coefficient. It corresponds to an effective correlation time Xl c (T,r 2 ~f (T,r ; b is the mean length of one skeletal bond.
The property of homogeneity of the CO0/T1 functions is well illustrated from polybutadiene chains, observing 13C of 1H nuclei [14]. Polybutadiene is a flexible polymer whose viscoelastic properties have been extensively studied over a very broad temperature-frequency range [11]. The first criterion of application of this analysis is the observat ion of a nearly constant value of the maximum of the co0/T1 function, whatever the Larmor
frequency. Then, the experimental c00/T1 curve must be drawn as a function of T-Tg of T-Ti, for a given Larmor frequency, with T i = Tg +55, introduced from viscoelastic measurements; it must be shifted- towards high temperatures when the Larmor frequency is increased. The second criterion of application is the possibili ty to superpose all CO0/T1 functions recorded by using several Larmor frequencies. The property of superposi t ion of all relaxation curves is well observed for pure polybutadiene, by using the variable T-Ti (Fig. 5). The broad shape of the relaxation curves near the maximum indicates that a spectrum of correlation times is involved in segmcntal motions; nevertheless, the maximum of efficiency corresponds to correlation time values near 4.x10 -9 s. A shift factor a(T-Ti) which applies to the friction coefficient is introduced; then the product
coo ~f(Ti)a(Ti ,T)is considered as reduced variable. Considering two Larmor
frequencies coO' and coO", the property of homogeneity reads
coo' a(T'-Ti) = r a(T"-Ti)
where T' and T" are determined from any two chosen equal amplitudes of
experimental curves. The value of the ratio co0'/co0" gives the relative evolution of the a(T-Ti) shift factor which is found to be in agreement with results of viscoelastic measurements
30
T-Ti Log [ a(Ti ,T) ] = -5.78 T- Ti + 94.8 (19)
This equation applies to any chain microstructure provided the Ti reference temperature is defined with respect to the glass transit ion temperature Tg, corresponding to the observed polybutadiene chain [11].
Such an analysis has been also extended to several solvent-polymer systems. A similar approach has been applied to polyisoprene in dilute solution in toluene [15]. The maximum value of the relaxation time is found to be independent of polymer concentrations ranging from 0.58 to 1 ; it is equal to 20 s- 1. This result shows that the effective number of interacting protons does not depend on the polymer concentration ; interactions between protons located on different chain segments can be neglected.
3 -
2 -
1 -
E~
o
-2.5
s
c~ rn
A 0
AFa
A O
@
E] DO
~ O
z~
v]
. l ! I ! 1 1 1
-2 - I .5 - 1 - . 5 0 . 5 1 1.5
log o~oa ( T - Ti)
Figure 5. Spin-lattice relaxation rates of protons, attached to polybutadiene chains, as a function of the variable m0 a( T-Ti); r 32 (O), 60 (O) and 100 (A) MHz.
31
The observation of the property of homogeneity of spin-lattice relaxation rates with respect to both the Larmor frequency and one basic dynamical segmental parameter is interpreted as reflecting average fluctuations which occur within short chain segments and not a single rotational isomerization process of skeletal bonds. Helical turns are known to form and to dissociate rapidly in any homopolymer ; the average fluctuation probably spreads over about 10 monomeric units since the spin-lattice relaxation rate is not affected by crosslinks between segments unless the contour length defined by two consecutive junctions is reduced down to 20 skeletal bonds [16]. Finally, it may be worth emphasizing that spin-lattice relaxation rates usually observed on high polymers are independent of the chain molecular weight. This property reflects the effect of dynamic screening ; it does not hold when polymer chains are too short. Then, variations of the polymer molecular weight correspond to variations of the concentration of chain ends which plays the role of solvent molecules.
To conclude, the ratio o~0/T1 may be considered, in most cases, as a homogeneous function with respect to the product of the Larmor fequency and one characteristic molecular time constant; this property is very useful to disclose relative variations of this time constant as a function of temperature or concentration. The coincidence of maximum values of the co0/T1 ratio with one another, indicates that the mean square dipole-dipole interaction is hardly dependent on temperature or polymer concentration. It is worth noting that spin- lattice relaxation rates are found to be insensitive to chain length variations unless molecular weights become lower than about 10,000 g/tool..
3.3. S o l v e n t d i f f u s i o n : f r ee v o l u m e e f fec t . Local motions which occur in macromolecular systems can be probed from
the diffusion process of small molecules in concentrated polymeric solutions. The translational diffusion is detected from NMR over a time scale which may vary from about 1 to 100 ms. Such a time interval corresponds to a very large number of elementary collisions and a long random path ; consequently, details about mechanisms of molecular jump are not disclosed from this NMR approach. However, the dynamical behaviour of small solvent molecules, immersed in a polymer melt and observed over a long time interval, permits the determination of characteristic parameters of the diffusion process. Applying the Langevin's equation, the self-diffusion coefficient Ds is defined as
Ds = kT/~s , (20)
where the ~s friction coefficient of solvent molecules is governed by Stokes'law
~s = 6~ dTIo ,
d is the molecular diameter and TIo is the liquid viscosity [17]. The purpose of investigations into diffusion properties is twofold.
i) First, starting from a free volume law which characterises a pure molten polymer, it is of interest to analyse the evolution of the free volume description upon addition of solvent.
32
ii) Secondly, the limiting value of the ~s((~) coefficient obtained when the solvent concentration goes to zero, gives a good estimate of the monomeric friction coefficient.
Since the pioneering work of Stejskal, the pulse field gradient method is currently used to characterize the diffusion process of small molecules or of macromolecules in dilute or semi-dilute solutions [18-20]. In this Chapter, the NMR approach is illustrated from the self-diffusion of cyclohexane molecules th rough polybutadiene. Variations of the Ds self-diffusion coefficient of cyclohexane in polybutadiene have been reported as a temperature function considering several concentrations [21].
The WLF way of analysis has been extended to polymeric solutions assuming that polymer and solvent fractional free volumes add linearly to each other [22]. More recently, a different way of analysis has been proposed by Vrentas and Duda; it includes several parameters such as a jumping unit volume [23].
It is worth emphasizing that these two approaches become equivalent to each other when the parameters introduced in the Vrentas-Duda analysis have appropriate values. This is the case in the cyclohexane-polybutadiene system. Measurements performed by varying both the polymer concentration and the temperature lead to the determination of free volume parameters with a good accuracy. For the pure polymer; the fractional free volume is
fp = 0.124 +_ 0.005 at 298 K,
and the thermal expansion coefficient is
ap = (7.0 + 0.5) x 10 -4 K -1 ,
while for the pure solvent the fractional free volume is
fs = 0.223,
and the thermal expansion coefficient is as = 11.8x 10 ~ K q at 298 K.
3.4 M o n o m e r i c f r i c t i on coe f f i c i en t There is an interesting feature about the limiting value of the trace
diffusivity of solvent determined from experimental curves when the solvent concentration is set equal to zero. The diffusion coefficient Ds(0,T) is found to obey the free volume law of the pure polymer when the above numerical values are assigned to the parameters. Furthermore, the ~s(0,T) translational friction coefficient, derived from the ratio kT/Ds(0,T), is equal to (1.64+_0,06) x 10 -1~ N.s.m -1, at 298 K ; this value is in good agreement with the result of viscoelastic measurements performed on polybutadiene (1.8 x 10 -1~ N.s.m-1).
The trace diffusivity clearly shows that there is no discontinuity when the solvent concentration goes to zero ; in other words, solvent molecules can be used to probe local properties of polymer melts directly.
33
4 . M O L T E N H I G H P O L Y M E R S : S E M I . L O C A L D Y N A M I C S
In molten high polymers, chain coupling junctions dissociate slowly; the full rotation of one chain occurs over the terminal relaxation time, hereafter called TR; this obeys a power law predicted from the reptation model proposed by De Gennes :TR = N 3 [6]. The transverse magnetic relaxation of protons, attached to polymer chains, behaves as they would in a pe rmanen t gel; a network structure is detected from the observation of well-defined pseudo-solid spin- echoes. This network structure is temporary and its TR relaxation time shortens rapidly when the chain molecular weight is lowered. Considering dipole-dipole interactions of nuclear spins located on one chain, the full motional averaging of these interactions occurs when TR satisfies the inequality
i HDTR IN <1 , (21)
o r i
HD nO IN G <1, (22)
where ~0 is the zero shear rate viscosity of the polymer and ~ is its modulus of temporary elasticity, associated with the t ime break of conformational
fluctuations; the estimate of ~ is 105rad.s -1, for protons. For example, for
polydimethylsiloxane ( N = 6x103 ), the G~ modulus is equal to 106 Pa. and ~o is
equal to 103 Pa.s; then, proton dipole-dipole interactions satisfy ttie above inequality and no anisotropy of random rotations of monomeric units is detected from NMR, for N = 6x103. The time dependence of the transverse magnetic relaxation of protons, observed in several polymeric systems corresponding to decreasing chain molecular weights, exhibits a progressive evolution from pseudo-solid properties associated with the presence of a temporary network structure to a liquid-like behaviour. The motional averaging effect is one of the most difficult problems left to solve in NMR of complex molecular systems. The analysis of NMR properties is based on the assumption tha t there exists a Gauss i an G(p)structure function which represents a temporary network structure. NMR becomes sensitive to the time dependence of the network structure when the chain molecular weight is varied toward low values. A scaling analysis of relaxation curves can be applied to the characterization of relaxation processes using a second order cumulant expansion to express the spin-system response [7]. Experimental results are represented according to the expression
Log [ Mx(t)] t 2 =- (AeT~/t) 2 F2 (t/T~) , (23)
with
1-'2 (t/T~)= j dul j du2 f2(ul,u2) , (24)
3 4
where Ae is the residual dipole-dipole interact ion associated with the mean segmental spacing defined by two consecutive entanglements along one chain; f2(ul,u2) is the correlation function of the square of one of the three components
of one p vector, considered at two reduced times U l and u2 with u = t ~ R. A T v , R
correlation t ime which may be specific to NMR observations is introduced instead of TR. The scaling analysis was applied to polybutadiene chains, observed at room tempera ture , in concentrated solutions in deuterated cyclohexane [24]. The polymer concentrat ion was 0.7 (w/w) and chain molecular weights were 70, 120, 170 and 360x103 g/tool.. The proton (Log [ Mx(t)]/t 2 ) function is represented as a function of t in Fig. 6, for several chain lengths. Then, experimental curves are shown to obey a property of superposition by applying a suitable factor to the time scale of observation of drawn curves. The relaxation curve corresponding to Mw = 70x103 g/tool, is used as a reference ; the superposition yields the relative
.v is found to be a var ia t ion of T~ as a function of the chain molecular weight- T R
l inear function of Mw. The molecular weight dependence of ~ shows tha t long
range chain fluctuations are involved in the relaxation process of the transverse magnet isa t ion of protons ;
0 ,,, ," ,'t. ~e oe oe o:e ~oe " *o : o
I- t, - ,TS~u_^oo o ~o_ _i : , , o o o , , - -
i_ ~
M Q
- 0 . 3 _,
- 0 . 4
- 0 . 5 - /~ - 0 . 6 , : ~ . . . . . . . . . . , ,
0 1 2 3 4 5
t ( m s )
Figure 6. Proton function Log [Mx(t)] / t 2) recorded from pure polybutadiene ; molecular weights are 103Mw = 70 (A), 120 (O), 170 (D) and 360 ( ) g/tool..
35
however, the value of the exponent (1 instead of 3, for reptation) reveals that fluctuations detected from NMR are faster than those which give rise to the
mechanism of reptation. The order of magnitude of T~ is Ae -1 (lms.).
CONCLUSION
The magnetic relaxation of protons linked to polymer molecules is sensitive to collective rotational isomerisations which occur along one chain. Orientational correlations of monomeric units give rise to relaxation mechanisms specific to polymeric systems, studied above the glass transition temperature. A deep insight into dynamic properties is provided by observing the irreversible behaviour either of the proton transverse magnetisation or of the longitudinal component. Most experimental results obey a time-scaling analysis ; the time scale is governed by a friction coefficient for local observations or by the presence and the life-time of entanglements for semi-local observations. Both NMR and viscoelastic responses of polymers manifest a dual behaviour ; this appears as liquid-like and solid-like magnetic relaxations while it corresponds to viscosity and elasticity for flowing systems. The general character of nuclear magnetic relaxation properties arises from the statistical average which applies to monomeric units along one chain.
The most striking feature about NMR investigations into dynamic properties of linear macromolecules concerns the possibility to explore the broad relaxation spectrum inherent to such complex systems without assuming that any frequency-temperature or frequency-concentration shift may apply to the analysis of the observed polymeric relaxation behaviour. Considering several NMR methods, the characterisation of the dynamics of a polymeric medium starts from the estimate of the monomeric unit friction coefficient; then, it concerns the gel-like behaviour of high polymer melts and segmental motions involving several skeletal bonds. Finally, long range fluctuations are taken into consideration. The domain of correlation times which can be probed from NMR spreads over a time scale going from 10 "1~ s to about 1 s.. The analysis which results from the NMR approach requires a deep insight into molecular mechanisms which underlie chain fluctuations. This work shows that molecular properties, disclosed from NMR observations made on polybutadiene, are closely related to effects evidenced from viscoelastic measurements; it may be stated now that the viscoelastic behaviour of other polymers can be investigated from NMR without making, necessarily, a step by step comparison between the two approaches.
R E F E R E N C E S
1. B.H. Zimm, J. Chem. Phys. 24 (1956) 269. 2. P.E. Rouse, J. Chem. Phys. 21 (1953) 1272. 3. A. Abragam, Principles of Nuclear magnetism, Oxford University Press,
London (1961). 4. J.D. Ferry, Viscoelastic Properties of Polymers (3rd edn), J. Wiley, New-York
35
(1980). 5. W.W. Graessley, Advances in Polymer Science, Springer-Verlag, Berlin (1974). 6. P.G. De Germes, Scaling Concepts in Physics, CorneU University Press, Ithaca
(1979). 7. J.P. Cohen-Addad, Progress in Nuclear Magnetic Resonance Spectroscopy (Ed.
J.W. Emsley, J. Feeney and L.H. Sutcliff), Pergamon Press 25 (1993) 1. 8. J.P. Cohen Addad, Phys. Rev. B 48 (1993) 1287. 9. M. Todica, Thesis - University of Genoble (1994). 10. A. Labouriau and J.P. Cohen Addad, J. Chem. Phys. 94 (1991) 3242. 11. J.M. CareUa, W.W. Graessley and L.J. Fetters, Macromolecules 17 (1984)
2775. 12. R. Dejean de la Batie, F. Laupr~tre and L. Mormerie, Macromolecules 22
(1989) 122. 13. C.K. Hall and E.J. Helfand, J. Chem. Phys. 77 (1982) 3275. 14. A. Guillermo, R. Dupeyre and J.P. Cohen Addad, Macromolecules 23
(1990) 1291. 15. D.J. Gisser, S. Glowinkowskq and M.D. Ediger, Macromolecules 24 (1991)
4270. 16. G.C. Munie, Jiri Jonas and I.J. Rowland, J. Polym. Sci. 18 (1981) 1061. 17. P. Langevin, Comptes Rendus de l'Acaddmie des Sciences (Paris) 146
(1908) 530. 18. E.O. Stejskal and J.E. Tanner, J. Chem. Phys. 42 (1965) 288. 19. E.D. von Meerwall, J. Grigsby, D. Tomich and R. Van Antwerp, J. Polym.
Sci. Polym. Phys. Ed. 20 (1982) 1037. 20. P.T. Callaghan and D.N Pinder, Macromolecules 17 (1984) 431. 21. A. Guillermo, M. Todica and J.P. Cohen Addad, Macromolecules 26 (1993)
3946. 22. H. Fujita and A. Kisbimoto, J. Chem. Phys. 34 (1961) 393. 23. J.S. Vrentas and J.L. Duda, J. Polym. Sci. Polym. Phys. Ed. 15 (1977) 403. 24. A. Labouriau- Thesis - University of Grenoble (1992).
Rheology for Polymer Melt Processing J-M. Piau and J-F. Agassant (editors) �9 1996 Elsevier Science B.V. All rights reserved.
37
C h a i n r e l a x a t i o n p r o c e s s e s of u n i a x i a l l y s t r e t c h e d p o l y m e r c h a i n s �9
a n i n f r a r e d d i c h r o i s m s t u d y
J. F. Tassin a, L. Bokobza b, C. Hayes b, L. Monnerie b
aLaboratoire de Physicochimie Macromol~culaire, URA CNRS 509, Universit~ du Maine, Avenue Olivier Messiaen, 72017 Le Mans Cedex, France
bLaboratoire de Physic~ St ructura le et Macromol~culaire, URA CNRS 278, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France
1. I N T R O D U C T I O N
Polymer processing usually involves deformation of macromolecular coils induc ing a preferen t ia l or ienta t ion of the polymer chain segments . The orientat ion is, of course, the result of a balance between shear and extensional deformat ion and molecular re laxat ion processes. The knowledge of these processes is a first but important step in the rheology of polymer melts, in order to obtain analytical expressions for the relaxation modulus for small as well as for large deformations. Mechanical measu remen t s can be carried out on a large variety of samples but are often difficult to interpret in terms of molecular re laxat ion mechanisms, al though exper iments on model b inary blends have been quite fruitful [1]. A better unders tanding is expected if information can be obta ined at a molecular level, for ins tance through segmenta l or ienta t ion measuremen t s . An extensive study has therefore been carried out on model mate r ia l s to probe molecular theories of chain relaxation. Infrared dichroism has proved to be a useful tool to observe the orientation at a segmental scale.
In this chapter , the resul ts of or ientat ion measu remen t s by inf rared dichroism will be presented, discussing three main topics : - Relaxat ion of chain segments as a function of the position along a l inear cha in - Comparison between relaxation of l inear and branched chains - Relaxation of long and short chains in binary blends.
In order to create a s trong molecular orientation, uniaxial ly s t re tched samples have been investigated. Large deformations (draw ratios up to 4) have been applied to enhance non linear relaxation phenomena tha t are dominant in most processing techniques. The predictions of molecular models have been compared wi th exper imen ta l da ta in order to identify various possible relaxat ion processes and eventually to introduce some improvements.
38
2. Theoret ical backg round in In f ra red Dichroism
Infrared dichroism is one of numerous methods used to character ize molecular orientation. The degree of anisotropy of the s trained polymers may also be accurately characterized by other techniques such as X-ray diffraction, birefringence, sonic modulus, polarized fluorescence and polarized Raman spectroscopy [2]. These techniques directly probe the orientational behavior of macromolecular chains at a molecular level, in contras t to the macroscopic information provided by mechanical measurements .
Birefringence and dichroism represent two optical methods which can be applied to mater ia l s under flow conditions, forming the basis of Optical Rheometry [3,4]. The aim of these two techniques is to measure the anisotropy of the complex refractive index tensor n = ~ ' - i v.". Birefringence is related to the anisotropy of the real part, whereas dichroism deals with the imaginary part. Recent applications of birefringence measurements to polymer melts can be found in Chapter III.1 of the present book.
Dichroism refers to a change in the a t tenuat ion of light which depends on the orientat ion of the electric field. Two mechanisms (either scat ter ing or absorption) may contribute to dichroism in anisotropic mater ia l s and the choice of the wave leng th range allows selection of the origin of l ight a t tenuat ion. In the infrared range, for homogeneous samples, the dichroism only arises from strong absorption effects. In this region of the electromagnetic spectrum, absorption originates from vibrat ional motions of atomic groups wi th in the molecules. These mot ions are classically t r ea t ed us ing the mechanical analysis of a vibrat ing system and can be represented by a set of normal vibrat ional modes associated wi th well-defined normal v ibra t ion frequencies. An infrared active normal mode leads to a change in the dipole moment of the molecule which can be figured as a t rans i t ion moment , M, which possesses a fixed and known position in the molecule. It is generally re la ted to the symmet ry e lements of the molecule. The absorpt ion of the infrared radiat ion is precisely caused by the interaction of the electric field vector of the incident l ight wi th the electric dipole t r ans i t ion moment associated with a particular vibration.
Segmental orientation in a material submitted to uniaxial elongation may be conveniently described by the average of the second Legendre polynomial <P2(c~ :
<P2(cos0)> = 1 <3 cos20 - 1> (1)
where 0 is the angle between the macroscopic reference axis (usually taken as the stretching direction) and the local chain axis or any vector characteristic of a given chain segment . The brackets indicate an average over all the segmental units.
The dichroic ratio R defined as R = A/// A• is commonly used to characterize the degree of optical anisotropy; A//and A• being the measured absorbances of the invest igated band for an incident radia t ion polarized respectively along and perpendicular to the stretching direction.
The orientation function <P2(cos0)> is related to the dichroic ratio R by [5]:
(R- 1) 2 = C((x) R+ 1 (2) <e2(cos0)> = (R + 2) (3cos2ct_ 1) 2
39
where a is the angle between the t ransi t ion moment and the local chain axis. It can be obtained from theoretical considerations aris ing from group theory analysis , or de te rmined exper imental ly from another absorption band since the segmental orientation is independent of the chosen absorption band.
Pract ical problems associated with infrared dichroism measu remen t s include the requ i rement of a band absorbance lower than 0.7 in the general case, in order to use the Beer-Lambert law; in addition infrared bands should be sufficently well assigned and free of overlap with other bands. The specificity of infrared absorption bands to par t icular chemical functional groups makes inf rared dichroism especially at t ract ive for a detailed s tudy of submolecular or ientat ions of mater ia ls such as polymers. For instance, information on the o r ien ta t ion of both crys ta l l ine and amorphous phases in semicrys ta l l ine polymers may be obtained if absorption bands specific of each phase can be found. Polarized infrared spectroscopy can also yield detailed information on the or ienta t ional behavior of each component of a polymer blend or of the different chemical sequences of a copolymer. Infrar-'~l dichroism studies do not require any chain labelling but owing to the mass dependence of the vibrational frequency, pronounced shifts result upon isotopic substitution. It is therefore possible to s tudy binary mixtures of deuterated and normal polymers as well as i so top i ca l l y - l abe l l ed block copolymers and t hus ob ta in i n f o r m a t i o n s imultaneously on the two types of units.
Our work has been focused on polystyrene, the infrared spect rum of which is wel l -documented . For normal po lys tyrene , d ichroism can be measu red using the 906 cm -1 and the 1028 cm -1 absorption bands. They are associated with an out-of-plane and in-plane deformation of the CH phenyl ring deformation modes respectively. The 1028 cm -1 band has a t ransi t ion moment which is perpendicular to the chain axis (a = 90 ~ whereas an angle of 35 ~ is accepted for the 906 cm -1 band [5]. For deuterated segments, the 2195 and 2273 cm -1 absorption bands have been used. They are respectively assigned to the asymmetr ic s t retching vibration of CD 2 groups and to stretching vibrations of the aromatic CD bonds. The 2100 cm -1 symmetrical stretching mode of the CD 2 groups has a weak in t ens i t y and cannot be re l iably used unless the concentration of deuterated units is quite high (40 wt%).
For deutera ted segments, a prel iminary s tudy involving binary mixtures of normal and deutera ted polystyrene of the same molecular weight has led to the following values for the calculation of the deuterated segment orientation : C = -3,8 for the 2195 cm -1 band and C = -2,7 for the 2273 cm -1 band [6].
3. E x p e r i m e n t a l
E x p e r i m e n t s have been car r ied out on essen t i a l ly monodisperse polys tyrene , e i ther homopolymers (normal or deu te ra ted) or isotopically- labelled block-copolymers. These mater ia ls have been used alone or blended. Sample nomenclature and average molecular weights, as determined by SEC, are presented in Tables 1 and 2. Isotopically-labelled block copolymers, either l inear or 6-arm stars, synthesized by anionic polymerization, were kindly provided by Dr. L. J. Fe t te rs (Exxon Research and Engineer ing Company).
40
Othe r m a t e r i a l s have been pu rchased from P r e s s u r e Chemica l , Po lymer Laborator ies or supplied by E. A. H. P. (Strasbourg, France).
Table 1 Sample character is t ics and nomencla ture of l inear and s t a r polymers Deutera ted Block Hydrogenous Block Position 10 -3 Mw Mw/Mn 10 -3 Mw Mw/Mn
PS DH 184 end 27 1.03 157 1.05 PS HDH 188 centre 30 1.03 79 1.03 PS DHD 500 ends 39 1.03 424 1.08 Star-a end 37 1.05 188 1.05 Star-b branch pt 30 1.04 215 1.05 PS 1190 - - - 1190 1.4 PS 2000 - - - 2000 1.3
Table 2 Sample character is t ics and nomencla ture of b inary blends Deutera ted chains (PSD) Mass fraction in 10 -3 Mw Mw/Mn Tg (~ the blend
PS 2000/PSD 72 (20%) 0.2 PS 2000/PSD 27 (20%) 0.2 PS 2000/PSD 18 (20%) 0.2 PS 2000/PSD 10 (10 %) 0.1 PS 2000/PSD 10 (20 %) 0.2 PS 2000/PSD 10 (30 %) 0.3 PS 2000/PSD 3 (20 %) 0.2
72 1.03 105 27 1.06 105 18 1.01 105 10.5 1.02 105 10.5 1.02 105 10.5 1.02 103 2.6 1.05 97
3.1. Preparation of the samples The polystyrene films (50 ~m thick) were obtained by cast ing a 6 to 10wt%
(depending on the molecular weight of the polymer) dist i l led benzene solution on a glass plate followed by drying at room tempera tu re . Str ips 2 cm wide were annealed under vacuum for at least 48h hours a t 140~ in order to remove any trace of solvent and in te rna l s t ress due to cut t ing. The films' suppor t dur ing the drying t r e a t m e n t was chosen to provide an i r regu la r surface for the films in order to min imize in t e r fe rence f r inges d u r i n g the record ing of FTIR spectra .
3.2. S t r e t ch ing condi t ions S t re tch ing exper iments were performed on labora tory-des igned hydraul ic
machines [7]. Uniaxial elongations were made a t cons tan t s t ra in r a t e and at t e m p e r a t u r e s above the glass t rans i t ion t e m p e r a t u r e (Tg = 106~ The draw ratio was l imited to k = 4 (~ being the rat io of the final to the ini t ial length) to avoid sample breakage. The t e m p e r a t u r e range was chosen be tween 107 and 14~0~ St ra in rates ranged from 8.10 -3 up to 0.1 s -1.
Two types of exper imen t s were performed. In the ini t ia l s tudies , the samples were s t re tched under given s t ra in ra tes and t e m p e r a t u r e conditions at a given draw ratio k ( 1.5 _< k < 4 ). As soon as the draw rat io was achieved,
41
the sample was rapidly quenched to room tempera tu re in order to freeze the s t a t e of o r i en ta t ion at the end of s t re tching. In a large major i ty of the exper iments , a l inear relat ionship was observed between <P2(cose)> and k, and
herea f te r the or ienta t ion obtained at ~. = 4 will be reported. A mas te r curve of the o r ien ta t iona l re laxa t ion is obta ined by t ime - t empera tu r e superposi t ion us ing exper iments carr ied out at var ious s t ra in ra tes and t empe ra tu r e s [6]. Obtent ion of a mas te r curve implies that , for a given draw ratio, the orientation m e a s u r e d at a given t e m p e r a t u r e , T1, and a s t ra in ra te , al, is, within exper imenta l uncer ta inty , identical to tha t obtained at a tempera ture , T2, and a
s t ra in rate, e2, such tha t ~:2 = ~:1 / aT2/T 1 where aT2/T 1 is the shift factor between t e m p e r a t u r e s T 1 and T2, given by the classical WLF equat ion [8]. The coefficients obtained by Plazek [41] have been used throughout this work (C 1 = 9,76 and C 2 = 64,8 for a reference tempera ture of 115~
In more recent exper iments , the samples were s t re tched at 0,1 s -1 at a given t empera tu re and at a draw ratio k = 4. The samples were then allowed to relax u n d e r cons tant conditions (length, t empera ture) for a given aging time rang ing from 3 to 103 s and subsequent ly quenched to room tempera ture . The res idua l or ienta t ion was thus frozen at a given aging time. The t ime range explored by the exper iments can be extended at both ends by changing the s t re tch ing tempera ture . The data points represent an average over 3 samples s t r e t ched u n d e r the same conditions. A m a s t e r curve of the or ienta t ional re laxa t ion , covering a lmos t 4 decades in time, can be obtained by applying t ime- tempera tu re superposition. On samples which have been subjected to both procedures it was checked tha t the mas te r curves coincide within the accuracy of our measu remen t s .
These s t re tch ing conditions can be compared with those of Miiller and Picot described in Chapte r 1.4 of the present book. Our samples were obtained at comparat ively shor ter times, using higher s t ra in rates. Deviations from the l i nea r v iscoelas t ic b e h a v i o u r in s t r ess g rowth are expected. As far as re laxa t ion is concerned, the shortes t t ime invest igated in neu t ron scat ter ing e x p e r i m e n t s cor responds roughly to the longest ones s tudied by infrared dichroi ' sm.
3.3. Orientat ion m e a s u r e m e n t s The inf rared spectra were recorded on a Nicolet 7199 or a Nicolet 205
Four ie r t r ans form infrared spectrometer at a resolution of 2 cm -1 with a total n u m b e r of scans of up to 128 for low-orientation samples. The infrared beam was polarized using a SPECAC gold wire-grid polarizer. Samples, r a the r than po la r i ze r , were r o t a t e d 90 ~ in order to obta in the two po la r i za t ion m e a s u r e m e n t s . The dichroic ra t io was ca lcu la ted from the m e a s u r e d absorbance at the max imum of the infrared band.
4. Theore t ica l basis of in t e rp re ta t ion
The molecular-level na ture of our orientation relaxation data has led us to carry out an in te rpre ta t ion in te rms of molecular motions. Therefore we have based our in te rpre ta t ion on the Doi-Edwards model [9]. We briefly recall the main qual i ta t ive features of the model and include other processes proposed by
42
various authors . The relaxat ion will be described from shor t to long t imes following a step strain. It is assumed that , at the end of the step s train, the Rouse chains and subchains be tween e n t a n g l e m e n t points are aff inely deformed.
4.1. l , i n e a r cha in s The first re laxat ion process (designated A hereaf ter) corresponds to a
Rouse-like re laxat ion of chain segments between en tang lemen t points. I t is assumed tha t the entanglement points remain fLxed dur ing the t ime-scale of this re laxat ion and tha t no diffusion of monomers th rough the slip-links is allowed in such short times. The associated relaxation time, ZA, is related to a
monomeric friction coefficient, ~0, and to the number of monomers between
entanglement points N e. ~A is given by :
3:A ~0 a2 = ~ Ne 2 (3) 6/1;2kBT
where a is the monomer length, k B the Boltzmann constant and T the absolute temperature . This relaxation time is thus independent of the molecular weight of the chain and as a consequence of its local character, the re laxat ion kinetics for this process are independent of the position of the segment along the chain.
The second process (B), characterized by a relaxation t ime x B proportional to the square of the molecular weight of the chain, is a re t rac t ion of the deformed chain inside its tube to recover its equil ibrium eurvi l inear length. This process has a strong nonlinear character since its ampl i tude is strongly increased by large deformations [10]. From the diffusion equat ion used to determine the kinetics of this process, it is clear that the retract ion motion first takes place at the ends of the chain (with short modes) and diffuses towards the centre. The re laxat ion is thus predicted to be non-uniform along the chain. However, at the end of the retract ion process, and a s suming tha t no other process has s tar ted, the orientation is constant along the chain and again independent of the position of the segment.
It has been shown by Doi [11] tha t on the time scale of XB contour length f luctuat ions may induce a rapid re laxat ion of chain ends, especial ly for modera te ly long chains. Indeed, wiggling motions involve forward and backward motions of the chain ends. Thus, chain length f luc tua t ions in oriented mater ia ls lead to the creation of isotropie par ts of tubes at each end. Their fractional length is roughly equal to 1.3(N/Ne)-I/2, where N is the number of monomers per chain. An elaborate expression for the re laxat ion due to this mechanism, based on the Pearson-Helfand picture for s tar polymers [12], has been proposed by Viovy [ 13].
The next relaxat ion stage (C) corresponds to the rep ta t ion process itself where the chain relaxes towards an equilibrium isotropie configuration. The corresponding re laxat ion time scales as the thi rd power of the number of entanglements per chain, and thus as the third power of the molecular weight. During this process, chain segments are assumed to lose thei r orientat ion as they leave the original tube, which induces a faster relaxation of chain ends.
This model involves only one adjustable relaxation time, XA, for instance, since the others are linked by the following sealing laws :
43
1: B = 2 (N/Ne)2 1: A (4)
xC = 3 (N/N e) 1: B (5)
4.2. Star polymers Following a step strain, the first relaxation motions of these polymers are
almost identical to those of l inear chains. Namely, a s tar polymer is assumed to relax locally by a type A process and a retract ion of its s tretched arms (B process). However, it has been shown that the reptative motion of l inear chains is quenched by branching [14]. The relaxation is supposed to involve, via large scale fluctuations, the "pushing back" of the end of an arm close to the branch point. The orientation, like the stress, is proportional to the portion of the tube still unvisi ted by a chain end. The relaxation time associated with these large- scale b rea th ing modes is given by :
Tm = q:A / Narm/3/2 exp(v Narm/Ne) (6) ~V Ne/
where Narm/N e is the number of entanglements per a rm and v a numerica l constant of the order of 0.6 [14]. Expressions for the kinetics of this large-scale f luctuat ion process have been given in the l i te ra ture [12,13,15] and allow calculation of the relaxation as a function of the position of a segment along the arm. Qual i ta t ive ly , a strongly delayed relaxat ion around the b ranch point compared to l inear chains is predicted due to the exponential dependence of the relaxation t ime on the length of an arm.
For both l inear and s tar polymers, the above-described theories assume the motion of a single molecule in a frozen system. In polymers melts, it has been shown, essential ly from the s tudy of binary blends, tha t a self-consistent t r ea tmen t of the relaxation is required. This leads to the concepts of "constraint release" whereby a loss of segmental orientation is permit ted by the motion of su r round ing species. Retract ion (for l inear and s ta r polymers) as well as repta t ion may induce constraint release [16,17,18]. In the homopolymer case, the main effect is to decrease the relaxation times by roughly a factor of 1.5 (XB)
or 2 (xC)- In the case of s tar polymers, the factor v is also decreased [15]. These effects are extensively discussed in other chapters of this book especially for binary mixtures. In our work, we have assumed that their influence would be of second order compared to the relaxat ion processes themselves. However, they may contr ibute to an unexpected relaxation of par ts of macromolecules which are assumed not to be reached by relaxation motions (central par ts of linear chains or branch point in s tar polymers).
4~3. B ina ry b lends The theory of the relaxation of binary blends is mainly limited to the linear
regime and devoted to mixtures of long and short chains, both species being sufficiently long enough to be entangled [19,20,21]. Another case corresponds to long en tang led polymer chains diluted with shor ter ones with molecular weights comparable to the en t ang lemen t molecular weight. It is usua l ly admi t t ed t h a t u n e n t a n g l e d shor t chains main ly act as d i luent species, reducing the number of en tang lements per unit volume and increas ing the average molecu la r weight be tween them. These conjectures have been
44
successfully tes ted exper imenta l ly in the l inear viscoelastic domain [22]. However, rheological measurements , in the linear as well as in the nonl inear regime, only give information on the overall behaviour of the mater ial . A bet ter ins ight into the re laxa t ion mechanisms of individual molecules is again obtained through spectroscopic techniques [23,24]. For instance, the individual r e l axa t ion of long and shor t chains in a b inary blend, in s h e a r step experiments, has been followed by infrared dichroism using deu te ra ted species [24,25]. These exper iments have shown tha t constraint release theory could qualitatively predict the behavior of long chains, but tha t the relaxat ion of short chains was retarded. Diffusion experiments on isotropic samples, on the other hand, had shown tha t the diffusion coefficient of short chains was independent of the molecular weight of the matr ix [26]. It has therefore been suggested tha t intermolecular orientat ional coupling between the relaxing species could take place leading to a res idual orientat ion of short chains even at t imes much longer than their individual relaxation times.
The aim of our study on binary mixtures was twofold : - checking, us ing in f ra red dichroism the existence of o r i en t a t iona l
coupl ing which had been observed by ano ther spectroscopic t echn ique (fluorescence polarization) [27]
- invest igat ing the influence of short chains, with length comparable to the entanglement length, on the relaxation of the long chains.
5. Resul ts a n d d i s c u s s i o n o n isotopically labeled c h a i n s
5.1. I n f l u e n c e o f t h e p o s i t i o n o f t h e segment We compare the orientational relaxation of central deutera ted pa r t of the
PS HDH 188 copolymer with tha t of the end part of the PS DH 184 copolymer. Both types of chains have almost the same molecular weight as well as deutera ted blocks of comparable length. The master curves of the orientationa] re laxa t ion average (calculated as the weight average of the m e a s u r e d orientation of each block) can be superimposed as shown in Figure 1.
0 , 3 L , , n , , ~ , , , , . . . . , . . . . , . . . . j
[ �9 I
. 1 ^ 0 , 2 [ ]
o o
0.1 a v [] Q
. J
~ i �9 vlo
_ , ,
-1 0 1 2 3 4 - log ~ aT
Figure 1. Relaxation of average orientation of PS DH 184 (e) and PS HDH 188 (~q) at a reference temperature of 120~
45
The comparison between the relaxation of the deuterated end block of the PS DH 184 and tha t of the central block of the PS HDH 188 is shown in Figure 2. The non-uniformity of the relaxation along the chain is clearly apparent. More precisely, whereas at short times (- log ~a T < 1) the orientation of both species is quite similar, the relaxation of the central par t is clearly slower than that of the end block at longer times.
0,3 ' ' ' ' I ' ' ' ' 1 ' ' ' ' I ' ' ~ ' I ' ' ' '
-~ �9
, . o
0,2 A .
o
- 0,1 V
"" . . ~ . . ~ ' ~
I I U . . . . . . . . . . . . ~ , , ,~.i . . . .
-1 0 1 2 3 4 o
- l o g e a T
Figure_2. Relaxation of the end block of PS DH 184 (e) compared to that of the central block of PS HDH 188 (o)
The orientat ion of the central block shows a plateau in an intermediate t ime scale. It can also be noted that at very short times, the orientation of the end part is slightly below tha t of the central block. This effect can be accounted for by the fact that, during the deformation process, no topological constraint is acting at the end of the chain to orient it. This behaviour is similar to dangling chains in elastomeric networks [28].
0,3
m
~= 0,2 A
r
o r
0 , 1
v
-1 4
. . . . I ' ' ' ' I ' ' ' ' I . . . . I ' ' ' '
a
2~. . tX
, , , "',",-~--"
0 1 2 3 - l o g ~: a T
Figure 3. Comparison between the relaxation of the short end block of PS DH 184 (o) and that of the long end block of PS HDH 188 (~)
The influence of the length of the end part on its relaxation is depicted in Figure 3 where the re laxat ion of an end par t of molecular weight 27000 is
46
compared to t h a t of an end pa r t of molecular weight 78000 for an ident ica l overal l cha in length . The longer the end par t , the slower the r e l a x a t i o n (especially a t long times).
52.~ In f luenee of t h e l e x ~ h of the c h a i n D a t a obta ined on the PS DHD 500 triblock copolymer can be used to
compare the re laxat ion of an end block of molecular weight 27000 wi th t h a t of an end block of molecular weight 39000 but located on a chain roughly 2.5 t imes longer. The resul ts are given in Figure 4 where it appears tha t the longer the overall chain, the slower the relaxat ion of the end block a t long t imes.
0 ~ 3 . . . . I . . . . I . . . . I . . . . I . . . .
A 0,2
O
v 0,1
O "�9176
"~'... "'"" :i , , " . . . . . o : . . : . . . a _ ~
0 I 2 3 4 - l o g ~: a T
F igure 4. Comparison between the relaxation of the end block of PS DH 184 (~) and tha t of PS DHD 500 (o)
5.3. D i scus s ion Previous studies dealing with the influence of the molecular weight on the
o r ien ta t iona l re laxat ion average have shown tha t the or ienta t ion re laxa t ion m a s t e r curves could be divided in two par ts [7]. At short t imes (for - log ~a T < 1) the re laxat ion is independent of the molecular weight whereas at longer t imes the r e l a x a t i o n is slowed down as the molecular we igh t inc reases . This qual i ta t ive agreement with the predictions of the Doi-Edwards model led to the conclusion t ha t the first s tages of the re laxat ion m e a s u r e d e x p e r i m e n t a l l y correspond to the A process, for which a strong increase in or ientat ion a t short t imes and a molecular weight independence are expected. A fi t t ing procedure wi th the theoret ical predictions was used to determine the re laxat ion t ime ~A,
giving ~A = 5.6 s at 120~
In order to d i sc r imina te be tween subsequent re laxa t ion processes, it is necessary to calculate the or ientat ional re laxat ion as a function of the location of a segment along the chain and to compare the resul ts of such predict ions to e x p e r i m e n t a l da ta . Theore t ica l express ions are given e lsewhere [29] and reference should be made for fur ther details. In this work, the more e labora te t r e a t m e n t of the chain f luctuat ion process has been used and equations used in the calcula t ion of or ien ta t ion have been extracted from reference [13]. The
47
var ious re laxa t ion t imes involved in the theore t ica l calculat ions have been related to 1: B using the theoret ical prefactors and scaling laws.
In order to avoid the use of ad jus t ab l e p a r a m e t e r s on resca l ing the theoret ical wi th the exper imenta l or ientat ion, and since we were in te res ted in differences in the re laxat ion kinetics, we chose to s tudy the or ienta t ion of the dif ferent blocks no rma l i zed by the ave rage or ien ta t ion , as a funct ion of r e laxa t ion t ime. The r e l axa t ion t imes have been ad imens iona l i zed by the re t rac t ion t ime, XB, which can be de t e rmined exper imenta l ly by applying the theoretical scaling law (see section 4.1).
The inf luence of var ious r e l axa t i on processes ( re t rac t ion alone, r ep t a t i on alone, chain-length f luctuat ion alone) on the relat ive re laxat ion of an end-block of size equal to tha t of PS DH 184 is compared to exper imenta l da ta in Figure 5. It can be seen tha t none of these processes alone can account for the observed behavior.
~0
;>
r
~9 c,4
0,8
0,6
0,4
0,2
- X -_
. . . . I . . . . I, ~ ~ " - - - - -4 -3 -2 - 1 0 1 2
log t / x B
Figure 5. Influence of various re laxat ion processes on the relative or ientat ion
of chain ends ; ( ...... ) " re t rac t ion ; ( - - ) " chain length f luctuat ion ; ( . . . . )" reptat ion alone; (~) �9 exper imenta l data of PS DH 184
wi th e s t ima ted error bars.
The data agree with the a lmost fiat re laxat ion at short t imes and show the impor tance of the chain r e l axa t ion process t a k e n into account but clearly disagree at longer t imes. Be t te r a g r e e m e n t is obta ined if the th ree re laxat ion processes are considered s imul taneous ly , as shown in Figure 6.
Figures 7 and 8 shows the same predict ions applied to the case of the PS DHD 500 and PS HDH 188 copolymers. Again, a g r e e m e n t at shor t t imes is reasonable whereas at longer t imes, the relat ive or ienta t ion of the end block is u n d e r e s t i m a t e d .
48
r
r
r r
0,8
0,6
0,4
0,2
. . . . I ' ' ' ~ 1 . . . . I . . . . I . . . . I . . . .
-4 -3 -2 - 1 0 1 2
l o g t / z a
Figure 6. Relative relaxation of a chain-end in PS DH 184 (o) with est imated error bars, compared to the prediction of fluctuation, retraction and reptation
processes .
0 , 8 . . . . I . . . . I . . . . J . . . . I . . . . I . . . . i . . . . 1
-
r ; > = 0,6
~q
- ~ 0,4
0 - - . . . . : . . . . I . . . . I . . . . I , _
-5 -4 -3 -2 - 1 0 1 2
log t / x e
Figure 7. Relative relaxation of chain-end of PS DHD 500 ( 0 ) w i t h est imated error bars compared to the prediction of fluctuation, retraction and reptation
processes .
These results , as compared to a previous study [29], i l lustrate a better agreement between theory and experiments . This can be attributed to a better a p p r o x i m a t i o n of the mode dis tr ibut ion involved in the chain f luctuat ion process and to which the end of the chains are quite sensitive. These data also prove that under our exper imenta l conditions, the relaxation of the chain is dominated by the chain retraction and chain- length f luctuation process. The
49
influence of reptation is very weak and only the first modes contribute to the relaxation of chain ends.
~9 ~0
;>
c,t
~9 eq
1,2
0,8
0,6
0,4
0,2
-5 -4 -3 -2 - 1 0 1 2
log t / gB
Figure 8. Relative relaxation of chain end of PS HDH 188 (~) with estimated error bars compared to the prediction of fluctuation, retraction and reptation
processes.
These experimental results which validate the basic processes of the Doi and Edwards model are in agreement with findings of Ylitalo et al. [30] who have been able to draw the same conclusions in step-shear flow experiments. Relaxation of a centrally deuterated chain in a high molecular weight matrix has led to the same conclusions [31]. However, other a t tempts have not been able to distinguish between the relaxation of different parts of chains [32,33]. Discrepancy between the results of the various experiments may be partly due to a different contrast between the relative length of the blocks. Clearly, most discr iminat ing results are obtained when the invest igated blocks are ra ther small, as it is particularly apparent in ref 30.
6. Resul ts and discuss ion on isotopical ly label led 6-arm stars
In this section, we will compare resul ts obtained on isotopically labelled polystyrene stars with those of the l inear copolymer. With regard to the characteristics of the star, we may regard the linear polystyrene of molecular weight 180 k as having roughly the length of one arm (PS DH 184), the linear polystyrene of Mw = 500 k as having roughly the lengths of two arms (PS DHD 500) and a pure homopolymer of molecular weight 1.19 106 g/mol as having roughly the same molecular weight as the overall s tar (PS 1190). Inside a star, we may differentiate between the segments at the branch point (deuterated monomers located close to the branch point in s tar b), those of the end of an arm (deuterated monomers located at the ends of the arms of star b), and those of the long central block of star-a (hydrogenated monomers of star-a).
50
The results are reported as a function of t/x A since x A is independent of the molecular weight and na ture (linear or s tar shaped) of the polymer. This avoids the arbi t rary choice of a reference temperature
6.1. O r i e n t a t i o n o f v a r i o u s p a r t ~ o f a s tar The or ientat ional relaxation of various parts of the s tar is depicted in
Figure 9. Segments close to the branch point appear initially more oriented than those of the central block. The end part of an arm is clearly less oriented at short times. As relaxation proceeds, no difference can be made between the re laxa t ion of the branch point and of the central block. The end block nevertheless relaxes faster.
6.2. Compar i s on be tween ~ and l inear chains The average orientation of the 6-arm star is compared to that of l inear
chains PS DH 184, PS DHD 500 and PS 1190 in Figure 10. It seems quite clear tha t the short-time behaviour is almost the same in linear and star materials, independent ly of the molecular weight. In contrast, the relaxation of the star, at longer times, is intermediate between that of the linear chain PS DHD 500 and tha t of PS 1190. It is therefore more rapid than tha t of a linear chain of the same overall molecular weight but slower than that of an unbranched chain of the same length (PS DHD 500). This suggests that long relaxation times are increased or even suppressed by branching.
A
O ~ 9
r
V
0,2
0,15
0,1
0,05
e, �9 "',0
�9 ,
( D . " "O. . .
~ ,
, -
- 1 -0,5 0 0,5 1 1,5 2 2,5 3
log t/XA
Figure 9. Relaxation of various parts of a s tar" branch point (o), central block (o) and chain end (~)
In order to test whether the chain ends or the central part are responsible for this behaviour, the orientation of both blocks has been compared to that of l inear chains. The relaxat ion of chains ends in the s tar polymer and in copolymers PS DH 184 and PS DHD 500 is compared in Figure 11. The short time orientat ion is similar for the 3 materials, but the end of an arm relaxes slower than that of the PS DHD 500 copolymer although the involvec( length is
51
slightly lower and the length of the arm also lower t han the half of the l inear chain.
A
r
�9 r
r
V
0,2
0,15
0,1
0,05
. . . . I . . . . I . . . . I . . . . 1 . . . .
x
:, \
n ~
m - - j �9
~ . _ X x X " ' n . �9 . -. . . . . . . . . X X . X -
�9 g
-1 0 1 2 3 4 log t/~ g
Figure 10. Comparison between the relaxation of average orientation of a s tar (x), PS DH 184 (o), PS DHD 500 (O) and PS 1190 (m)
As far as the orientation of the central block is concerned, a slower relaxation is also observed for the star as shown in Figure 12. At long times, the central block of the star remains more oriented t han tha t of the l inear chain. The differences are significant and cannot be explained by simple differences between length of chains and blocks. This effect must therefore be a t t r ibuted to the "pinning" effect of the centre of the chain and subsequent suppression of the reptation motion.
0,2
/k
(2:)
�9 r
V
0,15 ~ ~
0,1 , . , -,
0,05 -~
I
-1 0 1 2 3 4
log t/x A
Figure 11. Relaxation of chain ends ofPS DH 184 (~1), PS DHD 500 (O), Star (0)
52
A
o
V
0 , 2 , , , , i , , , , i , , , , , , , , , I , , , ,
f _ ~, _
�9 -
0 , 15 -
O
0,1 � 9
I - o , o 5 - o
L ' --,,.
-1 0 1 2 3 4
log t/x A
Figure 12. Comparison between the relaxation of the central block (o) of the PS DHD 500 and of the central block of star (~)
6 . 3 . D i s c u s s i o n Before going into a detailed comparison of experiments with theoretical
predictions, the main points of this study may be emphazised. The relaxation of the star is intermediate between that of the PS DHD 500
and PS 1190 polymers. In this range of molecular weights, the viscosity and the te rminal relaxation times are highest for the star. This indicates tha t under our expe r imen ta l condit ions, the re laxat ion is not domina ted by the disengagement of the branches. However, the effect of pinning the centre of the chain is clear from the slower relaxation of the s tar (or central block) as compared to the PS DHD 500 copolymer. The last point deals with the surpris ingly higher orientation of the ends of the arms. Since the length of this par t of the arm is on the order of one tenth of the length of an arm, we expect a quite rapid relaxation, at least at much shorter times than the arm fluctuation time. The strong orientation observed might be qualitatively interpreted as an effect of an orientational coupling with surrounding segments. The difference between s tar and linear species would be due to a higher average orientation in the s tar as seen in Figure 10.
As far as the fluctuation dynamics of an end-attached chain is concerned, a characterist ic length is defined as the size of the thermal fluctuation of the tube which is given by :
Leq (7) Xeq = # 2 v Narm/N e
where Narm/N e is the number of entanglements per arm and Leq is the equilibrium contour length of an arm, simply given by Leq = d Narm/Ne, d being the average contour length between entanglements. The associated relaxation time, ~eq, is given by :
53
d2 (Narm/Ne)2 (8) T, eq = 2 v k T
being the ave rage friction coefficient ac t ing on the d i s tance d (~ = N e ~0)- The re laxa t ion t ime assoc ia ted wi th the complete re laxa t ion of an a r m is r e l a t ed to
l:eq by "
---- ~eq( ll~ }112 exp(V Narm/Ne) (9) Tm V Narm/Ne !
All t he se r e l a x a t i o n t imes can be expressed as a func t ion of the local
r e l axa t ion t ime ~A. Sehemat iea l ly , a t any t ime following the s tep de fo rma t ion process, the s t ress is a s s u m e d to be re leased only on the l eng th of an a r m t h a t has been vis i ted by a f luctuat ion. A rapid re laxat ion of cha ins ends is expected
on the t ime scale of Zeq for d i s t ances s m a l l e r t h a n Xeq a n d the d y n a m i c s proposed by Viovy [13] can be used as for l inear chains . At longer t imes , the
fract ion {(t) of an a r m which has been re laxed by a f luc tua t ion can be computed as the solut ion of the following equa t ion"
t = l:eq (v Na~m/Ne)n I/2 1__~ exp(VNarm~Z/Ne) (10)
and the s t ress is propor t ional to (1 - ~(t)) for t imes such as Xeq < t < Xm �9 Since
our expe r imen ta l condit ions correspond to re laxa t ion t imes on the order of Zeq,
the app rox ima te express ion for the re laxa t ion a t t imes longer t h a n Xm has not been considered.
1,2
0,8
0,6 co . , . . ~
~9 0 4 0 ,
[" 0,2
, , l I ' ' ' I ' ' ' I ' ' ' I ' ' J ) ' ' '
mlmm dmNlm ml'Imm mm m ~ ........................................... �9
- 0 2 , J , I , ~ , I , , , I . . . . I , ,
-4 -2 0 2 4
log t / Z B
i i
I
6 8
F igure 13. Theoret ical or ientat ion of var ious pa r t s of a s t a r :
( - - - ) end of a rm, ( ) average , ( ....... ) b r a n c h poin t a s s u m i n g re laxa t ion by a rm f luctuat ion alone (l ight l ines)
or combined with re t rac t ion (heavy lines).
54
It is thus possible to compute the relaxation of different par ts of the s tars using this f luctuation process alone or combined with the retract ion process. As an illustration, the orientational relaxation of a short end-block (like in star- a) is compared to the average orientation and to the orientation of the branch point (like in star-b) in Figure 13. Since relevant relaxation processes occur on the time scale of XB, theoretical curves are presented with t/x B as abscissa. Although a very large difference can be expected if only the fluctuation process is taken into account, the retract ion of the arms increases significantly the overall relaxation and tends to induce less-differentiated behaviours.
As far as q u a n t i t a t i v e compar ison with the exper imenta l da ta is concerned, we decided to compare directly the exper imental ly de termined orientat ion with the theoretical prediction. This requires a rescaling of the theoretical orientation in order to match it with the experimental data at t imes of the order of XA. A sat isfactory agreement for the average orientat ion is observed as shown in Figure 14.
1,2 - , , , , , . . . . I . . . . I ' ' ' ' 1 ' ' ' ' 1 ' ' ' "
1 r
"~ 0,8 ~9
> 0,6
a, 0,4
0,2
[]
-4 -3 -2 -1 0 1 log t / x
B
Figure 14. Comparison between the experimental rescaled average orientation of a s tar (~) with theoretical predictions (full line).
Using the same rescaling, the orientation of the short end-block and of the central block of the s tar can be compared with theoretical predictions as shown in Figures 15 and 16. This quantitative comparison between the theoretical and experimental relaxation of star polymers shows that the main observations can be accounted for by the combination of the arm fluctuation and arm retraction processes. A reasonable order of magni tude of the orientation of chain ends can be theoretically predicted. However, the relaxation of the branch point is significantly more rapid than predicted theoretically. This discrepancy might be a t t r ibuted to constraint-release processes which tend to relax the branch point of the star. The influence of the lat ter processes is more easily observed on the branch point where the relaxation should appear at longer times.
55
I . . . . I ' " ' ' ' I . . . .
0 , 8
0,6
= 0,4 q:~
0,2
L []
0 . . . . . . , . . . . , . . . . , , . . . . . . . . .
-4 -3 -2 -1 0 1 2 log t / ~ B
Figure 15. Comparison between theoretical (full line) and experimental (el) relaxation of chain ends of star-a.
1 , 2 . . . . - , - , , ,D , , , , , 1 . . . . , . . . . i . . . .
~ 0,8 ~Ez
0,6
= 0 , 4
~ 0,2
0 . . . . ! . . . . I , ~ ~ , , I , , , , [ ~ , , , I . . . .
-z -3 -2 -1 0 1 2 log t / "tB
Figure 16. Comparison between theoretical (full line) and experimental (a) relaxation of the branch point of star-b.
7. Resu l t s and discuss ion o n b i n a r y b l e n d s o f l ong a n d shor t c h a i n s
Binary blends composed of deutera ted short chains and of en t ang led hydrogenated long polystyrene chains have been investigated. The re laxat ional behav iou r of the shor t chains has been ana lysed as a funct ion of t he i r molecular weight. Additionally, a t tent ion has been focussed on the role of the short chain concentration on the orientational relaxation of the long chains.
F igure 17 shows mas te r curves of the orientat ional re laxat ion of shor t deutera ted chains of various molecular weights present at 20 wt.-% in a mat r ix of molecular weight 2x106 g/mol (PS 2000). The master curves have been d rawn at a reference t empera tu re of Tg + 9 ~ using t ime- tempera ture superposit ion.
56
This choice rel ies on the a s s u m p t i o n t h a t a c o n s t a n t Tg+T cor responds to a cons tan t free volume s ta te . Such an approx imat ion p r e s u m e s t h a t the t h e r m a l
expans ion of free volume, af, as well as the f ract ional free volume, fg, a t Tg, are i ndependen t of the b lend composit ion. In the case of b ina ry blends wi th shor t
chains, x A m a y depend on the composit ion of the blend, so t h a t a no rmal i sa t ion of expe r imen ta l t imes wi th th is p a r a m e t e r would be mis leading .
0.20 ' " ' ' , . . . . u . . . . 1 . . . . I . . . . I ' ' ' ' J . . . .
A
r
C )
r
V
0.15
0.10
?
/b O \
~ ~ \
x o
\ %
0 \ �9
.
\ o \
0.05 - . ,~ ~ \ .,o~ ,-I.
�9 . ~ 5 - -~ ----- � 9 o- " * - " - - : c ~ . o. _ o - o r - ~ , b T M - ~
0.00 , . , t . . . . , , o , , , . . . . , . . . . . , . . . . . . . . . .
-1 0 1 2 3 4 5 6
log t (Tg+9~
Figure 17. Or ien ta t iona l re laxa t ion of the shor t d e u t e r a t e d chains PSD72 (o), PSD27 (O), PSD18 (i), PSD10 (~) and PSD3 (#),
b lended a t 20 wt.-% with PS2000.
While the or ien ta t ion of these deu te r a t ed chains shows a s t rong molecular weight dependence espec ia l ly a t shor t t imes , all cha ins exhib i t a r e l a t ive ly cons tan t r e s idua l o r i en ta t ion a t long t imes. Even cha ins of molecu la r we igh t as low as 3000 p r e s e n t a non-zero local o r ien ta t ion . This was conf i rmed by po la r i sa t ion m o d u l a t i o n i n f r a r e d d ichro ism - a t e c h n i q u e sens i t ive to ve ry small anisot ropies [34].
The shor t cha ins wi th M<M e can be considered as behav ing like Rouse chains wi th re laxa t ion t imes on the order of the e x p e r i m e n t a l t ime-scale . For example, the Rouse t imes of PSD3 and of PSD10 are app rox ima te ly 9s and 19s seconds, respect ively a t 115~ [35,36]. Consequent ly , the res idua l or ienta t ion a t long t i m e s for t he se c h a i n s can be a t t r i b u t e d to o r i e n t a t i o n a l coup l ing in te rac t ions wi th the long cha ins of the polymer mat r ix . S imi l a r o r i en ta t iona l corre la t ions have been observed on var ious sys t ems by 1H N M R spect roscopy s tudies of s t r e t ched e l a s t o m e r s whe re even dissolved solvent molecules and free cha ins were shown to possess a very shor t - l eng th scale local o r i en ta t ion [37].
The coupling coefficient, E, proposed by Doi et al . [38] has been ex t rac ted from the o r i en ta t ion da t a , y i e ld ing a va lue of c a . 0.26 [36]. However , the u n c e r t a i n t y in the o r i en ta t ion m e a s u r e m e n t s a t long t imes does not allow the
57
de te rmina t ion of a molecular weight dependence of E, as observed by Ylitalo et al. [39] in a s tudy of polybudadiene binary blends.
The cooperat ive na tu r e of the o r ien ta t iona l re laxa t ion in a s y m m e t r i c b inary blends has been fur ther invest igated through the analysis of the effect of the shor t chain concentration, for a given short chain length, Mw = 10 000.
M a s t e r curves of the o r ien ta t iona l r e laxa t ion of the long chains are p r e s e n t e d in F igure 18. The long chain re laxa t ion in the pure po lymer is compared to t h a t in the blends which conta in increas ing concent ra t ions of shor t chains .
0.30 . . . . I " " ' ' 1 . . . . I ' ' ' ' 1 ' ' ' ' 1 . . . . I . . . .
_
O \
^ 0.20 - r
OX VS. ~ _
o \ _ o',~,,o o -
' ~ " "o �9 v 0.10 - tb " O ~
.
m--. o o ~ ~ - a - - 0--10 ~ 0 Dq~- O
0.00 . . . . I . . . . I . . . . ~ . . . . I . . . . i . . . . I . . . .
-1 0 1 2 3 4 5 6
log t (Tg+9~
Figure 18. Mas te r curves of the or ientat ional relaxat ion of the long chains, at Tg+9~ for PS2000 (m), PS2000/PSD10(20 wt.-%) (o)
and PS2000/PSD10(30 wt.-%) (o).
In addi t ion to the usual charac te r i s t i c rapid decrease in or ien ta t ion at short t imes before reaching a p la teau at long t imes, a decrease in or ientat ion is observed as the concentrat ion of short PSD10 chains increases to 30%. Such a high concentra t ion of short chains is required to affect notably the re laxat ion of the long species. Indeed, Figure 19 shows a negligible difference be tween PS 2000 and blends containing 20 wt.-% of short chains, even of very low molecular we igh t s .
One convenien t s t r a t e g y to i n t e r p r e t these resu l t s is to rev iew the mo lecu la r charac te r i s t i cs of b inary blends as ex t rac ted from polymer mel t rheology [40]. The influence of short chains (M < M e) is to effectively decrease the p l a t e a u modulus and the t e rmina l re laxa t ion t imes as compared to the pure polymer . Consequent ly , the molecula r weight be tween e n t a n g l e m e n t s
0 0 which is re la ted to G N by G N - pRT/Me is increased. The short chains, in other words, cause a typical di luent effect and thus the " temporary e n t a n g l e m e n t network" is looser. Therefore, a lower or ienta t ion of long chains is expected.
58
However , an increase in M e also implies an increase in I:A, as given by equation (3) and, therefore, the relaxat ion kinetics should be slower.
0.30
^ 0.20
0
e , i
a, 0.10 V
0.00 -1
~
~~ �9 �9
o o 5 " - . ~
0 l 2 3 4 5 6
log t (Tg+9~
Figure 19. Master curves of the orientat ional relaxation of the long chains at T= Tg+ 9 ~ PS2000 (0), PS2000/PSD18(20 wt.-%) (o) and
PS2000/PSD3(20 wt.-%) (0).
Besides Me, the relaxation t ime XA is proportional to a monomeric friction
coefficient, ~0. This p a r a m e t e r , which descr ibes the fr ic t ional d r ag per monomer unit as it moves th rough its environment , depends on tempera ture , pressure and molecular weight (for low molecular weights) [8].
Rheological s tudies of the und i lu ted polymer (PS2000) and the blend (PS2000/PS10) containing 30 wt.-% short chains have confirmed a decrease in the pla teau modulus with the addition of the short chains. Mas te r curves of the s torage G'(co) and loss G"(co) moduli of the pure polymer and the blend are presented in Figure 20. The moduli were measured at t empe ra tu r e s rang ing from Tg + 100 to Tg + 30~ and data were reduced to a reference t empera tu re of 165~ The horizontal shift factors, aT, were used to evaluate the WLF coefficients C1 and C2 [8]. No a p p a r e n t var ia t ion in the product C1C2 was detected on dilution with the short chains. In fact, an average product of 650 + 30 was obtained for both the pure polymer and the blend, consis tent with various other reported values [41,42].
Apar t from a reduction in the width of the p la teau region and a lower
pla teau modulus (G~ 0 , N blend = 0.5 G N pure polymer), the blend exhibits a shift
in the G'/G" crossover modulus and f requency in the t r ans i t i on zone. G o N
corresponds to the value of G'(co) at the min imum of tan 5 = G"(c0)/G'(o~).
59
106
10 5
lO 4 f
103 ........ ' ........ i ........ l ........ , ........ , ........ l ........ . ........ i ........ I 10 .4 10 -2 100 102 104
T
Figure 20. Mas te r curves of G'(o)) (heavy lines), G"(co) at T = 165~ for PS2000 (full lines) and PS2000/PS10(30 wt.-%) (dotted lines).
Due to difficulties in m e a s u r i n g the ze ro-shear viscosity of such high mo lecu l a r w e i g h t po lymers , and thus deducing the monomer ic fr ic t ion coefficient f rom Graess ley ' s ' uncor re la t ed d rag model ' [43], the following equat ion adapted from the modified Rouse theory has been applied [8].
G'(o~) = G"(oJ)= (appNA/4Mo)(~kBT/3)Y2o) 1/2 (11)
In equa t ion 11 "ap" is a character is t ic length ( taken to be 7.4 .s [8]), p the dens i ty , NA Avogadro 's number , Mo the monomer molecular weight , k B
B o l t z m a n n ' s cons t an t , T the abso lu te t e m p e r a t u r e and co the a n g u l a r frequency. Al though this equat ion is based on the behaviour of low molecular weight , u n e n t a n g l e d polymers, only the difference, r a t h e r than the absolute va lue of fr ict ion coefficients be tween the pure polymer and the blend, is s ignif icant . F u r t h e r m o r e , the value of ~o ext rac ted for the pure polymer is comparab le wi th t h a t de t e rmined by en t i re ly i ndependen t e s t ima tes from forward-recoil spectroscopy measu remen t s [44].
The resu l t s have indicated tha t the monomeric friction coefficient is sens i t ive to the presence of the shor t chains, and is smaller , a t a given t e m p e r a t u r e , t h a n the homopolymer. This observed reduct ion in ~o has been shown to be only a t t r ibu ted to an increase in the fractional free volume. Indeed, the t e m p e r a t u r e dependence of the monomeric friction coefficient is described by a WLF-type equation :
~~W)._ =- CI(T-To) (12) lo ~To C2 + (T-To)
This equat ion can be rewri t ten as :
60
CzC2 _ log~o(T) - C g + 1 (13) log~o(T) = log~o(Tg)- C g + (T-T=)- 2.303 fw
where T~ = Tg-C~ is the Vogel temperature , tha t is, the t empera tu re at which the free volume vanishes. As mentioned above, the product C1C2 is the same
for both samples. Consequently, plots of log ~0 as a function of 1/(T-T~) present identical slopes for the homopolymer and the blends. In fact, as shown in Figure 21, not only are the slopes the same, but the intercept does not vary. Thus, the change in log ~0 upon dilution with shorter chains is only controlled by T~.
A fur ther step in the analysis of the orientation curves is to consider the role of ~0 in the re laxat ion processes t ak ing into account the rheological measurements . An alternative method of present ing the orientat ion-relaxat ion curves is to compare the pure polymer and the blend at t empera tu res at which ~o is the same (115~ for the homopolymer and l l0~ for the PS2000/PSD10 (30 %) blend). It is shown in Figure 22 that the relaxat ion curves of both systems can be superimposed at short times but tha t increased deviations appear at longer times. This result suggests that at short t imes (or short distances) the monomeric friction coefficient governs the local relaxat ion processes, while at longer t imes the relaxation dynamics depends on the environment . Recalling the Rouse and tube models, this explanation fits in with the well-established notion tha t the local relaxation processes can be described by Rouse modes [35]. Fur thermore , at a fixed ~o' the short t ime relaxat ional kinetics of long chains are essentially insensitive to the presence of the short species.
5
E 7
Z
~ 8 �9
9
~ , , , , i , , , , i , , , , i , , , , i , , V , l V , , , l ~ , , ,
B i
[]
D
~e~ e F.
g D []
- 1 0 L ~ , , l . . . . i . . . . z . . . . i . . . . ~ . . . . ~ . . . . 6 7 8 9 10 11 12
103 ( 1/T-T ) o o
13
Figure 21. Logarithmic plot of the monomeric friction coefficient as a function of 1/(T-Too) for PS2000 (0) and PS2000/PS10(30 wt.-%) (~).
It can thus be deduced tha t the "dilution effect" of short chains is a dynamical process which does not act at short times�9 On the contrary, at longer times, as reptat ion theory postulates, the topological constraints which govern the chain dynamics appear looser as the concentrat ion of short chains is increased. Indeed, even in an environment of equivalent local chain character is t ics , as
61
fixed by a given monomer ic friction coefficient, the PS2000/PSD10(30 wt.-%) blend, wi th a lower n u m b e r of effective constraints , tha t is, a higher Me, has thus an increased chain mobility, and, consequently, a reduced orientation.
0.30 . . . . I . . . . I . . . . , . . . . , . . . . , . . . . , ' ' '
0.25
^ 0.20 ~D
o 0.15
a," 0.10 V
0.05
g~ , . O �9
O o 00 �9
c~ �9 2 .~o�9 �9 LT- I~ l i e ~ 1 7 6 1 7 6 o
D
oc:b ~ ~ o
0 . 0 0 , , , , I , , ~ , l , J , , i , , , , I , , , , l , , , , l . . . .
-1 0 1 2 3 4 5
log t (~o = Constant)
Figure 22. Long chain orientat ional relaxat ion reduced to reference
t empera tu res at which log ~o = -3.54 N.s.m-l: PS2000 (o), PS2000/PSD10(30 wt.-%) (g).
8. C o n c l u s i o n
In f r a r ed dichroism has been successfully applied to charac ter ize the or ien ta t iona l re laxat ion of l inear and branched polystyrene chains as well as b inary blends of long and short chains. By deutera t ing some chains or par ts of chains, inf rared spectroscopy provides a method of analyzing the orientat ional b e h a v i o u r of the different species and consequent ly probe the molecular r e l axa t ion mechan i sms .
In the case of l inear polymers, the or ientat ion of a chain segment has been shown to depend on its location along the chain. A more rapid orientat ion of chain ends has been evidenced.
Similar ly , in the case of s ta r polymers, the re laxat ions of the branch point, a r m centre and chain end have been differentiated. The influence of b r anch ing has been detected by compar ing l inear chains wi th arms of star polymers. A slower relaxat ion is observed in the case of branched species.
The spect roscopic da ta have been compared wi th the theore t i ca l predict ions of the Doi- Edwards model. In the t ime scale of our experiments , a quan t i t a t i ve ag reement be tween exper iment and theory is obtained if chain length f luctuat ions , re t rac t ion and rep ta t ion are taken into account. In the case of s t a r polymers, the large scale f luctuat ion mechanism as proposed by Pearson and Helfand associated wi th the re t rac t ion process is accounting for
52
the observed relaxation, except in the case of the branch point of the star, where a more rapid relaxation is experimentally observed.
The infrared analyses of binary polystyrene blends have been discussed in the light of rheological measurements of the same systems. It is shown that, in addition to the existence of an orientational coupling which is essentially evidenced by the high orientation of short species, the behaviour of the long chains is sensitive at short times to the average friction coefficient in the blend. Addition of short species leads to a decrease of the friction coefficient and thus a decrease in the relaxation times. At longer times, topological effects (changes in the entanglement spacing) are superimposed to the change in the relaxation times.
R E ~ ~ C E S
1. See for instance Chapter 1.5 and references cited therein 2. I. M. Ward, Structure and properties of Oriented Polymers, Applied Science
Pub, London, 1975 3. H. Janeschitz-Kriegl, "Polymer Melt Rheology and Flow Birefringence",
Springer Verlag, Berlin, 1983 4. G. G. Ffiller, Ann. Rev. Fluid Mech., 22, 1990, 387 5. B. Jasse, J. L. Koenig, J. Macromol. Sci., Rev. Macro. Chem., C17, 1979, 61;
J. Polym. Sci., Polym. Phys. Ed., 17,1979,799 6. J. F. Tassin, L. Monnerie, Macromolecules, 21, 1988, 1846 7. R. Fajolle, J. F. Tassin, P. Sergot, C. Pambrun, L. Monnerie, Polymer, 24,
1983, 379 8. J. D. Ferry, "Viscoelastic Properties of Polymers", 3rd edition, Wiley, New
York 1980 9. M. Doi, S. F. Edwards, "The Theory of Polymer Dynamics", Oxford
University Press, New York 1986 10. M. Doi, J. Polym. Sci., Polym. Phys. Ed., 18, 1980, 1005 11. M. Doi, J. Polym. Sci., Polym. Lett. Ed., 19, 1981, 265; J. Polym. Phys.
Polym. Phys. Ed., 21, 1983, 667 12. D. S. Pearson, E. Helfand, Macromolecules, 17, 1984, 888 13. J. L. Viovy, Polymer Motions in Dense Systems, Springer Proceedings in
Physics, 29, 1988,203 14. P. G. de Gennes, J. Phys. Fr., 36, 1975, 1199 15. R. C. Ball, T.C.B. McLeish, Macromolecules, 22, 1989,1911 16. J. L. Viovy, L. Monnerie, J. F. Tassin, J. Polym. Phys. Polym. Phys. Ed., 21,
1983, 2427 17. W. W. Graessley, Adv. Polym. Sci., 47, 1982, 67 18. J. Klein, Macromolecules, 19, 1986, 105 19. M. Doi, W. W. Graessley, D. S. Pearson, E. Helfand, Macromolecules, 20,
1987, 1900 20. M. Rubinstein, E. Helfand, D. S. Pearson, Macromolecules, 20, 1987, 822 21. M. Rubinstein, R. H. Colby, J. Chem. Phys., 89, 1988, 5291 22. A. Schausberger, H. Knoglinger, H. Janeschitz Kriegl, Rheol. Acta, 26,
1987, 4668 23. J. F. Tassin, L. Monnerie, J. Polym. Sci., Polym. Phys. Ed., 21, 1983, 1981 24. J. A. Kornfield, G. G. Fuller, D. S. Pearson, Macromolecules, 22, 1989, 1334
63
25. G. G. Fuller, C. M. Ylitalo, J. of non crystalline Solids, 131, 1991, 676 26. P. F. Green, P. J. Mills, C. J. Palmstrom, J. W. Mayer, E. J. Kramer, Phys.
Rev. Lett., 53, 1984, 2143 27. J. F. Tassin, Th~se de doctorat d'Etat, Universit~ Paris VI, 1986 28. L. R. G. Treloar, "The Physics of Rubber Elasticity", Oxford Univ. Press.
London, 1958 29. J. F. Tassin, L. J. Fetters, L. Monnerie, Macromolecules, 21, 1988, 2404 30. C. M. Ylitalo, G. G. Fuller, V. Abetz, R. Stadler, D. S. Pearson, Rheol. Acta,
29, 1990, 543 31. A. Lee, R. P. Wool, Macromolecules, 20, 1987, 1924 32. K. Osaki, E. Takatori, M. Ueda, M. Kurata, T. Kotaka, H. Ohnuma,
Macromolecules, 22, 1989, 2457 33. W. J. Walczak, R. P. Wool, Macromolecules, 24, 1991, 4657 34. T. Buffeteau, B. Desbat, S. Besbes, M. Nafati, L. Bokobza, Polymer, 35, 1994,
2538 35. P. J. Rouse, J. Chem. Phys., 21,1953, 1272 36. J. F. Tassin, A. Baschwitz, J. Y. Moise, L. Monnerie, Macromolecules, 23,
1990, 1879 37. H. Toriumi, B. Deloche, J. Herz, E. T. Samulski, Macromolecules, 18,1989,
1488 ; B. Deloche, A. Dubault, J. Herz, A. Lapp, Europhys. Lett., 1,1986, 629
38. M. Doi, D. Pearson, J. Kornfield, G. Fuller, Macromolecules, 22,1989, 1488 39. C. M. Ylitalo, J. A. Zawada, G. G. Fuller, V. Abetz, R. Stadler, Polymer,
33,1992, 2949 40.J.P. Montfort, G. Marin, P. Monge, Macromolecules, 17,1984, 1551; 19,1986,
1979 41. D. J. Plazek, J. Phys. Chem., 69,1965, 3480 42. P. Lomellini, Polymer, 33,1992, 4983 43. W. W. Graessley, J. Chem. Phys., 54,1971, 5143 44. R. J. Composto, E. J. Kramer, D. M. White, Polymer, 31, 1990, 2320
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Rheology for Polymer Melt Processing J-M. Piau and J-F. Agassant (editors) �9 1996 Elsevier Science B.V. All rights reserved. 65
Chain conformation in elongational and shear flow as seen by SANS
R. Muller and C. Picot
Institut Charles Sadron (CRM-EAHP), 4-6 rue Boussingault, 67000 Strasbourg, France.
1. INTRODUCTION
The central problem in the understanding of the rheological properties of polymer melts is to achieve a description at a molecular level of such highly entangled and consequently interacting long flexible chains systems. This goal is clearly of fundamental interest as well as of practical importance with regard to polymer processing. A variety of molecular models have been proposed to represent the behaviour of polymer chains in interaction with their surroundings. Extensive reviews describing the molecular approaches of polymer viscoelasticity can be found in monographs discussing the limits of their applicability [ 1,2]. In these models, equilibrium and dynamical properties are described in terms of spatial distribution of the chain segments. Whatever the models invoked, from the simplest (i.e the Rouse model) to the more sophisticated (reptation model,...), the stress is directly related to the orientation of the bound vectors of the polymer chain. Therefore a direct knowledge of the spatial distribution of the chain segments is the key point in the molecular description of the macroscopic viscoelastic behaviour.
Among all experimental techniques, radiation scattering is the most appropriate to accede to this topological description. The scattering is indeed directly related through Fourier transform with the spatial correlation of fluctuations. More specifically, when observing a polymer system containing labelled chains, the distribution of chain segments in space can be analyzed [3]. With regard to polymeric condensed systems, small angle neutron scattering (SANS) is particularly well suited for studying chain conformations [4]. The advantages of SANS are twofold: - the available range of scattering vectors on low angle spectrometers (10 -3 - 1 A "t) matches particularly well the polymer chain dimensions. - neutron scattering is the result of neutron-nucleus interaction and hence different isotopic species can interact with neutron differently. In particular, hydrogen and deuterium have coherent scattering lengths (arr=-0.374x 10~2; aD=+0.667x 10 ~2 cm respectively) which differ in both sign and magnitude. Consequently, a deuterated molecule in a protonated matrix will be "visible" by neutrons and vice versa.
The SANS technique has already contributed to the analysis of polymer conformation in systems at equilibrium (i.e. dilute and concentrated solutions, bulk) [5-7] and to the molecular response of polymer systems under external constraints (i.e. polymer networks, polymer melts under uniaxial or shear deformation) [8-14].
The aim of the present work has been to establish correlations between bulk macroscopic response of polymer melts under flow and the behaviour at a molecular level as seen by SANS, and to discuss the results in the frame of molecular theories. Two simple and well defined geometries of deformation have been investigated: uniaxial elongation and simple shear. The
66
basic experiment consists in submitting the polymer melts containing deuterium labelled chains to different, well-defined regimes of flows (transient and steady flows, relaxation, creep) and then perform SANS measurements on probes where the chains have been frozen in by rapid quenching below the glass transition temperature Tg.
2. METHODOLOGY
2.1. Polymers used All samples consist of anionically synthesized PS containing deuterated polystyrene (PSD)
chains. Two hydrogenated (PSH1 and PSH2) and two deuterated (PSD 1 and PSD2) polymers were synthesized. Their weight-average molecular weight was determined by light-scattering measurements in benzene taking into account the refractive index increments for PSH and PSD. The polydispersity was characterized by gel permeation chromatography using the universal calibration. As shown in Table 1, PSH2 and PSD2 have a weight-average polymerization index which is about ten times that of PSH1 and PSD 1.
Table 1 Molecular WeiBht Characteristics of Polymers
Polymer Ms Mw Mz M,dM, Il~
PSH1 80 000 90 000 99 000 1.12 865 PSD 1 81 000 95 000 108 000 1.13 848 PSH2 865 000 970 000 1 057 000 1.12 9 330 PSD2 975 000 1 170 000 1 355 000 1.2 10 450
With these polymers, two narrow MWD samples (S 1 and $2) and seven binary blends were prepared for the SANS experiments. The composition of all samples is given in Table 2. They were obtained by freeze-drying a solution of all constituents in benzene [9]. For the narrow MWD samples, the weight fraction ~D of deuterated species is 10%. The weight fraction O2 of high molecular species in the binary blends ranges from 10% to 60%; they contain either PSD 1 03'20 and B'40) or PSD2 chains (B 10, B20, B40 and Br0), which allows one to investigate respectively the conformation of the low or high molecular weight species. As seen in Table 2, the total weight fraction ~ of deuterated species is lower for the binary blends than for the narrow M W samples. The concentrations were calculated so that the measured scattering intensity remains simply related to the form factor of the deuterated species (see section 2.3.).
The linear viscoelastic properties of all samples were characterized by dynamic shear measurements in the parallel-plate geometry. Experimental details have been previously published [9]. Using time-temperature equivalence, master curves for the storage and loss moduli were obtained. Fig. 1 shows the master curves at 140~ for the relaxation spectra and Table 3 gives the values of zero-shear viscosities, steady-state compliances and weight-average relaxation times at the same temperature.
67
Table 2 Composition in weight fractions of all samples
Sample ~PSm ~PSD~ q~Psm (~)PSD2 ~2
S1 0.9 0.1 0 0 0 0.1 $2 0 0 0.9 0.1 1 0.1
B10 0.9 0 B20 0.8 0 B40 0.6 0 B60 0.4 0
0.092 0.189 0.382 0.574
B'20 0.76 0.04 0.2 B'60 0.36 0.04 0.6
0.008 0.011 0.018 0.026
0.1 0.008 0.2 0.011 0.4 0.018 0.6 0.026
0 0.2 0.04 0 0.6 0.04
n
v
o --Im-- $1
BIO - - ~ - - B20 - -V - - B40 - - ~ - - $2
Figure 1. Relaxation spectra at 140~ of samples S 1, $2, B 10, B20 and B40.
-2 -1 0 1 2 3 4 5
log(x) (s)
Table 3 Linear viscoe!astic parameters at 140~ of all samples.
Samp!e 110 (106 Pa.s) Jr ~ (10 s Pa ~) X.(s)
S1 0.65 1 B10 9 60
B20, B'20 42 27 B40 290 7.8
B60, B'60 780 4.3 $2 3550 1.6
6.5 5 400 11 300 22 600 33 500 56 800
68
2.2. Experimental setup The data in this work were obtained by quenching either uniaxially stretched samples for
which the low thickness of the specimens allows rapid cooling, or samples deformed in simple shear where higher thickness of the specimens was required in order to perform the scattering experiments along the three principal shear directions. For both types of flow, special devices were developed to control the flow kinematics in the molten state as well as the quenching process which freezes the molecular orientation. These devices are briefly described in the next paragraphs.
2.2.1. Eiongational flow For the elongational flow experiments, all specimens were uniaxially stretched with the
extensional rheometer schematically shown in Fig. 2. A detailed description of this instrument has been published elsewhere [ 15]. Its main features are the following: (a) symmetric stretching with respect to the centre of the specimen (this allows measurement of flow birefringence at a fixed point of the sample), (b) temperature control with a double silicon-oil bath, which can be removed by vertical displacement. This oven provides a good temperature control (less than • ~ gradient in the whole bath), whereas buoyancy prevents the specimen from flowing under the effect of gravity. Homogeneous deformation is thus achieved even at very low strain rates for which the time that the specimen stays in the molten state is longer than the terminal relaxation time of the polymer sample. Stretched specimens are quenched by rapid removal of the surrounding oil bath.
Figure 2. Schematic drawing of extensional rheometer. (a)motors, (b)screws, (c) transducer, (d) sample, (e) oil bath, (f) laser, (g) lower position of bath.
69
2.2.2. Shear flow The experimental device constructed to orient uniformly thick samples in simple shear is
schematically represented in Fig. 3. It is basically a sliding-plate rheometer, the polymer sample being sheared between two temperature-controlled parallel plates. The upper plate is fixed whereas the lower plate can be displaced both horizontally and vertically with two pneumatic jacks.
The shearing experiment occurs in the following way: (a) the temperature of both plates is set to its value during the melt flow by a silicon oil circulation before the polymer sample is inserted between the plates; (b) once thermal equilibration of the sarriple is achieved, the gap between the plates is fixed by a mechanical stop; (c) a constant shear force Fs is applied to the melt with the horizontal jack during a given time. At the same time, the normal force Fr~ in the sample is recorded with a force transducer mounted below the lower plate. From the horizontal displacement, the shear strain is obtained as a function of time; (d) after shearing, the sample is cooled by stopping the oil circulation and starting the cold water circulation which is located close to the surface of the plates. Throughout the cooling stage, both the shear force Fs and the normal force F~ are kept constant, the vertical jack being now driven with the same compressive force which has been measured during the sheafing stage.
Keeping a constant compressive force on the sample during cooling allows compensation for thermal shrinkage and prevents the polymer from slipping away from the surfaces of the rheometer. Furthermore, if the temperature is assumed to be uniform in each polymer layer (at constant height) at any time during cooling, each polymer layer will experience the same constant shear and normal stresses during quenching. This in turn means that if steady flow is reached at the beginning of the cooling stage, the orientation will be uniform throughout the whole thickness of the sample, although the inner layers, which will be cooled later, will experience a higher final shear strain than the outer layers in contact with the plates which are quenched first. Birefringence provides an easy way to check the uniformity of orientation in the sample thickness: the observation with a polarizing microscope of small strips cut out vertically in the x-y sheafing plane shows that both the.birefringence and the extinction angle are uniform in the whole thickness (see also chapter III.1). Further experimental details have been previously published [ 13].
Figure 3. Schematic drawing of shear apparatus: (a) specimen, (b) oil circulation, (c) water circulation, (d) normal force jack, (e) shear force jack. x, y and z are the principal shear directions.
70
2.3. Small-angle neutron scattering (SANS) in bulk polymers
2.3.1. Theoretical background Detailed reviews about application of SANS to the study of polymer in bulk are available in
the literature [3]. These reviews provide information on the experimental technique as well on as on the fundamental theories related to scattering by polymers. Here, we will restrict ourselves to the description of some basic concepts to enable a better understanding of the experimental results that will be presented.
A schematic representation of an experimental device used for SANS measurements is shown in Fig. 4.
I x I
I Stretching direction
z IIIII ~ I \ \ \ x ~ , , X q
N e u t r o n s t
Y
%
�9 I
Sample
Y
Figure 4. Schematic representation of an experimental setup for SANS.
As mentioned before, the scattered intensity arises from the interaction between neutrons and nuclei. It decomposes into two terms, namely the coherent and the incoherent contribution. The coherent intensity depends on the scattering vector q which is the difference between the momentum of the incident and the scattered neutrons. Its norm is q=(4r~/k) sin(0/2), 0 being the
71
scattering angle and ~, the neutron wavelength. The incoherent intensity arises mainly from H nuclei and is not q dependent in the small q range of SANS. Its contribution results in a fiat background which can be taken into account by appropriate subtraction.
For a mixture of identically D-labelled chains in a matrix of H chains with the same polymerization index, the coherent scattered intensity is given by:
S(q) = N AP~2-2 (aD - all) 2 ~D(I- q)D)P(q)= KM~D (1- q)D)P(q) (1)
where aD and aH are the coherent scattering lengths of D and H monomer units, m and M are the molar weights of monomer unit and polymer chain, p is the density, ~D is the volume fraction of labelled chains, NA is Avogadro's number and P(q) the intra chain correlation function relative to the labelled coil:
1 P(q) = ~ Z (exp(iq. rkl))
k,1 (2)
where r~a is the vector joining subunits k and 1 of the chain formed by N segments, < > is an ensemble average over all chain conformations in the sample.
Isotropic samples. For an isotropic set of identical coils, the form factor only depends on 0 and is given by the
Debye function g(x):
g(x) = ~--
with x=q2xRg 2, Rg being the radius of gyration of the chains. In the Guinier range of q ( qR~<l ) if a distribution of chain length is introduced one obtains:
1 ) S-'(q)= KMw~D(I_~D ) 1---~-R2z (4)
where Mw and Rz are the weight and z averages for the molecular weight and radius of gyration respectively. The Zimm representation of the scattered intensity (S'l(q) versus q2) then allows the determination of Rz. In the intermediate range of scattering vector (1/R~< q <l/a ; a: length of the monomer unit of mass m):
12KCPD(1 - eD)P(q) m q2S(q) = 2 (5)
a
72
The Kratky plot (q2xS(q) versus q) leads to a plateau allowing the determination of a2/m. These two representations will be used as references for characterizing the anisotropy of the deformed samples.
In the case of binary blends of chains with different molecular weights (the case which will be considered in this work), Eq. 1 no longer holds. In order to take into account the mismatch between the lengths of the chains of different species (H and D chains) one has to reformulate exactly the expression of coherent scattered intensity according the theory of random phase approximation (I~A) [16]. After straightforward calculations one can show that the Zimm representation leads to an apparent radius of gyration which is the true radius of the labelled chains multiplied by a coefficient C. This coefficient depends on the ratio of the molecular weights of the two polymers, the composition of the blend and the volume fraction of labelled chains. For all polymer blends that have been investigated, the value of C was close to unity so that this correction could be neglected. In the intermediate range of q, the theory shows that whatever the composition, the effect of the difference in chain length of the blend components is negligible.
Oriented samples. For anisotropic samples, the space correlation function will depend on the orientation
imposed to the vector r~ and P(q) will vary both with the scattering angle 0 and azimuthal angle 6. One can then define scattering form factors along the reference axes as:
1 P ~ (q) = .-ST ~ (exp(iq act ld ) with cL = x, y, z
IN- k,1 (6)
Expanding the exponential for small q gives:
1 [ 1 (q2x(X2) + q2 ( y h ) + q2 (zh)) ] P(q) = - ~ ~ exp - ~ k,l
(7)
This defines the anisotropic radii of gyration along the reference directions :
g,x - 2 N 2 x 2 ) the same for y and z (8)
These quantities measure the average value of the square of the distance of chain units to the plane normal to the reference axis and going through the centre of mass of the chain. If one refers to the isotropic case, they are equal to one third of the radius of gyration.
2.3.2. SANS measurements and analysis of data The neutron scattering experiments have been carried out at the Institue Laue Langevin
(ILL) Grenoble, France and at the Laboratoire L6on Brillouin (LLB), Saclay, France. By using different spectrometers the following ranges of scattering vector amplitudes q were covered:
73
Spectrometer: ILL-D 11 ILL-D 17 LLB-PAXY q range(/l-1): 4 x 1 0 "3 . 3x10 -2 1 0 -2 _ 1.2x10 "l 1 0 "3 . 5 . 5 x 1 0 "2
Water calibration was used to convert the scattering data to an absolute coherent scattering cross-section, S(q), with units of reciprocal centimetres. A general result for all samples is the agreement between the data obtained on the different spectrometers in the overlapping q- domain. After appropriate incoherent background subtraction, the data analysis is carried out over the whole bidimensional detector by grouping at constant scattering angle 0 for isotropic samples. For oriented samples, grouping of data is performed at constant 0 in limited sectors around the reference axis or the main axis of deformation.
3. ELONGATIONAL FLOW
3.1. Narrow MWD samples
3.1.1. Results A first series of tensile experiments at constant strain rate has been carried out on sample
S 1: the specimens are stretched at constant elongational strain rate up to a given extension and immediately quenched. According to time-temperature superposition, a rheological material function at a given temperature To can be shifted to any other temperature T knowing the thermal shift factor aT-,To. For the tensile stress-growth coefficient, the following equation holds:
( ,) , = ~ - - tie ~ , t x aT_,T0, (9) pT aT~To aT_~T ~
which has been confirmed experimentally [ 15,17-18].
For these first experiments, a temperature relatively close to Tg, T=123~ was chosen with the intention of minimizing the relaxation of stress and chain orientation during the quenching: the weight-average relaxation time of sample S 1 at 123~ is calculated from that at 140~ and the thermal shift factor between 123~ and 140~ Lw(123~ On the other hand the cooling time of the stretched specimens can be estimated to a few seconds [ 19], which is very small compared to the polymer relaxation time at the temperature of the experiments.
For strain rates lower than 8• 10 .4 s ~, it was found that the rheological behaviour is nearly
linear viscoelastic: Fig. 5 shows the tensile stress-growth f u n c t i o n cr§247 at 123~ for three different strain rates in the linear range; after about 1000s, the stress reaches a
steady value close to 3ri0 x e where r10=3.8x107 Pa.s is the zero-shear viscosity at 123~ For each strain rate, samples for the scattering experiments were obtained by quenching specimens at different macroscopic extensions in the steady-state flow as shown by the arrows in Fig. 5.
74
Comparison of o § (~:, t)for different samples stretched at the same strain rate shows that the reproducibility of the tensile test is satisfactory.
Figs. 6 and 7 show the Kratky plots of these samples and Table 4 gives their radii of gyration t~ / and R~;. At this stage, several conclusions can be drawn on a qualitative level: (a) there is a good agreement between the results obtained on the two spectrometers (D 11 and D 17) in the overlapping q-domain, (b) as shown by Fig. 6, the anisotropic scattering intensity becomes time-independent like the stress. This result was expected and confirms the reliability of the experimental method. (c) The chain anisotropy increases with strain rate, but the decrease in R~. is less pronounced than the increase in R~/. The measured radius of gyration for an isotropic sample is close to 82 A and the data are represented by the solid line in Fig. 6; they are in close agreement with the Debye function for Cvaussian coils (Eq. 3).
A
v
u) I,OxIo s. (/)
._---m 7,5~- e -
cl ~ c 2
~ v ~ ~ v v vvv
4,V ~bl ~ b2
lit ~ Z X zxz~ zx "/x zx
~q~E)"O Q (!R)~p(P o o o o
2 , ~ 0 4-
0,0 , o , ~ " tom ~ ~ ~o 3oo0
~ m e (s)
Figure 5. Tensile stress-growth function for sample $1 at 123~
B,O,O: ~: = 3.7x10 "4 s "l
e,A: ~ = 5.3 x 10 .4 s "1
O,V:e = 7.4x 10.4 s a
~ " 0.15
E
, ~ 0,10"
l
0,05.
%-
0,05 0,00
1 o a2D17
'
o.m o.~ " o.~ o.~ "o.~o q(A -~)
Figure 6. Kratky plots for the three specimens of the (a) series. Data below the isotropic curve correspond to the parallel direction.
0,15 -.
-SBB'~'~ ~
�9 c?. Dl l
0 , ~ - . ,,,
o.oo o,~ 0,6 " o,~s " oN 0,10
q(A "I)
Figure 7. Kratky plots for specimens a2 and c2.
75
v
d3 d2
"2 T ~ 3%
log(t) (t in s)
Figure 8. Tensile stress-growth coefficient of sample S 1 at 123~
A:e=3.7xl0 -4 s -Z, I I : e = 1 0 -a s'~;
i-'1. s = 2.3x10 -2 s -1.
Table 4 Radii of G/ratio n of S 1 specimens.
Sample �9 Hencky Pw/(A) R~; (A) Rv/(A) R~. (A) e (s") ..strain (exp_erim.) (experim.) .... (Rouse). (Rouse)
al 3.7x10.4 0.37 91.5 76.5 95 77 a2 3.7x10.4 0.65 91 75.5 99 76.5 a3 3.7x10.4 0.925 91 75.5 101 76.5
bl 5.3x10 "4 0.53 95.5 75 102 75.5 b2 5.3 x 10 .4 0.93 95.5 74.5 110 75 b3 5.3 • 10.4 1.32 95 76.5 116 74.5
cl 7.4x 10.4 0.74 102 74.5 114 73.5 c2 7.4• 10.4 1.3 100.5 75.5 131 72.5
dl 10 -3 0.5 102 77 109 73.5 d2 10 .3 1 106 74 133 71 d3 10 "3 1.75 115 170.5 70
el 2.3x10 "2 0.3 100 63 108 72 e2 2.3• .2 0.69 125 60 153 62 e3 2.3 • 10 .2 1.29 164 56 262 52
Further tests have been carried out on sample S1 at two higher strain rates. The stress- growth coefficient corresponding to these experiments is represented in Fig. 8, where the
76
arrows indicate the specimens quenched for the SANS experiments. As can be seen in Fig. 9, which shows the Kratky plots of specimens e 1, e2 and e3, and in Table 4, the anisotropy of the chain conformation increases with the macroscopic elongation just like the stress.
,•o•
e 2 o 3
o,0o . . . . o,| o.'o5 r 0,15
q(A "1)
Figure 9. Kratky plot for the 3 specimens of the (e) series
stretched at e -- 2.3 x 10 -2 s "] (data from D 17 spectrometer).
For the high molecular weight sample $2 as well as for the binary blends, a higher temperature (140~ was chosen for the elongational flow. On the other hand, instead of being quenched at various extensions during a constant strain rate test, these samples were rapidly stretched (within about 3s) to a given extension (L/L0=3 for all specimens) and the stress was then allowed to relax for a certain time tR before quenching. In all cases, tx was very large compared to the stretching time of 3s. Fig. 10 shows SANS data for sample $2, which confirm that the chain anisotropy decreases with increasing tR.
o, o
o E ~ ,~l~'o o ~:~x~ ] ] ~ , L r A ~:3ooos
o.=q Z
o,oJ .
. . . . . . 0,00 0,05 0,10 0,1,5
q ( A "1)
Figure 10. Kratky plot for sample $2 stretched at 140~ The two specimens were allowed to relax during 200s and 3000s (data from D 11 and D 17 spectrometers).
77
It has to be mentioned that the radii of gyration of the high molecular weight chains (which are of the order of 300 A) could not be determined in the frame of the present study. Reaching scattering vectors as low as 10 -3 A "~, which would have been required, involves technical difficulties on the spectrometer and very long counting times.
3.1.2 Rouse model A more quantitative interpretation of the scattering data involves molecular models. As an
example, we turned to the Rouse model [20] developed for solutions and extended to melts [21]. As this model is unable to account for the molecular weight dependence of zero-shear viscosity (rl0--M 34) above the critical molecular weight (M,~35 000 for PS), the analysis will be extended as a next step to other models which are more realistic for entangled systems. A basic result of the Rouse model relates the monomeric friction coefficient Q0 and the zero-shear viscosity TI0:
~0 36r10m2 = (10) pa2MNA
m and a being the molecular weight and length of the statistical segment, M is the molecular weight of the whole chain and NA is Avogadro's number. In the linear viscoelastic limit, the Rouse model is equivalent to a generalized Maxwell model with relaxation times g~=Xl/n 2. The longest relaxation time k,1 (or Rouse time) is simply related to the zero-shear viscosity and molecular weight of the chain:
X, 1 = 6rl~ (11) x2oRT
If the Rouse time is determined from the experimental value of the zero-shear viscosity, one finds XI=615s for sample S 1 at 123~ which is not too far from the experimental value of the weight-average relaxation time (X~380s). Clearly, the determination of ~0 and ~,i from the experimental value of 110 implies the assumption of monomeric friction enhancement by entanglements [20], since for sample S 1 M is of the order of 3 •
To allow a comparison with rheological and SANS data, the stress and scattering intensity have been calculated for the Rouse model as a function of time and strain rate for simple elongational flow. L being the chain length, the n th normal mode is given by [1 ]:
L ( n x s ) ds R n = 2~ r(s) co~---~-) -L-
0
(12)
r being the position vector along the chain axis. If n,p, the n th and p th modes are uncorrelated (<R,Rp>=0). The spatial components X~, Y, and Zn of a mode are also uncorrelated and obey Gaussian statistics. If x is the stretching direction, it is easily shown that:
78
�9 E I " l nl 1 - 2 ~ , n exp - 1 - 2 ~ , n
Xn 2 (t) = Rg,iso ~n 2 �9 1 -2~;~ n
2 2 (Yn2 (t)) = (Zn2 (t)) = R g ~ ~2n2
�9 Ic �9 l nJ I+E:~L n exp - I+~:~L n
l + e k n
(13)
The tensile stress is shown to be [2]:
o - o ~ = X XR2)+ xx 2aM (14)
The rheological constitutive equation of the Rouse model is that of an upper-convected Maxwell model, with the consequence that steady-state elongational flow only exists for strain rates lower than 1/(2~,0. The steady-state elongational viscosity depends then on strain rate:
(~) 3pRT N ~, TIE M
" l - 2 e ~ . p l+e~.p
and obeys Trouton's relation for vanishing strain rates. The scattering intensities in the parallel and perpendicular directions:
S//(q) = ~ E ~ e x p { -~ q (x k - x 1 k 1
S• q (Yk-Yl k 1
(16)
where xk, yk and Zk are the components of rk, the position vector at monomer k, can also be calculated from the normal modes [22] since:
r k = R 0 + 2 ~ R p c o p=l
(17)
79
A first comparison between the model and our rheological data for sample S 1 shows that
the critical strain rate of the model (e = 8.1 • 10 "4 s "~) is close to the value corresponding to the departure from linear viscoelasticity in elongation (see Fig. 8). However, a fundamental discrepancy lies in the fact that no strain rate dependence of the steady-state elongational viscosity is seen in the experimental data below the critical strain rate.
Figs. 11 and 12 compare the stress data of Figs. 5 and 8, and the Rouse model predictions according to Eqs. 5 and 6. Close agreement is only found for the lowest strain rate corresponding to the (a) series, the model generally overestimates the tensile stress. The scattering intensity has been calculated according to Eqs. 16, 17 and 13 and the results are compared to the data in Fig. 13, 14 and 15 for samples a2 and c2, d l and d3 and el to e3 respectively. Table 4 compares the calculated radii of gyration with the experimental data.
m 1'5x10S
~ I +~ 1,t~o ~,
5,0x104
0,0 ~, . . . . . o 1 ~ 2 ~ 3o0o
time (s)
Figure l 1. Tensile stress-growth function of sample S1 at 123~ Symbols" data of Fig. 5. Solid lines: Rouse model.
--O--a ~ b "'<)me - - - - - R o u s e ;.-y
. . . . ~ , ~ ml d ~
d Rouse O .......... e Rouse
Jog(t) (t in s)
Figure 12. Tensile stress-growth coefficient of sample $1 at 123~ Symbols: data of Fig. 8. Solid and dotted lines' Rouse model.
~,--'- O,lS <
"T, E o
0,10 ,G, I.-
,E, ~ 0,05-
A, ~ R~ m~ I
c2 Rouse model I
.... , , 0 , ~ - ~ " . �9 - ~
0,00 0,02 0,04 0,06 0,08
q(A I)
Figure 13. Kratky plots for specimens a2 and c2. Symbols: data of Fig. 7. Solid and dashed lines: Rouse model.
0,15
& d3 Data d l Rouse
- - - - - cl3 Rouse
0 ,00 �9 " "
o,oo o,~ o.~ o,~ olin q(A "I)
Figure 14. Kratky plots for specimens d 1 and d3. Symbols: data. Solid and dashed lines Rouse model.
80
(13) <C 'E
0,15.
0,10.
_~ ~|
(tO0 (~oo 0.05 O, W10 " (~15
q(A "I)
Figure 15. Kratky plots for specimens el, e2 and e3. Symbols: data of Fig. 9. Solid lines: Rouse model.
The main conclusions coming out of these results are the following:
(a) the Rouse model systematically overestimates the parallel radius of gyration whereas the agreement is satisfactory in the perpendicular direction.
(b) For strain rates below the critical strain rate, the experimental values of stress and scattering intensity become time independent at a time shorter than that predicted by the model.
(c) These results should be compared to the fact that the model also overestimates the steady-state elongational viscosity with respect to the experimental value which obeys Trouton' s relation: VIE=3 Xrl0(shear).
(d) At higher q-values, an overall satisfactory agreement between model and experimental data is found over the whole range of strain rates, except for the highest strain rate in the transverse direction. For molecular weights up to a few times the molecular weight between entanglements (for S 1 about 5 times) the chain conformation in uniaxial elongation, especially at a local scale, is fairly well accounted for by the Rouse approximation.
We now examine the data for the high molecular weight sample with a molecular weight of about 50xM~. If the Rouse time for sample $2 at 140~ is estimated as before from the experimental value of the zero-shear viscosity according to Eq. 3, one obtains l~=6.3x 105 s, which is about 10 times the experimental value of L~. This discrepancy reflects the incapacity of the Rouse model to account for the molecular weight dependence of both rl0 and 7~, for highly entangled melts.
For this reason, we calculated the scattering intensity of the high molecular sample by taking for the Rouse time the measured terminal relaxation time (57 000s). Fig. 16 shows the result compared with the data of Fig. 10. It is seen that the model only agrees with the data in the small q-range, which is related to the choice of the Rouse time k~, taken here as the measured terminal relaxation time of the sample. On the other hand, the model strongly underestimates the chain orientation at a more local scale, which can be qualitatively understood by considering that the chain is actually trapped in a tube which keeps an orientation at the scale of the tube diameter for times close to the reptation time.
81
0,30 <
,
'E 0,25, o
0,20
~ 0 0 0 ~ 0 0 tR=200 S Data �9 tR=3000S Data
. . . . tR=200S R o u s e m o d e l
\ \ = ~ O O ~ ~ t R : 3 0 0 0 S R o u s e m o d e l
!
~-- 0,15 o ~. 0,10,
0,05
0,00, , , , 0 00 0,05 0,10 015
q(A "I)
Figure 16. Kratky plot for sample $2 stretched at 140~ Symbols: data of Fig. 10. Solid and dashed lines: Rouse model.
3.1.3. Temporary network model Rheological models using the concept of a temporary network of entanglements have been
very successful in describing the rheological behaviour of polymer melts [23-25] and especially the strain-hardening effect in elongational flow [26]. However, the assumption of affine displacement of temporary junctions with respect to macroscopic deformation leads to an overestimation of the stress. Fig. 17 shows the tensile stress during a constant strain rate experiment, plotted against the quantity (~2_~-~) where ~ is the recoverable strain. ;~ has been measured by cutting off a part of the stretched and quenched specimen (of length L) and annealing it at the surface of an oil bath at 150~ L, is then defined as the ratio L/L~ where Lf is the length after recovery.
Since for each experimental point a new specimen is required, the data in Fig. 17 were obtained with a commercial polystyrene sample, available in larger quantities than the anionic and deuterated polymers used for the SANS experiments. The curves in Fig. 17 suggest the following expression for the tensile stress:
t~+(;,t 1 =t~0(; 1 +G(;l(~r2_ ~r -1) (18)
It has been found that Equation 18 describes to a good approximation tensile stress data on
various polystyrene samples [27] . G(~) is an increasing function of strain rate and becomes close to the plateau modulus GN ~ at high strain rates. Eq. 18 amounts to the assumption that the deformation experienced by the entanglement network is just the macroscopic recoverable deformation of the specimen. With respect to the scattering experiments, this picture is of particular interest since it suggests to calculate the form factor of the chains in the melt as for a labelled multi-cross-link path in a strained network [28]. A slightly different calculation, taking
82
into account the dangling end segments has been proposed for uncrosslinked chains in a melt [14].
"~' 3.0x10 5. I:L
1/ ) 1@
~ 2 , ~ ,
0,0- - - . . , 0
&
0 0 0 o
0 i l l i l thai
A
~176 o o
0
- ! ' . - �9 -
5 10 15
& &
2'0' ~ 30 ~Lr2-;Lr "1
Figure 17. Tensile stress as a function of ~2_~-, for a commercial polystyrene at 140~ and 3 different strain rates, i : 0.005s'1; r'l: 0.01 s~; A: 0.02 ]-1.
The calculation only requires the knowledge of the network deformation, taken in our case as the recoverable strain which is known from experiment, and the number n of elastic subchains per chain which for high enough strain rates should be close to M/M~ where M is the total molecular weight of the chain and Mr the molecular weight between entanglements. The network model leads to the following expression of the radius of gyration in the directions parallel and perpendicular to the stretching direction:
Rg,j. = Rg,iso 1 + - 1 + n 2n 2
(19)
One may also estimate n from the scattering data by finding the best fit in the Kratky plots between the experimental and calculated form factor. This is illustrated in Fig. 18 for samples a2 and d2, in Fig. 19 for samples e2 and e3 and in Fig. 20 for sample $2 relaxed for 200s and 3000s. For the last sample, the sensitivity of the form factor to n is illustrated in Fig. 21.
The values of n determined by adjusting the SANS data in the intermediate scattering vector range are given in Table 5. Figs. 18 to 21 show that the network model satisfactorily describes the SANS data in the intermediate scattering vector range with reasonable values of the number of subchains, close to M/Mr for both S1 and $2 samples. As shown by the data in Table 5 for the radii of gyration, the agreement is also good in the Guinier range, for lower values of the scattering vector. It has to be noted that the radii of gyration of the long chains could not be measured, due to very long counting times on the spectrometer and to experimental problems arising for scattering vectors lower than 1 0 " 3 A "1. In the flame of the temporary network model, the number of elastic subchains can also be determined from the tensile stress according to the classical relation of rubber elasticity:
83
( cr = v c ~ r r n~ ~ r r
where v= is the number of elastic strands per unit volume and r~ the number of elastic strands per chain determined from the value of the stress. According to the network model, ~ should be equal to n (determined from the SANS data).
0,15 a:
,~3 0'10"
. ~ 0 , ~ .
O,O0. 0,00
z,,~=~ ....
0,02 0,04 0,06 0.08
q(A "1)
, • 0,20-
0,15.
,----~ 0,10 '
~ . 0,05,
o,o0. o
L~-~,-~-- ~ e2 ~ - O e 3 D a t a
, , v - - - - - - e 3 N e t w o r k
00 o,os O,lO o.ls q(A "1)
Figure 18. Kratky plots for specimens a2 and d2. Symbols: data. Solid lines: Network model for %=1.17 and n=5. Dashed lines: Network model for %= 1.4 and n=5.
Figure 19. Kratky plots for specimens e2 and e3. Symbols: data of Fig. 9. Solid lines: Network model for ~=1.64 and n =10. Dashed lines: Network model for 7~=2.21 and n=10.
A ~ 1~ 0,30 .r--
'E o v 025.
o~o-
'O' 0,15 u" --~" 0,10 %r
0,135
0,00 0 00 0,05
o 0 . 3 0 . . . . . . . . . IX I R = 3 ( X J O s D a l a ~.~'er 025-] " " . . . . . . . . n=40
/ . , ' ~ ------ n=60 - - - - - IR-- '200S N e l w o r k ~ 1 ~ \ \ - - - - - n=100
Ooooo~c ~
0,10 0, 5 ' 0,00 0,05 0,10 0,15 q(A -4) q(A "1)
Figure 20. Kratky plots for sample $2 with tR=200S and 3000s. Symbols: data. Dashed lines: Network model for L~=2.63 and n=60. Solid lines: Network model for ~=2.45 and n=60.
Figure 21. Kratky plots for specimens e2 and e3. Symbols: data of Fig. 9. Lines: Network model for ~=2.45 and 3 different values of n.
84
The data in Table 5 show that both values are of the same order of magnitude, though the number of elastic strands determined from the stress data are systematically lower than those determined from the form factor. The difference becomes even stronger if the viscous contribution o0 is subtracted from the stress according to Eq. 18.
Table 5 Recoverable strain and number of subchains determined from SANS (n) and stress (n,) data. Experimental radii of gyration compared to the predictions of Eq. 19.
Sample ~ n M o ~, (A) Pw, (A) ~I(A) ~ (A) pRT Z, 2 - kr -l (expefim.) (Network (experim.) (Network
model) model)
a2 1.17 5 2.5 91 92 75.5 78 b2 1.25 5 2.5 95.5 96 74.5 76 c2 1.35 5 2.5 100.5 102 75.5 74 d2 1.4 5 2.7 106 106 74 71 e2 1.64 10 6 125 129 60 67 e3 2.21 10 6 164 171 56 59
$2 tR=200S 2.63 60 42 757 181 $2 tR=3000S 2.45 60 30 705 188
3.2. Binary blends The chain conformation of the high molecular weight species in the binary blends has been
characterized in the same rheological test as for sample $2 (quick stretching up to L/L0=3, stress relaxation during a time tR at the same temperature, quenching). Fig. 22 shows the influence of the blend composition on the stress relaxation curve. For a given blend composition, the form factor can be measured as a function of tR, which is shown as an example for blend B 10 in Fig. 23. As expected, the chain anisotropy decreases with increasing time of the relaxation step. It is interesting to notice that the $2 chains have not completely relaxed even after tR=40 000S, which corresponds to about 8 times the terminal relaxation time
determined from linear viscoelasticity (see Table 3). Another possibility is to look at the influence of the blend composition on the anisotropy of
the long chains at a given value of tR. This is shown in Fig. 24 for tR=3000S and three blend compositions. The anisotropy of the high molecular weight chains increases by increasing their concentration in the blend. The SANS data in Figs. 23 and 24 are confirmed by the values of recoverable strain listed in Table 6.
We tried to extend the network model to the binary blends and to determine the number n of elastic strands per $2 chain by adjusting the form factor calculated from the recoverable strain to the SANS data. Fig. 25 gives an example of the type of agreement which could be obtained between the model and the data. The corresponding values of n are listed in Table 6.
85
(q-.
o �9 --A,-- BIO 0,15
~ ;L - - ,~_~- - .~ ._~ .. - - - ~ ~ ,--
o,o
e~. 0,05-
o i ~ ~ ;, ~ o.oo log(t) (s) o
Figure 22. Tensile stress as a function of time during relaxation at 140~ following a rapid deformation of L/Lo=3.
~z~'a~,zxzxzx
o %= 40(XX~
~0 o,~ 030 q(A "~)
Figure 23. Kratky plots for blend B 10 stretched and relaxed for various times.
020- �9 2~
0,15 - mll~OOOoom~ ' I ~ i 1~11~ m - m u
o.=t~ ~'~176176176176176 .~ 61~ o
0,1~ 0,02 0,04 0,06 0,08 0,10 0,12
q(A 1)
Figure 24. Kratky plots for blends B 10, B20 and B60 stretched and relaxed for 3000s.
<. 020-
~ 3 0,15
|
e'~5 o, Io
o,o5
0,1111. 0,000
/ 0 I E / \ r-1...
~tJ \ ~. 000, ~
g . _ _ - _ =~ o oo~o. j ~m_._==~._, _=.=._.= ..... . _
- - ~ a l l i ' - . . . . " - - - " = l n i l E
0,025 0,050 0,075 0,100 0,125 q(A "1)
Figure 25. Symbols: Data for blends B10 (tR=40 000S) and B60 (tR=3000S). Solid lines: network model with: L~ =1.19, n =10. Dashed lines: network model with ~=2.45 n=30.
To compare the values of n determined from the SANS data with those obtained from the stress values, a modified expression of the stress should be used instead of Eq. 20:
pRT (~r2 ~r_ 1 a - Ve(~r2 - ~ r - l ) - ~)2n~ - ~ - ) (21)
86
where ~2 is the weight fraction of high molecular weight species in the blend. Eq. 21 amounts to the assumption that the stress in the blends is only due to the high molecular weight chains.
The SANS experiments allow us to verify this assumption by using blends where the low molecular weight component S1 has been deuterated (blends B'20 and B'60). Fig. 26 shows the Kratky plots of these two blends, stretched in the same way as all the other blends and relaxed for 200s. Even for the shortest time of relaxation of the present study, the orientation of the low molecular weight component is found to be very low, and the assumption leading to Eq. 21 appears to be valid. However, the data in Fig. 26 show that the orientation of the short chains, like that of the long chains depends on the blend composition: the higher the content in high molecular weight species, the higher the short chain orientation.
It has to be noticed that due to the rather short relaxation times used for blends B'20 and B'60, no <<lozenge >> or <<butterfly>> scattering patterns [29] were observed for our experiments.
Table 6 Recoverable strain and number of elastic strands per $2 chain determined from SANS data (n) and tensile stress (n,,).
Sample ts (s) Z. n 1~-"
M r r Xr 2 - ~,r -1
B 10 200 2.29 20 16
B 10 1000 2.02 15 10
B10 3000 1.93 10 8
B 10 20000 1.23 10 6
B 10 40000 1.19 10 4
B20 200 2.59 20 15
B20 3000 2.08 15 8
B40 200 2.8 30 18
B40 3000 2.26 20 12
B60 200 2.84 40 23
B60 3000 2.45 30 18
87
0,15 a: 'E 0
0,10-
e,
e ~ , 0,o5. 0 "
= -
0,00 0
/ B'20 & B'60
o,~ .... o ,& " o ,= 0,08 q(A -~)
Figure 26. Kratky plots of blends B'20 and B'60 stretched up to L/Lo=3 and relaxed for 200s at 140~
4. SHEAR FLOW
4.1. Experimental results All experiments were carried out on sample S1. The radius of gyration of an isotropic
sample, determined from the Zimm plot in the low scattering vector range, is close to 82 A. Specimens were sheared by using the above described sliding plate rheometer at two
different shear stresses 07:0.05 and 0.2 MPa. For o~y=0.05 MPa, the behaviour of the melt was found to be nearly Newtonian. One sample was sheared at the highest shear stress up to a final shear strain of y=2.4. For the lowest shear stress, two samples with different shear strains were prepared: ?=2.8 and y=4. These samples are referenced in Table 7.
Fig. 27 defines the principal directions of molecular anisotropy (I, II and III) with respect to the principal directions of shear (x, y and z), x being the direction of velocity and y the normal to the plates. Symmetry of the flow leads to III-z and the relative orientation of (I,II) with respect to (x,y) is defined by the angle g.
Figure 27. Definition of the principal directions of molecular orientation (I,II) with respect to the principal shear directions (x,y).
88
At vanishing chain orientation (at either small strains or low shear rates) X is expected to be close to 45 ~ whereas for large affine chain deformation X tends to 0 ~ More generally, depends on shear rate or shear stress. It should be emphasized that the value of ~ depends theoretically on the way the orientation is measured. Birefringence measurements will give the extinction angle Xa, whereas SANS measurements in the range of low scattering vectors give slightly higher values (see Table 7). This can be qualitatively understood in the framework of molecular models where relaxation mechanisms with different relaxation times are associated with characteristic length scales on the chain [1]. It would be expected that the molecular anisotropy at a local scale on the chain is lower than the anisotropy of the end-to-end vector or radius of gyration. In the particular case of a shear flow, this will result in an orientation with respect to the shear axes which also depends on the considered length scale; in other words, in a scattering experiment may depend on the magnitude of the scattering vector [30]. In the following, the limit of ~ at zero scattering vector will be referred to by ~0.
Table 7 Parameters of the shear flow, characteristic angles of orientation and experimental root mean square chain dimensions in the various directions. Radius of gyratio n of isotropic sample: 82A.
Sample A B C
Shear Stress (Pa) 5 x 104 5 x 104 2x 10 ~ T(~ 130 130 120
Shear strain 2.8 4 2.4
X z~ 26~ 24~ 22"5~ Xaff. 18 ~ 13.5 ~ 20 ~ X 0 ~27 ~ ~27 ~ 19 ~
P-,g.i exp. (/~) 94 99 143 R ~ exp (A) 73 69 47
R ~ exp.x-z plane(A) 89 93 136 R ~ exp.x-z plane(A) 83 83 81 Rg, y exp.y-z plane(A) 77 R ~ exp.y-z plane(A) 83
R ~ calc. (A) 91 95 137 R~y calc. (A) 75
Fig. 28a shows typical isointensity curves on the multidetector for a specimen in the x-y shearing plane (for the sake of clarity, only three curves have been plotted). According to the above comments on a possible q-dependence of the chain orientation with respect to the shear directions, the scattering data were analyzed as follows: each isointensity curve was fitted with
89
an ellipse using a least square program which determines for each curve both the minor and major axes (which are the scattering vectors qI and qn along directions I and II, respectively) and their orientation ~ with respect to the shear axes.
:Y ~Z
b
._X.
'y
,
C s
._Z
Figure 28. Isointensity curves of sample B in the three principal shearing planes: x-y (a), x-z (b), y-z (c). All isointensity curves are fitted with ellipses. In the x-y plane, the minor and major axes define the scattering vectors qt and qa and their orientation with respect to the shear axes defines X. In the x-z and y-z planes, the axes of the ellipses coincide with the shear axes.
Fig. 29 shows the variation of 7~ as a function of the minor axis qi for the isointensity curves of sample C. Despite some scatter in the data, it appears that for low q values (typically qi<2.5 • -t which corresponds to qiI<5.5x 10-2A-~) X levels offto a value of the order of 19 ~
~3 v
25
20-
m m m m mm m
�9 �9 �9
~ mmmmm m~m �9
15 o,o o',s i',o
m
1000 x q2 (A-2) 2,0
Figure 29. Variation of ;~ with the square of the scattering vector in direction I.
90
The angle go can be compared on the one hand to the extinction angle gA of birefringence, and on the other hand to the orientation ~ . of the principal directions of the Cauchy deformation tensor, which would correspond to a molecular deformation purely affine with the macroscopic deformation shear strain. For a simple shear deformation y, %=r is given by:
1 arctg(2) (22) ;(aft '= 2
It appears that for sample C, the value of go determined from the SANS isointensity curves in the low q-range is lower than %A and close to ~ t (see Table 7). Furthermore Fig. 29 clearly shows that % increases with increasing magnitude of the scattering vector, thus confirming the idea that the direction of molecular orientation in shear flow depends on the considered length scale on the chain: the overall chain orientation is closer to that of an affine deformation than the local chain orientation [30]. The extinction angle of birefringence should then represent some average of g over the whole q-range. Since for the highest q-value of the present set of experiments, % is roughly equal to %A (see Fig. 29), this should be confirmed by complementary measurements in a higher q range, where it is expected that % will take values above 7,A.
Another interesting result is obtained from the comparison of the scattering of the sheared samples in the z direction to that of an isotropic sample. Fig. 30 shows the Zimm plot of sample C in the directions x and z for a specimen cut in the x-z plane. The corresponding curve for an isotropic sample, which has also been plotted in the same figure, is found to be identical to within experimental error to the curve in the z direction. This result indicates that the position correlations within the chain in the neutral direction of the shear flow are not affected by the flow, at least up to the values of stress and strain used in the present study. In particular, Table 7 shows that the mean square chain dimension in that direction, Rg z, which has been determined either in the x-z or in the y-z planes for the various samples, is found to be equal to the radius of gyration of an isotropic sample to within experimental error (Rg, z=82-d:I/~). The same result has been found by Lindner in dilute solutions [31 ].
=== == ~.~ i m
~ 3 =rl ==1~1 []
o; 0.0 o:s ~',0 ~',s 2:0 2:s looo x q2 (A-2)
3,0
Figure 30. Zimm plot of the absolute coherent scattering cross-section reduced to the concentration of deuterated species for sample B in the directions z (r'l) and x ( I ) compared to an isotropic sample (full line). The curves extrapolated at zero q are found to intersect at the same value. From their initial slope, one can calculate Rg,z and Rg,x.
91
4.2. Discussion For all three samples, it can be assumed that g becomes independent of q in the low q-
range (X=g0). As a consequence, one can determine a mean square dimension of the chain in the direction of gO and in the perpendicular direction g0+rd2, from a Zimm plot of the scattered intensity as a function of qi and qn respectively, according to Eqs. 7 and 8. The data for Rg,i and Rg,I I for the three samples are given in Table 7.
From specimens cut in the x-z and y-z sheafing planes, mean square chain dimensions similar to Rg,I and Rg,i I can also be determined in the principal shear directions x, y and z. Typical isointensity patterns in the three shearing planes are shown in Fig. 28 for sample B. It can be seen that the shape of these curves in the x-z and y-z planes is nearly elliptical like in the x-y plane. However, due to the symmetry of the shear flow, the z direction is a principal direction for molecular orientation at all scales and the major and minor axes of the isointensity curves are indeed found to be parallel to the {x,z} and {y,z} directions. Therefore, the analysis of the scattering data in the x-z and y-z planes has been carried out by fixing the direction of the axes of the ellipse (to 0 ~ and 90 ~ in the least-squares fit program. The major and minor axes of these ellipses then define the scattering vectors qx, qy and qz in the principal shear directions and allow us to calculate the mean square chain dimensions Rg,x, Rg,y and Rg,z.
Obviously, Rg,I and Rg,i I are not independent of Rg,x and Rg,y. From the relations between x, y and xx, xn:
{~ = x I COSX0 - Xli sin ~0
= X I sm g0 + x , cosz0 (23)
the following relations between the mean square coordinates are obtained:
<< x2 > = <xI 2 >COS2 X 0 + < X l I 2 >sin2 X0 - 2 <XlXi i >s inx0 cosg0
y2 > = < x i 2 >sin 2 Xo+<Xii 2 >cos 2 go +2 <xixii >s inxo cosgo (24)
If the principal directions for the chain orientation were X 0 and 7~ 0 +rt/2 at all scales, the mean value of the product XlX n would be zero (due to the symmetry of the conformation distribution with respect to the I and II axes). In this case, the following relations between the mean square chain dimensions would hold:
R 2 2 g,x = Rgj COS 2 X0 + Rg,II sin2 X0 2 2 n 2 2 Rg,y = Rgj si Zo + RgjI cOs2 X0
(25)
In Table 7, the experimental values ofR~. ,,x and Rg,y determined in the x-z and y-z shearing planes are compared to the values calculate~l from Rg,I, Rg,i I and g 0 according to Eq. 25. The observed agreement indicates that the q-dependence of ~ at local scales as shown in Fig. 29 has little influence on the mean square chain dimensions Rg,i and Rg, I I. As a matter of fact, these quantities are determined in the low q-range and involve mainly large scale position correlations for which the principal directions of orientation are equal to ~ 0 and X 0 +zc/2-
92
In section 3.1.3. we proposed a simple model to calculate the anisotropic form factor of the chains in a uniaxially deformed polymer melt. The only parameters are the deformation ratio Z, of the entanglement network (which was assumed to be identical to the macroscopic recoverable strain) and the number n e of entanglements per chain. For a chain with dangling end submolecules the mean square dimension in a principal direction of orientation is then given by Eq. 19. As seen in section 3.1.3. for low stress levels n can be estimated from the plateau modulus and the molecular weight of the chain (n~-5 por polymer S 1).
The molecular deformation ratio Z, in the directions I and II can be estimated in the following way: the difference Ao between the principal stresses in the x-y plane can be readily calculated from the birefringence A (measured parallel and perpendicular to the direction of extinction) and the stress-optical coefficient C for molten polystyrene (C = 4.8x10"gPa "1, see Chapter III. 1). According to the classical network theory, the stress tensor is proportional to the Cauchy deformation tensor which means that the network deformation along the principal directions of the stress tensor are Z, and I/Z, where:
k2 __}_.1 = Ao (26)
The values of Z, calculated from Eq. 26 are given in Table 8. It should be pointed out that the classical rubber theory does not account for a dependence of molecular orientation on the considered length scale, as found for sample C in Fig. 29. Therefore the simplistic approach presented here should in principle be restricted to samples A and B for which the principal directions for the low-q SANS data (I and II) are close to the principal directions for the refractive index. The data in Table 8 actually show a satisfactory agreement between calculated and experimental mean square chain dimensions for samples A and B. For sample C, the agreement is less satisfactory in the direction perpendicular to the chain elongation, but one must be aware that for this sample, the directions in which Rg I and Rg, i I have been measured are different from those corresponding to the calculation of Eq. 26.
Table 8 Extension ratio and root mean square dimensions in the principal directions of stress calculated from the network model (see text). . .....
Sample A B . . . . . C
A 9.4x 10 .4 1.08x 10 -3 3.8x 10 -3
X A 26~ 24~ 22"5~ AO (Pa) 2• 105 2.3• 105 8.1• 105
Z, 1.27 1.32 2.07 l/Z, 0.79 0.76 0.48
Rg(~) (A) 98 101 148 I/Z,) (A) 70 69 56 ,I (~) 94 99 143 ,II (A) 73 69 47
93
5. CONCLUSIONS AND PERSPECTIVES
The reliability of the experimental procedure (deformation in the melt, quenching, characterization of orientation at room temperature) could be verified in elongation (specimens quenched for different macroscopic extensions in steady elongational flow) as well as in simple shear (good correlation of the chain dimensions measured in all principal shear directions).
In the uniaxial elongational flow geometry, the proposed model of a temporary network leads to a satisfactory description of the SANS data both in the intermediate scattering vector range and in the Guinier range. The number of elastic strands per chain determined from both stress and SANS data is approximately the same. The agreement also holds for high molecular weight chains and binary blends. For these systems, SANS proves to be a powerful technique since selective deuteration allows one to characterize the orientation of the various components of the blends.
For the characterization of chain orientation in shear, specific experimental problems had to be solved. An apparatus was specifically designed for the purpose of quenching thick specimens oriented in simple shear with good homogeneity of orientation throughout the thickness.
The data obtained with the scattering vector in the x-y principal shearing plane showed that the principal directions of orientation depend on the magnitude of the scattering vector, and therefore on the considered length scale on the chain. On the other hand, the position correlations within the chain in the neutral direction of the flow have been found to remain unaffected by the shear deformation.
It should be mentionned that the result of q-dependent chain orientation in shear was also found by nonequilibrium molecular dynamics (NEMD) simulations by Kr6ger et al [32]. The increasing power of computational techniques will surely result in increased accuracy and usefulness of this type of numerical simulation.
REFERENCES
.
.
.
9. 10. 11.
Doi M. and Edwards S.F., The Theory of Polymer Dynamics, Clarendon Press, Oxford, 1986. Bird R. B., Armstrong R.C. and Hassager O., Dynamics of Polymeric Liquids, Wiley, New York, 1987. Higgins J.S. and Benoit H., Polymer and Neutron Scattering, Oxford Science Publications, 1994. Picot C., Static and Dynamic Properties of the Polymeric Solid State, NATO Advanced Study Series, Vol.94, Pethrick A. and Richards R.W. (Eds), Reidel Publishing. Cotton J.P., Decker D., Benoit H., Farnoux B., Higgins J., Jannink G., Ober R., Picot C. and des Cloizeaux J., Macromolecules, 7 (1975) 863. Kirste R.G., Kruse W.A. and Ibel K., Polymer, 16 (1975) 20. Daoud M., Cotton J.P., Famoux B., Jannink G., Sarma G., Benoit H., Duplessix R., Picot C. and de Gennes P.G., Macromolecules, 8 (1975) 804. Picot, C., Prog. Colloid Polym. Sci., 75 (1987) 83. Muller R., Picot C., Zang Y.H. and Froelich D., Macromolecules 23 (1990) 2577. Bou6 F., Bastide J. and Buzier M., Springer Proc. Phys., 42 (1988) 65. Lindner P and OberthOr R., Colloid Polym. Sci., 263 (1985) 443.
94
12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.
26. 27. 28.
28. 29. 30.
31.
Lindner P. and Oberth0r R., Physica B, 156 (1989) 410. Muller R., Pesce J.J. arid Picot C., Macromolecules, 26 (1993) 4356. Bastide J., Herz J. and Bou6 F., J. Phys. (Les Ulis, Ft.) 46 (1985) 1967. Muller, R; and Froelich D., Polymer, 26 (1985) 1477. De Cremes P.G., Scaling Concepts in Polymer Physics, Comell University Press, 1979. M0nstedt H., Rheol. Acta, 14 (1975) 1077. MOnstedt, H. and Laun H.M., Rheol. Acta, 18 (1979) 492. Delaby I., Ernst B., Germain Y. and Muller R., J. Rheol., 38 (1994) 1705. Rouse P.E., J. Chem. Phys., 21 (1953) 1272. Ferry J.D., Landel R.F. and Williams M.L., J. Appl. Phys., 26 (1955) 359. Rabin Y., J.Chem.Phys., 88 (1988) 4014. Green M.S. and Tobolsky A.V., J. Chem. Phys., 14 (1946) 80. Yamamoto M., J.Phys.Soc.Japan, 11 (1956)413; 12 (1957) 1148; 13 (1958) 1200. Lodge A.S., Armstrong R.C., Wagner M.H. and Winter H.H., Pure Appl. Chem., 54 (1982) 1349. Wagner M.H., Rheol. Acta, 18 (1979) 33. Muller R., Froelich D. and Zang Y.H., J. Polym. Sci., Polym. Phys. Ed., 25 (1987) 295. Bou6 F., Bastide J., Buzier M., Lapp A., Herz J. and Vilgis T.A., Colloid Polym. Sci., 269, 195 (1991). Ullman R., Macromolecules, 15 (1982) 1395. Muller, R. and Picot, C., Makromol. Chem., Macromol. Symp., 56 (1992) 107. Lindner P., Neutron, X-Ray and Light Scattering, Lindner P. and Zemb T. (Eds); North-Holland, Amsterdam, 1991. KrOger M., Loose W. and Hess S., J. Rheol., 37 (1993) 1057.
Rheology for Polymer Melt Processing J-M. Piau and J-F. Agassant (editors) �9 1996 Elsevier Science B.V. All rights reserved.
95
M o l e c u l a r Rheo logy a n d L i n e a r Viscoe las t i c i ty
G. Marin and J.P. Monffort
Laboratoire de Physique des Matdriaux Industriels, URA-CNRS 1494, Universit6 de Pau et des Pays de r Adour, 64000 Pau (France)
1. INTRODUCTION
Measurement of the l inear viscoelastic properties is the basic rheological characterization of polymer melts. These properties may be evaluated in the time domain (mainly creep and relaxation experiments) or in the frequency domain: in this case we will talk about mechanical spectroscopy, where the sample experiences a harmonic stimulus (either stress or strain).
One may use the linear viscoelastic data as a pure rheological characterization, and relate the viscoelastic parameters to some processing or final properties of the mate r ia l under study. Fur thermore , l inear viscoelasticity and nonl inear viscoelasticity are not different fields that would be disconnected: in most cases, a linear viscoelastic function (relaxation fimction, memory function or distribution of relaxation times) is used as the kernel of non linear constitutive equations, either of the differential or integral form. That means that if we could define a general nonlinear constitutive equation that would work for all flexible chains, the knowledge of a single linear viscoelastic function would lead to all rheological properties.
On the other side, the linear viscoelastic functions of polymer melts are directly related to their molecular structure, and these relations are now defined well enough to derive, with a reasonable accuracy, what would be the linear viscoelastic behaviour of a given species from its molecular weight distribution, at least in the case of linear polymers. Moreover, there is a definite trend to use rheology as a molecular characterization tool, in the same way as other popular spectroscopic methods such as N.M.R spectroscopy or Size Exclusion Chromatography. However, the inverse problem, i.e., getting molecular weight distribution from rheological measurements, is a difficult (and ill-defined) problem, and this is a very up-to-date area of research due to its obvious practical implications in polymer characterization.
The relationship between chemical structure and viscoelastic behaviour is established through molecular models considering that polymers relax or diffuse in the same way: they are considered as flexible statistical chains trapped between the topological constraints created by the surrounding chains.
96
2. LINEAR VISCOELASTIC BEHAVIOUR OF LINEAR AND FLEXIBLE CHAINS . BASICS AND PHENOMENOLOGY
We will begin with a brief survey of linear viscoelasticity (section 2.1) �9 we will define the various material functions and the mathemat ical theory of linear viscoelasticity will give us the mathematical bridges which relate these functions. We will then describe the main features of the linear viscoelastic behaviour of polymer melts and concen t ra ted solutions in a pure ly ra t iona l and phenomenological way (section 2.2)" the simple and important conclusions drawn from this analysis will give us the support for the molecular models described below (sections 3 to 6).
2.1. L i n e a r viscoelast ic functions The measurable l inear viscoelastic functions are defined either in the time
domain or in the frequency domain. The interrelations between functions in the frequency domain are purely algebraic. The interrelations between functions in the time domain are convolution integrals. The interrelations between functions in the time and frequency domain are Carson-Laplace or inverse Carson-Laplace transforms. Some of these interrelations will be given below, and a general scheme of these interrelations may be found in [1]. These interrelations derive directly from the mathematical theory of linear viscoelasticity and do not imply any molecular or continuum mechanics modelling.
In the various formula t ions of the ma thema t i ca l theory of l inear viscoelasticity, one should differentiate clearly the measurable and non- measurable functions, especially when it comes to modelling: apar t from the material functions quoted above, one may also define non-measurable viscoelastic functions which are pure mathemat ical objects, such as the distribution of relaxation times, the distribution of retardation times, and the memory function. These mathematical tools may prove to be useful in some situations: for example, a discrete distribution of relaxation times is easy to handle numerically when working with constitutive equations of the differential type, but one has to keep in mind that the relaxation times derived numerically by optimization methods have no direct physical meaning. Furthermore, the use of the distribution of relaxation times is useless and costs precision when one wishes simply to go back and forth from the time domain to the frequency domain. This warning is important, given the large use (and sometimes overuse) of these distribution functions.
2.1.1. Functions in the time domain The relaxation modulus G(t) is the value of the transient stress per unit strain
in a step-strain experiment. This type of experiment may be achieved with modern rotary rheometers with a limited resolution in time (roughly 10 -2 s). If one wishes to evaluate G(t) at shorter times, it is necessary to derive G(t) from the high frequency G*(co) data by an inverse Carson-Laplace transform.
The creep function J(t) is the transient strain per unit stress in a step-stress experiment. The resolution at short times is also limited from instr~ment response and sensitivity. J(t) at short times may also be derived from the high frequency complex compliance data.
97
One may also use a memory function m(t) within the integral formulation of l inear and nonl inear viscoelasticity; this memory function m(t), which is the derivative of the relaxation function G(t), is not a measurable function.
2.1.2. Functions in the frequency domain Three complex functions may be used to characterize the l inear viscoelastic
behaviour in the frequency domain"
- The complex shear modulus, which is the complex stress to complex strain ratio"
G * (co) = complex stress complex strain
c*(r y*(~) - ------- = G'(co)+ j G"(co) (2-1)
- The complex viscosity, which is the complex stress to complex rate of strain ratio"
11"(o)) = complex stress = ~* (co-----)) = Tl' (co)- j 11"(o)) = ~ _ _ G *(co) (2-2) complex rate of strain ~/* (co) jco
- The complex compliance, which is the complex strain to complex stress ratio" j .(co) = complex strain = 7*(co) = j,(co)_ j j,,(co) = 1 (2-3)
complex stress ~* (co) G* (co)
These three functions are related via simple algebraic relations"
1 G * (co) = ~ = jco T1 * (co) (2-4)
J*(r
Although the complex shear modulus is not the most appropriate function to use in all cases, we will describe the linear viscoelastic behaviour in terms of this last function, which is the most referred to experimentally; furthermore, molecular models are mostly linked to the relaxation modulus, which is the inverse Fourier t ransform of the complex shear modulus.
2.1.3. The distribution of relaxation times By analogy with a generalized Maxwell model, it is possible to write the
re laxat ion modulus or the complex modulus as a sum of the contributions of n individual Maxwell models"
t 12 12 - - - -
G ( t ) = ~ G k ( t ) = ~ G k e ~k k = l k = l
(2-5)
4 o o
and G * (co)= jco ~ G(t) e -j~t dt 0
(2-6)
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n n
hence G*(co)= ~ G k *(co)= ~ jOTlk k:l k = l l + f ~ k (2-7)
with Tlk=Gk ~k. In terms of molecular models (section 3), the set of individual times ~k and weighting factors Gk is imposed by the model.
It is also possible to derive a set of relaxation times and weighting factors numerically by optimization or approximation methods from the experimental data. In that case, the relaxation times have no real physical meaning and are simply numerical/empirical parameters which allows one to represent the viscoelastic behaviour as a sum of decaying exponentials which are handy to use for numerical analysis.
It is also possible to give integral forms of these sums"
o o
G * (co) = jr H(z) 0 ~+jcoz dz (2-8)
+oo t
and G(t)= ~ H(z)e ~ d ln(z) - - o o
(2-9)
H(z) is a continuous distribution of the logarithms of relaxation times and is called the "relaxation spectrum" by rheologists, whilst the "true" distribution of relaxation times is zH(z). We have r e p o ~ on Figure I the normalized distribution of relaxation times for 4 polystyrene samples with polydispersity indices ranging from 1.05 to 4.2 [2]. It is clear that the distribution of relaxation times broadens with the distribution of molecular weights; these features will be analyzed in terms of molecular models in sections 3 to 6.
v
o
I -3
A
-2 -I 0 +1 *2 Log z/Zo
Figure 1 �9 Reduced distribution of relaxation times for atactic polystyrene samples having different values of polydispersity index P=l.05 (A), 1.25 (B), 2.45 (C), 4.2 (D)[ref. 2]
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2.2. The m a i n f ea tu re s of the l inear viscoelast ic b e h a v i o u r of po lymer mel ts :
We will discuss in this section the variations of the viscoelastic parameters derived from linear viscoelastic measurements; all these parameters may be derived from any type of measurement (relaxation or creep experiment, mechanical spectroscopy) performed in the relevant time or frequency domain. The discussion will be focused however on the complex shear modulus which is the basic function derived from isothermal frequency sweep measurements performed with modern rotary rheometers.
log G* (W) - G, . - . . . - . .~ ~ - . - - . ~ G'
f -
- / / , , ' M 2 f ~ ~
G o'
Mu
G e
G ~
log UJ
t er min al plat eau transition glass y
POLYMERIC "BEHAVIOUR SMALL SCALE
MOLECULAR MOTIONS
Figure 2 �9 Schematic of the variations of the complex shear modulus of linear polymers (dotted lines: molecular weight M2>M1).
The main features of the linear viscoelastic behaviour of polymeric melts in the frequency domain are reported on Figure 2 : - the lowest frequency range describes the slowest relaxation motions of the macromolecules. The double logarithmic plot of G' and G" exhibits slopes of -respectively- 2 and 1, leading to two characteristic parameters.
The zero-shear viscosity"
~o = lim G"(co____~) (2-10) co-,0 CO
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The limiting compliance:
j 0 = l i m G'(r = 1 limG'(c~ ~-~o [G"(co)] 2 Tio z ~-,o r ~ (2-11)
which is the elastic parameter governing the main features of the elasticity of the melt (first normal stress difference, extrudate swell, etc...).
The zero-shear viscosity is the norm of the relaxation spectr~m :
Tlo = ~ ~H(z)d In z (2-12)
The product TIo jo is the characteristic relaxation time zo of the terminal region. In terms of molecular models, this time scales as the longest relaxation time. In terms of the distribution of relaxation times H(z), zo is the "weight-average relaxation time" which is the average relaxation time related to the second order moment of the relaxation spectm~m :
~o = no jo =< ~w >= ~ z2H(z)d In (2-13)
At intermediate frequencies, monodisperse polymers exhibit a well-defined "plateau region" where G'= constant G~ (Figure 2). For a given macromolecular species, the value of the plateau modulus is a characteristic feature tha t does not depend on molecular weight. The only way to lower the plateau modulus is to add small compatible molecules, either of the same species or not : this is, for example, what is done for Hot-Melt Adhesives (HMAs) when adding a "tackifying resin" which softens the polymer and improves the "tack". In terms of the distribution of relaxation times, the ratio TIo/G~ is an average relaxation time (we may call it "n-tuber-average relaxation time"), which is the first order moment of the normalized relaxation spectm, m :
< ZN >= no / G~ = i ~H(z)d In H(~)d In z
(2-14)
The ratio:
< Zw > o c o = J e (2-15)
<~N >
is a "polydispersity index of relaxation times" which characterizes the broadness of the distribution of relaxation times. For monodisperse species, the experimental value of this ratio lies between 2 and 2.5, whatever the polymer nature. This value increases largely as polydispersity increases. One of the direct practical
lO1
applications of molecular modelling will be to relate the distribution of molecular weights to the distribution of relaxation times.
2.2.1. The effects of chain length 2.2.1.1. The zero-shear viscosity Tl0
The variations of the zero-shear viscosity of monodisperse polymeric melts and concentrated solutions exhibit two domains, each being characterized as a first approximation by a power law exponent" - at low molecular weights, corresponding to less than 200 monomeric units, the exponent lies between 1 and 1.5. This domain will be analyzed in section 5. - above a critical molecular weight Mc, the power law exponent is 3.4-3.5. As far as viscosity is concerned, Mc defines the begining of a regime where the
macromolecular chains are viewed as "entangled", which explains the large molecular weight dependence of viscosity. The entangled regime will be modelled later using the "reptation" concept (see section 3).
i 0 K)
vi r I0 e -
n I ' - - I
(J o
c; . IO s -
~_ _
0
10 4 -
10 2 10 3
f ! i
_
o / + /
13 g / /,
o
/
,+ #o
o/
I I I 10 4 10 5 10 6
log M
Figure 3 : Molecular weight dependence of the structure factor of nearly monodisperse polystyrene samples [3].
Figure 3 illustrates this type of behaviour for various series of anionic polystyrene samples with a fairly narrow distribution of molecular weights.
In the case of entangled polydisperse materials (Mw>>Mc), the zero-shear viscosity follows approximately the same molecular weight dependence as for monodisperse species when the viscosity data is plotted as a function of the weight-average molecular weight Mw; i.e. :
110 = A(T) M~ 4 (2-16)
I02
2.2.1.2. The plateau modulus G~ The plateau region begins to be developed at molecular weights somewhat
above Mc; it is however a well-defined elastic parameter for a given chemical species of high molecular weight. Comparing different polymer species, its value increases with the flexibility of the chain (i.e: o o GNPolystyrene< GNPolyethylene). The use of the theory of rubber elasticity may give an order of magnitude of the average molecular weight Me between entanglements which create a temporary network:
G~ = pRT (2-17) Me '
p being the polymer density and T the absolute temperature. The critical molecular weight Mc is roughly two times the molecular weight between entanglements Me.
2.2.1.3. The limiting compliance jo
In the entangled regime, the limiting compliance jo of monodisperse samples is also an elastic constant characterizing a given polymer chemical species: contrary to the plateau modulus, its value depends on the distribution of molecular weights, i.e., on the polydispersity index as a first approximation. This is a very important point, as many elastic effects (first normal stress difference, extrudate swell, ...) of the melt are governed by the limiting compliance. For purely monodisperse samples, the product jo G~ has a value close to 2 for all flexible polymers.
Below the entangled regime, the limiting compliance follows approximately a linear dependence with molecular weight according to Rouse's theory (see section 6)-
jo = 0.4 M . (2-18) pRT
The molecular weight value M'c where the compliance becomes independent of molecular weight is larger than Mc (M'c--3Mc), which indicates that the "polymeric" regime seems to appear at higher molecular weights for elastic properties compared with viscous properties. So one has to keep in mind that the chain length (or molecular weight) at which "entanglements" effects begin to appear depends strongly on the physical property measured (melt viscosity, melt elasticity, self-diffusion, etc...) (see in particular chapter L1). In the case of polydisperse polymers, the limiting compliance increases strongly with the broadness of the distribution of molecular weights. The limiting compliance is not, however, a simple function of the polydispersity index, because its value depends on the shape of the distribution itself. There is indeed no simple correlation with any molecular weight moments (averages), and molecular models will be really helpful to describe the elasticity of the melt.
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2.2.2. The effect of temperature: For a given polymer, the viscoelastic curves (either moduli or compliances)
obtained at different temperatures in the plateau and terminal regions are simply afflne in the frequency (or time) scale, in a double logarithmic plot. The use of this time-temperature equivalence allows one to obtain "master curves" at a reference temperature, which enlarges considerably the experimental window. For glass-forming materials such as polystyrene, polymethylmetacrylate, polycarbonate, polymerists describe the shift factor aT in terms of the WLF equation:
-c~ -To) (2-19) In aT = (c o + T - T o ) '
T being the experimental temperature and To the reference temperature to which the data is shifted. The WLF may be reduced to the Vogel equation which describes the viscosity of molten glasses and supercooled liquids"
B/af B / a f (2-20) In a T = T0 _ - - - ~ - W-'-~_ '
where the limiting temperature Too may be related to the glass transition
temperature Tg by the approximate rule: Tg-Too = c2g -- 60 ~ C. The entropic nature of the elasticity of the melt implies also a slight vertical
sbJR in the plateau and terminal regions. This shi~"
b T = P ~176 (2-21) p T '
may be neglected when using the time-temperature equivalence in a limited range of temperatures.
The time-temperature equivalence implies that the viscosity (or relaxation times) of polymers may be written as the product of two functions :
no = $(P(M)). M (T) (2-22)
The mobility factor M (T) describes the segmental mobility of the chain : it depends mostly on temperature and pressure, but may be affected by the presence of small chains (such as solvent molecules or small chains of the same chemical species as the polymer). For concentrated polymer solutions, the addition of small molecules affects mostly the glass transition temperature (hence Too), and the value of B (eq.2-20) is essentially the same as for the bulk polymer. A plastifyer will decrease the value of Too, and hence increase the segmental mobility. On the contrary, the addition of a tackifying resin which has a higher Tg than the polymer will increase the segmental mobility of the polymer in the case of formulations of Hot-Melt adhesives.
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The structure factor $(P(M)) describes the topological relaxat ion of the macromolecular chains: t ~ s is the function which will be described by molecular models, P(M) being the distribution of molecular weights. Here lies a very i m p o ~ t point: if one wishes to "isolate" the topological effects in order to test molecular models, one has to use rheological functions defined at the same segmental mobility, and hence the same value of the mobility factor: as far as viscosity is concerned, the reduced function Tlo/M (T) will be used instead of the viscosity itself.
2.2.3 The effects of concentration (concentrated solutions): In the case of concentrated entangled solutions, the "elastic" parameters follow
power law dependences as a function of polymer volume fraction r :
( 0 0 {~{~ GN)sol=(GN)bulk (2-23)
( jO)sol=(jO)bulk ~-a (2-24)
with an exponent a - 2-2.3 for entangled chains, so the product jo G~(which reflects the polydispersity of relaxation times) remains the same whatever the concentration. That means that the effects of the addition of small compatible species on the elastic parameters are mainly topological, i.e., the nature itself of the solvent molecules has a very small effect on the melt elasticity and the shape of the distribution of relaxation times.
On the contrary, the effects of dilution on the polymer viscosity will be twofold : - a topological effect on the structure factor $ that will be described by molecular models; - a change of the mobility factor M, that may either increase or decrease, depending on the plastifying -or antiplasfif3dng- effect of the molecules added to the polymer.
2.2.4 The self-similarity of the viscoelastic behaviour of flexible chains The above phenomenological description of the viscoelastic behaviour of
polymer melts and concentrated solutions leads to the following impor tan t conclusions �9 if one focuses on the behaviour in the terminal region of relaxation, what is usually done for temperature (time-temperature equivalence) may also be done for the concentration effects and the effects of chain length; one may define a "time-chain length equivalence" and "time-concentration equivalence"[4]. For monodisperse species, the various shifts along the vertical (modulus) axis and horizontal (time or frequency axis) are contained in two reducing parameters: the plateau modulus G~ and a characteristic relaxation time, ei ther Zw = qo jo or
ZN = rio/G~.
A plot of G*(cOZo)/G ~ - where z o is either T~V o r 1; s - in the terminal relaxation region is a universal function independent of temperature, concentration, chain length, and independent also of the chemical nature of the polymer (Figure 4).
This self-similarity of the viscoelastic behaviour of monodisperse linear chains, whatever their chemical structure, may be extended to polydisperse species having the same shape of the molecular weight distribution (i.e., the same
105
polydispersity index as a first approximation). This implies some universality in the large-times relaxation processes of entangled polymers. As a consequence, the general features of the mechanical relaxation of long and flexible polymeric chains will be described by molecular models that do not "see" the local structure but describe the overall diffusion and relaxation of these chains in a universal way. Hence the power of the models described below lie in their universality:, it is easy to shift from one polymer to another, changing only a few parameters linked to the local scale structure of the polymer under study.
0 -
o
@
l I I , _ , I -2 -I 0 I 2
log (~qo G~
Figure 4 : Master curve for the linear viscoelastic behaviour of entangled polymers in the terminal region of relaxation : V Polystyrene, bulk (M=860000, T=190~ Q Polyethylene, bulk (M=340000, T=130~ A Polybutadiene solution (M=350000, polymer=43%, T=20~ [from ref.4].
3. THE CASE OF ENTANGLED MONODISPERSE LINEAR SPECIES : P U R E REPTATION
3.1. The basic reptation model The reptation concept was introduced by de Gennes [5] in 1971: it is based on
the idea that long and flexible entangled chains rearrange their conformations by reptation, i. e., curvilinear diffusion along their own contour. De Gennes considered the reptation of a linear chain among the strands of a crosslinked network which create permanent topological obstacles. First, the dynamics of the wriggling motion of the chain along its own contour (what Doi and Edwards called later the primitive path) was described by de Gennes in terms of a diffusion equation of a "defect gas": he showed that this motion is fairly rapid: its longest relaxation time Teq is proportional to M 2, where M is the molecular weight of the chain (that time Teq would be equivalent to the ~B relaxation time in the slip-link model; see text below and Fig.8).
�9 �9 �9 �9 �9 �9 �9 �9 Q �9
106
Figure 5 �9 The basic concept of P.G. De Gennes �9 reptation of a chain t rapped in a tube-like region by migration of "defects" along the chain.
At t imes t >Teq, the wriggling motion results merely in a fluctuation around the primitive path, so the chain moves coherently in a one-dimension diffusion process, keeping its arc length constant. The macroscopic diffusion coefficient of a reptat ing chain scales with chain length (molecular weight) as"
D o, M-2, (3-1)
and the time for complete rearrangement of conformation"
1; o~ M 3. ( 3 - 2 )
This time is the longest relaxation time of a linear chain. We will refer to it as the "reptation time".
e ~ - , . e
, /
. -
Figure 6 �9 The tube concept: The real chain is trapped between entanglements and is wriggling aroud the "primitive chain" (full line).
Doi and Edwards used this concept to derive the viscoelastic properties of polymer liquids from the dynamics of reptating chains [6]. They ass1~med that
107
reptation would be the dominant relaxation process for polymer melts, even in the absence of a permanent network. This is a very strong assumption, as the topological constraints are made by surrounding chains that also diffuse by reptation. This assumption was justified later when analyzing the constraints release (or tube renewal) process (section 4).
f ~ / I "- \ ? J / ~ \ \
_ _ k J
Figure 7 �9 Reptation: the chain disengages from its initial tube by back-and-forth motions; the time necessary for a complete renewal of its initial configuration is the "reptation time" which is the longest relaxation time of the polymer.
When analyzing the overall diffusion of a single chain due to Brownian motion, the topological constraints made by the surrounding chains confine the chain in a tube-like region. The centreline of the tube is called the "primitive path" and can be regarded as the curve which has the same topology as the real chain relative to the other polymer molecules; the real chain is then wriggling around the primitive path. The Doi-Edwards calculation is based on the theory of rubber elasticity. In order to calculate the time-dependent properties, the contribution of individual chains to the stress following a step strain is evaluated; then the relaxation of the stress is related to the conformational rearrangement of the chains by a reptation process. For this calculation, the topological constraints along the chain are represented by frictionless rings around the chain (Fig. 8), which is another way of describing entanglements. The succession of segments ("primitive segments") joining these "slip-links" along the chain is called the "primitive chain". As the slip- links are now local constraints, the continuous nature of the constraints is accounted for by assuming a natural curvilinear monomer density along the chain. So all three models (reptation among fixed obstacles/tube model/slip-links model : Figs. 5 to 8) are equivalent in essence. When submitting the polymer sample to a sudden deformation (step strain), the primitive path is distorted by affine deformation (Fig. 8b), and the curvilinear monomer density is perturbed from its equilibrium value.
108
B D F
A C E
a) t < O : e q u i l i b r i u m s t a t e
A C E
b) t = O : s t e p s t r a i n
c) t--_ 1; a ( A p r o c e s s )
C E
d) t _=_ ~s (B p r o c e s s )
C ~ E ~
B' D'
e) t =_ 1: c (C p r o c e s s : reptation)
Figure 8 " Relaxation of a polymeric chain after a step-strain deformation[6]: process A (8c): reequilibration of chain segments; process B (8d): reequilibration across slip links; process C (8e): reptation.
Then the chain will relax: The f i~ t relaxation process (called the A relaxation process; Fig. 8c) which
occurs at the shortest times will be a local reequilibration of monomers without slippage through the slip-links. In other words, it is basically a Rouse relaxation process between entanglement points which are assumed to be fixed in that time scale. The characteristic relaxation time of this process is rather short and is independent of the overall chain length (see below).
The second relaxation process (B process; Fig. 8d) is a reequilibration of segments along the overall chain, i.e., across slip-links. It is basically a retraction of the chain to recover its natural curvilinear monomer density, which may be depicted as a Rouse relaxation process along the entire chain.
In the last relaxation process (C process; Figs. 7 and 8e), the chain renews its entire configuration by reptation.
The viscosity, p la teau modulus, l imiting compliance and maximum (terminal) relaxation time derived from the basic D-E model are power laws of the molecular weight M:
Tlo or M 3,
G~ or M o,
jo or M 0,
to = ~o jo o~ M 3,
(3-3)
109
and j O G ~ = 6 . 5
All viscoelastic functions may be expressed in terms of a single reptation parameter (for example the plateau modulus or tube diameter) and the monomeric friction coefficient (or mobility factor in our terminology), in agreement with the above phenomenological presentation.
If the general features of the viscoelastic behaviour are well depicted, the experimental molecular weight dependence of these parameters is :
110 or M3.4,
G~ o~ M o,
jo o~ M o,
~0 = T10 jo or M3.4,
and j O G ~ = 2 .
(3-4)
In particular, the fact that the experimental "polydispersity of relaxation times" jo G~ is larger than the theoretical value indicates the presence of other relaxation processes.
In the following section we will describe a somewhat different analytical derivation of the Doi-Edwards model taking into account additional relaxation processes (in particular the Ta (glass transition) relaxation). Our derivation, which gives a good quantitative agreement with the observed l inear viscoelastic behaviour for a large number of linear polymers, keeps however the basic physical concepts of the reptation model. Hence this model is dedicated only to the case of entangled linear monodisperse species. The case of branched polymers, polydispersity and short chains effects will be presented in, respectively, sections 4,5 and 6. The same basic models are also used in chapter 1.1 and 1.3. In chapter 1.1, the mutual diffusion in polymer melts is related to the reptation and constraint release processes, whereas in chapter 1.3 the relaxation of chain segments along the polymer chain is investigated in terms of the some relaxation mechanisms as described below.
3.2. Detai led der iva t ion of l inear viscoelast ic p roper t ies of l inear s p e c i e s [7]
3.2.1. The (A) relaxation process That relaxation process may be defined as a Rouse diffusion between
entanglement points. The characteristic relaxation time of the (A) process is :
~ob2 N 2, (3-5) 1; A = 6~2ksT
where Ne is the number of monomers between entanglement points, ks Boltzmann's constant, T the absolute temperature, b the effective monomer length
110
(b=C~o 1,1 being the monomer length) and ~o the monomeric friction coefficient. The relaxation function associated to the (A) process is :
FA(t) = expF- P t], p=l L ~A J
(3-6)
where Ne is the number of Kuhn segments between entanglements. In order to define completely the relaxation function, we have to determine the initial modulus G'N SO:
O(t) = GN FA(t), (3-7)
G'N is a function of strain and may be written as :
GN = pRT 1 Me (~Ul>o , (3-8)
where p is the polymer density, E is the strain tensor, u a uni t vector corresponding to the vector linking two entanglement points (slip-link segment) [6];
1 the < ~ / o term expresses the fact that the molecule is outside its equilibrium
configuration" from Doi-Edwards theory G N = 4 pRT. 5 M e
We may then write the relaxation modulus corresponding to the (A) relaxation process (short times) as :
N~ [ 21 GA(t) = 4 pRT ~ e x p _ p t 5 M e p=l L ZA J"
(3-9)
3.2.2. The (B) relaxation process Doi and Edwards postulate that the chain recovers its equilibrium monomer
density along its contour by a retraction motion of the chain within the tube. That motion, induced by the chain ends, leads to a relaxation function :
8 exp[-p2t] FB(t) = poddZ p2/t;2 L -~-BJ' (3-10)
with:
~0 b2N2 ~B = 3g2kBT
(3-11)
where N is the number of segments along the chain.
111
Viovy [8] describes that re-equilibration process as an exchange of monomers between neighbouring segments : he calls that process "reequilibration across slip- links", and the corresponding relaxation function may be written as :
N/Ne Ne [ p2t] FB(t)= ~ - -~ -exp- - - - - .
p=l ~;B J (3-12)
This function corresponds to a Rouse spacing of relaxation times and gives a better fit of the experimental data than Eq. 3-10. Hence the relaxation modulus of the (B) process may be written as a hmction of the entanglement density N/Ne:
G B(t) = 4 pRT N~~ Ne [ p2t] 5 Me -'NeXPL- B j (3-13)
3.2.3. Reptation: relaxation process C In that final relaxation process the molecule recovers its final isotropic
configuration by a reptation motion. The characteristic time for the reptation process is (see also chapters L1 and L3):
1 ~ 0 a 2 N e ( N ) 3 ~:c =--g kB T "~e '
(3-14)
where a is the tube diameter (a 2 = Ne b2). The relaxation function is given in the original Doi-Edwards picture [5-8] by the equation"
exp[-p2t] Fc(t)= E p28~2 podd k "~'C J
(3-15)
The plateau modulus of that relaxation domain is :
= RT Me '
and hence the relaxation modulus is:
(3-16)
Go(t)= pRT 8 exp[-pat] podd k -~-cJ
(3-17)
The refinement introduced by Doi [9] who considers tube length fluctuations is more relevant to experimental scaling laws as far as the viscosity/molecular weight dependence is concerned. Following that concept, Gc(t) can be cast into the integral form :
112
Oc t':0RT[ ' / ' / Me JO4N/N~ e - ~ ( i i d~ + ~2v/s, exp - ~(2i ~, '
where
N ~4v4 2v
~(1) = Ne 16 Zc ~ < ~]N/Ne,
v
~(2) = ~- 4N / Se ~c
(3-18)
(3-19)
2 v 1 > ~ > ~~/'N/N~, (3-20)
The v parameter may be approximated to 1 for highly entangled chains.
3.2.4. The Ta high-frequency relaxation domain: the transition region between the rubbery and glassy regions
In order to give an analytical representation of the mechanical properties in the high frequency range characterizing local motions in the molecular chain, we used analytical forms derived from studies on dielectric relaxation : a Cole-Cole or Davidson-Cole equation generally gives a good fit in the transition region; we used in the present case a Davidson-Cole equation, that presents the advantages of being truncated at large times and to give analytical forms both in the time and frequency domains"
G *HF = Goo - Go. (3-21) (1+ j~HF) 1/2 '
with"
~o 12 1;HF - X2kBT
(3-22)
The inverse Fourier transform of Equation 3-21 gives the relaxation modulus"
] (3-23)
Other mathemat ical forms may be used to describe the high frequency relaxation. These various equations, either phenomenological or based on diffusion defect models lead to a characteristic relaxation time ~;HF of the glass transition (Ta) domain of the same order of magnitude.
113
As a s u m m a r y , the characteristic relaxation times of the various relaxation mechanisms presented here above are linked to each other by (see also Chapter 1.3):
1 2 I:.~ = ~ I: i Ne
( ~ ) 2 l N2 SB = 21;A = ~ 1;i
N N 3 1;c = 3ZB ~ee = I;i Ne
with ~i = ~0 b2 ~2kB T "
(3-24)
3.2.5. Viscoelastic function in the whole time/frequency domain Thus the relaxation modulus may be calculated from a very limited number of
physical parameters (G ~ Goo and ~i), with no "ad-hoc" parameters , in a time range covering the initial glassy behaviour down to the terminal relaxation region. For a typical polymer, this range exceeds ten decades of times. The complete expression of the relaxation modulus is :
- 2v exp ~ 2) G(t) = Me NINe exp zr 4N/N e
4 p R T I N ~ e [ p2t 1 N~eNe [ p2tll [ ( # - -~ ) ] + - e x p - - - + --~-exp- + G~ 1 - e f t .
5 Me L p=l I:AJ p=l ~BJJ (3-25)
The complex shear modulus is the Fourier transform of Equation 2-25 :
2,, jcozr ~1 jco~r ] G * (co) = G~ f04N/Ne d~ + 2v d~
1+ jco~(1) ~]N / Ue 1+ jC0Z~(2)
+ - G ~ e Ne jc0(1: B/p2) + 5 k p=l N l+jco(~ B/p2)
N Ne ] I 1 1 jc0(~A/p2) + Goo 1- 1 .(3-26) p=l 1+ jco(x A/p2) (1+ jO~HF)2
We have reported on Figs 9 through 12 a comparison between the experimental complex shear modulus and its theoretical calculation (full line) for two polymer species. The model fits reasonably well the linear viscoelastic properties of a large number of linear polymers ranging from polyolefins to glass-forming polymers. This calculation gives us the basic "long-chains monodisperse behaviour" which feeds the more complete derivation taking into account the effects of constraints release (section 5).
114
,oj. f
- 8 I
~-6
4
i l -5 0 5 I0
log t~ (sec -I]
E o
~,6
o =
4
1
8 . -
log ~ (sec-a l
Figures 9 and 10 �9 storage (G') and loss (G") moduli of nearly monodisperse polystyrene samples at 25~ �9 (A)M=900000; (Q)M=400000; (O)M=200000; (0) M=90000; full line ( )" theory (eq.3-25) [from ref. 7].
I ~ f : ~' I 1 i " ' l 1 1 Io l ~ ; ~ ~ ; J !i
/ 1
8~- 8
4 i
loq ~ (sec J ] log ~J (sec -~)
Figures 11 and 12: Storage (G') and loss (G") moduli of nearly monodisperse polybutadiene samples at 160~ (A)M=361000; (~)M=130000; (O)M=39400; full line ( ): theory (eq.3-25) [from ref.7].
4. ENTANGLED MODEL.BRANCHED POLYMERS
Branched polymers may be classified into two categories from the point of view of rheology : - polymers with short branches (Marm<<Me, Marm being the branch molecular weight and Me the molecular weight between entanglements for the l inear species). These polymers may be considered also as reptat ing species with a higher friction coefficient ~0, (hence a higher viscosity activation energy) and a
115
lower plateau modulus, so the formalism used for linear flexible polymers may still be used with a good approximation; - polymers with long entangled branches. In spite of the fact that the mlmber of branches per molecule is generally small, experimental data show a tremendous viscosity increase with branch length and a much wider distribution of relaxation times compared with linear polymers. On the other side, the plateau modulus values as well as the temperature dependence of viscosity and relaxation times are close to what is obtained for linear chains of the same species. Another difference with linear chains is the regular increase of the limiting compliance with molecular weight.
We will deal in this review article with monodisperse, model-branched polymers in order to describe the basic relaxation modes of branched polymers. The concepts described below are the source of current attempts to describe the viscoelastic properties of complex tree-like structures which are close to those found in low density polyethylene, for example. One may found interesting approaches of that problem in recent papers presented by Mac Leish et al [10].
If one considers that the reptation process is dominant for linear chains, one has to imagine additional processes of diffusion for polymers with long branches. The experimental data suggest strongly, however, that the basic kinetic unit of the chain (whatever it is) is the same as for linear chains : the Rouse-like A and B processes are still there, which are still strong imprints of the "tube".
. . . . . . : : - . :
(a) (b)
(c] (d)
Figure 13 :The picture of de Gennes: "reptation of stars": the branch has to go back to its attachment point to renew its configuration.
The basic models consider well-defined star-branched polymers. De Gennes [11] imagined in 1975 a simple relaxation mechanism of a branch based on the Brownian motion of an arm of a star-branched molecule in a network of fixed obstacles (Figure 13). From statistical considerations, the time necessary for a branch to renew its configuration is :
I; m or f ( N a r m ) exp(k N arm) (4-1)
where Narm stands for the number of chain segments per arm.
116
The ex~nent ia l dependence of the relaxation time with arm length is a constant feature for all models describing the renewal of configuration of long branches, and the debate has focused on the non-exponential term f(Narm).
Doi and Kuzuu [6] have proposed a somewhat different approach based on the tube concept. They s tar t with three basic assumptions :
- the segments of a branch are confined within a tube; - the tube deforms amnely with the macroscopic deformation; - the centre of the star is assumed to be fixed during the relaxation of
the branch. The relaxation of a branch occurs by a retraction process within the tube
and the branch end is assumed to be in a potential barrier. The max imum relaxation time of a branch is analogous to De Gennes' result (eq. 4-1) with a predicted value of v=8/15. Doi and Kuzuu [6] subsequent ly derived the the relaxation modulus of a star-branched polymer as :
G(t) 4---G~ ex - d~ 15
(4-2)
with z~ = Zrn ~2exp(a(~2-1))
8 Narm w i t h a = ~ . 15 N e "
(4-3)
i i
(a] [b]
" " ~ - A
�9 �9 �9 ~ e
�9 �9 �9 �9
�9 " ] ' X 2
:
Figure 14 �9 Another picture of the disengagement of a branch from its initial pa th is a '"oreathing" of the branch by fluctuations in path length.
Pearson and Helfand [12] used a somewhat similar approach to determine the characteristic relaxation time for a branch to disentangle; their calculation leads to a similar form, with a different exponent for the front factor :
~rn~ Ne ) exp v Ne ). (4-4)
I17
Ball and Mac Leish [13] used the same concept of the free end of a branch in a potential well, creating a process of"dynamic dilution" (Figure 15) which results in a var ia t ion of the molecular weight between entanglements during the disentanglement process of a branch. The mad'mum relaxation time of a branch is the same as that given in equation (4-4), but the relaxation modulus which takes into account the gradual disentanglement of the branch is given by"
G(t) = ~ Jo 1 exp - ds (4-5)
Mann being the molecular weight of a branch.
(a) (b) (c)
Figure 15 : The various models of Mac Leish lead rather to disengagement of a branch by fluctuations from its ends. The tree-like cloud is the trace of the agitation of chain segments over a period of time of the order of Zm.
Some general important remarks may be formulated regarding the characteristic viscoelastic parameters of these polymers with long chain branching :
- the maximum relaxation time does not depend on the total molecular weight, but depends essentially on the molecular weight of the branch, and more precisely on the entanglement density of the branch Narm/Ne.
- all models predict an exponential dependence of the zero-shear viscosity and terminal relaxation time as a function of the entanglement density on the branch. This is confirmed by the experimental data obtained on model-branched monodisperse samples. The v factor which appears in the exponential term is fairly close for all models, ranging from 0.5 to 0.625; this parameter is also related to the polydispersity of relaxation times jo G O (see section 2.2) as �9
jo G~ = 5 v Narm (Doi-Kuzuu) (4-6) 4 N e
Sarrn jo G~ = v - - - - (Pearson-Helfand) (4-7) Ne
118
1 Narm Je ~ G~ = ~ v Ne . (Ball-Mac Leish) (4-8)
which explains the broadening of the distribution of relaxation times as well as the linear dependence of the limiting compliance with respect to molecular weight (as opposed to linear polymers).
It is possible to recalculate the entire relaxation function, including the A, B, and glass transition processes in the same way as we did for linear polymers, with yet another remark: it can be shown by theoretical arguments that the equilibrium relaxation time of the entire branch along its own contour, which is the equivalent of the B process of a linear chain, is 4 times the value of ZB of a linear chain of same length:
4 ~o b2 Teq = ~/t2kB T N2arm" (4-9)
We have reported on Fig. 16 the complex shear modulus of two star-branched polybutadiene samples at 25~ The full lines have been calculated using the Ball and Mac Leish model for the terminal relaxation region, whereas the same relaxation functions as for the linear polymers have been used regarding the A, B and glass transition domains. Hence:
M SO 1- e x p - z - ~ ds
+ G~ e Ne jco(Te q/p2) e J~(~:A/p2) - + 1+ j - ~ 2 7~-2) +G.. 1 - 5 L p=l N 1+ j~(Teq/p2) p=l 1 - 1 1 .(4-10)
(I+jo HF)
109/ i i i i I I
~ I O s " 6 -
"~10 i i I 1
i0 3
I0 -'~ 10-2 I I0 2 I0 4 I0 6 I0 e ioIO r -~)
Figure 16 : Complex shear modulus of two nearly monodisperse star-shaped polybutadiene samples at 25~ ( - ) 3 branches, total M=164 000; (O) 4 branches M= 45 000.
119
The essential physics of the mechanical relaxation of star-branched polymers seems to be well-understood when the branches are highly entangled. However, the transition domain where the molecular weight of the branches goes down to Mc is not yet well-described : as molecular weight decreases, one observes a strong decrease of the plateau modulus as well as important effects of constraints release, along with maybe additional relaxation processes. Also, when one deals with polydisperse branched polymers, or blends of linear and branched polymers, constraints release (section 5) becomes rapidly the dominant relaxation process, and it is difficult at the present time to give a clear picture of the effects of polydispersity of molecular weights or polydispersity of branches as simply as it is done for linear polymers.
5. ENTANGLED POLYDISPERSE LINEAR CHAINS : DOUBLE R E P T A T I O N
5.1. Tube r e n e w a l The simple reptation concept proposed by de Gennes and developed by Doi and
Edwards deals with permanent entanglements creating a fixed tube around each chain. However, as in a melt, the tube is made of similar chains diffusing also by reptation, a self-consistent model should consider the motion of the surrounding chains. Then, each topological constraint or entanglement should be assigned a finite lifetime and the attached segment of the tube will be lost when it is visited by one end of the passing chain (Fig. 17). Therefore, the constraint release mechanism leads to the vanishing of internal segments of the initial tube due to simple reptation of passing chains whereas by pure reptation the initial tube disappears by losing its end segments. Both mechanisms could be combined by saying that a tube segment is lost when one or the other chain involved in the corresponding entanglement has one end in the close vicinity of the entanglement. In tha t sense, we are not thinking in terms of individual chains (simple or pure reptation) but in terms of coupled chains (double reptation [14]).
Figure 17 : The tube renewal concept : an internal tube segment is lost when the attached entanglement disappears.
Various authors [15,16,17] postulated that for a tube made of N/Ne segments the longest relaxation time ~ren accounting for contraint release is the same as the
120
Rouse time of an N-chain with an elementary time ~cr directly connected to the reptation time Zc of each passing chain.
~ren = (N / N e)2 Zcr -= ( S / N e)2 ZC" (5-~)
The so-called tube renewal time 1:re n can be compared to the reptation time zc if
the prefactor is known in the above relation. Zren / ZC = (N / N e)2 / 18~2 for Klein
[15] and Zren / ZC = 4~2(z-1)( N/Ne) 2 / z-12z with z = 2 to 4 for Graessley [17]. z is a coordination number accounting for the hypothesis that an internal tube segment is lost when the first constraint among z "active" constraints has been removed.
Both expressions predict that Sren is higher than zc for monodisperse samples with N higher than about 12 [15] or 4 [17]. This means that a pure reptation description is correct for highly entangled polymers as shown in section 3.1.
For weakly entangled polymers or long N-chains surrounded by shorter P- chains, a tube renewal time Zren shorter than Zc can be expected. If the two mechanisms are assumed to be independent of each other [17], the overall relaxation time z can be put as the harmonic average of the two times :
-1 z-1 = z51 + Z~n. (5-2)
This combination is equivalent to saying that the overall diffusion coefficient is the s:lm of the coefficients of the two processes. Watanabe and Tirrell [18] suggested that the rate constant of constraint release depends on the tube configuration, but the comparison between the two models [19] does not allow us to conclude in favor of either one of the models. Thus we will use the simplest one given by relation (5-2).
5.2. Effects of polydispers i ty for en tangled cb - ln s In a polydisperse sample, each N-chain is surrounded by chains of different
lengths. Therefore, the constraint release time Zcr varies according to the reptation time of the passing chain. Some entanglements can be considered as permanent (P>>N), while others will disappear quickly (P<N). The time distribution will affect the tube renewal time and consequently the overall relaxation time of each chain. Experimental evidence of this hypothesis is given in Fig. 18. Two monodisperse polystyrene samples (N = 50 and P = 5) are blended with different concentrations and the complex shear modulus is measured in a wide range of frequencies allowing us to scan the relaxation of the two components. The reduced imaginary part of the complex viscosity (Vl"= G'/co) shows, at intermediate concentrations, two maxima connected to the relaxation time of each component (~=l/(omax). The most striking point is the large decrease of the relaxation time of the longest N-chains (increase of (~)max).
.4 l N=50 P=5
.3
o
0
= .06
.I
O, -4 -3 -2 -I 0 i 2 log
121
Figure 18 : Variation of the terminal relaxation domains of the two components of a binary mixture of polystyrene [19] samples ( ~ : concentration of N-chains).
Their tube is made of N and P-chains (P<<N). As the ratio of N chains is decreasing, the number of constraints involving P-chains increases leading to a shorter average lifetime of entanglements. Conversely, the overall relaxation time of the P-chains increases with polymer volume fraction $ as the number of constraints involving N-chains increases. It tends to a pure reptation time. Quantitatively, we observed that the reptation time ~c (P) was about twice the relaxation time of the monodisperse P-sample. This is consistent with diffusion measurements (chapter 1.1) where the diffusion coefficient of short chains is lowered when the motion of the surrounding chains is frozen down. Moreover, the data follow the reptation power law: D or Along those lines, the infrared dichroism technique (chapter 1.3) applied to binary blends shows that contraint release theory could qualitatively predict the behaviour of the long chains.
5.2.1. N-chain isolated in a matrix of shorter chains The simplest case for testing the tube renewal models is to isolate a long N-
chain in a matrix of shorter entangled P-chains. A chain can be viewed as isolated from similar N-chains when the volume fraction ~N in the blend is such as N ~)S <1. Therefore, no entanglement can occur between N-chains and all constraints acting on an N-chain are assumed to come from the P-chains. The tube renewal time is expected to scale as :
~ren(N,p) _=_ (N/Ne)2~c(p) or N2p 3 (5-3)
and a comparison with the reptation time of the N-chain can be made through
122
Graessleys' expression with the coordination number z---3"
~ren(N,p) / ~ 4 p3 (5-4) %(N) = ~g) NN, 2
For N/N~ > (P/Ne) 3, the tube renewal meeh~_ism is dominant and can be
directly measured. O t h e r ~ s e , it can be derived from relat ion (5-2) and the experimental value of the overall relaxation time.
Experimentally, only average times are determined and the most commonly
used are ei ther (i) the weight-average relaxation time Zw = Tlo J~ or (ii) the
reciprocal of the frequency corn 1 at the maximum of TI" as shown in Fig. 18. Both times depend on the relaxation time distribution function H(z) and can be compared to the longest time zm-
We showed [19] tha t theoretically C~m 1 is quite insensitive to the width of the
distribution and depends somewhat on its shape" corn 1 = 0.73zm for a Doi-Edwards
spectr,,m and corn 1 = 0.63~ m for a Rouse distribution. The weight average time Zw ~2
is more sensitive to the shape of the distribution: Zw = ]-~ zm (Doi-Edwards) and
~2 ~:w - - ~ ~:m (Rouse).
For diluted chains, a Cole-Cole plot of the complex viscosity (Fig. 19) exhibits a relaxation domain well-separated from the matrix allowing one to measure the same average relaxation times as above. However, the weight average time Zw has to be corrected by the matrix contribution. Watanabe [20] cast it into the form :
G " Zw = lim -4)p Gp (5-5
where the subscript P stands for the matrix.
._.1.6
d o_ o~ 0
~ ' " +%.
/ ~ + + , + / . b+ q~ + +
. . . . I , , \ , , l 1.6 3 . 2
r l ' ( l O 3 P o , s.)
% / \ / \ +
/ \ + \++~+ + + ++
\ - - �9 , , i / L | t . 1
3 6 ~ ' ( I 0 2 p O . S.)
(b)
Figure 19" The terminal relaxation domain of diluted polymethylmethacrylate (a) and diluted polyisoprene (b) can easily be distinguished from the matr ix one (dashed lines) in a Cole-Cole representation of complex viscosities [19]-
123
As the data of co m are straightforward and experimentally connected to the
longest time by O~m 1 = 0.7z m [21], they will be used along with the reptation time ~c in order to calculate the tube renewal time Zren of the diluted chains:
- 1 1;re n (N,P) = zml(N,P) - ~cl(N) (5-6)
We have conducted experiments on different polymers [21, 22] (polystyrene, polyisoprene, polymethylmethacrylate) and used data in the l i tera ture on polystyrene [20] and polybutadiene [23] in order to check the scaling laws predicted by relation (5-3).
All the data confirm that the tube can be viewed as a Rouse chain (1;re n or N 2) but with an elementary time Zcr which is not the reptation time of the passing
chain. The experimental P dependence of the tube renewal time is ~ren or p2.5+o.1 and has been interpreted in terms of multiple contacts between the N-chain and each given P-chain [21, 24, 25]. For Klein [24], the N-chain can be divided in blobs containing p monomers with a volume scaling as p3/2. In tha t volume , the
number of passing chains is roughly proportional to p3/2 / p = p1/2. Therefore, the number of contacts between the N-chain and an identif ied P-chain is p /p1 /2 = p1/2. Assuming that the constraint release time Zcr is modified in the same way, we can write"
1:re n (N,P) o, N2~c(p)/p1/2 or N2p 5/2, (5-7)
which is observed experimentally. Furthermore, the chemistry of the polymer can be taken into account by means of the number N e of monomers between entanglements or the molecular weight Me.
-G.O
-6 .5
r'-i r Z
- 7 0 -
_.a.
- 7 5
- 8 0
8 5 3 O
!
3.2
PB 1 I I
.J
i ...... I J,. l ,,, l., 3.4 3.6 3.8 4 . 0 4.2 4 .4
log Me
I . . . . . I ' ' I "
Figure 20 �9 Experimental relation between the tube renewal time 1;re n and the reptation time z c of the matrix for different molecular weights of various polymers [ data from ref 19].
124
1
Fig. 20 shows that the ratio ~ren(N,P).M~p/Zc(P).M 2 scales as Me 2 and all the data can be east into the form"
4(MN ~2 zc(Mp) 1:ren(N,P) = ~,'~e J M - - - ~ ' (5-8)
which confirms that the tube is made of N/Ne segments and the constraint release 1
time 1:cr = 4z c/M 2 differs from the theory. For monodisperse polymers, the tube renewal time scales as"
M3/2 1;re n (N) = 41:c(N). ~
~e (5-9)
which implies that Zre n = Zc for M = (Me/2)}. This number of segments varies from 4.9 for polybutadiene to 10.4 for polystyrene and is consistent with the theoretical predictions[15,27]. Consequently, for highly entangled polymers, the pure reptation model holds as described in part 3.1. but it has to be corrected for weakly entangled samples according to the double reptation model. The main modification is to use the overall time z instead of the pure reptat ion time z c, defined by relations (5-9, 3-14 and 5-2). Furthermore, in Doi's formula (3-18) the numerical constant v accounting for the contribution of contour length fluctuations should vary as a function of N as v = 1-0 .5 /~]N/N e in order to recover the well-known scaling law for the zero-
shear viscosity 11o o, M 3-4. Nevertheless, the expression for Zren does not hold until N=Ne where we expect the overall relaxation time to merge with the Rouse time zB (relation 3-11). The entangled- unentangled transition remains to be clarified even though a recent approach by des Cloizeaux [26] looks very promising.
5.2.2. Polydisperse samples Let P(M) be the normalized molecular weight d is t r ibut ion function
( ] S P(M)dlnM = 1)g iven , for example, by the S.E.C. technique; the weight x , J
average molecular weight Mw is defined as:
M w = ~? P(M)dM (5-10 )
5.2.2.1. Relaxation times Each molecule is surrounded by the above distribution of chain lengths and the
tube renewal time has to take into account the distribution of the a t tached constraint release times. For a monodisperse sample, Graessley [17] defines the constraint release time ~cr from the Doi-Edwards relaxation ftmction F(t) such a s "
125
z
%r = ~[[F(t)] dt = f(z).~c (5-11)
where z is the coordination number . As the experimental connection between Zcr and zc is not so simple, we suggest using the simplest relaxation function :
F(t) = exp(-t/ZZcr) with Zcr = 4zc/~t-M as deduced from relations (5-1) and (5-9). For a polydisperse entangled polymer, the distribution of chain lengths malting
every tube is given by P(M) and an average relaxation function <F(t)> can be defined by:
< F(t) >= ~.'_ F(t,M).P(M)dlnM (5-12)
The subsequent average constraint release time < %r > is given by"
<%r >= ~o ~< F(t)>"dt = ~Z[P(M)exp(-~/z~r
which yields the following tube renewal time for each N-chain in the sample"
(5-13)
zt(M, P(M)) = < Zcr > (5-14)
The expression of < ~c, > has been checked for binary blends of monodisperse polystyrenes (Fig. 21). The tube renewal time of the high N-component is measured at different volume fractions CN and the molecular weight distribution is defined by two step functions : 1-CN at Mp and r at MN. The experimental data fit well the model with z=3.
f 0 _ - -
/ / / /
~ ." / ' . , " ,~ /( i [ / I I I
0 0.5 q>N
Figure 21: Variations of the tube renewal time of N-chains (MN = 2 700 000 g.mo1-1) in a matrix of shorter chains (Mp = 100 000 g.mo1-1) of polystyrene, as a function of concentration ~N [21].
126
Therefore, the longest relaxation time of a chain in a polydisperse somple is modified by a shift factor ~. defined by:
'r(M, P(M)) 'rtl(M:) + zc(M) -1 4M 3/-----~ § 1 ~'(M) = z(M) = -~ = 2 . . . . �9 (5-15)
'rren(M'P(M))+'rc(M)-I 're ( - - ~ ) + 1
<'rcr >
For chains such that 'rot(M)=< 'rcr > the relaxation time is unchanged. For longer chains, for which < 'rcr > is lower than Zcr(M), the relaxation time is decreased (~. < 1) whereas for shorter chains the relaxation time is higher in the blend than in a monodisperse environment.
5.2.2.2. Relaxation functions The double reptation approach allows us to visualize the blend of n different
species of the s~me polymer (molecular weights : M1, M2, ..., Mi, ... Mn ; volume fractions r as a network of (i, j) knots accounting for the entanglements between an i-chain and a j-chain [14, 19, 22, 27]. Therefore, the t ime-dependent density Fij (t) of initial knots in the blend is proportional to the relaxation function of each species involved in the knot, tha t is to say :
Fi5(t) a Fi(t) Fj(t). (5-16)
From tha t relation, it can be shown tha t the density of (i, j) knots is equal to the geometric average of the density of knots between similar chains :
Fij(t) = [Fii (t).Fjj(t)] 1/2 (5-17)
As the volume fraction of (i, i) knots is ~i 2, and that of (i, j) knots is given by 2 {~i.{~j, the overall average number density of initial knots can be writ ten as :
~ 1/2 2 F(t) = Z Z r (t) = [Z r (t)]. (5-18)
On the other hand, the relaxation function of (i~i) knots is directly connected to that of i-chains by a mere shift, e.g. : Fii (t) = F i (at). In other words, in a monodisperse polymer the knots are renewed at a ra te proportional to t ha t of the chain segments. The shift coefficient a is assumed to be length-independent, allowing the
same shift factor to be applied to the overall relaxation function F(t) = F(at). Then, we recast relation (5-18) into :
F ( t )= r " (t) . (5-19)
127
The individual relaxation function F i (t) is defined from Doi's expression (relation
3-18) where F i (t) = Gc(t) / G~ and % is replaced by:
x(M,,) [zj1 + (Me/M)2(%r)-l] -1 = accounting for the molecular weight distribution.
An experimental check of such a quadratic blending law is given by the storage modulus G'(m) of binary blends which exhibits a plateau G'N at intermediate
- I -i, frequencies, 2;(M1) < (D O < 1~(M2) corresponding to $(M2) < 1;0 < 2;(M1) for the relaxation function. Therefore, for blends such as M 1 >> M~, the blend relaxation function is given by
F(to)_--r 2 leading to G'N = ~12 G~, which is observed experimentally [28]. For a polydisperse polymer defined by its MWD function P(M), the relaxation function is given by"
F(t) = P(M) 2(t,M)d In M (5-20)
if only the reptation process is taken into account. But, for large polydispersities, the Rouse process (B) of the long chains overlaps the reptation process of the short chains. Consequently, the most general expression of the relaxation function (or relaxation modulus) must include all the relaxation processes described in part 3.2. As the Rouse dynamics is assumed to be linear with respect to the MWD and that the A and HF processes are mass independent, we define the relaxation modulus of a polydisperse linear polymer by :
[f +~ G~I2 G(t)= P(M) (t,M)dlnM + P(M)GB(t,M)dlnM+GA(t)+G~(t) , (5-21)
which is consistent with rel. (3-24) for monodisperse samples.
5.2.2.3. Viscoelastic behaviour The relaxation modulus is the core of most of the viscoelastic descriptions and
the above expression can be checked from experimental viscoelastic functions such as the complex shear modulus G*((D) for instance. In addition to the molecular weight distribution function P(M), one has to know a few additional parameters related to the chemical species : the monomeric relaxation time x o,
the rubbery plateau modulus G~ and the glassy plateau modulus G~. The temperature dependence is included in the relaxation time Xo and more precisely in the friction coefficient ~o- Expressed in terms of free volume fraction f which increases linearly with temperature and expansion coefficient af, the WLF
o o
equation gives the temperature dependence from two parameters C1 and C2
128
at a reference temperature To. The product C~ C2 = B / af is constant as long as the free volume expansion factor af can be considered as temperature and mass independent. An alternative description such as the Vogel equation introduces a temperature
T~ = T O - C 2 which is a constant for a given high polymer species. Therefore, the friction coefficient can be written as :
B ln~o = lnA + . (5-22)
af(T - T.)
The values of B / o~f and To. are tabulated for different polymers [29] and the value of A can be derived from the elementary relaxation time zi measured in the transition zone. The high-frequency domain does not depend on molecular weight value and distribution, and thus the tabulated values of ~o at a given temperature are applicable to commercial samples. Figure 22 gives two examples of the description of the viscoelastic data of commercial polypropylene and high-density polyethylene samples by the expression for the complex shear modulus derived from expression (5-21).The first term is dominant for highly entangled systems.
~3
2_
I i
+ . §
IoQ w
Figure 22 : Experimental data and theoretical curves (expression 4-21) of the complex shear modulus of commercial polypropylene (M w = 348 500, Mw/M N = 6.1) and high density polyethylene (M w = 210 000, Mw/M s = 11.7) [19]
The agreement is satisfactory but it is worth noting that the fit will be poorer if the high molecular tail is not described properly or more generally if the relaxation time shi~ function ~(M) is not correct. For example, we showed [19] that failure to take into account the shift; factor k leads to a large discrepancy between the model and the experimental data.
129
Another important point is that, when approaching Me, the tube consistency becomes weaker or in other words, the constraint release scaling law is modified and the rubbery plateau disappears whereas the steady-state compliance jo decreases. A self-consistent approach should predict that around Me, the reptation modes would be gradually replaced by Rouse modes in order to describe the non entangled- entangled transition.
6. EFFECTS OF NON-ENTANGLED CHAINS
6.1. The unentangled regime It is commonly admitted that a linear flexible polymer melt behaves as a dilute
solution as long as the molecular weight is sufficiently low so that entanglement effects do not occur. The Rouse formulation of the bead-spring model with no hydrodynamic interactions holds for such undiluted polymers because of the presence of segments belonging to other chains within the coil of a given molecule. The Rouse description predicts a relaxation modulus given by GB(t), Equation (3- 13), where the product MeN/Ne is replaced by the molecular weight M, so the longest relaxation time is :
~o R2N 6M "~Rouse - 6~2kT - 11o p~2NAR T, (6-1)
Rg being the radius of gyration and NA the Avogadro's number; it follows that :
~o R2N A 0 no = P 3-6~oo and Je = ~
0.4M. (6-2) pRT
The temperature dependence of Tlo is mainly included in the friction coefficient ~o (relation 5-22). Therefore, 11o can be expressed by :
B In 11o = ln($(M)) + , (6-3)
o~f[ T - Too(M ) ]
where the structure factor $(M) describes the variations of the radius of gyration. Furthermore, the temperature Too is no longer a constant. In the free volume models, T~ accounts for the var iat ions of the free volume fraction f (f = a f ( T - Too)) which is assumed to be mainly due to the concentration of chain ends. As the chains become shorter, the free volume fraction f increases, hence Too decreases.
130
2O
15-
_~I0 - I / ) 0 ( J
._. >
5 -
0 0
I 1 I i I
$
+
1 I I I 1 50 I00 150 200 250 300
temperoture (~
Figure 23 �9 Data of zero-shear viscosities of polystyrene fractions ranging from 900 g.mol -z to 30 000 g.mo1-1 as a function of temperature [29-37]. The master curve is obtained by experimental shifts from the data of a reference mass of 110 000 g. mole -z. It includes more than one hundred experiments lying within the exper imental bar error. A least squares analysis gives the parameters of the reference mass and the other ones are deduced from the shiit factors.
The plot of a mas te r curve of the thermal variations of 11o for various molecular weights and temperatures (Fig. 23) shows that the expansion coefficient af can be considered as a constant in a wide range of temperatures . The vertical and horizontal shift factors respectively describe the mass dependence of the radius of gyration and temperature T.. Polystyrene is a good example for analyzing the non-entangled regime because the molecular weights available are as low as Me/20. Consequently, the experimental data are significant in a range of molecular weights exceeding one decade.
/ , , ,
v v
4O
z::
0 1 - - N o - I 0 ~ I I ! " ~ I , J
0 0.2 0.4 0.6 0.8 i. 103/Mw
i.0
Figure 24" The horizontal shift factors of the master curve of Fig. 23 give the tempera ture Too as a function of molecular weight (reference Too = 49.4 ~ C).
131
The variations of T=. are derived from the horizontal shift factor and can be expressed by (Fig. 24) :
D T.. = (T..)= - - - (6-4)
M
For polystyrene, D = 83 500g.mol-t, which means tha t beyond a mass of
approximately D, the temperature T=. is fixed - (T=)=. = 49.7 _+ 0.3~ in the entangled region, the free volume fraction is constant at a given temperature and the iso-free volume state merges into the isothermal state. From the vertical shift factor of the master curve, we are able to describe the mass dependence of the zero-shear viscosity in the iso-free volume state which is directly connected to the radius of gyration of the chains. In the molten state, it is generally assumed that the chains exhibit a Gaussian conformation and therefore the viscosity should be proportional to the molecular weight. Unexpectedly, we observed (Figure 25) an unambigous different scaling law (% a M 6/5) which confirms previous results [31] and should be explained by the variations of the radius of gyration. This hypothesis is consistent with direct measurements of Rg by SAXS experiments on solutions of polystyrene in O conditions [38]; the same scaling law is found for molecular weights lower than about Me (Fig. 26).
10 4 _
I0 ~
~r 6 / 5 Io' . "" 4 t sO
,o~
I01 i / I 10 3 10 4
Mw IO s
Figure 25 �9 the vertical shift factor of the master curve of Figure 23 gives the structure factor A' as a function of molecular weight.
In addition, Pearson [39] made numerical simulations of the mean-square radius of gyration of polyethylene by using a rotational isomeric state method. For n- alkanes and low molecular weight polyethylenes below M c, he also found a stronger increase of the calculated radius of gyration with molecular weight than expected from Gaussian statistics. Therefore, Gaussian statistics does not seem to apply to short chains as shown by numerical simulations [40] but the observed
132
scaling law, which is the same as for solutions in a good solvent has no connection with an expansion of the chains due to excbaded volume effects as the absolute value of Rg is lower than the extrapolated gaussian value (Fig. 26).
i 10 4 -
_ / a
. y / O
-61' / ~ l / . _ j 6 1 5
I 0 2 - I~ E!
/
I 10 3 10 4
Nw
A r ..,~ IO s N ~
I a./-
I
IO s
Figure 26 : Molecular weight dependence of the mean-square radius of gyration of solutions of PS in cyclohexane at 34.5 ~ C [38].
The steady-state compliance Je ~ follows the Rouse expression until M'c --5 M e. Then, the longest relaxation time is expressed by :
15 Tlo j0 (6-5) ZRouse - ~2 "
From dynamic experiments and applying the time temperature superposition principle, the complex shear modulus is measured over about five decades and the Rouse model can be checked extensively [37].
I0 ~
I0 e
,,,,..,
iO s
a 10 4 n
b
I0 2
IOOi / i 1 I i 1 1 I 1 ~ J I0 ~ I0 2 I0 4 I0 e I0 e I0 I~
c~(s -4)
Figure 27 �9 Experimental complex shear modulus of unentangled polystyrene (M = 8 500 g.mol-1) compared to the Rouse model [37].
133
The agreement is good over the whole range of molecular weights below Me (Figure 27) provided that one adds the high frequency term (equation 3-21) accounting for the very local relaxation modes of the chains. The overlap between the two domains becomes significant at frequencies corresponding to 1 000 zi and 100 ziaccording to the molecular weight because of the very different orders of magnitude of the rubbery and glassy plateau moduli.
6.2. Effect of s h o r t cha in s in the e n t a n g l e d r eg ime When short M S chains are introduced into a sample of entangled M L - chains
wi th a volume fraction r of long chains such as r M L > M e, the blend can be viewed as a concentrated solution of the long chains, or in other words, the M s component is acting as a solvent at least in the terminal zone of relaxation of the long chains.
According to Doi-Edwards theory, the reptation of the long chains will occur in a
tube whose diameter a varies as r Thus the number of monomers between
entanglements will scale as r Accordingly, the reptation t ime zc (relation 3-14) should be proportional to r as a first approximation, the zero-shear viscosity Tlo
and the steady-state compliance j0 should respectively scale as r and r
Exper iments conducted on concentrated solutions of high molecular weight hydrogenated polybutadiene in a commercial oil [41] showed tha t the expected law
for 11o and the average terminal relaxation time corn 1 is satisfied (Fig. 28).
10 7
I0 e
0 1 0 5
1 0 4
1 0 3
I "1 I '
_ l , ~ m [ j j ! l IO 'J (JD I
I0 ~
,E iO-I
! I ' I
l n L n I n n n t J
I0 "j ~ I
Figure 28 �9 Zero-shear viscosity and average relaxation t ime of concentrated solutions of entangled hydrogenated polybutadiene (M/M c = 300) [from ref 28]. The
exper imental scaling laws are T10 a r and corn 1 a r in ag r eemen t wi th Doi- Edwards' theory.
134
For melts of long chains containing short chains, the contribution of the unentangled component is in some cases non-negligible and has to be removed from the data of the blend in order to isolate the contribution of the entangled component:
TIoL = ~lO - (1- r (6-6)
and JeOL =J~ TI_x_O )2, (6-7) 1]OL
as defined for concentrated solutions. Moreover, as far as relaxation times are concerned, the reptation mechanism in an enlarged tube should lead us to favour the tube renewal process. The expression of the tube renewal time Zren (relation 5-8) shows a ~3 scaling which implies that
~ren(ML,~)=zc(ML,~) for M=(Me/2dp) 4/3. Therefore, the double reptation approach applies in an extended range of molecular weights because the concentration is low. The overall relaxation time z(M L, ~) can be cast into the form
,~(ML ' ~)-1 = [lii ,I; c (M) ] - I + [{il3Zren (M)] -1
or 1;(ML,~)) = ~ zc(ML) [1+ 4~2ML 3/2M2 ]-1, (6-8)
whereas the zero-shear viscosity TIoL will vary as r z (ML, 4)). The experimental evidence of the importance of the tube renewal mechanism when short molecules are added to a high polymer is provided by blends of narrow polystyrene (M s = 8 500 and M L = 900 000) [28]. In a large range of concentrations (4)> 0.05) where the high component is assumed to behave as an entangled melt, the variations of the terminal relaxation time ~m-: in the iso-free volume state (Fig. 29) confirms relation (6-8). As the
steady-state compliance J~ L scales as ~-2 (Fig. 30), the zero-shear viscosity T10L
varies as expected and the plateau modulus G~ which reveals the entanglement
network is proportional to r The description of the relaxation modes including the behaviour of the entangled M L - chains and unentangled M S - chains remains to be done. At high concentrations, the long chains will dominate the viscoelastic behaviour in the terminal zone but, when approaching 4) M L = Me, the tridimensional diffusion of the short chains impeded by the surrounding long molecules will bring a noticeable contribution which could depart from the Rouse description.
135
10 3
._E i0 z !
I 0 i
I0 o 10-2
s
s
= = - s
. i . , , , i ~ i l . , l l l i i I I
I0 ~ I @
Figure 29 : Average relaxation time of a high molecular weight polystyrene (M = 900 000) in the presence of short chains (M = 8 500). The dotted line represents pure reptat ion and the full line stands for the contribution of tube renewal according to relation (6-8).[from ref. 28]
10-3
10-4
-3
lO-S
I 0 " 6
I I
l i t , , t i i i l
IO "~ ~ I
Figure 30 �9 Steady state compliance as a function of concentration (same blends as in Fig. 29).
7. PROBLEMS STILL PENDING
Along the same lines as described above, several questions arise as far as molecular weight distribution is concerned. The main question is : in the near future, will we be able to predict the viscoelastic behavior of l inear polymers whatever the molecular weight dis tr ibut ion, encompassing the broadest dispersion from oligomers to very long molecules ? The
136
answer could be positive provided that we know more about the environmental impact on the relaxation modes of each individual chain, whatever its molecular weight.
Remembering tha t three characteristic molecular weights are defined from viscoelastic parameters : Me, the molecular weight between entanglements,
M c _= 2 - 3 Me, the cross-over mass for viscosity and M' c = 5 - 7 M e, the cross-over mass for compliance, intermediate situations should be explored which could be, in the last stage, merged into a "universal" law : All the masses lie beyond M'c : tha t si tuation has been described in this chapter and it could easily be extended down to Mc by taking into account the decrease of the compliance, All the masses lie below M e : in the l i terature, the Rouse description of the relaxation modes of non-entangled melts or solutions is also used for polydisperse samples by means of a linear blending law. In order to consider the free volume variations of each mass in the blend, the relaxation times have to be shii~ed by a factor ~. which is the ratio of the monomeric friction coefficients of the blend ~Ob
and of each species (~o)- For a polydisperse sample [42], the relaxation modulus can be cast into :
G(t) = IZP(M)G(kt , M ) d l n M , (7-1)
with k = ~o ~Ob
As a consequence, the zero-shear parameters are :
Na l '~p(M) Rg2(M) d in M tlo b = P 36 m o ~0 b - ~ (7-2)
0.4 and jo = pRT rl--'-----~ K M P(M)R 2 (M)d In M. (7-3)
Therefore, for Gaussian molecules, the above parameters are functions of
moments of the molecular weight distribution �9 tl0 a M w and j0 a Mz.Mz+l /Mw. Otherwise, the mass dependence should be slightly different for tl0 and a large deviation from a combination of various average molecular weights is expected for the steady-state compliance. For mass distributions extending over the cross-over region between the non- entangled and entangled regimes, the situation is more complicated as anticipated in section 6. When the average molecular weight Mw - or other moment of the distribution - is lower than Me, a Rouse diffusion could be expected with relaxation times shifted according to the free volume variations, even for the high part of the distribution. Reciprocally, when the average molecular weight is higher than M c,
137
the reptative motion of the long chains will be made easier by the short chains according to the description given in section 6.
For weakly entangled monodisperse and polydisperse polymer melts, J. des Cloizeaux [26] proposed a theory based on time-dependent diffusion and double reptation. He combines reptation and Rouse modes in an expression of the relaxation modulus where a fraction of the relaxation spectrum is transferred from the Rouse to the reptation modes. Furthermore, he introduces an intermediate time ~i, proportional to M 2, which can be considered as the Rouse time of an entangled polymer moving in its tube. But, in the cross-over region, the best fit of the experimental data is obtained by replaced ~i by an empirical combination of 1;i , ~c a n d 1;Rous e .
In a more empirical way, Lin [43] establishes an expression of the stress relaxation modulus of monodisperse polymers including the Rouse motions between entanglements and a relaxation process related to the slippage of the polymer chains through entanglement links. The associated relaxation function is a single exponential with an empirical relaxation time ~x- Moreover, the reptation process is assumed to be accompanied by a fluctuation one relevant to a Rouse description and which is the only mechanism remaining at M = Me. Consequently, he qualitatively describes the entire range of molecular weights including the entangled and unentangled regimes.
Therefore, more work remains to be done in order to interpret the viscoelastic behaviour of very broad polymers with average molecular weights ranging from low values, lower than Me, to strongly entangled situations. A comprehensive MWD - viscoelastic properties relationship will be of great help for designing materials for specific applications but a new challenge has been around for the last few years.
It consists in developing numerical and analytical methods to invert a linear viscoelastic material function to determine the molecular weight distribution. There are several reasons to pursue such an objective - m a n y commercial polymers are slightly or not at all soluble in usual solvents, thus techniques like gel permeat ion chromatography or light scattering are inapplicable. Rheological characterization can be performed on-line and in real time and it is also a less cots-effective technique.
So far, the important issue on how to determine MWDs from rheological data has been addressed with limited success, mainly for three reasons. The monodisperse relaxation function F(M,t) must be described precisely in the terminal and plateau regions, one has to provide a correct blending law yielding the polydisperse relaxation modulus G(t) ; and even if these two obstacles are overcome, specific mathematical procedures are needed in order to solve the ill- posed problem of numerical inversion of integrals. Many different sets of solution parameters are not physically meaning~l and appropriate constraints have to be imposed in order to determine an acceptable MWD.
For entangled systems, the two first conditions are fulfilled in the framework of reptation theories : a comprehensive expression of the monodisperse relaxation modulus G(M,t) is given by expression 3-24 and the double reptation model generalized to a continous molecular weight distribution provides the integral relation between the MWD function P(M) and the polydisperse experimental
138
complex shear modulus G*(co) derived from the polydisperse relaxation modulus G(t) (equation 5-21).
A recent publication by D.W. Mead [44] investigates numerical and analytical methods involving the double reptation mixing rule used with specific relaxation functions. A single exponential or a step-function monodisperse relaxation function are relevant to analytical methods whereas more general multiple time constant monodisperse relaxation functions require numerical treatments. The first results sound encouraging, making that rheological measurements which imply bulk, macroscopic techniques are able to catch important molecular features such as chain length distribution. This is further evidence of the sensitivity of rheological techniques to nanoscopic changes, moving them into analytical tools.
R E F E R E N C E S
1. B. Gross "Mathematical structure of the theories of viscoelasticity", Hermann. Paris (1953).
2. G. Marin, J.J. Labaig, Ph. Monge. Polymer, 16 (1975) 223. 3. J.P. Montfort, Doctoral Thesis, Universitd de Pau (1984) 4. G. Marin, J.P. Montfort and Ph. Monge. Rheol. Acta, 21 (1982) 449. 5. P.G. de Gennes, J. Chem. Phys., 55 (1971) 572; see also "Scaling concepts in
Polymer Physics", cornell U. Press, Ithaca (1979). 6. M. Doi and S.F. Edwards ' ~ e theory of polymer dynamics" Oxford U. Press,
Oxford, (1986). 7. A. BenaUal, G. Matin, J.P. Montfort and C. Derail. Macromolecules, 26
(1993) 7229. 8. J.L. Viovy J. Polym. Sci, Physics, 23 (1985) 2423. 9. M. Doi J. Polym. Sci. Lett., 19 (1981) 265. 10. T.C.B. Mac Leish, Europhys. Lett., 6 (1988) 511. 11. P.G. de Gennes, J. Phys. (Paris), 36 (1975) 1199. 12. D.S. Pearson and E. Helfand, Macromolecules ,17 (1984) 888. 13 R.C. Ball and T.C.B. Mac Leish, Macromolecules, 22 (1989) 1911 14. J. des Cloiseaux, Europhys. Lett., 5 (1988) 437. 15. J. Klein, Macromolecules, 11, (1978) 852, ASC Polymer Prep., 22 (1981) 105. 16. M. Daoud, P.G. de Gennes, J. Polym. Sci., Phys-Ed., 17 (1979) 1971. 17. W.W. Graessley, Adv. Polym. Sci, 47 (1982) 67. 18. H. Watanabe, M. TirreU, Macromolecules, 22 (1989) 927. 19. P. Cassagnau, J.P. Montfort, G. Marin, Ph. Monge, Rheologica Acta, 32
(1993) 156. 20. H. Watanabe, T. Sakomoto, T. Kotaka, Macromolecules, 18 (1985) 1436. 21. J.P. Montfort, G. Marin, Ph. Monge, Macromolecules, 17 (1984) 1551. 22. P. Cassagnau, Doctoral Thesis, Pau (1988). 23. J. Roovers, Macromolecules, 20 (1987) 184. 24. J. Klein, Macromolecules, 19 (1986) 1. 25. G. Marin, J.P. Montfort, Ph. Monge, J - N. Newt. Fluids Mechan., 23
(1987) 215. 26. J. des Cloiseaux, Macromolecules, 25 (1992) 835. 27. C. Tsenoglou, Macromolecules, 24 (1991) 176. 28. J.P. Montfort, G. Marin, Ph. Monge, Macromolecules, 19 (1986) 393 and 1979.
139
29 J.D. Ferry Viscoelastic properties of polymers, Wiley, New York 3rd ed. (1980). 30. G.C. Berry, T.G. Fox, Adv. Polym. Sci., 5, (1968) 261. 31. R. Suzuki, Doctoral Thesis, Strasbourg (1970) unpublished results. 32. S. Onogi, T. Masuda, K. Kitagawa, Macromolecules, 3 (1970) 1098. 33. D.J. Plazek, V.M. O'Rourke, J. Polym. Sci., A9 (1971) 209. 34. R.W. Gray, G. Harrisson, J. Lamb, Proc. Roy. Soc., London, A 356 (1977) 77. 35. T.G. Fox, V.R. Allen, J. of Chem. Phys., 41 (1964) 344. 36. J.P. Montfort, Doctoral Thesis, Pau (1984) unpublished results. 37. J.C. Majestd, A. Benallal, J.P. Montfort, submitted to Macromolecules. 38. T. Konishi, T. Yoshizaki, T. Santo, Y. Einaga, H. Ysmakawa, Macromolecules,
23 (1990) 290. 39. D.S. Pearson, G. Ver Strate, E. Von Meerwall, F.C. Schilling, Macromolecules,
20 (1987) 1133. 40. C. Degoulet, J.P. Busnel, J.F. Tassin, Macromolecules, 35 (1994) 1957. 41. V.R. Raju, E.V. Menezes, G. Matin, W.W. Graessley, L.J. Fetters,
Macromolecules, 14 (1981) 1668. 42. J.C. Majestd, J.P. Montfort, submitted to Macromolecules 43. Y.H. Lin, Macromolecules, 17 (1984) 2846. 44. D.W. Mead, J. of Rheology, 38, 6 (1994) 1797.
This Page Intentionally Left Blank
Rheology for Polymer Melt Processing J-M. Piau and J-F. Agassant (editors) �9 1996 Elsevier Science B.V. All rights reserved. 141
Experimental validation of non linear network models
Christian CARROT, Jacques GUILLET, Pascale REVENU, Alain ARSAC
Laboratoire de Rhdologie des Mati~res Plastiques.
Universitd Jean Monnet. Facultd des Sciences et Techniques.
23 Rue Dr Paul Michelon. 42023- Saint-Etienne. Cedex 2. FRANCE
1.INTRODUCTION
1.1.Constitutive and model equations for computer simulation of complex flows.
Nowadays, industrial and research applications of computers are widely
used in the field of polymeric materials. Computer aided rheology and
simulation of processing operations involving the flow of polymeric liquids
have received growing attention for the understanding of the physics of the
processes, for the design of the equipments and for control purposes.
The theoretical foundation of this software is provided by principles of continuum mechanics, together with improved numerical methods for the
solution of the mathematical equations and by the use of pertinent constitutive
equations for the description of the theological behaviour of molten polymers.
Most flows encountered in processing or experimental problems are
characterized by complex kinematics involving different geometries combining
shear and elongation, different time dependences and amplitudes of the
deformations. Moreover, the behaviour of polymeric materials under such
conditions exhibits a large variety of specific features such as the shear rate
dependence of the viscosity, appearance of norma| stresses, memory effects and high resistance in elongation.
Among the large m~nber of existing models, only a relatively few equations
can predict the variety of phenomena encountered in the flow of polymer melts
and can give a fair description of the physics involved in the theological
142
behaviour of these materials, both from a microscopic and a macroscopic point of view.
From the physicist's and chemist's points of view, the ideal constitutive
equation should be able to describe the behaviour of polymers in any flow
situation without any adjustable parsmeter. Linear and non linear viscoelastic
characteristics of the material should be theoretically described from
molecular dynamics, knowing the basic properties of the polymer molecules,
from the monomeric unit to the molecular weight distribution as obtained from
the polymerization process. Many advances have been performed in this sense
in the past few years (see Section 1.5), however the complexity of the equations
that might be obtained to take into account these various features may lead to
numerical difficulties. Indeed, the computation of complex flows, such as
those encountered in typical processing conditions, sets its own practical
requirements on the constitutive equation, especially when reasonable
computation times and memory requirements are expected. The numerical
properties of the equation generally do not match those of the physicist. In this
sense, for example, the use of a large number of relaxation modes or of a
continuous spectrum to describe the memory function is time and memory
consuming and the economic spectrum using only a few contributions, though
unsatisfactory from the molecular dynomics point of view, remains the rule.
Thus, this constitutive equation is bound to be replaced by an unsatisfactory
but easy to handle model equation which involves a minimum of violation of
basic principles of material physics. This equation will necessarily contain a
few adjustable material parameters, which have to be easy to determine in a
limited number of well defined flow experiments.
Obviously, the minimum requirement that could reconcile the two stand
points should be the ability of the model equation to describe properly the
response of the polymer to simple viscometric flows (simple shear, uniaxial,
biaxial and planar extension) in which quite perfectly controlled conditions can
be obtained. From this, it can be hoped that the situation might be at its best in the combined complex flow.
The aim of this section is to perform comparisons between the predictions of
some constitutive equations and experimental results in simple shear and
uniaxial elongation on three polyethylenes. In addition, this is expected to
provide well-defined sets of material parameters to be used in the model equations for the computation of complex flows.
143
L2.Network theories for polymer melts and related models.
Among the various approaches in use for the depiction of the interactions of
the polymer molecules in the melt, these being known to be at the origin of the
observed theological behaviour, the network theories enable the building of
reasonable models tha t fulfill the previous requirements for the sake of
simplicity.
These theories are based on the classical theories of rubber elasticity of
maeromoleeular solids, wherein permanent chemical crosslinks connect
segments of molecules, forcing them to move together. This central idea can be
applied to polymeric liquids. However in this case, the interactions between
molecules are assumed to be localized at junctions and are supposed to be
temporary. Whatever their nature, physical or topological, these crosslinks are
continually created and destroyed but, at any time, they ensure sufficient
connectivity between the molecules to give rise to a certain level of cooperative
motion.
The stress is considered to be the sum of the contributions of segments
between junctions that are still existing at the present time. By the way, these
segments, tha t were created in the past, may be of different ages and
complexities. This time dependence is generally described by a relaxation
spectrum tha t gives rise to the linear rheologieal behaviour. Strain
dependence, at the origin of nonlinearity, can be described either by changing
the motion of the network relative to the continuum or by special rates of
creation and loss of junctions.
Owing to their relatively fair tractability and because they retain some
physical consistency, network models are widely used in computer simulation
of the flow of polymer melts. Thus, the attention of the present article is focused
on constitutive equations of this type.
l~.Integral and differential forms of the models. At this point, the question of the use of either an integral or a differential
equation arises. Integral forms are closer to those obtained by the recent
molecular dynamics concepts for entangled polymer melts. Unfortunately,
their use requires the knowledge of the Finger strain tensor in complex
kinematics, which together with the fluid memory, involve the description of
the material history. This, in turn, sets the difficult task of particle tracking.
Though it has been difficult to cope with, alternative descriptions (Protean
144
coordinates) and new ideas (such as those of the streAm-tube, see Section III-2)
enlighten the subject and bring new hope for this kind of equation.
In this sense, differential equations appear more tractable since they do not
require particle tracking. Indeed, the solution of the coupled equations of mass,
momentum and energy balance including the material equation, properly
described on a suitable finite element mesh, theoretically provides the material
lines. Nevertheless, the correct description of the basic experiments ot~en
requires the use of strong nonlinear terms. Such improvements may be
unsatisfying from the numerical point of view since they can lead to stiff
systems of nonlinear equations and to many convergence related problems.
Considering these previous remarks, two network models, thought to be
representative of each class of equation, have been investigated, namely the
Wagner model and the Phan-Thien Tanner model,
1.4.Experimental validation of network model~
Par t 2 presents a summary of the theoretical considerations and basic
assumptions that lead to the model equations. Part 3 discusses some
experimental aspects and focuses on the measurements in various shear and
uniaxial elongational flow situations. Part 4 and 5 are devoted to the
comparisons between experimental and predicted rheological functions.
Problems encountered in the choice of correct sets of parameters or related to
the use of each type of equation will be discussed in view of discrepancies
between model and data.
2~THEORETICAL ASPECTS
2.1.The basic integral and differential non-linear constitutive equation~
2.1.1.The l inear M~xwell model and its limits.
Constitutive equations of the Maxwell-Wiechert type have received a lot of
attention as far as their ability to describe the linear viscoelastic behaviour of
polymer melts is concerned. From a phenomenological point of view [1-4], these
equations can be easily understood and derived using the multiple spring-
dashpot mechanical analogy leading to the linear equation :
145
~i(t) + ~ ~ = Th ~ and ~=(t) = .~ ~(t) 1
(1)
where ~i(t) is the contribution of the ith Maxwellian assembly (spring and
dashpot in series) to the extra-stress tensor ~(t),
d ~-~ is the tensor time derivative for small displacements,
~ = Vu + (Vu) t is the rate of strain tensor (u being the fluid velocity),
is the relaxation time of the ith element,
Tli = gi ~ is the viscosity contribution of the ith element,
gi is the modulus contribution of the ith element.
This differential form can be integrated to give the integral form of the model
which can also be derived from the Boltzman superposition principle using the
concept of fading memory of viscoelastic liquids:
t ~(t) =- j re(t- t') dt(t') dt'
. - O O
o r
t ~=(t)= j G( t - t ' )~ dt'
- 0 0
(2a)
(2b)
where Th t
m(t) = ~ ~.2 exp{-~.} is the memory function,
Th t din(t) G(t )= ~ ~. exp{-~.} =- dt is the relaxation modulus,
dt(t') is the infinitesimal strain tensor (t being the reference of the
deformation, so that the strain at time t' is relative to that at time t ).
Satisfactory agreement is achieved from these equations for depiction of the
main features of l inear viscoelastic properties tha t can be obtained with the
experimental tools, either in t ransient or in oscillatory rheometry. These equations are all the more attractive in tha t similar mathematical
forms can also be obtained from molecular considerations for the description of
146
the flow behaviour of non-entangled [5] and entangled [6-10] melts at least in
the case of narrow molecular fractions, so that any parameter in equations (1)
or (2) becomes physically meaningful. However, the latter approach leads to
complex relaxation spectra with a large number of Maxwellian modes (and
maybe even more complex if polydispersity and branching are taken into
account). Recognizing that only one or two modes per decade are generally
sufficient to get a proper description of the linear viscoelastic behaviour [11],
such an economic spectrum can be numerically adjusted [11-17] and is often
used, while keeping in mind that, in this case, one loses the physical meaning of such modes.
Though a simple Maxwell model in the form of equations (1) and (2) is
powerful to describe the linear viscoelastic behaviour of polymer melts, it can
do nothing more than what it is made for, that is to describe mechanical
deformations involving only infinitesimal deformations or small perturbations
of molecules towards their equil ibrium state. But, as soon as finite
deformations are concerned, which are typically those encountered in
processing operations on polymers, these equations fail. For example, the
steady state shear and elongational viscosities remain constant throughout the
entire rate of strain range, normal stresses are not predicted.
2.1.?~The ~ e and UCM models.
One major discrepancy of the previous model can be attributed to the use of
the infinitesimal strain tensor and to derivatives restricted to time changes.
Indeed, in the case of large deformations, one has to refer to finite strain tensors, such as the Finger c t ' l ( t ') or Cauchy Ct(t') strain tensors (t being the
present time as a reference of the deformation), and to derivatives involving
time and space changes known as convected derivatives.
Making these modifications, the integral form of the equation of state becomes:
t ~(t) =-p'~ + ~ r e ( t - t ' ) C t l ( t ') d t ' = - p ' ~ + ~=(t) (3)
- O O
where ~ is the total stress tensor,
5= is the unit tensor,
147
p' is an arbitrary isotropic term.
For an incompressible liquid, because of the arbitrary term, normal stresses
are known only up to a constant using the constitutive equation.
This equation was proposed by Lodge [18, 19] in the case of entangled
polymer molecules, considering that the response of polymers to flow is that of
a temporary network of junctions. The strands of the network can be of
different topology (i) but they are considered as Gaussian springs that deform
like the macroscopic continuum. This la t ter assumpt ion of affine motion
means tha t any s t rand end-to-end vector, ini t ial ly coincident with a
macroscopic vector embedded in the continuum, remains parallel to and of
equal length with it during deformation. The s t rands are continuously
destroyed and rebuilt by thermal effects only. Assuming tha t the rates of
creation and destruction of the ith segment are constant, an equation for the
memory function similar to that of equation (2) can be written, with viscosity contributions Tli connected to the stiffness of any ith spring and to its rate of
creation, and relaxation time ~i connected to the survival probability of the ith
junction. This formulation provides the way to cope with the economic
spectrum discussed previously as a crude physical description of the entangled melt.
The Lodge equation can also be obtained in a differential form known as the Upper Convected Maxwell equation (UCM):
~i(t) + ~ ~i(t) = Tli ~ and ~=(t) = .~ ~(t) (4) 1
where ~i(t) is the upper convected derivative of the contribution of the ith mode
to the extra stress tensor ~=(t).
It should be noted that this is the exact derivation of equation (3). These
models are usually referred to quasi-linear models and display qualitatively
correct predictions of typical phenomena of elongational flows such as the
occurrence of the strain-hardening effect in t ransient extension. Nevertheless
the predicted elongational viscosity is never bounded in the long time range
and a steady state value can only be expected for small elongation rates.
Moreover, the shear behaviour remains unrealist ic as compared to the
experiment, especially because of constant predicted viscosity and first normal
148
stress coefficient. The cure of these discrepancies either requires much more
complexity in the formulation of the constitutive equation or an additional
attention to some characteristics of the viscoelastic material that might only be displayed in nonlinear flows.
2.2.An in tegra l consti tutive equation: the Wagner modeL
2.2.1.The general K-BKZ model.
Additional complexity can be brought to the constitutive equation in its
integral form. Indeed, the idea of rubber elasticity that is inherent to the Lodge
model has been generalized by Kayes, Bernstein, Kearsley and Zapas [20-23] in
a large class of constitutive equations. In a perfect body, the strain energy W
may be linked to strain and stress by:
8W - be (5)
1 where e= is a diagonal strain tensor (Hencky strain tensor) such as ~= = ~ In C "1.
This can also be rewritten in terms of the Cauchy and Finger strain tensors as:
1: =2~IW 1 C -t 8W = = "~I2 C=} (6)
Since W is a scalar value, it depends on scalar characteristics of the strain tensors, namely on the invariants I1 and I2 defined as:
I1 = tr[C 1] (7a)
1 -1])2 I2 = ~ {(tr[C - tr[C2]} = tr[C] (7b)
In rubber and viscoelastic fluids, these two quanti t ies are sufficient since, when incompressibility is taken into account, I3 = 1.
For viscoelastic fluids, both strain energy and stress can be assumed to depend
on the strain history through the strain invariants:
149
t W = ~ u ( t - t', I1, I2) dt'
- O O
(8)
and thus:
~(t) = ~ 2 { C t l ( t ' ) " ~22 Ct(t')} dt' - 0 0
(9)
o r
t ~(t) = ~ {Ml(t- t', I1, I2) c t l ( t ' ) - M2(t- t', I1, I2) Ct(t')} dt'
--OO
(10)
where M l ( t - t', I1, I2) and M2(t- t', I1, I2) are ma te r i a l functions.
The K-BKZ has the interes t ing property being the rmodynamica l ly consistent
because one can theoretically derive two mater ia l functions which depend on a
single potent ia l function u in the general form:
Mi( t - t', I1 I2) = 2 8u( t - t', I1, I2) (11) ' ~Ii
The Lodge model is a special case of this class of models, where the mater ia l
functions are selected as:
M l ( t - t', I1, I2) = m( t - t') (12a)
M2(t- t', I1, I2) = 0 (12b)
1 u ( t - t', I1, I2) = ~ m( t - t') (I1- 3) (13)
However , a l though it has some the rmodynamic consistency, the l a t t e r model
failsto describe the non l inear viscoelastic behav iour propert ies , especial ly in
shear, where in the shear- thinning behaviour of the viscosity and of the normal
s t ress coefficients are not predic ted . As a consequence , more complex
150
nonl inear functions have to be introduced to get a proper description of the
observed behaviour.
22. .2.Time-strain separabil i ty.
Some exper imental features might simplify the problem. Considering tha t in step shear s t rain of constant ,mpl i tude 7 start ing from the mater ia l at rest,
the K-BKZ model leads to the following equation for the shear stress:
t �9 (t, 7) = ~ {Mi(t - t', Ii, I2) - M2(t- t', Ii, I2)} 7 dt'
0
(14)
and since, in this case:
I i = I 2 = 3 + T 2 (15)
one can write"
t �9 (t, 7) = ] M(t- t', Y)7 dt'
0
(16)
The nonl inear relaxation modulus is then obtained as:
G(t , T) = Z(t, 7) (17) 7
As a l ready mentioned by various authors [24, 25], it is found experimentally
tha t a t various shear strains, for polydisperse polymers, the logarithmic plots of G ( t , 7) versus t ime are only shifted vertically. This indicates tha t the
nonl inear relaxation modulus might be factorizable:
G(t , 7) = G(t ) h(7) (18)
where h is a s t ra in-dependent function, between zero and unity, called the
d a m p i n g function. This t ime-s t ra in separabi l i ty seems to hold in the
exper imental range of shear strains. From a theoretical point of view, this can
only be achieved if u ( t - t', Ii, I2) is also factorizable:
151
u(t - t', I1, I2) = m ( t - t') U(I1, I2) (19)
As a consequence, the K-BKZ model is now writ ten in the less general form of
the factorizable K-BKZ model:
t 8U 8U ~(t) = ~ 2 m( t - t') {~-~1 ct ' l(t ' ) " ~ 2 Ct(t')} dt'
- 0 0
(20)
o r
t ~(t) = ~ re(t- t') {hl(I1, I2) c t l ( t ' ) - h2(I1, I2) Ct(t')} dt'
- O O
(21)
It is worth mentioning that the strain function is not t empera ture dependent
and tha t the influence of temperature is only applied on the memory function
or relaxation modulus through the shortening of the re laxat ion t imes with
increasing tempera tures .
2.2.3.The Wagner m o d e l
Since second normal stresses are generally difficult to obtain from the
experimental point of view, it may seem attractive to cancel the Cauchy term of the K-BKZ equation setting h2(I1, I2) = 0 and to find a suitable material function
hl(I1, I2). Wagner [26] wrote such an equation in the form:
t �9 =(t) = ~ m( t - t') hl(I1, I2) C t l ( t ')
- O O
(22)
Cancellation of the Cauchy term may bring some discrepancies, the more
evident one being that, whatever h is, it leads to a zero second normal stress
difference. A more subtle one concerns the loss of the t he rmodynamic
consistency of the model. Indeed, it is not possible to find any potential function in the form U(I1, I2) with h2(I1, I2) = 0 unless hi only depends on I1. As
mentioned by Larson [27, 28], this can induce violation of the second principle
in complex flows such as those encountered in processing conditions.
152
22~4.The genera l ized invar ian t
The form of the function h can be described from shear experiments. Indeed, since in these experiments"
I1 = I2 = 3 + 7t2(t ') (23)
The mathemat ical form of the function can be derived simply from a fit of the
experimental h(T) as obtained in step shear strain for example. However, the
problem is fur ther complicated if one now takes into account flows where the
two invariants differ from each other as, for example, in uniaxial elongational flows where:
I1 = exp{2et(t')} + 2 exp{-et(t')} (24a)
I2 = exp{-2et(t')} + 2 exp{et(t')} (24b)
In order to conciliate these features, Wagner [29, 30] proposes the use of a generalized invariant:
I = ~ I1 + (1-~) I2 (25)
which enables the derivation of a single form of the damping function, since in
shear flows I = I1 = I2. It should be pointed out tha t one may find other
generalized invar iants which satisfy this condition. These have been proposed
by various authors but are less frequently used, for example:
I = (I1)a(I2)1~ (26)
In any case, the new parometer of the generalized invariant has now to be
obtained from additional experiments in elongational flows. But the Wagner
equation can now be written in the unified form:
t ~=(t) = ] m( t - t') h(I) c t l ( t ') (27)
- O O
153
In terms of network description, Wagner considers t ha t the damping
function may reflect an additional process of destruction of network junctions
by strain effects, as described by the generalized invariant, thus involving some
peculiarities of the flow such as its geometry and the s t ra in intensity. As
regards the rate of creation of network junctions, it is assumed to remain constant as in the Lodge model.
2~2.&Interpretation and mathemotical forms of the damping funct iom
In its form, the model of equation (27) is very useful since using a proper
damping function enables a correct description of various shear or uniaxial
elongational basic experiments.
Going back to the Lodge model, Wagner keeps the assumption of constant creation rate (connected with Tli for the ith segment in the memory function) but
assumes tha t the loss probability of junctions is now a combination of two
independent mechanisms. One is related to Brownian motion as in the case of
the Lodge model and depends on the time elapsed between the creation of an
entanglement and the present time ( t - t'), the survival probability being related
to 1/ki. The second reflects the network breaking by deformation and depends
on the kinematics of the flow and not on the segment configuration, it is
expressed as a survival probability between times t' and t. Since the processes
are assumed to be independent, the total loss probability is jus t the sum of the
former and this leads to a separable nonlinear memory function :
M(t- t') = m(t - t') h(I). (28)
Wagner proposes a single exponential form of the damping function:
h(I) = exp(-n~] I - 3). (29)
The exponential form is interest ing because, in shear, the response of the
model can be analytically derived. However because of the exponential, it
decays very rapidly, even at low deformation and therefore it cannot take into
account the linear viscoelastic domain, which is sometimes found to extend to
relatively high values of the strain (typically 0.5).
Another interesting form was used by Papanastasiou and al. [31]:
154
1 (30) h ( I ) = l + a ( i . 3 ) "
Using a factorizable K-BKZ equation (21), Wagner and Demarmels [32, 33]
showed that an equation of the damping function such as �9
h(I1, I2)= 1 (31) 1 + a~](I1 : 3)(I2 - 3)
may be suitable for shear and uniaxial extensional flows. Though it is not
wri t ten in term of a generalized invariant, it degenerates to equation (30) in
shear . The in t e r e s t of equat ion (30) also lies in its s imi lar i ty with
approximat ions of nonl inear functions obtained from the Doi-Edwards
constitutive equation [8] for the reptation theory, at least in shear. Indeed, the
later authors have developed a simplified equation in the form of the K-BKZ
model considering the "independent alignment" ass,lmption, which states that
the strands contract back after the strain is imposed and before relaxation
occurs, so that they are only oriented and not deformed. Currie [34] found an accurate approximation for the related potential function U(I1, I2) which, in
shear, leads to:
h(7) = 5 N~4~+ 25 + 10 (32)
(~2+ 2)~J4r~+ 25 + (4r~+ 25)
Equation (30) is an approximation of equation (32) in shear. Larson [35] has further simplified the Currie potential to obtain U(I1) and he derived h(7) in the
form of equation (30) ~ i th a = 0.2.
In terms of network analogy, the damping function may be viewed as the
expression of the retraction of the strands as compared to the continuum. The
Lodge model thus corresponds to no retraction (affine deformation, a=0 in
equation (30)), the Doi-Edwards equation corresponds to complete retraction
(a=0.2), whereas incomplete retraction makes the damping function more
softly decreasing (0 < a < 0.2). In the later cases, the deformation is non-affine
since there is a difference between tha t of the continuum and tha t of the
network strands. Wagner [33] showed that the Doi Edwards strain function
155
exaggerates the strain dependence and that obviously complete retraction is not
consistent with the data.
Another form which, due to a third parameter, enables a slightly better
approximation of equation (32) and of the experimental data was used by Soskey
and Winter [36] in the form:
1 h(I) = 1 + a (I- 3) b~2 (33)
Parameters a (and b) can be associated with the completion of the retraction
process together with the strain amplitude.
2,2.6.Integral temporary network models and molecular theories.
Wagner and Schaeffer [37-39] made an interesting attempt to reconcile the
different aspects of the temporary junction network model together with the
Doi-Edwards model. They proposed a simple picture of the effects of large
deformation on the stress in terms of a slip links model. Assuming that the
entanglements can be thought as small rings through which the chain may
reptate freely, in addition to equilibration and reptation, two deformation
processes are assumed to give rise to the observed nonlinear behaviour after a
step strain. The first process is connected to equilibration of the monomeric
units between the entanglements by a slippage of the chains and is described by
a normalized slip function S, giving the number of monomers in a deformed
s t rand relative to equilibrium. The second process is related to a loss of
junctions at the chain ends or along the chains by constraint release described
by a normalized disentanglement function D giving the mlmber of strands for a
deformed chain relative to equilibrium. These functions are connected to the
tube dimensions (relative length of the strands or tube segments u' and tube
diameter a). Since the number of monomers on a chain is balanced, at
equilibrium, the average over the configuration space of their product is
unity : <D . S> = 1. Calculating the stress with various assumptions on the
functions leads to different types of equations with different strain measures :
-No slip and no disentanglement (D = 1 and S = 1) leads to the Lodge model.
-Isotropic slip and disentanglement (D = <D>, S = <S>, D . S = <D . S> = 1) leads to the Wagner model with h(I1, I2) = D 2.
-Slip related to relative strand extension but constant tube diameter (S = u',
the molecular tension in a deformed chain is equal to its equilil~rium value)
156
and anisotropic or isotropic disentanglement (D. S = 1 or D . <S> = 1) provides
the Doi-Edwards model with or without the " independent al ignment"
assumption.
-Slip related to relative strand extension but constant tube volume (S = u 'y2,
the molecular tension in a deformed chain depends on the individual
segmental stretch) and anisotropic d isentanglement (D . S = 1) slightly
improves the predictions [40].
-Wagner and Schaeffer ass~lmed that the tube diameter is a function of the
average stretch <u'> through a molecular stress function (S = u' f'l[<u'>], the
molecular tension in a deformed chain depends on the average segmental
stretch) and anisotropic disentanglement (D . S = 1). The function f can be
theoretically derived from the experimental damping function.
2~7J r reve r s ib i l i ty assumptiom
As mentioned by several authors, there is experimental evidence that the
process of loss of junctions, whatever its nature and whatever the domping
function, may be irreversible. This led Wagner and Stephenson [41, 42] to
consider that their original equation is only valid in experiments wherein the
deformat ion is monotonical l ly increasing. To t ake into account the
irreversibility of the loss of junctions, which means tha t these are never rebuilt
in a decreasing deformation following an increasing one, they suggested the
use of a functional ra ther than a damping function. So that, in the original
model, the damping function should be replaced by:
I ' C = t I " ' H(II(t,t'), 2(t,t )) = ~ f m t . _ t, h(Ii(t',t '), 2(t, t )) (34)
In a monotonically increasing deformation, H(I1, I2) = h(I1, I2).
2.3.A differential constitutive equation: the Phan-Thien Tanner modeL
2~3.12VIodifications of the UCM m o d e l
Many improvements or modifications to the UCM model can be found in the
literature. These can lead to various classes of constitutive equations keeping
the differential nature of the equation [2, 3, 35]. As pointed out by Larson [43], a
systematic classification of these can be performed by rewri t t ing the UCM
model as:
157
~(t)- ~ 1 _
(35)
The various changes that may be carried out can be either on the convected
derivative or in the right term of equation (35) or both; these imply the removal
of some assumptions of the initial model. Such a possible modification, that
was claimed to give a correct description of the essential phenomena of the
nonlinear viscoelastic behaviour of polymer melts, is that proposed by Phan
Thien and Tanner [44-46] involving the use of a special convected derivative and special kinetics of the junction.
~ 2 ~ o n ~ mot ion. The Gordon-Schowal~ derivative.
The first kind of modification to the UCM model that may be conceivable is
tha t of the convected derivative. This leads one to consider that the motion of the
network junctions is no more tha t of the continuum and thus, the affine
assumption of the Lodge model is removed. Among the various possibilities,
Phan Thien and Tanner suggested the use of the Gordon-Schowalter derivative
[47], which is a l inear combination of the upper- and lower-convected
derivatives, instead of the upper-convected derivative:
~(t) a a = (1 - ~) ~i(t) + ~ (t) 0 < a < 2 (36)
This is equivalent to the assumption that there is some slippage of network
junctions with respect to the continuum during the deformation. The local
velocity field is thus different from the macroscopic one. The adjustable
parameter a materializes the slippage and is called the slip parameter. To
avoid confusion, it should be noticed that equation (36) is often written as:
~i(t) = 1 2 - - ~ ~(t)+ ~ ~(t) -1 < ~ < +1 (37)
Using this derivative in the former UCM model, Johnson and Segalman
proposed a model [48] that improves the predictions especially in shear, leading
to normal stress differences and shear viscosity which are now shear rate
dependent. Unfortunately, although it appears to be attractive in shear, the use
of such a derivative can lead to some physical paradoxes that will be discussed
158
later. Moreover, although it brings some improvement in shear flows towards
the UCM equation, it can be shown easily tha t such a modification does not
basically change the elongational behaviour. For example, the elongational
viscosity remains unbounded at large times at high elongation rates (and a
steady state value cannot be obtained). This could be expected since the Gordon-
Schowalter derivative mainly takes into account rotational effects which are
absent in elongation flows.
~ T i m e - d e p e n d e n t segment kinetic~ This led Phan Thien and Tanner to consider an additional change of the
constitutive equation, removing the ass,lmption of constant creation and loss
rates for the junctions. This could be done in many ways, for example by
making these quantifies functions of the invar iants of the stress or s train
tensors. Phan Thien and Tanner assumed that, for each type of junction, these
rates might be dependent on the average internal extension of the surrounding
s t rands towards their equilibrium length. They wrote this in the form of viscosity contributions ~'i and relaxation times ~'i tha t are now dependent on the trace of the extra-stress tensor (tr[~=i]), together with the equil ibrium
characteristics of the segment (Tli, ~-i) in the form:
X~,X4 L~J) and Tl'i V(l~i, ~i, tr[T_i]) ~.'i = y(1]i , t r . _ ~ . , = (38)
where Y is some increasing function of the trace.
The complete Phan Thien Tanner model is:
�9 �9
Y(rli, ~, tr[l:~(t)]) 1;i(t) + ki ~i(t) = Tli ]~,
Two suggested forms of Y are"
and ~(t)= = ~. ~i(t).=
Y(Tli, ki, tr(1;i))= 1 + e tr(1;i),
Y(Tli, ~, tr(~i)) = exp{ e ~i tr(Ti)} ,
(39)
(40a)
(40b)
159
where e is a positive adjustable parameter.
Provided that a suitable Y function is chosen, this is claimed to give a correct
p ic ture of many phenomena displayed in simple shear and uniaxial
elongational flows. One should note that the original model of Phan-Thien and
Tanner uses the non-affine derivative together with nonlinear stress term.
3.EXPERIMENTAL ASPECTS
3.12Vlaterials.
Three different polyethylenes referred to as HD, LD, LLD have been
investigated. They mainly differ in their molecular weight distribution and in
the structure of the chains which can be either very linear as in the case of HD
(high density polyethylene) or short-branched in the case of LLD (linear low
densi ty polyethylene) or long-branched in the case of LD (low density
polyethylene). The weight average molecular weight and the polydispersity
index of the samples are summarized in Table la.
3.2.IAnear viscoelastic behaviour and memory function.
3.2 .1J)ynamic measurements .
In order to get a good description of the linear relaxation modulus, which
remains the basis of any nonlinear model, experiments have been carried out
in the l inear viscoelastic domain in oscillatory shear flow. Dynamic moduli
have been measured in a parallel plate rheometer (Rheometrics RDA700) for
frequencies ranging from 0.01 to 500 rad/s. The frequency window at a
reference temperature of 160~ has been slightly expanded by the use of time-
t empera tu re superposition from experiments performed between 130 and 200~ Though in the case of polyethylene, the activation energy (Ea) of flow is
often so low tha t no appreciable gain can be expected (see Table lb), the
achievable frequency range at 160~ covers about five decades in the terminal
zone from 0.005 to 1000 rad/s.
Some terminal viscoelastic parameters have also been evaluated at 160~
using a Cole-Cole expression for the dynamic viscosity. Table lc shows the zero shear viscosity (T10), the characteristic relaxation time (~.o, corresponding to the
160
maximum of the imaginary part of the complex viscosity) and the relaxation
time distribution parameter (h, obtained from the eccentricity of the Cole-Cole
plot towards the real axis).
~ t e re laxat ion SlmCWUnL
From the dynamic moduli, discrete relaxation t ime spectra have been
calculated by a nonlinear minimization procedure [17], which lets the number
of relaxation modes and the time value of these modes freely adjustable. The
spectra of Table ld have been obtained and their use in a Maxwell model
enables a very good recovery of the experimental values of the storage and loss
moduli in the explored frequency window (within 4% error) as shown on Fig.1.
It is worth noticing that the spectrum of LLD for the terminal zone seems to be
ra the r complete whereas that of the two other polymers are more or less
sharply cut. This occurs respectively in the long time range for LD and in the
short time range for HD. This peculiarity should be kept in mind when dealing
with nonl inear models since it can lead to some discrepancies for the
predict ions in nonl inear situations. These problems might be expected
especially when the spectrum is truncated in the long time range, wherein
these modes are known to be very important for the viscoelastic behaviour of
polymers.
Table la: Molecular weight characteristics of the s,,amples. Material Mw [g/mol] Mw / Mn
. . . . .
HI) 90000 4.6
LD ~ 4.5
LLD 120000 6.3 , ,
Table lb" Thermal characterist ics of the samples (WLF coefficients and
activation energy for flow at 160~ , . . . . . . . _~6o _16o _'~6o
Material C 1 C2 [o] E a [kJ/mol]
HD 2.38 311 27.5
LD 4.81 308 56.0
LLD 2.49 311 28.9 = , , , , , , �9 | |
161
Table !c: Pa r -mete r s of the terminal zone, at 160~ from a Cole-Cole plot.
Mater ia l 110 [Pa.s] ~o [s] h ,,, . . . . . ,, , , , =
HD 3500 0.088 0.47
LD 426(D 8.82 0.41
LLD 154O0 0.38 0.49 . . . , , . ,, . , . , . . . . . . . , . , , , �9
Table 1. d:, Discrete relaxation spectra at 160~
[s]
0.00104
0.00573
0.0258
0.113
0.485
2.30
11.2
80.3
HD LD LLD
71i [Pa.s] ki [s] Tli [Pa.s] ~ [s] Tli [Pa.s] . . . . , , , . ,
234 0.000645 98.1 0.000128 236
514 0.(D535 285 0.00612 1350
842 0.0285 768 0.041 3360
765 0.155 2620 0.277 4 { ~
576 0.891 6460 2.01 3730
312 4.58 ~ 15.7 2010
164 23.4 13200 135 956
85.5 118 7 ~ ) , , , , , , , ,
,...I 106
~.~ 105 - , = 1
~ 104 " 0
o
~ o ~ 103 [
J lO 2 ~ c -
m 101 c
~ I ~_ lO ~ ,~ 0
10-1 10 ~ 10 z 102 103 Frequency [rad/s]
Figure la: Storage (o) and loss (Q) moduli of HD at 160~ (full line indicating a
fit obtained from the spectrum of Table ld).
162
10 6 ~ [
n
'' lO s " -,=4 ,---4
0 4 -u 1 0
tn 10 3
o j 10 2 ~1 13 e- ra 1 0 1 .
L ~ 0 o L 0
o~ 10 -1 10 -2 10 -I i0 o 101 102 103
Frequency [ r a d / s ]
Figure lb: Storage (o) and loss (Q) moduli of LD at 160~ (full l ine indicat ing a
fit obtained from the spectrum ofTable ld).
106 F n ~-~ 105r -,-4
104 r 0
I 103
o G~ " 102
~ 0 c 01 I
~ I ,~ lO ~ L 0
4-J m I0 -1
10-2 10 - I i0 o 101 10 2 10 3 Frequency [ rad /s ]
Figure lc: Storage (o) and loss (o) moduli of LLD at 160~ (full l ine indicating a
fit obtained from the spectmlm of Table ld).
163
3~3.Experimental data in ~mple shear.
3~. l~teady state shear flow. Steady state shear flow experiments have been performed using a capillary
rheometer (Instron 3211) at 160~ with different dies (diameter range: 0.25-
1.25mm; length range: 2.25-25mm) so that Bagley corrections were taken into
account. Other experiments have been performed using a cone and plate
geometry (Rheometrics RMS800 rheometer) at different temperatures. The
time temperature superposition has been applied on both the shear stress (1:(~))
and the first normal stress difference (NI(~)). Combining the two sets of results
for each material, the viscosity data TI(~)extends to nearly five decades from
0.01 to 1000 s -1. The first normal stress results only cover a limited window
from 0.1 to 10 s -1 for HD and about three decades for LD and LLD. The main
measurement problem is linked to the low value of the normal force at low
shear rate and to instabilities at high rates.
The "extended" Cox-Merz rule [49] can be successfully applied for HD and
LLD. This rule states that the viscosity and elasticity coefficients for oscillatory
and steady state shear flows are related, according to:
Tl*(co) = Tl(~) and 2G'(co)/a~ 2 = VI(~) when co = T, (41)
where ~1(~) is the first normal stress coefficient,
is the shear rate,
co is the frequency.
Whenever it is applicable, such a comparison can lead to a considerable
widening of the shear rate range, this is especially interesting in the case of
normal stresses which are generally difficult to measure on a broad window of
rates. However, significant differences were noted in the case of LD preventing
the use of the previous rule and any enlargement of the data set.
3~3~Stress g r o w ~
Using the cone and plate geometry, stress growth experiments have also
been performed using different temperatures and different shear rates. �9 �9 § �9 § �9
Correct tangential (1:+(t,7); Tl+(t,7)) and normal stress (N i(t,7); ~i(t,7)) data were
154
obtained from 0.1 s until the steady state flow is obtained.The shear ra te range
which enables satisfying measurements covers about one decade typically from
0.2 to 2 s-1.
~ t r e s s re laxa t ion af ter a s tep st~_in-
Measurements of the nonlinear relaxation modulus G(t,7) have also been
carried out using the plate-plate geometry. Various step strains were applied
on the sample and the stress relaxations were recorded. Since the shear strain
is known to be inhomogeneous in such a geometry, a correction of the apparent relaxation Ga(t,T) modulus has to be taken into account to get the real
relaxation modulus for the maximum strain in the disk sample. This
procedure is very similar to that proposed by Rabinowitch in Poiseuille flow,
wherein the shear rate is also non-homogeneous, and has already been
described by Soskey and Winter [36]. The correction factor is:
dlnGa(t,T) G(t,7) = 4 + n Ga(t,T) with n = . (42)
n dln7
In this way, the nonlinear relaxation modulus was measured from the linear domain to strains up to 10.
3.4.Experimental data in ~ elongation.
3A.1.Stress growtl~
Measu remen t s of the elongational da ta during stress growths have been
obtained on a constant s t ra in rate , exponentially increasing length, tensi le
rheometer (Ballman type) at 160~ The elongation ra tes achievable in this type
of rheometer for the three mater ia ls range between 0.05 and 2.6 ~-1. The largest
window was obtained for the more viscous mater ia l (LD) ana in this case
correct measurements were possible in the shorter t imes (0.01 s) whereas this
limit is slightly grea ter for the other materials. In any case, the m a x i m u m
Hencky strain does not exceed 2.5. This range could not be extended because of
the force t ransducer sensitivity, ei ther because of the low viscosity (short times)
or because of the vanishing sample cross area (long t imes) or because of
f i lament breaking and inhomogeneous deformation of the tested sample.
165
3.42~to~dy state viscosity.
Since any elongational stationary viscosity can hardly be obtained in the
t ransient experiments unless in the Troutonian regime (low strain rates),
determination of a steady state viscosity has been performed on LD and LLD by
indirect methods using the Cogswell analysis of convergent flows and entrance
pressure losses [50, 51], as derived from the Bagley plots of the capillary
experiments. According to Cogswell, the polymer, during its flow from the
reservoir to the die, will form its own streamlines, leading to the lowest
pressure loss. In this situation, for an axisymmetric geometry, the entrance
pressure loss is mainly due to elongational stresses and thus one can calculate
an average elongational stress as:
3 ~E = ~ (n + 1) APE , (43)
and an average elongational strain rate on the flow axis as:
�9 4~/1; e = (44)
3 ( n + 1)APE '
where n is the flow index,
APE is the entrance pressure loss,
is the shear stress (using the Bagley correction),
is the apparent shear rate.
Though the Cogswell analysis was derived for 180 ~ entrance angle, the
pressure losses were found to be independent of the angle between 90 and 180 ~
provided that the die length is short enough to avoid material compressibility.
It should be noticed that in such cases, exit pressure was also claimed to be the
origin of important errors, nevertheless the entrance pressure loss was found
to be insensitive to the die diameter, and conversely to the contraction ratio, in
the range of 4.76 to 12.7. A question arises concerning the residence time of
particles in the convergent part of the die. Indeed, it is important to evaluate
and check this residence time in order to assess whether it is sufficiently long
to get a stationary value of the viscosity or not. At low shear rates (high
residence times), for LLD, a good agreement was found between the calulated
values of the converging flow analysis and those of the transient experiments.
166
But at high shear rates, this might be questionable in some cases since in the
case of LLD for example, at elongation rates greater than 17s -z, the residence
time was found to fall in the order of the maximum relaxation time of the
spectrum.
3 , 5 ~ t a l rauges of data for n o - H m ~ r viscoelasticity exper iment~
The experimental ranges of strain rates (or strains) are summarized in
Table 2 for the various types of experiments. Time-temperature superposition
was successfully applied on the various steady shear flow and transient shear
flow data. The shift factors were found to be exactly the s_~me as those obtained
for the dynamic data in the linear viscoelastic domain. Moreover, these were
found to be also applicable in the case of entrance pressure losses leading to an
implicit application to elongational values.
Table 2a: Experimental ranges for shear data (shear rates or shear strains, (*)
indicates that the Cox-Merz rule was applied).
Material G', G" 11 TI (*) Nz N1 (*) Tl + , Nz + G
and
Function co [rad/s] ~ [s-l] ~ [s-l] ; [s-l]
HD 0.01-1000 0.15-2000 0.01-2000 0.15-7 ,
0.01-1000 0.5-5 0-10
LD 0.005-1000 0.01-300 Non valid 0.01-7 Non valid 0.2-0.8 0-10
LLD 0.005-150 0.01-600 0.005-600 0.01-3 0.005-150 0.2-2 0-6 ,, , , . , = | . . , , . . . .
Table 2b" Experimental ranges for elongational data (elongation rates or
Hencky str~ns) .
Material TIE
and
Function ~ [s-Z]
HD
LD 0.09-9
LLD 0.2-100
TIE +
[s-q (c)
0.125-1 (2.0)
0.05-2.6 (2.5)
0.05-1.4 (2.5)
167
4 . E X P E R I M E N T A L V ~ A T I O N O F T H E W A G N E R M O D E L
4.1.Time-strahl separability.
Figure 2 shows some results obtained in step shear strain experiments, in
terms of nonlinear relaxation modulus, in the case of LD. On logarithmic
scales, the curves are only shif ted vertically, indica t ing tha t they are
superimposable in the experimental t ime range and tha t the separability,
expressed by equation (18), seems to hold.
10 5 -
n
1 0 4
o ~E
t- O "~ 10 3
x t~
e~
10 2 . . . . . . . . .
i0-2 lO-i
",,,..,.
",,,,~.
"X'.'.. \
I 0 ~ 10 i 10 2
Time [s]
Figure 2: Linear and nonlinear relaxation modulus of LD obtained from step
shear strain experiments at 160~ (-----): linear, ( ..... ): T = 2, ( - - - ) : T = 3.
4,2.The damping function.
4 . 2 . 1 ~ r i m e n t a l determinat ion.
Damping fimctions in shear can be obtained experimentally by two different
methods. Firstly, by direct exper iments in step shea r as indicated in
paragraph 2, and secondly by a derivation from the tangential stresses (l:+(t,~))
in t rans ien t experiments as proposed by Wagner in elongation [52] and
Fulchiron et al. in shear [53]:
158
�9 +(t,~) t fl+(t',~) m(t')
7t dt'}. (45a)
This la t ter method can also be used in uniaxial elongation using the t ransient § �9 + �9 + �9
elongational stress (oE(t,e)= ~zz(t,e)-Ve2(t,e)):
§ �9 + . ~@. 1 aE(t,s t oE(t,e) m(t')
= " { G2(t') dt'} (45b) h(e) expl2~t} - exp{-~t} { G(t) 0 ~
Figure 3 shows a comparison between damping functions obtained from step
shear and t ransient data for LD.
1 . 0
= 0 8 :..k
o .r-i
u ( - 0 . 6
LL
( ' 0 . 4 -w-4 (:1. E
r ' 0. 2 -
�9 O
A ~p &
A ~
0 0 ' ' ' ' 0 2 4 6 8 I0
Shear Strain
Figure 3" Exper imenta l damping function of LD obtained from t rans ien t
experiments either in step shear (filled symbols) or in step shear rate a t 0.2 and 0.5 s -1 (open symbols).
The shear damping function obtained from t rans ient experiments remains
unchanged whatever the shear rate is in the experimental window, so tha t one
may assume tha t it is not shear-rate dependent. More surprising is the large
discrepancy between the results between the two methods, wherein the step
169
shear experiment leads to smoothly decreasing functions. One could argue
that the delay time for application of strain in a classical rheometer is a
possible origin of this discrepancy, however this is in contradiction with the
higher value of h. More probably, this might contradict the former conclusions
concerning the shear rate independence for very high initial shear rates such
as those encountered at the start of step strains.
4 . 2 ~ t h e m a t i c a l form of tho. d~mping function-
In this study, the mathematical form of h, equation (33), is chosen to fit the
experimental function obtained from transient data. In the case of LD, it is
found to be very close to equation (30) since the value of the parameter b is
nearly equal to 2.
As far as the elongational damping function is concerned, it was found to be
independent of the elongational rate for the experimental range. Figure 4
shows a plot of the damping function in shear and in elongation for LD
together with the fit using equation (33) with invariant (25). Indeed, from the elongational data, the last parameter ~ describing the
generalized invariant can be theoretically obtained. However, it should be noted
that this might be sometimes difficult, especially when strain hardening is not
very marked as it is the case for LLD and HD for example. Indeed, in this case,
equation (45), which implies simultaneously the difference of two terms and
numerical integration, can lead to large errors mainly due to uncertainties in
the experimental data. In the short time range, this can lead to physically
unrealistic values of the domping functions (greater than unity), because of the
slight difference between the Lodge (h = 1) and the Wagner model and because
of the finite starting time of rheometers. On the other hand, in the long time
range, there may be large uncertainties on the calculated functions and thus
on the fitted parameters because of the difficulty to obtain the steady state in
constant elongation rate experiments. As a consequence, when strain
hardening is not marked, one has to refer to the transient function itself to fit the ~ parameter and even more, doing that, one may be faced with multiple
choices unless the steady state viscosity data can be obtained from indirect
experimental methods (converging flow analysis for example).
170
1.0
c 0 . 8 o
-J- I
o c 0 .6
c 0 . 4 . , ~
E rl$
o 0 .2
O o
0.0 L , , ~ J _. i . . . . 0 2 4 6 8 10
Strain
Figure 4: Damping functions of LD in shear (o) and in elongation (o), fitting is
done according to equation (33) with invariant (25) (a = 0.084, b = 2.06, ~ = 0.019).
In spite of uncertainties in ~ in the case of LLD and HD, the values of Table 3
were obtained for the three polyethylenes. The corresponding shear damping
functions are plotted on Figure 5. It can be seen that the decrease of the
function is smoother in the case of LD and this might indicate some resistance
of junctions to s t rain as described by the retraction assumption. This is
consistent with the branched nature of LD where long side branches might
hinder complete retraction of the polymer chains and lead to a conservation of
network junctions. Though the value of ~ is ra ther uncertain, its observed
constancy was already mentioned by Papanastasiou et al. [31]. However, this
might be questionable since one may think that this parameter should reflect
an additional influence of the geometry towards the retraction process.
Table 3" Fitte d Ear .ameters of the damping function for the three polyet_hylenes
Material a b
HI) 0.106 2.32 0.020
LD 0.084 2.06 0.019
LLD 0.086 2.56 0.020
171
1 01
c 0 8 o
. ~ 1
U c 0 6
c 0 4 -.=4
E
~
O0 0
. ~"~--.z..z.... - - z . z_ -_L :_ - : : : .~ .=
| I = I I , , I
2 4 6 8 10 Shear Strain
Figure 5: Damping functions of the three polyethylenes calculated with
equation (33) and invariant (25). (----): LD; ( ..... ): HI); ( - - - ) : LLD
~ r i m e n t a l validation of the Wagner model in simple flow~
This section presents different results obtained using the Wagner equation
with the form of the damping function of equation (33) together with the
generalized invariant of equation (25). It is primarily devoted to comparisons
between predictions and experimental resul ts in shear and uniaxial
elongational flows described in paragraph 3 for LD.
4 ~ l . S t e a d y state f low~
Figure 6 shows the predictions of the Wagner model compared to
experimental data for elongational viscosity, shear viscosity and first normal
stress difference of LD. These have been calculated according to:
o o
Tl(~) = J re(s) h(~/s) s ds , (46a)
172
o o
~1(~) = J re(s) h(~ s) s 2 (:Is, (46b)
o o
~]E(~) = 1 g m(s) h(~ s) {exp(2~ s)- exp(-~ s)} ds (46c)
L _ j m--J
~-~ 10 5 "~. o o
u)
> 0 4 1 O0
2~ ' . ' o L 10 3 f _ Z o ~
0 2 ~ _ 1 "- 10 CO 10 -2 10-1 10 o 101 10 2 10 3
Shear or Elong. Rate Is -1]
Figure 6: Steady state functions for LD at 160~ (experimental and calculated). (o): elongational viscosity, (o)- shear viscosity, (A): first normal stress difference.
4.3~.Transient s h e a r f low~
Figures 7 and 8 show the predictions of the Wagner model compared to experimental da ta for t r ans ien t shear viscosity and first normal stress coefficient of LD. These have been calculated according to:
t Ti(t,~) = ~ re(s) h(~ s) s ds + t G(t) h(~ t),
0 (47a)
t vl(t,~) = j" re(s) h(~ s) s 2 ds + t 2 G(t) h(~ t).
0 (47b)
173
~ T r a n s i e n t elongational flows
Figure 9 shows the predictions of the Wagner model compared to experimental data for transient elongational viscosity of LD. These have been calculated according to:
qz(t,~) = _.1 t re(s) h(~ s) {exp(2~ s) - exp(-~ s)} ds -d 8
1 + ; G(t) h(~ t) (exp(2~ t)- exp(-~: t)}.
E
(48)
10 5 -
r
>, ~
m 10 4 0 U
-0-,l
L
e"
_ n . o ~ o o o o o o o o o
1 0 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 -1 10 o 101 10 2
Time Is]
Figure 7: Transient shear viscosity for LD at 160~ (experimental and calculated). (o): 0.2 s -z, (o): 0.5 s -1, (A): 1 s o1.
174
0 CJ
U'; U~ q,; L. r - -1
,,,%
m n
L. 0
Z
m f,_
~ b _
10 6 -
10 5
1 0 4
O o O O O O
, - j O ~ n n n n n n n m n n
& & A ^ e
I0 ~ 101 10 2 Time Is]
F i g u r e 8: T r a n s i e n t f irs t n o r m a l s t r e s s coe f f i c i ent for LD at 160~ (experimental and calculated). (o): 0.2 s -1, (o): 0.5 s -1, (A): 1 s -1.
1 0 6 -
>, lO s -~..4
-r.t
> 1 0 4
0
Ld
1 0 3 10-2
�9 , , . . . . . I , . . . . , , , i ' ' . . . . . . ! , . . . . . . i I
10 -1 10 o 101 10 e Time [s]
Figure 9: Transient e longational viscosity for LD at 160~ (experimental and calculated). (o): 0.05 s -1, (~): 0.5 s -1, (A): 1 s -z, (0): 2 s -1.
175
4.4.Conclusion.
The previous set of comparisons shows that the Wagner model enables a
good description of the experimental data in various experiments in simple
shear and uniaxial elongation. However, it is worth pointing out that in most
cases, there is a major difficulty concerning the determination of the parameters of the damping function.
Firstly, it has been shown that there may be many experimental problems in
a direct determination of the experimental function. In shear, damping
functions obtained from step strain and from step strain rate experiments do
not match each other. This poses an important question on the validity of the
separability assumption in the short time range. Significant departures from
this factorization have already been observed in the case of narrow polystyrene
fractions by Takahashi et al. [54]. These authors found that t ime-strain
superposition of the linear and nonlinear relaxation moduli was only possible
above a certain characteristic time. It is interesting to note tha t this is
predicted by the Doi-Edwards theory [10] and according to this theory, this
phenomena is attributed to an additional decrease of the modulus connected to
a tube contraction process and time-strain separability may hold after this
equilibration process has been completed. Other examples of non-separability
were also reported by Einaga et al. [55] and more recently by Venerus et al. [56] for solutions.
Another explanation of the observed discrepancy can be found in the
numerical treatment of the stress growth experiments, either in shear or in
elongation, which sometimes gives rise to physically unrealistic values of the
damping function because of uncertainties in the measured data. This is
especially the case for transient elongational data on low viscosity materials or
on polymers having nearly Troutonian behaviour in the experimental range of elongation rates (LLD or HD).
To avoid these problems, one can prescribe a mathematical form of the
equation of the damping function and of the generalized invariant. Then,
minimization of the experimental rheological functions is expected to provide
the values of the required parameters. Unfortunately, the adjustable
parameter of the generalized invariant, which can only be obtained from
elongational data, cannot be found accurately in many cases. Indeed, accurate
determination of this value requires both a pronounced strain hardening
behaviour and a steady state viscosity in the long time range, which is seldomly
176
achieved in current experiments. The lat ter problem can be avoided by
resorting to indirect methods for the determination of the elongational steady
state viscosity such as the convergent flow analysis, while keeping in mind that
this technique has not yet received sufficient theoretical justification.
Combined flows, involving both simple shear and uniaxial elongation could
be of great help in providing additional tools to select the parameters of the
damping function with better accuracy. Other strain histories may also be used
for this purpose such as that proposed by Giacomin et al. [57], who proved that
large amplitude oscillatory shear flows are a useful tool to obtain the
parameters of the damping function reliably. Unfortunately, the authors also
showed that the concept of irreversibility described by equation (34) is not
sufficient to get an appropriate depiction of the rheological phenomena in reversing flows.
Predict ion of the second normal s tress difference in shea r and
thermodynamic consistency obviously requires the use of a different strain
measure including of the Cauchy strain tensor in the fo re of the K-BKZ model.
With the ratio of second to first normal stress difference as a new parometer,
Wagner and Demarmels [32] have shown that this is also necessary for
accurate prediction of other flow situations such as equibiaxial extension, for example.
5 J~PERIMENTAL VALIDATION OF THE PHAN THIEN TANNER MODEL
5.1s o ~ n ~ f i n g from the use of the Gordon-Schowalter derivative.
The use of the Gordon-Schowalter derivative, equation (36), brings some
discrepancies that can be easily pointed out considering the limiting case of a = 0 [58]. This case is known as the Johnson-Segalman equation:
~(t) + ki ~(t) = qi ~ , and ~=(t) = .~ ~i(t). (49) 1
The slip parameter can be easily determined from various experiments in
shear situations by some fit of the steady state shear viscosity and primary
normal stress coefficient. Analytic expressions are easily derived in steady
state and transient flows in the form:
177
i 1 + a(2- a)(X i })2 (50a)
2Thki ~(~) = 2
i 1 + a(2- a)(X i ~)2 (50b)
Tl+(t,~/) =v. + a ( 2 - a ) ( ~ ) 2 i 1
{1- e-t&i (cosP4a(2 - a )~ t ] - s ~4a(2 - a) sin[~/a(2 - a)~t])} , (50c)
+ �9
~l(t,7) = Z i 1 + a(2- a ) ( ~ 1 2
{1- e-t/~ (cos ['4a(2 - a)~t] + sin [~/a(2 - a)~t])} .
~]a(2 ' a)~s (50d)
Second normal stress difference is predicted with the following dependence:
N__~2 _ _a (51) N 1 - ' 2 "
Alternat ively, in t rans ien t flows the slip pa rame te r can also be determined using the t ime position of the max imum of the experimental functions (tw for
t a n g e n t i a l s t ress and tN for normal stress). However, this can only be
pe r fo rmed accura te ly if the s t ress overshoot is large enough to avoid
uncer ta int ies in these values and this can only be achieved at high shear rates.
In this case, according to equations (50c) and (50d) :
/ t
trr = , (52a) 2 "4a(2- a)
178
t N ' -
~]a(2- a)~ (52b)
The values of the calculated slip factors are listed in Table 4.
Table 4: Slip parameters of the various mater ia l s
Mater ia l aT aN
HD 0.25 0.8O
LD 0.08 0.24
LLD 0.24 0.77
As can be seen from these resu l t s and from Fig.10 (for HD), sa t i s fac tory
ag reement cannot be achieved by the use of a single value of the slip pa ramete r
for both tangent ia l and normal stresses.
~-~ lO s 1:1.
r L_.J
ID
~--~ 10 4
-,"4
o 10 u ~
~
c L o 102 ~ z l -
ffl L
~. I01 . . . . . . . . . 10 . 3 1 0 - 2
~1 ,~ ,El
o
10 -1 i0 o 101 10 2 10 3 Shear Rate Is -1]
Figure 10: Viscosity (o) and first normal stress coefficient (~) of HD at 160~
(experimental data and fit using ( ~ ) : a = 0.25 or ( . . . . . ): a = 0.80)
Using the corresponding value of a obtained by a fit on ei ther the tangential
or normal stresses (named aT and aN), equat ions (50, a and b) give a r a the r
good fit of the experimental curves on a large range of shear ra tes . The same
179
conclusion is also valid in t rans ient shear flows. Two different values are
obtained from the maxima of either the tangential or the normal stresses and a
single value of the slip parameter cannot describe simultaneously the maxima of both the viscometric functions. Nevertheless, the values aT and as , as
determined from each function, remain independent of the shear rate for a
pa r t i cu la r mater ia l . Moreover, similar values of the slip pa rame te r are
obtained for a particular material whatever the time-regime of the shear flow
experiment is. The fact that a single value of the parameter is not able to
describe both tangent ial and the normal stresses can be considered as a
manifestat ion of the violation of the Lodge-Meissner rule [59] by the Johnson-
Segalman model, as previously described by many authors [43, 60]. This
relationship states that, after a step strain, the ratio of the first normal stress
difference to the shear stress should be equal to the strain magnitude �9
Tli - T22 = 7 . (53)
It requires that the principal stress axes should coincide with the principal
strain axes. This rule has been experimentally checked by many authors
[24, 56] Actually, the use of the Gordon-Schowalter derivative involves the
violation of the Lodge - Meissner rule, indeed when a equals 0 or 2, either the
upper or the lower convected derivatives implies that the relationship is
respected. In the general case, the double value of the slip parameter is a
natural way to accommodate this rule.
The connection between the double value of the slip parameter obtained from
the viscometric functions and the violation of the Lodge-Meissner rule becomes
more evident when the time-strain separability of the model is considered. For
this purpose, the Johnson-Segalman model can be rewritten under the form of
a single integral equation, cancelling the Cauchy term, which gives the
following form in simple shear flows:
t
=~(t) = / m ( t - t ' ) h ( y ) Ctl( t ') dt ' . (54)
- 0 0
180
From this equation, derivation of equations (50) cannot be obtained with a
unique form of the damping function. Derivation of equations (50a) and (50c) requires:
s i n [ ~ T ] h(T)= qa(2'a)Yy " (55a)
Derivation of equations (50b) or (50d) requires:
h(~,) = 2 (1, c o s [ ~ ~ / ] ) a(2- a) T 2
(55b)
Because of the sine form of equations (55), physically unrealistic values of h(7) 2x
occur at strain higher than . ~]a(2 - a)
Nevertheless, a rather good fit of the experimental damping function (as determined by equation (45) of the preceeding section) can be obtained until T = 4
for all the materials with equations (50), provided that the corresponding slip parameters are used (aT for equation (50a) and aN for equation (50b)).
The significance of the double value of a is clearly shown in equations (50)
since respect of the Lodge-Meissner rule or of material objectivity requires the
identity of equations (50a) and (50b) which can only be achieved by two different values for a:
sint~]aw(2 aT)T] 2(1-cos[~]aN(2-aN)T])
~]aT (2- aT)T - aN (2- aN) T 2 (56)
Solving equation (56) numerically gives the following relation between aT and
aN :
aT = 0.3 aN, (57)
which is valid whatever the material is and for strains lower than 4. This
equation is consistent with the experimental values derived previously from the steady and transient experiments.
181
However, it is now worth pointing out that the difference in aT and aN imply
tha t the initial model should be replaced in shear by a pair of two independent
correlations for shear stress (eq. 50a and 50c) or for first normal stress
coefficient (eq. 50b and 50d). But at this point some questions arise concerning
the choice of the proper value (aT or aN) to be used in any other flow situation.
Though it is possible to imagine equation (49) including some variation of a
with flow history or invariants, it could hardly be different in two equations for
the same flow kinematics.
The sine form of the "damping function ~ leads to another major problem,
which lays in the occurrence of undesirable oscillations in t ransient shear
flows (Figure 11). This phenomenon may be misleading for example when
modelling instabilities in complex flows, since it is then hardly possible to
distinguish between real phenomena and those generated by the model itself.
3500
~aooo ~
-~
> 2 5 0 0
2 0 0 0 '~ ' '' ' - ' 0 2 4 6 8 10
Time [s]
Figure 11: Transient viscosity of HD at 160~ (experimental data (o): 0.5 s "1, (~):
5 s -1 and fit with a = 0.25)
Keeping in mind the previous remarks, it must be recognized that the lower
the value of the slip parameter, the smaller the deviation to the Lodge-Meissner
relation. This may be especially interesting, since in such a case, one can
expect a single value of a tha t enables some kind of compromise for an
acceptable depiction of the rheological shear functions. This can be expected,
182
for example in the case of LD (see Table 4) for which ra ther low values of the slip parameter are obtained.
At least, even if the Gordon-Schowalter derivative is obviously an improper
tool for the description of non-affine deformation, its basic meaning retains
some consistency with experimental observations, especially concerning the
loss of junctions in materials with very different molecular structures. Indeed,
going back to the significance of a, its low value in the case of LD is in
agreement with the assumption of some kind of resistance to slip of junctions
in branched materials, whereas the opposite trend is observed in the case of
linear polymers (LLD and HI)), for which higher and ra ther similar values are found (Table 4).
As a final comment, it is worth mentioning tha t the Johnson-Segalman
model predicts an elongational behaviour that is very close to tha t of the UCM
equation. Indeed, the t ransient elongational viscosity remains unbounded in
the long time range at high elongation rates and no steady state value can be
obtained in the general case. In this type of flow, the slip pa ramete r only has
an influence on the occurrence of the strain hardening effect, whose time
position changes towards the UCM equation (Figure 12). Divergence of the
t ransient elongational viscosity occurs in the Johnson-Segalman model when :
1 t > if0 < a < 1, (58a)
2 ~ (1- a) (~)max
o r
1 t> if1 < a < 2, (58b)
e (a- 1) (ki)max
instead of (UCM model):
1 t > . (58c)
2 ~ (~)max
183
10 6
Q_ i __1
>, lO s ~ t n 0 t ~ 01
-e-i
> 1 0 4 ..
0
i , i 1
10 3 - - 10-2
// o /
10 -1 10 o 101 10 2 Time I s ]
Figure 12: Trans ien t elongational viscosity of LD at 160~ (experimental data
(o):0.5 s -1, (Q):2 s -1 and fit using the Lodge model (--): a = 0 and the Johnson-
Segalman equat ion (- -): a = 0.24).
5.2.Choice of the segment idnet ic~
Since the slip parameter does not basically change the divergence behaviour
of the elongational viscosity of the UCM model, the l imiting case of the Phan
Thien Tanner model with an upper convected derivative (a = 0) may give an
indication of the influence of the function Y tha t governs the segment kinetics
and the choice of its mathematical form. The linear equation for Y is obviously
improper since, as can be seen from Fig.13, an unrealis t ic behaviour of the
steady elongational viscosity is predicted (constant value a t high elongation
rates). In th i s sense, the exponent ia l form is more realistic. Using the
exponential form, the description of the pronounced maximum of elongational
viscosity requires ra ther low values of paramete r e, of the order of 0.01 when
the upper convected derivative is used (a = 0, Table 5).
As a general rule in this case, it is obvious that the higher the value, the less
pronounced the maximum, the extreme case being tha t of s = 0 (Lodge model
and divergence of the viscosity).
184
This is unfortunately contradictory with the value that is needed to describe
the shear functions. Indeed, although the Lodge-Meisner rule is valid when
the upper-convected derivative is used, the shear thinning effect on shear viscosity and primary normal stress coefficient requires rather high values of e
(Table 5 and Fig.14) which are, in any case, one order of magnitude greater than in elongation, the extreme case being again that of e = 0 (Lodge model and
no shear rate dependence of the shear functions).
Table 5: Parameter e for the various materials
Material e (elongational data) . . . . . . _ . ,=.
HD O.085
e (shear data)
0.7
LD 0.030 0.15
L [ ~ 0.105 0.7
106 f - - - i
e@
EL
>, 10 5
t n
o u
. r . 4
> 10 4
r -
0
t ~
. . j . . _ . . . . . . . | , . . 1 , . . , a . . . . . . . . i . . . . . . !
10 3 . . . . . . . . . . . 0'- 1 0 o 1 0 e 0 3 10 - 3 10 -2 1 1 10 1 1 Elongational Rate [ s - t ]
Figure 13: Steady state elongational viscosity for LD at 160~ (experimental
data and fit using ( .... ) l inear form and (-----) exponential form of Y).
185
r---i
t'-J t--=J ~ K,~ 4~a"
>'"- 1 s ~ ~ 8 ~ ~'" 4-J -,-4 ~.gx me- '~
o 2 .1 m
-F,4 >~.~mL 10 4 I ~u=L~-u~ ""'""-. o~
L L ~ 4/ ~ "~ 10 2 [ . . . . . . . . . . . . . . . . . ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . r u~ 1 O- 3 1 O- 2 1 O- 1 10 ~ 101 10 2 10 3
Shear" or Elong. Rate [ s - l ]
Figure 14: Steady s tate functions for LLD at 160~ (experimental d a t a
(o): elongational viscosity, (Q): shear viscosity, (A): first normal stress
difference and fit (--) e = 0.105, (- -): e = 0.7)
It is thus impossible to find a single value that enables correct description of
both shear and elongational data. This may be understood considering the
efficiency of the Y function in describing the shortening of the junction life-
times. In the model, this shortening is all the more important since the stress
magnitude is higher. Since it is generally observed that materials which
exhibit the highest stress in elongation (elongation thickening) also show the
opposite trend in shear (shear thinning), the weighting by the function can hardly be achieved in any coherent way with a single value of e in different flow
geometries. Once more, as in the Johnson-Segalman equation, this sets an important limitation for the easy handling of such an equation.
5~Combinat ion of the two modifications-Experimental vah'dation of the Pban
Thien Tanner model
The original P h a n Thien Tanne r equat ion was w r i t t e n using
simultaneously both modifications: Gordon Schowalter derivative and segment
kinetics term. The segment kinetics term (exponential form) enables a more
186
realistic description of the steady elongational behaviour, giving rise to a
bounded viscosity in the long time range. The Gordon Schowalter derivative
has its major influence on the shear properties and additionally predicts a
second normal stress difference as in the case of the Johnson-Segalman model,
equation (53). Unfortunately it also introduces, conversely, the infringement of
the Lodge Meissner rule. Considering the previous remarks, one must keep in
mind the following important points. The smaller the value of a, the lower the
deviation to the Lodge-Meissner rule. However, in this case the flow behaviour of the model is primarily described by e and the simultaneous description of
shear and elongational data using a single value of this parameter is impossible. The smaller the value of e, the higher the value of the steady
elongational viscosity can be. However, in this case the shear flow behaviour of
the model is described by a so that the violation of the Lodge Meissner rule may
become important. The predictions are then very close to those of the Johnson-
Segalman model with the associated discrepancies such as spurious oscillations in transient shear. The determination of a couple of values (a,e) is
then bound to be a compromise obtained from a simultaneous fit of the elongational and shear functions (Table 6).
Table 6: Parameter a and e for the various materials
Material a
HD 0.50
LD 0.15
LLD 0.35 , . . , . . , . �9 . , , , .
E , , , , , = =
0.050
0.025
0.060
Figures 15 to 18 show the predictions of the model for LD at 160~ in steady
state and some transient flows in shear and uniaxial elongation.
187
~ ~ ~'--' I06 o o
~ i0 s
U
> ~ 1 0 4
E o~ ~Zo~b 10 3. ///:"
1 0 2 . ~ . . . . . . . . . . . . . . . . . . . . " ~ . . . . . . . I . . . . I . I - i l l . . . . . . . . I . . . . . . . .
I0 -3 I0 -2 I0 -I I0 ~ I0 x I(
Shear or Elong. Rate [s - I ] 0 2 10 3
Figure 15: Steady state functions for LD at 160~ (experimental and calculated). (o): elongational viscosity, (u): shear viscosity, (A): first normal stress
difference.
10 5 -
U~
o ~
. E I O 1 3 0 1 3 1 " 1 0 0 r l
1 0 3 . . . . . . . . j . . . . . . . . , . . . . . . . . , I0 - I 10 ~ 101 10 2
Time [s]
Figure 16: Transient shear viscosity for LD at 160~ (experimental and calculated). (o): 0.2 s -1, (u): 0.5 s -1, (A): 1 s -1.
188
o
L
p-g
E ( . 0
Z
L o,-g LL
m
10 6
105
1 0 4
103 10-1
O 0
0
10 ~ 101 10 2 Time [s]
F i g u r e 17: T r a n s i e n t p r imary s t ress coetticient for LD a t 160~ (exper imenta l a n d calculated). (o): 0.2 s -1, (~): 0.5 s -1, (A): 1 s -1.
1 0 6 -
105 -,-4
-a=4
> 10 4
o
W
103 10-2
. . . . . . ,,I �9 , . . . . . . i . . . . . . . . ,i i i l ..... i
10 -1 10 o 101 10 2 Time [ s ]
Figu re 18: T r a n s i e n t elongational viscosity for LD at 160~ (exper imenta l and calculated). (o): 0.05 s -1, (Q): 0.5 s -1, (A): 1 s -1, (0): 2 s -1.
189
5.4.Conclusion.
The ability of the Phan Thien Tanner equation and related models for the prediction of data in shear and elongation has been investigated. Attention has
been focused on special simplified cases of the original equation which enable
the understanding of the influence of each parameter. The use of a single parameter equation, removing the affine assumption of
the UCM model by replacement of the upper-convected derivative by the
Gordon-Schowalter derivative, is the case of the Johnson-Segalman model. This kind of modification does not significantly improve the elongational prediction towards the UCM equation. Moreover, in shear, though the improvement is obvious, discrepancies remain, especially concerning the non-
uniqueness of the slip parameter for tangential and normal stresses. The change of kinetics of the junctions also leads to a single parameter
equation in the form of the original PTT equation but using the upper-convected
derivative. Only the exponential form of the stress term gives a realistic
description of the steady elongational behaviour in the long time range. Though this is shown to improve the behaviour in elongation, this was conversely found to be contradictory with a better description of the trend in
simple shear because of opposite requirements on the value of the parameters.
Indeed, the parameter that controls the kinetics promotes both an elongation thickening behaviour together with a shear-thinning trend which is in
contradiction with experimental data on LD for example.
At least, using the complete Phan Thien Tanner equation, with non-affine
motion and modified kinetics enables a correct description of the data in shear
and in elongation. However, the parameters that can be determined for this
model are bound to be some compromise. This is necessary in order to
minimize the deviation to the Lodge-Meissner rule, due to the use of the
Gordon-Schowal ter derivative. This is also requ i red to give adequa te
description of both the shear and uniaxial elongational behaviour. Additional
undesirable phenomena in some flows have also been pointed out such as
oscillations in transient flows.
At least, it is worth noticing that the Phan Thien Tanner model is, in its
mathemat ical form, a non-separable equation. However, it has been pointed
out tha t , for some special forms of the re laxat ion spectrum, appa ren t
separability may be displayed [61].
190
6.CONCLUSION
Two different constitutive equations, namely the Wagner model and the
Phan Thien Tanner model, both based on network theories, have been
investigated as far as their response to simple shear flow and uniaxial
elongational flow is concerned. This work was primarily devoted to the
determination of representative sets of parameters, that enable a correct
description of the experimental data for three polyethylenes, to be used in
numerical calculation in complex flows. Additionally, advantages and
problems related to the use of these equations have been reviewed.
Both these models find their basis in network theories. The stress, as a
response to flow, is assumed to find its origin in the existence of a temporary
network of junctions that may be destroyed by both time and strain effects.
Though the physics of time effects might be complex, it is supposed to be
correctly described by a generalized Maxwell model. This enables the recovery
of a representative discrete time spectrum which can be easily calculated from
experiments in linear viscoelasticity.
Though the Wagner and Phan Thien Tanner equations seem to give
adequate description of the observed behaviour either in shear or in uniaxial
elongation, it is worth mentioning some peculiarities and key points that
should keep the attention of the user to avoid misleading conclusions.
These constitutive equations differ in their mathematical form: the Wagner
equation is an integral equation whereas the Phan Thien Tanner model is a differential one.
This induces important differences in the way they might be treated for
calculations in complex flows, since integrals will require particle tracking
whereas differential equations will not. These numerical t rea tments are
generally mutually exclusive since, in the general case, the problem of
correspondence between integral and differential forms is not solved. Attempts
at finding such correspondences may be found in various papers by Larson [62,
63] especially concerning the Wagner model and the Phan Thien Tanner
equation with upper-convected derivative.
On the other hand, integral forms are closer to the results of our knowledge
of molecular dynamics in entangled polymers and hybrid theories combining
191
simplified molecular models and temporary network equations are worth
thinking over.
The Wagner equation finds its theoretical basis in the derivation of the more
general K-BKZ equation. Unfortunately, it loses pa r t of its original
thermodynamic consistency since, for simplification purposes, only the Finger
strain measure is taken into account. Doing so, it is no more derivable from
any potential function and additionally it does not predict second normal stress
differences any more.
The equation leads to the definition of a time and strain-dependent memory
function which can be further factorized into a time-dependent part (the linear
memory function) and a strain-dependent damping function. Though on one
hand, there is some experimental evidence for this in limited time ranges, on
the other hand, a few experiments might question this strong hypothesis since,
for example, the damping function obtained from step shear rate data is found
to be different from that in step shear strain.
The construction of a single mathematical form of the damping function in
shear and uniaxial elongational flows requires the use of a generalized invariant, which includes the effect of both strain invariants I1 and I2 through
a proper combination of them. The simplest one being a linear combination can
be used in various equations for the damping function, including a limited
number of adjustable parameters.
Including these features, the Wagner model can give a proper description of
experiments in shear and in uniaxial elongation for increasing deformations.
When deformation is non-increasing, since the damping function reflects the
loss of junctions under the influence of strain, and since it should obviously be
an irreversible process, a functional damping term has to be introduced.
Nevertheless, this key point for any use in complex flow calculations has to be
improved.
In its general form, the Phan Thien Tanner equation includes two different
contributions of strain to the loss of network junctions, through the use of a
particular convected derivative which materializes some slip of the junctions
and through the use of stress-dependent rates of creation and destruction of
junctions. The use of the Gordon-Schowalter der ivat ive brings some
improvement in shear and a second normal stress is predicted, whereas the
192
influence of the kinetics through the trace of the stress tensor is much more
important in elongation.
Unfortunately, the use of the Gordon-Schowalter derivative brings large
discrepancies, especially as far as material objectivity is concerned. Indeed,
using it, the principal axes of strain and stress do not rotate together during
shear flows and this induces the violation of the Lodge Meissner rule.
Consequently, the slip parameter of the derivative is found to be different for
tangential and normal stress functions. This becomes evident in the limiting
case of the Johnson-Segalman model which, for representative parameters, is
found to be a good approximation of the Phan Thien Tanner model in shear.
Moreover, this kind of derivative induces spurious oscillations for transient
rheological functions.
One may try to avoid the problem by the use of the upper-convected
derivative, which ensures the coincidence of the principal axes of stress and
strain. But doing that, it appears that any kinetics based on the stress
amplitude is improper, since materials which exhibits thickening behaviour in
elongation are, to the contrary, shear-thinning. Consequently no unique
dependence can be expected for these two kinematics. The determination of a
single set of parameters in various flows in then bound to be a compromise.
Table 7 gives a summary of qualitative performances and problems
encountered for simple shear and uniaxial elongational flows, using the
Wagner and the Phan Thien Tanner equations or more simple models as
special cases of the former. Additional information may also be found in
papers by Tanner [46, 64]. All equations presented hereafter can be cast in the
form of a linear Maxwell model in the small strain limit and therefore are
suitable for the description of results of the linear viscoelasticity in the terminal
zone of polymer melts.
193
Table 7: Wagner and Phan Thien Tanner equations- Problems in simple shear and uniaxial extension.
Model Comments
UCM-Lodge
Eqs.(3) or (4)
Wagner
Eqs.(33 ) and
(25)
Johnson-
Segalman
Eq.(49)
Phan Thien
Tanner
with UCD
Phan Thien
Tanner
Eq.(39)
cD
~D
o
o p==r
bb o ~=~
Constant viscosity and first normal stress difference.
Second normal stress difference is zero.
No overshoot and linear limits in transient stress growth.
Linear relaxation modulus in step shear strain.
Unbounded transient viscosity at high rates.
Strain hardening at short time.
Linear results or infinite value for steady state viscosity.
Second normal stress difference is zero.
Inaccuracy on the generalized invariant parameter .
Exaggerates shear-thinning.
Lodge Meissner rule unsatisfied (2 slip parameters).
Oscillations in transient stress growth. Negative relaxation modulus in large step shear strain.
Unbounded transient viscosity at high rates.
c~ cD
o
c~
o
Linear results or infinite value for steady state viscosity.
Non-separable equation.
Linear form of the junction kinetics unsuitable.
Parameter of the junction kinetics differs from shear.
Non-separable equation.
Lodge Meissner rule unsatisfied.
Oscillations in transient stress growth.
Linear form of the junction kinetics unsuitable.
Compromise is necessary for the parameters.
194
R ~ ' ~ C ~ .
1. J.D.Ferry,~Viscoelastic Properties of Polymers ~, 3rd edition, JohnWiley &Sons, 1980.
2. R.B.Bird, R.C.Armstrong, O.Hassager,~Dynamics of Polymeric Liquids" Vol.1, Fluid Mechanics, 2nd edition, John Wiley & Sons, 1987.
3. R.B.Bird, C.F.Curtiss, R.C.Armstrong, O.Hassager,~Dynamics of Polymeric Liquids ~, Vol.2, Kinetic Theory, 2nd edition, John Wiley & Sons, 1987.
4. N.W.Tschoegl, ~The Phenomenological Theory of Linear Viscoelastic Behavior- An Introduction ~, Springer Verlag, 1989.
5. P.E.Rouse, A theory of linear viscoelastic properties of dilute solutions of
coiling polymers, J. Chem. Phys. 21 (1953), 1272-1280.
6. M.Doi, S.F.Edwards, Dynamics of concentrated polymer systems- Part 1: Brownian motion in the equilibrium state, J. Chem. Soc, Faraday Trans / / 74 (1978), 1789-1801.
7. M.Doi, S.F.Edwards, Dynamics of concentrated polymer systems- Part 2: Molecular motion under flow, J. Chem. Soc, Faraday Trans H 7..44 (1978), 1802-1817.
8. M.Doi, S.F.Edwards, Dynamics of concentrated polymer systems- Part 3: The constitutive equation, J. Chem. Soc, Faraday Trans H 74 (1978), 1818- 1832.
9. M.Doi, S.F.Edwards, Dynamics of concentrated polymer systems- Part 4: Rheological properties, J. Chem. Soc, Faraday Trans H 75 (1979), 38-54.
10. M.Doi, S.F.Edwards, ~The Theory of Polymer Dynamics", Clarendon Press, 1986.
11. M.Baumgaertel, H.H.Winter, Determination of discrete relaxation and retardation time spectra from dynamic mechanical data, Rheol. Acta 28 (1989), 511-519.
12. M.Baumgaertel, H.H.Winter, Interrelation between continuous and discrete relaxation time spectra, J. Non-Newt. Fluid Mech. 4.~4 (1992), 15- 36.
13. M.Baumgaertel, A.Schausberger, H.H.Winter, The relaxation of
polymers with linear flexible chains of uniform length, Rheol. Acta 29 (1990), 4(D-408.
195
14.
15.
16.
17.
18.
19. 20.
21.
22.
23.
40
25.
26.
27.
28.
J.Honerkamp, Ill-posed problems in rheology, Rheol. Acta 28 (1989), 363- 371.
J.Honerkamp, J.Weese, Determination of the relaxation spectrum by a
regularization method, Macromolecules 22 (1989), 4372-4377. J.Honerkamp, J.Weese, A non linear regularization method for the calculation of relaxation spectra, Rheol. Acta 32 (1993), 65-73. C.Carrot, J.Guillet, J.F.May, J.P.Puaux, Application of the Marquardt-
Levenberg procedure to the determination of discrete relaxation spectra, Makromol. Chem., Theory Simul. 1 (1992), 215-231.
A.S.Lodge, A network theory of flow birefringence and stress in concentrated polymer solutions, Trans Faraday Soc. 52 (1956), 120-130.
A.S.Lodge, "Elastic Liquids", Academic Press, 1964. B.Bernstein, E.A.Kearsley, L.J.Zapas, A study of stress relaxation with finite strain, Trans Soc. Rheol. 7 (1963), 391-410.
B.Bernstein, E.A.Kearsley, L.J.Zapas, Thermodynamics of perfect elastic
fluids, J.Research Nat. Bur. Stand., B: Mathematics and mathematical
physics 68B (1964), 103-113.
B.Bernstein, E.A.Kearsley, L.J.Zapas, Elastic stress strain relations in perfect elastic fluids, Trans Soc Rheol. 9 (1965), 27-39.
A.Kayes, An equation of state for non-newtonian fluids, Brit. J. Appl. Phys. 17 (1966), 803-806.
H.M.Laun, Description of the non-linear shear behaviour of a low density polyethylene melt by means of an experimentally determined strain dependent memory function, Rheol. Acta 1_/7 (1978), 1-15.
H.M.Laun, Prediction of elastic strains in polymer melts in shear and elongation, J. Rheol. 30 (1986), 459-501.
M.H.Wagner, Analysis of time-dependent non-linear stress-growth data
for shear and elongational flow of a low-density branched polyethylene melt, Rheol. Acta ~ (1976), 136-142.
R.G.Larson, Convection and diffusion of polymer network strands, J.of
Non-Newt. Fluid Mech. 13 (1983), 279-308.
R.G.Larson, K.Monroe, The BKZ as an alternative to the Wagner model for fitting shear and elongational behavior of a LDPE melt, Rheol. Acta 23 (1984), 10-13.
196
29.
30.
31.
32.
33.
40
35.
36.
37.
38.
39.
40.
41.
42.
M.H.Wagner, Elongational behaviour of polymer melts in constant
elongation rate,constant tensile stress, and constant tensile force experiments, Rheol. Acta 18 (1979), 681-692. M.H.Wagner, J.Meissner, Network disentanglement and time dependent flow behaviour of polymer melts, Makromol. Chem. 181 (1980), 1533-1550. A.C.Papanastasiou, L.E.Scriven, C.W.Macosko, An integral constitutive
equation for mixed flows: viscoelastic characterization, J. Rheol. 2_.7. (1983), 387-410.
M.H.Wagner, A.Demarmels, A constitutive analysis of extensional flows of polyisobutylene, J. Rheol. 34 (1990), 943-958.
M.H.Wagner, The nonlinear strain measure of polyisobutylene melt in general biaxial flow and its comparison to Doi-Edwards model, Rheol_Acta. 29 (1990), 594-603. P.I~Currie, Constitutive equations for polymer melts predicted by the Doi-
Edwards and Curtiss-Bird kinetic theory models, J. of Non-Newt. Fluid
Mech. 11 (1982), 53-68. R.G.Larson ~Constitutive equations for polymer melts and solutions ~,
Butterworths Publishers, 1988. P.R.Soskey, H.H.Winter, Large step shear strain experiments with parallel disk rotational rheometers, J. Rheol. 28 (1984), 625-645. M.H.Wagner, J.Schaeffer, Nonlinear strain measures for general biaxial extension of polymer melts, J. Rheol. 36 (1992), 1-26. M.H.Wagner, J.Schaeffer, Constitutive equations from Gaussian slip-link network theories in polymer melt rheology, Rheol. Acta 31 (1992), 22-31. M.H.Wagner, J.Schaeffer, Rubbers and polymer melts: Universal aspects of nonlinear stress-strain relations, J. Rheol. 37 (1993), 643-661. G.Marucci, B.de Cindio, The stress relaxation of molten PMMA at large deformations and its theoretical interpretation, Rheol. Acta 19 (1980), 68- 75. M.H.Wagner, S.E.Stephenson, The irreversibility assumption of network disentanglement in flowing polymer melts and its effect on elastic recoil
predictions, J. Rheol. 23 (1979), 489-504. M.H.Wagner, S.E.Stephenson, The spike strain test for polymeric liquids and its relevance for irreversible destruction of network connectivity by deformation, Rheol. Acta 18 (1979), 463-468.
197
48.
49.
43. R.G.Larson, Convected derivatives for differential constitutive equations,
J. of Non-Newt. Fluid Mech. 24 (1987), 331-342. 44. N.Phan Thien, R.I.Tanner, A new constitutive equation derived from
network theory, J. of Non-Newt. Fluid Mech. 2_ (1977), 353-365. 45. N.Phan Thien, A non-linear network viscoelastic model, J. Rheol.
(1978), 259-283.
46. R.I.Tanner, Some useful constitutive models with a kinematic slip hypothesis, J. of Non-Newt. Fluid Mech. 5 (1979), 103-112.
47. R.J.Gordon, W.R.Schowalter, Anisotropic fluid theory: a different
approach to the dumbbell theory of dilute polymer solutions, Trans Soc. Rheol. 16 (1972), 79-97. M.W.Johnson, D.Segalman, A model for viscoelastic fluid behavior which allows non-affine deformation, J. of Non-Newt. Fluid Mech. 2 (1977), 255- 270.
W.P.Cox, E.H.Merz, Correlation of dynamic and steady flow viscosities, J. Polym.Sci. 28 (1958), 619-622.
50. F.N.Cogswell, Converging flow of polymer melt in extrusion dies, Polym.
Eng. Sci., 12 (1972), 64-73. F.N.Cogswell, Measuring the extensional rheology of polymer melts, Trans. Soc. Rheol. 16 (1972), 383-403. M.H.Wagner, A constitutive analysis of uniaxial elongational flow data of a low density polyethylene melt, J. of Non-Newt. Fluid Mech. 4 (1978), 39- 55.
R.Fulchiron, V.Verney, G.Marin, Determination of the elongational behavior of polypropylene melts from transient shear experiments using Wagner's model, J. Non-Newt. Fluid Mech. 4.~ (1993), 49-61. M.Takahashi, T.Isaki, T.Takigawa, T.Masuda, Measurement of biaxial and uniaxial extensional flow of polymer melts at constant strain rates, J. Rheol. 37 (1993), 827-846.
Y.Einaga, K.Osaki, M.Kurata, Stress relaxation of polymer solutions under large strain, Polym.J. 2 (1971), 550-552. D.C.Venerus, C.M.Vrentas, J.S.Vrentas, Step strain deformations for
viscoelastic fluids: experiment, J. Rheol. 34 (1990), 657-682. A.J.Giacomin, R.S.Jeyaseelan, T.Samurkas, J.M.Dealy, Validity of separable BKZ model for large amplitude oscillatory shear, J. Rheol. 37
(1993), 811-826.
51.
52.
53.
54.
55.
56.
57.
198
58.
59.
00
61.
62.
63.
4.
A. Arsac, C. Carrot, J.Guillet, P.Revenu, Problems originating from the use of the Gordon-Schowalter derivative in the Johnson Segalman and related models in various shear flow situations, J. Non-Newt. Fluid Mech. 55 (1994), 21-36.
A.S.Lodge, J.Meissner, On the use of instantaneous strains, superposed on shear and elongational flows of polymeric liquids, to test Gaussian network hypothesis and to estimate the segment concentration and its variation during flow, Rheol. Acta 11 (1972), 351-352.
C.J.S.Petrie, Measures of deformation and convected derivatives, J. of Non-Newt.Fluid Mech. ~ (1979), 147-176. R.G.Larson, A critical comparison of constitutive equations for polymer melts, J. of Non-Newt. Fluid Mech. 23 (1987), 249-269.
R.G.Larson, Derivation of strain measures from strand convection models for polymer melts, J. Non-Newt. Fluid Mech. 17 (1985), 91-110.
S.A.Khan, R.G.Larson, Comparison of simple constitutive equations for
polymer melts in shear and biaxial and uniaxial extensions, J. Rheol. 31 (1987), 207-234.
R.I.Tanner,'Engineering Rheology ~, Clarendon Press, 1985.
Rheology for Polymer Melt Processing J-M. Piau and J-F. Agassant (editors) �9 1996 Elsevier Science B.V. All rights reserved. 199
Mathematical Analysis of Differential Models for Viscoelastic Fluids
J. Baranger a, C. Guillop6 b. and J.-C. Saut c
aLaboratoire d'Analyse Numfrique, Universit6 Claude Bernard and CNRS, 43 Boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France
bMathfmatiques, UFR de Sciences et Technologie, Universit6 Paris X I I - Val de Marne, 61 avenue du Gfnfral de Gaulle, 94010 Crfteil Cedex, France
C Analyse Num~rique et Equations aux D~rivfes Partielles, Universit6 Paris-Sud and CNRS, Bs 425, 91405 Orsay Cedex, France
1. I N T R O D U C T I O N . T H E MODELS
The complexity of viscoelastic flows requires a multidisciplinary approach including modelling, computational and mathematical aspects. In this chapter we will restrict our- selves to the latter and briefly review the state of the art on the most basic mathematical questions that can be raised on differential models of viscoelastic fluids. We want to em- phasize the intimate connections that exist between the theoretical issues discussed here and the modelling of complex polymer flows (see Part III) and their numerical simulations (see Chapter II.3).
As a matter of fact we do think that a better understanding of the mathematical properties of the models for viscoelastic fluid flows is fundamental in order to select a "good" constitutive equation, and to develop and implement numerical codes of practical u s e .
This chapter will be organized as follows. After a brief review of the classical differential models we will emphasize two important features of "Maxwell type models" (i.e., models without Newtonian viscosity), namely the instability to short waves and a "transonic" change of type in steady flows.
Then we review the existence results of solutions known for steady flows and for unsteady flows. Afterwards we discuss the important topic of stability of flows. This issue is still wide open, in particular because of the presence of memory effects for viscoelastic fluids, contrary to the case of Newtonian fluids.
Finally we conclude by presenting a few numerical schemes appropriate for simulating viscoelastic fluid flows, and we give some error estimates related to these schemes.
For simplicity and because they are widely used in the numerical simulations, we will restrict ourselves to the class of differential models. Actually they display (at least qu~li-
*And Analyse Num~rique et Equations aux D~riv~es Partielles, CNRS and Universit~ Paris-Sud.
200
tatively) most of the striking phenomena observed in viscoelastic flows. These models obey the constitutive equations
n
r = r ~ + rP, 7-" = 2r/,D, rP = ~ r i , i = 1
- ~ a T - i r i + A i ~ + [3i(D, 7-i) = 2r/iD , 1 < i < n,
(1)
where 7" is the (symmetric) extra-stress tensor, v is the velocity field, and D = D[v] =
1 ( Vv + Vv T) is the rate of deformation tensor. The coefficients r/j(> 0), 1 < i < n, and 2 - -
ris(> 0) are viscosities; A/(> 0), 1 < i < n, are relaxation times. The tensor 7" s corresponds to a Newton/an contribution or to a fast relaxation mode. It plays a fundamental r61e in the mathematical and the numerical analysis; models with r/~ = 0 will be called "of Maxwell type", those with r/, > 0 will be called "of Jeffreys type".
In the 7-i equation in (1) the symbol ~ - represents an objective derivative of a sym-
metric tensor, and is given by
9 ~ r 0 = ( ~ . + ( v - V ) ) 7 " + r W - WT- - a ( D T + 7-D), (2)
Dt
1 where a, - 1 < a < 1, is a given real parameter, and W = ~(XTv - V v T) is the vorticity
tensor. Finally ~ i (D , vi) is a tensor-valued smooth function, which is at least quadratic in its two arguments in a neighborhood of 7- = 0, and submit ted to restrictions due to objectivity.
Most differential models of viscoelastic fluids reduce to (1), with an appropriate choice of the funct ions/3 i. Here are some examples, where for simplicity we only consider the case of o n e relaxation time, i.e. n = 1,/~11 - / 3 , )~1 ~- )~, and r h -- r/p.
1. 13 _= 0 corresponds to a version of Oldroyd models. When in addition r/~ = 0, the particular values a = - 1 , 0, and 1 correspond respectively to the lower-convected, corotational (or Jaumann), and upper-convected Maxwell models.
2. ~ (D, 7-) = 2Tr (7-D)a(Tr T)(T + I) and a = 1: Larson's model. Here a(TrT-) is a scalar function of Tr 7-.
3. /3(D,T) = a rTr(7-) , with a constant: this is a version of the Phan-Thien and Tanner model. Larson generalized it to /3(D, v) = a7 "2 + ~7-, where a and ~ are scalar functions of Tr 7" and v. Note that the most general version of the Phan- Thien and Tanner model is when /3(D, 7 - )= v ( - 1 + exp(aTr 7-)) with a positive constant.
4. f l(D, v) = a 7 "2, cr constant" Giesekus' model. It is a part icular case of 3.
5. ~ ( D , r ) = ~o(Tr r ) D - p l ( r D + D r ) + vlTr ( r D ) I + #2D 2 + v2Tr (D2)I, where #0, #1, Vl, #2, v2 are some constants: S-constant Oldroyd model.
201
6. White--Metzner's models correspond to/3 = O, A and % being some given functions n - - 1
of the second invariant II = I I D = ~Wr (D2). Typically, r/p = 70(1 + (AoII)2"} "~'~ ,
n > 0, which corresponds to Carreau's law.
Of course this list is not exhaustive. (See other models in [1,2].) Also models with internal variables (order parameters) as those of [31 can be put in a similar (though more complicated) framework. In particular, there are additional constitutive equations of differential type for the order parameters [4].
Equations (1) are to be solved together with the equations of conservation of momentum and mass:
0v + (v. v),,) + vp
div v = 0,
= div r + f, (3)
and appropriate initial and boundary conditions. Compressibility effects cannot always be neglected for polymer flows (see e.g. [5]), equations (3) could be replaced by the corresponding equations for slightly compressible fluids. (See section 3.3 below.)
Solving the previous set of equations, especially with realistic boundary conditions, is a formidable task and a lot of issues are still unanswered. This is not surprising because of the complexity of the equations, and because of their recent derivation, around 1950 for the first nonlinear models, the Oldroyd models. On the other hand, the mathematical theory for the Euler and the Navier-Stokes equations for incompressible Newtonian fluids is still not complete though these equations were derived in 1755 and 1821 respectively!
2. M A X W E L L T Y P E M O D E L S : LOSS OF E V O L U T I O N A N D C H A N G E OF T Y P E
Maxwell type models (r/$ = 0) display two striking phenomena which are not present in Jeffreys type ones (r/$ > 0), and which will be described now.
2.1. Loss of evolut ion: H a d a m a r d instabi l i t ies or instabil i t ies to shor t waves This section concerns the Cauchy problem or initial value problem, where initial data
at time t = 0 are given. It was noticed by Rutkevitch [6,7], and systematized by Joseph et el. [8], Joseph and Saut [9], and Dupret and Marchal [10] that Maxwell type models can present Hadamard instabilities, that is, instabilities to short waves. (See [11] for a recent discussion of more general models.) Then, the Cauchy problem is not well-posed in any good class but analytic. Highly oscillatory initial data will grow exponentially in space at any prescribed time. An ill-posed problem leads to catastrophic instabilities in numerical simulations. For example, even if one initiates the solution in a "stable" region, one could get arbitrarily close to an "unstable" one.
For instance, the Maxwell model
202
"~a "/" A---~-- + r = 2r/D,
p( v (4) + v > ) = div v, ,
div v - 0,
is ill-posed whenever
1 - a h m ~ , - 1 +_.._.~a A~t, > .~, (5) 2 2
where hm~ (resp. hmtn) is the largest (resp. smallest) eigenvalue of r . Inequality (5) is actually equivalent to the fact that the (second order in t and x) equation for the vorticity is ill-posed.
It can be shown [8] that Hadamard instabilities are possible for admissible motions if a is in the interval ( -1 ,1 ) , e.g., in extensional flows. On the other hand, restrictions on the eigenvalues of r prevent Hadamard instabilities for a = +1. This is immediately seen from the integral forms of (4)-(5) for the upper- and lower-convected Maxwen models, which imply constraints on the eigenvalues of the Cauchy-Green tensors. (See, for instance,
[121.)
2 .2 . C h a n g e o f t y p e in s t e a d y flows" a t r a n s o n i c p h e n o m e n o n in v i s c o e l a s t i c
f lu ids The partial differential equations system for steady flows of Maxwell type (i.e., with
o _ 0) is of composite type, neither elhptic, nor hyperbolic. This is not surprising, the same being true for instance for the stationary system of ideal incompressible fluids. The new feature, discovered in [8], is that some change of type may occur. In fact an easy but tedious calculation shows that three types of characteristics are present:
- complex characteristics (elhptic part) associated with incompressibility; - real characteristics (hyperbolic part) associated with the propagation of information
along streamlines (double in 213, of multiplicity 4 in 3D); - characteristics which change type: complez if and only if the equation for the vorticity
is elliptic, real if it is hyperbolic. The change of type (analogous to the well known situation in gas dynamics) occurs
t - - - -
when the modulus Iv(x)l of the velocity exceeds ~/-~, which is the speed of propagation v r - -
o f shear waves in the fluid at rest. In other words, if one introduces a viscoelastic Mach number M = Re We (see Section 3 below for the definitions of Re and We), the flow goes from sub- to supercritical as M crosses 1. Then the vorticity equation changes from elliptic to hyperbolic, and there are waves of vorticity. This, of course, implies a qualitative change in the nature of the flow, which is supported by some experiments (e.g., [13]). It is interesting to note in this context that the experiments of Metzner et hi. [14] can be interpreted as suggesting a discontinuity in the vorticity. This change of type leads to drastic changes in the boundary conditions. (See Section 3.) Very few (mathematical or numerical) results are known in the supercritical case. (See [15-17].)
Among models without Newtonian contribution, Maxwell-like models possess the nice feature that the change of type is associated with a change of type in the vorticity. A
203
general class of differential models sharing this property was exhibited in [8]. This paper (and [9,18])also contains a classification of classical flows (Couette, Poiseuille, extensional, . . .) for various Maxwell models with regard to type. To illustrate the generality of change of type in elastic fluids (i.e., fluids whithout Newtonian contribution) we present here a brief analysis of the linearization around a uniform flow v = (U, 0, 0), using the Coleman- Noll theory of fading memory [19]. Stemming from the rather general concept of a simple fluid (Noll), this theory provides a systematic way to derive constitutive laws for special flows (e.g., perturbations of rigid motions). The idea is to choose a ]L2-weighted space for the history of deformations (see [20] for other choices of function spaces), and to use the Riesz theorem and isotropy to express the derivative of the stress at a given motion.
One gets in this fashion constitutive laws of the type
~0 ~ r = x(s)[Ct(t - s) - I]ds,
where x is a scalar kernel (the relaxation kernel) satisfying ~/h 2 ~. L2(O, cx~), Ct is the right relative Cauchy-Green tensor, and h is the weight function associated to the space of histories of deformations. For example, the upper-convected Maxwell model, where
77 exp(-s /A). a = 1, corresponds to x(s) -
It can then be shown that the vorticity ~ of the linearized flow around the uniform flow v = (U, 0, 0) satisfies the equation
02'~ x(0)A~ = lower order terms. pU2 b-~z 2 - (6)
0 2 0 2 Define the differential operator in the plane perpendicular to the flow by A• = ~ + az----~,
the Mach number M = U/c, and the speed of shear waves c = ~x(0)/p. Then equation (6) reads
02( - A• - lower order terms, (M 2 - 1) ~'x2
which shows that the vorticity changes type from elliptic to hyperbolic when M crosses 1.
For the upper-convected Maxwell model, the full equations for ( reads
02~ (M 2 - 1)~-7x 2 - A • +
M 0~ --0~
cA Ox
with c : ~/~o" (See [21].) u i
A change of type analysis for more complicated models is performed in [22].
3. S T E A D Y F L O W S
First equations (1)-(3) are written in a nondimensional form (see [23] for the details),
204
Ov Re (-~- + ( v - V ) v ) - (1 - e)Av + Vp = d iv 1" + f,
Or We(-~- + (v-V)1" + f l(Vv, r ) ) + v = 2eD,
d ivv = 0.
(7)
The parameters in these equations are the Reynolds number Re = pUL/r/(U and L are a typical velocity and a typical length of the flow, and 7/= r/s + r/p is the total viscosity of the liquid), the Weissenberg number We = AU/L, and the retardation parameter e = r/p/r/. Obviously, 0 < e < 1; e = 1 corresponds to Maxwell-type fluids, and 0 < e < 1 corresponds to Jeffreys-type fluids. Observe the change of notation in equation (7), where /3(Vv, r ) denotes now all the nonlinear terms in Vv and r other than the term (v . V ) r . f denotes some given body forces.
To start with, we consider steady flows of Maxwell-type fluids in a bounded smooth domain 9t of ]R N, N = 2, 3, and with simple boundary conditions, namely the system
Re (v- V)v + Vp = div r + f,
div v = 0, (s)
We( (v - V)7" + ~ (Vv , 1")) + r = 2D,
Vl0a = v0, v �9 nloa = 0,
where n denotes the outward unit normal vector to the boundary Oft of ~, and where v0 is a prescribed velocity field satisfying div v0 = 0 in 9t and v0 �9 nl0a = 0.
The following result is due to Renardy [24].
Theorem 3.1 (Ex i s t ence of slow flows) Let Ilfll~ a~d Ilvoll.~/~(oa) b~ sufficiently small. Then there exist v E I-I3(a), 1" E l-I2(f~),
and p E H2(f~) solution of system (8), unique among all small solutions. If moreover f E I-Ik(ft), Vo E l-Ik+X/2(Of~), for some integer number k > 2, then v E
,- e p e
Above and throughout this chapter [[-ilk, for k integer, will stand for the norm in the Sobolev space Hk(9/) (space of real functions defined in ~) or l-Ik(Ft) (space of vector or tensor valued functions defined in ~). The idea of the proof of Theorem 3.1 is to use an iterative scheme which alternates between a perturbed Stokes system (corresponding to the elliptic part of the system (8)) and a hyperbolic equation whose characteristics are the streamlines.
This result has been extended in several ways.
3.1. W h i t e - - M e t z n e r m o d e l s Hakim [25] has shown that Theorem 3.1 is also true for White-Metzner models, where
r = ~.s + rv satisfies 1 "s = 2r/~D, r/~ _> 0 (constant),
~a q'p 7"P "4- "~II ~ --= 2r/IID,
205
1 provided that )~II and r/u are smooth functions of II = ~Tr (D 2) such that )~II > 0 and
r/II > 0.
3.2. Weak ly elast ic fluids Let ~ be a steady solution of the Navier-Stokes equations (with prescribed body forces
and zero boundary velocity). Note that ~ is not assumed to be "small". Then, there exists a steady solution (vc, re, pc) of any Jeffreys model with a sufficiently small Weissenberg number We, and with a sufficiently small retardation parameter e, such that (vc, re) is close to (~r O) and e close 0. (See [26].)
3.3. Sl ight ly compress ib le fluids In the case of sligthly compressible fluids, and with the hypotheses that the flow stays
at low pressure and isothermal, system (7) is replaced by
0v Re ( ~ - + (v . V)v) - (1 - e)(Av + Vdiv v) + Vp = div 7 + f,
+ (v . + (Vv, + = 2 o, (9)
+Iv + aivv- 0,
where fl > 0 is a large parameter characterizing the (slight) compressibility. In the case of steady flows of slightly compressible fluids, R. Talhouk [27,28] considers a weak version of (9), where the conservation of mass equation is replaced by a conservative steady equation,
r I ( (v , v )p) + div, , = 0,
and II denotes the projection on zero mean value functions, defined by H(g) = g - (fn9 dz)/lfll for a scalar function g. He shows that the corresponding system admits a unique steady regular solution (v~, r~,pz) in both cases, Maxwell (c = 1) and Jeffreys (0 < c < 1), provided that the parameter/3 is large enough. Moreover, for fixed ~, 0 <
< 1, the slightly compressible solution converges to the incompressible one (satisfying the steady equations associated to (7)) when/3 goes to oo.
The proof of the existence results is based on the study of a linearized system and on the application of the Schauder fixed point theorem. The convergence to the steady incompressible limit is obtained by proving that the solution is bounded independently of /3 large. (See [27,281.)
3.4. P r o b l e m s with inflow b o u n d a r i e s It is well known that, for the Navier-Stokes equations, the prescription of the velocity
field or of the traction on the boundary leads to a well-posed problem. On the other hand, viscoelastic fluids have memory: the flow inside the domain depends on the defor- mations that the fluid has experienced before it entered the domain, and one needs to specify conditions at the inflow boundary. For integral models, an infinite number of such conditions are required. For differential models only a finite number of conditions are necessary (more and more as the number of relaxation times increases, ...). The number
206
of conditions depends on the model--Maxwell- or Jeffreys -type--and on the flow---sub- or supercritical. Determining the nature of the boundary conditions is by no way a trivial matter and no complete answer is known so far.
We treat briefly two different approaches.
3 .4 .1 . P e r t u r b a t i o n s o f a u n i f o r m flow. The first example, due to Renardy [17,29], deals with a special nonlinear situation,
namely that of a small perturbation of a uniform flow v = (U, 0, 0) transverse to a strip, as shown on Figure 1.
v=(U, O, O)
x=O x=1
X
Figure 1" Uniform flow in the strip {0 < x < 1}
The problem consists in finding the appropriate conditions to prescribe for the extra- stress tensor at the inflow boundary {z = 0}, so that the steady problem is well-posed. The method of analysis is a variant of the algorithm leading to Theorem 3.1.
For Jeffreys models, where r/$ > 0, all the components of the elastic part r p of the extra-stress can be prescribed at {z = 0}.
For Maxwell models, where r/$ = 0, we need to distinguish the sub- and the supercritical
cases. In the subcritical case (i.e., U < v/rl/(p,~)) and in two space dimensions, one can prescribe the diagonal components ~rP and 7 p, whereas in three space dimensions a correct choice of boundary conditions for rP is not simple. A possible choice of four boundary conditions is a nonlocal one (in terms of the Fourier components of rP--see [29]). An alternative approach leading to first order differential boundary conditions at the inflow boundary is described in [301 .
For Maxwell models in the supercritical case (i.e., U > x/rl/(P~)), the previous choice of boundary conditions leads to an ill-posed problem (as does the Dirichlet boundary
v
condition for a hyperbolic equation), as shown in [17]. In addition to the normal velocities at both boundaries (inflow and outflow) and to the previous inflow conditions on the stresses, one can prescribe the vorticity and its normal derivative in two space dimensions, or the second and third components of the vorticity and their normal derivatives in three
207
space dimensions. A discussion of the traction boundary conditions--where the total normal stress is
prescribed on the inflow and outflow boundaries--for Jeffreys-type fluids is given in [31], and for Maxwell-type fluids in [32].
a.4.2. Absorb ing b o u n d a r y condit ions for viscoelastic fluids We briefly present here some results taken from Tajchman's doctoral thesis [33]. Interest
is focused on models described by (see equations (7) and (9))
Re ( - ~ + (v -V)v) - ( 1 - e ) A v + Vp = div r + f,
Or We (-~- + (v. V)r +/3(Vv, r)) + r = 2eD, (10)
0p d i v v = 0 , or - ~ + / 3 d i v v = 0 ,
according to whether we consider the incompressible case (/3 = oc) or the slightly com- pressible case (/3 > 0).
The geometry of the flow is supposed "infinite" (i.e. very large in one direction), such as in the case of a flow around an obstacle, or in a long die. The flow "at infinity" is assumed to be known (uniform, Poiseuille flow, ...). For computational purposes, one introduces artificial boundaries, at a finite, hopefully not too large, distance. The problem is to define which conditions to impose on the artificial boundaries in order to obtain a solution of the truncated problem, which is as close as possible to the solution of the original problem.
Such considerations where first carried out by Engquist and Majda [34] for wave equa- tions (linear hyperbolic equations), and developed by many authors later on. In particular Halpern [35] considered the case of parabolic perturbations of hyperbolic systems.
From physical considerations (plane waves travelling through the fluid), one gets bound- ary conditions which make the artificial boundaries transparent to the waves leaving the computational domain and which absorb the waves entering the domain (other than those generated by the solution at infinity).
We suppose that the artificial boundaries are far enough away in order to justify the linearizations around the uniform solution at infinity. The linearized problem reads
0v Re (-~- + (v~ . V)v + (v. V)v~) - ( 1 - e)Av + Vp = div r,
vo~ (11)
Op d i v v = 0 , or ~ - ~ + / 3 d i v v = 0 .
Looking for plane wave solutions amounts to testing nontrivial solutions of the type
N
U(s. J)exp[ir + ~(s, w/ )xl + ~ ~bkxk)], k = 2
where ~b = ( ~ 2 , ' " , ~bN), and where X 1 is the normal direction to the artificial boundary, outwards pointing.
208
The sign of the real part of these solutions determines the directions of propagation of the corresponding wave. For each wave entering the domain of computation one imposes a boundary condition which eliminates it:
= 0 ,
where V is the corresponding left eigenvector, and ~ denotes the Laplace transform of v with respect to t and the Fourier transform with respect to x' = (x2,. . . ,XN).
These conditions are not local (they are integral relations difficult to incorporate in a numerical code). By perturbation techniques one gets local approximations which are par- tial differential equations on the boundary. Higher order approximations can be obtained at the price of increasing difficulty in the computations.
To be of any practical use, the artificial boundary conditions have to be stable: in particular, they should not depend on rounding errors. The stability analysis is made on the linearized problem by computing the time evolution of some solution norms.
Finally one can compare the absorbing conditions which are obtained in different cases, e.g., the limit as e goes to 1 or as ~ goes to infinity. The results crucially depend on the subcritical or supercritical nature of the flow. We refer to [33] for a precise description.
3.5. T h e r e - e n t r a n t corner s ingula r i ty So far only domains of the flow with smooth boundaries have been considered. However,
re-entrant corners as in a sudden 4:1 contraction are well-known to give rise to numerical difficulties in the numerical simulation of viscoelastic flows. (See e.g., Chapter I1.3.)
The analysis of the corner singularity is delicate. We refer to the recent works of Hinch [36] and Renardy [37,38], who have contructed a matched asymptotic expansion for the steady solution to a Maxwell fluid flow near the corner.
4. U N S T E A D Y F L O W S
Existence results for unsteady flows are important in two ways. First, the (local) well-posedness of the initial and boundary value problem proves the adequacy of a given model to describe (at least locally) dynamical situations. Second, global well-posedness is preliminary to any nonlinear stability study.
4.1. F ixed g e o m e t r y We first consider Jeffreys-type models, namely system (7) with e < 1, which is comple-
mented with the Dirichlet boundary condition
vl0~ = 0, (12)
and initial values
V [ t = 0 --" V0~ 7"It=0 - " 7"0-
Note that here v0 and 7"0 are not assumed to be "small". In what follows we shall denote the $obolev spaces of real functions, vector or tensor
valued functions previously defined, by H k = Hk(~) or l-I k = l-Ik(~t). To start with we state a local existence theorem [23].
209
T h e o r e m 4.1 (Local existence of u n s t e a d y flows) Let Ft be a bounded domain of R N, N = 2, 3 with C, 3 boundary. Le t f e L~oc(R+; tta),
f' E L~oc(]R+; I-I-a), v0 E I-I 2 R tt~, with div v0 = 0, and ro E I-I 2. Th~. t h ~ ~i~t T" > 0 ~.d ~ ..iqu~ ~ol.tio. (v. r.p) of ~u~t~m (~), (1~). ~.d (lS),
which satisfies v e L2(0, T*; I-I 3) nC([0, T*); I-I 2 N IH~),
v' e L2(0, T*; I-I~)N C([0, T*);L2),
p e L2(0, T*; H2), r e C([0, T*); H2).
A similar result has been proven by Hakim [39] for a class of White-Metzner models (still with e < 1) under suitable assumptions on the constitutive functions $(II) and y(II). These assumptions are satisfied in particular by the Carreau and the Gaidos-Darby laws [40].
R e m a r k 4.1 The results of Theorem 4.1 do not depend on the precise nature of the term fl(Vv, I") and thus are model independent. In particular, these results are still valid for differential models with internal variables [4].
R e m a r k 4.2 If more regularity is assumed on the data, then more regularity of the solution is obtained provided an additional compatibility condition on the initial values is imposed.
We now turn to local existence of solutions for Maxwell-type models. The situation is much trickier here since these models can display Hadamard instabilities (see Section 2.1), and no general results seem to be known so far. One has, in any case, to restrict initial data to "Hadamard stable" ones. A possible way to overcome the difficulty is to consider models satisfying an ellipticity condition, which will imply well-posedness. This approach was followed by Renardy [41], whose results are briefly described below.
The extra-stress tensor 7" = (rij) is assumed to satisfy an equation of the form
Ovk + v . = +
where (due to frame indifference)
1 A o k t ( v ) = -~(,Sikr,3 - ~Silrk3 -- rik~Sl3 + ri,6kj) + B,jkt(~');
the tensor (Bijkt) is symmetric in k and l (and of course in i and j) , and satisfies Bi3kz = Bklij. We set
1 C~jkz = Bokl + ~(*ikrU -- *~rkj -- r~k&j -- T~t6kj),
so that Aip:t = 7it6kj + Cijm, and C i j k l "-- Cklij. The crucial hypothesis is the strong ellipticity condition,
o~,(~-)r162 >_ ,~(~-)1<1~1,71 ~ v4,,7 ~ R ~. (15)
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where ~;('r') > 0.
Note that , for Maxwell models with -1 _< a < 1, relation (15) is satisfied locally in time provided it is satisfied at t ime t = 0. For a = + l , relation (15) is equivalent to relation (5), which insures that the initial value problem is well-posed: this is a natural condition to impose on the stress. But, for a 5r + l , condition (15) reveals that the model is not always of evolution type, which means that Hadamard instabilities can occur. (See Section 2.1)
Under hypothesis (15) and under some smothness conditions on the (bounded) domain of the flow, and on the functions Aijkl and gij, Renardy [41] proves the local existence and the uniqueness of a I-I~ fl ][-I 4 solution of equations (3) and (14), provided that the initial data v0 and r0 are smooth, and satisfy a compatibility condition at time t = 0.
In a recent work [42], Renardy characterizes a set of inflow boundary conditions which leads to a locally well-posed initial boundary value problem for the two-dimensional flow of an upper-convected Maxwell fluid transverse to a domain bounded by parallel planes.
Another local in time existence result concerns the case where the domain of the flow is unbounded, but satisfies certain uniform regularity conditions [27]. This result extends the result of Theorem 4.1, but in a context where there is non-compactness.
Talhouk [27,43] also has studied the local existence of flows in a bounded two dimen- sional channel f~ = (0,1) x (0, L), for which the inflow v_ and the outflow v+ are given and satisfy the following: there exists an a > 0 such that v_ -n < - a on F_ = Off_ , and v+ �9 n > 0 on F+ = 0fl+. (See Figure 2.)
F
V
0 1 x
v + r
F +
Figure 2: The domain D, = (0, 1 ) • (0, L) with inflow and outflow boundaries, I'_ and F+
The flow is assumed to be periodic in the y- and the z-directions. Moreover the extra- stress components are given on the inflow boundary. Assuming that the data on the inflow boundary satis~- a compatibility condition. Talhouk shows the existence of a local in time solution in the bounded channel.
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Concerning global (in time) existence of solutions, the first general result is the fol- lowing [23], which is established for Jeffreys models with a sufficiently large Newtonian contribution to the extra-stress.
T h e o r e m 4.2 (Global exis tence of flows wi th small da ta ) Let fl be a bounded domain in 1~ N, N = 2,3, with a C 4 boundary. There exists a
parameter eo depending on f~, 0 < eo < 1, such that if 0 < e < eo and if v0 E I-I2N I-I~, with div v0 = 0, f E L~ I-IZ), f' E L~(R+; I-1-1) are small enough, then system (7), (12), and (13) admits a unique solution
v e c~(R+; H ~ n ~ ) n L~or ~ ) ,
v' e C~(l%; L ~) n L~or ~I~),
The proof of this result is based on an energy method: one shows that the quantity
Y(t) = (1 - e ) 2 - - - - -~~e 2 Jlv(t)ll~ + Re IJv'(t)flo 2 +
We(1 - e) 3
Re a e3
W e f]r(t)r[~ +--][r'(t)JJ~ s
satisfies the inequality
Y' + aoY <_ al (Y2 + y3 + y6) + a~, ao > O, a l > O,
where a2 > 0 is small when the data are small. The conclusion is reached when noticing that a function Y which satisfies the above differential inequality and has small initial value Y(0) stays bounded for all times.
A similar result for White-Metzner models is proven in [25] under the same hypotheses on the constitutive functions as for the local existence result mentioned earlier, plus the hypothesis A(z) _< M, for all x in IR+.
R e m a r k 4.3 The restriction 0 < e < e0 is due to the treatment of the boundary condi- tions in the linear coupled terms.
R e m a r k 4.4 No result such as Theorem 4.2 seems to be known for Maxwell models. We however have to mention the result [44], where the upper-convected Maxwell model in the whole space IR 3 is considered.
In the context of Theorem 4.2, one can prove that two solutions, which are sufficiently small at t = 0 and correspond to sufficiently small body forces, will get asymptotically exponentially close as t goes to oc. Using this stability result and classical arguments ([4.5,46]) one obtains the following result.
Coro l la ry 4.1 (Exis tence of small s table T - p e r i o d i c or s t e a d y flows) Let ~ and f~ be as in Theorem 4.2.
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(i) Let f E L~176 ]H1), f ' E L~176 ]H -1) be time periodic of period T > O, and small enough. Then there exists a T-periodic solution of system (7), (12), unique among the small T-periodic solutions.
(ii) I f moreover f is time independent, there exists a steady solution, which is unique among small steady solutions.
(ii O The aforementioned solutions are non-linearly stable. (See also Section 5.)
R e m a r k 4.5 Corollary 4.1 obviously implies the stability of the rest state (the zero solution) for e not to close to 1. Using a different approach, Renardy [42] has recently removed this restriction and showed the Liapunov stability of the rest state for all e's, 0 < e < 1. This fact is not known for Maxwell-type models (e = 1).
R e m a r k 4.6 A number of problems are still open. For example the existence of (small) periodic solutions for Maxwell models is unknown, as is that of arbitrary (not small) periodic solutions for Jeffreys models. For example too, nothing is known concerning the global existence of unsteady solutions for differential models in two or three space dimensions (say, weak solutions, singularities in finite time, . ..). See below, for some examples in one space dimension.
More specific results can be obtained in some one dimensional situations, which we describe now. Following [47], we consider shearing motions of an Oldroyd fluid, such as Couette or Poiseuille flows. The dimensionless equations are easily reduced to a system for the shear component of the velocity v(x,t), the shear stress r(x, t) , and a linear combination of normal stresses a(x, t), x E I, t >_ O,
Re vt - (1 - e)v** = r . - f ,
T s + =
For Couette flow, one has I = (0,1), and f = 0, while for Poiseuille flow, I = ( - 1 , 1 ) , and f = 1 is the constant pressure gradient in the flow direction. The boundary conditions are
v ( - 1 , t ) = v(1,t) = 0, t _> 0, for Poiseuille flows, or (17)
v(O,t) = 0, v(1,t) = 1, t _> 0, for Couette flows.
The parameter a will be assumed to satisD -1 < a < 1. The crucial observation is the following a priori bound on the stress components:
- + (1 - aZ)v2(x, t) ( is )
Using the bound (18), we can then prove the following result.
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Theorem 4.3 (Existence and uniqueness of one-dimensional global flows) Let 0 < e < 1. (i) Uniqueness. There exists at most one solution (v, a, T) of system (16)-(17) in the
space [L~(]R+; L 2) N L~or H1)] x [L~ x ]R+)] 2. (ii) Existence. Let v0, a0, TO E H ' (I) such that Vo satisfies the boundary conditions (17).
Then for any T > O, there exists a unique solution (v , a , r ) of system (16)-(17) in the space [C([0, T]; H ~) N L2([O,T];H2)] x [C([0, T); H~)] 2. Moreover v e Cb(R+; L 2) and the bound (18) holds true.
Another proof of this result is obtained by Malkus et al. [48].
R e m a r k 4.7 In the case e = 1 (Maxwell models), system (16)-(17) is not always of evolution type. (See section 2.1.) Indeed, Renardy et al. [49] have constructed initial data in the hyperbolic domain, with steep gradients, such that the velocity and the stress develop singularities in their first space derivatives in finite time. The idea is to reduce the system, by a clever change of variables, to a degenerate system of three nonlinear hyperbolic equations.
R e m a r k 4.8 The results of Theorem 4.3 depend crucially on the model (the Oldroyd model). It would be interesting to know what happens for one dimensional flows of general differential models with a Newtonian contribution.
Similar results can be obtained for Couette or Poiseuille flows of several fluids in parallel layers: these flows are important in particular in the modelling of coextrusion experiments. Le Meur [50] has studied the existence, uniqueness and nonlinear stability with respect to one dimensional perturbations of such flows. The behaviour of each fluid is governed by an Oldroyd model such as (16)-(17), where the nondimensional numbers Re and We are defined locally in each fluid. On the rigid top or bottom walls, the velocity is givenPzero on both walls for Poiseuille flow, and zero or one depending on the wall for Couette flow.
The interface conditions on the given interfaces are
~7"aim(-pI + 2(1 - e)D + r I = 0,
where [-] denotes the jump of a quantity across the interface, and "/'dim = ~U/L is defined in each fluid.
One can also show that all one dimensional time-dependent perturbations of a steady multifluid flow exist for all times, and stay boundedmas in the case of one fluid. Similar results can be obtained for axisymmetric Poiseuille flows of several fluids. A similar study is also made for plane Poiseuille or Couette flows of several fluids having a Phan-Thien- Tanner constitutive equation [50].
4.2. Free b o u n d a r y p rob lems In many practical situations the geometry of the domain occupied by the fluid is not
given. In particular, there are free boundaries such that interfaces between air and liquid, interfaces between several l iquids, . . .
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Mathematical results for these flows are not simple, even for Newtonian fluids. In [50,51], Le Meur has considered unsteady flows of an incompressible fluid of Jeffreys-type submitted to surface tension above a fixed rigid bottom. He proves that there exists a unique local solution with a free surface, which is close to a flow with a given flat interface.
5. S T A B I L I T Y I S S U E S
Instabilities are one of the main challenging problems in the mathematical theory of viscoelastic fluids. We shall only consider bounded geometries. Let us briefly review the goal and the main difficulties of this subject. One wants to explain (predict, avoid, . . .) the instabilities occurring in polymer processing. When polymer melts are extruded from a pipe, instabilities often occur at a critical value of the wall shear stress. They are known as spurt flows, shark skin defects, melt fracture, . . . They manifest themselves in a jump of the flow rate for a given pressure gradient, irregularities on the surface of the extrudate, pressure oscillations, chaotic behaviour, . . . (See [52-54], and Chapter III.4.) Their physical explanation is not well understood. Possible causes could be slip at the wall (interaction of the fluid with the wall of the pipe), or propagation of defects in the pipe, . . . Mathematical explanations could be change of type, or constitutive instabilities (non-monotone shear stress / shear rate curve), . . . In any case viscoelastic instabilities differ from the classical "hydrodynamic instabilities" which occur in Newtonian flows at high Reynolds numbers. Those instabilities happen at moderate Reynolds numbers, and are basically due to the elastic effects. (See a very good review in [55].) Note that the viscoelastic analogue of the classical hydrodynamic instabilities (B6nard and Taylor experiments) can in principle be studied, at least formally, by the methods of bifurcation theory. (See the recent work of Renardy and Renardy [56] on the B~nard problem, and of Avgousti and Beris [57] on the Taylor-Couette problem.)
From a fundamental point of view, none of the theoretical results necessary to justify rigorously bifurcation or stability studies have been fully established so far for the equa- tions governing viscoelastic flows (contrary to the Newtonian case). Such results concern, for instance, the relations between linear stability and the spectrum of the associated operator, between linear and nonlinear stability, or the reduction of the dynamics to a centre manifold.
In order to explain and clarify the mathematical issues we recall some classical facts.
5.1. Genera l i t ies The equations for a perturbation u of a steady solution u5 of an incompressible vis-
coelastic fluid can be written as an abstract equation in a Hilbert space X,
= A~ + f(~) , (19) u(o) = Uo,
where u C X is a vector whose components are the velocity and extra-stress components. (The pressure can be eliminated by projection onto a space of divergence free vectors.) A is the linearized operator around u~. f(u) the nonlinear part (at least quadratic in u), and u0 the perturbation at time t = 0. The boundary conditions are taken into account
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in the space X. The problem of the stability of the steady flow u, is expressed by the three following notions.
1. ($1) Nonlinear stability (or Liapunov stability). For any 51 > 0, there exists 52 > 0 such that for every u0 with Ii~011x < ~=, the corresponding solution u(t) of (19) exists for all t > 0, and satisfies II~(t)llx _< 5x, for ~11 t > 0.
If furthermore Ilu(t)llx goes to 0 as t goes to ~ , u, is said asymptotically stable.
2. ($2) Linear stability. For any v0 in X, the solution v(t) of the linear problem
~t =Av , v ( o ) = vo,
(20)
satisfies lim I Iv ( t ) l l x = o.
3. ($3) Spectral stability. The spectrum a(A) (i.e., the set of complex numbers ~ such that the operator )~1 - A: D(A) ~ X is not invertible, where D(A) is the domain of A) is contained in the left half plane { ~ < 0 }.
The strongest notion (and the most "physical" one) is (Sl). On the other hand (S3) is the easiest to check: it "suffices" to compute, numerically in general, the spectrum of the linear operator A. This is essentially the classical Orr-Sommerfeld approach. (See Section 5.5 for an example.)
Unfortunately, in general, there is no relation between these three notions. The link between ($2) and ($3) is in relation to a formula of the type
a(S(t)) = exp(ta(A)), t >_ O, (21)
where S(t) is the semi-group generated by A (i.e., the family of operators in X, which associate v0 to the solution v(t) of equation (20)). If relation (21) holds true, then it is clear tha t (S3) implies ($2).
In general, relation (21) is false: one only has exp(tcr(A)) C e(S(t)). The corresponding formula is true for the point spectrum (constituted of eigenvalues), and for the residual spectrum, but it is false for the continuous spectrum: there exist abstract operators A with empty spectrum, but such that the continuous spectrum of S(t) is the circle of radius exp(Trt); thus the solutions of equation (20) grow exponentially in time.
Other than for the classical case of the finite dimension, it is known that relation (21) is true for compact or analytic semi-groups. This covers the case of Newtonian flows, where the associated system of partial differential equations is parabolic. In this case, one also has the implication "($3) ==~ (S1)". (See [58] for instance.)
The main question concerning the implication "($3) ~ ($2)" is to know whether or not the pathological situations of the aforementioned abstract examples can occur in "concrete" situations, in particular in those occurring in the study of linear stability for viscoelastic flows. In fact, because of memory effects, the underlyingsystem is (at least)
216
partially hyperbolic, and we cannot hope that S(t) be compact or analytic (except for the one dimensional flows described in Section 5.3, where the hyperbolic part degenerates.)
A good situation to first at look is the one of linear hyperbolic equations or systems. Actually the implication "($3) ~ ($2)" has been proven in [59] for certain hyperbolic systems in one space variable. A more general (and simpler) proof has been given by Renardy [60]. On the other hand, Renardy [611 has constructed a simple example (namely the wave equation utt = u** + u,~ + eiYu,, with periodic boundary conditions), showing that the linear stability of hyperbolic partial differential equations in two or more space dimension is not necessarily determined by the spectrum of the operator.
We now turn to the case of viscoelastic fluids.
5.2. Linear s tabi l i ty There are very few results concerning the linear stability (with respect to two or three
dimensional perturbations) for viscoelastic flows, essentially by lack of a general result "(S3) ~ (S2)'.
In [62] Renardy proves the linear stability of Couette flow of an upper-convected Maxwell fluid under the assumption of creeping flow. This extends a result of Gorodtsov and Leonov [63], who showed that the eigenvalues have negative real parts (i.e., condi- tion ($3) holds). That result, however, does not allow any claim of stability for non-zero Reynolds number, however small. Also it uses in a crucial way the specific form of the upper-convected derivative in the upper-convected Maxwell model, and does not general- ize so far to other Maxwell-type models.
In another paper M. Renardy [64] considers the plane Couette flow of an Oldroyd-B fluid (i.e. a Jeffreys model associated to the upper-convected derivative). He proves linear stability under the (theoretical) assumption that the eigenvalues have negative real parts, condition ($3) implying condition ($2) in this context. Note that the Couette flow is always stable in the Newtonian case [65], and numerical calculations (on the Orr- Sommerfeld side) suggest that this may be so for the Maxwell and Oldroyd-B models as well [66]. Again the structure of the model and of the flow is made use of in a crucial way (in fact one uses a factorization of a differential operator which was discovered by Gorodtsov and Leonov [63]). This result of stability has been extended by Renardy [61] to any parallel shear flow with a strictly monotone profile--thus excluding Poiseuille flows-- for an arbitrary fluid of Jeffreys-type.
Finally we comment briefly on weakly elastic fluids. (See [26] and section 3.2.) We assume that the given Newtonian solution ~r satisfies ]1~r < (ClRe)-X, where Cl is some constant depending only on the domain of the flow: this condition ensures that ~ is asymptotically Liapunov stable. (See e.g., [67].) Then, the viscoelastic solution (v,, ~',) close to (~, 0) is linearly (asymptotically) stable for e > 0 small enough.
5.3 . One dimensional flows It turns out that one dimensional perturbations to a viscometric flow lead to a system
for which the implication "($3) ~ ($2)" is true. For plane laminar flows, such as Couette or Poiseuille flows (see Figure 3), the velocity
and the stress depend only on the first coordinate x E I, where I is the interval (0,1) for
217
Couette, and ( - 1 , 1) for Poiseuille, and have the form
( ) o ~(x, t) = v ( x , t ) = v(~, t) ' ~-(~,t) , ( ~ , t ) ) ,,/(~, t) "
- , / / / / / / 0
/ / / +1 ~
/ / / - 1
X / / /
,,,
_ j J / / /
Couet te f low Poiseuille flow
Figure 3: Plane Couette and Poiseuille flows
The system for a Jeffreys-type fluid, which is to be solved in the interval I, reads as follows,
Rev, - (1 - r = r . - f, (7"
at + ~ e - ( I + a)Tv~: + #l(V~, T) = 0, ,~ (z2)
"7, + ~ e + (1 --a)rv:~ +/32(v~, 7-) = O,
r (@ee l + a 1 - a )) r ) 0, ~ ' + ~ - 7 - + ( 2 ~ - 2 " ~ + # ~ ( ~ ' =
with boundary conditions
v ( - 1 , t ) = v(1, t ) - 0, for Poiseuille flow, or (23)
v(0, t) = 0, v(1, t) = 1, for Couette flows.
The functions ~i, i - 1,2, 3 depend on the model. (See the description of different models in Section 1). The Oldroyd model corresponds to ~qi = 0, i = 1,2,3, from which we
l + a - - ~ ' - ~ O". deduce system (16) for (v, c~, r) where c~ = -7-7 For the Giesekus model, one has
#1 = ~ + ~ , #~ = ~ + ~ , ~ = (~ + z)~. Let (vs, a s ,%, r s ) be a steady solution to system (22)-(23) (e.g., a Poiseuille flow or a
Couet te steady flow) and consider the linearized system around this state:
218
R e vt - (1 - e ) v ~ - r . = 0 ,
a rOY, 4 Ov~ at + ~ee - ( 1 + a)(T,V~ + OX ) + ~-~0i~(-~X' %)Ui = 0,
i=1 7 Ova. 4 "- '0 Ov~
7t + ~ee + (1 - a ) ( r s v , + r--~z ) + ~.~ i~/2(-~z , r , )ui = 0, (24)
r (@ee 1 + a 1 - a a . ) ) v ~ _ ( l + a 1 - a a ) O v , T t + W e e - - + ( 2 % - 2 2 7 - 2 Ox
4 O V s
= o , i --1 ""
with homogeneous boundary conditions. Above 01, 02, 03 and 04 denote the partial derivatives with respect to the variables Ul =. v,, u2 = a, u3 = 7 and u4 = 7". Pro- ceeding as in [47], one can show that the linear operator A associated with system (24) generates an analytic semi-group if 0 < e < 1 (non-zero Newtonian contribution), so that the implication "($3) ~ ($2)" is true in this case. Note that we only consider one-dimensional perturbations, but that the constitutive law is arbitrary.
Following [47] we restrict now the study of stability to Oldroyd models (where/3~ = 0). It is easy to check that the steady Couette flow, solution of the steady equations corresponding to system (16)-(17), is given by
~k 2
v s ( x ) = x , a , = W e( l +k 2 ) ' r , = l + k 2' 0 < x < l ,
where k 2 = We2(1 -a2) .
Concerning the Liapunov (nonlinear) stability of the Couette flow under one dimen- sional perturbations, we have for instance the following result [47].
T h e o r e m 5.1 (Nonl inear s tabi l i ty to 1D p e r t u r b a t i o n s ) (i) Let e 6 (0, 8/9). Then the Couette flow is stable in H ~. (ii) Let e G (8/9, 1). Then there ezists some function k~(e) such that the Couette flow
is stable in H 1 for all k < kl (e).
This result is proven by making use of an energy method. We refer to [47] for a proof and for other related results, e.g., sufficient conditions on e and k for unconditional stability in L 2. These results have been extended by Le Meur [50] to the case of multifluid flows.
The situation for the plane Poiseuille flow for Oldroyd models is not as simple, as shown by the following result.
T h e o r e m 5.2 (Exis tence of s t eady Poiseui l le flows) (i) Let e 6 [0, 8/9). Then there exists a unique steady solution v~ = v~(x), which is very
smooth on (0, 1), actually in the class of C ~ functions having continuous derivatives at all orders.
5 0 Let e 6 (8/9,1). Then there exists a critical Weissenberg parameter k~ > 0 such that
(a) if k < k~, the conclusion of 5) holds; (b) if k > k~, there does not exist any C ~ solution, but a continuum of C o solutions,
which are C ~ except at a finite number of points.
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Some typical half profiles of Poiseuille flows are drawn on Figure 4.
o v(x) o 0 < ~ < 8/9
v(x) 8/9 < e < 1
E k > k c
Figure 4: Profiles on (0,1) of the shear velocity for Poiseuille flows
Results of stability at small Reynolds numbers for the nonregular Poiseuille flows de- scribed in Theorem 5.2 are obtained in [48], where it is shown that the steady Poiseuille flows for which the velocity takes its values in the increasing part of the S-shaped curve (see Figure 5) only--excluding a neighbourhood of the max and the minmare nonlinearly stable. Stability holds when the perturbation from a steady state satisfies the following: the total shear stress is small in I-I 1, the normal stresses are small in L a, and bounded pointwise by some large constant. Moreover, if Re is small enough, every unsteady Poiseuille flow converges, as t --~ oo, to some steady state, possibly nonregular, possibly unstable.
The proofs of these results rely on the geometric study of the approximate dynamical system obtained at zero Reynolds number. This system has been thoroughly studied in [68], while [69] was devoted to a model system with only one equation for the stress.
We go back to the linear stability of Couette flow of an Oldroyd fluid. The results are better understood if we draw (Figure 5) the curve of the total dimen-
sional shear stress ~'(~/) versus the shear rate ~, = U/L, where U is the velocity of the upper plate and L the distance between the two parallel plates. We have the following stability result [47].
T h e o r e m 5.3 (Linear s tab i l i ty of the C o u e t t e flow) (i) I f 0 < e < 8/9, the Couette solution (v,, r~, a,) is linearly stable for all k 's. (ii) I f 8/9 < ~ < 1, the Couette solution (vs, r~,e~s) is linearly stable if and only if
0 < k < k_ or k > k+, where k_ and k+ are the solutions of
en 2 + (2 - 3~)n + 1 = O.
The stability regions are exactly those where the curve in Figure 5 is monotone, i.e.,
220
correspond to the values of/7 satisfying 0 < 4/ < ,~ or "} > "7~. The stability is lost when the spectrum of the linearized operator crosses the imaginary axis at O, eigenvalue of infinite multiplicity.
The proof of Theorem 5.3 consists in showing that the underlying semi-group is analytic (because of the degeneracy of the equation for c~ and r in (22)), and then in localizing the spectrum by the Routh-Hurwitz criterion.
~=0 ~*~ c < 8 / 9 = 9
> 8/9
, E = I
! 2
Figure 5: Curves (-},'?*(-})) for different values of e E (0,1)
R e m a r k 5.1 It seems natural to conjecture that the linear stability regions in Theorem 5.3 coincide with the nonlinear stability regions. Theorem 5.1 gives the answer for part of the first increasing portion of the S-shaped curve in Figure 5.
R e m a r k 5.2 The instability quoted in Theorem 5.3 is due to the aforementioned S- shaped curve. This constitutive instability--inherent to the model--has been questioned as unphysical by some authors.
5.4. Non l inea r s tab i l i ty Nonlinear stability results for viscoelastic fluids are very few. They essentially concern
Jeffreys-type fluids. We have already mentioned those of [47] for the one-dimensional stability of Couette flows (see Section 5.3), and for the stability of flows of Jeffreys-type fluids which are small perturbations of the rest state (see Corollary 4.1).
In a recent paper [70] Renardy has investigated the nonlinear stability of flows of Jeffreys-type fluids at low Weissenberg numbers. More precisely, assuming the existence of a steady flow (~, ~), he proves that this flow is linearly and Liapunov stable provided the spectrum of the linearized operator lies entirely in the open plane {~/~ < 0} and that the following quantity is sufficiently small
We(1 + (-1/2) sup (IVl + 0 B(V, + 01B( , f~
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where B(cr ~) = (~. V)~ +/3(V~, @), and 01 and 02 denote the partial derivatives of B with respect to re and ~ respectively.
5.5. Spectral studies As was mentioned before, no general theoretical result exists, which would justify the
implication "($3) ~ ($2) ' , i . e . , the usual procedure of calculating the eigenvalues of the linearized system to assert the stability of viscoelastic flows. Despite this fact, a lot of studies have been devoted to such computations, and it is impossible to review them all. We have therefore selected significant examples, and quoted a few others.
The general idea is to look for plane waves perturbations having the form e " t e k'x leading to spectral problems in a, which can be solved numerically. We emphasize once again that this kind of study does not imply the nonlinear--even the linear!--stability of the flow, if the assertion "($3) ~ ($2)", for instance, has not been proven mathematically.
It is worth noticing that Tlapa and Bernstein [71] have proven that the Squire theorem holds true for the Poiseuille flow of an upper-convected Maxwell fluid. It means that any instability, which may be present for three dimensional disturbances, is also present for two dimensional ones at a lower value of the Reynolds number. This property is not true, in general, for non-Newtonian fluids [72].
Renardy and Renardy [66,73] have investigated the stability of plane Couette flows for Maxwell-type models involving the derivative (2). The flow lies between parallel plates at x = 0 and x = 1, which are moving in the y-direction with velocities -t-1, such as in Figure 6.
Y
1
O
Figure 6: Plane Couette flow with velocity +1 on the plates
The steady Couette solution has velocity v~ = (0, x, 0) and stress with components
(1 + a)W'e 1 (1 -- a)We or,= l + k 2 , r s = l + k 2' % = l + k 2 '
222
with k 2 = We2(1 - a2). One adds to the basic flow a perturbation for which the velocity has a component only in the z direction, v(x, y, z) = xey + r where here denotes a small parameter.
The equations of motion and the constitutive relation are then linearized with respect to ~. To avoid some possible unphysical instabilities, the range of parameter a, which a priori belongs to ( -1 ,1 ) , is restricted to the set {a >_ 1/2, k < 1 }. The first restriction ensures that the model is consistent with rod climbing, the later that one stays on the increasing part of the curve of the shear stress as a function of the shear rate, which has a maximum at k -- 1.
The eigenvalue problem is solved numerically by the tau-Chebychev method [66]. It can be determined analytically that there is a continuous spectrum given by
1 1 - a2)We 2 = [2(2 + - -< -<
In addition to the continuous spectrum there is an infinite number of discrete eigenvalues which are essentially lined up along a parallel to the imaginary axis--this is another illustration of the fact that the underlying semi-group is not analytic.
Various computations, involving different values of the parameters Re, We, a, and a, are performed in [66]. None of them leads to instability.
Many studies have been devoted to the Taylor-Couette problem (flow between two concentric cylinders with radii R1 and R2, R1 _< R2, of infinite length, and rotating with angular velocities ~1 and -Q2 repectively). For instance Zielinska and Demay [74] consider the general Maxwell models with - 1 < a < 1. They show that the axisymmetric steady flow (the Couette flow) does not exist for all values of parameters where the steady state exists; moreover all models, except for a very close to -1 , predict stabilization of the Couette flow in the spectral sense, for small enough values of the Weissenberg number. (See also [55].)
On the other hand, Muller et al. [75-78] have reported on a purely elastic Taylor- Couette instability for models with or without Newton[an contribution (Jeffreys or Maxwell). The conclusion of their studies is that negative second normal stresses are stabilizing, es- pecially for very small gap ratios, and that the Newton[an relative contribution has a stabilizing influence.
We now discuss the study of stability of two-dimensional flows of two viscoelastic fluids by a spectral approach. (See Figure 7.)
After linearization around the steady solution, one is led to a system, symbolically written as :
P(O~, O~)V = at Q(O~, O~)U,
where P and Q are polynomials, and U collects all the fields involved here: the velocity (u, v), the pressure p and the extra-stress (a, ~,, r) for both fluids. It is usually assumed (see [79] for a discussion) that the perturbation fields can be decomposed into exponential terms of either e .q(~-ct) or e ̀qx+~ type. This leads to the Orr-Sommerfeld equations, respectively
P(iq, 0~)0" = - iqc Q(iq, c3y)U, or P(iq, coy)(] = s Q(iq, O~)b ~. (25)
223
,,///
F l u i d 1
F l u i d 2
///
~Y III _
X
Figure 7: The two-fluid basic flow and its perturbation
In [80], as in previous works (e.g. [81])on viscoelastic flows, Chela assumes disturbances of the form e iq(~-ct) for a steady Couette flow of two upper-convected Maxwell fluids, and obtains the Orr-Sommerfeld equations. Then, he carries out a long wave asymptotic analysis (q ~ 0) and gives the explicit formula characterizing the asymptotic long wave stability. The corresponding short wave asymptotics of the same flow is performed by Renardy [82]. In this article, she proposes a spectrally stable arrangement, in which the less viscous fluid is in a thin layer, to stabilize the long waves, and a good choice of the elasticities, to stabilize the short waves. In [83], the authors investigate under the same exponential perturbations as in [80], both the asymptotic stability (q ~ 0) and the numerical stability (for q E l~ fixed) of the plane Poiseuille flow of two viscoelastic fluids in a slit and in a converging channel. (See Figure 8.)
The constitutive equations are the Oldroyd-B model and a modified Oldroyd-B model in which the viscosity depends on the rate of strain. In [79], Laure et al. study the spectral stability of the plane Poiseuille flow of two viscoelastic fluids obeying an Oldroyd-B law in two configurations: the first one is the two layer Poiseuille flow; in the second case the same fluid occupies the symmetric upper and lower layers, surrounding the central fluid. (See Figure 9.)
The investigations being very similar for these two flows, we only report here on what was done for the two-layer Poiseuille flow. In [79], the Orr-Sommerfeld equations are rigorously derived, for the second type of perturbations e ~q~+st, via Laplace and Fourier transforms. The difference between both types of exponential perturbations is investigated in [50]. It is not clear whether or not both give the same conditions on the long wave asymptotic instability.
A careful long wave (q ~ 0) asymptotics is also performed in [79] and gives, in special cases, very simple formulas. For instance, if the jump across the interface in the basic velocity is zero (case of the neutral asymptotic stability of a Newtonian flow), then a
224
simple condition on the sign of the jump in the normal extra-stresses gives the asymptotic stability or instability.
/ , /Z . . i l l . .
Fluid 1 v (y ) 1
Fluid 2 " ~ v2(Y) _
' l t l - ] i l
Y
_ _ _ / / / , / / / _ ~ _
Fluid 1 v(x, y) 1
Fluid 2 v2(*, y) --- //-/ ,;'II
(A) (B)
Figure 8: Slit and converging channels (resp. (A) and (B))
The numerical study of the Orr-Sommerfeld equations requires to discretize the 0 v
operators in equation (25). As in [84], the spectral tau-Chebychev approximations are often used, though pseudo-spectral [85] or finite element techniques [86] may be chosen too.
Whatever the discretization in y is, the resulting equations are a generalized eigenvalue problem of the form A U = s B U , A and B being two complex matrices, s the eigenvalue characterizing stability, and U the discretized vector of velocity, pressure, and extra-stress for each fluid.
Concerning the assembling of the matrix A, Herbert [87] has noticed that, upon a mere change of variables in the evaluation of the polynomials involved in A, the round-off errors could be significantly reduced. This shows how stiff the problem is.
For the numerical study of the whole spectrum (for q e R fixed), [79] uses a spectral tau- Chebychev discretization in y and the Arnoldi method (see [88]) to solve the generalized eigenvalue problem (see [89]). This numerical method is based on the orthonormalization of the Krylov space of the iterates of the inverse of the matrix A -1B. This method has been used more recently in [90]. It has been proven efficient in the stiff problems arising in the study of spectral stability of viscoelastic fluids.
In [91], Chen studies the long wave asymptotics of the concentric Poiseuille flow of two upper-convected Maxwell fluids under axisymmetric perturbations. He concludes that "stability can generally be achieved by increasing the volume of the more elastic component", in agreement with the lubrication effect. The corresponding short wave asymptotic study of the same flow is done in [92]. In the case of coextrusion flows a study for arbitrary wavelengths and low Reynolds numbers is done in [93].
Let us finally quote some more references where the flow is neither the Poiseuille, nor the Couette one. In [94], Oztekin and Brown study the spectral stability of the flow of
225
an Oldroyd-B fluid between rotating parallel disks" they find that the nonaxisymmetric perturbations are the most dangerous ones. Some other studies have been carried out for the cone and plate flow [95,96], or for the flow down an inclined plane [97].
/ / (
Fluid 1
Fluid 2 / / /
g / (
I gravity
X
/ / /
Fluid 2
Fluid 1
Fluid 2 , n,
/ I / / / /
X
/ / /
(A) (B)
Figure 9: Two-layer (A) and three-layer (B) symmetric geometries
6. N U M E R I C A L ANALYSIS OF V I S C O E L A S T I C F L O W S
6.1. Introduction In order to build up and implement efficient numerical schemes for partial differential
equations, it is necessary to have informations on the mathematical properties of the system of equations--this has been done in the previous sections--as well as on the stability and the convergence properties of the schemes: this is the purpose of numerical analysis.
In the context of viscoelastic fluid flows, numerical analysis has been performed for differential models only, and for the following types of approximations: finite element methods for steady flows, finite differences in time and finite element methods in space for unsteady flows. Finite element methods are the most popular ones in numerical simulations, but some other methods like finite differences, finite volume approximations, or spectral methods are also used.
Moreover although the results we present here are only valid for regular and/or slow flows, they give some confidence in the use of these numerical schemes in more realistic situations, where for example the domain of the flow is not regular.
A typical result of numerical analysis is an estimate of the error U - Uh between the solution U of the "continuous" problem (i.e., the solution of the initial boundary value problem) and the solution Uh of the "discrete" problem (also called approximate problem). In what follows the error estimates are obtained with the assumption that U is sufficiently regular. In many realistic situations the geometry of the flow has singularities (corners for example), the solution U is not regular, and these results do not apply. (As a matter of fact existence of a solution has not been shown yet in those singular situations.)
In order to simplify the presentation we make the following assumptions: - the flow is steady;
226
- inertia is neglected in the stress balance equation; if necessary, the ( v - V ) v term could be handled as in the case of the Navier-Stokes equations;
- the velocity is zero on the boundary of the domain; this hypothesis is rather strong, implying in particular that the entry part Off_ of 9t is empty;
- the fluid obeys a Jeffreys-type constitutive law (i.e., the Newtonian viscosity is not zero).
We write the steady problem corresponding to equations (7)--where inertia is neglected-- in the following form:
Find ( r , v, p) such that
r + We ((v- V)v +/3(Wv, r ) ) - 2e D[v] = 0 in fl,
- d i v r - 2(1 - e) div D[v] + ~7p = f in 12,
div v = 0 in 12,
v = 0 o n 0ft.
(26)
The solution ( r , v, p) to Problem (26) is supposed to exist in a space of regular functions (T, X, Q). (See, for instance, Theorem 3.1, and Theorem 6.1 below.)
6 . 2 . F i n i t e e l e m e n t a p p r o x i m a t i o n
We assume that the domain of the flow ~ C ]R 2 is a polygonal domain equipped with a uniformly regular family of triangulations Th made of triangles K, 12 = UgeTh K, satisfying
voh ~ hK ~ b~lpg,
for some constants ~0, ua > 0. Here hK is the diameter of K, PK the diameter of the greatest ball included in K, and h = maxgeTh hK.
The first equation in system (26) is, for v fixed, a transport equation in ~-, so that some upwinding is needed for the practical computation of the solution. This fact was first recognized in [98] where streamline upwinding methods were used, and in [99] where discontinuous Galerkin methods were implemented. We describe, in the following, a finite element approximation of system (26) using discontinuous approximations of r .
The integer number k >__ 1 will be fixed throughout this section. Pk(K) denotes the space of polynomials of degree less or equal to k on K E Th. For the approximation of (v,p), we use finite element spaces Xh and Qh satisfying:
C~ 2 D Xh D {u E X(-]C~ 2 " u]A- E Pk+I(K)~,VK E ~ } ,
Qh D {q E Q; qi~ E Pk(K), Vii E Th},
or Qh D {q E Q ~ C ~ qlK E Pk(K), VK C Th}.
We assume that the spaces Xh and Qh satisfy the usual "inf sup condition" for the Stokes problem:
(q, div w) inf sup >/3 > 0, for some constant/3. (27)
q~Q,, wexh -]qllD[w]] -
227
Here, (-,-) denotes the scalar product in L 2 or 1[, ~, and I" I the norm in these Hilbert spaces. For the approximation of r , we consider the finite element space:
Th = {IT E T ; ITIK E Pk(K) 4, VK (5 Th}.
In order to describe the approximation of the r equation of Problem (26) by the dis- continuous Galerkin method of [100], we introduce the notation:
0 K - ( v ) = {x E OK, v ( x ) - n ( x ) < 0},
r e (v ) (x ) = lim r ( x + ev(x)) for x E OK-(v) v...+O • ~ ,
where OK is the boundary of K and n is the outward unit normal. We define the scalar products:
(T, IT)h -- E ('r, tT)K , KETh
(T'4"' IT4")h v -- K ~ T h f 0 ' K-(v) ~-~(v)- ~ ( v ) Iv �9 n, d~,
where (., ")K denotes the scalar product in L2(K) or in L2(K). We also define the trilinear form bh on Xh x Th x Th by
bh(v, r , a ) = ((v- V ) r , IT)h + 1/2((div v) 1", a)h + ( r + - 7"-, IT+)h,v-
Problem (26) is approximated by the following system:
Find (rh, vh,ph) E Th x Xh x Qh such that
(rh, a)h + We (bh(vh, rh, IT) + (r rh), a)h) - 2e (D[vh], IT)h = 0 VIT E Th,
(rh, D[w])h + 2(1 -- e)(D[vh], D[w])h -- (Ph, divw)h = (f, w)h Vw E Xh,
(divvh, q)h = 0 Vq E Qh-
(2s)
6.3. A convergence t h e o r e m All known results are of the following type. If the continuous problem (26) admits a
solution which is sufficiently smooth and small, then:
1. the approximate problem (28) (which is also nonlinear) admits a solution close to the continuous solution;
2. one gets an error estimate;
3. the approximate solution is unique and can be obtained by a fixed point iterative scheme.
In other words a typical theorem is the following.
228
T h e o r e m 6.1 (Exis tence of con t inuous and approximate flows. E r r o r estimate) Let e < 1 and k >_ 1. There exist two constants Co > 0 and ho > 0 depending on k, such
that if Problem (26} admits a solution (v, r , p ) E t t k+2 • ][-I k+l • H k+l satisfying
M = m~x{ll v Ilk+2, II 1" IIk§ II P IIk+x } ~ Co,
then, for all h < ho, Problem (~8} admits a solution (Vh, ~rh, Ph ) E Xh X Th • Qh, satisfying the following error estimate: there exists a constant C > 0 independent of h such that
I~ - rh l + I D [ v - vh] l + Ip - phi < C h k+~/2.
Moreover, there exist two constants C~ > 0 and h' o > 0 such that M < C~ and h < h'o imply that the solution of Problem (28) is unique and is the limit of the fixed point iteration scheme (we omit the subscript h}"
v n pn ,Tn+l vn+l pn+l For aiven ( r n, , ), f ind( , , ) E Th • Xh • Qh such that
( r ~+x, a) - 2, (D[vn+']), a) + We bh(v n, r ~+1 , a) = - W e (/3(Vv ~, v ~), a) Va e Th, (29)
(r~+l,D[w]) + 2(1 - e)(D[v~+'], D[w]) - (p"+l,div w) = (f ,w) Vw e Xh,
(div v ~+~, q) = 0 Vq e Qh.
The proof of the first part of this theorem is given in [101] for k = 1, and in [102] for k > 2. Uniqueness and fixed point aspects are studied in [103].
6.4. Miscellaneous r emarks 1. An analogous result is vahd for continuous approximations of 7" when upwinding
is performed by the streamline upwinding Petrov-Galerkin method (SUPG) [104]. The same is true for finite element methods based on a quadrangular mesh [105].
2. For cost reasons, k is most often equal to one in practical implementations. Com- monly used finite elements are:
(a) the Hood-Taylor finite element: /92 for the velocity, continuous P1 for the pressure, and discontinuous P1 for the stress;
(b) P2 plus bubble (also called P+) for the velocity, discontinuous Px for the pres- sure, and discontinuous P2 for the stress;
(c) on a quadrangular mesh, (image of) Q2 for the velocity, discontinuous Q1 for the pressure, and discontinuous Q2 (or even incomplete Q2 with eight nodes) for the stress.
3. When We = 0, the Oldroyd-B model (26) reduces to a three-field version of the Stokes problem. For e < 1, this problem is stable under condition (27). It was proven in [106] that, in the case of the Maxwell-type problem (where e = 1), one has to add a second inf sup condition to obtain stability:
229
inf sup (t r, D[w] ) > 7 > 0 for some constant 7- ~ e x . ~ T . I~l ID[w]l -
(30)
As explained in [1061 condition (30) may be satisfied either by imposing D[Xh] C Th,
suggesting the use of discontinuous Th, or by giving a sufficient number of interior nodes in each K of Th in case of continuous Th. This last fact was first observed numerically in [98], where a sixteen node Q1 finite element approximation of r is used.
For efficiency reasons it seems preferable to use finite element satisfying (30) even for e < 1, but near one. Inexpensive finite elements satisfying (27) and (30) have been recently developed (see [107-109]).
Most of the above comments apply to three dimensional problems, but corresponding 3D simulations are still exceptional for cost reasons.
4. Theorem 6.1 is still valid for models with several relaxation times, as well as for models with a quadratic 13 in r , such as the Giesekus and the linearized Phan- Thien-Tanner models. (See [110])
5. For White--Metzner-type models, the associated Stokes problem obtained for We = 0 is then nonlinear. This is related to quasi-Newtonian models and is studied in [111]. Numerical analysis for We > 0 has not been done yet.
6. Nothing seems to be known concerning the numerical analysis of Maxwell-type mod- els.
7. Problem (26) can be viewed as a transport equation in r for given v, and a Stokes system in (v, p) for given 1-. The fixed point iteration scheme described in Theorem 6.1 does not use this fact. A more natural iterative scheme, which uncouples the r and the v equations, reads as follows:
For given (7" ~, v ~, p'~), find (r~+l , vn+l , pn+l) E Th x X h X Q h such that
( r ~+1 , or) + We bh(v ~, T n+l , tT)
= - W e (fl(Vv ", r~), (r) + 2e (D[v~], ~r) Vet E Th,
2(1 - e + 7)(D[v~+~], D[w]) - (p~+X, divw) = ( f , w ) + 27(D[v"], D [ w ] ) - (rn+l,D[w]) Vw e Xh,
(divv TM, q) = 0 Vq E Qh,
(31)
where g > 0 is a fixed parameter. The convergence of this iterative scheme is proven in [112].
8. Most of the numerical analyses so far concern steady flows. For unsteady problems, the convergence of the following scheme has been proven in [113]"
230
For given (r" , v'~, p=), find (v"+l,v"+i,p n+l) E Th x Xh x Qh such that
Re ((v ~+1 - v")fiSt, w) + 2(1 - e)(D[vn+l], D[w]) - (pn+~, div w) +(vn+',D[w]) = (f( t ,+i) ,w) Vw E Xh,
We((7 "n+l - - r")l,St, ,,-) + we bh(v", I" TM, a) + ( r ''+l, a') -2e (D[v"+'], tr) = -We (fl(Vv", ~'"), tr) Vtr E Th,
(divv '~+1, q) = 0 Vq e Qh,
(32)
where (St denotes the time step. This result is extended in [114] to an uncoupled unsteady scheme based on (31).
7. Conclusion
Mathematical and numerical analyses of differential models for viscoelastic fluid flows are highly challenging domains, which still need a lot of effort. However, significant progress has been made during the last decade, and mathematical results have shown to be quite useful for the modelling.
Classification of differential models with respect to typical properties such as change of type or loss of evolution has been done. Robust results--it means that they are in- dependent of the models--for existence of regular flows have been obtained in different situations: slow steady flows, or steady flows perturbing a uniform flow; unsteady flows on a short time interval, or unsteady flow for all times, but small data; flows with inflow boundary values for certain simple geometries, . . .
The important problem of stabihty of viscoelastic flows is far more involved than for Newtonian flows, and many problems still remain open. However, it is clearly established that the difficulty lies in the relationship between various mathematical notions of stabil- ity. Some results have been obtained in this direction for restricted classes of flows and/or of models. Moreover several important studies of spectral stability have been performed.
Last, numerical analyses of schemes for approximating simple flows have been devel- oped: the convergence results give some confidence in the most often used methods for the computation of flows in realistic situations.
Acknowlegments . We thank H. Le Meur for his valuable comments during the prepa- ration of this paper and for his drawings of all the figures.
R E F E R E N C E S
1. R.G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworths, Boston, 1988.
2. R.B. Bird and J.M. Wiest, Constitutive equations for polymeric liquids, Ann. Rev. Fluid Mech., 27 (1995) 169-193.
3. T.H. Kwon and S.F. Shen, A unified constitutive theory for polymeric liquids, I and II, Rheol. Acta, 23 (1984) 217-230, and 24 (1985) 175-188.
4. C. Guillop6 and J.-C. Saut, Mathematical analysis of differential models with internal variables for viscoelastic fluids, in preparation.
231
.
.
10.
11.
12.
13.
14.
15.
16.
17.
18. 19.
20. 21.
22.
23.
B.J. Edwards and A.N. Beris, Remarks concerning compressible viscoelastic fluid models, J. Non-Newtonian Fluid Mech., 36 (1990) 411-417. I.M. Rutkevich, Some general properties of the equations of viscoelastic incompressible fluid mechanics, J. Appl. Math. Mech. (PMM), 33 (1969) 30-39. I.M. Rutkevich, The propagation of small perturbations in a viscoelastic fluid, J. Appl. Math. Mech. (PMM), 34 (1970) 35-50. D.D. Joseph, M. Renardy and J.-C. Saut, Hyperbolicity and change of type in the flow of viscoelastic fluids, Arch. Rat. Mech. Anal., 87 (1985) 213-251. D.D. Joseph and J.-C. Saut, Change of type and loss of evolution in the flow of viscoelatic fluids, J. Non-Newtonian Fluid Mech., 20 (1986) 117-141. F. Dupret and J.-M. Marchal, Loss of evolution in the flow of viscoelastic fluids, J. Non-Newtonian Fluid Mech., 20 (1986) 143-171. Y. Kwon and A.I. Leonov, Stability constraints in the formulation of viscoelastic constitutive equations, J. Non-Newtonian Fluid Mech., 58 (1995) 25-46. C. Guillop~ and J.-C. Saut, Mathematical problems arising in differential models for viscoelastic fluids, in Mathematical Topics in Fluid Mechanics, J.F. Rodrigues and A. Sequeira (eds.), Longman Scientific and Technical, Pitman, 1992, 64-92. D.D. Joseph, J. Matta and K.P. Chen, Delayed die swell, J. Non-Newtonian Fluid Mech., 24 (1987) 31-65. A.B. Metzner, E.A. Uebler and Chang Man Fong, Converging flows of viscoelastic materials, AIChE J., 15 (1969) 750-758. L.E. Fraenkel, On a linear partly hyperbolic model of viscoelastic flow past a plate, Proc. Roy. Soc. Edinburgh, 114 A (1990) 299-354. M.J. Crochet and V. Delvaux, Numerical simulation of inertial viscoelastic flows with change of type, in Nonlinear Evolution Equations That Change Type, B.L. Keyfitz and M. Shearer (eds.), IMA Volumes in Mathematics and its Applications 27, Springer- Verlag, Berlin, 1991, 47-66. M. Renardy, A well-posed boundary value problem for supercritical flow of viscoelastic fluids of Maxwell-type, in Nonlinear Evolution Equations That Change Type, B.L. Keyfitz and M. Shearer (eds.), IMA Volumes in Mathematics and its Applications 27, Springer-Verlag, Berlin, 1991, 181-191. D.D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer-Verlag, Berlin, 1990. B.D. Coleman and W. Noll, Foundations of linear viscoelasticity, Rev. Modern Phys., 33 (1961) 239-249. J.-C. Saut and D.D. Joseph, Fading memory, Arch. Rat. Mech. Anal., 81 (1983) 53-95. J.S. Ultman and M.M. Denn, Anomalous heat transfer and a wave phenomenon in dilute polymer solutions, Trans. Soc. Rheol., 14 (1970) 307-317. G. Schleiniger, M.C. Calderer and L.P. Cook, Embedded hyperbolic regions in a non- linear model for viscoelastic flow, in Current Progress in Hyperbolic Systems: Rie- mann Problems and Computations, W.B. Lingquist (ed.), Contemporary Mathemat- ics 100, American Mathematical Society, Providence, 1990. C. Guillop6 and J.-C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal., Th. Meth. Appl., 15 (1990) 849-869.
232
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
M. Renardy, Existence of slow steady flows of viscoelastic fluids with differential constitutive equations, Z. Angew. Math. Mech., 65 (1985), 449-451. A. Hakim, Analyse math~matique de modules de fluides visco~lastiques de type White--Metzner, Th~se de l'Universit~ Paris-Sud, Orsay, 1989. C. Guillop~ and J.-C. Saut, Existence and stability of steady flows of weakly viscoelas- tic fluids, Proc. Roy. Soc. Edinburgh, 119 A (1991) 137-158. R. Talhouk, Analyse math~matique de quelques ~coulements de fluides visco~lastiques, Th~se de l'Universit~ Paris-Sud, Orsay, 1994. R. Talhouk, Ecoulements stationnaires de fluides visco~lastiques faiblement compress- ibles, Comptes Rend. Acad. Sc. I, 320 (1995) 1025-1030. M. Renardy, Inflow boundary conditions for steady flows of viscoelastic fluids with differential constitutive laws, Rocky Mount. J. Math., 18 (1988) 445-453, and 19 (1989) 561. M. Renardy, An alternative approach to inflow boundary conditions for Maxwell fluids in three space dimensions, J. Non-Newtonian Fluid Mech., 36 (1990) 419-425. M. Renardy, Existence of steady flows of viscoelastic fluids of Jeffreys-type with trac- tion boundary conditions, Diff. Int. Eq., 2 (1989) 431-437. M. Renardy, Existence of steady flows for Maxwell fluids with traction boundary conditions on open boundaries, Z. Angew. Math. Mech. 75 (1995) 153-155. M. Tajchman, Conditions aux limites absorbantes pour des fluides visco~lastiques de type diff~rentiel, Th~se de l'Universit~ Paris-Sud, Orsay, 1994. B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simula- tion of waves, Math. Comput., 139 (1977) 629-651. L. Halpern, Artificial boundary conditions for incompletely parabolic perturbations of hyperbolic systems, SIAM J. Math. Anal., 87 (1985) 213-251. E.J. Hinch, The flow of Oldroyd fluid around a sharp corner, J. Non-Newtonian Fluid Mech., 50 (1993) 161-171. M. Renardy, The stress of an upper-convected Maxwell fluid in Newtonian field near a re-entrant corner, J. Non-Newtonian Fluid Mech., 50 (1993) 127-134. M. Renardy, A matched solution for corner flow of the upper-convected Maxwell fluid, J. Non-Newtonian Fluid Mech., 58 (1995) 83-89. A. Hakim, Mathematical analysis of viscoelastic fluids of White-Metzner type, J. Math. Anal. Appl, 185 (1994) 675-705. R.E. Gaidos and R. Darby, Numerical simulation and change of type in the developing flow of a nonlinear viscoelastic fluid, J. Non-Newtonian Fluid Mech., 29 (1988) 59-79. M. Renardy, Local existence of the Dirichlet initial boundary value problem for in- compressible hypoelastic materials, SIAM J. Math. Anal., 21 (1990) 1369-1385. M. Renardy, Initial value problems with inflow boundaries for Maxwell fluids, SIAM J. Math. Anal., (1996) to appear. R. Talhouk, Unsteady flows of viscoelastic fluids with inflow and outflow boundary conditions, Appl. Math. Letters, (1996) to appear. J.U. Kim, Global smooth solutions for the equations of motion of a nonlinear fluid with fading memory, Arch. Rat. Mech. Anal., 79 (1982) 97-130.
233
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55. 56.
57.
58. 59.
60.
61.
62.
63.
64.
65.
J. Serrin, A note on the existence of periodic solutions of the Navier-Stokes equations, Arch. Rat. Mech. Anal., 3 (1959) 120-122. A. Valli, Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method, Ann. Sc. Norm. Sup. Pisa, 10 (1983) 607-647. C. Guillop6 and J.-C. Saut, Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type, Math. Mod. Anal. Numer., 24 (1990) 369-401. D.S. Malkus, J.A. Nohel and B.J. Plohr, Analysis of a new phenomenon in shear flow of non-Newtonian fluids, SIAM J. Appl. Math., 51 (1991) 899-929. M. Renardy, W.J. Hrusa and J.A. Nohel, Mathematical Problems in Viscoelasticity, Longman Scientific and Technical, Burnt Mill, Harlow, 1987. H. Le Meur, Existence, unicit6 et stabilit6 de plusieurs types d'6coulements de fluides visco61astiques avec interface, Th~se de l'Universit6 Paris-Sud, Orsay, 1994. H. Le Meur, Existence locale de solutions des 6quations d'un fluide visco61astique avec fronti~re libre, Comptes Rend. Acad. Sc. I, 320 (1995) 125-130. J.P. Tordella, Unstable flow of molten polymers, in Rheology: Theory and Applica- tions 5, F. Eirich (ed.), Academic Press, New York, 1969. N. E1 Kissi and J.-M. Piau, The different capillary flow regimes of entangled poly- dimethylsiloxane polymers: macroscopic slip at the wall, hysteresis and cork flow, J. Non-Newtonian Fluid Mech., 37 (1990) 55-94. J.-M. Piau, N. E1 Kissi and B. Tremblay, Influence of upstream instabilities and wall slip on melt fracture and sharkskin phenomena during silicones extension through orifice dies, J. Non-Newtonian Fluid Mech., 34 (1990) 145-180. R.G. Larson, Instabilities in viscoelastic flows, Rheol. Acta, 31 (1992) 213-263. M. Renardy and Y. Renardy, Pattern selection in the B6nard problem for a viscoelastic fluid, Z. Angew. Math. Mech., 43 (1992) 154-180. M. Avgousti and A.N. Beris, Viscoelastic Taylor-Couette flow: bifurcation analysis in the presence of symmetries, Proc. Roy. Soc. London A, 443 (1993) 17-37. G. Iooss, Bifurcation et stabilit6, Publications Math6matiques d'Orsay 31, 1974. A.F. Neves, H. de Souza Ribeiro and O. Lopes, On the spectrum of evolution operators generated by hyperbolic systems, J. Funct. Anal., 67 (1986) 320-344. M. Renardy, On the type of certain Co semi-groups, Comm. Part. Diff. Eq., 18 (1993) 1299-1307. M. Renardy, On the linear stability of hyperbolic partial differential equations and viscoelastic flows, Z. Angew. Math. Phys., 45 (1994) 854-865. M. Renardy, A rigorous stability proof for plane Couette flow of an uppe-convected Maxwell fluid at zero Reynolds number, Eur. J. Mech. B, 11 (1992) 511-516. V.A. Gorodtsov and A.I. Leonov, On a linear instability of a plane parallel Couette flow of a viscoelastic fluid, J. Appl. Math. Mech. (PMM), 31 (1967) 310-319. M. Renardy, On the stability of parallel shear flow of an Oldroyd B fluid, Diff. Int. Eq., 6 (1993) 481-489. V.A. Romanov, Stability of plane-parallel Couette flow, Funct. Anal. Appl:'7 (1993), 137-146.
234
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
M. Renardy and Y. Renardy, Stability of shear flows of viscoelastic fluids under per- turbations perpendicular to the plane of flow, J. Non-Newtonian Fluid Mech., 32 (1989) 145-155. G. Prodi, Teoremi di tipo locale per il sistema di Navier-Stokes e stabilitk delle soluzioni stazionare, Rer.d. Sem. Mat. Univ. Padova, 32 (1962) 374-397. J.A. Nohel, R.L. Pego and A.E. Tzavaras, Stability of discontinuous steady states in shearing motions of a non-Newtonian fluid, Proc. Roy. Soc. Edinburgh, 115 A (1990) 39-59. J.A. Nohel and R.L. Pego, Nonlinear stability and asymptotic behaviour of shearing motions of a non-Newtonian fluid, SIAM J. Math. Anal., 24 (1993) 911-942. M. Renardy, Nonlinear stability of flows of Jeffreys fluids at low Weissenberg numbers, Arch. Rat. Mech. Anal. 132 (1995) 37-48. G. Tlapa and B. Bernstein, Stability of a relaxation type viscoelastic fluid with slight elasticity, Phys. Fluids, 13 (1970) 565-568. F. J. Lockett, On Squire's theorem for viscoelastic fluids, Int. J. Eng. Sci., 7 (1969) 337-349. M. Renardy and Y. Renardy, Linear stability of plane Couette flow of an upper- convected Maxwell fluid, J. Non-Newtonian Fluid Mech., 22 (1986) 23-33. B.J.A. Zielinska and Y. Demay, Couette-Taylor instability in viscoelastic fluids, Phys. Rev. A, 38 (1988) 897-903. S.J. Muller, R.G. Larson and E.S.G Shaqfeh, A purely elastic transition in Taylor- Couette flow, Rheol. Acta, 28 (1989) 499-503. R.G. Larson, E.S.G. Shaqfeh and S.J. Muller, A purely viscoelastic instability in Taylor-Couette flow, J. Fluid Mech., 218 (1990) 573-600. E.S.G. Shaqfeh, S.J. Muller and R.G. Larson, The effects of gap width and dilute solution properties on the viscoelastic Taylor-Couette instability, J. Fluid Mech., 235 (1992) 285-317. R.G. Larson, S.J. Muller and E.S.G. Shaqfeh, The effect of fluid rheology on the elastic Taylor-Couette instability, J. Non-Newtonian Fluid Mech., 51 (1994) 195-225. P. Laure, H. Le Meur, Y. Demay and J.-C. Saut, Linear stability of multilayer plane Poiseuille flows of Oldroyd-B fluids, Pr6publication 96-12, Universit~ Paris-Sud, Math~matiques, submitted. K.P. Chen, Elastic instability of the interface in Couette flow of viscoelastic liquids, J. Non-Newtonian Fluid Mech., 40 (1991) 261-267. C.H. Li, Stability of two superposed elastoviscous liquids in plane Couette flow, Phys. Fluids, 12 (1969) 531-538. Y. Renardy, Stability of the interface in two-layer Couette flow of upper-convected Maxwell liquids, J. Non-Newtonian Fluid Mech., 28 (1988) 99-115. Y.Y. Su and B. Khomami, Interfacial stability of multilayer viscoelastic fluids in slit and converging channel die geometries, J. Rheol.. 36 (1992) 357-387. S.A. Orszag, Accurate solution of the Orr-Sommerfeld stability equation, J. Fluid Mech., 50 (1971) 689-703. Y.Y. Su and B. Khomami, Numerical solution of eigenvalue problems using spectral
235
techniques, J. Comp. Phys., 100 (1992) 297-305. 86. S. Yiantsios and B.G. Higgins, Analysis of superposed fluids by the finite element
method: linear stability and flow developement, Int. J. Numer. Meth. Fluids, 7 (1987) 247-261.
87. T.A. Herbert, Die neutrMe Fls der ebenen Poiseuille-StrSmung, Habilitation, Uni- versits Stuttgart, 1977.
88. Y. Saad, Variations on Arnoldi's method for computing eigenelements of large un- symmetric matrices, Lin. Alg. Appl., 34 (1980) 269-295.
89. S. Scotto, Calcul num~rique des modes critiques de l'6quation d'Orr-Sommerfeld visco~lastique, Rapport de DEA, Universit~ de Nice Sophia-Antipolis, 1994.
90. R. Sureshkumar and A.N. Beris, Linear stability of viscoelastic Poiseuille flow using an Arnoldi based orthogonalisation algorithm, J. Non-Newtonian Fluid Mech., 56 (1995) 151-182.
91. K.P. Chen, Interracial instability due to elastic stratification in concentric coextrusion of two viscoelastic fluids, J. Non-Newtonian Fluid Mech., 40 (1991) 155-175.
92. K.P. Chen and D.D. Joseph, Elastic short wave instability in extrusion flows of vis- coelastic liquids, J. Non-Newtonian Fluid Mech., 42 (1992) 189-211.
93. K.P. Chen and Y. Zhang, Stability of the interface in coextrusion flow of two vis- coelastic fluids through a pipe, J. Fluid Mech., 247 (1993) 489-502.
94. A. 0ztekin and R.A. Brown, Instability of a viscoelastic fluid between rotating parallel disks: analysis for the Oldroyd-B fluid, J. Fluid Mech., 255 (1993) 473-502.
95. N. Phan-Thien, Cone and plate flow of the Oldroyd-B fluid is unstable, J. Non- Newtonian Fluid Mech., 13 (1985) 325-340.
96. D.O. Olagunju, Asymptotic analysis of the finite cone-and-plate flow of a non- Newtonian fluid, J. Non-Newtonian Fluid Mech., 50 (1993) 289-303.
97. K.P. Chen, The onset of elastically driven wavy motion in the flow of two viscoelastic liquids films down an inclined plane, J. Non-Newtonian Fluid Mech., 45 (1992) 21-45.
98. J.-M. Marchal and M.j. Crochet, A new finite element for calculating viscoelastic flows, J. Non-Newtonian Fluid Mech., 26 (1987) 77-114.
99. M. Fortin and A. Fortin, A new approach for the FEM simulation of viscoelastic flows, J. Non-Newtonian Fluid Mech., 32 (1989) 295-310.
100P. Lesaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, in Mathematical Aspects of Finite Elements in Partial Differential Equations, C. de Boor (ed.), Academic Press, New-York, 1974, 89-123.
101J. Baranger and D. Sandri, Finite element approximation of viscoelastic fluid flow: existence of approximate solutions and error bounds I - Discontinuous constraints, Numer. Math., 63 (1992) 13-27.
1022. Baranger and D. Sandri, High order finite element methods for the approximation of viscoelastic fluid flow, Proceedings of the Tenth International Conference on Com- puting Methods in Applied Sciences and Engineering, R. Glowinski (ed.), Paris, 1992, 185-194.
103J. Baranger and D. Sandri, Some remarks on the discontinuous Galerkin method for the finite element approximation of the Oldroyd-B model~ submitted.
236
104J). Sandri, Finite element approximation of viscoelastic fluid flow: existence of ap- proximate solutions and error bounds. Continuous approximation of the stress, SIAM J. Numer. Anal., 31 (1994) 362-377.
105A. Bahar, J. Baranger and D. Sandri, Quadrilateral finite element approximation of viscoelastic fluid flow, Rapport de l'6quipe d'analyse num6rique Lyon-Saint-Etienne 162 (1993), submitted.
106.M. Fortin and R. Pierre, On the convergence of the mixed method of Crochet and Marchal for viscoelastic flows, Comput. Meth. Appl. Mech. Engrg., 73 (1989) 341-350.
107.V. Ruas, An optimal three field finite element approximation of the Stokes system with continuous extra-stresses, Japan J. Ind. Appl. Math., 11 (1994) 103-130.
108.V. Ruas, J.H. Carneiro and M.A. Silvaramos, Approximation of the three-field Stokes system via optimized quadrilateral finite elements, Math. Mod. Anal. Num., 27 (1993) 107-127.
109.D. Sandri, Analyse d'une formulation k trois champs du probl~me de Stokes, Math. Mod. Anal. Num., 27 (1993) 817-841.
l l0J. Baranger and D. Sandri, Finite element method for the approximation of viscoelas- tic fluid flow with a differential constitutive law, First European Computational Fluid Dynamics Conference, Bruxelles, 1992, C. Hirsch (ed.), Elsevier, Amsterdam, 1993, 1021-1025.
111J. Baranger, K. Najib and D. Sandri, Numerical analysis of the three-field model for a quasi-Newtonian flow, Comput. Meth. Appl. Mech. Engrg., 109 (1993) 281-292.
112.K. Najib and D. Sandri, On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow, Num. Math., 72 (1995) 223-238.
113J. Baranger and S. Wardi, Numerical analysis of a finite dement method for a transient viscoelastic flow, Comput. Meth. Appl. Mech. Engrg., 125 (1995) 171-185.
114.S. Wardi, Convergence of an uncoupled algorithm for a transient viscoelastic flow, submitted.
Rheology for Polymer Melt Processing J-M. Piau and J-F. Agassant (editors) �9 1996 Elsevier Science B.V. All rights reserved. 237
Computa t ion of 2D viscoelast ic f lows for a different ial const i tut ive equat ion
Y. Demay
Institut Non Lin6aire de Nice, 1361 route des Lucioles, 06560 Valbonne, France.
1. INTRODUCTION
We will consider in the following chapter numerical difficulties encountered in the
numerical simulation of two-dimensional viscoelastic flows described by a differential
constitutive equation. As a particular emphasis is put on numerical simulation of molten
polymer flows existing in rheological laboratory experiments or polymer processing, only the
finite elements method is considered. Our goal is to help the unfamiliar reader to understand the
considerable mathematical and numerical difficulties encountered and we refer to several
reviews for more detailed studies. Let us first recall some mathematical considerations
concerning fitting of numerical methods to general properties of the considered equations.
1.1. The type of an equation and links with modelling
It was recalled that the mathematical type of a system of equations is characterized by
notion of characteristics associated with a direction. If all the characteristics are complex (resp.
real) the problem is elliptic (resp. hyperbolic). EUipticity and hyperbolicity are the mathematical
notions associated with diffusion and propagation. The reader is possibly not familiar with
these mathematical notions, but he is surely used to the consequences. Ellipticity has a
regularizing effect on a singularity while hyperbolicity propagates it. If a phenomenon is
controlled by an elliptic system of equations, effects at long distance depend only on averaged
quantifies. Let us give some examples. For an elastic rod loaded at an extremity, stress and
displacement values far enough from this extremity are only determined by mean values of
loading (Saint-Venant principle). In the same way the flow of a Newtonian fluid at the exit of a
pipe (Poiseuille flow) depends only on the flow rate at the entry and the velocity profile at exit
is regular no matter what it is at entry. Other examples include those with thermal diffusivity.
On the contrary, for a phenomenon controlled by a hyperbolic system of equations, shocks and
singularities are transported. Several examples such as shockwaves, water waves or interfaces
of immiscible (and non reacting) fluids are familiar. Let us come back to the example of pipe
238
flow. For a Newtonian fluid, the Poiseuille flow is fully developped (in stress and velocity) at a
distance of two diameters of the entry. As shown previously, viscoelastic effects are introduced
through convected derivatives of the viscoelastic part of the extra stress tensor and as a
consequence, the entrance length is largely increased. This is in good agreement with
experiments as can be seen on the experimental and numerical flow birefringence pattern of Fig.
40 in III.2. The problem now, for a viscoelastic fluid, is that this transport equation is also able
to transport shocks and, by the effect of equilibrium equations, shocks in extra stress severely
pollute the velocity field. This is why the introduction of a geometrical singularity, such as the
re-entrant comer, induces stress singularity and hence both mathematical and numerical
problems.
In the previous chapter, the mathematical specification of viscoelastic flows governed by a
differential law was given out for a general class of models including the Oldroyd-B or White-
Metzner model. Viscoelasticity introduces a coupling between a transport system of equations
and a Navier-Stokes equation. The problem was then to study the mathematical nature of this
set of equations. It was found in section 2.2, that for the steady creeping flow of a Maxwell
fluid, both types of characteristics exist. Furthermore the characteristics associated with the
vorticity equations can change of type for some values of the Mach number M =Re We. In the
following we will restrict our study to low Reynolds number.
1.2. Realistic geometries
The mathematical existence of the viscoelastic flow for an Oldroyd B fluid was proved in
chapter II for steady flows and for sufficiently small data by theorem 3.1. It was proved for
unsteady flows and locally in time by theorem 4.1. In both cases the frontier of the domain is
assumed to be smooth enough (smoothness required by definition of Hs/2(Of2) space in
theorem 3.1 and C 3 in theorem 4.1). In industrial processes as well as in laboratory
experiments, molten polymer flows in complex geometries. The polymer is generally molten in
a screw extruder, then distributed in a flat die and stretched in air (cast film process), or
templated in air (tube), or pushed in a mould (injection moulding) and finally cooled and
solidified. Various kinematical or geometrical singularities, such as re-entrant corners or change
of boundary conditions at extrusion, are present. These processes are commonly used in
industry and a mathematical simulation are relatively easily performed with a Newtonian fluid at
low Reynolds number (due to the high viscosity, the Reynolds number in polymer processing
is low). We will point out in the following that viscoelasticity is to be introduced carefully in
such a geometry. More precisely, classical viscoelastic constitutive equations, such as the upper
convected Maxwell or Jeffreys models, used to analyse rheological experiments on a cone-plate
apparatus introduce considerable numerical problems.
239
1.3. The Finite Element Method
The finite element method was fast introduced (in the 60s) to compute numerically elastic
deformations of solids. It can be easily proved that this system is elliptic. As a consequence of
no particular space direction, boundary conditions on displacement or stress vector are given on
the whole frontier of the domain. In the 70s, this method was extended to computation of
purely viscous flow by a convenient treatment of the compressibility condition. In this case
again, the system is elliptic in the velocity components and hence boundary conditions on the
velocity field or stress vector are given on the boundary. These conditions can be, for example,
vanishing velocity components on some part of the boundary (if the fluid sticks at the wall) or
vanishing stress vector on another part (for a free surface with air). Mixed conditions involving
a component of velocity and the associated condition on a component of the stress vector can
also be used on a symmetry axis. The important point is that on each point of the boundary two
and only two convenient conditions are associated with two-dimensional Stokes flow. Unless
for large value of Reynolds number, this is not changed by introduction of inertia which adds
derivative of lower order.
Because of the success encountered by finite elements in the solution of elliptic problems,
it was extended (in the 80s) to the advection or transport equation which is a hyperbolic
equation with only one real characteristic. This equation can be solved naturally for an analytical
velocity field by solving a time differential equation. It appeared important, when the velocity
field was numerically obtained, to be able to solve simultaneously propagation and diffusion
equations at low cost. By introducing upwinding in test functions or in the discretization
scheme, the particular nature of the transport equation was considered. In this case, a particular
direction is given at each point (the direction of the convecting flow) and boundary conditions
are only considered on the part of the boundary where the flow is entrant.
1.4. The computation of a viscoelastic flow: a mathematical problem
So far, numerical techniques for a purely hyperbolic (transport) or a purely elliptic
equation are clearly established and the problem is now to manage with a mixed type problem.
As shown in section 3.4.2 of the previous chapter, the natural problem of boundary conditions
mathematically associated with this system of equations is difficult to solve rigorously. In fact,
in the considered geometry, it is assumed that established extra stress tensor values are given in
the entry section of the boundary and velocity or stress vector are given on the whole frontier.
Due to nonlinearity, it was not realistic to consider a rigorous finite element method introducing
an adapted treatment of each characteristic. For example, results of Renardy ((16), (32) of
previous chapter) are obtained for small perturbations of a uniform flow. As the numerical
240
method was not able to take into account the particular nature of the system of equations
governing a viscoelastic flow, it was split into an elliptic problem for velocity and pressure
more or less coupled with an advection equation in the extra stress. It is important to notice that
this splitting technique is present in all numerical methods, at least through a particular treatment
of the constitutive equation and the choice of test functions. Very soon it appeared that
significant nurnerical difficulties are observed in realistic geometries such as convergent or
extrusion geometries and generally convergence with mesh refinement was not obtained. A non
closed controversial debate began to decide if the failure was due to weak or non adequate
numerical approximation, to the mathematical nature of this system of equations unable to
accept geometrical singularities or to the use of a non-realistic viscoelastic law. Simultaneously,
more sophisticated approaches were developed in all of these ways. The approximation space
was improved and the nature of the viscoelastic part of the constitutive equation was modified at
high values of the stress. Up to now the situation is not completely clear, despite the great
number of studies devoted to this subject by authors from a large range of fields such as applied
mathematics, mechanical engineering or physics. In recent years, important progress has been
made in the numerical simulation of viscoelastic fluid flows for increasing elasticity. Very
detailed reviews have been devoted to numerical computation of viscoelastic flows ([ 10], [24]
and [39]). In the following sections, we recall the main mathematical results concerning the
numerical solution of the Stokes problem (16)-(14) by a finite element (Galerkin) method.
2. COMPUTATION OF A PURELY VISCOUS FLOW
The main difficulty is to conveniently satisfy the incompressibility condition. In the following
we will first recall the continuous mathematical formulation of the Stokes problem. Then it is
recalled that a compatibility condition between pressure and velocity elements is necessary to
prove convergence. Finally several possible strategies to solve the discretized system are
developed.
2.1. The weak formulat ion of the Stokes problem
Let us consider the following Stokes problem on domain s with boundary conditions on
O~ = FlkgF2 (with FlnF2=O)
(1) o = 2nD(U)- p I d ,
(2) div(c) + F=0,
(3) div(U)=0,
(4) U = U o x ~ U1. c.n=O x E F2.
241
where n is a vector normal to the boundary at x. Multiplying equations (2) and (3) respectively
by test functions V (vanishing on F1) and q, and integrating by parts, we obtain the following
"weak" problem:
(5) ITID(U):D(V)- I V p . V = IF.V
(6) .[ q div(U) - 0 f~
Equations (5)-(6) hold for any test function V in a subspace of Hl(f~) including homogeneous
boundary conditions on 1"1 and for any q in the L2(f~) space. A continuous inf-sup inequality
ensures the existence and uniqueness of the solution of the weak problem (5)-(6).
The considered sohtion is a saddle point (minimum in V and maximum in q) of the Lagrangian: 1 (7) L(V,q) - ~ .f TID(V):D(V)- j'F.V - J" q div(V) .
f2 ~ f~
2.2. The numerical solutions
2.2.1. The elements
If a discrete inf-sup inequality is satisfied by the velocity and pressure elements, then the
discrete problem has a unique solution converging to the solution of the continuous one (see
equation (27) of the chapter Mathematical Analysis of Differential models for Viscoelastic
Fluids). This condition is referred to as the Brezzi-Babuska condition and can be checked
dement by element. Finite dement methods for viscous flows are now wen established and
pairs of elements satisfying the Brezzi-Babuska condition are referenced (see [6]). Two
strategies are used to compute the numerical solution of these equations involving velocity and
pressure.
2.2.2. The penalty method
Let us define the augmented Lagrangian as: =1 r )2 (8) Lr(V,q) ~ ~ rlv D(V):D(V)- ~F.V- ~ qdiv(V) + ~ I div(V
f~ f~ f~ f~
The solution is also a saddle point of the augmented Lagrangian: ~Lr ~gLr (9) ~ ( U , p ) = 0 , ~ ( U , p ) = 0
In order to solve equations (9), a generalization of the Uzawa algorithm is generally chosen for
its simplicity: 1 (10) ~ I rlv D(un) 'D(V) + r f div(Un) d iv(V)= ~F.V + f pn-1 div(V)
f~ f2 f~ f~
(11) pn = pn- 1 + r div(U n)
242
Equation (10) holds for any function V vanishing on F1. The last term of the augmented
Lagragian (for r----0, Lr is a Lagrangian) introduces a penalty of the incompressibility condition
and the Uzawa algorithm allows us to satisfy equation (3) as precisely as we wish using
moderate values of r.
2.2.3. The two-field solution
The linear system obtained by the discretization of equations (5)-(6) can also be solved
directly. Notice that this system is symmetric but not definite positive. A three-field version of
the Stokes problem was considered in [ 17] and a second inf-sup condition is then necessary to
obtain stability (equation (30) of 6.4).
2.3. Effect of a non-smooth geometry
We first recall some basic properties of Newtonian flows in two classical singular
geometries.
2.3.1. The stick-slip singularity
. . . . . . . . . .
0
Figure 1.a: The angular sector f2 of angle co and boundary F1 and F2
Let us consider as flow domain, the angular sector f~ of angle co and boundary F1 and F2
as represented in Fig. 1.a). The boundary of the domain is regular except at point O. The
boundary conditions are vanishing velocity on F1 (the fluid sticks to the wall on F1), vanishing
normal velocity and tangential stress vector on F2 (the fluid slips on the wall on F2). The
particular case of the half plane (co = x) is referenced as the stick-slip problem and represents a
local analysis of a fluid slipping on the wall downstream of point O. This problem has been
largely studied theoretically for a Newtonian fluid. It was proved that solutions of poor
regularity (presenting infinite values of pressure and viscous stress) at discontinuity exist.
These singular solutions have the form (in polar coordinates): A
(12) U(r,0)= r a U(O), p(r,0)= r a-1 ~(0),
and ct is computed as a solution of an eigenvalue problem for an elliptic equation in 0 (0<0_<co).
It can be proved (see for example [29] and [21 ]) that ct is solution of the following equation:
(13) sin(o~co) 2 = 1
243
However, the viscous energy is finite if ct>0 and the stress at the singularity is finite if 0t_>_ 1. 1 . For to = ~, ct = ~ is a solution. These solutions can be considered as similarity solutions,
eigenvector of the dilation r --~ ar.
2.3.2. The re-entrant comer singularity
Let us now consider the same flow domain (represented in Fig. 1.a) with the boundary
conditions of vanishing velocity on F] and 1-'2 (the fluid is sticking at the wall on F1 and F2).
This problem too has been largely studied for a Newtonian fluid. In this case, singular
solutions of the homogeneous Stokes problem exist if t~ is a solution of the following equation:
(14) (sin(otto))2 = (sin(to))2
Otto CO
It can be easily verified that there is at least one solution of equation (14) satisfying 0<ct<l if
3~ < to < 2~. For co = -~--, the smallest value of tx is approximatively 0.5445.
2.3.3. A realistic geometry
I
I r e - e l ~ l : I 1 ~ t c o 1 ' 1 ~ 1 " , , , "" I I I I I I I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 1.b: Capillary flow.
A classical laboratory experiment is viscosity measurement using capillary rheometry
(described and analysed in chapter V). The polymer is forced through a convergent geometry
(with a re-entrant angle) in a capillary and swells in air (Fig. 1.b in axisymmetric or plane
geometry). The relationship between the pressure drop in the capillary and the flow rate gives a
measure of the viscosity and the importance of swelling in air gives an estimate of the normal
stress difference. This classical geometry presents two geometrical singularities: the re-entrant
angle at the contraction and a change of boundary conditions at extrusion. Two particular
geometries, the so called 4:1 contraction and the stick-slip (an extrusion without swelling) are
considered as benchmark problems to test numerical codes. It should be noticed that both the
re-entrant angle at the contraction and the loss of contact with the apparatus at extrusion induce
high local stresses. The reader is referred to figure 34 of III.1 to have a physical idea of the
stress singularity at the corner. They show birefringence patterns at different flow rates
(;?=-25 s -1 and �89 s-l). It can be seen that an increase of the ray number at the corner is
244
induced by an increase of the flow rate. This leads to an increase in stress concentration. These
stress concentrations at the re-entrant comer or die exit, are considered as a possible explanation
of surface (melt fracture) or volume defects observed on the extrudate.
This notion is commonly used in solid mechanics. Investigation of the singular behaviour
of the stress at a singular point of the boundary of an elastic solid (particularly for a composite
material or in presence of a crack) is a tool with to wich study the limit of elasticity, even if the
stress is obviously not physically infinite and the same behaviour was observed for a
Newtonian flow in the presence of a singularity (it can be extended to a general elliptic
problem). In this case, values of the stress components are integrable despite the theoretical
infinite values of the stress components at the singularity. Hence this problem is easily
overcome by using mesh refinement and the computed values of the stress components increase
as the size of the mesh decreases at the re-entrant corner. These difficulties are largely
exacerbated by viscoelasticity.
3. FINITE ELEMENTS METHOD FOR VISCOELASTIC FLOWS
Let us consider the set of equations governing the flow of an Oldroyd-B fluid:
(15) o = T v + T N - p l d , 8Tv
(16) Tv + ~ ~ = 2 rlv D(U),
(17) TN = 2 tin D(U),
(18) div(U)=0,
(19) div(o)=0.
Substituting equation (15) in equation (19), and using equation (17), we obtain:
(20) div(Tv) + div(2 tiN D(U)) - Vp =(3
3.1. The general ideas
The non-homogeneous Stokes problem (18)-(20) in velocity and pressure is
mathematically coupled to transport equations (16) through Tv. In this case the elimination of
the tensor Tv is not possible, it has to be considered as a primitive variable. Two basic ideas
(introduced by Marchal and Crochet) guide these developments.
3.1.1. Upwinding
The viscoelastic constitutive equations are of hyperbolic type and it is well-known that
numerical solutions require special care. The classical Galerkin method was very shown to be
inadequate for such problems. Si~ecial techniques were developed based on the very general
245
idea of "upwinding" wich had been widely used finite differences. The hyperbolic nature of
these equations leads to introduce upwinding in order to make a difference between past and
future on the trajectory. If this is not done, the numerical scheme will be unstable. This can be
done in several ways. The Streamline-Upwind/Petrov-Galerkin (SUPG) method, introduced by
Brooks and Hughes ([23]), used modified test functions (a non consistent version is the SU
method). The streamline integration of the extra stress equations has been suggested by Shen
and largely developed by Tanner and co-workers ([26]). The discontinuous Galerkin method,
now largely used, introduces upwinding by the jump of discontinuous variables at the
interfaces.
3.1.2. The second inf-sup condition
The three-field formulation should reduce to a convenient approximation of the Stokes
problem when applied to a Newtonian flow. Hence a second inf-sup condition is necessary to
obtain stability. If the approximation (Tv)h of the extra-stress tensor is continuous, this
supplementary condition can be satisfied by using a sufficient number of interior nodes in each
element. On the contrary if this approximation is discontinuous, this can be done by imposing
that the derivatives DUn of the approximated velocity field are in the space of (Tvha. Various
possible choices concerning the satisfaction of the inf-sup condition and the introduction of
upwinding have been explored since 1987. In the following we will recall the basic steps (see
[ 10], [24] and [38] for details).
3.2. The elements of the three-field problem
3.2.1. The continuous approximation
An element for the stress components composed of 16 sub-elements (4x4) on which
bilinear (continuous) polynomials are used, was introduced by Marchal and Crochet in [28].
This leads to a continuous C o approximation of the three variables. The velocity is
approximated by biquadratic polynomials while the pressure is linear. Fortin and Pierre ([ 17])
made a mathematical analysis of the Stokes problem for this three-field formulation. They
conclude that the polynomial approximations of the different variables should satisfy the
generalized inf-sup (Brezzi-Babuska) condition introduced by Marchal and Crochet and they
proved it was the case for the Marchal and Crochet element. In order to take into account the
hyperbolic character of the constitutive equation, Marchal and Crochet have implemented and
compared two different methods. The first is the Streamline-Upwind/Petrov-Galerkin (SUPG).
Thus a so-called non-consistent Streamline-Upwind (SU) is also considered (already used in
[ 13]). As a test problem, they selected the "stick-slip" flow. With SUPG method applied to this
problem, wiggles in the stress and the velocity field were obtained. In the SU method, the
modified weighting function only applies to the convective terms in the constitutive equations.
246
This method is highly stable and results have been obtained at large values of the Weissenberg number for various flows. Figure 2 from [28] show the evolution of velocity and Tyy stress
component in a "stick-slip" geometry along the line x=l at De=10 for SU and S U t ~ method
with the 4x4 elements. The disadvantage of the SU method is that it introduces an artificial
diffusivity in the discrete form of the constitutive equations. In [3], a continuous biquadratic
approximation is used for the extra stress components. A method of characteristics is used for
the integration of the constitutive equation and this is time consuming. Difficulties arise with the
interpolation if deformed elements are used. However this method seems to be competitive
compared to many other algorithms.
V
1.'
0.75.
0.5'
0.25
8 O. 2.5 5. 7.'5 I(I.-Y'
0.75 "
0.5"
0.25i
O. C
o. 2.5 5. r:5 i 6 . ,
300.'
20(1.'
100,
O.
T lyy qlYw
200.'
loo:
T lyy .__... qlVw
- 2 . 5 O. 2.5 5. Y - 2 . 5 l d O. 2]5 5. Y
Figure 2: Evolution of velocity and Tyy stress component in a "stick-slip" geometry along
the line x= 1 at De= 10 for SU (a, b) and SUPG (c, d) 4x4 elements (Marchal & Crochet [28]).
3.2.2. The discontinuous approximations
A discontinuous Galerkin Q~ method based on the Lesaint-Raviart techniques was used in
[16]. An interesting point in this formulation is the use of an element by element basis if a
247
proper ordering of the elements is given. The ordering is such that when solving on an element,
the stress tensor has already been computed on elements adjacent to the inflow part of the
boundary. If this condition is not totally satisfied, this can be seen as an iterative process similar
to a block Gauss-Seidel method. This Q82 element for extra stress was coupled with a Q9-Q 1
(discontinuous) pair of elements for velocity and pressure. Maxwell, Oldroyd-B and PTT
model are considered in [ 19]. It is considered by these authors that numerical convergence with
the mesh size is impossible to reach when Oldroyd-B or Maxwell model are used on the "stick-
slip" geometry. Basombrio et al. ([4]) used continuous piecewise linear approximation for
velocity and pressure. In order to satisfy B.B. condition, each pressure element contains four
velocity elements. Viscoelastic stresses are approximated by piecewise linear and discontinuous
triangles (corresponding to the velocity mesh). This discretization satisfies the criterion
introduced by Marchal and Crochet. An important point is that linear interpolation allows one to
reach high Weissenberg numbers with no need for additional upwinding (SU). A change of
variable introducing a splitting of the extra-stress into the viscous part and the elastic part (S =
Tv - 2 TIv D(U)) has been suggested in [27]. This method, referenced as the Elastic Viscous
Stress Splitting method (EVSS) enhances numerical properties of the numerical scheme. This
method has been used in [31]. The rate of strain D tensor is introduced as an additional
unknown, leading to a four-field (U, p, S, I)) problem. In [18], the three-field problem (U, p,
S) is solved using discontinuous Galerkin Finite elements for S components. As a convective
derivative of the rate of strain tensor D(U) is introduced, an integration by parts is used.
3.3. The four-field method
The EVSS-G method introduced by Brown et al. uses the velocity gradient as an
additional unknown ([7]). In order to come back to primitive variable, Gutnette and Fortin
([20]) have introduced a (U, p, t~, D) method where no explicit change of variable is performed
in the constitutive equation. Hence this method is easier to implement. The elements used by
these authors are continuous Q92 for velocity, discontinuous P1 for and pressure and continuous
Q1 for a and D. This method was tested on the 4:1 contraction and the stick-slip problem. This
method seems robust and no limiting Weissenberg number was reached when using the PTI'
model for the stick-slip problem.
3.4 The solution of the discretized system
A difficult task is now to compute the solution of the' non-linear system of equations. In
this case the Newton iterative procedure gives the best convergence. However, due to the huge
number of equations, iterative methods were investigated.
3.4.1. The fixed point algorithm
248
The simplest coupling between the elliptic problem in pressure and velocity and the
hyperbolic equation in the viscoelastic part of the stress tensor is the Picard algorithm (fixed
point) and in this case each problem is solved successively. Let us first consider a three-field
method. For a given mesh, the convergence of the Picard algorithm is not reached even at
moderate values of viscoelasticity. For a given grid, the fixed point iteration does not converge
above a critical value of the Weissenberg number (the rate of convergence is then equal to 1).
Furthermore this critical value tends to zero with the mesh size. That means that convergence is
not reached if the mesh is f'me enough at the singularity. Concerning the four-field method, a
careful analysis of the fixed point iterative method is performed in [20].
3.4.2. The Newton algorithm
On the contrary, the Newton method gives robustness and quadratic rate of convergence.
The main drawback of this algorithm is the computation of a Jacobian matrix and the resolution
of a large linear system at each step. This method has been largely described and used
successfully. The reader is refered to [9] for more details.
3.4.3. The quasi-Newton algorithm
As the size of this matrix can easily reach 30 000 equations for two dimensional
applications on a rather coarse mesh, Fortin and Fortin ([ 16]) have used a Generalised Residual
Method (GMRes). This iterative solver uses a Krylov space of dimension k much smaller than
the dimension of the non linear system. The number of unknowns is the same but the storage
requirement is considerably reduced. This method is now largely used by various authors (see
for example [12]). Baaijens introduced ([2]) a Newton like algorithm to use with the
discontinuous Galerkin method. Using an implicit temporal discretization, linearization of the
set of discretized equations, except the integral over the input part of the element boundary, is
performed. This method is well adapted to any number of relaxation times.
3.4.4. The time dependent method
When studying the stability of the steady-state, time-dependent calculations are needed
(see [7]). It can also be used as a simple method to compute the steady-state solution. A time-
dependent approach using the Lesaint-Raviart technique for the normal stress components and
the Baba-Tabata scheme for the shear stress component is developed by Saramito and Piau
([34]). This method allows one to obtain rapidly stationary solutions of the PTT models.
Convergence with mesh refinement is obtained as well as oscillation-free solutions.
3.5. Numerical results in a smooth geometry
In a recent survey, it was concluded by Tanner ([39]) that smooth viscoelastic flows are
now computable. In such a geometry, the use of highly accurate numerical techniques (based in
[30] on a pseudospectral and finite difference approximation) allows the computation of the
249
converged solution at high Weissenberg numbers. From a theoretical point of view, the
introduction of inertia in viscoelastic flows can induce a "change of type" (see I1.2) in the
governing steady-state equations and it has been suggested that this change of type may be
responsible for the loss of convergence observed in finite differences and Galerkin finite
elements calculations. However, oscillation-free solutions have been obtained ([36] and [30]) in
an undulating tube even under conditions where the steady-state equations change of type. For
creeping flow, it has been shown by these authors that, even at relatively high values of
elasticity (De=10), the flow resistance does not deviate more than 5% from the Newtonian
value. A very precise study of numerically induced bifurcation is performed. The consistent
SUPG 4x4 method was revisited by Crochet and Legat [ 11] for the flow around a sphere in a
tube and an undulating tube. These authors have shown that the SU method is slowly
converging with mesh refinement, compared to the SUPG method. A precise comparison
performed in [35] leads to the same conclusion. In this paper the flow around a shere in a
periodic geometry is computed using EVSS-G/SUPG, EVSS-G/SU and spectral elements.
3.6. Numerica l results in a non smooth geometry
When the stress for the Newtonian flow is infinite at a singularity, the corresponding
viscoelastic flow problem becomes singular. Let us consider f'trst the stick-slip singularity.
3.6.1. Influence of viscoelasticity on the stick-slip singularity
Lipscomb et al. ([25]) established that the second order fluid theory leads to unphysical
stresses in the neighbourhood of a boundary discontinuity where a Newtonian singular solution
exists. They consider the existing descriptions of viscoelastic flow to be inadequate to describe
the physical behaviour near a geometrical singularity. Apelian et al. ([ 1]) show normal stress
behaving roughly like r -1 for this model. Tanner and Huang ([37]) have generalized the use of
singular solutions to Newtonian power-law fluid. These results gives estimates of the stress
concentration from simple boundary integrals. For the F I T model, numerical experiments
using special r 1/2 elements shows little improvement in convergence.
These difficulties can be understood by the following not rigorous computation. Let us
consider in this domain, the flow of a Maxwell fluid of relaxation time k. The velocity field U
and the stress tensor t~ are solutions of the following equations (as commonly accepted, inertia
and mass forces are neglected):
(21) c = T-p I d , ~T
(22) T + ;~ --~ = 2 rl D(U),
(23) div(U)=0,
(24) div(~)=0.
250
Let us notice U0 and o0 (resp. U 1 and ol) velocity and stress tensor at ;L=0 (resp.
derivative according to ~, of velocity and stress tensor at k---0 ). The velocity field U0 is the
solution of an homogeneous Stokes problem with a "stick-slip" singularity (co = g) and a
singular solution (U0(r,0)= r 1/2 U0(0)) exists. As it is easily verified U1 is the solution of the
following inhomogeneous Stokes problem:
(25) rl AU1 - Vpl = 2 div(F),
(26) div(U1)---0,
(27) F= UoV(D(U0))- V(U0).(D(U0) - D(U0).V(U0) t .
The rate of strain tensor D(U0) behaves r-1/2 like at the singularity and hence F behaves as r-
1. That means that the Stokes problem (25)-(26) has no solution in the classical variational
formulation using the Hilbert space of finite energy solution (the Sobolev Hi/bert space Hl(f~)).
The derivative of higher order according to the relaxation time ~. at ;L = 0 will be more singular
again. That proves that the solution of the Maxwell problem in the "stick-slip" configuration (if
it exists) is not regular according to the rheological parameters. This cannot be accepted from a
rheological point of view because it means a strong influence of measurement errors. This
analysis can be extended to the Oldroyd B model.
3.6.2. The re-entrant comer singularity
The same analysis as in the "stick-slip" case can be performed. In the neighbourhood of a
3g re-entrant-~-- comer, a similarity solution has been found for the Oldroyd B fluid ([22]) with a
velocity varying like r 5/9 and a stress singularity like r 2/3. A stress behaving between r -~ and
r -1 was reported in [25] and [8] from numerical behaviour. Renardy ([33]) has investigated the
stress of an upper convected Maxwell fluid in the neighbourhood of a re-entrant ~ comer,
assuming that the velocity field is Newtonian. This analysis shows boundary layers near the
wall corresponding to a transition from viscometric behaviour at the wall to a core region
dominated by the convected derivative. The stresses behave approximatively like r -~ in a core
region and like r -0-91 at the wall (the latter is simply the square of the Newtonian shear rate).
These computations show large spurious stresses downstream resulting from numerical errors.
3.7. The convergence with mesh refinement
The "High Weissenberg Number Problem" concerns numerical computation of
viscoelastic flows for an Oldroyd-B or a Maxwell fluid in presence of a geometrical singularity
(see [24] and [ 10]). As observed in section 3.3 and according to various authors, convergence
with mesh refinement is not obtained for several choices of the numerical scheme used for the
transport equation. Then, even for moderate value of the Weissenberg number, convergence of
251
the numerical coupling is not reached (the numerical solution cannot be found) and the velocity
field presents spurious oscillations. Due to the great complexity of the numerical techniques
involved, a categorical answer is not possible. However, it seems that, for an Oldroyd B
model, if the numerical scheme is not diffusive enough, the shock due to the re-entrant comer is
transported too sharply and pollutes drastically the velocity field. This mathematical difficulty
has been overcome in two ways. The first one is to stabilize this problem numerically. This can
be done by using large enough mesh at singularity, poor elements or diffusive schemes for the
transport equation. In the second way, the locally singular viscous stress is decoupled from the
non singular viscoelastic extra stress. This can be done by the choice of a constitutive equation
limiting the influence of the convected derivative at large values of the stress. All these solutions
have been considered.
3.8 The limitation of the extra stress singularity
This difficulty can be overcome by the use of a viscoelastic model limiting the effect of the
singularity in the transport equations. In the Modified Upper Convected Maxwell (MUCM)
proposed by Apelian et al. (see [ 1]), the relaxation time ~. is a function of the trace of the
deviatoric part of the extra stress tensor:
(28) ~,(tr(T))= ~.0 1 + (Ftr(T))a- 1
This model, for c~>l, reduces to a Newtonian behaviour near the comer but modifies the
difference of normal stress in a shear flow. Coates et al. present in [8] numerically stable and
accurate algorithms for the MUCM model in a contraction flow. However these results are still
constrained by numerical instabilities related to approximating the stress behaviour near the
singularity. Good convergence and oscillation-free solutions are obtained in [2] for high value
of the Deborah number (stick-slip problem) with the use of a Phan-Thien-Tanner (PTr) model
with bounded elongational viscosity by employing a fairly high value of the non linearity
parameter (E=0.25). Decreasing this parameter to E=0.001 leads to failure of convergence. The
influence of this parameter is studied for the linear and the non linear version of the ~ model
in a 4:1 contraction in [34]. Their conservative finite element method guarantees non artificially
oscillating solutions. Studying the "stick-slip problem", Fortin and Zine [18] obtained
numerical results showing that convergence with the mesh size is impossible with Oldroyd-B
and Maxwell model. They observe a limiting value of the Weissenberg number for ~--0 and
E--0.02 but none was reached for E=0.25.
252
4. APPLICATION
II
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3 J ,
-20 -I0 0 I0 20 30 40 50
Figures 3: Numerical plot of the u componant on the symmetry axis obtained by Fortin
and Zinc [ 18], SU, De = 2 ; 6 ; 11.
Despite the difficult numerical problems introduced by the computation of a viscoelastic
flow by using a differential law the main expected results such as important swelling or longer
stress relaxation after a convergent geometry are obtained. The 4:1 convergent geometry is a
classical test. An overshoot of velocity is observed after the contraction. Figure 3 presents a
numerical plot of the u component on the symmetry axis respectively obtained by Fortin and
Zinc ([ 18]). The relaxation of Txx and Tyy is longer and longer as the Weissenberg number is
increased. This is shown in the chapter devoted to comparison between numerical results and
experiments. This relaxation is reached on half of the outflow width in the Newtonian case and
this length is substantially increased by viscoelasticity. In the Newtonian case and in the
absence of inertia and gravity, the swelling of a Newtonian fluid is about 18%. It is a well
known experimental effect that swelling can be considerably increased by viscoelasticity.
Numerical results are compared with experiments in HI. 1.
5. CONCLUSION
Severe difficulties have been encountered for several years in the numerical simulation of
viscoelastic flow for differential constitutive equations. Let us now give a summary of the
numerical problems previously presented.
As described in 3.2, the solution of the discretized non linear system of equations is
difficult to compute. This is now well known and various adaptations of the Newton method
253
have been used with success. Given that numerically computed viscoelastic velocity fields are
not very different from the viscous ones it is surprising that sophisticated algorithms are
required for convergence.
Non-convergence with mesh refinement is reported by various authors. This is possibly
connected with the mathematical non existence of the solution for a singular geometry. The
theorem 3.1 of chapter 11 proves the existence of solutions in HS(f~) in a smooth geometry. As
the Newtonian solution does not satisfy this condition, this obviously fails in a singular
geometry. A stabilisation of the numerical method can be obtained by introducing diffusion in
the numerical upwinding or by a convenient projection appearing for example in the four-field
computation. The differential constitutive equation has been modified in various ways to avoid
transport of singularity. A strategy is to limit the growth of the viscoelastic part of the
singularity by using for example a P I ~ or a MUCM model. This allows convergence (for large
enough values of e) with mesh refinement but seems insufficient to obtain an easy computation
in a realistic geometry. These difficulties have not been, up to now, clearly understood from a
mathematical point of view. It seems that for a differential constitutive equation, the relaxation
time introduces a hyperbolic characteristic and hence a singular coupling between the elliptic
part and the transport equation.
Because of these difficulties, integral constitutive equations were used with success by
Tanner and co-workers. These allow one to introduce more complex relaxation behaviour
giving a better description of the complex flow around a reentrant comer (and more generally
on a complex trajectory presenting important variations on the rate of strain tensor). However
as Maxwell, Oldroyd B and ~ models have integral formulations, the use of an integral
constitutive equation is not sufficient to ensure computability of the solution. The computation
principles are, in fact, not so different. Basically the two major steps if a decoupled method is
used, are the computation of the viscoelastic stress and the solution of the momentum and
continuity equations with the stress treated as a body force (see [32]). The problem in this case
is the importance of the required storage due to the integral kernel. Hence precise numerical
studies concerning convergence with mesh ref'mement are not available.
Due the present storage and computation possibilities, the differential constitutive
equations are at this time largely used. However, the very precise studies developed in this
context will be useful to compute the history of the deformation on a trajectory. Hence
considering the great progress made in recent years concerning the numerical solution of a
transport equation on a given velocity grid, the computation of a transient 3D viscoelastic flow
in a complex geometry using an integral constitutive equation is possibly not so far away.
254
REFERENCES
[ 1] M.R. Apelian, R.C. Armstrong and R.A. Brown, J. Non Newtonian Fluid Mech. , 27
(1988), 299-322.
[2] F.P.T. Baaijens, J. Non Newtonian Fluid Mech., 51 (1994), 141-159.
[3] F.G. Basombrio, J. Non Newtonian Fluid Mech., 39 (1991), 17-54.
[4] F.G. Basombrio, G.C. Buscaglia and E.A. Dari, J. Non Newtonian Fluid Mech., 39
(1991), 189-206.
[5] A.N. Beds, R.C. Armstrong, R.A. Brown, J. Non Newtonian Fluid Mech, 22 (1987),
129-167.
[6] Brezzi and M. Fortin, Mixed and Hybryd Finite Elements Method, Springer, New-York,
1990.
[7] R.A. Brown, M.J. Szady, P.J. Northey and R.C. Armstrong, Theor. Comp. Fluid Dyn.,
5, (1993) 77.
[8] P.J. Coates, R.C. Armstrong and R.A. Brown, J. Non Newtonian Fluid Mech , 42
(1992), 141-188.
[9] M.J. Crochet, A.R. Davies and K. Waiters, Numerical Simulation of Non Newtonian
Flow, Elsevier (Ed.), Amsterdam (1984).
[ 10] M.J. Crochet, Rubber Chemistry and Technology, 62 3 (1989), 276-305.
[ 11] M.J. Crochet and V. Legat, J. Non Newtonian Fluid Mech., 42 (1992), 283-299.
[ 12] T. Dumont, Proc. 'Elasticitd, Viscodlasticit6 et controle optimal', Lyon, Dec. 95.
[13] B.A. Finlayson in G.F Carey and J.T. Oden (Eds), Proc. Fifth Int. Symp. on Finite
Elements and Flows Problems, Austin, Texas (1984) pp. 107-111.
[14] M. Fortin and D. Esselaoui, J. Num Meth. in Fluids, 7 (1987), 1035-1052.
[15] M. Fortin and A. Fortin, J. Non Newtonian Fluid Mech., 32 (1989), 295-311.
[ 16] M. Fortin and A. Fortin, J. Non Newtonian Fluid Mech., 36 (1990), 227-228.
[17] M. Fortin and R. Pierre, Comput. Methods Appl. Mec. Eng., 73 3 (1989), 341-350.
[18] A. Fortin. and A. Zine, J. Non Newtonian Fluid Mech., 42 (1992), 1-18.
[19] A. Fortin, A. Zine and J.F. Agassant, J. Non Newtonian Fluid Mech., 45 (1992), 209-
229.
[20] R. Gu6nette, M. Fortin, J. Non Newtonian Fluid Mech., 60 (1995), 27-52.
[21] P. Grisvard, EdF Bulletin de la direction des dtudes et recherches-Sdrie C, 1 (1986), 21-
59.
[22] E.J. Hinch, J. Non Newtonian Fluid Mech., 50 (1993), 161-171.
[23] T.J.R. Hughes and A.N. Brooks, Comput. Methods Appl. Mec. Eng., 45 (1984),.
255
[24] R. Keunings, Simulation of Viscoelastic fluid flow, in Computer Modeling for Polymer
Processing, C.L. Tucker III (Ed.), Hanser Verlag (1989) p. 403.
[25] G.G Libscomb, R. Keunings and M.M. Denn, J. Non Newtonian Fluid Mech., 24
(1987), 85-96.
[26] X.L. Luo and R.I. Tanner, J. Non Newtonian Fluid Mech., 22 (1986), 61-89.
[27] X.L. Luo and R.I. Tanner, J. Non Newtonian Fluid Mech., 31 (1989), 143-162.
[28] J.M. Marchal and M.J. Crochet, J. Non Newtonian Fluid Mech., 26 (1987), 77-114.
[29] H.K. Moffat, J. Fluid Mech., 18 (1963), 1-18.
[30] S. Pilitsis and A.N. Beds, J. Non Newtonian Fluid Mech., 39 (1991), 375-405.
[31] D. Rajagopalan, R.C. Armstrong, and R.A. Brown, J. Non Newtonian Fluid Mech., 36
(1990).
[32] D. Rajagopalan, R.A. Brown, and R.C. Armstrong, J. Non Newtonian Fluid Mech., 46
(1993), 243-273.
[33] M. Renardy, J. Non Newtonian Fluid Mech., 50 (1993), 127-134.
[34] P. Saramito and J.M. Piau, J. Non Newtonian Fluid Mech., 52 (1994), 263-288.
[35] M.J. Szady, J. Non Newtonian Fluid Mech., 59 (1995), 215-243.
[36] R.I. Tanner and X. Huang, J. Non Newtonian Fluid Mech., 27 (1988), 299-322.
[37] R.I. Tanner and X. Huang, J. Non Newtonian Fluid Mech., 50 (1993), 135-160.
[38] R.I. Tanner and H. Jin, J. Non Newtonian Fluid Mech., 41 (1991), 171-196.
[39] R.I. Tanner, Theoretical and applied rheology, P. Moldenaers and R. Keunings (Eds),
Proc. XI Int. Congress on rheology, Vol.1 Elsevier, Amsterdam, 1992, pp. 12-15.
This Page Intentionally Left Blank
Rheology for Polymer Melt Processing J-M. Piau and J-F. Agassant (editors) �9 1996 Elsevier Science B.V. All rights reserved. 257
Validity of the stress optical law and application of birefringence to polymer complex flows
R. Muller a and B. Vergnes b
aEcole d'Application des Hauts Polym~res, 4 rue Boussingault, 67000 Strasbourg, France
bCEMEF, Ecole des Mines de Paris, URA CNRS 1374, BP 207, 06904 Sophia-Antipolis, France
1. INTRODUCTION
The purpose of the present chapter is to introduce the techniques of birefringence, to define the validity of the stress optical law (i.e. the linear relationship between stress and birefringence) in both simple (simple shear and uniaxial elongation) , n d complex (flow in abrup t contraction) deformations, and to propose some examples of the interest of these techniques for the study of the flow of molten polymers.
Natural birefringence (or double refraction) has been known for some time for some particular materials, such as Iceland spar; it is due to a difference between the refractive indices along two principal directions. Most t r a n s p a r e n t isotropic mater ia ls are not birefringent, but can exhibit "accidental" birefringence when subjected to deformation or stress. This was first observed by Brewster [1] in 1816 and led to the development of solid photoelasticimetry, which allows one to establish relationships between stress and birefringence. Flowing liquids may also exhibit such birefringence [2]. Lodge [3] and Philippoff [4] first suggested that classical photoelastic relationships could also be applied to these materials.
2. GENERAL REI~TIONSHIPS AND USEFULNESS OF BIREFRINGENCE MEASUREMENTS
2.1 General relationships Flow birefringence is due to the optical anisotropy created by the
orientation of the macromolecules within the flow field. In a plane flow, if we denote by I and II the principal axes of the refractive index tensor (principal axes are those for which the tensor is diagonal; they are defined by the eigenvectors of the tensor), the local birefringence ~ is defined as:
A = nI- nil, (1)
258
where the refractive index n is the ratio of the speed of the light in a vacuum to tha t in the considered medium:
C n = - . (2)
V
After passage th rough a birefr ingent mater ia l of thickness e, the two components of linearly polarized light show a phase difference 5 equal to:
2 x e = - T - (hi - ni l) , (3)
where x is the wavelength of the light. A typical optical ar rangement for the measurement of birefringence is
given in Fig. la. A light beam (generally monochromatic) is linearly polarized. It then passes through the medium to be characterized before finally passing through an analyzer. The medium retards one component of the incident beam according to Eq. (3), and, when exiting the analyser, the intensity received by the observator may be expressed as [5]:
I = I0 sin 2 (213) sin 2 (~), (4)
where I0 is the intensi ty of the incident light, 13 is the angle between the polarization direction of the incident beam and the faster principal direction of the refractive index tensor and 5 is the phase difference.
monochromatic light polarizer test sample analyser
observation
(a)
monochromatic light polarizer test sample analyser
quarter ~ quarter wave wave plate plate
(b)
observation
Figure 1. Classical arrangements for birefringence observations
259
From Eq. (4), it may be deduced that extinction bands will be observed for two different conditions:
-13 = 0, x/2, ~, 3~/2 ... These bands are related to the direction of the principal refractive indices. They move when the polarizer and analyser are rotated simultaneously and are called "isoclinic" fringes.
- 5 = 0, 2x, 4~ ... These bands are called "isochromatic" fringes. They appear black for monochromatic light, but coloured for white light. From Eq. (3), it may be seen that they are related to the difference of the principal refractive indices:
kX (5) nI -nII = e '
where k (k = 0, 1, 2 ...) is the order of the fringe. This order is determined s tar t ing from the zeroth order fringe, which always appears black in white light and remains stationary when rotating the polarizers.
In flow birefringences studies, we are principally interes ted in the isochromatic fringes, because they are directly connected to the stress field (see paragraph 2.2). Moreover, observation of isoclinics are often difficult in slit geometries, due to parasitic effects as reported by Wales [6] and Mackay and Boger [5]. It is possible to avoid these isoclinic fringes by inserting two quar ter wave plates along the trajectory of the light beam, as presented in Fig. lb. More sophis t i ca ted techniques can be used for de t e rmin ing di rect ly and simultaneously the parameters of the local birefringence (for example, the polarization modulated flow birefringence [7, 8]), but they will not be used in the present study.
2.2. Usefulness of birefringence measurements As explained previously, flow birefringence is due to the optical
anisotropy created by the orientation of the macromolecules within a stress field. It will thus be an interesting tool for studying orientation and stresses.
a. Orientation: Birefringence is a simple way to measure the average orientation of
polarizable molecular units, e.g. monomer units or statistical segments along the chain axis in a polymeric material. An electric field E induces on each unit a dipole p = _a.E with the molecular polarizability tensor _a. The components of p along the principal directions of ~ are ~-i E cos(~i), where ~i are the angles between the direction of E and the principal directions of ~. The contribution of this dipole to the macroscopic polarizability along the direction of the field therefore involves trigonometric functions of no grea ter complexity t han cos2(~0i).
If one assumes fibre symmetry of the sample (like in uniaxia l deformation), transverse isotropy for the molecular units (for instance a l r a2 = ~3), and additivity of polarizabilities, it is easily shown [9] that the difference in macroscopic polarizabilities along and perpendicular to the fibre axis of the sample is simply:
260
PI- P" = N(a~-a2 ) ( 3(c~ q~)-2 1) (6)
where ~ is the angle between the fibre axis and the principal axis of each molecular unit, and the average is taken over all N units. For non-absorbing systems, the Lorentz-Lorenz equation converts polarizabilit ies to refractive indices:
n 2 - 1 4 r~P = ~ , (7) 3 n2+2
which by differentiation gives the expression of birefringence:
0i 2 + 2) 4 ~(p, _ Pn ) (S) A = n~ - n H = 6n 3
where fi is the average refractive index of the material. Combining Eqs. 6 and 8 shows tha t the birefringence is simply related to the second moment of the
orientation distribution function P2(cos v)=l (3<cos2 0>_1):
3(cos 2 q0',> - 1 A = A~ x (9)
2
where Area x is the intrinsic birefringence obtained for full orientat ion of all uni ts .
It should be noted that , apar t from orientation birefringence considered above, two other phenomena may contribute to optical anisotropy: deformation birefringence related for instance to distortion of bond angles or bond lengths in glassy polymers, and form birefringence arising in mul t iphase systems when the phase dimensions are comparable to the wavelength of light.
To i l l u s t r a t e the use fu lness of b i re f r ingence m e a s u r e m e n t s in or ientat ion studies, we now briefly discuss two simple models of orientation leading to different expressions of the second moment of the or ientat ion function: the affine deformation model for rubbers and the pseudo-affine model more frequently used for semi-crystalline polymers.
Affine deformation network model: For a single Gaussian chain with n statistical links and end-to-end vector r, the distribution function for the angle between each link and the direction of r has been derived by Kuhn and Griin [10] as a simple expression of the inverse Langevin function of the relative chain extension r/nl, 1 being the length of a link. From the second moment of this distribution, the optical anisotropy of the single chain with respect to its end-to-end vector is shown to be:
261
3r
(PI- Pii)chain = n(O~l- {X2)1- n l (10) L-l(Br) "~-~
For a network of Gaussian chains having the same number n of links, un iaxia l ly s t re tched by an amount L/L0 = ~, the a s sumpt ions of affine displacement of junction points and initial Gaussian distr ibution of end-to-end vectors allows one to calculate the optical anisotropy of the ne twork by in tegra t ing Eq.10 over the distribution of end-to-end vectors in the stretched state. By taking Treloar's expansion [11] for the inverse Langevin function, the orientation distribution function for the network can be put into the form of a power series of the number of links per chain:
(cos / , , , / / - ( Z - + 1 ;~4 + _ _ (11) 25n ~ 3 3~f "
A similar series can be obtained for the tensile stress, the first te rm of which is given by:
(J()~) = vekT(~ 2- )~-1) + O(~2 ), (12)
showing tha t for small strains and a large enough value of n, the linear stress- optical rule comes out as a consequence of the s tat is t ical theory of rubber elasticity. The stress-optical coefficient C is simply obtained from Eqs. 6, 8, 11 and 12:
C = 2 :n: (n2+2) 2 45 kT fi (~ - a2 )" (13)
Pseudo-affine model: the deformation process of polymers in cold drawing is very different from that in the rubbery state. E lements of the s tructure, such as crystallites, may retain their identi ty dur ing deformation. In this case, a ra ther simple deformation scheme [12] can be used to calculate the orientat ion distr ibution function. The material is a s sumed to consist of t ransversely isotropic units whose symmetry axes rotate on s t re tching in the same way as lines joining pairs of points in the bulk mater ia l . The model is s imilar to the affine model but ignores changes in length of the uni ts tha t would be required. The second moment of the orientat ion function is simply shown to be:
0 v
f I3/ arccos ~ 1 3X, 3
r'2(cos( )) = i 4 z - 1 "
k (14)
Fig.2 compares the predictions of the pseudo-affine model and the a f fne model with different values of the number n of finks per chain. It is clearly seen tha t the pseudo-affine scheme gives a much more rapid initial orientation than the affine network model.
1,0
0,8
0,6
0,4
02
0,01 2 3 4 5
262
x=Uto
Figure 2. Second moment of the orientat ion dis t r ibut ion as a funct ion of un iax ia l s t rain. Broken line: affine ne twork model with n=30 (Eq. 11). Fu l l l ine: p s e u d o - a f f i n e model (Eq. 14).
b. Stresses: In solid photoelas t ic imetry, birefringence is related to local s t resses
through the s t ress optical law, which expresses tha t the principal axes of stress and refractive index tensors are parallel and tha t the deviatoric parts of the refractive index and stress tensors are proportional:
n = C a , (15)
where C is the stress optical coefficient, which is a function of the material and the t e m p e r a t u r e . More general ly , in a two dimensional s i tua t ion , the birefringence a is related to the the principal stress difference:
A = nI - nII = C (($I - o'II ) . (16)
The stresses and the refractive indices are expressed here in the principal axes (noted I and II), in which the tensors are diagonal. These axes generally do not coincide with the laboratory coordinate system. They are rotated with an angle x, which corresponds to the extinction angle determined by the isoclinic fringes.
263
The validity of the stress optical relationship for flowing liquids has been proved for ideal rubbers and flexible macromolecules at low extension. Many au thors have shown tha t this re la t ionship holds for a broader range of materials, providing the stresses are lower than 10 4 Pa in shear flow or 1 MPa in elongational flow (see Refs. [6, 13, 14] and part 3 of the present Chapter). Table 1 summarizes values of the stress optical coefficient C for some classical polymers.
Table 1 Values of the stress optical coefficient C (10 -9 m2/N)
Polymer Temperature (~ C References
PVC 100 0.23 / 0.35 [30] 210 0.5 [6]
HDPE
LDPE
LLDPE
PP
PS
150 2.4 [15] 190 1.8 [6] 210 1.5 [16]
150 2.0 [6]
145 3.o [is] 205 2.5 [18]
180 0.6 [13] 210 0.9 [6]
140 - 5 [30] 190 - 4.1 /-4.8 [6] 210 - 4.9 [13]
PDMS 25 0.16 / 0.18 [8] 25 0.13/0.17 [6]
PB 23 3.9 [17]
PC 140 5.8 [31]
If we consider a two-dimensional Poiseuille flow (x: flow direction, y: shear direction), the principal stresses may be expressed as:
~ = ~ (~xx + %y) + + (~xy2, (17)
1 ~ (O'xx 40"yy) 2 O'II = ~ ((~xx + Oyy) - + ~xy2. (18)
254
The angle x of the principal axes wi th the laboratory frame defined jus t above is given by:
2 ($xy tan 2 z = �9 (19)
(~xx - (Yyy
For a Newtonian fluid, ~xx - ~yy = 0 and )~ is equal to 45 ~ wha tever the position. For a viscoelastic fluid, x is equal to 45 ~ along the centrel ine, but its va lue decreases in the y-direction to reach a m i n i m u m at the die wall.
F rom Eqs. (16) to (18), we can deduce a r e l a t i onsh ip be tween the birefr ingence and the s t ress components:
~ (~xx - ~yy)2 a = 2 C 4 § (Yxy 2 �9 (20)
Along the centre l ine , whe re the s h e a r s t r e s s is zero, the b i re f r ingence is directly proportional to the normal s t ress difference:
a = C l axx- ~yyl" (21)
More genera l ly , the n o r m a l s t r e s s difference and the s h e a r s t ress can be de te rmined anywhere in the flow field us ing the following relat ionships:
A OXX - Oyy = ~ COS 2X' (22)
A ~ x y - 2 C sin 2~- (23)
3. VALIDITY OF THE S T R E S S O P ' H C A L LAW
3.1. Simple flows
a. E x p e r i m e n t a l techniques: Two different t echn iques were used for uniaxia l elongation and s imple
shea r . The stress-optical behaviour in uniaxia l elongation was character ized by
using the extensional r heome te r described in section 1.4. The s t re tching of the specimen can be performed a t cons tant elongat ional s t ra in ra te or a t cons tant tens i le s t ress . Typical in i t ia l d imens ions of the spec imens are 50 m m in length , 20 m m in w i d t h and 0.5 m m in th ickness . The b i re f r ingence is measu red during the whole tes t us ing the method described in Fig. l a wi th ~ = ~/4. The oil ba th used for h e a t i n g is equipped wi th glass windows al lowing a HeNe laser beam to cross the sample. Symmet r ic d i sp lacement of the c lamps allows the m e a s u r e m e n t s to be carr ied out nea r to the cent ra l pa r t of the specimen, where the de format ion is not affected by any end-effects due to
265
clamping. The th ickness of the specimen required for the calculat ion of birefringence according to Eq. 5 is calculated as a function of t ime by assuming tha t the deformation occurs at constant volume and that the ratio of thickness to width remains unchanged during the elongation. This last assumption may be checked at the end of the test when the sample is removed from the oil bath. More details about this equipment have been published previously [19].
In s imple shear , real t ime bi refr ingence m e a s u r e m e n t s are more difficult [20-22]. The sliding plate rheometer described in section 1.3 has been used to charac ter ize the birefr ingence in samples where the or ien ta t ion induced by the flow has been frozen-in by quenching below the glass t ransi t ion tempera ture . This rheometer allows one to control both the shear and normal forces dur ing simple shear flow in the mel t and to main ta in constant shear and normal s t resses dur ing cooling of the specimen. In this way, uniform shear or ientat ion can be induced in plates of up to 5 mm thickness. These plates are typically 8 cm in length and 5 cm in width. As for the S.A.N.S. experiments , 1 mm thick specimens have been cut in the principal shear ing planes from the sheared and quenched polymer plates, as shown in Fig.3.
If x, y and z are respectively the flow, gradient and vorticity directions, birefringence measurements in the y-z and x-z planes also lead to the second and th i rd normal stress differences, providing the material verifies the l inear stress-optical law:
) N2 = G y y - ~ z z " - ny - n z , (24)
1 N3 = ~xx -~ = ~(nx - n~). (25)
The first normal stress difference is simply obtained since N1 = N3-N2. In the x- y plane, the extinction angle and birefringence are related to the shear s t ress and first normal stress difference according to Eqs. 19 and 20.
266
b. Experimental results in uniaxial elongation: Two polymers have been used: polys tyrene (PS) from Elf-Atochem
(Lacqrene 1241H) and polycarbonate (PC) from Bayer (Makrolon 2805). The glass transit ion temperatures of these two polymers as determined by DSC are 95 and 145 ~ for PS and PC, respectively. The weight average molecular weight of PS is of the order of 250 000 g/mole and the polydispersity MwfMn is close to 2.5.
For tempera tures typically above Tg + 20~ a one-to-one relat ionship between the true stress ~ and the birefringence a is found for both polymers. In this tempera ture range, the stress-optical law is independent of t empera ture and strain rate [19, 29]. As shown in Fig. 4, the birefringence versus stress curves are l inear at low stresses: c < 2 MPa for PS and ~ < 4 MPa for PC. A stress-optical coefficient C in the rubbery or molten state can be determined as the initial slope of these curves: for polystyrene, we find C ~ - 4.8 10 -9 Pa -1 and for polycarbonate C -- 5.9 10-9 pa-1, in good agreement with the l i terature data [30, 31]. Our results confirm that C is positive for PC and negative for PS, due to the fact that the polarizability of the backbone of the PS chain is lowest in the chain direction. The deviation from the l inear stress-optical rule observed at high stresses can be qualitatively explained by finite chain extensibility which results in a saturat ion of the orientation [32]. This result is in agreement with birefringence data in complex flows (see Fig. 18). It is important to notice tha t if the t e m p e r a t u r e is high enough, the re la t ionsh ip be tween a and a is independent of temperature and strain rate, even in its non linear part. Due to the closely related influence of t empera tu re and pressure on the viscoelastic properties of polymer melts, it can be assumed that the same relation will also hold at elevated pressures like those of the complex flows considered below (sections 3.2 and 4.). The curves in Fig. 4, which are characterist ic of the behaviour of the melt at temperatures which are high enough with respect to the Tg, will therefore be called "equilibrium" curves.
5O o o o
x 40
<3
30
20
10
Figure 4. Equilibrium stress- optical curves for PS(Q,m) and PC ( - ) . T e m p e r a t u r e and s t ra in rate are: (~) 120 and 0.5; (m)126 and 0.1; ( - ) 157 ~ and 0.025 s "1.
5 10 15 20
o" (MPa)
In a second step, the stress-optical behaviour of the two polymers has been studied at temperatures closer to Tg, namely in the range 98.5-117 ~ for
267
PS, and in the range 142-155 ~ for PC. The following mechanical test has been adopted: the specimens are stretched at constant elongational strain ra te (0.05 s -1 for PS and 0.025 s -1 for PC) up to a final extension ratio L/L0 close to 2.5 for all of the samples. The stress is then allowed to relax at constant deformation.
The curves in Fig. 5 show the time evolution of the t rue tensile stress dur ing elongation and relaxation at various tempera tures for PS. It is observed t h a t dur ing elongation the stress rapidly reaches a threshold value and then inc reases more progressively. Similar ly , du r ing the re laxat ion , ~ first decreases rapidly by an amount close to its initial value during elongation, and re laxes more slowly af terwards . The main effect of t empera tu re on these curves is a pronounced increase of the initial threshold stress with decreasing tempera ture . A similar behaviour has also been observed for PC [33].
13_ ~ 6 v oo or) u.I n," 4 I-- fJ~
ELONC~TICN RELAXA'rION 15
-! ! i ,, ! | , i
2o 4o 6o 2'o ~ ~o TIME (s) T IME (s)
Figure 5. Tensile stress as a function of t ime at var ious t empera tu res . (~) 103; (i) 106; (~) i i i and ( . ) 117 ~
Figure 6. Birefringence as a function of t ime for the experiments of Fig. 5.
The birefr ingence for the same exper iments has been plot ted as a function of t ime in Fig. 6. In contrast to the stress curves, the time evolution of birefr ingence shows no initial threshold value : ~ increases and decreases smoothly during both elongation and relaxation.
If the above resul ts for PS are plotted on a stress-optical d iagram (i.e. b i refr ingence versus stress) as shown in Fig. 4, the curves in Fig. 7 are obtained. The data show that the equilibrium stress-optical law is obeyed for the elongation at 117 ~ and tha t there is only a slight deviation towards higher s t ress values at 111 ~ For still lower tempera tures , the deviation becomes more and more pronounced and indicates t ha t the above discussed initial s t ress (around 1 MPa at 106 ~ and 3 MPa at 103 ~ does not apparent ly contr ibute to the birefringence. The same type of deviation from the l inear s t ress-opt ical rule has a l ready been observed for PS [29]. The s t ress and birefringence data at the lowest temperature, which have been plotted in Fig. 7 for the elongation as well as for the relaxation, clearly show a hysteresis. In s tage (a), the s t ress first increases up to the initial value of 3 MPa, almost without any birefringence; this first step corresponds to a macroscopic defor-
268
o o 15 o
x ILl 0 Z 10 I.U
u . 5 w
m
!
1 2 3 4 5
STRESS (MPa)
Figure 7. Birefringence as a funct ion of s t r e s s for the experiments of Figs. 5 and 6 (see text ).
mation which is typically of the order of al./L0 = 10 %. Stretching the specimen fur ther (stage (b)) leads to an increase in both a and ~, but this occurs with stress values much higher than for the equil ibrium behaviour. As soon as the elongation is stopped, the stress rapidly relaxes by an amount corresponding to its initial value during stretching, whereas the relaxat ion of birefringence is a much slower process over this same time-period (stage (c)). At the end of this th i rd process, the represen ta t ive point in the ~-a d iagram is now on the equi l ibr ium curve. In the final stage (d), the re laxat ion of both s t ress and birefringence follows the equilibrium relationship.
A very similar behaviour is also found for PC in the t empera tu re range 142-155 ~
c. Exper imenta l results_insimDle shear: - 7 _
The exper iments in shear were carr ied out on PS samples us ing the sliding plate rheometer mentioned in Section 1.3. A constant shear stress axy is applied to the sample during a given time. The sample is then cooled down below Tg, and the birefr ingence in different di rect ions is m e a s u r e d as described in Fig. 3. By varying the creep t ime for different specimens, the birefr ingence a and the extinction angle x were obtained as a function of the shear s t ra in ~. One should notice tha t a new specimen is required for each value of ~. The whole set of experiments was repeated for different values of the shear stress. Fig. 8 shows the evolution of a and ~ as a function of ~ for axy = 5
104 Pa. The data show tha t both the birefringence and the extinction angle reach a steady value at high strains. For higher shear stresses, the s teady state was not obtained for shear s trains lower than about 5, which is the max imum value before end effects in the experiment become too important.
As will be shown below, the experiment in the sliding plate rheometer does not allow one to determine N1, since the normal force is in fact re lated to the second normal stress difference. For this reason, we studied the stress- optical law in shear by assuming tha t the principal directions of shear and re f rac t ive index are close to each o ther in the x-y plane. I t is then s traightforward to express the difference of principal stresses in the x-y plane
269
90
U. I 85 _ , , I
z ,c[: 75
70
65
i= 6o LLI 55
50
m
1,5
1,0
0,5
, n , t , I , t , I , 60 ,0 450 1 2 3 4 5
SHEAR STRAIN
0 0 0
x Figure 8. Ext inct ion angle (m) and birefringence ( a ) i n the x-y plane during a creep
test at ~xy = 5 104 P a.
as a function of the shear stress ~xy and the extinction angle z"
2(~xy ~ I - - ~ I I - - (26)
sin(rt- 2X) "
Fig. 9 shows the evolution of a as a function of ($I- ~II calculated according to Eq. 26 for different values of the shear stress. A deviation from the l inear relat ionship, increasing with shear stress, is observed in the initial stage of the flow, which is in agreement with the results obtained in uniaxial e longat ion (the birefringence increases more slowly than the stress). In the l inear par t of the curves, the slope corresponds to a good approximation to the stress-optical coefficient determined in uniaxial elongation.
o 10 8 x
I ~ I
, , 1 , 5 2,0
cr a (MPa)
Figure 9. Birefr ingence in the x-y plane during a creep t es t as a funct ion of the d i f f e r e n c e of p r i n c i p a l stresses calculated from ~xy and z for different values of the shear stress: (*) 5 104; (u) 105; (n) 2.4 105; (~) 4 105 Pa.
270
Birefringence measurements in the x-z and y-z p l a n e s a l s o allow one to answer the question whether the measured normal force FN is related to ei ther N1 or N2. Since the outer surfaces of the specimen normal to the x and z directions are both free surfaces (~xx = 0, ~zz = 0), one may indifferently write:
FN = 0"~ =-(~xx-Cyy) =-N1 (27> S
S = ~yr = ~ -ozz = N2 ,
(28)
where S is the surface of the specimens. Obviously, both conditions cannot be satisfied simultaneously. Figure 10 shows a comparison of FN/S, -N1 = N2-N3 and N2, N2 and N3 being determined from the birefringence measured in the y- z and x-z planes according to the l inear stress optical rule. I t appears clearly tha t the best agreement is obtained with the second normal s t ress difference, which means tha t the free surface conditions hold ra the r on the x-y than on the y-z surface. It should be noticed tha t the values of birefringence measured in the different planes were homogeneous within 10 % in the whole specimen, except close to the free surfaces normal to the x direction, where end effects were present.
The difference observed at small strains in Fig. 10 may be a t t r ibuted to a small compressive force ar is ing when the specimen is placed between the plates of the rheometer, and which has not entirely relaxed at the beginning of the experiment.
-(2),4
0-
-0,3 03
UJ n," I-- -02. ,._.1
.@.._.14 , I /
J
~ @@ m~m ~ ~ ~
_ -. , . .~ | | | 0'01 3 4
SHEAR STRAIN
Figure 10. FN/S, N2 and -N1 as a function of shear s t ra in during a creep test (see text). (~) FN/S; (w) N2; and ( . ) -N1.
d. Decomposition of stress: The comparison between the time evolution of s t ress and birefringence
during an elongational test at constant s t ra in ra te suggests t ha t the total tensile s t ress is made up of two contributions: (i) an entropic contr ibut ion arising from chain orientat ion and (ii) a second contr ibut ion re la ted to the
271
initial stress value, which is strongly temperature dependent and vanishes at t e m p e r a t u r e s h igher than Tg + 20 ~ If we assume that the entropic contribution is related to the birefringence through the equilibrium stress- optical law, we can calculate its time evolution from that of the birefringence and the curves in Fig. 5. This has been done for stretching experiments carried out at 98.5 ~ and 0.05 s -1 for PS and 142 ~ and 0.025 s -1 for PC.
~ 2 0
ft.
v
O9 15 (/) Figure 11. Total (n,,u) and entropic par t ( o , o ) of the tensile stress as a function of ~2_ ~-1 for PS (re,o) and PC (u,o) (see text).
0,0 0,5 1,0 1,5 2,0 2,5 3,0 ~2~-1
In Fig. 11 the total stress and its entropic part calculated from a have been plotted for PS and PC as a function of the quantity ~2 _ ~-1, where ~ is the extension ratio L/L0. An almost linear behaviour is found for both PS and PC for the entropic part of the stress, up to values of ~ - 2. A neo-Hookean shear modulus can be calculated as the slope of these curves: values of approximately 1 and 3 MPa are found for PS and PC, respectively. Although these values are higher than the respective plateau moduli of the two polymers (i.e. 0.2 MPa for PS and 1 MPa for PC), their ratio is close to that of the plateau moduli. Actually, the neo-Hookean moduli defined from the entropic part of the stress are closer to the values at the crossover between the G'(co) and G"(~) curves corresponding to the high frequency limit of the rubbery plateau (i.e. 0.5 MPa for PS and 2 MPa for PC). The non-entropic contribution to the stress appears in Fig. 11 as the difference between total and entropic stress. It is found that the non-ent ropic s t ress r emains a lmost cons tant dur ing the s t r e t ch ing experiment, apart from the initial overshoot observed only at temperatures very close to the Tg.
An a t t empt has been made to describe the part icular stress-optical behaviour observed close to Tg. The idea was to associate the entropic part of the stress with relaxation times corresponding roughly to the rubbery plateau and the terminal zone (see Fig. 12), whereas the non-entropic part is assumed to be related to shorter time relaxation phenomena (glass t ransi t ion and glassy state). This approach is similar to that proposed by Inoue et al. [34] who considered two contributions to the stress with different associated stress- optical coefficients.
272
10
10 A
4 n
% . . . . 0 0=5 1,0 1,5 2,0 log(co) ~-a= (MPa)
Figure 12. Master curves for G' (m) and G"(~) of PS at 173 ~ Full lines r e p r e s e n t the con t r ibu t ion of long r e l a x a t i o n t imes a s soc i a t ed wi th entropic stress.
Figure 13. Full lines: model predic- tions for the data of Fig. 9 (see text).
An upper-convected Maxwell model has been used wi th the full re laxat ion spectrum for the calculation of the stress, but for calculat ing the birefringence this spectrum has been restricted to long relaxat ion t imes as shown in Fig. 12. The model predictions for the data of the Fig. 9 are shown in Fig. 13. The deviations from the linear stress-optical rule are well accounted for by this very simple model. However, the model does not describe the stress da ta in simple elongation, and in par t icular the init ial s t ress values at temperatures close to the Tg.
3. 2. Complex flows
a. Experimental techniques: The abrupt contraction is a very interesting flow situation, which allows
one to characterize s imultaneously shear flow near the downstream die wall, elongational flow along the centreline, mixed flow in the converging section and exit effects. Fig. 14 shows a typical experimental set-up for a p lanar contraction [18, 23]. The die with t ransparent glass walls is continuously fed by an ext ruder . It consis ts in a large reservoir in which p r e s s u r e and tempera ture may be measured, and a final adjustable slit section, which may be changed to modify the flow geometry.
One important problem in such a design is to ensure tha t the flow in the die can be considered as two-dimensional . Wales [6] showed t h a t the perturbat ion introduced by the presence of the die walls was negligible as soon as the aspect ratio of the die (width over depth) was greater than 10. Moreover, this influence is more pronounced for isoclinic fringes than for isochromatics.
273
Figure 14. E x p e r i m e n t a l set-up for slit die birefr ingence (from [18]).
Table 2 shows t h a t the major par t of exper imenta l devices used in the l i t e ra tu re respec ts th is o rder of m a g n i t u d e , which allows a good compromise be t w een qual i ty of resul ts and facility of observation.
Table 2 Dimens ions (in ram) of cont rac t ion flow exper iments from the l i t e ra tu re
Reservoir Die land Aspect ra t io Contract ion Ref. (width x depth) (width x depth) ( w i d t h / d e p t h ) rat io 10 x 12 10 x 2 5 6 [24]
30 x 10 10 x i 10 10 [6] 20 x 1 20 10 [6] 30 x 1 30 10 [6]
19.9 x 25.4 19.9 x 1.99 10 12.8 [13]
7.6 x 15 7.6 x 1 7.6 15 [15]
9.5 x 14.8 9.5 x 2.5 3.8 5.9 [23]
25.4 x 10.2 25.4 x 2.54 10 4 [25]
16 x 20 16 x 2.5 6.4 8 [18]
24 x 8 24 x 2 12 4 [26]
45 x 20 45 x 2 22.5 10 [17]
57.2 x 5.08 57.2 x 1.27 45 4 [8]
10 x 3 10 x 1.2 8.3 2.5 [27]
14.7 x 14.7 14.7 x 2.3 6.4 6.4 [28]
274
The results presented in sub-sections 3.2.b and 4 were obtained for a 8/1 contraction, using the experimental set-up described in Fig. 14 (die land: 16 x 2.5 ram; aspect ratio: 6.4), with a sodium monochromatic light (wavelength: x = 0.589 ~m) and different die lengths (5 and 20 ram) and converging angles (45 and 90 o).
b. Experimental results: These results will be focused on the flow of a linear polyethylene (LLDPE
Lotrene FC 1010) previously characterized (Chapter II-I). In Fig. 15 are presented the general changes observed in patterns of isochromatic fringes when increasing the flow rate. When looking at a particular picture, different areas may be distinguished:
- the reservoir: upstream of the contraction, the stress level is minimal. When moving downstream along the centreline, we observe the first order fringe, then the second, third and so on, with quasi-circular shapes.
- the restriction: the birefringence level is very high here. On the centreline, birefringence reaches a maximum just before the land entry and then decreases. In the perpendicular direction, the order of the fringes continues to increase, to reach a maximum in the corner. This value is however very difficult to determine due to the very high number of fringes.
- the die channel: the order of the fringes decreases when the polymer relaxes along the channel. It may be seen on some pictures that the zeroth order fringe is not observed before the exit, which means that the stress relaxation is not achieved during these flows.
- the exit: a drastic change is observed just a few millimetres before the die exit, with an abrupt local increase, followed by a total relaxation in the extrudate.
A global quantification in terms of stresses of such a pa t tern would require the knowledge of the extinction angle x everywhere, to apply the expressions given in sub-section 2.2.b (Eqs. 22 and 23). In fact, the de terminat ion of x using the set-up presented in Fig.14 is very difficult, because, due to side effects, isoclinic fringes are very wide and imprecise [5]. For that reason, it is easier to follow the change in birefringence along the centreline, where its value is directly proportional to the normal stress difference (Eq. 21). An example corresponding to the different pictures of Fig. 15 is given in Fig. 16.
When increasing the flow rate, the number of fringes increases regularly. The stress build-up in the reservoir appears sooner and the relaxation in the channel is lower at high flow rate. The general evolution along the centreline remains similar, whatever the flow rate, and the maximum value measured at the entry of the downstream flow increases very regularly (Figures 16 and 17).
c. Validity of the stress optical law: The stress optical law has been carefully checked by Wales [6] in steady
shear conditions for a wide number of molten polymers. However, the experimental set-up limited the range of stresses to a maximum of 105 Pa. In elongational situations, a departure from the linearity was previously observed
275
Figure 15. Change in birefringence patterns with the flow rate (LLDPE Lotrene FC 1010, 205 ~
276
60 tO o 50 X
v
LU 40 o z LJ.J 3O 0 Z n" 20 Li.. ILl rr"
10
10 0 10 20
AXIAL POSITION (mm)
Figure 16. Change in birefringence along the centreline (Lotrene FC 1010, 205 ~ u 20 s -1, �9 84 s -1, o 133 sq).
60 A
o 50 T~
X
u.J 40 o z I.LJ
30 Z
LL. 20 ILl CC
10 !
/
/
~9 / ~
J
j O O s
1 0
I " . . . . I '~'
0 50 100 150
APPARENT SHEAR RATE (s-l)
Figure 17. Change in maximum birefringence on the centreline with the apparent shear rate (Lotrene FC 1010, 205 ~
for polystyrene, above 1 MPa by Matsumoto and Bogue [29] and 2 MPa by Muller and Froelich [19] (see also Fig. 5 and sub-section 3.1.b).
At h igher stresses, the number of fringes in an abrupt contraction makes it very difficult to determine the birefringence level, except along the centreline. It could thus be interesting, as proposed by Beaufils et al. [18], to evaluate the max imum value, measured jus t before the die entry, and to compare it to the corresponding theoretical value of oI- (~II, computed using a
277
flow simulation software. This method allows one to define correctly the order of magnitude of the stress optical coefficient, but its main drawback is that the result will depend on the constitutive equation chosen for the simulation. Effectively, as explained in Chapter III.2, it involves also the validity of the chosen constitutive equation and the computation method. However, the results obtained are similar to those issued from more specific techniques. For example, for the LLDPE studied here (Lotrene FC 1010), we obtained values ranking from 2.07 10 -9 m2/N (K-BKZ) to 2.21 (Phan-Tien Tanner), 2.57 (Generalized Oldroyd-B) and 2.88 (Inelastic Carreau-Yasuda).
In Fig. 18 are presented the results obtained for generalized Oldroyd-B and multimode PTT simulations (see Chapter III-2). It can be seen that, in this transient elongational situation at high strain rate (the elongational strain rate can reach approximately 120 s-l), the proportionality between birefringence and stress difference is only valid up to 0.2 MPa.
60
~ 50 o
X
v 40 W
Z LU 30 (.9 Z n" 20 LL LU EE
10
/
o ~ /
�9 ,0 O 0
/
f f
/ O ~ 0 0
0 . - , 0 ~
' .... I ' . I ' ' . I ' " '
1 2 3 4
PRINCIPAL STRESS DIFFERENCE (X 10 5 Pa)
Figure 18. Validity of the stress optical law for the LLDPE at 205 ~ (o" PTT, e" Generalized Oldroyd-B).
4. APPLICATION TO COMPLEX FLOW STUDIES
4.1 I n f l u e n c e o f d i e g e o m e t r y In Fig. 19 are presented, for the same flow rate, the isochromatic
patterns obtained for different channel geometries (long (20 ram) and short (5 mm) dies, 45 ~ and 90 ~ converging angles). If the shape of the fringes appears different for the 45 ~ convergent, the change along the centreline is very similar: stress build-up is identical, and the relaxation mechanism also, until reaching the proximity of the die exit, where it becomes very rapid for the shorter dies. It is clear that this similarity is only valid along the centreline. Other flow parameters (velocity field, streamlines ..) are obviously dependent on die entry geometry.
278
Figure 19. Change in birefringence patterns with the die geometry
(Lotrene FC 1010, 145 ~ ~/a = 36 s-l).
279
Figure 20. Change in birefringence patterns with the flow rate (LDPE FN 1010,175 ~
280
4.2 Influence of molecular s t ruc ture We study here the case of a branched polyethylene (LDPE FN 1010),
characterized by long chain branching, whose rheological behaviour has been previously described in Chapter II-1. Compared to linear LLDPE, it exhibits s t ra in-hardening in elongational situations and higher values of first normal stress difference.
Figure 20 presents birefringence patterns obtained for a long die (20 ram) at 175 ~ If the birefringence along the centreline changes similarly to the preceding cases, the general aspect of the pat tern is different, principally around the re-ent rant corner. Characteristic W shapes are observed for the LDPE, whereas LLDPE exhibited more regular evolution. These shapes result in a non-monotonic variation of birefringence between the centreline and the die walls (see Chapter III-2). The area affected by this particular behaviour is c oncen t r a t ed a round the channel en t ry and develops progress ive ly downstream when increasing the flow rate. For the long die, the pat tern far downstream appears regularized even at high flow rate, whereas, for the short die, the perturbed area progresses until the die exit. Similar observations were made by Checker et al. [15] on a polypropylene and explained by a high level of elasticity combined with a strain hardening elongational behaviour, which is also the case for the LDPE.
4~3 Flow ~n.c~abflities We have seen that flow birefringence is a powerful tool for studying
steady and t ransient flows of molten polymers. This technique can also be used to assess local flow conditions during flow instabilities. These problems are specifically t reated in Chapter III-4. We would just like to present here the potential of birefringence techniques for such studies.
Tordel la [35] was probably the first to show HDPE and LDPE birefringence pat terns highly perturbed after the onset of flow instabilities. Discontinuities in the extinction bands and non stationary patterns were then observed, which was confirmed la ter by Vinogradov et al. [24] on polybutadienes or Oyanagi et al. [36] on HDPE and PS. Piau et al. [17] observed on polybutadiene slight pulsations at the die exit, correlated to the frequency of the sharkskin defect.
Figure 21. Discontinuities in the birefringence pattern in the die land
at high flow rate (LDPE FN 1010, 145 ~ �89 = 170 s-l).
281
We can see in Fig. 21 that, for the LDPE (FN 1010) above a given flow rate, bands of discontinuity appear in the birefringence pattern. These bands are observed parallel to the flow direction, at an intermediate position between centreline and die wall. Such "defects" may also be detected in the converging flow near the die entry (Fig. 22). For these flow conditions, the extrudate begins to exhibit distorded volume shapes, characteristic of melt fracture. By comparison with rheological studies carried out on a capillary rheometer, we may assume that these defects belong to the family referred to as "upstream instablities" in Chapter III.4.
Figure 22. Discontinuity bands at the die entry
(LDPE FN 1010, 145 ~ ~/a = 287 s "1)
Fig. 23 illustrates another type of unstable behaviour for a polystyrene. In this case, an alternative discharge of the large entry vortices in the main flow leads to a periodic asymmetric pattern, both in reservoir and die land, resulting in an helical type defect at the die exit. Such observations were also reported by Oyanagi et al. [36].
5. CONCLUSION
We presented some applications of the birefringence techniques for the characterization of orientation and stress field in both simple and complex deformations. The validity of the stress optical law in these particular flow situations was checked and assessed. Birefringence techniques appear to be very powerful for the study of polymer melt processing and for the understanding of polymer flow behaviour.
282
Figure 23. Oscillating sequence for a polystyrene
(PS Gedex 1541, 180 ~ �89 = 200 s -1)
283
REFEPd~qCES
I o
2. 3. 4. 5.
.
.
8. 9.
10. 11. 12. 13.
14.
15.
16.
17.
18. 19. 20.
21.
22. 23.
24.
25. 26.
27.
28 29.
30. 31.
D. Brewster, Trans. Roy. Soc. London (1816) 156 M.E. Mach, "Optisch-Akustische Versuche", Calve, Prague (1873) A.S. Lodge, Nature, 176 (1955) 838 W. Philippoff, Nature, 178 (1956) 811 M.E. Mackay and D.V. Boger, in: "Rheological Measurement", A.A.
Collyer, D.W. Clegg Eds., Elsevier, London (1988) J.L.S. Wales, "The Application of Flow Birefringence to Rheological
Studies of Polymer Melts", Delft University Press (1976) P.L. Frattini and G.G. Fuller, J. Rheol., 28 (1984) 61 S.R. Galante and P.L. Frattini, J. Non Newt. Fluid Mech., 47 (1993) 289 I.M. Ward (ed.), Structure and Properties of Oriented Polymers, Applied
Science Publishers, London (1975) W. Kuhn and F. Grtin, Koll. Zeit., 101 (1942) 248. L.R.G. Treloar, Trans. Faraday Soc., 50 (1954) 881. O. Kratky, Koll. Zeit., 64 (1933) 213. C.D. Han, "Rheology in Polymer Processing", Academic Press, New
York (1976) H. Janeschitz-Kriegl, "Polymer Melt Rheology and Flow Birefringence",
Springer Verlag, Berlin (1983) N. Checker, M.R. Mackley and D.W. Mead, Phil. Trans. R. Soc. London,
A 308 (1983) 451 S.T.E. Aldhouse, M.R. Mackley and I.P.T. Moore, J. Non Newt. Fluid
Mech., 21 (1986) 359 J.M. Piau, N. E1 Kissi and A. Mezghani, J. Non Newt. Fluid Mech., 59
(1995) 11 P. Beaufils, B. Vergnes and J.F. Agassant, Int. Polym. Proc., 2 (1989) 78 R. Muller and D. Froelich, Polymer, 26 (1985) 1477. J.A. Van Aken, F.H. Gortemaker, H. Janeschitz-Kriegl and H.M. Laun,
Rheol. Acta, 19 (1980) 159. K. Osaki, S. Kimura and M. Kurata, J. Polym. Sci., Polym. Phys. Edn
19, (1981) 151. R.M. Kannan and J.A. Kornfield, Rheol. Acta, 31 (1992) 535. G. Sornberger, J.C. Quantin, R. Fajolle, B. Vergnes and J.F. Agassant,
J. Non Newt. Fluid Mech., 23 (1987) 123 G.V. Vinogradov, N.I. Insarova, B.B. Boiko and E.K. Borisenkova,
Polym. Eng. Sci., 12 (1972) 323 S.A. White and D.G. Baird, J. Non Newt. Fluid Mech., 29 (1988) 245 H.J. Park, D.G. Kiriakidis, E. Mitsoulis, K.J. Lee, J. Rheol., 36 (1992)
1563 M. Van Gurp, C.J. Breukink, R.J.W.M. Sniekers and P.P. Tas, SPIE,
2052 (1993) 297 R. Ahmed and M.R. Mackley, J. Non Newt. Fluid Mech., 56 (1995) 127 T. Matsumoto and D.C. Bogue, J. Polym. Sci. Polym. Phys. Ed., 15 (1977)
1663 W. Retting, Colloid Polym. Sci., 257 (1979) 689. R. Wimberger-Friedl and J.G. De Bruin, Rheol. Acta, 30 (1991) 419.
284
32.
33. 34. 35. 36. 37.
L.R.G. Treloar, The Physics of Rubber Elasticity, Clarendon Press, Oxford, 1949.
R. Muller and J.J Pesce, Polymer, 35 (1994) 734. T. Inoue, H. Okamoto and K. Osaki, Macromolecules, 24 (1991) 5670. J.P. TordeUa, J. Appl. Polym. Sci., 7 (1963) 215 Y. Oyanagi, K. Kubota and K. Ohji, 4th PPS Meeting, Orlando (1988) M.R. Mackley and I.P.T. Moore, J. Non Newt. Fluid Mech., 21 (1986) 337
Rheology for Polymer Melt Processing J-M. Piau and J-F. Agassant (editors) �9 1996 Elsevier Science B.V. All rights reserved. 285
C o m p a r i s o n b e t w e e n e x p e r i m e n t a l d a t a and n u m e r i c a l mode ls
J. Guillet, C. Carrot, B.S. Kim a J.F. Agassant, B. Vergnes, C. Bdraudo b
J.R. Clermont, M. Normandin, u B~reaux c
a Laboratoire de Rh~ologie des Mati~res Plastiques, Facultd de Sciences et Techniques, Universitd Jean Monnet, Saint-Etienne 23 rue du Docteur Paul Michelon, 42023 Saint-Etienne Cedex 2, France
b CEMEF, Ecole des Mines de Paris, URA CNRS 1374 BP 27, 06904 SOPHIA-ANTIPOLIS Cedex, France
c Laboratoire de Rhdologie, URA CNRS 1510 Domaine Universitaire, BP n ~ 53 X, 38041 GRENOBLE Cedex 9, France
1. INTRODUCTION
A complete understanding of flow phenomena occurring in entry and exit regions of complex geometries is still elusive at the present time for viscoelastic fluids. Understanding entry and exit flows of non-Newtonian fluids like polymer melts is of importance in polymer processing operations, such as extrusion and injection moulding.
Typically, Newtonian fluids flow radially through the entry region in a contraction geometry, sweeping out the corners of the upstream channel and exhibiting small size vortices. They give a constant extrudate swell ratio for the jet emerging from the downstream channel. At the opposite, large corner vortices may appear for many viscoelastic fluids. Concerning the extrudate jet emerging from the die, the swell ratio is not only higher than that corresponding to a Newtonian fluid, but also depends on flow conditions, such as the range of strain rates, the temperature and the length-to-diameter ratio of the die.
An illustrative example of the previous considerations may be given for polyethylene melts. It is admitted that low density polyethylene (LDPE) melts develop rapid vortex growth in an abrupt contraction, and that high density polyethylene (HDPE) and linear low density polyethylene (LLDPE) melts do not. However, in exit flows, all these polyethylene melts can swell notably, and, for many years, there has been no clear understanding about differences in entry and exit flows of these polymer melts.
286
Observation, discussion and u n d e r ~ d i n g of recircttlating vortices in flows of polymer solutions and melts through abrupt contraction geometries have been the subject of many papers over the last 30 years. (e.g. review papers by Boger [1], White et al. [2], McKinley et al. [3])
The question of the presence, origin and growth of vortex remained unanswered until the works by Cogswell [4] and White and Kondo [5]. In both papers, it was suggested that materials involving an increasing extensional viscosity versus the extensional rate exhibit vortex enhancement, whereas vortices are small or absent for materials of constant or decreasing extensional viscosity. In further papers [6-7], the magnitude of the t ransient elongational stress growth was emphasized to be the most significant, whether vortices occur or not. The combined effects of strain rate and molecular characteristics such as molecular weight distribution and long chain branching [8-9] may imply quite different behaviour for the stresses through the entrance of the contraction. Thus, it would not be unusual to observe differences in entry flow patterns of polymers of a given type. This definitive conclusion may also be considered as relevant to the exit flow behaviour of molten polymers, since it is generally admitted that the extrudate swell phenomenon originates in the main from extensional strain recovery.
From a numerical viewpoint, rapid progress has been made in the last few years in studies generally devoted to the entry flow problem, together with the use of more and more realistic constitutive equations for the fluids. Consequently, more complexity was involved for the numerical problem, in relation to the non- linearity induced by the rheological model in the governing equations. The use of nonlinear constitutive models required approximate methods for solving the equations, such as finite element techniques, even for isothermal and steady-state conditions related to a simple flow geometry.
In the story of numerical flow simulation, the ability to predict observed and significant viscoelastic flow phenomena of polymer melts and solutions in an abrupt contraction has been unsuccessful for many years, in relation to the incomplete rheological characterization of materials, especially in elongation. The numerical t reatments have often been confined to flow of elastic fluids with constant viscosity, described by differential constitutive equations as the Upper Convected Maxwell and Oldroyd-B models. Fortunately, the recent possibility to use real elastic fluids with constant viscosity, the so-called Boger fluids [10], has narrowed the gap between experimental observation and numerical prediction [11].
Various numerical studies indicated a loss of convergence at relatively low values of the Weissenberg number. The so-called ~high Weissenberg number problem" relates the inability to obtain numerical solution for flows in complex geometries involving singularities, for high or even moderate values of the Weissenberg number [ 12]. In particular, it was found tha t the greater the mesh was refined, the lower the maximum Weissenberg number for convergence.
Several research groups have concentrated on the problem of computational instability. The failure of calculations has often been attributed to approximation errors, which refer to the inability of the numerical method to fit the set of governing equations. So far, mlmerical studies have revealed that the usual computational instability, occurring with simple but unrealistic models, was overcome to a great extent by the use of new integral models, which have enabled
287
more realistic Weissenberg numbers and extrudate swell ratios to be reached [13- 16]. Nevertheless, it was shown recently that the convergence problems were at least partially due to the use of a constitutive equation which did not exhibit a steady solution at the Weissenberg number attempted, rather than to numerical difficulties [17].
In pioneering work, Keunings and Crochet [18] used the Phan-Thien Tanner model, with an additional purely viscous retardation term to correct the unrealistic rapid viscosity decrease predicted by this model in simple shear flow. The improvement of convergence with a refined mesh was found to be opposite to that observed in previous numerical studies. This improved convergence might be due to the Phan-Thien Tanner model itself or to the additional retardation term, which appears to stabilize the solution.
It should be pointed out that the improvement of convergence might also be related to realistic preditions of shear and elongational viscosities by the Phan- Thien Tanner model, when compared to the Upper Convected Maxwell, Oldroyd-B and White-Metzner models. Satisfactory numerical results were also obtained with multi-mode integral constitutive equations using a spectr~_~m of relaxation times [7, 17, 20-27], such as the K-BKZ model in the form introduced by Papanastasiou et al. [ 19].
Although different numerical approaches were used by these research groups, the calculations revealed large vortex growth of LDPE with increasing Weissenberg number, similar to that observed experimentally in an axisymmetric abrupt contraction. However, the differences in vortex growth of LDPE and HDPE related to their elongational behaviour, presented in the paper by Luo and Mitsoulis [7] have to be pointed out. Their computed results indicate an increasing vortex size and intensity versus the Weissenberg number for strain-thickening extensional viscosity of the melt under consideration. On the other hand, for a fluid which exhibits a strain-thinning extensional viscosity, the weak vortex size and intensity observed at low flow rates are insensitive to the increase of the Weissenberg number. In addition, the authors came to the conclusion that strain- hardening apparently enhances the vortex, but is not responsible for its presence or absence in the way that the level of elasticity is.
Feigl and Ottinger [28] presented an extensive comparison of numerical and experimental results for the entry flow problem. They used a Rivlin-Sawyer integral constitutive equation, with a spectrum of relaxation times and a non-zero constant ratio of the second to first normal stress difference. The general performance of the model was tested by comparing the numerical predictions of vortex growth and entrance pressure loss at different flow rates with the experimental results related to the axisymmetric contraction flow of a LDPE melt reported by Knobel [29]. The numerical results were found to be consistent with experimental results and in some cases, in quantitative agreement. A good consistency was obtained for the vortex size and the increase of shear and extensional strain rates in the upstream channel. However, small or even large discrepancies still remained concerning the increase of the entrance pressure loss with flow rate and the evolution of shear strains in the flow geometry.
Birefringence measurements are often performed to compare theoretical and experimental stress distributions in an abrupt contraction (see Section III-1). Such comparisons have been already published for the White-Metzner [30], K- BKZ [27, 31] and Wagner [32, 33] constitutive equations. Generally speoking,
288
stress increase in the contraction entry is well described, but the relaxation along the downstream channel is often more difficult to capture.
Considering now the extrudate swell problem, a few papers deal with the flow simulation of memory integral fluids at the exit of long and short dies [22-25, 27, 34-35]. Some of the papers previously cited present comparisons between experimental and numerical results. Goublomme et al. [23] presented numerical results for the extrudate swell problem for a long capillary die. In their calculation, they used a K-BKZ constitutive equation involving a spectrum of relaxation times and a damping function, either of the Wagner [36] or of the Papanastasiou type [19]. To avoid the rapid viscosity decrease of the model at high shear rates, the authors introduced a weak Newtonian contribution, ass,~med to be equivalent to the effects of the smallest relaxation times of the true spectr~m, which are not available because they relate very few data accessible at very large frequency values in dynamic experiments. If the calculation provided consistent numerical results for a long capillary die, the assumption of a short die, which required the inclusion of the upstream converging section in the calculations, resulted in largely overestimated values of the swell ratio. The numerical data presented in a second paper [24] proved that, if extrudate swell was practically unaffected by non- isothermal flow conditions, it was considerably modified by change of parameters in the K-BKZ equation. A non-zero constant ratio of the second to first normal stress difference led to significantly lower computed extrudate swell ratios towards the experimental data, as a consequence of modification of the extensional behaviour of the fluid.
The experimental and numerical results reported by Luo and Tanner [22] are of particular interest since they are related to a thoroughly characterized IUPAC LDPE melt. In order to ensure good predictions in mixed flow situations, they relaxed the separability assumption in the K-BKZ form adopted in their study, at least in elongation, allowing a parameter of the damping function to vary according to each relaxation time of the memory function. Predictions close to experimental data were obtained at low shear rates (~ = 1 s-l), even with very short dies. However, when increasing the shear rate, the m~merical results were found to be larger than experimental ones. As pointed out later by Goublomme et al. [23], neither the non-isothermal flow assumption, nor the arbitrary decrease of normal stress ratio can affect the numerical swell predictions, to eliminate completely the discrepancy between numerical and corresponding experimental swell ratios. It should also be underlined that the calculated extrudate swell was found to be very sensitive to the wall slip boundary condition. This states a problem for the choice of a suitable criterion for wall slippage.
A similar study is reported in the recent paper by Barakos and Mitsoulis [25]. It is worth mentioning that simulations for the same IUPAC LDPE melt have been undertaken, using the same constitutive equation with parameters determined by Luo and Tanner [22]. Comparison are made with extrusion experiments given by Meissner [37]. Concerning calculations for short and long dies, the trends are similar to those observed by Luo and Tanner [22]. If the agreement is generally good for short dies and low shear rates, meaningful overestimation becomes larger and larger as the die length-to-diameter ratio and the apparent shear rate increase. It should also be noted that quantitative discrepancies still remain between calculated and experimental entrance pressure drops (or Bagley end corrections).
289
In the purpose of comparisons between theory and experiments, this short review points out the need for significant experimental studies related to flow pat terns , pressure drops and extrudate swell ratios for polymer melts characterized in shear and elongation. Streamlines, velocity components and stress fields related to flow birefringence studies should be used to validate numerical simulations of entry and exit flows. Progress in flow computations in domains involving the upstream and downstream channels together with the exit region are also necessary.
In order to get a better insight into these major issues, the purpose of the present work is to report some experimental and numerical results on flows in planar and axisymmetric contractions for two polyethylene melts, namely a linear low-density polyethylene (LLDPE) and a low-density polyethylene (LDPE). Various constitutive equations are set up to model more or less thoroughly the rheological properties in simple flow, such as a memory-integral Wagner-type model, and the differential multimode Phan-Thien Tanner (m FI~) and generalized Oldroyd-B (GOB) models. From a numerical viewpoint, comparisons with experiments will be made between results provided by two numerical methods, the first one involving a discontinuous Galerkin finite element method [38, 39] and the second one considering the stream-tube analysis developed by Clermont [40].
2. MATERIALS
The polymers used in the present work are two commerdal polyethylenes, supplied by Elf Atochem and previously characterized (see Section II - 1) : a linear low density polyethylene (LLDPE FC 1010) and a low density polyethylene (LDPE FN 1010).
The LLDPE material is an ethylene-butene copolymer, containing 4 % of butene. The LDPE polymer is known to involve long chain branching while LLDPE has a ra ther low content of branches, which are known to be short. These structures explain why large differences could be expected in their rheological behaviour in shear and elongation and, consequently, in complex flow situations. The molecular characteristics of both samples are given in Section II - 1.
3. CONSTITUTIVE EQUATIONS
The description of flow patterns in entry and exit flows requires one to provide an adequate mathematical and mechanical description of the behaviour of polymer melts, by means of realistic constitutive equations. These equations must describe the melt behaviour from linear viscoelasticity conditions to nonlinear flow situations. Though it should be emphasized that the constitutive equation chosen for any process should provide the information required, particular rheological equations of state are generally considered as more realistic for polymer solutions and melts. These models involve integral forms of constitutive equations of the K- BKZ type, the advantage of which lies in their connection to molecular aspects of viscoelasticity. With this type of equation, numerical calculation are often difficult to perform, due to the particle tracking problem and computing time requirements. These difficulties can be overcome to a great extent using stream-tube analysis.
290
Differential equations of the Oldroyd-type are easier to implement for flow computation. According to these considerations, we selected an integral constitutive equation of the Wagner-type, and two differential constitutive equations, namely the multimode Phan-Thien Tanner (mPTT) and a generalized Oldroyd-B (GOB) models. It should be noted that two of these models, the Wagner- type and the mPTT, predict bounded extensional viscosities as well as shear- thinning viscosities, whatever the strain rate is; conversely, the GOB model predicts infinite extensional viscosity in the long time range.
The LLDPE and LDPE polymers have been thoroughly characterized in simple shear and extensional flows in a previous part of this book (see Section II - 1). To enhance the u n d e ~ d i n g of the present section, we recall rapidly the basic features of these models.
3.1. The Wagner - type in tegra l constitutive equation Derived from the Lodge's rubberlike liquid theory, the Wagner model is based
on a concept of separability, since it is assumed that the memory function is the product of a t ime-dependent linear function by a s t rain-dependent nonlinear function.
When considering an increasing deformation, this model, which is a particular case of the BKZ model, is written as follows :
t
g(t) = m(t- 1:). h(Ii,I2). (~ t (1:)d2: - o o
(1)
where m ( t - z) is the memory function of the linear viscoelasticity, h(II,I 2) the -1
damping function and (~ t is the Finger relative strain tensor.
3.1.1. Memory function From the concept of separability, the memory function of the l inear
viscoelasticity is required. This memory function can be related to a discrete relaxation time spectra, m, available from dynamic experiments, given by:
N re(t-z) = Z ~ exp(" ( t -z ) ) (2)
i-1 ~ ~'i where 1~i and ~i are the viscosities and the discrete relaxation times, respectively.
These parameters were calculated for the two polyethylenes, at 160 ~ using a procedure given by Carrot et al. [41], which enables the recovery of a rnirlimnm number of relaxation times (Table 1).
Using these values, the loss modulus G" and the elastic modulus G' versus frequency curves can be recovered with maximum errors lower than 4 % (see Section II- 1).
291
Table 1 Discrete relaxation time spectra of LDPE and LLDPE at T = 160 ~
, , , , . . . . . - _ _
LLDPE LDPE _ , , , ,
, ~ * i [ S ] , , ~ i [Pa.s] . . . . ki[s] . ~ i [ p a . s ]
1.28 E-04 2.367 E+02 6.45 E-04 9.810 E+01
6.12 E-03 1.346 E+03 5.35 E-03 2.855 E+02
4.10 E-02 3.363 E+03 2.85 E-02 7.679 E+02
2.77 E-01 4.691 E+03 1.55 E-01 2.619 E+03
2.01 E+00 3.726 E+03 8.91 E-01 6.464 E+03
1.57 E+01 2.007 E+03 4.58 E+00 1.219 E+04
1.35 E+02 9.563 E+02 2.34 E+01 1.325 E+04
1.18 E+02 7.296 E+03
3.1.2. D_~mping function Different forms have been proposed for the damping function but many of
them are not acceptable for both shear and elongation. In this work, we retained for the damping function a modified form already proposed by Soskey et al. [42]"
1 h(II' I2) = i + a(I- 3) v~ (3)
where I is the Wagner's generalized invariant [36], written as"
I = ~I 1 + (1 - ~) I2 (4)
with"
11 = I2 = 72 + 3 in shear (5)
11 = exp(2e) + 2 exp(-e) and 12 = exp(-2s + 2 exp(s in elongation (6)
Parameters a and ~ are relevant to the shear and extensional behaviour, respectively, while parameter b acts in both flow situations.
292
The adjustable parameters of equation 3 are listed in Table 2. To get an appropriate set of values, a minimization procedure was carried out, where both shear and extensional data were used.
Table 2 Parameters of the damping function for LDPE and LLDPE.
Parameters a ~ b
LLDPE 0.086 0.02 2.56
LDPE 0.084 0.019 2.06
3~2. Differential constitutive equations: the GOB and m P T r mode l s The Cauchy stress tensor ~ can be written as follows :
~ = - p ' S + ~ v + ~ (7)
where p' is the isotropic part of the stress tensor, 8 the unit tensor, ~ v the
viscoelastic par t of the extra-stress tensor and ~ the purely viscous part , usually refered to as solvent behaviour.
In this paper, the GOB model is defined as follows :
~r = ~r~ + ~r . (8)
V ~ +~(~)~v = 2 n v ( ~ ) ID (9)
'Jl~i~ = 2 TIN( ~ ) ]]~) (10)
The triangular superscript denotes the upper-convected derivative and (
T ) is the second invariant of the strain rate tensor ID. To simulate the behaviour of the LLDPE melt, we retained the following
Carreau-Yasuda relationships for the viscosity and the relaxation time:
(i) viscosity:
11( ~ )= 'I1N (~ ) -t- 'I~v ( ~ )= 'I10 [1 +( ~0 ~)a] (m'lya (11)
where m is the flow index (m = 0.19), ~o the zero-shear viscosity (TIo = 12500 Pa.s), ko the characteristic time (~.o = 0.0796 s) and a = 0.7 at 175 ~
293
(ii) relaxation time:
x( )= (12)
The parameter p = 0.8 defines the relaxation time dependence upon the strain rate and C = 0.9 s is a time constant. The characteristic time X r is 0.4 s and the parometer b is 0.48 at 175 ~
The ratio of the solvent viscosity to the total viscosity retained in the model is 1/9, generally used in most of the papers on the subject.
For the P2~ model with a single relaxation time, we adopt the general form for the extra-stress tensor:
Q
f(Tr(~ v )) ~rv + ~ ~T v = 2 ~vo ]D (13)
Q
The time derivative ~ v is the linear combination of the upper and lower-
convected derivative of ~ v , namely the Gordon - Schowalter derivative [43]:
v a ~
~r~ = ( 1 - 2 ) ~Tw+ ~ V with O < a < 2 (14)
where a is the slip parameter, introduced by Johnson and Sego!man [44]. The function f (Tr ( ~Tv )) has an exponential form, given by Phan Thien [45]:
f ( T r ( ' T v ) ) = e x p ( ~ v ~ ) T r ( ~ v ) ) (15)
where E: is a material constant. If the slip parameter a is a non-zero constant, the requirement tha t the
shear stress be a monotonically increasing function of the shear rate in simple shear flow imposes a constraint upon the viscosity ratio. Using a spectr~m of relaxation times loosens this constraint and allows for more realistic fitting of the rheological data.
Therefore, using multiple relaxation times, Equation 13 yields"
E XPLT]vi T r ( ~ ) ] ~ + ~i~vt = 2Tl~ with ~v = Z ~rv~ (16) 1
The values of parameters a and E for both samples are listed in Table 3.
10 6
Table 3
Parameter a and e of the P2~ model for LLDPE and LDPE. ,, , , m ,
Parameters a
LLDPE 0,35 0,06
LDPE 0,15 0,025
I02 L-- 10-3
l- - - t u}
10 5
c r 0 4
m Z
o. 10 3
%
294
10-2 10-I Shear or Elongational Rate Is
10 ~ 101 10 2 10 3 - 1 ]
Figure 1. Steady shear and extensional viscosities and normal stress difference versus strain rate for LLDPE at T=160~ ( o , ~ ) experimental, ( .... ) F I ~ model, (--) Wagner model.
The comparison of the predictions given by these constitutive equations in steady-state shear and extensional simple flows are summarized in Figs. 1 and 2.
The multimode Wagner-type and PTT models fit the experimental data ra ther well, that justifies the use of these constitutive equations to simulate the entry and exit flow of LLDPE and LDPE. The GOB model is not represented because of the prediction of infinite extensional viscosities in the long time range. Nevertherless, the parameters of equations 11 and 12 have been determined from the shear experiments shown in Figs. 1 and 2.
295
10 6
r - ' - i tD ,; l0 s
w ~ c a . 04 u 1
f f l z
~ 103
o
o r 0
I0 2 10-3 10-2 i0 - I I0 0 I01 I0 2 10 3
Shear or Elongational Rate [s - I ]
Figure 2. Steady shear and extensional viscosities and normal stress difference versus strain ra te for LDPE at T= 160~ (o,o~) experimental, ( .... ) PTT model, (--) Wagner model.
4. FLOW GEOMETRIE.S AND E X P E R I M E N T S
Two types of flows were investigated. At CEMEF, experiments on a plane geometry were carried out. Birefringence measurements were performed, the results of which are presented in Section III-1. Comparison with computed results will be presented in paragraph 6.2. At LRMP, flows in aYisymmetric contractions were studied. We present in the next pages the experimental results obtained for pressure drops, entrance pressure losses and die swell.
4.1. F l o w c u r v e s a n d e n t r a n c e p r e s s u r e l o s s e s Capil lary experiments were performed at 160 ~ with a constant speed
capillary rheometer Instron ICR 3211. The axisymmetric converging test section of half-angle 45 ~ is defined as follows : the upstream and downstream radii are 4.76 mm and 0.65 ram, respectively, so tha t the entrance contraction ratio is 7.32. In addition, the capillary length-to-diameter ratios range from 1 to 20 and the shear rate can be varied from 3 to 150 s -1 (Figs. 3 and 4), so tha t flow measurements were generally done in the shear- thinning region, as shown in Figs. 1 and 2. It should be pointed out that the measurements were limited at shear ra te of 150 s -1 by the onset of melt fracture.
In addit ion, en t rance pressure losses were e s t ima ted us ing Bagley's procedure. The set of values presented in Figs. 5 and 6 were obtained from flow measurements with two different rheometers, one equipped with a load sensor, the
296
other with a pressure sensor. Scattering of the data is observed, especially for LLDPE for which the average values of entrance pressure losses are given with errors as high as 50%.
I0 s -
EL.
EL o 1 0 7 k.
a
{D
:::} ID
L 1 0 6 I:1.
0 I - -
IO s - 10 ~
@
0
A
E! A
El 0
o Apparent Shear Rate [s - I ] �9 i | | �9 " ' l | | | �9 �9 �9 . . . | ,L i �9 . ! . . . I
10 a 10 2 10 3
Figure 3. Total pressure drop versus apparent shear rate for LLDPE at T=160 ~ with various die length-to-diameter ratios: (o) 0.96; (~) 2.88; (A) 4.81; (0) 19.23
lO s -
Q.
o 10 7 L _
Q
L
6 I0 O.
4-J
10 5 . . . . . . . . I0 ~ 101
Apparent Shear Rate [s - I ] , . . . . . . i | t i . , . , . . !
10 2 10 3
Figure 4. Total pressure drop versus apparent shear rate for LDPE at T=160 ~ with various die length-to-diameter ratios: (O) 0.96; (O) 2.88; (A) 4.81; (0) 19.23
297
0.
In (i 0 L- C~
Q) L. D
0.
0 C
L
(-
ILl
10 7
I0 s
10 5
~A
A~ A
A A A AA
A
A A
A
1 0 4 . _ �9 . . . . . . . ' �9 �9 �9 . . . . . , � 9 . . . . I . . . . . . . . . . . , , . . , - . . . . ,
10 - I I0 o 101 10 2 10 3 10 4 Apparent Shear Rate [s - I ]
Figure 5. Entrance pressure losses versus apparent shear rate for LLDPE at T=160 ~
(I.
Q. 0 t_ C~
D t~
k. O-
~D o c
L -bJ
tD
10 7
10 6
iO s
10 4 - 10-i
A
~A
i �9 �9 . . , . . I . , | | . , . , , I . , . . . . . . i . . , , , . . . L . . , I . , . , !
I0 ~ I0 i 10 2 10 3 10 4 Apparent Shear Rate [s - I ]
Figure 6. Entrance pressure losses versus apparent shear rate for LDPE at T=160 ~
298
4.2. E x t r u d a t e swel l The polymer melt was extruded at 160~ downwards directly into air. The
diameter of the passing extrudate was then measured at 6.5 mm from the die exit, using an electro-optical device ZIMMER OGH. All measurements were made approximately 5 mm back from the lower end so that sagging under gravity could be n e g l e ~ [46].
1.4--
1.3
�9 f "~ 1.2 -o ~ o ~ L_ i .~ 4-~ O x 2 0 o L,J I.I ~o
~o
1.O "n~ , 0 l
a & & & & & & & & & ~ '
0 0 0
& A & a
I = l . _1 , i , ! , ! . i
2 3 4 5 6 7 8 Z/Ro
Figure 7. Evolution of the free sin-face at the die exit;
L/d =19.23, ~a = 3.3 s -1 (A) LDPE, (o) LLDPE.
2 . 2
"A&&a & & A A & 2.0 AA" ="AAAA
1.8 &AA& A&&A&"~A&&A'&'&'&&A &&& A
1.6
1.4 o o o o t 0 0 0
1.2
1.0 0 1 2 3 4 5 6 7 8 9 I0 I i 12 13
Z/Ro
Figure 8. Evolution of the free surface at the die exit.
L/d = 0.96, ~s = 33 s -1 (A) LDPE, (o) LLDPE.
299
In order to get complementary information on the evolution of the shape of the extrudate at the die exit, photographs were taken simultaneously for each flow condition. Examples of free surfaces build establishments are presented in Figs. 7 and 8, respectively for smooth (long dies and low shear rate) and for stronger flow conditions (short die and high shear rate). It is clear that strong flow conditions enhance the differences between LLDPE and LDPE. While LLDPE exhibits rapid evolution towards a constant diameter, instabilities occur for LDPE, which correspond to the onset of melt fracture.
In Figs. 9a and 9b, the extrudate swell ratios of the two polyethylenes are plotted as a function of the die length-to-diameter ratio, for various shear rates. The extrudate swell values for a given shear rate are higher for short dies and then decrease to reach asymptotic values as the length-to-diameter ratio increases. This enhanced swelling for short dies and high shear rates as opposed to swelling obtained for long dies and low shear rates is related to the fading memory of the viscoelastic fluid, when its residence time inside the capillary becomes shorter. For long dies (e.g. L/d = 20), the =equilibrium" extrudate swell values of LLDPE and LDPE are not so different, as it can be seen in Fig. 10, but due to the strong viscoelastic behaviour of LDPE, its extrudate swell values for short dies and high shear rates are eonsiclerably higher than those of LLDPE.
1.6 2.2 -
A
o
o | Q
1.4 ~ A ~ 1.8 CO D n Or)
(P ~ O
"O I::] "~ O L. O O L.
x 1.2 x 1.4 t d I,J
1 . 0 . . . . . I I I " I I I 1 t . 0 - - I I I - - '
0 5 10 15 20 0 5 10 15 20
L/d C/d
A
0 A 0
O 0
a- LLDPE b- LDPE
Figure 9. Extrudate swell versus die length-to-diameter ratio, at various apparent shear rates. (T = 160 ~ (o) 3.3 s-i; (O) 11 s-l; (h) 33 s-l; (0) 110 s -1
300
2.2-
OO
"0
t_ .&a x
bU
2.0
1.8
1.6 �9
1.4 g t o
8 o 0
1.2 o
~n ........ , ........ , ........ ,
I0 ~ I0 t 10 2 10 3 Shear Rate Is - I ]
Figure 10. Extrudate swell of LLDPE and LDPE versus apparent shear rate at T = 160~ Open symbols- L/d = 19.23; filled symbols" L/d = 0.96. (O,o) LLDPE; (A,A) LDPE
5. NUMERICAL MODELS
5.1. The s t r e a m . t u b e m e t h o d 5.1.1. Introduction
Major advances in numerical simulation of complex flows of viscoelastic fluids were particularly significant in the recent years for differential constitutive equations. For fluids with memory, more theoretical and numerical difficulties arise. The approximation of kinematic tensors leads one to consider the particle's kinematic history along a streemline, the points of which do not pass in general through the mesh nodes of the physical flow domain, in the context of classic finite-difference or finite-element techniques. Different approaches were proposed in the literature, in two-dimensional flow situations [21,22,47,48]. The fact that, to our knowledge, few numerical simulations [49] of memory-integral viscoelastic 3D flows have been attempted up to the present time is due mainly to significant problems in defining accurate parametric equations for warping curves. A different technique for computing the flow of memory-integral fluids was proposed by Papanastasiou et al. [20], where a Protean coordinate system introduced by Duda and Vrentas [50] and developed by Adachi [51-52], was used. In the Protean system, one coordinate is the stream function.
The stream-tube method is more closely related to the Protean coordinate approach. It refers to the flow analysis introduced some years ago by Clermont [40,53], which may be applied to the study of two- or three-dimensional duct or free surface flows [54-56] and pure circulatory or vortex flows [57]. In this analysis, the unknowns of the problem are, in addition to the pressure p, a one-to- one transformation between the physical flow domain D (or a subdomain D* of D)
301
and a transformed domain D', used as computational domain, where the mapped streamlines are parallel and straight. Although this analysis means considering only open streamlines, main flows in ducts involving recirculations can still be calculated with the stream-tube analysis, as was done in recent numerical works [26,55]. The stream-tube method is presented here in the scope of numerical calculation of flows of fluids obeying viscoelastic (differential or integral) constitutive equations. 5.1.2. Stream-tube method in two-dimensional flows
The main features of the stream-tube method in 2D and 3D flows, discussed more extensively elsewhere [40, 53, 55], are now summarized for two-dimensional situations. Flows with open streamlines are considered. The main flow region D* of the physical domain D (D ~ D*) is mapped into a domain D' such tha t the transformed streamlines are straight and parallel to an assumed main direction Oz of the flow. An example corresponding to 2D flows is illustrated in Fig 11.
~ L Stream tube
PHYSICAL I . . . . . . . . . ' - I
(r, z) zo ; !
y , I !
I
k ' t
MAPPED ...... " ..... "'" "\ . . . . . . . . . . . . "~ \ , DOMAIN D' " ' j Stream band (R, Z) ; _
I
I
,, i . i~ ~7/,.//i .- 111 J ,.111 i~,~., 11 J 1111..-+~i~
7-o
Figure 11. Physical and mapped domains in a two-dimensional problem
In two-dimensional flows, the domains D' and D are related by a transformation '% by means of a mapping function f , a priori unknown, such that"
x = f ( X , Z ) , y=Y, z = Z , (17)
with the following upstream boundary condition"
x = f ( X , Zo)=X; y = Y ; z=Zo. (18)
These equations refer to cartesian coordinates X i (X 1-- X; X 2 =y, x 3 = z) and XJ (Xl= X; X 2 =Y, X 3 = Z) in domains D and D', respectively. The mapped domain D' is a cylinder of generators parallel to OZ, of basis identical to the cross section So of the physical domain D at section Z0= z o. The following assumptions are made :
302
(i) At sections z~< Zo, the kinematics are known.
(ii) The flow in domain D* corresponding to the mapped domain D' does not involve secondary motions. This implies that the transformation Tis one-to-one, which leads to the condition of the Jacobian of the transformation :
A = ~(x,y,z)/~(,Y~Z) = fix # 0. (19)
Equations related to the velocity components verify the incompressibility condition. The function f is to be determined from the governing equations, by considering domain D 'as computational domain.
In axisymmetric flows, the transformation q 'between the physical flow domain D (or D*) and its mapped domain D' may be expressed by r = f ( R , Z ) ; 0 = O ; z = Z , (20)
using (r, O, z) and (R, O, Z) as cylindrical variables in the physical and mapped domains, respectively. The Jacobian of the transformation, assumed to be one-to- one, is given by the following equation
13(r, O, z)/O(R, O, Z) I = f 'R(R, Z). (21)
The respective na tura l and reciprocal bases b i and xJ, related to (R, O, Z) may be expressed in terms of the orthonormal frame ci corresponding to the variables (r, 9, z) by the equations :
b l = f l R C l ; b 2 = f c 2 ; b 3 = f ' Z C l + C 3 ; (22)
~l=[1/f 'R]Cl-[f 'z / f iR]C 3 ; ~2=[1/f]C2 ; %3 = c3 ' (23)
and the velocity components are given, in the incompressible case, by
u ; v = o ; w f a ] (24)
In equations (24), ~~ stands for the R-derivative of the transformed function of the s t ream function ~Y(r, z) , given at the upstream section z o of the flow domain, by
r
z = Zo, R = r ; ~~ = d~(r, Zo)/dr J mR = d [-~o ~ w(~, z o) d~] /d r . (25)
5.1.3. Viscoelastic constitutive equations in the stream-tube method For differential viscoelastic models, the derivative operators are evaluated on
the rectilinear mapped streamlines. It is not necessary to determine the particle history.
303
For fluids with memory, the evolution in time of a particle X which occupied positions Xto (R, O, Zto) and X~ (R, O, Zx) at respective times t o (reference time related to the position Zo = Zo of the particle) and t, is readily given by the equation
z~
= to- [l~*(R)]fZo f(R, ~) f'R(R, ~) d~. (26)
In the corotational formalism related to rate equations, the kinematic tensor is the rate-of-deformation tensor ID(z). In order to satisfy the concept of
objectivity, the tensor ID(~) and the corresponding stress tensor are required to be
written first in a corotational frame (at each time ~), which is relatively simple to determine in the two-dimensional case. The velocity derivatives with respect to variables of domain D' are then evaluated in terms of functions f and g by derivative operators deduced from equations (17) or (20). Details on the use of the stream-tube method with a corotational memory-integral equation are given in [54,56].
In codeformational equations, the basic kinematic quanti t ies are the displacement functions. This generally means using the respective Cauchy and Finger tensors ~t(~) and (~t-l(~), related to the deformation gradient tensor
~t(l:) = [~ X~/~ X~ ] by the following equations
(~t(~) =T ~t0:). ~t0:) ; (~t-l(z) = [ T ~t(~:) "~t0:)]-I. (27)
In equations (27), the symbol "T" denotes the transposition of a tensor. Starting from Adachi's work on Protean coordinates [51,52], it can be shown [59,60] that ~t(z) may be expressed, in the natural frame corresponding to variables (Xj) of the mapped domain D', by the following matrix components
Fll=F22 = F13=1; F 12 = F 21 = F 23 = F 32 = 0; F 31 = ~ Z~/~R t ; F 33 = ~ Z~/OZ t . (28)
The Cauchy and Finger tensors may be expressed in terms of the mapping function f and its derivatives [26,60], using the vector positions X t and X~ and
equations (22-23) related to natural and reciprocal bases e i and xJ for variables (R, O, Z) and the following equations
OZz/OR t = w((R, Z~) t [Ow(R, Z~)/ORt]d ~/w(R, Z~) ; ~ Z~/~ Z t = w(R, Z t) / w(R, Z~). (29)
5.1.4. Governing equations in stream-tube method The relevant equations to be written down are only the momentum conservation equations, under isothermal conditions. These equations are given in differential
304
forms, using the function f related to the spatial variables (R, O, Z) of mapped domain D' and derivative operators, by
-( 1/f'R)~}p/~}R + (1/f'R)~}Tll/~}R + (T 11 - T22)/f
- ( f ' z / f ' R ) ~T la/~}R + bT z3/~}Z = 0
"f'z/fiR ~}p/~}R - ~}p/~}Z + (1/fiR) c}T13/~R + T13/f
, (30)
- ( f ' z / f ' R ) ~T33]~R + ~}T33]~Z (31)
In these equations, the body forces and the inertia terms are ignored. Here the stress tensor is given by ~=-p II + ~P.
Given a simply-connected geometry, the main flow field may be computed by considering successive stream tubes in the mapped computational domain D', from the wall to the central flow region. This interesting property, which was discussed in previous papers, means taking into account the action of the complementary domain B c of the stream tube under consideration (Fig 12). This is done by writing the global form of the equations of conservation of momentum. If R and M 0 denote the resultant and moment vectors at a given point 0, we get
R = [f~Kl r �9 n ds] = 0 , (32)
MO = lID Bc O M ^ ~ ( M ) . n ds ] = 0 . (33)
UPSTREAM SURFACE n
D O W N S T R E A M SURFACE
n
Bc
n
STREAM
T U B E B LATERAL
SURFACE
Figure 12. Stream tube B and its complementary domain B e considered for the global momentum conservation equation.
305
In equations (32-33), ~B c denotes the limiting surface of the complementary
domain B c = So w S 1 u S 2 (Fig 12). ~ and c are the total stress tensor and vector,
respectively, n is the outward unit vector normal to the surface. So and S 2 are upstream and downstream surfaces limiting the flow domain under consideration, perpendicular to the z-axis. In the cases under consideration (ducts involving symmetries), it can be shown that equations (32-33) reduce into one scalar equation [55]:
[~Bc ~ . n d s ] . e3 = 0. (34)
The quantities involved in the non-linear boundary condition equation (34) are expressed as functions of the variables defined in domain D'.
The stress equations are considered together with the dynamic equations, leading to a set of equations involving the primary variables f, g and the pressure p or a mixed formulation with the stress components Tij. Both cases were considered in numerical applications of the stream-tube method.
5.1.5. Solving the equations The numerical procedure requires app ro , mating the unknowns in the simple
computational domain defined by a stream band B in the mapped domain D'. The s t ream bands are divided into rectangular elements built on two rectil inear streamlines. Meshes are generally refined in the vicinity of contraction sections of the flow domains, as is usually the case. The discretization schemes adopted depend on the problem under consideration. We find it of interest to underline the following points"
(i) In stream-tube analysis, a weak formulation is still not written at the present time, mainly due to the fact that the dynamic equations are related, even in the simple Newtonian case, to a third-order nonlinear set of differential equations;
(ii) Regarding the validity of our assumption to compute the numerical solution of the problem under consideration in the total flow domain by considering e l emen ta ry subdomains (the s t ream-tubes) and the global momentum conservation equations, it is not possible at the present time to provide a mathematical proof. This should be considered as an open theoretical problem. However, for validation of the method, we can rely at the moment on the results provided with our experiments related to various flow problems considered for duct and free surface flows.
The discretized governing dynamic equations of the problem define a dosed set of equations :
A~ (Y1, Y2, �9 �9 �9 , YN) = 0 , i = 1, 2 , . . . , N (35)
When considering an elementary stream-tube, the numerical computation of the unknowns for a stream tube may thus be related to the following problem
306
(P) : rain { ~ : Y e RN } (36)
under the constraints related to the global momentum conservation equations (action of the complementary domain of the s tream tube) formally written as :
Cj (Y1, Y2, . -- , YN) = 0, j = l , . . . Jo, (37)
G denotes the following quadratic function
i=N
G(Y) = TA(Y). A(Y) = ~ =I Zi2(YI, Y2,- �9 -, YN) �9 (38)
In optimization, it is well-known tha t non-linear constraint problems are generally "hard" to solve [61] particularly for a priori non-convex constraints. To overcome this difficulty, the equations of problem (P) are formulated as problem (P~ defined as :
(P') : rain {E(Y) : Ye R M }. (39)
To this problem corresponds the following set of equations
)~i (Y1, Y2, . . . , YN) = 0 , i= 1,2 . . . . . M, (40)
which represents equations (35) and (37) in such a way tha t the quadratic form defined in (38) is given by :
i=tVI
E(Y) = Tz(Y). z(Y) = ~-=1 Zi2(Y1, Y2 , . - . , YN)- (41)
To solve the equations of problem (P'), the optimization algorithms used are Levenberg-Marquard t and Trust-Region procedures. These methods enable computation of the solution by using the Jacobian matrix and the Hessian matrix (or its approximation) related to the objective function E(Y) [57].
5.1.6. General remarks on stream-tube method Despi te the fact tha t the s t r eam- tube method was l imited to the
computat ion of main flows (the flow may involve secondary flows, but the recirculations are not simulated), the present analysis is still considered to be attractive for the following reasons:
1) Various two- and three-dimensional flow situations can be considered by the method. Integral constitutive equations may be involved in 3D flows ;
2) For flow domains involving singularities such as sharp angles and those involving free surfaces, the geometrical discontinuities are not explicitly taken into
307
account. However, the singularity effects may be highlighted by considering streamlines of the main flow region close to the boundaries ;
3) Regarding numerical aspects, analysis of a single stream tube allows us to consider relative differences between the problem variables which are less important than those that could be observed in a computational method involving the total flow domain. This limits convergence problems ;
4) The computing time and the storage area are reduced, when elementary stream tubes are considered.
5.1.7. Application of the stream-tube method to duct and free-surface flows Under assumption of the existence of a main direction Oz of the flow, the
stream-tube method may be applied to computation of flows in ducts, even those where recirculations are encountered and flows involving free surfaces, as those corresponding to the die-swell flow problem. We present here the procedures related to those flow situations.
5.1.7.1. Flow in axisymmetric contractions The mesh is refined in the vicinity of the contraction section, as is usually the
case (Fig 13). The boundary is not explicitly taken into account, because of the existence of secondary flows close to the wall : the limiting wall only provides known boundary values for the function f. This function is approximated in the peripheral stream band using a modified Hermite element (Fig 13) [26] such that :
i=M
f(e) (~,rl) = Zi =1 Hj (~,r I) f*(e)j (42)
In equation (42), the quantities f,(e)j denote the values of the function f or those of its derivative f ' R and f ' z- The details of the basis functions, the quantities f.(e)j and the nodes of an element have been presented elsewhere [26]. The first and second derivatives of the mapping function f can be obtained from derivatives of the basis functions. To compute the pressure p and tensor components, the four nodal values at points A, B*, C* and D of each element are chosen as unknowns. Their derivatives are evaluated by using finite-difference formulae. In Fig 14 we present computed s t reamlines for 4/1 and 8/1 axisymmetric contractions, and the computed shear stress along the computed streamlines, for a K-BKZ fluid of the form given by Papanastasiou et al [19].
From the determination of flow in the peripheral stream tube, the other stream tubes may be computed successively, from the wall to the central region of the main flow, provided the action of the complementary domain is taken into account. An example of the results is given in Fig 15 for the flow of a corotational memory-integral (Goddard-Miller) fluid, where the recirculating flow zone is not explicitly calculated. However, the description of the computed main flow region illustrates the i m p o ~ c e of the vortex flow zone.
308
PHYSICAL DOMAIN PHYSICAL DOMAIN
"•-•.• STREAM-TUBE .._ ~ ~" STREAMLINES
D , ~ D ; r ' - - ' - " ~ - - , - : -
' l l ' ' STRF.AMLINES i j r STREAM-BAND , f
= I / ~ , . . . .
1 ..... i l !-~I 11111I I'I, II1 D' [,I~.I'~11I I i 111II !_1 D' - x , \ / / :
_ _ , | i i | i
MA"PPED STREAMLINES MAPPED DOMAIN MAPPED DOMAIN
B C {~) A, D (f , fi~ , f ~ )
B* C* O B*, C* (f~ , f~)
�9 B, C (f) A D
Figure 13. S t r e a m tubes and s t ream bands for flows in a converging and a contraction flow - Local mM_jfied Hermite element.
12
0.0
LDPE (160 ~ C) SR= 1.84
- - ~ . - ~ ~ 8/1 contraction
0 20 40 60
Pa Trz 4--- 8/1 contraction
-5E+4
-1E+5 LDPE (160 ~ - .
-2E+5 0 10 20 30
Figure 14. Computed streamlines and corresponding shear s t resses along the s t reamlines , when considering the per ipheral s t r eam tube, for a 4/1 and a 8/1 axisymmetric contractions (K-BKZ memory-integral fluid).
309 �84
Figure 15. Example of computed stream tubes for a memory-integral corotational Goddard-Miller equation (We = 4.73) in a 4/1 axisymmetric contraction.
5.1.7.2. The extrudate swell problem The unknown surface and the corresponding swell ratio can be determined
by considering only the peripheral stream tube involving the wall and the free surface. Fig 16 illustrates the peripheral stream tube and its corresponding stream band in the mapped computational domain. In relation to the particular shapes of the free surface and stream lines in the physical flow domain, we use, for approximating the mapping streamline function f in the peripheral stream tube, analytical forms derived from the equation proposed by Batchelor and Horsfall [66] and already used in previous papers [54,58] :
jr (S, Z) - C 1 (R, Z ) + A (R). q (Z) {1- exp [- (Z - Z o) B (R)]} (43)
in the tube, for Zp < Z< Zo, where Z o = z o denotes the exit section (Fig 5) and
f (R, Z) = f(R, Z2) - A (R) exp [- ( Z - Z0) B (R)]} (44)
in the jet, for z o < Z < Z 2.
The final jet radius is given by the equation
f(Ro, 712) = Po (45)
The functions A(R), B(R), C(R,Z) and q(Z) involve unknown coefficients [54,56]. The approximating expressions in the peripheral stream tube B may be found
in papers by Clermont et al. [55,56].
Swelling criterion In this work, a procedure similar to that already adopted in previous studies
on swelling with stream-tube analysis is adopted. The criterion for determination of the unknown swell surface Z o, by the condition (C) -
( C ) . I = ~ 2: o I ( - p ]] + ~ ) n I ds] = 0 (49 )
310
n .;
Z stream tube B o Wo n \ ~ ......... ~ ........ ./.... .....
g--.... . \ . : -
streamline L
. . . . , _ str.eam, tube B c ,i
Zp z 0 z2!,
---~n
mapped streamline
X \ \ \ \ \ ' \ �9 -. \ ~ ~ \ k \ ~ \ ~. ,. "X=
L' o
J mapped streamline L' 1
Figure 16. The peripheral stream tube and its mapped computational domain in the extrudate swell problem
The pressure p and the stress components ~J, unknown at points Mo of the boundary surface S O = W 0 u2:0 (wall and free surface), are determined from corresponding values of p and ~ on an internal stream surface S of the stream
tube B using the following approximating equation
E(M* o )=E(M*) + (Ro-R) bE/bRIM, + O ( ( R o - R ) 2 ) ,
given Z, M* o (R, Z) e S*o, mapped surface of S0, and M* (R,Z)e S*.
(5O)
In equation (50), E(M* O) and E(M) denote functions evaluated at points M* o and M* of the mapped peripheral stream tube B* (Fig 17).
Taking into account the approximate relations for the relevant equations and unknowns in the numerical procedure, the zero-stress condition (49) is to be expressed by the following condition, using the swell ratio Z as a parameter :
{rE I ( - p ] I + ~r)n I ds] }ZER is minimum I (Z) = o (51)
The free surface is then determined iteratively in order to satisfy equation (51).
311
5.2. L e s a i n t - R a v i a r t (d i scont inuous Galerkin) quas i -Newton m e t h o d 5.2.1. Introduction
As pointed out in Section II-3, numerical modelling of viscoelastic flows leads to numerical difficulties related to the mixed character (elliptic - hyperbolic) of the constitutive equation, to the propagation of "stress singularities" and to the so- called "High Weissenberg N,!mbef' problem.
Upwinding methods as well as discontinuous Galerkin (Lesaint-Raviart) methods have been introduced to account for transport terms in the differential constitutive equation [62].
Decoupled methods as well as direct Newton methods have been used to account for the nonlinearities in the constitutive equations. Those methods exhibit limited convergence, especially when flows with singularities are considered. Newton methods are efficient but need a high storage area and powerful computational facilities.
In this paragraph , a Lesaint-Raviart method is presented. A Newton algorithm allowing fixed values of the viscoelastic extra-stress components outside the finite elements is used. A fixed-point algorithm on those external extra-stress components is also involved. This quasi-Newton method needs a storage requirement of the same size as that related to a classical decoupled method, but allows improved convergence [39].
5.2.2. Constitutive equations and boundary conditions. The molten polymer is assumed to be incompressible; mass and inertia forces
are neglected:
Vu = 0 (52)
V ~ = 0 (53)
The stress tensor is the sum of an isotropic component, a viscous component and a viscoelastic component :
(54)
Two equat ions have been selected for the viscoelastic ext ra-s t ress component ~ : a generalized Oldroyd-B model (GOB) and a multimode Phan-Thien Tanner model (mPTT). The values of the corresponding parameters are given in sub-section 3-2
As pointed out in section 4, several plane and axisymmetric convergent geometries have been considered :
312
(a) , ,
Ib)
Figure 17. Typical plane convergent geometry: (a) long die, (b) short die
- For long die geometries (which implies the existence of a Poiseuille flow downstream of the contraction), fully developed velocity and stress profiles are imposed at the entry section ; only the velocity profile is imposed at the final section.
- For short die geometries, the same boundary conditions are considered in the entry section but free surface equations are written in the downstream flow region (see paragraph 5.2.4).
In each flow situation, a zero-velocity condition is assumed along the die wall.
5.2.3. Numerical resolution Triangular Crouzeix-Raviart dements [63] were selected, which verify the so-
called Babuska-Brezzi compatibility conditions (quadratic approximation for the velocity ; discontinuous linear approximation for the isotropic part of the stress tensor ; discontinuous quadratic approximation for the stresses) (Fig. 18).
o , o
(a) (b) Figure 18. Triangular Crouzeix-Raviart elements. (a) velocity, (b) pressure, (c) extra stress components
(c)
The Lesaint-Raviart (discontinuous Galerkin) method [62] is based on discontinuous approximations of the extra-stress components and is characterized by an element per element t reatment of the constitutive equation. This leads to the introduction of a stress step in the Galerkin formulation, from the preceding
313
element to the next one. The weak form of the constitutive equation will be written on each element, for example for the generalized Oldroyd-B formulation :
~K[~V + k( ~ )~rv-2 Tlv( ~ ) ~(u)]:(p*d~
-S~iK- X( ~ ) r an (~v- ~vext):(P*dY = O, (55)
cp* is a test function. The first term in equation (55) represents the constitutive equation for the element K considered, the second term represents the jump in the stress along the ent rant border 5K" of element K (~1~, ext is the stress in the entrant neighbour element).
The quasi-Newton method consists in searching X = (u, p', ~v ), solution of
F(X) = 0 (equations 68) for fixed values of~v ext.
f2q* div u d~ = 0
k
(56a)
(56b)
-~SK ~.(~) u.n (~,r ~ * dT= 0} (56c)
After elimination of the pressure p' in equation (56a) using an Uzawa algorithm [64], equations (56) may be linearized on each element K as :
All(U) Au + A12(u) A~v = - Rl(u , ~v, P') (57a)
A21(u,~ v) Au + A22(u,~ v ) A~, =- R2(u,~,) (57b)
From equation (57b), it is possible to express A~v, as a function of hu. Substituting in equation (57a) leads then to a non-symmetric system (58) with the velocity as unknown, and for which the storage requirement is equivalent to that for a generalized Stokes problem :
314
A (u, ~v) Au =- R(u, ~v, p') (58)
A fixed point algorithm permits to adjust, at each iteration, the value of ~ e x t . The same kind of method is also used for the multimode Phan-Thien Tanner model.
5.2.4. Computational algorithm for the extrudate swell problem
The algorithm is presented here in a plane flow situation (Fig. 19). Z o is a free surface if the two following conditions are respected along Z o �9
@ . n = O u . n = O where n is the vector normal to the free surface.
T Y y = h(x)
I .
, v ~.-
L x
(59a)
(59b)
Figure 19. Extrudate swell in plane flow situation.
The second free surface condition (59b) expresses also that E o is a s t ream function ; in a planar flow, the stream function ~ is defined by"
u = ~-~ and v =- ~ (60)
Here ~0 is deduced from the finite element velocity field (u, v) by searching the minimum of the following functional"
J (~1 = ( ~ + v) 2 + ( ~ - u d~ (61)
The value of the stream function along the free surface ~(x, h(x)) is equal to the volumetric flow rate"
~(x, h(x)) = Q = h(x) fih(X) (62a)
315
h 1j with: U h(X) =~ u(x,y) dy. (62b)
The following iterative scheme has been used : (i) For a given value of Zo (y = hn(x)), the numerical resolution presented in Sub- section 5.2.3 is performed using boundary condition (59a) along the free surface and an uniform velocity at x = L ; (ii) The stream function ~n is computed by minimizing equation (61) on the finite element domain ~ ;
(iii) The mean velocity fih(X) is computed from equation (62b) ; (iv) A new value of the free surface hn+l(x) is deduced from equation (62a).
5.2.5. Examples of results We present here numerical results obtained in a planar short die geometry
with both constitutive equations for the two differential models (GOB and ~ ) . The shear rate at the wall of the downstream channel is equal to 19.25 s 1.
Figure 20. Streamline patterns. (a) generalized Oldroyd-B ; (b) multimode Phan-Thien Tanner.
316
The streamline patterns are quite identical for both constitutive equations. However, the vortex is more pronounced for the multimode Phan-Thien Tanner model, whereas the swelling is greater for the generalized Oldroyd-B model.
The velocity along the symmetry axis is significantly different : an overshoot at the contraction as well as an undershoot downstream of the die exit are observed with the generalized Oldroyd-B model, but only a smooth overshoot is indicated by the multimode Phan-Thien Tanner model. The final value of the velocity after swelling is more important for the PTT, which is consistent with the lower value of swelling observed in Fig 20.
A
E E
>,, m 0 0
m Q
15
10
0 qnO, O 0 �9 �9 O
o - - o o o o O 0 O
�9 o o o o o o e o O 0
O - I I i l . i I I
- 20 -15 -10 -5 " 0 5 10 15 2'0
Axial distance (mm)
Figure 21.Velocity along the symmetry axis. (o) generalized Oldroyd-B (o) multimode Phan-Thien Tanner.
The changes in the viscoelastic extra-stress components along the symmetry axis are markedly different (Fig. 22): Tvxx is much more important at the contraction for the GOB model; on the contrary, Tvyy is more pronounced for the PTT model ; at the die exit, Tvxx is equivalent ibr the two constitutive equations, but relaxes more quickly for the mPTT model. Tvyy is much more important for the GOB model, which is again consistent with the more important value of the swelling observed in Fig. 20a. Obviously, Tvxy remains equal to zero along the symmetry axis for the two constitutive equations.
317
A
x x >
A m
v
w 0 L_
| m
qmo X
W
0,075
0,05
0,025
a) i o 8: o
o
-0,02 ' -20 -10 0 10 20
A x i a l d i s t a n c e ( r a m )
O, b)
0 o
0
0 �9
0,02 ~ e D e �9
d "o o ~ 0 . . . . ~ e b ~ 1 7 6 1 7 6 Q o a .
-20 -10 0 10 20
A x i a l d i s t a n c e ( r a m )
Figure 22. Viscoelastic extra-stress components (a: Tvxx, b" Tvyy) along the symmetry axis. (O) generalized Oldroyd-B model ; (0)multimode Phan-Thien Tanner model.
6. C O M P A R I S O N BETWEEN NUMERICAL R E S U L T S A N D E X P E R I M E N 2 ~
Flow experiments were carried out at LRMP for axisymmetric contractions and at CEMEF for plane geometries. Numerical simulations were performed at Laboratoire de Rh~ologie, with Wagner memory-integral constitutive equations, with the stream-tube method (sub-section 5.1) and at CEMEF, where the finite-
318
element method was combined with the use of differential constitutive equations (GOB and mPTT models).
At Laboratoire de Rh6ologie, different procedures were used for the converging flows and extrudate swell:
(i) For converging flows, the peripheral stream-tube was considered, with modified Hermite elements (Sub-section 5.1). The solution of the equations was performed with a Trust-Region optimization algorithm. The use of a stream-tube analysis associated with this resolution algorithm did not require, for a given flow rate, a prior solution at a close value of the flow rate under consideration. The code was implemented on an Apollo 425 work station and no limitation based on the flow rate was found in the flow calculations. The rectangular mesh involved 24 to 36 elements. The lengths Lu and Ld upstream and downstream of the contraction section were taken as L u = 13 r 1 , 30 r 1 < Ld~ 45 r 1 (rl denotes the downstream tube radius), in order to obtain relaxation of stresses given by the memory-integral equations. For all runs, the number of iterations was found to be lower than 25. Given a flow rate, the CPU time required for convergence was found to be of order of 8 hours.
(ii) For the extrudate swell problem, the code was implemented on a HP 9000 work station.The peripheral stream-tube involved a computational domain in the tube and the jet domains such that ( z 0 - Zp)/r 1 = 5 ; ( z 2 - Zo)/r 1 = 10 (Sub- section 5.1). The computational algorithm used for solving the equations was the Levenberg-Marquardt optimization algorithm [67]. Given a flow rate, the CPU time for obtaining a swell surface was found to be of order of 1 hour. No limitation of convergence was found for the flow rates investigated.
At CEMEF, the numerical results were obtained by using a generalized Oldroyd-B model and a multimode Phan-Thien Tanner model, with a Lesaint- Raviar t quasi-Newton finite element formulation (Sub-section 5.2). The computations were run on an IBM Risk 6000/580 work station. Convergence was obtained for most of the experimental flow situations presented in section 4, depending on the flow rates and on number of relaxation times accounted for in the mPTT model. The CPU time varies from 10 to 70 hours, but the storage requirements remained limited. It should be pointed out that the method has provided velocity and stress fields for all types of 2D planar and axisymmetric geometries investigated experimentally (contraction and converging flows for short and long dies, extrudate swell).
319
6.1. Axisymmetric flows 6.1.1. Flow curves and entrance pressure losses for LLDPE and LDPE
Figure 23 compares, for various L/d ratios, the experimental pressure drop and the numerical radial stress difference along the wall (between 10 mm upstream from the contraction and die exit), as a function of the apparent shear rate (in the reservoir, the wall radial stress C~rr is quite equivalent to the mean
stress in the flow direction, Dzz, which is the so-called pressure drop measured). For the longest die, the GOB model as well as the multimode P2~ and the WAGNER models give similar results, in close agreement with experimental data on LLDPE and LDPE (within 15%). Numerical results using the GOB and the mPTT models were also available for short dies. Here again, the agreement is generally good.
IO s -
a.
o. o 10 7 L_ C~
L.
Q~ 01
L 10 6 Q.
4-'
10 5 I0 ~
3
1
. . , . . . . . , I . �9 �9 | | �9 . . l l i l . , , . , , I
101 10 z 10 3
Apparent Shear Rate Is - I ]
Figure 23. Total pressure drop as a function of the shear ra te at the wall of the downstream channel.
Curve 1 : experimental data for LLDPE (L/d = 0.96). Curve 2 : experimental data for LLDPE (L/d = 4.81). Curve 3 : experimental data for LLDPE (L/d = 19.23). Curve 4 : experimental data for LDPE (L/d = 19.23).
(A) : GOB model; (o) : mPTT model; (O, 0) : mWagner model.
Fig 24 compares experimental entrance pressure losses of LLDPE obtained using Bagley correction and numerical results obtained with the three models, but using different methods.
320
- When considering the difference between the %r component of the stress tensor at the wall just upstream and just downstream the conical convergent, one may observe tha t the computed art loss for GOB and mPTT models is far smaller than the experimental one (a factor 5) ; a similar observation has recently been indicated in the work reported by Feigl and 0tfinger [28].
- To the contrary, the performance of the numerical computation using stream-tube analysis combined with a multimode Wagner-type constitutive equat ion is closer to the experimental results, especially for LLDPE. Unfortunately, quantitative disagreement still remains for LDPE, as shown in Fig. 25.
- In numerical simulations involving the GOB or the ~ models, we rebuilt a numerical Bagley correction, deduced from the drop of the %r component, for the three capillary lengths at the same flow rate. We observe that this numerical entrance correction remains smaller than the experimental one, but of the same order of magnitude (30 % difference). Similar results were presented recently by Barakos and Mitsoulis [25].
10 7 -
(g
ffl
0 L o 10 6
L. D U~ U) L
o. i0 s
U e-
$- 4.J e- Ld
l O 4
lO o
/
i o
~
A G 6
�9 . . . . . . . I �9 . . . . . . . I | , . . . . * * I
101 10 2 10 3 Apparent Shear Rate [s - I ]
Figure 24. Entrance pressure loss of LLDPE as a function of the shear rate in the downstream channel. (--) Experimental curve from Bagley correction. Open symbols: numerical computation of A~rr; Filled symbols: numerical Bagley corrections; (O) :mWagner model (~3) : mPTT model; (A) : C~B model.
0
10 7 r - - i
In Q. 0 I . . s
(D t.. Z3 ~) 10 6 ly)
I,,. 12.
0 e-
L 4~ e-
Ld
10 5 10 ~
321
L . | I . . . I , i I �9 �9 . | . . I i
101 10 2 Apparent Shear Rate [s -1]
,I I �9 �9 �9 i , I
10 3
Figure 25. Entrance pressure loss of LDPE as a function of the apparent shear rate in the downstream channel. (--) Experimental curve from BAGLEY correction. (o) numerical computation of AOrr (mWagner model).
6.1.2. Extrudate swell Computations were performed for the long dies using the three constitutive
equations. For short dies, only differential models (GOB and reFIT) were tested.
6.1.2.1. Long dies (L/d = 19.23) At low shear rate (3.3 s l ) , experimental and computed free surfaces are
compared in Fig. 26. The final extrudate swell values are equivalent, bu t the experimental shape of the free surface is different from the computed ones. In addition, a slight overswelling at the die exit is observed for the GOB model, which is observed neither experimentally, nor using a classic Oldroyd-B model.
Another example with the Wagner model at a higher shear rate (~ = 33s "1) is presented in Fig. 27, for LDPE and LLDPE. For both materials, the final extrudate swell values are well-predicted. However, the computed radius close to the die exit increases faster than the experimental one and a constant value is reached in a shorter time than in the real case, especially with LDPE.
322
1.3
1.2 / o o o Q -----
+ L,'f: Z
,,, l: 1.0 .o
0 1 2 3 4 5 6 7 8 9 10 11 12 13 Z/Ro
Figure 26. Free surface computation for LLDPE; L/d = 19.23 ; ~ = 3.3 s "1. (0)experimental measurements ; ( .... ) GOB model; (--) ~ model.
3~ CO
"I3
L_ 4J x
I , I
1+ F 0 0 0 0 0 ~ 0 0 ~ O , +
1.3 ~ 1 7 6 1 7 6 1 7 6 A
1.2
A A
1.1
1.0 0 1 2 3 4 5 6 7 8 9 I0 II 12 13
Z/Ro Figure 27. Free surface computation for LDPE and LLDPE with stream-tube analysis and m Wagner model.
(a) LLDPE; (o) LDPE; (--) computation; L/d = 19.23, ~ = 33 s "1.
All the models point to an increase in the extrudate swell value with the apparen t shear rate. The mPTT model underes t imates this effect, whereas the GOB model overestimates it, especially in the high shear rate range (Fig. 28).
323
Or}
"0
L 4-J X Ld
1.6 "
1.5
1.4
1.3
1.2
1.1
Apparent Shear Rate [s -I] . , . - - . , �9 ,
-o I0 2 0 3 0 40
Figure 28. Influence of the shear rate on the extrudate swell of LLDPE. (--) experimental ; (o), GOB model ; (e) PTT model.
Wagner computed values are in better agreement with experimental results, even for LDPE, provided the apparent shear rate is higher than 20 sec -1. At low shear rates, the predicted extrudate swell ratio is largely underestimated, for both LDPE and LLDPE. (Fig. 29)
1.6-
1.4 (D
"U Z) l,- 4-J X LtJ
1.2
~ o
~ o �9 ./..~~--'o"
0
1 0
0
Apparent Shear Rate [s -I] 1 . 0 0 , I . . , ~ �9 I , ! , I . ~ . _. i �9 J
0 20 40 60 80 100 120 140 160 Figure 29. Extrudate swell versus apparent shear rate. (---) Experimental data; computed values for LLDPE (O) and LDPE (e).
324
6.1.2.2. short dies Fig. 30 presents the comparison of free surfaces, for a shor t capillary
(L/d = 0.96) for a s train rate of 33 s "1. In this case, the computed final je t radius value given by the GOB model is very close to the experimental one, while the computed m P 2 ~ value underest imates experiments (about 10 %). Nevertheless, the computed shape of the free surface is very far from the experimental one for the GOB model (a large overswelling at the die exit is observed) as it seems to be homothetic when considering the ~ model.
1.5
1.4 1 I I / - " " " - o o ~ - - "* 0" -*
Z t.2
w
1.1
10 0 I 2 3 4 5 6 7 8 9 I0 i i 12 13
Z/Ro
Figure 30. Free surface computation for LLDPE; IJD = 0.96 ; ~ = 33 s -1. (0): experimental measurements ( .... ) :GOB model (--) : mPTT model.
The influence of the L/d ratio on the equilibrium extrudate swell is presented in Fig. 31, for three values of shear rate (3.3 s -1, 11 s -1 and 33 sl) . The GOB model seems to be insensitive to the L/D ratio at low shear rate and gives a tendency opposite to the experiments at higher shear rate. On the contrary, the mPTT model captures the decrease of the extrudate swell ratio with increasing die length- to-diameter ratio, but underest imates the values at high shear rates, mainly for the shortest dies.
325
(a)
4-D
"1o
L 4-m X
w
1.6-
1,5
1.4
1.3
1.2
1.1 0
o ~ o
A A A
I I I
5 10 15 L/d
I
2O
Co) 1.6-
1.5
u) 1.4
1.3
w
1.2
1 . 1 I I ! I
0 5 I0 15 20 L/d
Figure 31. Influence of the L/d ratio on the extrudate swell ratio of LLDPE. (a), GOB model ; (b) F ] ~ model.
(--) experimental ; (A) ~ = 3.3 S "1 ; ((:3) ~ = 11 s -1 ; (O) ~ = 33 S "1
Fig. 32, which gathers together all the computations and the corresponding experimental results for the long die (L/D = 19.23), confirms that , generally speaking, a better description of extrudate swell experiments is obtained with the multimode PTT and Wagner models instead of the GOB model. For the latter, the agreement becomes poorer and poorer when the shear rate increases. On the
326
contrary, the stream-tube analysis using the Wagner model provides satisfactory agreement with experimental data particularly in the high shear rate range.
1 . 6 - 0
r
1.4 0
~
C
1.2
0
1 . 0 . rn , ! . . . 1.o 1.2 1.4 1.e
Experimental Die Swell
Figure 32. Comparison between theoretical and experimental extrudate swell.(o) GOB model; (e) F I ~ model; (0,c}) Wagner model.
6.2. P l a n a r flows In this Section, the numerical results compared to experimental data were
obtained at CEMEF, with the differential constitutive equations already presented.
6.2.1. Linear low density polyethylene Experimental birefringence patterns at 205 ~ and numerical simulation
with a GOB model and a mPTT model are compared in Fig. 33. The apparent wall shear rate related to the downstream flow is 25 s "1 and the optical stress coefficient was selected to be 2.1 10 -9 m2/N (see Section III-1).
Both computations provide a good qualitative description of the birefringence patterns, but the ~ model captures more accurately the shape of the fringes in the downstream region.
A more precise comparison can be made by looking at the birefringence changes along the flow axis (Fig. 34). The birefringence increase in the reservoir is similar for both models and very close to experimental values, as already observed with results from other numerical computations [27, 30, 65]. The relaxation along the die land is very consistent for the mPTT model, where the stresses do not relax totally before the channel exit. Results with the GOB model indicate a less realistic relaxation, with a local overshoot near the exit, which is not observed experimentally.
327
Figure 33. Experimental and computed birefringence patterns.
(a) GOB model ; (b) ~ model ;~/= 25 s -1.
25 A
to
o 20- )r
�9 15 0 c 0 o~ 10_ c
m L .
L _
~ 5 - ~'ql~m qb ~ ~
0 ! !
-20 -10 10 !
20 30
Axial d is tance (mm)
Figure 34. Birefringence along the flow axis (T = 205 ~ ~ = 25 s-l).
(0) experimental measurements ; ( ~ ) GOB model ; (- - -) mPTT model.
328
At higher shear rates, similar results are obtained (Figs. 35 and 36). We observe again that a total relaxation is never obtained as in the experiment, where the fringe of order zero is never visible in these flow conditions.
Fig.35 " Experimental and computed birefringence patterns; ~ = 40 s'l. (a) GOB model ; (b) ~ model.
tO o 3 0
x
~p o 2 0 (P
C L .
N ' = '
~ 1 0 L .
co
0 - 2 0 - 1 0 0 10 20
Axial distance (mm) Figure 36. Birefringence along the flow axis (T = 205 ~ ~f = 40 s l ) . (e) experimental measurements ; ( - - ) GOB mode] ; (- - -) mP2~ mode].
30
329
The flow computations in a short die (here T= 145 ~ obviously imply free surface determination corresponding to the swell phenomenon. It may be seen in Fig. 37 that the general birefringence patterns are similar, even if the results of the mPTT model seem more realistic in the downstream region than those of the GOB model, for which discontinuities appear along the centreline, apparently due to an incorrect determination of the relaxation time at very low shear rate.
Figure 37. Flow birefringence patterns :
T = 145 ~ ? = 35 s -1.
(a) P2~ model (b) experiments (c) GOB model.
A
o T -
x
o o c Q
c t . . .
Q k .
= u
m
2 0 -
60
4 0 -
0 -20
- - I - - | |
- 15 -10 -5
330
_ _
; e
%0 : I \ e ~
0 5 10 15 20
Axial distance (mm)
Figure 38. Birefringence along the flow axis (T = 145 ~ 7 = 35 s'l).
(O) experimental measurements
( - - ) GOB model (- - -) ~ model
This is confirmed by the evolution along the centreline (Fig. 38), where the relaxation of the GOB model appears to be too slow, whereas the mPTT results are in very nice agreement with the experiment.
6.2.2. Low density polyethylene. LDPE is a highly branched material, whose flow behaviour exhibits some
peculiarities, which constitute a good test for a numerical simulation. As presented in Section III-1, birefringence pat terns are perturbed around the re -en t ran t corner, which leads to the appearance of W-shaped fringes at the entry of the die land. It can be seen in Fig. 39 that the mPTT model allows on to obtain these characteristic shapes (the computation is related to experiments carried out at 175 ~ and 21 s -1, for an est imated value of the stress optical coefficient of
331
1.9 10 -9 m2/N). By comparing with Fig. 33, obtained under similar conditions for a LLDPE melt, the difference is really significant.
If we look at the principal stress difference profile ~ - o~I within the die gap in the downstream section (Fig. 40), we observe for the LDPE a non-monotonic evolution near the land entry, which tends towards a more regular shape when moving downstream. For LLDPE, the corresponding profiles are totally monotonic and only change slightly along the flow.
Conversely, the evolution of birefringence along the centreline is very regular and shows a slow mechanism of relaxation, which is also very well captured by the model (Fig. 41).
From these results, it may be pointed out that the mPTT model is able to predict differences in flow behaviour between linear and branched polymers, which proves both the quality of the computation and the efficiency of this constitutive equation.
I
Figure 39. Flow birefringence pattern
( ~ model, T = 175 ~ ~ = 21 sl) .
332
A r a. :S
m B u |
m
u
0,25 ~ .~ymmetry
0,2-~
0,15
0,1
0,05
l ine (a) d le wall
A
r n_
v
m n
t:) |
m
t:)
I I
0 0,5 1 Flow depth (ram)
0,25
0,2
0,15
0,1
0,05
s y m m e t r y l i ne (b) d ie wa l l
0 0 0,5 1
Flow depth (mm)
Figure 40. Computed profile of principal stress difference across the die gap a) LDPE
b) LLDPE �9 (o ) - 2 m m downstream of the section of contraction (e) �9 8 mm downstream of the section of contraction
333
A 3O
>r v
2O o c
c ":" 10
I , . , m
m
�9 I I ~ ~ O
0 -15 -10 -5 0 5 10 15 20
Axia l d is tance (mm)
Figure 41. Birefringence along the flow axis (T = 175 ~ q = 21 s-l).
(o) experimental measurements ; ( - - ) mP2~ model.
25 30
7. CONCLUSIONS
This paper reports experiments and numerical simulations related to a linear low-density polyethylene (LLDPE) and a low-density polyethylene (LDPE), in a significant number of axisymmetric and planar mixed flows. Converging and abrupt contraction geometries involving short and long dies were considered as well as extrudate swell flows occurring at the exit of the ducts under investigation.
The rheological behaviour of the two polymers was determined using classical techniques of rheometry, already described in Chapter II.1 (rotational and capillary rheometers for shear viscosity and first normal stress difference measurements; Cogswell method for the elongational viscosity).
Different viscoelastic constitutive equations were adopted for modelling the experimental data of both fluids:
- a memory-integral K-BKZ constitutive equation, using a damping function of the Wagner type depending on the invariants of the Finger strain tensor;
- two differential constitutive equations : a generalized Oldroyd-B model (GOB), with viscosities and relaxation times depending on the second invariant of the rate-of-strain tensor, and a multimode Phan-Thien Tanner model, with a spectrum of relaxation times and viscosities and two adjustable parameters.
Two numerical methods, very different in essence, were carried out to simulate the flows :
- a stream-tube method, - a finite element method.
334
Since significant comparisons were made between experimental results and those from both numerical techniques, no global comparison exists between all the experimental measurements, the constitutive equations and the two numerical methods. For example, the stream-tube method has only been tested with a Wagner memory-integral equation, and the finite element method only with differential GOB or mPTT equations. The Wagner equation has been investigated to simulate flows in axisymmetric geometries and for long dies for a wide range of flow rates. The differential constitutive equations have been tested for all kinds of geometries, but concerned a narrower range of flow rates. Nevertheless, the work reported in this paper provides systematic tests on different constitutive equations, with the same polymers and over a wide range of flow conditions.
In planar flow situations, the qualitative agreement between computed stress fields and experimental flow birefringence patterns is to be underlined. Concerning the constitutive differential equations used to model the polymers, the
model provides a better quantitative agreement, especially along the flow axis, than the GOB model. The mPTT model is able to capture the differences in stress patterns which are observed between LLDPE and LDPE. We think that this difference is mainly due to a poorer description of the relaxation times of the polymers in the GOB model.
In axisymmetric flow situations, the global pressure drop in a capillary rheometer is well described by the three constitutive equations. If one focuses on the entrance pressure drop, the numerical entrance pressure drop related to Bagley correction is found to be less important than the corresponding experimental data for the differential models for LDPE and LLDPE melts. For the Wagner integral constitutive equation, the computed entrance pressure drops are found to be lower for both fluids, but the computed values are closer to the experimental data for LLDPE than those related to the LDPE melt. This descrepancy, previously reported in the literature, needs fugher investigation.
When considering the extrudate swell, none of the three constitutive equations provides a good estimation in all processing conditions. The integral Wagner model gives good predictions at high flow rates, but underestimates extrudate swell at low flow rates. Concerning the differential models, a good agreement is obtained at low flow rates, but numerical convergence remains difficult at higher flow rates. The influence of the L/D ratio is opposite to experimental observations for the GOB model and more realistic for the mPTT model. As a consequence, without consideration of numerical schemes defined for solving the equations, the extrudate swell flow appears to be one of the more discriminating experiments for testing constitutive equations.
335
REFERENCES
1. D.V. Boger, Ann. Rev. Fluids Mech. 19 (1987) 157. 2. S.A. White, A,D. Gotsis and D.G. Baird, J. Non Newt. Fluid Mech. 24 (1987)
121. 3. G.H. Mc Kinley, W.P. Raiford, R.b, Brown and R.C. Armstrong, J. Fluid
Mech. 223 (1991) 411. 4. F.N. Cogswell, Polym. Eng. Sci. 12 (1972) 64. 5. J.L. White and A. Kondo, J. Non Newt. Fluid Mech. 3 (1977) 41. 6. S.A_ White and D.G. Baird, J. Non Newt. Fluids Mech. 20 (1986) 93. 7. X.L. Luo and E. Mitsoulis, J. Rheol. 34 (1990) 309. 8. H. Miinstedt, J. Rheol. 24 (1980) 847. 9. H. Miinstedt and H.M. Laun, Rheol. Acta 20 (1981) 211.
10. D.V. Boger, Pure Appl. Chem. 57 (1985) 921. 11. P.J. Coates, R.C. Armstrong and A. Brown, J. Non Newt. Fluid Mech. 42
(1992) 141. 12. R. Keunings, J. Non Newt. Fluid Mech. 20 (1986) 209. 13. J.M. Marchal and M.J. Crochet, J. Non Newt. Fluid Mech. 26 (1987) 77. 14. R.C. King, M.R. Apelian, R.C. Armstrong and R.A, Brown, J. Non Newt.
Fluids Mech. 29 (1988) 147. 15. D. Rajagopalan, R.A. Brown and R.C. Armstrong, J. Non Newt. Fluids Mech.
36 (1990) 159. 16. F. Debae, V. Legat and M.J. Crochet, J. Rheol. 38 (1994) 421. 17. B. Berstein, K.&Feigl and E.T. Olsen, J. Rheol. 38 (1994) 53. 18. R. Keunings and M.J. Crochet, J. Non Newt. Fluid Mech. 14 (1984) 279. 19. A.C.Papanastasiou, L.E. Scriven and C.W. Macosko, J. Rheol. 27 (1983)
387. 20. A.C. Papanastasiou, L.E. Scriven and C.W. Macosko, J. Non Newt. Fluid
Mech. 22 (1987) 271. 21. S. Dupont and M.J. Crochet, J. Non Newt. Fluids Mech. 29 (1988) 81. 22. X.L. Luo and R.I. Tanner, Int. J. Num. Meth. in Eng. 25 (1988) 9. 23. A. Goublomme, B. Draily and M.J. Crochet, J. Non Newt. Fluids Mech. 44
(1992) 171. 24. A. Goublomme and M.J. Crochet, J. Non Newt. Fluids Mech. (1993). 25. G. Barakos and E. Mitsoulis, J. Rheol. 39 (1995) 193. 26. Y. Bdreaux and J.R. Clermont, Int. J. Num. Meth. Fluids Vol 21, 5 (1995)
371. 27. D.G. Kiriakidis, H.J. Park, E. Mitsoulis, B. Vergnes and J.F. Agassant, J.
Non Newt. Fluid Mech. 45 (1992) 63. 28. K.A. Feigl and H.C. ()ttinger, J. Rheol. 38 (1994) 847. 29. B. Knobel, Ph.D. Thesis, Diss. Zurich Eth N ~ 9480 (1991) 30. H. Maders, B. Vergnes, Y. Demay and J.F. Agassant, J. Non Newt. Fluid
Mech., 47 (1992) 339. 31. H.J. Park, D.G. Kiriakidis, E. Mitsoulis and K.J. Lee, J. Rheol. 36 (1992)
1563. 32. R. Ahmed, R.F. Liang and M.R. Mackley, J. Non Newt. Fluid Mech. 59, (1995)
29. 33. M. Van Gurp, C.J. Breukink, R.J.W.M. Sniekers and P.P. Tas, Spie,
2052 (1993).
336
34. E. Mitsoulis and H.J. Park, Theor. and Appl. P~eol. Proc. XIth Int. Cong. on Rheology, Brussels (1992). 385.
35. D.G. Kiriaskidis and E. Mitsoulis, Adv. in Polym. Tech. 12 (1993) 107. 36. M.H. Wagner, Rheol. Acta. 18 (1979) 681. 37. J. Meissner, Pure Appl. Chem., 42 (1975) 551. 38. C. Beraudo, T. Coupez, B. Vergnes, C. Peiti and J.F. Agassant, Les Cahiers
de Rh~ologie, XI, 3-4 (1993) 303. 39. C. B~raudo, Th~se de Doctorat (1995) 40. J.R. Clermont, C. R. Acad. Sci. Paris, s~rie II-1 (1983) 297. 41. C. Carrot, J. GuiUet, J.F. May and J.P. Puaux, Makrom. Chem. Theory
Simul. 1 (1992) 215. 42. P.L. Soskey and H.H. Winter, J. Rheol. 2 (1984) 625. 43. R.J. Gordon and W.R. Schowalter, Trans. Soc. Rheol. 16 (1972) 79. 44. M.W. Johson and D. Segalman, J. Non Newt. Fluids Mech. 2 (1977) 255. 45. N. Phan Tien, J. Rheol. 22 (1978) 259. 46. J. Guillet and M. Seriai, Rheol. Acta. 30 (1991) 540. 47. M. Viriyayuthakorn and B. Caswell, J. Non-Newtonian Fluid Mech, 6
(1980) 245. 48 X.L. Luo and E. Mitsoulis, J. Num. Meth. Fluids 11 (1990) 1015. 49 K.R. Rajagopal, Theor. Comput. Fluid Dynamics, 3 (1992) 185. 50. J.L. Duda and J.S. Vrentas, 22 (1967) 855. 51. K. Adachi, Rheol. Acta, 22 (1983) 326. 52. K. Adachi, Rheol. Acta, 25 (1986) 555. 53. J.R. Clermont, Rheol Acta 27 (1988) 357. 54. P. Andr4 and J.R. Clermont, J. Non-Newtonian Fluid Mech. 39 (1990) 1. 55. J.R. Clermont and M.E. de la Lande, Theoret. Comput. Fluid Dynamics,
4 (1993) 129. 56. J.R. Clermont and M.E. de la Lande, J. Non-Newtonian Fluid Mech. 46
(1993) 89. 57. J.R. Clermont, M.E. de la Lande, T. Pham Dinh and/~ Yassine, Int.
J. Num. Meth. Fluids 13 (1991) 371. 58. M. Normandin and J.R. Clermont, J. Non-Newtonian Fluid Mech. 50
(1993) 193. 59. Y. B~reaux, Rapport de DEA, Universit~ de Grenoble (1992). 60. J.R. Clermont, Rheol Acta 32 (1993) 823. 61. A. Yassine, Th~se de Doctorat de Math~matiques, Universit~ de
Grenoble, 1989. 62. P. Lesaint, P.A. Raviart and C. de Boor Ed., Academic Press, (1974) p. 89. 63. M. Crouzeix and P.A. Raviart, Rairo, A3, 3 (1973) 33. 64. M. Fortin and A. Fortin, Commun. Appl. Num. Methods, 1 (1985) 205. 65. P. Beauffls, B. Vergnes and J.F. Agassant, Int. Po, Report 189 (1971) 66. J. Batchelor and F. Horsfall, Rubber Plast. Res. Assoc. (1971) 67. A Gourdin and M. Bouhmarat, M~thodes Num~riques Appliqu~es, Tec et Doc
Ed. Lavoisier, Paris (1989).
Rheology for Polymer Melt Processing J-M. Piau and J-F. Agassant (editors) �9 1996 Elsevier Science B.V. All rights reserved.
337
Slip at the wal l
L. L6ger, H. Hervet and G. Massey
Laboratoire de Physique de la Mati6re Condens6e, URA C.N.R.S. 792, Coll6ge de France, 11 Place Marcelin - Berthelot, 75231 PARIS Cedex 05, FRANCE
1. I N T R O D U C T I O N :
It has been suspected for a long time that flows of high molecular weight polymers couJ exhibit a non-zero boundary condition for the velocity at the wall, contrary to what is usual J simple liquids. If such is the case, this is of great practical importance, as for example in extrusion process, the shear rate and the stress experienced by the polymer molecules at tt interface will condition the properties of the extrudate. A number of investigations have tht attempted to characterise wall slip in polymer systems, first through rheological macroscop: characteristics such as pressure drop as a function of flow rate, for various flow geometries [i 8], then trying to characterise the flow behaviour as a function of the thickness of the liquJ
[9,10], or determining the velocity gradient by the use of tracer particles or by velocimeu [ 11,12], and more recently, measuring directly the local velocity in the immediate vicinity the wall (within 1000A from the wall) [ 13,14,15]. At the same time, and because instability J an extrusion process and extrudate defects have been related to the onset of flow with sli boundary conditions [2,16,17], several attempts have been made to model flows with slip, an to relate the onset for strong slip with the molecular characteristics of the polymer [ 18 - 25].
The macroscopic investigations show that the appearance of wall slip depends on tt system polymer/surface under investigation. For some systems, and some geometries of flo~ wall slip seems to always be present, whatever the stress level, in the range accessible in tk macroscopic experiments [2] (but the data of reference [2] may have been misinterpreted, entry corrections, at capillary heads, have not been introduced), while in many other cases limiting shear rate has to be reached before slip can be macroscopically detected [26,4,3,12]. detailed analysis of the onset of slip is not however easy to conduct from macroscopi experiments: if the extrapolation length of the velocity profile to zero remains small compare with the thickness of the flowing liquid, the resulting change in the relation pressure drc versus flow rate may be hardly detectable. The direct investigation of the velocity at the wal] thus appears as a key step in the understanding of the flow behaviour of molten polymers.
We present here an investigation on the role of polymer/surface interactions on tk existence of wall slip. Direct measurements of the local velocity at the wall allow one t investigate how wall slip is influenced by polymer chains anchored to the surface, b adsorption or grafting [14,15]. Such local measurements are limited to a small number polymers, due to the technique used, but they allow unambiguous comparisons with tk molecular models which have been recently developed and can thus be validated. The~, experiments and models demonstrate that for surfaces with a weak roughness, the ke parameters which govern the existence of a shear rate threshold for the onset of strong slip~ the wall are the deformability of surface anchored polymer chains and their degree interdigitation with the bulk liquid. Several regimes of wall slip can thus be identified. The should, of course, also show up in macroscopic rheological measurements, each regirr corresponding to a particular friction law. The understanding of how the polymer surfac
338
interactions govern the structure of the surface anchored chains and as a consequence the slip regime should open the way to the design of tailored surfaces adjusted for a particular application: efficient extrusion, controlled friction ....
2. L O C A L D E T E R M I N A T I O N OF T H E V E L O C I T Y AT T H E W A L L :
2.1 Measuring technique- Unambiguous determination of the conditions under which slippage occurs requires a
technique able to measure the velocity of the fluid in the immediate vicinity of the solid wall over a thickness comparable to the size of a polymer chain, i.e. a few tens of nanometers. Classical laser Doppler velocimetry does not meet this requirement even if it allows for the determination of velocity profiles which clearly reveal a non-zero velocity within typically a
few 10 ~m from the wall. We have developed a new optical technique, Near Field Velocimetry (N.F.V.) [14], which combines Evanescent Wave Induced Fluorescence (E.WF.) [27] and Fringe Pattern Fluorescence Recovery After Photobleaching (F.P.F.R.A.P.) [28]. The former technique gives the spatial resolution normal to the solid wall, while the latter one enables the determination of the local velocity of the fluid. A major constraint of the technique is that it needs polymer molecules labelled with an easily photobleachable fluorescent probe.
The sample cell is schematically presented in Fig. 1. A drop of fluorescently labelled polymer melt (refractive index n2) is sandwiched between two plane silica surfaces (refractive
index n2 < n l) held at a distance d (d = 8 ~tm in all the experiments presented here) by two mylar spacers. Care is taken to maintain the two limiting surfaces parallel when adjusting the mechanical clamps which hold them at the distance d, checking for no or only one or two visible interference fringes in white light between the beams partially reflected by the two surfaces. In order to avoid parasitic hydrodynamic effects, the drop is never in contact with the mylar spacers nor with the edges of the silica plates, as schematically presented on the top view of the cell in Fig. 1 b. The upper surface can be translated with respect to the bottom one along a
!ar spacer silica plate Vt
laser beams
PDMS n2
s ~
I silica pri , i
!
top face of the pris PDMS
. ~ r / , v n t !
top
la lb
Figure 1. Schematic representation of the sample cell, a) Side view of the cell, b) Top view of the cell.
339
direction perpendicular to the fringes, at a controlled velocity, Vt, imposing a simple shear flow
to the drop of liquid. The apparent shear rate 4/~pp = Vt/~dd can be varied in the range
0.01s -1 ___ ?'app -< 40s-1. As flow tracers we use polymer molecules which are chemically labelled
with one fluorescent molecule at both ends. Typically one per cent by weight of the total polymer molecules are labelled, maintaining the concentration of fluorescent probes below 10 ppm for the high molecular weights used. To get a flow tracer, one needs to create in the sample a non- uniform concentration of fluorescent molecules, and then, to follow the deformation of this non-uniform distribution of probes under the effect of the flow. This is achieved through a photobleaching reaction which locally destroys a fraction of the fluorescent molecules if they are irradiated whith a high intensity beam of light with a wavelength in the absorption band of the probe. The photobleaching reaction is performed before turning on the flow. Several bleaching patterns can be used. In the N.F.V. technique, the photobleaching pattern is obtained by interference fringes localised in the immediate vicinity of the bottom surface. These fringes are
formed by the crossing of two laser beams of equal intensity and wavelength )~3 (obtained by a
polarisation based interferometer described in more detail below). The angle ot between the two incoming beams defines the fringe spacing i"
i - ~,0 The angle of incidence on the bottom surface, 0i, defines the vertical spatial 2 sin a/~ 2"
resolution of the experiment. If 0i is larger than 0c, total internal reflection occurs"
0c = sin-'(n~-//~ ). Then, anon-propagative evanescent wave exists in the liquid, with an \ /LI j I
exponentially decaying profile along the z direction, and a characteristic decay length A, the penetration depth. The investigated area has an approximately elliptic shape, with a typical size of .5 x 3 mm (The useful part of the sample in which interferences are formed has in fact the shape of the intersection of the two laser beams with the bottom surface, i.e. the intersection of two ellipses (see figure l b)). To increase the sensitivity, we have used a spatial modulation of the position of the fringes as described by J. Davoust and L. L6ger [28]. To understand how this modulation is produced, we need first to describe the interferometer which produces the fringes. An Argon laser beam is split into two beams with crossed polarisations by passing it through a Wollaston prism. The angle between these two beams is fixed by the Wollaston prism. Their relative intensity depends on the angle between the polarisation of the incoming laser beam and the directions of the neutral lines of the prism. For an angle of 45 ~ , the two outcoming beams have the same intensities. These two beams are rendered parallel to each other by passing them through a biprism, then the two polarisations are set parallel by letting one beam cross a half wave length plate. Finally the two beams are recombined by passing through a converging lens: as they are parallel, they recombine at the focus of the lens, which is adjusted to be exactly at the botom surface of the sample.In order to modulate the position of the fringes a phase shift is introduced on one of the two interfering beams via a Pockel cell placed in front of the Wollaston prism, with its neutral axis parallel to those of the Wollaston prism. The amplitude of the modulated phase shift is controlled by adjusting the modulated voltage applied to the Pockel cell, thus one can make the fringes oscillate around their equilibrium position with an amplitude of half a fringe spacing, at the frequency F.
The experiment is performed in two steps: 1) the position of the fringes is held constant I no spatial modulation) and the full power of the laser is shined for the bleaching period
(typically 50 ms), thus printing into the sample, over the thickness A, a periodic distribution of fluorescent probes, with the periodicity i 2) the power of the laser beam is attenuated by a factor 5x 103 , so that no appreciable bleaching of the fluorescent probes can occur during the whole measuring period, and the spatial modulation of the fringes is turned on. The
340
fluorescence intensity, collected with a photomultiplier, is then the superposition of a DC. level, an F and a 2F components[28]. The two modulated components are proportional to the product of the bleaching and the reading intensities both having a vertical distribution in e -z/A, we are
thus probing a slab of thickness A/2 above the solid surface.In the conditions of the
experiments presented here, A is typically 1000,~ and the vertical distance probed is comparable to the radius of high molecular weight polymer chains.
The F and 2F signals exhibit various features depending on the flow pattern: i) without flow, the spatial distribution of fluorescent probes produced by the bleaching pulse
relaxes towards equilibrium through diffusion, leading to an exponential decay of the 2F signal and to a zero F component; ii) with pure shear flow and no slippage, the original bleached fringe pattern is progressively
tilted, with a tilt angle increasing linearly with time. This distortion of the bleached pattern leads to a monotonic decrease of the 2F signal, and to the onset of the F signal. At long times both components go to zero; iii) if slippage occurs, with a slip velocity Vs, both the 2F and the F signals oscillate at the
frequency v = Vs/i, with the same amplitude, and in phase quadrature [28]. These oscillations in both the F and the 2F signals are a clear signature of slippage. Vs is obtained by either
measuring the time period of the oscillations (Vs = i/T) or their frequency, v, through a Fourier transform analysis of the signal. The technique is limited at high velocity, when the characteristic time of the data acquisition
system is comparable to l/v, leading to a decrease of the signal to noise ratio. It is also limited at low velocities, when the bleached pattern relaxes by diffusion faster than the appearance of the
oscillations due to the flow. The easily available range is typically l O-2~trn/s <Vs < 10~m/s. When the high velocity limit is reached, we switch to a bulk method which uses another bleaching pattern: only one beam (width - 50 ktm) is shined in the sample, at an angle smaller
2000 Vt _
top plate ~ , , , <~ I
[;/; ,'V.- V. �9 ,,., ,., ~ __>,
c .. ._x "E 1000 , , , , . .
Ax r prisme =
o r~ 2a
o
0
_ profile after shear ~'~ profile
before shear
-400 0 distance (~tm)
400
2 b Figure 2. �9 2.a) Principle of the bulk method �9 a line is first bleached through the sample (a), and then deformed by the flow. It is tilted without slip (b) or both tilted and translated with slip (c). The translation distance and the velocity profile are characterised through the deformation of the fluorescence intensiW profile monitored when the distorted bleached line is translated back and passes in front of the laser beam (2.b).
341
than 0c. During the writing period, this beam photobleaches a line through the sample (see figure 2). Under the action of the shear, without slip, this line tilts, while its origin on the bottom surface stays immobile. With slippage, the line is tilted and its origin is translated in the
x direction over a distance zXx equal to Vs times the shearing time. After turning off the shear, a backward translation of the entire sample brings the reading beam in partial coincidence with the bleached line, as evidenced by a decrease in the collected fluorescence intensity. Comparing the shape of this decrease as a function of the translated distance to that observed without shearing the sample, one can estimate both the effective shear experienced by the polymer and the slip velocity. Two such typical signals, one corresponding to a flow with slip, the other one corresponding to a test experiment in which the whole sample has been translated and not sheared (profile before shear) are presented in Fig. 2b.
2.2 Materials and surfaces: The near field velocimetry technique has been applied, up to now, to the investigation of
the flow behaviour of high molecular weights polydimethylsiloxane (PDMS). The samples are mixtures of labelled (less than 5% by weigth) and unlabelled PDMS, with an index of refraction
n2 = 1.410 at ~,0 = 457 nm. Several molecular weights have been investigated, their characteristics are reported in Table I. The corresponding weight average index of polymerisation is obtained by dividing Mw by the molar mass of the monomer m = 74 g/mol. The labelled chains are end terminated with 4-chloro-7-nitrobenzo-2-oxa-1,3-diazole (N.B.D.
for short, )~exc = 457 nm, )~em- 510 nm) [29]. The solid surface is the upper face of a
fused silica prism (index of refraction n2 = 1.463 at )~0), carefully polished (r.m.s. roughness -- 10A as characterised by X rays reflectivity). Then, the minimum penetration depth, obtained
for 0i -- 90 ~ is 94 nm. The experiments have been performed with a higher value of 140 nm, i.e. an explored thickness of 70 nm, typically twice the radius of the PDMS chains used.
Table I.
Polymer Mw (g/mole) Polydispersity Index Unlabelled polymer
Labelled polymer
32100 497000 608000 786000 962000
321000
1.18 1.14 1.16 1.22 1.27
1.18
Three types of surface state have been investigated, characterised by different polymer surfaces interactions: i) surfaces with weak polymer-surface interactions, obtained by chemical modification of the silica by grafting an almost dense monolayer of octadecyltrichlorosilane (O.T.S.). The packing of the O.T.S. monolayer can be controlled by the conditions (temperature and degree of hygrometry) during the chemical reaction [30], and can be estimated through the measurement of the contact angle of a reference liquid such as dodecane [30]. On a fully dense OTS layer, the interactions between the underlying silica and the PDMS are totally screened out, and PDMS no
longer spreads [31,32] or adsorbs. Then the advancing contact angle for dodecane is 0a = 34 ~ at
23~ On the almost densely packed monolayers used in the present investigation, the 0a values
were smaller, 24 .4~ 0a < 33.5 ~ A weak density of surface sites remain available for polymer adsorption.The adsorption of the PDMS chains takes place through hydrogen bonds between oxygens of the backbone of the chain and silanol sites of the surface, and is thus a
342
rather strong adsorption, but due to the surface treatement, the number of adsorbed chains per unit area remains small. ii) surfaces with strong polymer surface interactions, obtained by cleaning the silica surface by UV irradiation under oxygen flow [33]. This oxidising treatment burns all the impurities spontaneously adsorbed on the silica surface. When put into contact with this bare silica surface, many PDMS chains strongly adsorb, forming a quasi immobilised layer of chains anchored to the surface by many hydrogen bonds [29]. The structure of this quasi immobile dense adsorbed layer is expected to be very close to the pseudo-brush structure analysed theoretically by O. Guiselin [34], for irreversible adsorption from a melt or a concentrated solution. This situation should be representative of what happens when polymer melts are put in contact with common solid surfaces which are usually high energy surfaces on which polymers strongly adsorb. iii) surfaces covered by a dense simultaneously adsorbed and grafted layer of PDMS (Mw = 250 000, Mw/Mn = 1.1), having the structure of a pseudo-brush [35,34]. The grafting reaction performed at high temperature [35] reinforces the irreversibility of the anchoring of the chains on the surface, but the structure of the layer is expected to be very similar to that formed spontaneously when high molecular weights PDMS melts are put into contact with a clean silica surface.
2.3 Results: On Fig. 3, traces of the F and the 2F signals obtained using the N.F.V. experiment are
reported for different experimental situations. On Fig. 3a, no shear is applied to the polymer, and the whole sample is translated in the x direction at a velocity V. For the fluorescence intensity, this is equivalent to a strong slip. On Fig. 3b, the sample is sheared by moving the top surface at the velocity Vt. The oscillations in the F and the 2F signals are a clear signature of
the fact that the polymer layer within the distance A/2 from the solid surface is flowing in the x direction at the velocity Vs. The progressive decrease in the amplitude of the oscillations is due partly to the finite size of the laser beam, to the tilting of the bleached pattern and to self
3a 3b
F'-'" V / I I i I ~ I /
20 60 100 140 20 60 100 140 t(s) t(s)
Figure 3.Typical traces of the F and the 2F signals obtained in the N.F.V. method for: a) a translation of the whole sample in front of the reading beams; b) a sheared sample. The
oscillations in phase quadrature in the two signals indicate a motion of the fluid within the region of the evanescent wave.
343
diffusion of the labelled polymers which tends to relax the photobleached distribution of fluorescent polymer molecules.
2.3.1. Results for surfaces with a low density of surface anchored chains: On Fig. 4, a typical curve of the measured slip velocity is reported as a function of the
top velocity, for a surface with an advancing contact angle for dodecane 0a = 31 ~ A transition is
g3
E :=L
>
1 0 4
10 3
1 0 z
101
1 0 0
10 -I
i i i i l l l . " . . . . . . . . I . . . . . . . . I . . . . . . . . I . . . . . . . . I ' =
.
S
I
/ /
/
/
/
/ /
S
s4l'
/
/ /
1 0 -z ~ [
/
1 0 -3 / . . . . . . . . I . . . . . . . . I , , , . . . . . I . . . . . . . . I
1 0 -1 1 0 ~ 101 1 0 2 1 0 3
o
i l I l i l l |
10 4
V (pm/s) t
Figure 4. Evolution of the slip velocity, Vs, as a function of the top plate velocity, V t, for a PDMS melt of molecular weigt 9.6 105 in contact with a silica surface pretreated by grafting an
almost dense monolayer of OTS. The two dotted lines are respectively Vs =Vt and Vs= VtA/d. Clearly, slip is always present, as indicated by the fact that the measured Vs is always above the
average velocity one would have inside a layer of thickness A, in the case of a linear velocity gradient and no slip (lower dotted line). At very high shear rates V~ becomes comparable to Vt and the flow is almost a plug flow. The experimental relative uncertainty on Vs is larger for the low slip velocities (close to 20%) than for the larger ones (10%).
3 4 4
clearly visible between a regime of weak slip for V t < 1 Bm/s and a regime of strong slip for
Vt > 2 ~tm/s. A similar behaviour has been observed on all the surfaces grafted with O.T.S. monolayers, with variations in the location of the threshold. The data of Fig. 4 are reported in
V t - V s Fig. 5 as a function of the effective shear rate experienced by the polymer, T = ~ . d
E
V
>
=
=.
10 3 r
=
=
.
101 -
1 0 -1 _
e
10 .3
10 .2
. . . . . . . . I . . . . . . . . I . . . . . . . ~-
I
I = ',
I -. = ',
= I -
.=
l e l Ill
1 z
Isl =
=
. . . . . . . . I . . . . . . . . I . . . . . . . .
10 -I 10 0 101
(s )
Figure 5. Same data as on Fig. 4, now reported in terms of the slip velocity Vs as a function
of the shear rate y. A shear rate threshold is clearly visible, above which Vs strongly increases
with y, in a nonlinear manner, and then tends to saturate. The uncertainties in the slip velocity imply an uncertainty in the shear rate, which becomes very large at high slip veloiSities when the difference Vt - Vs becomes small, even if the relative uncertainty on Vs is smaller for large slip than for low slip.
The threshold is even more visible on this representation, as when the strong slip starts, the shear rate experienced by the polymer is no longer proportional to Vt. In fact, at the onset of strong slip, the shear stress remains locked, while the velocity at the wall strongly increases.
345
The slippage of the polymer melt can also be characterised by the extrapolation length b:
b = V ~ d r A logarithmic plot of b versus Vs is shown in Fig. 6.Three different regimes of S l dz "
flOW can easily be distinguished on this curve. At low shear, b is constant, independent of the shear rate, and rather small (even if larger than the radius of the polymer chains) indicating a constant (solid/like) friction between the polymer melt and the solid surface. Then, above a thresholdslip velocity, V*, b increases progressively, following a power law, with, for this particular molecular weight, an exponent 0.8 _+ 0.04, over three and a half decades This power law dependence indicates a nonlinear slip regime in which the friction between the polymer melt and the solid wall is no longer constant, but decreases progressively. Finally, at high Vs values, b tends to saturate, and one enters again into a linear slip regime, with a low but constant friction. This plot allows us to define quite accurately the slip velocity threshold for the entrance in the nonlinear slip regime, V*.
1 0 3
E
_Q
10 2
101
10 0
V
1 4~--- b
1 0 -~ . . . . . . . . I . . . . . . . I . . . . . . . I . . . . . . . . I . . . . . . . . I . . . . . . . ,I . . . . . .
10 -3 I0 -I 101 10 3
V (Bm/s) S
Figure 6. Slip length b deduced from the data of Fig. 4 and 5 as a function of the slip velocity Vs. The threshold for the onset of strong slip appears as a kink in the b (Vs) curve, at the critical velocity V*. Above V*, b increases with Vs, following a power law with an exponent 0.8 _+ 0.04. At very high shear rates, b deviates from the power law and the Vs dependence tends to saturate, indicating that a new regime of linear strong slip is approached.
These features are typical of surfaces on which the polymer is able to adsorb with a weak
surface density Z. We have not been able up to now to quantify the surface density of adsorbed chains, because of sensitivity problems: the total area of the drop of liquid deposited on the prism is too small to allow the use of the X ray reflectivity technique to measure the thickness of the adsorbed layer remaining on the surface after washing all the unattached chains with a good
346
solvent. But for comparison this has been done on similar surfaces the thickness of PDMS left on the surface is always small, comparable to a monolayer of monomers (thickness of order 10A), at the limit of quantitative analysis by X ray reflectivity. This means that the surface anchored chains are in the so called "mushroom" regime [36], with a surface density smaller or
comparable to Z0 = (afRN) 2 = N -1 : the surface anchored chains are independent of each other (RN = N l/2a is the average distance between the extremities of an ideal chain containing N monomers of size a).
We have investigated the way the typical features of Fig. 4 to 6 were dependent on the molecular characteristics of the system, i.e. the molecular weight of the polymer melt, Mw(P), the molecular weight of the surface anchored chains, Mw(N), and the surface density of surface attached chains, a parameter which is only qualitatively under control through the cbntact angle of dodecane with the surface.
N (adsorbed)" N=13000 10 o
" ' ~''"~1" ' ' ' ' ' ' "
\ *p323
-
I:1
10 "2 7 . . . . . . . I I , , , , ,
mira
10 3 10 4 10 5
P
E 0 I 4,
>
101
10 0 ... o r }
E -
" > 10 1 -
10 .2
10 a
10 4
~ -
E .
>
=,
10 .2 10 a
P (bulk) N =1300C i i k l l l ill i s i m ml aa
-1. 9
. . . . . , , I , ~. , , ,
10 4 10 5
N ' ' ' ' ' " I ' ' ' ' ' ' ' ~ -
i �9 ! !
10 4 10 5
N = P
Figure 7.Dependence of the critical velocity, V*, versus the weight average polymerization index of a) the flowing melt, P; b) the surface chains, N" c) surface chains,N, in the case where it is identical to that of the melt (N = P).
347
The experimental procedure to obtain a surface layer with a molecular weight different from that of the flowing melt is the following: first a layer of a given molecular weight is adsorbed on the surface, then all the unattached chains are washed out with a good solvent, and finally the drop of the polymer melt to be investigated is deposited. If one takes care to start by adsorbing the largest molecular weight, no important exchange takes place between the surface anchored chains and the melt chains. The results are gathered in Fig. 7 a, b, c, where the threshold velocity V* is reported as a function of the different molecular weights of the surface chains and of the melt chains.
To summarise, when a low density layer of surface anchored chains forms, three different wall slip regimes exist, depending on the shear rate" 1) at low shear rates, a linear weak slip regime is observed, (i.e. Vs is proportional to Vt in a simple shear experiment), with an
extrapolation length of the velocity profile b smaller than lgm, and thus not detectable in a macroscopic experiment. 2) above a shear rate threshold, which depends on both the molecular weights and the surface density of the surface attached chains, a strongly nonlinear slip regime appears, with an extrapolation length of the velocity profile depending almost linearly on the slip velocity, over more than two decades. In terms of shear rates, this nonlinear slip regime occupies a rather narrow shear rate range. The result is the appearance of strong slip, with macroscopic extrapolation lengths. The threshold and the extension of this regime depend on both the surface chains and the melt molecular weights. 3) at high shear rates, a linear slip regime is recovered. The slip velocity is then large, and the flow is almost a plug flow, with a macroscopic extrapolation length b depending on the molecular weight of the flowing molecules, and no longer of the surface attached chains. Part of the nonlinear slip regime and
this high slip regime, with b--- 1001am to 1 mm, are certainly visible in macroscopic investigations.
2.3.2. Results on surfaces with a high density of surface anchored chains" In Fig. 8 we have reported the slip velocity, V s, as a function of the top plate velocity,
Vt, for a melt (molecular weight 9.6 105) deposited on a pseudo-brush [34,35] formed by adsorption of a melt of molecular weight 2.4 105 on clean silica. The thickness of the pseudo- brush is 24 nm, comparable to the size of the chains which have been used to make it. The flow behaviour is now quite different from what has been observed on low adsorbing surfaces: Vs increases linearly with Vt (the slip length b is constant) and remains small compared to Vt, up to high shear rates. Above a shear rate threshold, one can usually observe, using the bulk detection technique, a transition towards a nonlinear slip regime in which b increases almost linearly with Vs. In a few experiments, a fracture of the fluid has been observed instead of the progresive nonlinear slip regime, with an immobile layer located close to the bottom surface, and the rest of the fluid moving along the shear direction, at the top plate velocity. In the low shear regime, the slip velocities and the b value are comparable (slightly smaller) to what is obtained with the same molecular weight of the flowing polymer on the weakly adsorbing surfaces, indicating that the friction between the surface chains and the melt is not drastically different in the two cases. The surface density of chains in the pseudo-brush is however much larger than that in the layers formed on the weakly adsorbing surfaces (thickness of the pseudo- brush: 24 nm, thickness of the weakly adsorbed layer --1 nm). This relatively weak friction is surprising at first sight. It could be a consequence of the fact that the interdigitation between a pseudo-brush formed from a melt and a melt is not expected to be very important [37] (and indeed such a situation has been observed to onlv induce a weak adhesion enhancement when the melt is replaced by an elastomer, [38]).
348
10 4
10 2
E ::I.
> " 10 0
1 0 -2
' ' " " q ' ' ' " h ' 1 ' ' " " ' I ' ' " " ' I I , , , , , a
- ~ -
=
o~ -, [ qo
. 0
~ o "
O 0 O "~
[ 0 ~ " l n , innnni , n nnlml~ n , t l l n ~ I l . . m l ' u ,
10 3
" ~ 101
10 1
' ' " " I ' ' " ' = I ' ' " = I - - ' ' ' " ' I ' ' " "5 ' ' ' "~
b : -
o
V* .
. 1'/'" �9 .
~
_ i , . h id ' 1 �9
~ O ~ / = ,
, , l i n d , , i n n | , ,n ,n "I , i n . , - ] , . u . "I n inn,n,
10 -1 101 10 3 10 .2 10 0 10 2 10 4
V (l.tm/s) V (pm/s) t s
Figure 8. Results obtained for a polymer melt of molecular weight 9.6 105 flowing on a silica surface covered with a pseudo brush made from a melt of molecular weight 1.93 105. a) Vs as a function of Vt; b) slip length b as a function of Vs. The friction in the low shear regime is comparable to that obtained on a low density surface layer, but the critical velocity V* is much larger.
3. MOLECULAR MODELS AND DISCUSSION:
3.1 The ideal surface- de Gennes's conjecture: Fifteen years ago, de Gennes suggested [18] that high molecular weight polymers, when
in contact with a smooth non adsorbing wall, should always display a slip boundary condition, with an extrapolation length b depending only on the molecular weight of the polymer chains, and able to become very large, i.e. to render the slip easily visible macroscopically. This relies
on the following remarks: monomers are in contact with the wall; the constraint at the wall, a, is
related to the friction, k, between these monomers and the wall, by k = o/Vs; k is characteristic of monomers/wall friction, and is expected to be the same for a liquid of small molecules
identical to the monomers or for the polymer liquid. The stress G can also be evaluated in the
liquid: cr = 1] dV/dz z=O = q Vs/b if slip occurs with the extrapolation length b. Then, comparing the situation of the liquid of monomers, for which b = a, and the polymer, one
expects b = a q/rio, for the polymer case, with 1"1 the bulk viscosity of the polymer melt, and
qo the viscosity of the equivalent liquid of monomers. In the reptation approach, the viscosity is
q = q0 p3/Ne2, with P the polymerisation index of the polymer chains, and Ne the average number of monomers between entanglements [40]. For high molecular weights polymers, with P = 104, Ne = 10 2 . and the size of a monomer a = 3A, one gets b = 3 cm, i.e. huge slippage.
Thus. high molecular weights polymers should slip whatever the shear rate. in a linear slip regime, i.e. with an extrapolation length b independent of the flow regime. The extrapolation length, b, and the magnitude of the slip velocity at a fixed shear rate, should strongly depend on the molecular characteristics of the polymer. These features do not seem to be observed experimentally. It has been progressively recognised that practical experimental situations could be very different from the ideal surface postulated in the simple de
349
Gennes'analysis, and that many additional effects could complicate the description. In many systems where the polymer melt is not far from its glass transition temperature, attractive interactions between the solid surface and the polymer can lead to the formation of a thin glassy layer close to the surface. It was first thought that such a glassy layer could block slippage at low shear rates [41 ], due to entanglements between chains pertaining to the glassy layer and chains in the liquid melt. The role of entanglements between chains immobilised at the surface and chains in the melt on wall friction has thus been investigated, starting from simple systems where the immobile chains are independent of each other to go towards more realistic and more complicated situations where the surface chains are interacting with each other and with the melt [23 - 25]. Obviously, the roughness of the solid surface can also play an important role, modifying the flow pattern and possibly deforming the polymer chains, thus reacting on the flow characteristics. This is certainly an important issue, from a practical point of view, but it has not yet been investigated in detail theoretically.We shall concentrate here on the effect of polymer molecules attached to an otherwise smooth and fiat wall, and analyse successively the two situations of independent adsorbed chains and of dense immobilised layers.
3.2. Role of polymer chains anchored to the wall at low surface densities A few polymer chains attached to the surface have a drastic effect on the boundary
condition for the flow velocity: they strongly reduce the magnitude of slip at low shear rates [23], as they increase the surface friction compare to an ideal surface. However, polymer chains are easily deformable objects, and under the action of the shear forces, the surface chains tend to elongate and to disentangle from the melt at high enough shear rates. This aptitude to deform in the flow is responsible for the appearance of a shear rate threshold above which strong slip, comparable to what is expected on ideal surfaces, is recovered. The magnitude of this shear rate threshold has been related to both the molecular weights of the surface attached chains and of the flowing polymer, and to the surface density of anchored chains by Brochard and de Gennes [23], modelling the friction between the surface chains and the flowing melt, and the deformability of the surface chains.
One surface chain (polymerisation index N), anchored on the solid by one extremity, experiences a friction force due to all the other flowing chains from the melt (polymerisation index P) which are entangled with it. The N chain elongates under the action of this friction force. Two delicate questions have to be answered in order to estimate the friction force exerted by the melt on one surface chain: first, one needs to evaluate the relative velocity of one melt
v (-xD
Figure 9. Schematic representation of the molecular process allowing the flow of the P chains when entangled with the surface N chains.
350
chain and of the surface chain when the local velocity of the melt at the surface is Vs, and secondly, one has to estimate the number of independent free chains from the melt which indeed are entangled with the surface chain. In the reptation picture, the process is schematically represented in Fig. 9: a P chain is entangled with the N chain fixed at the wall; in order to allow for their relative motion with velocity Vs, one extremity of the P chain has to pass close to the N
chain and relax the topological constraint, in a time shorter than the time 1: it takes for the N chain to travel parallel to the surface over a distance comparable to the tube diameter, i.e.:
1: = (aNel/2)/Vs .
The P chain has thus to move inside its own tube at a velocity Vtube, higher than Vs"
PaN , Vtube Ltub-------~e - Ne P -
Vs The amplification factor P/Ne is responsible for a strong dissipation and this is the origin of the suppression of slip at low shear rates when chains anchored to the solid wall are present. Knowing Vtube , the friction force associated with one P chain can be estimated" the dissipation
in the relative motion of the P chain along the N chain is TS = P~Vtube 2 = fvVtube, with ~ the local monomer-monomer friction and fv the friction force between the two chains. One has thus
fv = arlpVs, with rip the bulk viscosity of the melt of P chains. The total friction force experienced by the N chain depends on the number of P chains
which are entangled with it. This is one of the delicate problems of the reptation picture: the total number of entanglements between the N chain and its neighbours is N/Ne, and is fixed as soon as the average number of monomers between entanglements Ne is known, but it is not obvious whether or not some P chains have more than one entanglement with the N chain, with the result that the number of P chains trapped by one N chain may be different from the number of entanglements. This point has been strongly debated in the recent years (see the corresponding paragraph in the chapter on the reptation model in this book), and recently reanalysed and applied to the slippage problem by C. Gay et al [24,251: two different regimes in the way the number of trapped P chains, X, depends on N have to be distinguished" i) if N < Ne 2 ,
X = N//N , and each P chain has on the average one entanglement with the N chain; ii) if e
N > Ne 2, X = N 1/2 and one P chain has more than one entanglement with the N chain. With a typical Ne value of 100, only very long grafted chains are in the regime with more than
one entanglement per trapped P chain. In the simple regime with N < Ne 2, the total friction force experienced by the N chain is
N Fv = r l p a ~ e Vs.
Following the Pincus picture [42], an elastic restoring force develops, kT
Fe] - - ~ , D
with D the diameter of the associated Pincus blob. Equating the friction and the elastic forces, one gets kT N
= arle V s �9 D 7 e" above the velocity V 1 for which Fv = kT/RN, D decrease with Vs and the N chain elongates along the flow. When the velocity is increased, the N chain elongates more and more and the
351
diameter of the Pincus blob progressively decreases, down to a value comparable to the average distance between entanglements in the melt. One thus enters into what has been called the marginal regime: if the velocity is further increased, the N chain tends to disentangle. Then the friction decreases and can no longer balance the elastic force associated with the large elongation" the N chain thus tends to recoil. In fact, above the critical velocity V* for which D becomes comparable to the average distance between entanglements, the elongation remains locked at the value it had at V*, for a large domain of surface velocities.
These ideas can be used to model the flow behaviour in the presence of surface anchored chains at low surface densities ( i.e. for N chains in the "mushroom" regime of grafting
characterised by a number of surface anchored chains per unit area,v, related to the
dimensionless surface density of chains Z by Z = va 2, such that VRN2<<I, with RN the radius of the N chains),
Three regimes of wall slip are expected: 1) a linear low slip regime, at low shear rates, in which the N chains are only weakly
elongated. The number per unit area of P chains trapped by the surface layer of N chains is
vp = vN/Ne. The constraint at the surface is • = vN/NeaYlpV s, and the extrapolation length of 1
the velocity profile is b = - - - -N- v a ~
Ne 2) the marginal regime, above the critical slip velocity V* --- kT/rlpNa 2. In this regime,
the constraint is locked at ~* = vkT/aNe I/2 and the slip length increases linearly with Vs:
b = Vs rl_..~p aN e,~. This is a nonlinear slip regime, with the appearance of strong slip. If v kT
N = P, V* -- N -4, while at fixed N, V* --- P-3.V* is independent of the surface density of N chains.
3) at high shear rates, a regime of strong linear slip, comparable to what is expected on
ideal surfaces, in which the surface chains no longer play any role, with boo = arlp/rl0 (110 is the viscosity of a fluid of monomers, and a the monomer size).
In terms of shear stresses, the onset of the marginal regime corresponds to the stress
threshold o*-- vkT/Nel/2a, independent of N and P but proportional to the surface density of
surface chains. At ~*, the slip velocity increases drastically. The corresponding shear rate
threshold y'::= o*/~p --- v / P 3 is independent of N. All these predictions are in quite good agreement with the experimental results obtained
by direct measurements of the local surface velocity for PDMS melts in contact with modified silica surfaces on which only a small surface density of polymer chains can adsorb (Fig. 3 to 7). Three slip regimes can be clearly distinguished in Fig. 5 or 6, and the salient features of Fig. 6 are: 1) a critical slip velocity V* value indeed independent of the surface density of adsorbed chains which is not known quantitatively, but which increases when the contact angle of dodecane on the surface decreases, and" 2) a nonlinear slip regime, in which b follows a power law as a function of the slip velocity, over more than three decades. The experimental exponent 0.8 +0 .04 is smaller than the predicted value 1, but this may be due to the fact that at the corresponding shear rate, one is no longer in the Newtonian regime for these high molecular weight polymers. The evolution of the experimental exponent with the melt molecular weight is qualitatively in agreement with this idea [43]. Moreover, the data reported in Fig. 7 appear clearly compatible with the predicted evolution of V* with both the molecular weights of the adsorbed chains and of the flowing melt. These experiments, even if not fully quantitative, because the surface chain density is not quantitatively under control, strongly suggest that the above molecular description of the friction between a flowing polymer melt and chains anchored
352
to the surface with a weak surface density (so that the effects of these surface chains are additive) is correct.
The consequences for the macroscopic flow experiments are of three types: 1) on surfaces on which only a small number of adsorption sites are available, the threshold for the
onset of the marginal regime is small, especially for high molecular weights (qt s --- v/rip), and usually out of the range accessible in pressure loss versus flow rate experiments. On such surfaces, usually only the end of the marginal regime, and the final linear high slip regime should be detectable. This may well be what has been observed on the surfaces treated with fluorinated molecules [44]. 2) it would be interesting to investigate on such surfaces the molecular weight dependence of the extrapolation length b, which should be proportional to the bulk viscosity of the polymer melt. No instabilities are expected for the flow under these conditions. 3) As the extrapolation length may become very large, in the mm range for high molecular weights, the flow is expected to become a plug flow, and the elongational stress on the chains at the surface of the extrudate should decrease. The extrudates should not exhibit sharkskin defects, which have indeed been observed [4].
3.3 Role of surface chains anchored to the wall at high surface densities: Common solid surfaces are usually high energy surfaces on which polymer molecules
should strongly adsorb, forming a dense layer of surface anchored chains. The usual practical situation is thus not that analysed in 3.2. We do not have yet a complete description of the friction between a dense surface anchored layer and a polymer melt, at all shear rates, because this is a delicate question. Such a description would necessitate 1) the understanding of the interdigitation between a surface anchored layer and a melt, under static conditions, in order to evaluate the zero shear rate friction, and 2) the modelling of the deformation of the surface chains under the action of the shear stress associated with the flow, in order to describe the disentanglement between the two and a possible onset of strong slip.
We have begun to understand the first point. The structure of surface layers strongly adsorbed from concentrated solutions or melts has been analysed theoretically by O. Guiselin [34]. These layers, named pseudo - brushes, are expected to be very similar to highly polydisperse brushes, with a surface density which depends on the polymerisation index of the chains and on the concentration in the solution which has been used to form them. The Guiselin analysis appears to be in good agreement with the experimental investigations of those layers [35,45]. The interdigitation between a pseudo - brush and a polymer melt has been investigated by M. Aubouy and E. RaphaEl [37]. They predict a rather complicated diagram with several interdigitation regimes, depending on the relative polymerisation index of the layer chains, N, and of the melt, P, but if the surface layer is formed by adsorption of the melt itself (N = P), the expectation is only weak interdigitation. This is coherent with the fact that such pseudo - brushes lead to only weak adhesion enhancement when in contact with an elastomer of the same chemical species [38]. This is also coherent with the fact that from the direct local measurements of the slip velocity when the polymer melt has been put into contact with bare clean silica surface, a condition for which we know that a dense pseudo-brush forms [35], at low shear rates the extrapolation length b is of the same order of magnitude as for weakly dense surface layers, i.e. the friction is not at all that of a fully interdigitated system, with the actual surface
density of the surface layer (b0-- 1/vR0, with v-- 1/N1/2a 2, i.e. b0 = a). We do not know a lot however about the second point, because the deformability of a
highly polydisperse brush is not easy to model. We have begun to understand how a monodisperse brush responds to a shear stress [25], and qualitatively we expect these dense structures to be far more rigid than the weakly dense surface layers investigated in 3.2. It is thus plausible that the threshold for the onset of strong slip appears at higher shear rates for dense surface layers than for weakly dense ones, as observed experimentally. Up to now we do not have predictions for the molecular weight dependence of these thresholds.
The consequences for the macroscopic experiments are clear however: the onset of strong slip falls now in a range accessible macroscopically, and as the extrapolation length in
353
the weak slip regime remains small, in the micron range, it cannot be detected macroscopically, except eventually in especially designed experiments in which the flow characteristics are analysed as a function of the width of the channel, down to very thin channels. This situation has thus been identified macroscopically as a situation where increasing the shear rate, one was going from a flow with no slip to, above a threshold, a flow with strong linear slip. We know now from the local measurements that weak linear slip occurs at low shear rates. It is important to notice that the onset for strong slip observed in the local measurements is quantitatively the same as the onset for slip observed in the macroscopic experiments performed on similar polymer melts [4,44]. But the local experiments are able to determine if above the threshold a fracture appears inside the polymer melt. When such a fracture occurs, it becomes difficult to relate the pressure drop, the average flow rate and the extrapolation length, because the size of the flowing part of the liquid is not known. The hysteresis observed in the macroscopic experiments can however be qualitatively understood: if after fracture, the flow rate is decreased, weak slip will only reappear at low enough flow rates so that reentanglement between chains from the two edges of the fracture has time to occur. This reentanglement process is expected to take place on a time scale comparable to a reptation time of the chains of the melt on static conditions, i.e. increasing with molecular weight like p3, but we do not know yet how to describe this process when one side of the fracture is moving with respect to the other.
The situation of a dense layer of surface anchored chains thus appears far less understood than the weak surface density case. It is, however, far more important practically and work is in progress to model it in details.
4. C O N C L U S I O N S
The flow velocity at the solid wall (in a slab of typical thickness 700A) has been measured directly for polymer melts submitted to a shear stress, using a novel near field velocimetry technique. The control of the polymer-surface interactions, and especially of the density of polymer chains able to adsorb strongly at the wall to form an anchored surface layer, has allowed a systematic investigation of the molecular mechanisms at the origin of the appearance of slip boundary conditions for the flow. For smooth surfaces with a very weak roughness, the key parameters which control the boundary condition for the flow velocity are 1) the degree of interdigitation between the surface anchored polymer layer and the flowing melt and 2) the aptitude of the surface chains to deform under the sollicitation of the shear force. These two effects are competing" interdigitation and entanglements between the surface chains and the bulk liquid lead to a large friction, and tend to suppress the slip that high molecular weight polymers should exhibit on an ideal plane and nonadsorbing surface, while the deformation of the surface chains under the effect of the flow tends to decouple the surface layer from the bulk fluid and to decrease the friction. The consequence is the appearance of a shear rate threshold which separates a regime of low slip at low shear rates from a regime of high slip at high shear rates.The location of this threshold depends on the molecular characteristics of the system, i.e., the molecular weights of the surface and bulk chains and on the number of chains per unit area in the surface layer. When the density of surface anchored chains is weak enough so that the surface chains act independently of each other, the whole process can be modelled, and the transition from the weak to high slip is progressive, through a nonlinear slip regime which has been called the marginal regime, in which the friction is no longer constant. For the high molecular weights investigated in the present study, the corresponding extrapolation length of the velocity profile passes from small values (slightly smaller than one micron) to macroscopic values (several hundred of microns), following in the marginal regime a power law as a function of the slip velocity, with an exponent close to one, over more than two decades. For surfaces with high densities of anchored chains, we do not vet have a complete picture, but things are in progress. Obviously these results have important ~mplications in all practical situations where a polymer melt is flowing along solid walls, and
354
should open the way to the design of surfaces with adjusted friction. Further efforts are needed to incorporate the effects of surface roughness.
R E F E R E N C E S
1. J. J. Benbow, P. Lamb, SPE Trans. 3 (1963) 7 2. A. V. Ramamurthy, J. Rheol., 30 (1986) 337 3. G. V. Vinogradov, V. P. Protasov, V. E. Dreval, Rheol. Acta, 23 (1984) 46 4. N. El Kissi, J. M. Piau, J. Non Newtonian Fluid Mech. 37 (1990) 55 5. D. S. Kalika, M. M. Denn, J. Rheol. 31 (1987) 815 6. K. Funatsu, M. Sato, Adv. Rheol., (Mexico), 4 (1984) 465 7. W. Knappe, E. Krumb6ck, Adv. Rheol., (Mexico), 3 (1984) 417 8. C. De Smedt, S. Nam, Plastic Rubber Process Appl., 8 (1987) 11 9. S. G. Hatzikiriakos, J. M. Delay, J. Rheol., 35 (1991) 497 10. R. H. Burton, M. J. Folkes, K. A. Narh, A. Keller, J. of Material Sci., 18 (1983) 315 11. J. Gait, B. Maxwell, Modern Plastics, Mc Graw Hill, New York, (1964) 12. B. T. Atwood, W. R. Schowalter, Rheol. Acta, 28 (1989) 134 13. K. B. Migler, H. Hervet, L. Leger, Phys. Rev. Lett., 70 (1993) 287 14. K. B. Migler, G. Massey, H. Hervet, L. Leger, J. Phys. Condens. Matter, 6 (1994) A 301 15. L. Leger, H. Hervet, Y. Marciano, M. Deruelle, G. Massey, Isra~l J. of Chemistry, 35 (1995) 65 16. C. J. S. Petrie, M. M. Denn, AIChE J., 22 (1976) 209 17. J. M Piau, N. E1 Kissi, B. Tremblay, J. of Non Newtonian Fluid Mech., 34 (1990) 145 18. P. G. de Gennes, C. R. Acad. Sci. Paris serie B, 288 (1979) 219 19. A. I. Leonov, Rheol. Acta, 23 (1984) 591 20. Yu B. Chernyak, A. I. Leonov, Wear, 108 (1986) 105 21. H. C. Lau, W. R. Schowalter, J. Rheol., 30 (1986) 193 22. A. I. Leonov, Wear, 141 (1990) 137 23. F. Brochard-Wyart, P.G. de Gennes, Langmuir, 8 (1992) 3033 24. A. Ajdari, F. Brochard-Wyart, C. Gay, P. G. de Gennes, J. L. Viovy, J. Phys. II France, 5 (1995) 491 25. F. Brochard-Wyart, C. Gay, P. G. de Gennes, to be published 26. J. J. Benbow, P. Lamb, SPE Trans., 3 (1963) 7 27. D. Ausserr6, H. Hervet, F. Rondelez, J. Phys. Lett., 46 (1985) L929 28. J. Davoust, P. Devaux, L. Ldger, EMBO J. 1 (1982) 1233 29. L. L6ger, H. Hervet, P. Silberzan, D. Frot, in "Dynamical phenomena at Interfaces, Surfaces and Membranes", Ed. D. Beysens, N. Boccara, G. Forgacs, Les Houches Series, Nova Science, 1993; D. Frot, Thesis, University Paris VI, 1991 30. P. Silberzan, L. L6ger, D. Auss6rr6, J.J. Benattar, Langmuir, 7 (1991) 1647 31. P. Silberzan. L. L6ger, Phys. Rev. Lett., 66 (1991) 185 32. J. B. Brzoska, N. Shahidzadeh, F. Rondelez, Nature, 360 (1992) 719 33. J. R. Vig, Treatese on Clean Surfaces Technology, K. L. Millal Ed., Plenum Press, New York, 1987 34. O. Guiselin, Europhysics Letters, 17 (1992) 225 35. M. Deruelle, L. L6ger, to appear 36. P. G. de Gennes, a) J. Phys. Paris, 37 (1976) 1443" b) Macromolecules, 13 (1980) 1069 37. M. Aubouy, E. Raphael, Macromolecules, 27 (1994) 5182 38. Y. Marciano, Thesis University of Orsay, France (1994) 39. M. Aubouy, H. Brown, L. Leger, Y. Marciano, E. Raphael, C. R. Acad. Sci. Paris, to appem 40. P. G. de Gennes, J. Chem. Phys., 55 (1971) 572" MRS Bulletin, (1991) 20 41. F. Brochard-Wyart, P. G. de Gennes, P. Pincus, C. R. Acad. Sci. Paris. 314 (1992) 873 42. P. Pincus, Macromolecules, 9 (1976) 386
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43. G. Massey, H. Hervet, L. Leger, to be published 44. N. E1 Kissi, L. Leger, J. M. Piau, A. Mezghani, J. Non Newtonian Fluis Mech., 52 (1994) 249 45. L. Auvray, M. Cruz, P. Auroy, J. Phys. II France, 2 (1992) 1133
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Rheology for Polymer Melt Processing J-M. Piau and J-F. Agassant (editors) �9 1996 Elsevier Science B.V. All rights reserved. 357
Slip and friction of polymer melt flows
N. E1 Kissi, J-M. Piau
Laboratoire de Rh~ologie, Domaine Universitaire, BP 53X, 38041 Grenoble Cedex (France)*
1. I N T R O D U C T I O N
The appearance of extrusion defects and instabilities [1-3] in polymer melts flowing through sudden contractions is usually related to fluid slip at the wall [4, 5]. It is therefore understandable tha t this phenomenon has been widely investigated in many studies. However, it is important to s tar t with, to note the existence of various types of slip. In the field of polymer melt rheology, the following are referred to: Slip with additives containing ~ e . This shows the role of the additives, the wall material and the surface state of the wall. It has been studied by various authors [6-8] who have shown that the additive migrates towards the wall, forming a thin lubricating film and/or a thin film of low viscosity between the wall and the polymer. The polymer may then slip on this fine lubricating layer. Linear slip between polymer and wall. The main characteristic of this type of slip is that it occurs at all stress levels. It has been modelled by de Gennes in 1979 [9], but is difficult to observe experimentally since it supposes the use of ideal surfaces: perfectly smooth and developing no interaction with the flowing polymer. Cohesive slip with weak interacti.o.ns. Considering tha t the occurrence of slip is highly dependent on the interaction developing between the polymer and the wall, Brochard and de Gennes in 1992 [10] described the case where there is only few sites of adsorption of the polymer chains at the surface of the dies. The initial situation, prevailing before flow, is called the mushroom regime and the friction law controlling slip has been deduced theoretically:
- For low flow regimes, the absorbed chains at the wall are entangled with the bulk polymer. Slippage is then very small and wall stress increases with slip velocity.
- When flow regimes are increased, the marginal state could be observed. It corresponds to a t ransi t ion zone, with chains which are still unable to disentangle, because the balance between elastic and friction forces tends to ma in ta in a coiled structure. This regime is characterized by impor tan t slippage, during which the wall stress is no more dependent on the slip velocity. Variations in extrapolation length [10] as a function of slip velocity then follow a power law.
* Grenoble University (UJF and INPG) - UMR 5520 of CNRS.
358
- Finally, for higher flow regimes attached chains are completely elongated and disentangled from the bulk polymer. Slippage is then easy, wall stress is again proportional to slip velocity and friction is similar to that which could be obtained with ideal surface.
It should be underlined tha t it was possible to obtain slip at low stress values, by considering the flow of a polydimethylsiloxane (PDMS) through a silica die with walls graf ted by a f luorinated monolayer [11]. It can also be found in the exper imenta l s tudy published in 1993 by Migler et al. [12], which moreover validates the model prediction in term of extrapolation length, for sufficiently high slip velocities. Macroscopic slip at the wall with strong interactions. It is observed when considering high surface energy dies (stainless steel die ...). It is comparable to that which would occur between a soft solid polymer and another strongly interacting solid. This was observed by Benbow and Lamb in 1963 [13], with a silicon of high molecular weight. It has also been demonstra ted by means of pressure loss m e a s u r e m e n t s and photographs [14, 15]. Exper imental ly , macroscopic slip occurred at the wall for the flow of sufficiently entangled polymers, when a critical stress value has been exceeded [14, 16]. In spite of the experimental results mentioned above, there is at present no accepted theory for macroscopic slip with polymer fluids. Analytical modelling results have indeed already been published [9-10, 17-20]. Generally, the laws assume a linear relation between stress at the wall and slip velocity [9], which can then be extended by a non-linear relation to high stress values [10, 17-20]. These often have the advantage of being simple to use, but nevertheless have serious drawbacks. The main feature is that they consider the existence of slip for any stress value 1;R at the wall. As a consequence, a polymer of fixed molecular structure slips for any stress level, which is in contradiction with the experimental observations referred to [13, 14, 16]. Recently, in 1989, Atwood and Schowalter [21] studied the slip of a high-density polyethylene. In agreement with the works mentioned above [13, 14, 16], they demonstra te the existence of a threshold stress, below which the fluid adheres to the wall. Beyond, variations in stress at the wall as a function of slip velocity would appear to follow a strictly increasing linear relation. For their part, Hatzikiriakos and Dealy in 1992 [22] also take into account the existence of a static friction stress. They propose a friction relat ion tha t takes into account various pa rame te r s and in par t icu lar the existence of a normal stress. The results obtained for the flow of a polyethylene show that, beyond the critical shear stress, variat ions in slip velocity as a function of shear stress at the wall follow a strictly increas ing power law. Thus these slip relations tha t take into account the existence of a critical stress [21, 22], exclude the possibility of the shear stress at the wall decreasing when the slip velocity increases, though this is a common si tuat ion in tribology. It is obvious that these relations are presented as being consistent with the experiments to which they refer. However, because of the difficulties inherent in this type of experiment (choice of fluids, l imitat ion of t ransducers, etc.) they usually cover only a small range of slip velocities, and are unable to offer an overall view of friction with slip for polymer melts [23, 24].
Exper imenta l and theoretical studies propose a physical explanat ion of the mechanisms governing slippage phenomenon, based on the interaction likely to
359
develop between the flowing polymer and the wall of the die. Thus, in the case where no polymer chain is adsorbed at the wall, the flow of a highly entangled polymer takes place with a positive slip velocity whatever the wall stress value considered: this is the linear slip [9]. In practice, such a flow configuration needs the use of an ideal surface, perfectly smooth and developing no interaction with the fluid. It is clear that this situation is not possible to obtain experimentally, and a theoretical study shows that the combined effects of roughness and polymer-wall interactions reduces slippage by promoting the adsorption of polymer chains at the wall [10]. This result has been validated by experimental studies [11, 12, 25] considering the flow of highly entangled polymers through dies characterized by small roughness and/or small surface energy. When the polymer is strongly attached to the die wall, adhesion prevails at low flow regimes, and slippage occur only above a critical wall stress value [10, 12, 14]. It should be underlined that it is this situation which is usually encountered in industrial polymer processes, where the dies used are machined in classical materials (stainless steel ...) and with no special control of the wall roughness. In the present study, after having described the experimental means used, this later flow configuration will be described and characterized first. To do this, pressure loss measurements and appropriate visualization will be analyzed in section 3. They permit the determination of the friction law of polymer melts flowing and slipping along a solid wall. Secondly, the effect of wall material on the flow properties will be examined. This will be done using fluorinated dies which are characterized by their particularly low surface energy.
2. MEANS U S E D
2.1. F l u i d s u s e d The results reported here concern highly entangled polydimethylsiloxanes (PDMSs), polybutadienes (PBs) and polyethylenes (PEs) containing no additives [3, 14, 15]. Their main rheological characteristics are set out in Table 1. Two linear PDMSs (LG2 and LG3) provided by Rh6ne-Poulenc were used. The advantage of these fluids is that they are molten at ambient temperature, which means that the results of flow experiments can be interpreted without having to worry about problems of heat regulation. They are also reputed for their thermal, chemical and mechanical stability. In addition, they are transparent, which is vital for observing results and, lastly, they have low surface energy. Two PBs of different molecular weight provided by Michelin were also studied. One is linear (PB) and the other star-branched (PBb). These fluids are also molten at ambient temperature. Moreover they are reputed to be well-defined polymers. Finally, they are highly birefringent, making it possible to study the local stress field during flow, thanks to the use of flow birefringence methods. Lastly, two linear polyethylenes were considered. The LLDPE provided by Enimont is a linear low-density fluid, and the HDPE provided by Du Pont Canada a high- density one. As these are commercial polymers that melt at high temperatures, they enabled the study to be performed under quasi-industrial conditions. Being linear or branched and melting at ambient or high temperatures, these fluids provide a comprehensive view of the stable or unstable phenomena likely to occur during the extrusion of polymer melts with slip.
360
Table 1 Fluids used Fluid Mw (g/mol) Mw/Mn Tl0(Pas)
PDMS LG2 758 000 3.9 540 000 23~ PDMS LG3 1 670 000 7.9 ? 23~ PB 220 000 1.05 1 260 000 23~ PBb 170 000 1.38 23~ LLDPE 143 000 3.9 9 800 190~ HDPE 216 700 18.09 185~
All the experiments were performed in an air-conditioned room. In the case of the PDMSs and PBs, the temperature was 23~ and in the case of the LLDPE and HDPE, 190~ and 185~ respectively.
2.2. E x p e r i m e n t a l apparatus Two types of installation were used for the experiments, depending on whether the upstream pressure or upstream mean flow rate were controlled. In the first experimental installation, the fluid was placed initially in a reservoir and then forced out under pressure exerted by nitrogen into an axisymmetrical or two-dimensional capillary. In this system, the total pressure loss is monitored [11, 14]. This is determined by using a Bourdon-type pressure gauge. The mass flow is measured by weighing and timing. The second installation considered is a capillary rheometer (GSttfert 2001). In this apparatus, the fluid is contained initially in a reservoir, into which a piston slides. It is then forced into an axisymmetrical capillary, with the piston moving at a controlled speed. In this system, the mean flow rate is monitored, with the pressure being measured by sensors with a measuring capacity ranging from 50 to 2000 bar.
Two types of die were used, namely axisymmetrical capillaries and narrow slits. The dimension of the capillaries used are shown in Table 2. Most are made of metal ( tungsten carbide, stainless steel) and received no particular surface t reatment . However, in order to study the effect of the wall material on polymer friction, capillaries coated with an industr ial fluoride t rea tment by Isoflon were also considered. Two axisymmetrical narrow orifices of stainless steel, 0.5 and 2 mm in diameter, were also used. The ratio of their length to diameter is less than 1. For a given diameter, pressure measurements for a fluid flowing through one of these orifices enables entrance effects to be estimated for the same fluid flowing into a capillary [15, 26]. It should be noted that another and more precise way of determining the order of magni tude of entrance effects is to use the Couette- Bagley correction. This is the method that was used for the PEs.
361
The aim of using the narrow slits was to produce quasi-2D extrusion. The slits have a gap (2e) of 2 ram, are 45 mm wide and 20 mm long. Here again, flow is considered in both stainless steel and fluorinated dies. Fluorination was obtained by coating the wall of the dies with strips of commercial PTFE, which is normal ly used in plumbing for sealing pipes. An industrial t rea tment process (by Isoflon) was also used. Lastly, low-roughness silica surfaces could be treated in the laboratory by grafting fluorinated trichlorosilanes to the wall [11]. A narrow orifice die made of stainless steel and characterized by its short length (0.05 mm) in comparison with the gap (2e = 2 ram) will also be considered. As with flow in the axisymmetr ical capillary, flow through this type of orifice can be used to evaluate pressure losses for a given fluid flowing into a slit with the same gap and width (45 mm).
3. F L O W IN H I G H S U R F A C E E N E R G Y DIES
These feature essential ly metal walls (stainless steel, tungs ten carbide). The results reported may also apply to dies made of Plexiglas [14] or silica [11].
3.1. V i s u a l i z a t i o n a n d f l ow c u r v e s For each of the fluids tested, var ia t ions in pressure loss dur ing flow were represented as a function of flow rate. The general run of the flow curves obtained [14, 15, 27] is shown in Fig. 1. With low flow rates, the flow is stable and takes place without any slip occurring [14]. Above a certain flow regime, which varies depending on the dimensions of the dies and the degree of entanglement of the polymer considered, macroscopic slip appears at the walls. This was demonstrated by observing the s treamlines within a t ransparen t axisymmetrical capillary. To do this, a tracer was injected near the wall, perpendicular to the direction of flow, and its movement observed. With sufficiently high regimes, Fig.2 shows that the tracer injected into the flow moves very distinctly near the wall, thus clearly demonstrating the existence of a definite slip velocity in this area.
On the flow curves, this slip is accompanied by a d i scont inu i ty in the instantaneous flow values (Fig. 1). Its occurrence differs depending on whether or not the instal lat ion involves any storage of elastic energy connected wi th the compressibility of the polymer upstream of the die. In the case of controlled-pressure flow, once a critical pressure value AP* is exceeded, it was shown that slip was seen on the flow curves in the form of a jump in flow rate, reaching a factor of as much as 100 (portion (BC) of the flow curve) [15]. Moreover, by carrying out tests with decreasing controlled pressure, it can be seen tha t the slip persists with pressures of less than AP* (portion (FD) of the flow curve). As the pressure continues to decrease, Fig. 1 shows that there is a sudden decrease in flow rate (portion (DA) of the flow curve) and there is once again the portion of curve corresponding to flow with adherence to the wall. Thus, in this flow configuration, macroscopic slip is accompanied by a hysteresis. The h igher the molecular weight of the polymer, the greater the hysteresis value [14, 16]. In the case of flows with a controlled mean flow rate, the fluid compressibility plays an important role and the appearance of slip is accompanied by oscillations in the instantaneous pressure and flow rate as a function of time.
Table 2 Dies used.
Dies D or 2e (mm) LID or LJ2e Wall material Axisymmetric 0.5 0.1 Stainless steel
10 Stainless steel or tungstene carbide 20 I1
30 I1
4 0 II
2 0.1 Stainless steel 5 Stainless steel or tungstene carbide
10 Stainless steel, tungstene carbide, Plexiglas or Isoflon 15 Stainless steel or tungstene carbide
Two-dimensional 2 0.1 Stainless steel (1 = 45 mm) 10 Stainless steel, commercial PTFE, Isoflon, silica or fluorinated
silica.
363
/
Log Pu
B
�9 f . . . . 4 . . . . . . . . j
�9 D
H
Log q y
v
F igure 1 �9 Typical flow curve for highly entangled polymer melt extrusion through two-dimensional or axisymmetric dies.
During the compression phase, section (AB) of the flow curve applies, and the fluid adheres to the wall. During the relaxation phase, portion (CD) of the flow curve is found and there is slip [14, 22, 28]. When the controlled mean flow ra te is sufficiently high to correspond to a rising branch of the flow curve, there is once again a steady pressure regime with permanent slip. When sufficiently high flow regimes with slip can be reached, there is a further discontinuity in the slope of the flow rate-pressure curve, followed by a slight drop in the pressure (Fig. 1). In fact, in agreement with other works [29], this discontinuity is connected with the appearance of a second area of oscillation in the instantaneous pressure, indicating the existence of a second hysteres is loop between two pressure values. In comparison with the first loop, the second zone of oscillations occurs in a slip regime, during both the compression phase (portion (EF) of the flow curve) and relaxation phase (portion (GH)). Indeed, the minimum pressure reached in this loop (points E and H of Fig.l) remains sufficiently high to be permanent ly outside the zone corresponding to stable flow with adherence. After this second zone of oscillations, there is once again a s teady pressure regime and the flow curve represented in Fig. 1 once again has a slope discontinuity.
The flow curves characterizing the extrusion of highly entangled polymer melts in conventional dies (metal, with any reasonable degree of roughness and having no special prior treatment) thus show that such flows are governed by:
- adherence to the wall at the lowest flow regimes; this property must obviously be taken into account for modelling this type of flow,
- the existence of a critical stress beyond which flow takes place with slip. This is therefore a type of slip which occurs under high stress and which has been identified as macroscopic slip. Visualization of the velocity field shows tha t it occurs very close to the wall (less t han 10 ~m) in the case of the flows with heterogeneous shear rates and stress fields considered here.
Figure 2 : Observation of LG3 flow's into an acrylic capillary die, 2 mm diameter and 20 mm long, under stress controlled conditions at AP = 18 105 Pa. Tracer injected near the wall, perpendicular to the direction of flow, moves very clearly near the wall, thus showing the existence of slip in this area.
365
3 .2 . F r i c t i o n c u r v e s Let us consider the flow of a highly entangled polymer in an axisymmetrical die of d iameter D and length L. In the case of polymers, when the shear rates are sufficiently high or within a certain shear rate domain, variaff~ns in viscosity as a function of shear rate may be represented by a power law [30]:
T! = k ~n-1. (1)
If it is assumed that flow is established in the capillary, the exact equation taking into account entrance effects and enabling slip velocity to be calculated is writ ten as follows [26]"
n R [ R /kpe)] l/n, (2) - UR = 3n + 1 2 - - ~ (z~Pt-
where ~r is the mean velocity through the section, UR the slip velocity at the wall, R the radius of the capillary, APt the total pressure loss and APe the pressure loss at the entrance to the capillary. The apparen t shear stress at the wall Xa and the real shear stress at the wall XR are thus defined by:
APt APt- APe x a = and X R = . (3)
4L / D 4L / D
To determine the rate of slip, three different methods may thus be used:
Local measurements by using laser velocimetry, marking the flowing polymer or by optical techniques [12].
A direct method. According to relations (1) to (3), this requires knowledge of the following:.
- the rheometric behavior of the fluid during shear, in order to determine the power law parameters,
- the flow curves, giving variations in total pressure loss and entrance pressure loss as a function of flow rate.
This method requires great precision in determining "n", otherwise it can lead to contradictory results [5, 27]. In particular, the present authors have shown that it may be necessa ry to resor t to t i m e - t e m p e r a t u r e superposi t ion to obtain variat ions in viscosity over a sufficiently wide range of shear rate [15]. It is also obvious t h a t when pressures reach sufficiently high levels, the dependence of viscosity on pressure must also be taken into account [27].
A method based on Moon ey's diagram. This involves representing the variations in wall shear gradient defined by
3 n + 1 r162 i ,=
n R'
356
as a function of l/R, with a constant real shear s tress at the wall. According to relations (2) and (3), the curve obtained is then a straight line of slope
3 n + 1 UR.
This method requires the use of dies of various diameters and precise knowledge of the flow curves for each one. R a m a m u r t h y [4] applied this method by plott ing variat ions in apparent shear rate at the wall as a function of I]R, wi th Xa constant.
Relation (2) shows that the plot is only equivalent to tha t with constant x'R in two cases:
- the dies considered are very long (L/R-> ~). - entrance effects are very weak and can be neglected.
If this is not the case, neglecting entrance and exit effects can lead to erroneous interpretat ions [27].
The direct method was used here to determine the friction curves for a series of polymers in axisymmetrical capillaries of various dimensions. The characterist ic flow curves for the flow considered here are represented by the curve in Fig. 1. It can be seen tha t this curve, on log-log scale, consists of a series of s t ra igh t segments, with a discontinuity in slope and flow rate once a certain flow regime is reached. Let q* be the flow rate corresponding to the first discontinuity in flow rate and q** the flow ra te for which oscillating plug flow disappears . The curve s h o w n schematically in Fig. 1 can then be represented by the following equations:
q < q * , ZXP t = Ktc q ntc, (4a)
q > q** > q * , A P t - Ktd q ntd, (4b)
where q is the mean flow rate, APt the total pressure loss in the capillary, "Ktc", "Ktd", "ntc" and "ntd" coefficients tha t depend on the fluid and capi l lary under consideration (cf. Fig. 3a). It should be noted that, in the case of flow involving two zones of instability, these will be distinguished.
With a given diameter D, and for each flow rate value considered, the flow curves for an orifice die or the Couette-Bagley correction give an indication of the entrance pressure losses APe, for flow in a capillary of the same diameter D. On log-log scale, allowing for measurement errors, these curves are segments of s t ra ight lines. They all display a discontinuity in slope, corresponding to the appearance of unstable pu l sa t ions ups t r eam of the die and to the p h e n o m e n o n of me l t f rac ture downs t ream [3]. It was established that , for all the polymers considered in this s tudy and within the accuracy of the measurements , the flow rate corresponding to this break in the slope of entrance pressure loss curves is equal to the flow rate q* characterist ic of the discontinuity in flow rate occurring on the total pressure loss curves for the same polymer and with the same diameter [14]. The curves relat ing to the entrance pressure losses can thus be represented by the equations �9
I I Capillary LID I I Orifice D
A log AF'
n
Power law viscosity L
log q
Figure 3 : Principle of determination of friction curve during macroscopic slip of the polymer a t the wall.
368
n q _< q* , AP e = Kec q ec, (5a)
ned (Sb) q > q * , AP e = Ked q ,
where q is the mean flow rate, APe the entrance pressure loss, "Kec", "Ked", "nec" and "ned" coefficients depending on the fluid and orifice considered (cf. Fig. 3b).
With a given d i ame te r and for each polymer, the above resu l t s can be used for each flow ra te value considered to calculate the pressure loss APc in the capil lary itself, defined by:
/~Lp c = /~kp t - A p e . (6)
Now:
AP '~R = c = r l , i t "
4 L / D
It is therefore possible to deduce the following:
= Kcq~ with Kc = 4_.LL kF3n + 1 aPe D L 4n
32 ]~ (7) D 3 ,
where n is the power law coefficient defined in (1). It should be noted that, to determine APc using relation (7), it is necessary for:
- q < q*. In this case the flow is stable and it is possible to measu re variat ions in viscosity coefficient,
- q to be sufficiently high to consider the variat ions in viscosity coefficient as a power law of the shear rate.
In the field of flow ra te var ia t ions defined in this way, i.e., outs ide the area of Newton]an behaviour and the t ransi t ion zone between Newton ian and power law behavior, Kc and n may also be de termined from the flow curves by compar ing relations (6) and (7) (cf. Fig. 3c).
Hence, in order to m e a s u r e the friction of flowing po lymers wi th slip in axisymmetr ical capillaries, it is simply a question of measur ing total and entrance pressure losses. The curves obtained in this way and re la t ions (4a) and (5a) can then be used to determine the values of Kec, Ktc, nec and ntc. It is then possible to deduce the values of Kc and n by applying relations (6) and (7). In addition, these same curves and relat ions (4b) and (5b) can be used to de t e rmine the values of Ked, Ktd, ned and ntd.
Let us consider a value q of the mean flow rate corresponding to a flow with slip, i.e., such t h a t UR > 0, or again q > q*. It is then possible to de t e rmine the corresponding values of:
- APt (cf. Fig. 3a) by means of relation (4b) ; - APe (cf. Fig. 3b) by means of relation (5b) ; - APc (cf. Fig. 3c) by means of relation (7).
369
By introducing the pressure loss APc obtained in this way into relation (3), it is possible to calculate the stress I; R associated with the flow rate q considered. Relation (2) can then be used to determine the value of UR corresponding to this same flow rate, thus indicating variations in TR as a function of UR, and hence the friction relation for the fluid under the flow conditions considered.
This method was applied to two PDMSs (LG2 and LG3), to the PB and the LLDPE (cf. Table 1). The values of"Kec", "Ked", "Ktc", "Ktd", "Kc", "nec", "ned", "ntc", "ntd", "n", and the value of q*, determined from the experimental curves obtained for these fluids are given in Table 3. This table also contains the value of the shear stress at the wall for flow rate q*, denoted x*. By choosing these fluids, it was possible to obtain a comprehensive view of the friction curves for polymer melts with slip, over a significant range of slip ra te at the walls. Figs. 4a to 4d demonstrate:
- The existence of a static friction stress, corresponding in fact to the flow rate q* on flow curves. Thus in the conditions used in this study, the polymer melts only slip above a critical stress threshold. They adhere when at rest.
1,o
R
0,8
0,6 -
0,4
0,2
0 , 0 , 1 0 4
5 ( 1 0 P a )
j - - - " i , . 5 - J - ' -
A A
G u m _ L G 2 �9
A L/D = 10/0.5 Q L/D = 20/0.5
�9 L/D = 20/2
R e l a t i o n (10) V R ( m / s )
�9 ' ' ' ' 1 ~ 1 - ' " ' ' ' " 1 ' ' ' ' ' ' ' 1 , , 1 - , , , , ,
1 0 -3 10 -2 10 "1 1 0 ~
Figure 4a " Friction curve of PDMS LG2. x* is the static friction stress (Table 3).
- At low slip velocities, the stress at the wall decreases when the slip velocity increases, which is a current situation in tribology. This decrease corresponds to the flow regimes preceeding the minimum obtained on the flow curve. Consequently, it is not possible to deduce shear stress value in the decreasing par t of the friction curve from the flow curves, since they correspond to the oscillatory flow regimes. Following this decrease, the shear stress at the wall increases again with slip velocity for flow regimes corresponding to portion
370
(DF) on the flow curves. Note that in the case where the flow curves exhibit two instability zones, the corresponding friction curve exhibits two minima, the las t one being followed by a rising branch with a high slip velocity corresponding to portion (HG)on the flow curves. The friction curve depends slightly on the degree of en tanglement in a given family of polymers (comparison of Figs. 4a and 4b). In addition, in the case of the LLDPE, the friction curve does not appear to depend on tempera ture (Fig. 4d).
9 - - _ .
0,8 ~, (5/0.5)
0,6 x, (20/2)
0,4
0,2
0,6 10 -4
( 1 0 5 P a ) R
G u m L G 3 "
o I A ) = 5/0.5
�9 I./D = 20/2
mo ll E l l �9 o
0 0
�9 ' , , , ' , �9 "v , " , �9 �9 , i ' 1
10 -3 10 -2
ooOOO
U R (m/s)
10 -1
Figure 4b �9 Friction curve of PDMS LG3. ~* is the static friction stress (Table 3).
Thus it appears that, generally speaking, the run of the friction curves obtained for the various polymer melts is characterized by significant non-linearity. On a semi- log scale, these curves have in particular a bell shape, which is reminiscent of the work carried out by Moore in 1972 [31] on friction using slightly re t icula ted elastomers . These also show tha t adherence and hys teres is are the main mechanisms governing slip phenomena. This may be explained by the fact tha t the polymer melts used in this study are highly entangled. In addition, these curves have two minima corresponding to the two oscillation regimes in the instantaneous pressure. It should be noted that, in the case of the PDMSs, the small quantit ies of fluid avai lable for the s tudy were insufficient to reach flow ra t e values corresponding to the second minimum, if it exists.
It is also possible to characterize the polymer-wall slip by represent ing variations in the extrapolation length b as a function of the slip velocity. It should be recalled that b is defined as the ratio of the slip rate to the shear rate at the wall [9]. With the assumptions made in this study (power law, etc.) relation (2) can be used to determine the shear rate at the wall for flow regimes including slip, i.e.:
371
3,0
2,5
$ O
2,0
1,5
1,0"
0,5
0,0 10 -3
R ( 1 0
�9 8 o
5 P a )
0 o~ �9 �9 @ 0
O
o L/D = 5/0.5
�9 ][.JD = 20/2
O
...... . ' ' ' ' " �9 ' I ' '
10 .2
F igure 4c" Fr ic t ion curve of PB.
U R
O O
(m/s) �9 �9 �9 .=, .
10 "1
5 I; R ( 1 0 P a )
< D - 05 ]] D ~ i ~
I:* ( D = 2 m m )
L L D P E ; T = 1 6 0 ~
+ I / D = 30/2 x I / D = 20/2
�9 I.JD = 10/2
| i i I
I i
i �9 , �9 , | I I I I , I �9 �9 I ' � 9 �9 ' �9 ' | " | l l
10 -2 10 "] 10 0 10
L L D P E ; T = 1 9 0 ~
�9 I / D = 2O/O.5 �9 I / D - 15/0.5 �9 I / D - 10/0.5
o L/D = 30/2 A I / D = 2O/2 o L/D -- 10/2
U R ( m ] s )
1 10
F igure 4d" Fr ic t ion curve of LLDPE. x* is t he s tat ic fr ict ion s t ress (Table 3).
372
~/S = 3 n + 1 ~r UR [ R ]l/n n R = ~ ( A P t - APe) , (8)
and hence:
b = UR (o)
The resul ts obtained are presented in Fig. 5. It can be seen that , wi th a given polymer, the extrapolation length hardly depends on the geometric characterist ics of the dies. In addition, the results obtained with the PDMSs show that , wi th the same family of polymer, b increases with the molecular weight. Lastly, wi th the polymer-wall pairs envisaged here and the range of slip velocities achieved in this study, b appears to be an increasing function of the slip velocity. Fur thermore , on log-log scale, variations in b may be represented by a portion of a s t raight line with a slope of about 0.9 for all the experiments carried out, in the case of flow in high surface energy dies.
10
b (m) Gum LG3 Stainless steel die P__BB
L/D = 20/2 , . , . ~ ~ Stainless steel die ' o L/D = 20/2
El I]D=5/0.5 E! = L/D=5/0.5 Q
&
�9 PBb �9 .& �9 . o �9 cod ~11 r Slippery_ die e ~ LLDPE
4" & Stainless steel die
Gum LG2 Stainless steel die
�9 L/D = 20/2 & L/D = 10/0.5 �9 L/D = 20/0.5
A j "~~9- a IJD=10/2 f �9 I.]D = 20/2
i L/D = 30/2
= x L / D = 10/0,5 �9 L/D = 15/0,5 -t I_]D = 20/0,5
U R (m/s )
. _ 4 . . . . . . , . . ~ . . . . . . , . . . . . . , . . . . . . . . , . . . . . . , . . . . . . , . . . . . . , . . . . . . , . . . . . 10 10 '-5' 10 -4 10 ~3 10 -'2" 1 0 : 1 10 "0' 10 "1 10 2
Figure 5 �9 Extrapolation length as a function of slip velocity at the wall for different polymer-wall pair.
4. F L O W IN DIES WITH LOW S U R F A C E ENERGY
The effect of polymer-wall interactions is an essential parameter tha t has to be studied in order to understand the conditions under which slip is triggered. To do so,
Table 3 Fitting parameters of flow curves. Fluid Cap* q* Z*
LJD gls (bar) Kec (1) Ked (1) Ktc (1) Ktd Kc n e ~ ned ntc ntd nc mmlmm (cm3ls)
LG2 1010.5 1.6 10-4 0.60 802 102 2160 153 1690 0.56 0.32 0.42 0.14 0.41 2010.5 I , 0.60 802 102 2310 346 1990 0.56 0.32 0.36 0.18 0.35 2012 5.4 10-3 0.55 93.8 27.9 122 57.4 78.7 0.53 0.31 0.28 0.16 0.24
LG3 510.5 3.8 10-6 0.69 13734 145 5.8 105 184 3199 0.64 0.29 0.73 0.24 0.37 2012 1.6 10-4 0.49 137 50 673 52.4 529 0.43 0.32 0.38 0.15 0.37
PB 510.5 9.3 10-6 7.9 81600 377 2.8 106 427" 2.7 106 0.76 0.28 0.80 0.15* 0.81 561** 0.20**
2012 3.7 10-4 4.6 2750 234 3.4 105 327* 3.7 105 0.73 0.42 0.95 0.19* 0.96 332** 0.24**
LLDPE 1010.5 (4.1103) 3.65 2060 299* 7740 477" 7210 0.92 0.57* 0.59 0.13* 0.58 173** 449**
1510.5 11 3.6 2060 299* 10500 638" 10020 173"" 586**
2010.5 0, 3.7 2060 299" 15000 762* 14500 173** 718**
1012 (8.6 10-2) 2.3 60.6 29.8 323 105 282 2012 I t 2.3 60.6 29.8 579 176 539 3012 11 2.3 60.6 29.8 808 252 77 1
(1) : in bar (gls) -net for PDMSs and PB ; in bar (cm31s) -net for LLDPE. * : first oscillating region ** : second oscillating region
374
numerous studies have analyzed polymer flow in capillaries made of different materials or by using polymers containing various types of additive [4, 5, 8, 16, 32]. The particular effect of fluorine is emphasized in all cases, but the results obtained for similar polymers may be contradictory [32]. Indeed, not all commercially available fluorinated materials are equivalent [25]. The flow of various polymers was analyzed using axisymmetrical or two- dimensional fluorinated dies (cf. Tables 1 and 2). Several types of fluorine were tested [11, 25, 33]: solid PTFE in various forms (strips, massive die, etc.), industrial surface treatment (Isoflon), or treatment in the laboratory enabling fluorinated trichlorosilanes to be grafted on to the surface of silica objects of low roughness [34, 35]. The characteristics of the wall layer (chemical composition, surface energy, etc.) of fluorinated surfaces obtained by grafting on trichlorosilanes could be more thoroughly controlled in comparison with the surfaces obtained using commercial fluorine. The flow of a given polymer in such dies is thus a valuable way of studying the effect of wall material when extruding polymer melts with slip. Flow curves were plotted for all the polymer-wall pairs envisaged, and the influence of the wall material on the appearance of the extrudate was observed. In the case of the PBb flowing in the 2D dies treated by Isoflon, the results were supplemented by stress charts obtained by birefringence and by measurements of the velocity profile determined by laser velochnetry [25].
4.1. G e n e r a l resul ts In the case of all the polymer-wall pairs considered, analysis of the section of the flow curves corresponding to stable flow shows that flow rate at a given pressure is greater when the fluid flows in fluorinated dies. This can be seen in Fig. 6 for the particular case of the HDPE [11, 25, 36]. This increase, which may reach a factor of 5, demonstrates the combined effect of roughness and the wall material. It may be explained by the occurrence of polymer slip in the fluorinated dies even at low flow rates. In the case of regimes corresponding to unstable flow, the flow curves obtained with steel dies and fluorinated dies are virtually the same (Fig. 6). Therefore, they appear to be independent of the characteristics of the die wall. In fact, these relatively high flow regimes are governed by bulk instability upstream of the contraction [3, 33]: volume phenomena appear, therefore, to play a leading part in comparison with interface phenomena.
Wall material also appears to have a decisive effect on the appearance of the extrudate. Indeed, the classic succession of the various extrusion regimes can be observed in the steel dies [11, 25]. When the same polymers flow through fluorinated dies, the occurrence of sharkskin cracks or plug flow could be significantly delayed or completely eliminated [11, 33]. Thus, depending on the characteristics of the fluorine available (in particular in terms of surface energy), and the flow configuration considered, it was possible to multiply extrusion rates by 16 in comparison with situations where flow takes place in a conventional die made of stainless steel, while at the same time keeping the same qualities of transparency in the extrudate [11, 25, 33]. Consequently, by provoking slip at the wall, fluorination of the dies is also likely to prevent sharkskin defects (Fig. 7). These surfaces, characterized by their low energy and referred to as slippery surfaces [33], produce a different type of slip from that occurring with a high stress level along the high surface energy and relatively rough walls described in section
375
2.2. An attempt was made to identify and characterize this type of slip by studying the flow of the PBb in detail.
1 0 3
1 0 2.
5 AP (10 Pa)
�9 Stainless steel capillary Isoflon capillary
A A
�9 A A A
A
1 1 0 0 1 . . . . . . . . ' . . . . . . . . ' 1 1 0 2 103
�9 A
~/ (S "I )
�9 �9 , , , , ,
Figure 6 �9 Flow curves for the flow of HDPE through axisymmetrical capillaries, 2 ram diameter and 20 mm long.
4.2. C o m p l e t e a n a l y s i s o f P B b f l o w The PBb was chosen for various reasons. In comparison with PEs, it melts at ambient tempera ture and thus avoids any problems of hea t regulation. In comparison with the PB, it is t ransparent , making it possible to observe the changes that occur and to carry out velocity measurements by laser velocimetry. Lastly, in comparison with the PDMSs tha t were available, it is properly birefringent, making it possible to analyze the local stress field. The flow of this polymer under a controlled nitrogen pressure was examined in sudden, fiat- bottomed contractions. The extrusion dies considered are two-dimensional, 45 mm wide and characterized by a gap (2e) of 2 mm. One was made of stainless steel and was 20 mm long. The second had the same geometry as the first but the walls were coated with PTFE by Isoflon. The third was a narrow stainless steel orifice characterized by its short length (only 0.05 ram) in comparison with the gap. It should be recalled that flow in such an orifice gives a correct representat ion of entrance effects in a given fluid flowing into a capillary die with the same gap and width.
4.2.1. Flow curve- Visualization of extrudate In Fig. 8, flow in the 20 mm long stainless steel die is represented by a straight line with no slope discontinuity. This results indicates that flow is stable for all the regimes considered [3]: there is no melt fracture or macroscopic slip for the flow conditions reproduced here. The flow curve for the orifice die has the same features, though the overall energy losses are lower than in the long die. In the case of flow in the Isoflon die, Fig. 8 shows that the flow curve plotted on
Figure 7 : Elimination of sharkskin using fluorinated die. (a) LLDPE flowing in an axisymmetric capillary 2 mm diameter and 20 mm long; (b) PDMS flowing in a two-dimensional die 2 mm gap and 20 mm long. In both cases the polymer flows downwards, issuing from the upper part of the figure where the die is located. On figure (a) the lower half of a capillary section has been made slippery. On figure (b) the central part of both sides of the slit has been made slippery. Polymer which has been flowing along a slippery surface appears smooth and transparent, whether the same polymer appears to be cracked after flowing along a classic surface a t the same flow regime.
377
log-log scale consists of three portions of straight line. The first portion is shined by about 30% with respect to the flow in the steel die at a pressure of 5 bar. The second portion corresponds to a transition zone before reaching high-slip regimes, which are represented by the third portion of s t raight line, where the difference in flow rate for a given pressure between flow in the steel die and tha t in the Isoflon die may reach 500%. This difference may be a t t r ibuted to the triggering of slip at low stress levels along the fluorinated walls [25, 36]. Fur thermore , at high flow rates, the pressure loss curve for the Isoflon die appears to be superimposed on the curve obtained for the steel orifice die. As this die gives a good order of magni tude for the ent rance pressure losses, this result demons t ra tes the fact tha t energy losses in the Isoflon die occur almost exclusively in the inlet area, and are thus considerably reduced inside the capillary due to slip along the fluorinated walls. The ext rudate in the stainless steel die is first smooth and t r anspa ren t and then sharkskin cracks gradually invade the surface as the flow regime increases. No extrusion defects were observed with flow in the Isoflon die and with any of the regimes tha t could be obtained in this study, which are up to 24 times higher than the regime in which sharkskin occurs in the stainless steel die [25].
10 2
101
5 AP (10 Pa)
� 9 0
n
0 0 0 0
�9 0 o �9 0
o
O O O&
�9 Stainless steel capillary o Fluorinated capillary �9 Orifice die
Ick.l.vO . . . . . . . . , . . . . . . . . ,
1 0 "1 1 0 o 101 " 1 0 2
q (g/h) m
. �9 , . . , .
1 0 3
Figure 8 �9 Flow curves for the flow of PBb through two dimensional dies, 2 mm gap and 20 or 0.2 mm long. The orifice die is made of stainless steel.
Consequently, by considerably reducing the shear s tresses in the capillary and hence s t resses in the outlet area, slip modifies the conditions of flow for the polymer, which can then be extruded under low stresses and without any surface defects [36].
4.2.2. Measurement of velocity at the wall Velocity measurements were taken by using a laser doppler velocimeter with a green laser beam [25, 36]. Fig. 9 shows the velocity profiles measured with this
378
system for both the steel and PTFE dies, in an axially symmetrical plane and at 10 mm from the inlet. Fig. 9a confirms tha t flow in the steel die is similar to a fully developed Poiseuille- type flow, with adherence at the wall, even at the highest flow ra tes studied. Indeed, no slip was detected near the wall with the apparatus used. Fig. 9b shows that , at high flow rates in the PTFE die, there is a plug flow at a velocity close to the mean velocity. At 50 ~tm from the wall, the velocity is practically equal to the maximum velocity in the section. These resul ts show that , with flow in the PTFE die, energy is dissipated in a very th in wal l layer. However , it has a l ready been seen from the p ressure measu remen t s tha t there is relatively little dissipation in the same experimental conditions. The results of Figs. 8 and 9b obtained at high flow rates with the PTFE die thus show that the conditions for slip with practically no friction are achieved.
4.2.3. Birefringence patterns Quali tat ive information concerning the distribution and zones of concentration of stresses in the various areas of flow and the different dies considered was obtained by birefringence. Like many other authors [37-39], we shall assume tha t the photo-elastic law, which expresses the relation between the optical tensor and s t ress tensor , is l inear for the experiments performed here. Thus, at least in principle, this relat ion may be used to obtain an experimental measu remen t of stresses at all points of the flow [37, 38, 40, see also chapter III.1 in this book]. The isochromes, lines where the t ransmit ted intensity is cancelled by interference, provide isovalue curves for the difference in the principal stresses. Along the die axis, the maximum order of the fringes in the inflow area will be denoted Ke and the extreme order in the outflow area denoted Ko. The order of the fringes at the wall before the outflow area will be denoted Kw.
-3 O AP=25bar V R A AP = 35 bar
+ o,oo t ~ 0 - Theoretical profile 0 . , ~
\\A ,, .o,o % %
, , , . ' , < , ,
-0,06
y ( ram) -0,08 . , "
0 1 2
Figure 9a : Velocity profile for the flow of PBb through the 2D stainless steel die.
379
-0,1
-0,2
-0,3
~176 ~ o o o 0
-3 U R ( 1 0 m / s )
0 0 0 0 0 0 0
o o o o o o o
o AP = 25 bar A AP = 35 bar
+ AP = 40 bar A
, .
A - Mean velocity A A
A A A A A A A A A A A A A A A A A
-e + + + + + + + + + -~-
y ( m m ) �9 ! .
0 1 2
Figure 9b : Velocity profile for the flow of PBb through the 2D fluorinated capillary.
a- Visualization of birefringence in the stainless steel die The results obtained are in conformity with those of many other au thors [25, 37]. The birefr ingence pa t t e rn is thus symmetr ical with respect to the flow axis. Moreover, the n u m b e r of fringes increases with pressure , indica t ing t ha t the isochromes depend on the flow regime. Lastly, it is possible to observe the concentrat ion of s t resses around the capillary inlet. In the capillary, following a t ransi t ion zone of the order of 2-3 times the die gap, the isochromes are parallel to the walls [36]. This resul ts demonstrates the fact tha t for a given flow rate, shear depends only on the direction perpendicular to the axis of flow. In part icular , the shear stress at the wall remains constant along the capillary. As flow approaches the outlet section, the central fringes converge towards the sides of the die, showing tha t there is a new zone of stress concentration at this point [36].
b- Visualization of birefringence in the Isoflon die At low flow rates, the appearance and development of the isochromes are similar to those described for the steel die. However, with ups t ream pressures of between approximately 6 and 13 bar, pulsations in the number of fringes were observed throughout the flow field.
Thus, as shown in Fig. 10a, at a pressure of 6.4 bar, the order of fringe Ke is 9 with brief pulsat ions at 11. It should be noted tha t with a similar m e a n flow rate, the order of fringe Ke corresponding to flow in the steel die was 9 [36]. The oscillation in the ups t ream birefringence image corresponds to disturbance of the fringes (Fig. 10b) along the capillary and throughout the flow, occurring with a period of 16 s. At the wall, the order of fringe Kw is thus 9 and 8.5, depending on the side of the die, and it decreases by 0.5 of a fringe the moment Ke increases. In the case of steel, Kw was 13 with the same flow rate, and 8 bars were required to produce the same
380
flow rate (fig. 8). In the outflow area, the maximum order of fringes Ko increases from 3 to 3.5 when Ke increases from 9 to 11. In the steel it was 5 for the same mean flow rate. In addition, figure 10b shows that the entrance length, aider which the fringes become parallel to the walls of the die, is distinctly longer than when the fluid flows in the steel die [36]. At 6.4 bar, this transit ion zone is 13 mm long, i.e. more than 6 times the die gap. When the u p s t r e a m pressure is increased, the number of fringes oscillates throughout the field of observation in a similar way to tha t described above, and the period of the phenomenon decreases. Simultaneously, the entrance length increases until it occupies the entire length of the capillary. With an ups t ream pressure of 12 bar (fig. 11), Ke = 28, with peaks of 29 every 2 seconds, while Kw = 10 and 9.5 depending on the side of the die, and Ko = 0.5. The orders of Kw and Ko oscillate slightly. In addition, the last fringe on the axis at the outlet from the die is of order 4 and not 0 as in the case of the previous fully developed flows [36]. Thus, when the flow regime increases, the number of fringes observed between the axis of flow and the wall decreases, as shown in Figs. 10b and 11. However, the
Ke = 9 Ke = 11
Figure 10a : Birefringence pat terns for the flow of PBb at the entrance of a two- dimensional fluorinated die, 2 mm gap and 20 mm long. AP = 6.4 105 Pa; qm = 1.1 g/h.
order of the axial fringe is no longer 0 and it increases with the flow rate. The result is tha t Kw, which is the sum of the order of the axial fringe and this zmmber of fringes, increases slowly. It is 11 at 5 bar, drops to 9 between 6.4 and 10 bar, rises
381
to 10 for 12 bar and 13 for 14 bar [25]. It may thus be deduced tha t the s tress at the wall varies little during the transi t ion regime, and tha t i t even reaches a max imum and then a min imum before s tar t ing to rise, when the flow ra te increases from zero. Hence, as suggested by the pressure loss measurements, the distribution of energy spent between the wall of the die and the two geometric s ingular i t ies is very different, depending on whether the fluid flows through a steel or f luorinated die. Indeed, the stress field in the upstream zone is not affected directly by the coating. In contrast , for a given flow rate, the wall and exit area are subjected to lower stresses when the fluid flows through a PTFE die.
4.2.4 Fr ic t ion curve An a t t empt was made to determine the effect of wall mater ia l on friction with polymer melt slip, in terms of variations in the extrapolation length "b". To do this, the results represented in Fig. 8 were used to determine var ia t ions in pressure loss in the stainless steel and Isoflon capillaries, with the necessary corrections being made for entrance effects. These are obtained at a given flow rate by the equation:
APc = APt - APe,
where APt is the total pressure loss in the capillary and APe the pressure obtained at the same flow rate in the orifice die. Hence, for each value of APc, it is possible to determine from Fig. 8 the flow rate ql in the steel capillary and the flow ra te q2 that would occur in the Isoflon capillary. The slip rate can then be deduced by the equation:
UR = q 2 - ql 2e.1 '
and the corresponding extrapolation length by:
b = UR
where ~/is the apparent shear rate given by:
~, = 6ql (2e)2.1
Fig. 5 shows the variations in extrapolation length "b" as a function of the slip rate UR for PBb flow in the fluorinated dies. On log paper, the increase in "b" for low velocities is a portion of s t ra ight line with a slope of about 0.8. It was 0.9 for polymers flowing in high surface energy dies. This slope increases for h igher regimes, and variat ions in "b" occur as the square of the slip ra te a t the wall. Lastly, for the highest regimes that could be reached, "b" continues to increase on a logarithmic graph along a straight line, though with a very low slope. Fig. 5 thus shows tha t the fluorinated dies enable much lower slip regimes to be obtained than those possible with the stainless steel dies. This demonstrates the fact tha t slip along low surface energy walls is associated with wall friction tha t is ext remely weak compared with cases where the fluid flows along conventional walls.
The fringes are not parallel to the wall Unstable birefringence pattern into the capillary
Figure lob : Birefringence patterns for the flow of PBb into a two-dimensional fluorinated die, 2 mm gap and 20 mm long. AP = 6.4 105 Pa; q, = 1.1 g h .
The fringes are not parallel to the capillary wall The axial fringe is 4 and not zero
Figure 11 : Birefringence patterns for the flow of PBb into a two-dimensional fluorinated die, 2 mm gap and 20 mm long. AP = 12 105 Pa; q, = 9.2 g h .
384
5. DISCUSSION AND CONCLUSION
By using polymers with various molecular characteristics and analyzing the flow curves, it has been possible to propose a comprehensive view of friction curves with polymer melt slip over a significant range of slip velocities.
5.1. H i g h s u r f a c e e n e r g y d i e s - m a c r o s c o p i c s l ip The slip regimes corresponding to and then succeeding the first area of oscillations may be described as cohesive, i.e., they indicate a fracture within the polymer itself [33]. Indeed, owing to the roughness and interactions likely to occur between the wall and flowing fluid, a layer of the polymer is considerably adsorbed and is thus t rapped at the surface. When a sufficient level of stress is reached, the flowing polymer can become disentangled from this wall layer, and slide along it. In practice, it is then possible to describe the second portion of the pressure loss curves (portion (DF) on figure 1). With higher flow regimes, it has been seen that a second zone of instability is likely to occur. This is s i tuated entirely within the flow-with-slip regime and thermal effects should be taken into account for this zone. In addition, beyond a certain stress level and depending on wall properties, the adsorbed layer can become detached from the wall. This is another type of slip, which may be described as adhesive, i.e., indicat ing a break between the polymer and the wall. More experiments are clearly needed to identify the several possible slip regimes
The results set out in 3. show tha t a friction law must make allowance for the remarks made in 3.2 in order to represent friction with macroscopic slip in the case of polymer melts. An initial approach was made by Chernyak and Leonov in 1986 [18], and then Leonov in 1990 [20]. They proposed relations for modelling the bell- shaped curve with its maximum and minimttm/mi_nima. It appeared worthwhile to adapt these relations to take into account the existence of a positive stress at rest and the decrease in stress at the wall when the slip velocity increases, for low slip regimes. With given tempera ture and pressure, these relations are wri t ten as follows:
XR (UR) = 1:s + Ae-XU* + B , ( l o )
/ Ua - -U:-: 1 - l + m + e I:R (UR) = "r + Ae-~'u* + B U---K UR (11)
1 + " g n - e
Relation (10) is used to model friction laws that have a single minimum [18] while relation (11) makes allowance for the existence of a second minimum [20]. These relations involve various parameters:
385
- three stress parameters: Xs, A and B. The value of xs governs the mean stress level. The value of A fixes the stress level for low values of UR. The value of B governs the maximum stress level of the friction curve. In addition, when UR tends towards 0, the stress calculated by relations (10) and (11)tends towards the sum Xs + A, which thus corresponds to the s tress tha t t r iggers macroscopic slip for the case in question;
- a velocity parameter: Ua. Its value fixes the position of the maximum along the slip rate axis;
- ~,, an inverse velocity parameter. This governs the slope of the friction curve at low slip rate values;
- two dimensionless parameters "m" and "y'. The value of "m" determines the
curvature of the friction curve around the maximum, while t ha t of "~" determines the position of the second minimum in terms of stress and velocity at the wall.
On the basis of these remarks, relations (10) and (11) could be used to model friction with slip for the polymers under consideration.
For the PDMSs, only one zone of instability could be observed. Relation (10) was therefore used. Fig. 4a shows that, with the set of parameters given in table 4, the friction curves fit to within 5%. In addition, this same set of parameters can be used to give a faithfixl representation of flow regimes with oscillations [15]. In this regard, it should be underlined that the friction curves obtained experimentally and represented in Fig. 4a initially appear to be independent of the diameter and length of the capillary. It is therefore tempting to use a single set of parameters in order to model them with relation (10). However, it is impossible to represent all the results obtained in the oscillating regime with this single set of parameters [15]: the model se t t ing is hardly affected by the length of the capillary but it is necessary to choose slightly different values for the parameters, depending on the diameter of the dies used [15].
Table 4 Constants.of relation (10) for gum LG2. D (mm) 0.5 Xs (bar) 0.495 0.47
. . . . . . . . . . .
A (bar) 0.085 0.08 B (bar) 0.6 0.48 Ua (cm/s) 4.7 2.8
. . . . . . . .
(cm/s)-I 6 6.5 m 0.009 0.01
With the flow conditions envisaged here, extrusion of the PB and LLDPE displays two zones of instability. An at tempt was therefore made to model the curves of Figs. 4c and 4d using relation (11). In fact, owing to the considerable curvature of the friction curves around their respective maximum values (Figs. 4c and 4d), relation (11) cannot be applied. Indeed, the parameter governing the curvature is
386
"m" and, as "m" is assumed to be lower than unity [18], the work of Leonov in 1990 [20] showed that relation (11) would give little curvature around the maximum. This present study thus shows tha t this relat ion is incompatible wi th the slip behaviour of all polymer melts. In the present s ta te of knowledge, only a completely empirical relation may be envisaged with a view to modelling the friction curves represented on Figs. 4c and 4d. Eventually, it will be necessary to draw up models based on physical considerations and to incorporate additional experimental results based on careful investigations. Numerical modelling could also be a valuable tool for analyzing a complex situation in which it is difficult to check all the parameters precisely, by varying them over the entire range.
5.2. L o w surface en ergy dies The effect of polymer-wall interactions was studied by considering the flow of various polymers along fluorinated walls using various techniques. The results obtained show that, in the case of steel dies, the fluids flow while adhering to the walls, which have a high surface energy, and may crack as they leave the die. In the case of geometrically identical dies with fluorinated walls, the surface cracks observed at the die outlet may be delayed or even eliminated. In the part icular case of the PBb flowing in two-dimensional dies, the experimental observations obtained by the various mechanical, optical and physical techniques used enable this result to be put down to the triggering of slip at the die wall when the fluid flows in the fluorinated die. In fact, two different slip regimes occur, separated by a fluctuating transition zone. This result, with a second regime at high flow rates tha t resembles what slip might be along an ideal surface (Fig. 5) should be compared with the theoretical forecasts and measurements recently published on chain dynamics and their coil-stretch transition near low surface energy walls [10, 12]. Friction curves for slippery surfaces differ from the curves for high energy surfaces in the low stress-low slip velocity region. Instead of a critical stress value, stress can increase progressively with UR at low values of UR, before a friction curve similar to that for a high energy surface is obtained. However, it has not been possible yet in the type of experiments described here to reach sufficiently low regimes to describe friction curves for slip velocities tending towards zero, and a full comparison with the results by Migler et al. [12] cannot be made the more so as techniques used in [12] concern velocities smaller than lmm/s and did not give stress level. In addition, identification of the slip velocity measured in [12] with UR is not necessarily trivial. Fur ther experiments using appropriate measurement techniques are then necessary.
R E F E R E N C E S
1 : Petrie C. J. S. and Denn M. M., "Instabilities in polymer processing," AIChE Journal 22, 209-236 (1976).
2 : Larson R. G., "Instabilities in viscoelastic flows," Rheol. Acta 31, 213-263 (1992).
387
3 : Piau J.M., E1 Kissi N. and Tremblay B., "Influence of upstream instabilities and wall slip on melt fracture phenomena during silicones extrusion through orifice dies," J. Non-Newtonian Fluid Mech. 34 145-180 (1990).
4 : Ramamur thy A.V., "Wall slip in viscous fluids and influence of material of construction," J. of Rheol. 30, 337-357 (1986).
5 : Kalika D.S. and Denn M. M., "Wall slip and extrudate distortion in linear low- density polyethylene," J. of Rheol. 31, 815-834 (1987).
6 : Funatsu K. and Sato M., "Measurement of slip velocity and normal stress difference of polyvinylchloride,' Adv. in Rheol., 4 (Mexico 1984) 465-472.
7 : Knappe W. and E. KrumbSck, "Evaluation of slip flow of PVC compounds by capillary rheometry," Adv. in Rheol., 3 (Mexico 1984) 417-424.
8 : de Smedt C., Nam S., "The processing benefits of fluoroelastomer application in LLDPE," Plast. Rubber Process. Appl. 8, 11 (1987).
9 : de Gennes P. G., "Ecoulements viscosim~triques de polym~res enchev~tr~s," C. R. Acad. Sci. Paris, 288 s~rie B, 219-220 (1979).
10 : Brochard F., de Gennes P. G., "Shear dependant slippage at a polymer / solid interface," Langmuir 8, 3033 (1992).
11 : E1 Kissi N., L~ger L., Piau J. M. and Mezghani A., "Effect of surface properties on polymer melt slip and extrusion defects," J. Non-Newtonian Fluid Mech., 52, 249-261 (1994).
12 : Migler K. B., Hervet H. and L~ger L., "Slip transition of a polymer melt under shear stress," Physical Review letters 70, 287-290 (1993).
13 : Benbow J. J., Lamb P., "New aspects of melt fracture," SPE Trans. 3, 7-17 (1963).
14 : E1 Kissi N. and Piau J. M., "The different capillary flow regimes of entangled polydimethylsiloxanes polymers: macroscopic slip at the wall, hysteresis and cork flow," J. Non-Newtonian Fluid Mech. 37, 55-94 (1990).
15 : Piau J.M., E1 Kissi N., "Measurement and modelling of friction in polymer melts during macroscopic slip at the wall," J. Non-Newtonian Fluid Mech. 54 121-142 (1994).
16 : Vinogradov G.V., Protasov V. P. and Dreval V. E., "The Rheological Behavior of Flexible Chain Polymers in the Region of High Shear Rates and Stresses, the critical process of spurting, and supercritical conditions of their movement at T > Tg," Rheol. Acta 23, 46-61 (1984).
17 : Leonov A. I., Rheol. Acta, "A linear model of the stick slip phenomena in polymer flow in rheometers," Rheologica Acta, 23 (1984) 591-600.
18 : Chernyak Yu. B., Leonov A. I., "On the theory of adhesive friction of elastomers," Wear, 108 105-138 (1986).
19 : Lau H. C., W. R. Schowalter, "A model for adhesive failure of viscoelastic fluids during flow," J. of Rheol., 30 (1986) 1, 193-206.
20 : Leonov A. I., "On the dependance on friction force on sliding velocity in the theory of adhesive friction of elastomers," Wear 141, 137-145 (1990).
21 : Atwood B. T., Schowalter W. R., "Measurements of slip at the wall during flow of high-density polyethylene through a rectangular conduit," Rheol. Acta 28, 134-146 (1989)..
22 : Hatzikiriakos, S.G. and J. M. Dealy, "Wall slip of molten high-density polyethylene. II. Capillary rheometer studies," J. of Rheol. 36, 703-741 (1992).
23 : Hill D.A., Hasegawa T. and Denn M. M., "On the apparent relation between adhesive failure and melt fracture," J. of Rheol. 34, 891-918 (1990).
388
24 : Hatzikiriakos S. G. and Kalogerakis N., "A dynamic slip velocity model for molten polymers based on a network kinetic theory," Rheologica Acta, 33 (1994) 38-47.
25 : Mezghani A., "Interface polym~re-paroi et stabilit~ des ~coulements de polym~res fondus," Th~se (Universit~ de Grenoble) France, (1994).
26 : E1 Kissi N. and Piau J. M., "Ecoulement de fluides polym~res enchev~tr~s. Mod~lisation du glissement macroscopique ~ la paroi," C. R. Acad. Sci. Paris, 309 s~rie II, 7-9 (1989).
27 : E1 Kissi N. and Piau J. M., "Adherence of LLDPE on the wall for flow regimes with sharkskin," J. of Rheol. 38, 1447-1463 (1994).
28 : Derm M.M., "Issues in viscoelastic fluid mechanics," Annu. Rev. Fluid Mech. 22, 13-34 (1990).
29 : White J.L. and Yamane H., "A collaborative s tudy of the stability of extrusion, melt spinning and tubular film extrusion of some high-, low- and linear-low density polyethylene samples," Pure & Appl. Chem. 59, No. 2, 193-216 (1987).
30 : E1 Kissi N., Piau J. M., Attan~ P. and Turrel G., "Shear rheometry of polydimethylsiloxanes. Master curves and testing of Gleissle and Yamamoto relations," Rheologica Acta, 32, 293-310 (1993).
31 : Moore D. F., The friction and lubrication of elastomers (Pergamon Press, 1972).
32 : Hatzikiriakos, S.G. and J. M. Dealy, "Wall slip of molten high-density polyethylenes. I. Sliding plate rheometer studies," J. of Rheol. 35, 497-523 (1991).
33 : Piau J.M., E1 Kissi N., Toussaint F. and Mezghani A., "Distortions of polymer melt extrudates and their elimination using slippery surfaces," Rheologica Acta, 34 (1995) 40-57.
34 : Sagiv J., "Organized monolayers by adsorption. 1. Formation and structure of oleophobic mixed monolayers on solid surfaces," J. of Ame. Chem. Soc., t. 102 (1980) 92-98.
35 : Silberzan P., L~ger L., Ausser~ D., Benattar J. J., "Silanation of silice surfaces : a new method of constructing pure or mixed monolayers," Langmuir 7 1647-1651 (1991).
36 : Piau J.M., E1 Kissi N. and Mezghani A., "Slip flow of polybutadiene through fluorinated slits," J. Non-Newtonian Fluid Mech. 59, 11-30 (1995).
37 : Han C.D. and Drexler L.H., "Studies of converging flows of viscoelastic Polymeric melts. I. Stress birefringent measurements in the entrance region of a sharp-edged slit die," J. Appl. Pol. Sci., 17 (1973) 2329-2354.
38 : Brizitsky V.I., Vinogradov G.V., Isayev A.I. and Podolsky Yu. Ya., "Polarization-optical investigation of normal and shear stresses in flow of polymers," J. of App. Polymer science, 20 (1976) 25-40.
39 : Sornberger, G., J. C. Quantin, R. Fajolle, B. Vergnes and J. F. Agassant, "Experimental study of the sharkskin defect in l inear low density polyethylene," J. Non-Newtonian Fluid Mech. 23, 123-135 (1987).
40 : Fields T.R. and Bogue D.C., "Stress birefrigent patterns of a viscoelastic fluid at a sharp edged entrance," Transactions of the society of Rheol, 12 (1968) 1, 39-55.
Rheology for Polymer Melt Processing J-M. Piau and J-F. Agassant (editors) �9 1996 Elsevier Science B.V. All rights reserved. 389
Stability phenomena during polymer melt extrusion
N. E1 Kissi and J-M. Piau
Laboratoire de Rhgologie, Domaine Universi taire, BP 53X, 38041 Grenoble Cedex, France*
1. I N T R O D U C T I O N
During the extrusion of polymers different defects and flow instabilities occur at very low Reynolds numbers . The commonly known ones are sharkskin , melt fracture, slip at the wall and cork flow. These defects are of commercial importance, since they often limit the production rate in polymer processing. Many researchers have been interested in the subject, and thorough reviews on flow stabili ty and melt fracture have been wri t ten in the last 30 years [1-4]. More recently, two review papers dealing with viscoelastic fluid mechanics and flow stability, were published by Denn [5] and Larson [6]. However, a l though much work has been done in the field of extrusion distortions, controversy still exists regarding the site of initiation and physical mechanisms of the instabilities.
At the observations of various PDMSs extrudates Piau et al. [7, 8] consider that the first visual ly detectable extrusion defects appear locally in the form of superficial scratches. These grow as the flow rate is increased, leading to a more dramatic aspect of the extrudate which becomes rough and is then identified by the term sharkskin. It should be noted that roughness quantification by Beaufils [9], using a profilometer, shows tha t the extrudate is indeed rough before visual observation of sharksk in can be made. Nevertheless, it seems tha t sharkskin results from the cracking of the fluid as it leaves the die, under the action of high value s tresses which develop in this zone. Such an interpretat ion has already been suggested by Cogswell [10] and it may be noted that the existence of these stresses located in the exit zone is confirmed by birefringence [11-13] and by numerical modelling results [14, 15]. In a series of recent articles, Ramamur thy [16, 17] as well as Kalika and Denn [18], Hill et al. [19], Hatzikiriakos and Dealy [20], show t h a t for var ious polyethylenes (PEs), the flow curves show a discontinuity and a slope change for regimes in which sharkskin can be observed visually. These results agree with those of Kurtz [21] and lead them to associate the visual appearance of this defect with a loss of adhesion at the polymer-wall interface. However, a study by E1 Kissi and Piau [22], carried out with an LLDPE similar to tha t used by Ramamur thy [16] and Kalika and Denn [18], and based on the same methods, shows tha t during flow regimes with sha rksk in the
*Grenoble University (UJF and INPG) - UMR 5520 of CNRS.
390
polymer essentially sticks to the wall. This is in agreement with the conclusions of studies by Bartos and Holomek [23], Uhland [24] and Vinogradov et al. [25].
Melt fracture appears to be an unstable phenomenon, which occurs beyond a critical stress in a die and which may involve different and independent mechanisms. The possibility that melt fracture is initiated by slip at the capillary, or die wall has often been supported by different experiments. Thus, Benbow and Lamb [26] contended this hypothesis. Their main argument was based on a series of photographs of the motion of a coloured disk embedded in a silicon gum flowing through an acrylic die. Wall slippage was also investigated by Den Otter [27] in a series of experiments where velocity profiles were measured at the surface of an acrylic slit both below and above melt fracture for a variety of liquid polymers and in particular for a silicone g~lm. Den Otter measured very low velocities as low as 0.02 mm/s (2% of the maximum flow velocity at a distance of 0.01ram from the wall of the slit) during melt fracture, therefore indicating that wall slippage was not observed for these silicon gums of lower viscosities than those studied by Benbow and Lamb. Thus, Den Otter clearly shows that slip at the wall cannot explain melt fracture instabilities although it may accompany it in some circumstances and modify the appearance of the extrudate. Most researchers [2-4, 7] suggest that the region upstream of the contraction is the site of initiation of melt fracture instability. They base their opinion on experimental observations using essentially two techniques : observation of the motion of flow tracers in the fluid, and flow birefringence. This latter method for observing and investigating polymer flow through capillaries has been thoroughly studied by Vinogradov et al. [11] and the appearance of melt fracture has been characterized by the observation of birefringence pa t te rns and isochromes before and after the critical regime [2]. However the precise mechanism of the instability has not been completely elucidated yet and seems to be influenced by various properties such as the rheology of the fluid, the geometry of the capillary die entry, thermal effects... One early explanation suggested by Bagley [28] and Tordella [2] was that the liquid polymer is fractured by elongational stresses. Bagley [28] gave a critical capillary wall recoverable shear strain criteria for melt fracture. White [3] gave a different explanation based on experimental observations for the flow of a viscoelastic fluid into a contraction. He argued that a hydrodynamic instability is initiated in the form of a "spiral" flow when a critical Weissenberg number is reached and argued that this instability is the initial mechanism of polymer melt extrudate distortion.
Thus, polymer melt die flows appear highly complex and it is difficult to describe all of their many facets. Any die has an entrance which is a contraction, a die land, and an exit. It thus combines two geometric singularities with a variable length of fully developed flow. A limiting case is obtained with orifice dies, the die land of which is short. High stress levels are generally applied in the die to the flowing polymer melt, due to the upstream head, to the walls, and to the geometric singularities. Basic knowledge of polymeric fluid bulk properties, failure of polymeric liquids under stress, as well as surface properties of the polymer at the polymer-wall interface may be simultaneously involved.
391
There appear to be three crucial points in examining and attempting to throw light on these types of flow : the general relevance and specific properties of the fluids used in the experiments, the experimental techniques and the care with which experimental pitfalls are meticulously avoided, the caution with which interpretations are given of the experimental findings [29]. It is highly profitable to use various experimental techniques simultaneously, either because of their individual limitations or because of the cross-checking procedures allowed. Major advantage can be taken of the fact that many aspects of extrudate distortions reveal the same physics of entangled chains, when changing from one polymer species to another. This physics depends little on the polymers chemical family, and on the temperature at which they melt [30, 31]. Better choices of experimental conditions may then result. Comparisons between results obtained for different chemical families of polymers must be made, and general laws and classifications appear.
In chapter 2 the framework of this study will be defined and advantage will be taken of Bernoulli's theorem to define and understand clearly the interest of flow curves. They constitute an important tool and are characteristic of the instabilities likely to occur.
By appropriate and different visualization technique, we will then identify: - in chapter 3 the stability and the nature of the recirculating zone developing
in the upstream flow, - stable flow regimes and sharkskin phenomenon in chapter 4, - unstable flow regimes for slightly to moderately entangled polymers, in
chapter 5. It will be seen that those flow regimes are essentially controlled by the adhesion of the flowing polymer to the die wall. The hydrodynamic instability characteristic of melt fracture type phenomenon will be also described in this chapter,
- slip flow of highly entangled polymers will be presented in chapter 6.
Finally, the conclusions and some perspective of this work are detailed in chapter 7.
2. EXPERIMENTAL FACILITIES AND FLOW CURVES
2 .1 . T h e p r o b l e m s t u d i e d . The following study considers the flow of polymer melts in sudden contractions. A flow configuration of this kind involves many problems from both the theoretical and experimental standpoints. Polymer melts are complex fluids. Their viscoelastic properties during flow depend not only on their molecular structure but also on the interactions they are likely to develop at the walls, depending on the physical and chemical features of the interface and the flow conditions. In addition, not all their properties can be determined and the constitutive equations used are in practice often limited to considerations on the shear viscosity. From a theoretical point of view, considerable difficulties are involved and the problem to be studied here has not been solved. In particular, even though the boundary conditions considered in
392
this study are stat ionary for viscoelastic fluids as during turbulence, this does not guarantee the existence of a stat ionary solution. The flow may become uns teady at a low Reynolds number. It will be seen that various defects and instabil i t ies may indeed occur there. It thus appears necessary to perform experimental studies to analyze and elucidate the physics of this type of flow. The experimental conditions mus t therefore be carefully controlled to avoid introducing fur ther difficulties tha t would make interpretat ion of the results even more complicated. Hence it mus t be possible to characterize the fluids by their specific properties ( rheometry, molecular characteristics, etc.). The experimental techniques (visualization, laser velocimetry, etc.) mus t also be var ied or adapted so as to b reak free of technological limitations and widen the field of investigation as much as possible. Lastly, a min imum of care must be taken when interpret ing the exper imental results, in order to provide consistent interpretations of the phenomena observed and the mechanisms involved in their appearance and evolution.
Consequently, the following will be considered: Simple polymers: In particular, they must contain no filler and have no chemical reaction dur ing flow. The resul ts reported in this s tudy were obtained with polydimethylsiloxanes (PDMSs), polybutadienes (PBs) and polyethylenes, whose main rheological characteristics are shown in Table 1 [7, 8, 32]. They are ei ther l inear or branched, melt when heated or at ambient tempera ture and cover a wide range of molecular weights.
Table 1 Fluids used
Fluid Mw (g/tool) Mw/Mc Mw/Mn T10 (Pas)
PDMSs 23~
LO 131 000 5 1.9 158 BO 156 000 6 2.8 158 LG 1 418 000 17 3.2 15 000 BG 428 000 17 2.9 55 000
LG2 758 000 31 3.9 540 000 LG3 1 670 000 68 7.9 ?
Polybutadiems 23~
PB 220 000 44 1.05 (Linear)
PBb 170 000 34 1.38 (Branched)
1 260 000
Polyethylenes LLDPE 143 000 36 3.9
190~ HDPE 216 700 54 18.09 185~
9 800
Simple geometries" The dies used for the exper iments can be e i ther two- dimensional or axisymmetrical (Fig 1). In the first case they are characterized by their length L, gap 2e and depth 1. In the second case they are characterized by their length L and diameter D = 2R. Unless otherwise indicated, the material forming the die wall has a high surface energy (stainless steel, tungsten carbide, Plexiglas, etc.).
393
Axisymmetric die
1
Two-dimensional die
Figure 1 : Typical dies used.
Dies
Axisymetric
D or 2e (ram) IYD or L/2e
0.5 0.1 10 2O 30 4O
2 0.1 5
10 15
Two- 2 0.1 dimensional 10
(1 = 45 ram)
Sta t ionary boundary conditions: Both controlled mean rate and controlled pressure flows will be considered. Depending on the polymer-wall pa i r considered, there may be adherence or slip at the walls. In installations where the mean flow rate is controlled, the fluid is contained at the outset in a reservoir , into which a pis ton slides. The fluid is then forced into an axisymmetrical capillary, with the piston moving at a controlled speed. In this system, the pressure is measured by sensors with a measuring range that varies from 50 to 2000 bar. In the other types of experimental installation considered, the fluid in the reservoir is forced out under pressure exerted by nitrogen into an axisymmetrical or two-dimensional capillary. In this system it is thus the total head loss tha t is checked. This is determined by using a pressure gauge of Bourdon type, with the mass flow rate being measured by weighing and timing.
For each type of flow considered, a head loss curve was plotted, representing the variations in total head loss as a function of mean flow rate [7, 8, 32]. These curves are a valuable tool in analyzing and classifying the phenomena likely to occur in this type of flow.
2.2. F l o w c u r v e s . Bernoulli 's general theorem applied to the field consisting of the ups t ream reservoir, the die and the free surface of the extruded rod, shows tha t [33] the head loss in the isochoric flow is the sum of two terms. The first term is the usual volume term, responsible for the pressure loss in classical fluid mechanics. For purely viscous materials , this te rm represents the power dissipated due to viscosity, in the whole volume of the flowing fluid. The second term is representative of the energy dissipated along the surface of the walls. Its value is
394
zero if the fluid adheres to the wall. In contrast, when the extruded polymer slips at the die walls, the second t e rm becomes impor tan t . It m a y become preponderant , since the flow field differs considerably from tha t obtained for adherence conditions, and thus, the relative size of the volume term decreases significantly. The var ia t ions in head loss (approximated by the ups t r eam pressure value) as a function of flow rate provide valuable information on the effect of volume properties and on polymer-wall interactions. They are thus a valuable aid in analyzing the various flow situations [33].
The characterist ic curve of extrudate flow including adherence to the walls, and hence representat ive of slightly to moderately entangled polymer flow in sudden two-dimensional or axisymmetrical contractions [7, 32], is represented in Fig 2. It shows a slope discontinuity above a certain pressure level, which depends on the polymer-die pair considered. With low flow rates, the flow is stable. Indeed, for these reg imes , al lowing for en t rance effects, the flow curve is in fact representat ive of the shear rheometry of the polymer under consideration, at low shear r a tes [34]. The slope discontinuity of the head loss curve indicates a modification in the s t ructure of flow. It will be seen tha t this corresponds to the triggering of a hydrodynamic instability upst ream of the contraction.
/
Log Pu STATIONARY SOLUTION _ . �9149 �9
UPSTREAM
CRACKED
SLOPE 1/1 Log qv.~ v
Figure 2 �9 Typical flow curve for slightly to moderately entangled polymer melt extrusion through two-dimensional or axisymmetric dies.
Considering now the flow of polymers that are sufficiently entangled to slip along the die walls, the flow curve shown in Fig 3 is obtained. This is characterized by a break in the flow rate value, followed by a slope discontinuity of the flow curve. However, it should be noted that in rare cases [33], an initial slope discontinuity s imilar to tha t obtained for the flow of modera te ly entangled polymers is observed before the break in flow rate occurs. For low flow regimes, the branch corresponding to stable flow is obtained. Here it may be assumed that the fluid adheres to the wall [22]. After the break in flow ra te value, the observed regimes correspond to unstable flows tha t appear differently depending on the type of installation used.
395
Log p U
B
E
F
H
D
Log q
Figure 3 : Typical flow curve for highly entangled polymer melt during extrusion through two-dimensional or axisymmetric dies.
In the case of a controlled pressure instal lat ion, when the critical regime is reached, a very slight increase in the pressure level causes a jump in flow rate (from B to C) which may reach a factor of as much as 100, indicating the sudden triggering of slip at the walls. This slip is accompanied by a hysteresis tha t may be demonst ra ted by carrying out tests wi th decreasing pressure, following successively C, D and A [8]. The greater the molecular weight of the polymer, the more accentuated the jump in flow rate and hysteresis zone [8, 25]. The curve in Fig 3 also shows that in a controlled pressure installation there is a range of flow rates tha t cannot be described (contained between flow rates of points B and D). This becomes possible when the mean flow rate is controlled. The compressibility of the fluid then plays an impor t an t role and the occurrence of slip is accompanied by oscillations in the ins tan taneous pressure and flow rate as a function of time (Fig 4). During the compression phase, portion (AB) of the flow curve applies and there is adherence to the wall. During the relaxation phase, portion (CD) of the flow curve is found and there is slip [5, 8, 20]. After this area of oscillations, a new steady pressure regime is reached (Fig 4). When sufficiently high flow regimes with slip can be obtained, there is a further discontinuity in the slope of the flow rate-pressure curve, followed by a slight drop in the pressure (Fig 3). In fact, in agreement with other works [35], this discontinuity is related to the appearance of a second zone of oscillations in the ins tantaneous pressure (Fig 4), indicating the existence of a second hysteresis loop between two pressure values. In comparison with the first zone, the second zone of oscillations occurs in a slip regime, during both the compression phase (portion (EF) of the flow curve) and relaxation phase (portion (GH)). Indeed, the minimum pressure reached in this loop (points E and H of Fig 3) remains sufficiently high to be permanent ly outside the zone corresponding to stable flow with adherence. After this second zone of oscillations, there is once again a steady pressure regime (Fig 4) and the flow curve represented in Fig 3 shows once again a slope discontinuity.
4 1 I I I I I I
I I I I I I I t 0 2 4 6 8 10 12
Scale of Time (cm)
Figure 4: Variation of the instantaneous pressure as a function of time for the flow of the LLDPE through an axisymmetric capillary 2 mm diameter and 20 mm long (Stainless steel capillary, T = 190°C). At low flow regimes the pressure increases and then reaches a stable permanent value (portion (AB) and below of Fig. 3). For flow regimes between points B and D of Fig. 3, we can observe oscillations in the instantaneous pressure value. The fluid
\O m adheres to the wall during the compression phase and slips during the relaxation phase.
n For higher flow regimes, pressure reaches again a stable permanent value (portion (DF) of Fig.3). For flow regimes between points F and H of Fig. 3, a second zone of oscillations in the instantaneous pressure value is observed. It occurs in a slip regime during both the compression and relaxation phase.
397
3. V I S U A L I Z A T I O N OF U P S T R E A M FLOW
Ups t ream flow of the moderately entangled polymers was examined for regimes covering the various portions of the Figure 2 type flow curve. The observations were made in an axial plane, obtained by us ing a laser. A video film and photographs were used to analyze the structure of the flow. The photographs presented in Fig 5a and 5b for the l inear oil flowing in an orifice die, show tha t two types of vortices coexist even in low flow regimes [36]: a corner vortex, characterist ic of viscous effects (Fig 5a) ; a lip vortex, which develops at the inlet to the die and is characteristic of viscoelastic effects (Fig 5b). When the flow regime increases, the two recirculating areas expand. The lip vortex gradual ly invades the corner vortex, giving rise to a single area of recirculation ups t ream of the contraction as can be seen on Fig 5c. Lastly, it should be noted on Fig 5c t ha t the pressures are sufficient and the flow is, in fact, unstable. This property is totally masked by the visualization technique used.
Figure 5b
398
Figure 5c
Figure 5 :S t reaml ines generated upstream by the flow of PDMS LO through an orifice 0.5 mm diameter. (a) AP = 2 105 Pa, exposure time 120s ; (b) AP = 2 105 Pa, exposure time 5s; (c) AP = 16 105 Pa.
The resul ts for the LG gum, a linear PDMS with a higher molecular weight than the LO oil, are shown in Fig 6. No viscoelastic lip vortices could be observed regardless of the flow rate considered. Moreover, the viscous corner vortices hardly change when the flow regime increases, and remain very similar to what they would be for the flow of a Newtonian fluid. However, with sufficiently high regimes, these vortices may be pushed into the corner upstream of the capillary (Fig 6b). This resul t is consistent with the works of Kim et al. [37] and may be at tr ibuted to the effects of shear- thinning and normal stresses [36, 38]. It should also be noted tha t these observations were made for regimes corresponding to the unstable branch of the flow curves.
The role of elasticity can be analyzed by observing the flow of branched polymers. With a slightly entangled PDMS, the branched oil (BO) having the same viscous behaviour as the linear oil LO [34], large lip vortices could be observed (Fig 7) right from low flow regimes. In addition, at high flow regimes, the single recirculating area m a y become large and invade much of the ups t r eam flow (Fig 7b). Fundamental ly , however, there is no major difference between this and the flow of the linear oil LO.
The observations made with the branched gum BG, which has a high molecular weight, confirm all the above results. Indeed, Fig 8a shows that , with low flow regimes, it is essentially the lip vortex which develops, with the corner vortex remaining very similar to what it would have been for a Newtonian fluid. As with the l inear gum, this recirculat ing system is then pushed to the corner of the ups t r eam flow (Fig. 8b). Moreover, the lip vortex gradually invades the corner vortex, giving rise to a single area of recirculation upstream of the contraction as can be seen in Fig 8c. Here again, it will be noticed that Fig 8c is obtained in an
399
unstable flow regime. This property could be demonstrated easily in the case of the branched gum BG, the characteristic times of which are longer than those of the oils. The photograph in Fig 8d, which was taken in the same regime as photograph 8c but with a shorter exposure time, clearly shows that the fluid is not moving on the left pa r t of the picture, while it is moving on the r ight par t , which well demonstrates the existence of an instability in flow.
Figure 6 : Streamlines generated upstream by the flow of PDMS LG1 through an orifice die 0.5 mm in diameter. (a) AP = 4 105 Pa; (b) AP = 16 105 Pa.
In the light of these results, it is clear that the origin of the instabilities is difficult to observe, and this may cause serious errors in interpreting the results. In fact, as has been shown [7, 36] by carefully choosing the fluids and m e a s u r e m e n t techniques, these instabil i t ies occur at regimes corresponding to the slope discontinuity of the flow curves. By filming the four PDMSs when they are filled
400
with tungsten carbide particles, it was possible to observe the appearance and evolution of these instabilities in detail [7]. They originate along the axis of the upstream elongational flow in the form of pulsations in the particles along the streamlines. This explains why they are completely overlooked by photographs with a long exposure time (Fig 5c, 6b, 7b and 8c) that only show perfectly stable streamlines. The amplitude and frequency of these pulsations increase with pressure and, above a certain flow regime that depends on the fluid under consideration, reverse flows, directed upstream clearly appear along the axis. With even higher flow rates, pulsations with locally reversed flow progressively invade the entire upstream flow, which gradually loses its symmetrical properties [7], as can be seen in Fig 8d.
Figure 7 : Streamlines generated upstream by the flow of PDMS BO through an orifice die 0.5 mm in diameter. (a) AP = 5 10 4 Pa; (b) AP = 16 10 5 Pa.
402
Figure 8d
Figure 8 : Trajectories of tungsten carbide particles for the upstream flow of PDMS BG through an orifice die 0.5 mm in diameter. (a) AP = 1 104 Pa; (b) AP = 6.75 105 Pa; (c) AP = 10 105 Pa; (d) AP = 10 105 Pa.
Finally, to conclude this section, it should be noted that in the case of highly entangled polymers, the flow is very slow and the observation techniques described above reach their limits. No a t tempt was made, therefore, to take pictures of the recirculation zones obtained with this type of polymer. Moreover, no photograph has been taken for the upst ream flow obtained with PEs, due to problems caused by extrusion at high tempera ture , through t r anspa ren t dies. However, it is important to underline the generality of the results shown above. In particular, the possible existence of lip vortices, during the extrusion of linear polymers, is now established (Fig 5b).
4. OBSERVATION OF STABLE F L O W - SHARKSKIN
This section analyses the appearance of the ex t ruda te under regimes corresponding to the stable branch of the flow curves [8, 22], i.e., before the first slope discontinuity and, in certain cases, before the jump in flow rate value.
The extrudate leaving the die is at first smooth and t ransparent (Fig. 9a), and may be very swollen. As the flow regime progressively increases, and irrespective of the polymer used [7, 8, 22], scratches appear on the surface of the extrudate, situated in longitudinal bands tha t become increasingly wider and numerous, gradually invading the entire surface of the extrudate, as shown in Fig. 9b. The extruded rod thus loses its t ransparency and becomes increasingly mat t and opaque. With slightly entangled polymers, this appearance remains as long as the flow regime is
403
stable [7]. With moderately to highly entangled polymers, the scratches may evolve. Indeed, stresses with these polymers may reach sufficiently high levels to produce cracks in the fluid as it leaves the die [7]. These cracks penetrate deeply into the extruded rod just where it leaves the die, as can be seen in Fig. 9c and 9d. They close downstream of the outflow section owing to the relaxation of the polymer and the extrudate then has the characteristic appearance of sharkskin (fig. 9c). By using highly entangled fluids, hence with very long characteristic times, it was possible to observe the formation of these cracks in detail [8]. Figure 9d to 9g indeed clearly show that, during the formation of the crack, the fluid wraps around the capillary outlet and sticks against the wall, forming a ring just where it leaves the die. As this ring develops, the fluid continues to flow, producing a scratched rod. Above a certain thickness, the wrap-around is detached and it is possible to observe the formation of the following crack. It should be noted that the size of the crack in this precise case is marked by two successive rings. Fur thermore , as the characteristic times are long in comparison with the flow time, it is no longer possible to observe the stress relaxation that produces a rough unconfined surface with the appearance of sharkskin. However, the mechanisms involved are the same as those governing the phenomenon commonly known as sharkskin, which is therefore not simply a small-scale effect. In particular, it can be seen in figure 9d to 9g that, with gum LG2, the transition from a flow regime with sha rksk in cracks to one with the formation of rings takes place very progressively. This result confirms that the rings are simply an advanced aspect of the sharkskin phenomenon. Finally, it is important to notice that the results are similar in the case of two- dimensional extrusion [13, 39]. Figure 9h clearly shows a two-dimensional sharkskin for a PDMS.
In order to determine the stress field in the outflow section, a birefringence experiment was performed. The results obtained for a PBb (cf. Table 1) flowing in a two-dimensional steel die, which is well suited to this type of experiment, are given in Fig.10. Generally speaking, the birefringence pictures show that, in the inlet area, birefringence is maximum at the entrance of the capillary and decreases near the ups t r eam and downst ream limits of this area. In addition, the birefringence fringes are parallel to the wall in the area of established flow in the capillary. This suggests that with a given flow rate, the stress at the wall remains constant along the capillary. Lastly, a concentration of stresses at the outlet is seen by the appearance of additional fringes and by the convergence of the central fringes towards the die outlet (Figs. 10a and 10b). In the case of flow with sharkskin, birefringence patterns show that the number of fringes varies in time at a given regime. Indeed, Fig. 10c and Fig. 10d show clearly that at a pressure of 25.6 bar, the central fringe in the die outlet section is successively black and then lit. In the regime considered here, this pulsation in the number of fringes occurs every 30 seconds. In these regimes, the sharkskin effect has in fact totally invaded the surface of the extrudate. The period of crack formation could be measured. It is identical to that of the pulsations in the number of fringes. It therefore appears that, as the fluid cracks, the stresses relax at the die outlet and the central fringe in the die outlet section is black. The stress level then changes simultaneously with the formation of the next crack, which results in the development of this fringe and the start of a new lighted fringe of higher order.
Figure 9 : Extrudates of PDMSs at the exit of an orifice die. (a) Smooth extrudate for PDMS LO flowing through a 0.5 mm diameter die. AP = 1 105 Pa. (b) Scratches at the surface of PDMS BO flowing through a 0.5 mm diameter die. AP = 5 105 Pa. (c) Cracks a t the exit of a 0.5 mm diameter orifice for PDMS LG1. AP = 2 105 Pa. (dl to (g) Evolution of cracks a t the exit of a 2 mm diameter orifice for PDMS LG2.
(dl AP = 13 lo5 Pa; (el AP = 15 105 Pa; (0 AP = 21 105 Pa; (g) AP = 25 105 Pa. (h) Two dimensional cracks a t the exit of a 2D capillary for PDMS LG1. AP = 10 105 Pa
405
r~
c~
olll~
o
(€9 Figure 9 (continued).
(€9 Figure 9 (continued).
Figure 10 : Birefringence patterns for the flow of PBb through a 2D capillary, 2 mm gap and 20 mm long. (a) and (b) AF' = 8 105 Pa. (c) and (d) AF' = 25.6 10 Pa.
408
In conclusion, it seems tha t by using appropriate polymers and techniques, it can be demonstrated that sharkskin:
- is an outflow phenomenon tha t is produced by the concentrat ion of high stresses in this area; these may reach sufficiently high levels to cause the fluid cracking just as it leaves the extrusion die. It is thus at this point that observations should be made and not far downstream when the fluid has relaxed and the cracks have closed. It should be noted that this explanation was already suggested by Cogswell [10] and the existence of these high stresses has been confirmed by birefringence measurements [11, 13] and the results of numerical modelling [14, 15, 40].
- is a progressive phenomenon [7, 9, 22] tha t appears firstly in the form of scratches on the free surface of the extrudate. These then evolve towards a more severe form: a wrap-around develops at the die out le t and accompanies the formation of the crack. This was demonst ra ted by using appropriate polymers.
- is not limited to a small-scale phenomenon. It has been seen tha t the size of the crack could be of the order of magnitude of the capillary diameter.
Lastly, it must be underlined that it is possible to significantly delay or eliminate the appearance of sharkskin, by considering the polymer flow through fluorinated dies [33, 39, chapter III.4. of this book]. Characterized by their par t icular ly low surface energy, these type of dies cause the polymer slipping at the wall even in small flow regimes. Thus, the fluid could be extruded under low stresses and it doesn't crack.
5. O B S E R V A T I O N O F U N S T A B L E F L O W F O R S L I G H T L Y T O M O D E R A T E L Y E N T A N G L E D P O L Y M E R S - M E L T F R A C T U R E
It is now necessary to examine the appearance of the ext rudate corresponding to the second branch of the Figure 2 type flow curves in the case of sl ightly to moderately entangled polymers (LO, BO, LG1, BG, etc.). It was seen in section 3 that the flow becomes unstable at the moment there is a slope discontinuity in the flow curves. The extruded rod is excited by these instabilities, which tr igger the well-known phenomenon of melt fracture [7].
For the oils, which are slightly entangled, the stresses do not reach sufficient levels and the fluid does not crack at the die outlet (cf. section 4). Therefore, melt fracture is not disturbed by sharkskin defects and it is possible to observe it in isolation. In the case of the linear oil, it is revealed first by a helical extrudate. It has already been shown [7] tha t the helix pitch and pulsation frequency ups t r eam of the contraction are identical. As the flow regime increases, the helical oscillations become increasingly large and the extrudate progressively loses all regulari ty until it becomes completely chaotic, as seen in Figure l la. It should be noted that , at these regimes, the reverse flows described in chapter 3 appeared ups t ream of the contraction [36]. In the case of the branched oil, i.e., where the role of elasticity is greater , melt fracture appears in the form of a helical rod as soon as the critical flow regime is reached (figure l lb). Here again, it has been shown that the helix pitch is identical to the frequency of the upstream pulsations [7]. However, in this precise case, several successive modes occur and are superimposed on the first one as the flow
I-: a
AP= 15 10aPa AP = 17 106 Pa AP=%l@Pa
Figure l l a : Evolution of melt fracture at the exit of an orifice die 0.5 mm diameter for PDMS LO.
Figure l l b : Evolution of melt fracture at the exit of an orifice die 0.5 mm diameter for PDMS BO.
411
regime increases (Fig. 11b, AP = 20.6 bar, 21 bar and 22 bar). It is only in the case of sufficiently high regimes (Fig. l l b , AP = 25 bar or more) that the extrudate once again has a really chaotic appearance, corresponding as in the case of the LO to the development of reverse flows upstream of the orifice.
In the case of the two gums, LG and BG, melt fracture is superimposed on the sha rksk in effect once the critical regime is reached. As with the oils, the development of ups t ream instabilities governs the appearance of the extrudate. Figures l l c and l l d show the helical structure of the extruded rod for regimes where upstream pulsations occur. Once again, the helix pitch and frequency of the ups t ream pulsat ions are identical [7]. With higher flow rates, the extrudate gradually loses any regular appearance until it becomes totally chaotic (Fig. l lc , AP = 12 bar; Fig. l l d , AP = 10 bar) as reverse flows appear ups t ream of the contraction.
These observations lead to the conclusion tha t melt fracture is the result of a hydrodynamic instabi l i ty tha t occurs in the elongational flow ups t ream of the contraction. This conclusion concurs with the forecasts made in many other studies [2-4]. In particular, it should be noted that inertia, which produces similar phenomena in classical turbulent regimes, is not involved here as the Reynolds numbers for the flows considered in this study are very low (less than unity). Moreover, for the moderately entangled polymers considered here flow takes place with adhesion at the wall, and melt fracture could be observed independently of the occurrence of slip [7, 41].
Figure l l b (continued).
412
Figt~ 11d: Evolution ofmett fmct t~ at the exit ofan orifice die 0.5 mm diameter for PDMS BG.
413
6. I -HGHLY E N T A N G L E D P O L Y M E R S - F L O W W I T H S L I P
In the case of the highly entangled polymers (LG2, LG3, PB, LLDPE, etc.), the Figure 3 type pressure loss curves show a jump in flow rate, indicat ing the tr iggering of slip along the die walls. When one observes the appearance of the extruded rod, this phenomenon is superimposed on melt fracture. I t occurs in various ways, depending on the type of installation used.
Flow with controlled pressure Extrudate flow at the die outlet accelerates suddenly and the sharkskin cracks disappear immediately (Fig. 12a). This may be a t t r ibuted to the fact t ha t slip considerably reduces the stretch stresses at the die outlet and the fluid no longer cracks. Thus, as has been shown in numerous works [18, 24, 25] the extrudate is then opaque and irregular and has practically no h ~ h e r swelling (Figure 12a). The appearance of the extrudate is the resul t of slip effects superimposed on melt fracture effects.
Figure 12a : Ext rudate distortion of PDMS LG3 at the exit of a capillary 2mm diameter and 20 mm long. Slip under stress controlled conditions. AP = 26 105 Pa
Flow with controlled mean flow rate By analyzing the variations in ins tantaneous pressure, it was possible to show tha t slip in this configuration is accompanied by oscillations in pressure between two regimes (fig. 4). During the compression phase, the pressure increases in time and the ins tantaneous flow rate is low. The polymer sticks to the wall of the extrusion die and cracks at the outlet. During the relaxation phase, the pressure decreases in time and the instantaneous flow rate is high. The polymer slips along the die wall and the surface of the extrudate is more or less smooth.
414
It is this succession of relatively smooth portions and cracked portions tha t give the extrudate the characteristic appearance of cork flow [8, 18, 20, 42]. Figure 12b clearly shows tha t the respective size of the cracked zones and smoother zones depends on the amplitude and period of the compression and relaxation phases for a given mean flow rate. The longer the periods with respect to the characteristic times of the sharkskin and slip phenomena, the more the cracked and smooth sections are differentiated and the easier it is to observe cork flow [8, 20]. Fur thermore , when the flow regime increases, the sharksk in areas , which correspond to the compression zone, gradual ly decrease in size unt i l they completely disappear (fig. 12b). The appearance of the extrudate is then governed completely by melt f rac ture and slip, in pa r t i cu la r dur ing flow regimes corresponding to the second zone of oscillations in pressure and instantaneous flow rate. Indeed, it can be seen in Figures 4a and 4b, which show the changes in pressure as a function of t ime for the controlled mean flow of a PB and PE, that during the pressure oscillations corresponding to the second zone of instability, the polymer slips along the walls irrespective of the phase in question. The minimum pressure in fact remains sufficiently high so tha t sharkskin-type flow, i.e. stable flow with the fluid adhering to the walls, never occurs.
i) qm = 0.45 10 -3 g/s ii) qm = 1.13 10 -3 g/s
Figure 12b : Extrudate distortion for PBb at the exit of a capillary 2mm diameter and 20 mm long, during (i to iii) and just after (iv) the first oscillating regime.
It should be noted tha t this second zone of ins tab i l i ty had a l ready been demonst ra ted with similar polymers [35]. However, the authors assume the existence of another area of cork flow, i.e., an area where the polymer adheres to the wall, even at these high regimes, which is in contradiction with the analysis of
415
the flow curves [22] and the results shown on Figure 12c [33]. Indeed, this figure shows that the extrudate consists of a succession of portions more or less distorted by the ups t ream instabilities : along the branch (GH) of the flow curve it is essentially the chaotic flow that governs the aspect of the extruded rod.
Figure 12b (continued)
7. CONCLUSION
In conclusion, it appears that by: - using polymers with various molecular characteristics, linear or branched,
and covering a wide range of molecular weights, - systematically analyzing the flow curves, - using appropriate observation techniques and birefringence,
it is possible to give a new, consistent and complete interpretation of the stability phenomena occurring during extrusion of homogeneous and monophasic polymer melts.
It has been demonstrated that two types of vortices - viscous corner vortex and viscoelastic lip vortex - exist separately or s imultaneously ups t ream of the contraction. It has also been shown whether they are stable or unstable. The role of shear-thinning and elasticity in the development of upstream recirculations has also been demonstrated. It should be noted that in a previous s tudy [36], a correlation was established between the development of these vortices, the velocity profile along the upstream axis of flow and the elongational properties of the fluid.
Figure 12c : Extrudate distortion for LLDPE at the exit of a capillary 0.5mm diameter and 20 mm long, during the second oscillating regime. The polymer flows downwards. The largest instantaneous flow rate is observed for iii). Photograph ii) shows the transition from small (photograph i)) to large instantaneous flow rate.
417
Lastly, the succession of the various stable or unstable phenomena likely to occur is now well understood and the physical mechanisms governing their appearance clearly identified. In particular, distortions in the extrudate are described taking into account the physics of the phenomena that cause them, and no longer simply the appearance of the free rod surface. One good reason for this is given by the photographs represented on Figure 13.
Figure 13: Sharkskin and melt fracture for the flow of HDPE through an orifice die lmm diameter (Stainless steel die, T = 185~
Indeed, only by enlarging the photograph sufficiently was it possible to observe the rough sharkskin effect in Figure 13a. To the naked eye the extrudate appears to be smooth! Furthermore, in this particular case, melt fracture corresponds to a small- scale helix (Figure 13b and 13c)) that might have been confused with sharkskin roughness if only the appearance of the extrudate had been taken into account (Figure 13c). By analyzing the flow curves and carrying out observations under high magnification, the result of which is presented in Figure 13b, it is possible to
418
show that the appearance of the extrudate is in fact the result of superimposition between the sharkskin phenomenon and melt fracture type instability. Figure 13d shows the chaotic aspect of the extrudate for sufficiently high flow rates.
Lastly, the validity of the qualitative results obtained in this study, independently of the polymer chemical family considered (PDMS, PB, PE...), should be underlined. However, it is clear tha t the value of stresses for which the different defect and instabilities occur, varies from one polymer to another, and, for a given polymer, changes with molecular weight. For a given polymer family, general trends about the influence of the ratio of mass and entanglement molecular weights Mw/Me on stress levels can be tentatively indicated as in Fig. 14.
W
1 4
Lip
Upstream Instability Macroscopic Slip
10 40 60
Sharkskin !
Mw / Me
Mw / Mn = 1 Mw / M n > 1 Branched
Figure 14" Influence of molecular weight on critical stress levels.
REFERENCES
1 : Dennison M. T., "Flow instabili ty in polymer melts: a review," Plastics Polymers, 35, 803-808 (1967).
2 : Tordella J. P., "Unstable flow of molten polymers," in Rheology, F. R. Eirich, ed., Vol. V, Academic Press, New York (1969).
3 : White J. L., "Critique on flow patterns in polymer fluids at the entrance of a die and instabilities leading to extrudate distortion," Appl. Poly. Symp. 20 (1973) 155-174.
419
4 : Petrie C. J. S. and M. M. Denn, "Instabilities in polymer processing," AIChe Journal 22 (1976) 209-236.
5 : Denn M.M., "Issues in viscoelastic fluid mechanics," Ann. Rev. Fluid. Mech., 22 (1990) 13-34.
6 : Larson R.G., "Instabilities in viscoelastic flows," Rheologica Acta, 31 (1992) 213-263.
7 : Piau, J.M., N. E1 Kissi and B. Tremblay, "Influence of upstream instabilities and wall shp on melt fracture phenomena during silicones extrusion through orifice dies," J. Non-Newtonian Fluid Mech. 34 145-180 (1990).
8 : E1 Kissi, N. and J. M. Piau, "The different capillary flow regimes of entangled polydimethylsiloxanes polymers: macroscopic slip at the wall, hysteresis and cork flow," J. Non-Newtonian Fluid Mech. 37, 55-94 (1990).
9 : Beaufils, P., B. Vergnes and J. F. Agassant, "Characterization of the sharkskin defect and its development with the flow conditions," Intern. Polymer Processing IV, 78 (1989).
10 : Cogswell F.N., "Stretching flow instabilities at the exits of extrusion dies," J. Non-Newtonian Fluid Mech., 2 (1977) 37-47.
11 : Vinogradov, G.V. and A. Ya. Malkin, Rheology of Polymers (Mir, Moscow, Springer, Berlin, 1980).
12 : Sornberger, G., J. C. Quantin, R. Fajolle, B. Vergnes and J. F. Agassant, "Exper imenta l s tudy of the sharkskin defect in l inear low density polyethylene," J. Non-Newtonian Fluid Mech. 23, 123-135 (1987).
13 : Mezghani A., "Interface polym~re-paroi et stabilit6 des 6coulements de polym~res fondus," Th~se (Universit6 de Grenoble) France, (1994).
14 : Tanner R. I., Engineering Rheology (Oxford, New York, 1985). 15 : Tremblay, B., "Sharkskin defects of polymer melts: The role of cohesion and
adhesion," J. Rheol. 35, 985-998 (1991). 16 : Ramamurthy, A.V., "Wall slip in viscous fluids and influence of material of
construction," J. of Rheol. 30, 337-357 (1986). 17 : Ramamur thy , A. V., "Extrudate irregularit ies and the polymer-metal
interface connection," Proc. Xth Int. Congr. on Rheol., Sydney, Australia, 85- 90 (1988).
18 : Kalika D.S. and Denn M. M., "Wall slip and extrudate distortion in linear low- density polyethylene," J. of Rheol. 31, 815-834 (1987).
19 : Hill, D.A., T. Hasegawa and M. M. Denn, "On the apparent relation between adhesive failure and melt fracture," J. of Rheol. 34, 891-918 (1990).
20 : Hatzikiriakos, S.G. and J. M. Dealy, "Wall slip of molten high-density polyethylene. II. Capillary rheometer studies," J. of Rheol. 36, 703-741 (1992).
21 : Kurtz, S.J., "Die geometry solutions to sharkskin melt fracture," Advances in Rheology 3, Univ. Nat. Auton. Mex., Mexico city, 399-407 (1984).
22 : E1 Kissi N. and Piau J. M., "Adhesion of LLDPE on the wall for flow regimes with sharkskin," J. of Rheol. 38, 1447-1463 (1994 b).
23 : Bartos O. and Holomek J., "Unstable flow of amorphous polymers through capillaries. I. Velocity profiles of polymer having discontinuous flow curve," Polymer engineering and science 11 (1971) 4, 324-334.
24 : Uhland, E., "Das anomale fleissverhalten von polyathylen hoher dichte," Rheol. Acta 18, 1-24 (1979).
25 : Vinogradov, G.V., V. P. Protasov and V. E. Dreval, "The Rheological Behavior of Flexible Chain Polymers in the Region of High Shear Rates and Stresses,
420
the critical process of spurting, and supercritical conditions of their movement at T > Tg," Rheol. Acta 23, 46-61 (1984).
26 : Benbow J.J. and Lamb P., "New aspects of melt fracture," SPE transactions (1963) 7-17.
27 : den Otter J.L., "Mechanics of melt fracture," Plastics and polymers, (1970) 155-168.
28 : Bagley E. B., "The separation of elastic and viscous effects in polymer flow," Trans. Soc. Rheol., 5 (1961), 355-368.
29 : Piau J.M., E1 Kissi N., "The influence of interface and volume properties of polymer melts on their die flow stability," Proceedings of the XIth international congress on rheology, Brussels, Belgium, August 1992, 70-74.
30 : Doi M. and Edwards S. F., Theory of polymer dynamics (Oxford University Press 1986).
31 : de Gennes P. G., "Ecoulements viscosim~triques de polym~res enchev~tr~s," C. R. Acad. Sci. Paris, 288 s~rie B, 219-220 (1979).
32 : Piau J.M., E1 kissi N., "Measurement and modeling of friction in polymer melts during macroscopic slip at the wall," J. Non-Newtonian Fluid Mech. 54 121-142 (1994).
33 : Piau J.M., E1 Kissi N., F. Toussaint and A. Mezghani, "Distortions of polymer melts extrudates and their elimination using slippery surfaces," Rheologica Acta, 34 (1995) 40-57.
34 :E l Kissi N., Piau J.M., Attan~ P. and Turrel G., "Shear rheometry of polydimethylsiloxanes. Master curves and testing of Gleissle and Yamamoto relations," Rheologica Acta, 32 (1993) 293-310.
35 :White , J.L., and H. Yamane, "A collaborative study of the stability of extrusion, melt spinning and tubular film extrusion of some high-, low- and linear-low density polyethylene samples," Pure & Appl. Chem. 59, No. 2, 193- 216 (1987).
36 : Piau, J.M., N. E1 Kissi and B. Tremblay, "Low Reynolds number flow visualizations of linear and of branched silicones upstream of orifice dies," J. Non-Newtonian Fluid Mech. 30 (1988) 197-232.
3 7 : Kim M. E., R. A. Brown and R. C Armstrong, "The roles of inertia and shear- thinning in flow of an inelastic liquid through an axisymmetric sudden contraction," J. Non-Newtonian Fluid Mech., 13 (1983) 341-364.
38 : Boger D. V. and K. Walters, Rheological phenomena in focus (Elsevier 1993). 39 :E l Kissi N., L~ger L., Piau J. M. and Mezghani A., "Effect of surface
properties on polymer melt slip and extrusion defects," J. Non-Newtonian Fluid Mech., 52,249-261 (1994 a).
40 : R. C. King, M. R. Apelian, R. C. Armstrong and R. A. Brown, "Numerically stable finite element techniques for viscoelastic calculations in smooth and singular geometries," J. Non-Newtonian Fluid Mech. 29 (1988) 147-216.
41 : Bergem N., "Visualization studies of polymer melt flow anomalies in extrusion," Proceedings of the VIIth international Congress on Rheology, Gothenburg, Sweden, August 1976, 50-54.
42 : Lim F.J. and Schowalter, "Wall slip of narrow molecular weight distribution polybutadienes," J. of Rheol, 33 (1989) 8, 1359-1382.
421
SUBJECT INDEX
Adherence 393-397, 414 Affine motion assumption 147, 154, 157, 159, 182 Axisymetric geometry 287,289, 295, 302, 307-309, 311,319, 333,334, 357,391 Bagley procedure 288,320, 321,334 Binary blends 7, 37, 43, 44, 57-59, 63, 66, 72, 76, 83-84, 121,137 Birefringence 257-260, 262, 264-273,275- 276, 278-281,287, 289, 295, 326, 327-330, 331,333-334, 359, 374-375, 378-383, 389- 390, 403,408, 415 BKZ model (see K-BKZ model) Boundary condition 207-208, 213, 337, 348-349, 353, 357,361,372
Inflow boundary 205-206, Branched polymers 37, 109, 114-115, 117, 119, 159-162, 172-174, 400-402, 410-412 Change of time 202, 203 Chaotic 408, 411, 415 Characteristic ratio 21 Collective motion 6, 15 Complex shear modulus 97, 99, 113, 118, 120, 127-128, 132, 237 Complex viscosity 97, 120, 122 Compressibility 361,395, 413-414
Slightly compressible fluid 201,205, 207 Computing (CPU) time 289, 301,307, 318 Concentration 26, 104, 120-121,125, 129, 134-135 Constitutive equation 286-287,289, 302, 311,313, 332-333
Codeformational 303 Corotational 307,309 Memory function 145, 153, 159, 288,
290 Differential models 286, 289-290, 292,
300, 302, 311,315,318,321,326, 333-334 Integral models 286-290, 300-301,
303, 306-309, 318, 333-334 Constraint release 10-12, 15, 109, 119-120, 123-125, 139 Continuous approximation 244 Contraction, converging geometry 165, 252, 257, 277,285-289,295,305,307-309, 311-312, 316-320, 397
Cogswell analysis of converging flows 165
Convected derivative 147, 157-158, 176- 183, 185, 189, 238, 248-249 Cork flow (see oscillating flow) 389, 414 Corner singularity 208,250
Re-entrant corner 242, 243, 248 Couette flow 203, 212-213, 216-223 Couette-Taylor flow 214, 222 Cox-Merz rule 163 Crack (see sharkskin) 389, 403,408, 413- 414 Critical molecular weight 76 Critical velocity 345-346, 348, 351 Cross over 5-6, 10, 12-15 Damping function 150-156, 167-171,175, 180-181,288-290, 292, 333 Deuteration 65-66, 80, 85 Dies 277,288,296, 299, 312, 315,319, 324-326, 328, 332, 361,372, 391
Die swell 285-286, 295, 298,309-310, 314-315,318,321,333-334, 402, 413 Diffusion 1, 3-12, 31, 102, 105-107, 109, 112, 115, 120-121,134, 136-137 Dipole-dipole interactions 20, 34 Doi-Edwards model 43, 48, 154, 156 Dynamic measurements 159, 163 Elasticity 100, 102-104, 107, 398,408, 415 Elongation 257,264, 266, 269 Elongational flow (see contraction) 40-41, 68, 73, 76-77, 80, 203,264, 390, 402, 411, 415 Energy Losses (see pressure losses) Entanglement 1, 5, 10-11, 13-15, 18-19, 102, 106-107, 110, 114, 117, 119-121,123, 126, 133, 136-137, 155-156, 348-350, 353 Error estimate 227-228 Evanescent Wave 338-339, 345 Extinction angle 69, 87, 89 Extinction band 259, 280 Extra stress 238,240, 244-245,249 Extrapolation length 337,345, 347-348, 351-354, 357-358, 370, 372, 381 Extrudate swell (see die swell) Fading memory 203 Finite element 224, 226, 239, 243,298, 300, 305,307,309-310, 312, 314-315, 318, 321-322,324, 329 Flow curve 295,361,363,365-366, 368- 370, 373-375,377,384, 389, 391,393-399, 402,408 Flow rate jump 395,402, 413
422
Fluor 358-362, 374, 376-377, 379-383, 386 Fluorescence Recovery 6, 338 Form factor 66, 71-72, 81, 83-84 Four-field method 245 Fractional free volume 32 Free surface 213, 287-289, 299-300, 310, 321-326, 329, 402-417 Friction 337,345, 347-354, 357-360, 365- 371,378, 381,384-386 Friction coefficient 3, 29, 31-32, 42, 60-63, 109-110, 114, 127-129, 136 Galerkin method 286, 289, 311, 314-315, 318, 333-334 Galerkin method discontinous 226, 227, 245-247 Giesekus model 200, 217,229 Glass transition 265, 271 Guinier range 71, 82 Hadamard instabilities 201-202, 209-210 Helix 408, 411, 417 Hermite element 307-308, 318 Hydrodynamic instability 390-391,394, 411 Hysteresis 267,361,363, 370, 395 Independent alignment 154-156 Inf sup condition 227,229, 241,244 Infrared dichroism 37, 39-40, 45, 58 Instability 201,202, 208, 214, 220, 280- 282, 389, 397,402, 408 Interdigitation 337, 347, 352-353 Irreversibility 156, 176 Isochromatic 259, 272, 274, 277 Isoclinic 259, 262, 274 Jeffreys model 200-201,204-206, 208, 211- 212, 214, 216-217,220, 222, 226 Johnson-Segalman model 157, 176-183 K-BKZ model 145, 148-151,154, 287-289, 307-308, 333 Kratky plot 71, 73-76, 79-82, 84-86 Laser velocimetry 365, 374-375,377 Length fluctuations 43, 63 Lesaint-Raviart method 227,248, 311-312, 318 Levenberg-Marquardt algorithm 306, 318 Liapunov or nonlinear stability 208, 212- 214,218-221 Limiting compliance 100, 102, 108, 115, 118 Linear polymers 1, 37, 42, 65, 95, 99, 109, 113, 115, 118-119, 135, 159, 257,337, 357, 389 Linear stability 214-216, 219, 220 Linear viscoelasticity l, 4-5, 17, 67, 95-97, 99, 145, 159-161
Modulus of temporary elasticity 25-26 Lodge model 146-147, 149, 157, 169 Lodge-Meissner rule 179-181, 186 Mapping functions 302-303, 307, 309 Marginal regime 351-352, 354 Maxwell model 77, 200-206, 208-212, 215, 216, 221-224, 229
Linear 144, 145, 146 Upper convected 77, 146-147, 156-
158 Melt fracture 389-391,408, 411, 413-414, 417-418 Mesh refinement 240, 243, 246, 247, 248, 250 Molecular models 1-6, 20, 41, 77-81, 95- 98, 100, 102, 104-105, 155-156, 260-262, 348-353 Molecular weights distribution 98, 101, 102, 104 Near Field Velocimetry 338,341,353 Network model 25, 81,143, 147, 153, 155, 260 Normal stress difference 151, 176-177, 264-265, 270, 280
From model 147, 172-174, 177, 178, 183-185, 188
Data 147, 157, 163, 249 Nuclear magnetic relaxation 19, 20, 22, 25, 28, 31, 33-34 Numerical analysis 225,229, 230 Numerical convergence 286-287, 311, 318, 334 Numerical scheme 228-230 Oldroyd model 200-201, 212-213, 216-219, 223,225, 229, 243, 286-287,289-290, 292, 313,287-318, 321,326-330, 333-334 Optimization 292, 306, 315, 318 Orientation 37, 39-40, 42-46, 48-49, 52-63, 257,259-260, 262, 265, 270 Orr-Sommerfeld equations 215-216, 222- 224 Oscillating flow 389, 395, 413-414 Particle tracking 290, 300, 302-303 Pearson-Helfand 43 Phan Thien-Tanner model 156-159, 184- 193, 200, 213, 229, 247-248, 251,253, 289- 290, 292, 315-334 Photoelasticimetry 257 Planar geometry 277, 289, 295, 312, 314- 315,318, 326, 333-334, 337, 357, 391 Plateau modulus 5, 10 100, 102, 104, 108- 109, 111,113, 119, 127, 134 Poiseuille flow (see dies) 203, 207,212- 213,216-219,221,223,224, 238
423
Polarizability 259, 266 Polybutadiene 23-26, 29-30, 32, 34, 105, 113-114, 133, 359, 380-383, 392, 407,414- 415 Polycarbonate 266, 271 Polydimethylsiloxane 1, 6, 307-402,404- 406, 409-413 Polydispersity 98, 100, 102, 104-105, 117, 119-120 Polyethylene 105, 128, 159-166, 275-281, 285-300, 319-334, 359, 392, 416-417 Polymer-wall interactions 338, 357-359, 372, 384, 386, 389-390, 394 Polymethylmethacrylate (PMMA) 122-123 Polypropylene 128 Polystyrene 40, 66, 101,105, 114, 121, 125, 130, 132, 266-272, 282 Pressure losses 287, 289, 295, 319, 375, 377-378, 381
Entrance pressure drop 297,320-321, 334
Total pressure drop 296, 319, 334 Principal shear directions 90, 92 Protean coordinates 300, 303 Pseudo-brush 342, 347,352 Pseudo-solid spin-echoes 23 Pulsation 400, 408, 411 Pulse field gradient NMR 7-8, 12 Quasi-Newton method 311, 318 Quenching 68-69, 73, 76, 83 Radius of gyration 71-73, 75-76, 79, 81-83, 86-87 Recoverable strain 80-81, 83-85 Refractive index 257,258,260 Relaxation 37, 42-46, 48-60, 62-63 Relaxation function 95, 97, 110-111, 118, 124-127, 137-138 Relaxation modulus 96-97, 110-113, 116- 117, 127, 129, 136, 145, 150, 164, 167 Relaxation phase 395, 413, 414 Relaxation spectrum 98, 100, 137, 145, 147, 160-162 Relaxation time 66, 68, 73-77, 80, 84, 87, 95-98, 100-101,103-104, t09, 111, 113, 115, 117-118, 122, 124 Relaxation times distribution 95-98, 100- 101,104, 115, 118 Reptation 1-6, 9-11, 13-15, 43-44, 48-51, 53, 62-63, 65, 80,101,105-109, 111,115, 119-121,123-124, 126-127, 129, 133-134, 135, 137-138, 154 Reptation time 15 Retraction 43-44, 48-51, 56, 63-64, 154- 155
Reverse flow 400, 408, 411 Rotational isomers 18 Rouse 4, 12, 43, 58, 61-63, 65, 76-80, 132 Scattering vector 65, 70-72, 76, 82-83, 86- 88 Scratch 389, 402-403,408 Self diffusion 1, 4, 6-8, 10, 12, 15 Sharkskin 374, 376-377, 386, 389, 391, 402-403,408, 411, 413-414 Shear flow 69, 86-87,265 Shear rate threshold 336, 344, 347, 349, 351,353 Slip 340-341,343-345,347, 351-352, 354, 357-359, 361,363, 366, 374-375, 384-385, 389-391,393-395, 408, 411,413-414
Low slip regime 351 Weak slip regime 347, 353 High slip regime 347,352
Slip of junctions 155-157 Slip parameter 177-182, 293 Slippery surface 374 Small angle neutron scattering 65, 70-72, 74-77, 79-80, 82, 84, 88-89 Spectral stability 215, 221-225 Squire theorem 221 Star polymers 40, 41, 43-44, 51-55, 118 Stick-slip singularity 242, 245-247, 249 Stiffness parameter 21 Stokes problem 241 Strain energy 148 Strain hardening 147, 158, 183-185 Stream-tube method 289-290, 300, 317- 318,320, 322, 333-334 Stress levels 286-289, 294-295, 306-308, 311, 319-320, 326, 331-332, 334 Stress optical coefficient 261-263, 271,277 Stress optical law 257,264, 266-268,271, 274 Stress overshoot 177-178 Subcritical flow, supercriticaI flow 202, 206, 208 Surface anchored chains 337, 343, 346-347, 351-354 Surface energy 358-359, 361,372, 374, 381,384, 386 Swell (see die swell) Tau-Chebychev approximation 222, 224 Temperature 29, 66, 73, 102-104, 109, 115, 127-132 Temporary junctions 143, 147, 153, 155- 156, 158, 183, 189 Temporary network 70, 83 Thermodynamic consistency 149 Time periodic flow 211-212
424
Time strain separability 150, 167,175, 179 Trust Region algorithm 306, 316 Tube 2-5, 10-12, 14-15 Tube diameter 4, 11, 14 Tube renewal 107, 119-125, 134-135 Type of equation 238 Uniform flow 205,207 Upwinding 226, 228,244-245 Velocity profile 337-338, 347, 351,354, 378 Viovy 43, 55, 62 Viscoelastic flow 240, 243 Viscosity
Elongational viscosity from models 78, 146-147, 172-174, 182-185, 187-188,286-287,290, 333
Elongational viscosity data 75, 164- 166, 294-295
Shear thinning 149, 185 Shear viscosity from models 1, 4-5,
10, 12-14, 73, 77, 80, 99-104, 108, 111, 114, 115, 117, 124, 131,134, 136, 146-147, 171-173, 177-178, 183-187,286-287,290, 333
Shear viscosity data 133, 159, 161, 163-164, 166, 293-295 Visualization 391-392, 397 Vortex 285-287, 391,397-398,402, 415 Vorticity 200, 202-203, 206 Wagner model 151-156, 167-176, 190-193, 287-295, 315-323, 326, 333-334 Weakly elastic fluid 204, 216 Weissenberg number 204, 250, 286-287, 311 White-Metzner model 200, 204, 209, 211, 229 Wrap around 403,408 X ray reflectivity 341,345,346 Zimm plot 71-72, 86
