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Korea-Australia Rheology Journal March 2000 Vol. 12, No. 1 1 Korea-Australia Rheology Journal Vol. 12, No. 1, March 2000 pp. 1-25 Non-Newtonian fluid mechanics for polymeric liquids: A status report L. Gary Leal* and James P. Oberhauser** Departments of Chemical** and Materials Engineering University of California, Santa Barbara Santa Barbara, California 93106, USA Abstract In this paper, we review recent progress in the development of constitutive models for both dilute and entan- gled polymeric liquids. The status of recent applications of these models for fluid dynamics predictions is then discussed, as well as possible future research directions. 1. Introdution The development of a framework for theoretical predic- tion of the behavior of polymeric liquids in a flow has now occupied scientists, mathematicians, and engineers for at least the past fifty years. Today, it remains only partially resolved and is still the subject of intense investigation, spurred on by the development of new computational tools, new experimental techniques, and a major shift about twenty years ago in the way that we model and think about entangled polymer dynamics. It is probably useful to draw a distinction between what is commonly seen as the subject of rheology as opposed to non-Newtonian fluid mechanics. Rheology is generally concerned with the deformation of materials, which of course includes flow; however, rheologists have tended to confine their interest in polymeric liquids to the develop- ment of constitutive models and the experimental char- acterization of material properties via relatively standard tests using shear flow rheometers. The purpose of such experimental studies is primarily to distinguish one batch of a material from another rather than to incorporate results into constitutive theories. Indeed, the flows used in rhe- ometers are purposefully intended to be simple enough that the interpretation of measured results does not require a fluid dynamics analysis. This means that the flow is known a priori, and one need only measure data like stresses at boundaries that reflect on the material response to that flow. Although the theoretical aspect of non-Newtonian fluid mechanics must certainly be predicated upon con- stitutive models of material behavior, the aim is generally broader: to predict the flow field, as well as the material response, in systems in which the flow itself is not known a priori and is significantly modified from that of a New- tonian fluid. In an ideal world, one could anticipate a logical pro- gression from rheology to non-Newtonian fluid mechanics. The rheologist would derive constitutive models and test them in the fluid characterization experiments of standard rheometers (at the same time, establishing the values of any material parameters that may appear in the models as unknowns). The fluid dynamicist would then combine this constitutive model with the Cauchy equations of motion and, together with boundary conditions, predict the behav- ior of the fluid in some more complex flow situation. Unfortunately, things do not work in quite such an orderly fashion. The conventional rheological experiments are often insufficient to distinguish one constitutive model from another or even to unambiguously determine all of the constitutive model parameters. Thus, at least for the present, the attempt to predict more complex flows inev- itably becomes a part of the process of developing and evaluating models, and studies in rheology and non-New- tonian fluid mechanics are often intertwined in a way that makes distinctions between them meaningless. In this paper, we attempt to provide a progress report as we move into the next century. The viewpoint is neces- sarily a personal one, and we recognize that the field is rap- idly evolving. Inevitably, there will be some omissions, for which we apologize. 2. Continuum mechanics versus molecular mod- eling in the development of constitutive equa- tions The necessary prerequisite to any theoretical study of non-Newtonian fluid mechanics is a constitutive equation, as already noted. Recent developments have focused largely on molecular modeling as a source of these equa- tions. Historically, however, the emphasis had been almost exclusively on the classical continuum mechanics approach. The goal in both cases is the same; the difference is in the *Corresponding author: [email protected] 2000 by The Korean Socity of Rheology

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Korea-Australia Rheology JournalVol. 12, No. 1, March 2000 pp. 1-25

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Non-Newtonian fluid mechanics for polymeric liquids: A status report

L. Gary Leal* and James P. Oberhauser**Departments of Chemical** and Materials Engineering

University of California, Santa BarbaraSanta Barbara, California 93106, USA

Abstract

In this paper, we review recent progress in the development of constitutive models for both dilute and entangled polymeric liquids. The status of recent applications of these models for fluid dynamics predictions isthen discussed, as well as possible future research directions.

1. Introdution

The development of a framework for theoretical predic-tion of the behavior of polymeric liquids in a flow has nowoccupied scientists, mathematicians, and engineers for atleast the past fifty years. Today, it remains only partiallyresolved and is still the subject of intense investigation,spurred on by the development of new computational tools,new experimental techniques, and a major shift abouttwenty years ago in the way that we model and think aboutentangled polymer dynamics.

It is probably useful to draw a distinction between whatis commonly seen as the subject of rheology as opposed tonon-Newtonian fluid mechanics. Rheology is generallyconcerned with the deformation of materials, which ofcourse includes flow; however, rheologists have tended toconfine their interest in polymeric liquids to the develop-ment of constitutive models and the experimental char-acterization of material properties via relatively standardtests using shear flow rheometers. The purpose of suchexperimental studies is primarily to distinguish one batchof a material from another rather than to incorporate resultsinto constitutive theories. Indeed, the flows used in rhe-ometers are purposefully intended to be simple enough thatthe interpretation of measured results does not require afluid dynamics analysis. This means that the flow is knowna priori, and one need only measure data like stresses atboundaries that reflect on the material response to thatflow. Although the theoretical aspect of non-Newtonianfluid mechanics must certainly be predicated upon con-stitutive models of material behavior, the aim is generallybroader: to predict the flow field, as well as the materialresponse, in systems in which the flow itself is not knowna priori and is significantly modified from that of a New-tonian fluid.

In an ideal world, one could anticipate a logical prgression from rheology to non-Newtonian fluid mechanicThe rheologist would derive constitutive models and tthem in the fluid characterization experiments of standrheometers (at the same time, establishing the values ofmaterial parameters that may appear in the modelsunknowns). The fluid dynamicist would then combine thconstitutive model with the Cauchy equations of motiand, together with boundary conditions, predict the behior of the fluid in some more complex flow situationUnfortunately, things do not work in quite such an ordefashion. The conventional rheological experiments aoften insufficient to distinguish one constitutive modfrom another or even to unambiguously determine allthe constitutive model parameters. Thus, at least for present, the attempt to predict more complex flows ineitably becomes a part of the process of developing aevaluating models, and studies in rheology and non-Netonian fluid mechanics are often intertwined in a way thmakes distinctions between them meaningless.

In this paper, we attempt to provide a progress reporwe move into the next century. The viewpoint is necesarily a personal one, and we recognize that the field is idly evolving. Inevitably, there will be some omissions, fowhich we apologize.

2. Continuum mechanics versus molecular mod-eling in the development of constitutive equa-tions

The necessary prerequisite to any theoretical studynon-Newtonian fluid mechanics is a constitutive equatioas already noted. Recent developments have foculargely on molecular modeling as a source of these eqtions. Historically, however, the emphasis had been almexclusively on the classical continuum mechanics approaThe goal in both cases is the same; the difference is in

*Corresponding author: [email protected] 2000 by The Korean Socity of Rheology

Korea-Australia Rheology Journal March 2000 Vol. 12, No. 1 1

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L. Gary Leal and James P. Oberhauser

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starting point and framework for development. In the con-tinuum mechanics approach, the starting point is essen-tially a guess of the form of the relationship between stressand independent dynamical or microstructural variables.The latter might include a vector to represent the mean ori-entation of the macromolecular ensemble, or a second-ranktensor to represent the statistical “shape” of a microstruc-tural distribution function. From this guess, irreversiblethermodynamics, tensor calculus, and such basic principlesas material frame indifference are invoked to deduce themost general constitutive equation consistent with the orig-inal assumptions. Of course, the class of fluids that theresulting constitutive model can describe is completelydetermined by the initial guess and has nothing to do withthe subsequent manipulations of form. If the guess is suf-ficiently general, one may hope that the class of materialsdescribed is large. On the other hand, it is possible that aninitial guess represents no real material. The resulting mod-els also inevitably contain a large number of constitutivecoefficients or constants that are difficult (or impossible) todetermine experimentally. From a hypothetical point ofview, it is quite possible that a model is derived which hasno solutions or only unstable solutions for flows in whicha real fluid quite happily exhibits a stable flow structure.The conclusion is that the classical continuum mechanicsapproach to the description of non-Newtonian fluids, suchas polymeric fluids, is largely unsuccessful on its own. Thepotential of coupling the continuum mechanics approachwith molecular modeling, to provide insight into the formof the initial constitutive hypothesis (or guess), is relativelyunexplored at the present time but may prove to be animportant component in the overall picture. In effect, this isthe source of the toy models that are described in the lastsection of this paper. A more general, and potentially use-ful coupling of micromechanical/molecular modeling andcontinuum mechanics in the framework of non-equilibriumthermodynamics is the recent work of Ottinger, Grmelaand coworkers (Dressler et al., 1999; Grmela and Ottinger,1997; Ottinger, 1999; Ottinger and Grmela, 1997).

The molecular modeling approach to the derivation ofconstitutive equations offers some distinct advantages,though it too has limitations. The basic starting point in thisapproach is a mathematical model of the material at thelevel of either individual polymer molecules or at least astatistical ensemble of such molecules. Principles of clas-sical and/or statistical mechanics are then used to derive adynamical equation (or equations) whose solution describesthe micromechanical response of the model fluid to flow,as well as a second equation to relate the macroscopic/con-tinuum stress to the microstructural state of the polymer.

The limitation in this approach is that the rheologicalbehavior of the model material is completely determinedby the initial model. One might think that this should notbe a serious limitation given the existence of a sophisti-

cated molecular dynamics framework for the descriptionmolecules at an atomic or sub-atomic scale. However, level of description is not a viable approach for modelihigh molecular weight polymer molecules in either a mor a solution. Thus, the starting point for molecular the-ories of polymeric liquids is an intermediate scale modthat is intended to mimic the behavior of key moleculevel features in a flow. If the fluid is a polymer solutionthe solvent is treated as a continuum and the polymer mecules are viewed as deformable hydrodynamic particlesthat interact with the solvent via both deterministic (hydrdynamic) forces and stochastic thermally-driven fluctutions that are modeled as Brownian motion. The latintroduces a mechanism that drives the system towarthermodynamic equilibrium state.

The fact that one begins with a model of the polymer,which is intended to incorporate all of the essential micstructural features but to delete or smear out unnecesdetail, means that the theory is inevitably one that descr(perhaps exactly) the behavior of an idealized model mate-rial rather than the actual fluid. If the model is too simplistic, the predictions will deviate from the behavior of threal fluid. However, though the molecular level descriptiis certainly oversimplified in virtually every case compareto a specific polymeric liquid, the idealized model matrials are still real in the sense that they are all physicarealizable. We might, for example, decide that an adequmodel of some polymer molecule in solution is as a rigrod in a suspending liquid subject to rotational Brownimotion. Although this picture may be far too simple whecompared to a specific polymer molecule in solution, itstill a fluid that could exist as a physical entity. If thmicrostructural characteristics that generate macroscostress are dominated by the overall or average orientaof the polymer molecules (e.g., the orientation of the ento-end vector of such a molecule in solution), the simpmodel fluid will replicate the behavior of the real materialIn any case, it is a real viscoelastic fluid, albeit one thavery simple at the microstructural level, from which onmay expect to learn a number of things.

First, the predicted flow (and/or rheological) behavior fthe model fluid may provide useful insights into thresponse of general viscoelastic fluids. Even today, thhave been relatively few fluid mechanical studies doneeither experimental or model fluid behavior in generflows. Consequently, the kind of intuitive understandinthat one often has of Newtonian fluid behavior does not exist. The molecular-based models are particularly advtageous in this regard since, as we shall see, a naturalof the solution process involves not only obtaining the cotinuum velocity and pressure fields, as in classical Netonian fluid mechanics, but also a representation for microstructural state of the fluid and its evolution as a funtion of position and time. This can provide direct insig

2 Korea-Australia Rheology Journal

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Non-Newtonian fluid mechanics for polymeric liquids A status report

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into the connection between macroscale flow behavior andthe underlying changes in the material from its equilibriumstate. Of course, in the ultimate application of viscoelasticflow theory to polymer processing operations, the micro-structural state of the material induced by the processingflow may also be extremely important since it controls theproperties of whatever is being processed.

Secondly, the governing equations for the model fluidprovide an opportunity to develop solution techniques forviscoelastic flow problems, especially numerical methodsof solution. In this field, there has been a history of extremedifficulty in the solution of flow problems for virtually anyviscoelastic constitutive model. When numerical methodsfail with a phenomenological constitutive model derivedusing the continuum mechanics approach, it is often un-clear whether the problem is numerical or simply a reflec-tion of some pathology of an unphysical model. The sit-uation is generally much clearer when using constitutivemodels derived from molecular theory since, despite theirlevel of simplicity, they are physically realizable and un-likely to exhibit bizarre behavior. In fact, under some cir-cumstances it may be possible to utilize the microstructuralmodel to develop rigorous mathematical proofs for theexistence of solutions, and this has been done for flows ona bounded domain using a nonlinear dumbbell model ofdilute polymer solutions (El-Kareh and Leal, 1989).

Thirdly, the form of the constitutive model for a sim-plistic or idealized model of a polymeric liquid can provideguidance on appropriate hypotheses for the continuum-based approach. For example, it is unlikely that the con-stitutive model for a real polymeric liquid will be simplerin form than that associated with the idealized model sys-tem. We shall see that the constitutive model for a poly-meric liquid modeled as an infinitely dilute solution ofnon-interacting dumbbells with a linear spring can be writ-ten as the well-known Oldroyd-B model from continuummechanics. If we include a nonlinear spring to mimic thefinite extensibility expected of a real polymer molecule, theconstitutive model derived from molecular theory becomessignificantly more complex than Oldroyd-B. Hence, it isunlikely that the Oldroyd-B model will be generally usefulfor polymer liquids, almost all of which will have a micro-structure that is more complex than a solution of non-inter-acting linear dumbbells.

Lastly, the constitutive equations from molecular modelsare fully specified in the sense that all material coefficientscan be calculated from parameters of the underlying micro-structural model.

A word of caution, however, is in order regarding the roleof a molecular model as a guide to the behavior of genericclasses of polymeric fluids. It may be supposed that thesimplifications in the molecular scale descriptions wouldlead only to quantitative differences between the behaviorof the idealized model material and real fluids. This is not

true in general. The differences may be qualitative as wellas quantitative. When this occurs, it clearly means tthere are essential elements of the molecular scale behaof the real fluid that are not incorporated at all in the idalized model. For example, consider a system in whchanges in both the mean macromolecular orientation the length scale are important contributors to the rheogical behavior of some fluid and these evolve on fudamentally different time scales. In this instance, it is likethat an idealized model, perhaps one which includes oan orientable but not individually deformable microstruture or one which includes both orientation and deformtion but assumes that they evolve on the same time scwould predict qualitatively different rheological behaviothan a real fluid. Again, this does not mean that thereanything wrong with the molecular-based constitutive thory or that the material it describes is not a real viscoelafluid. The initial physical model is merely too simple tcapture the physics of the particular real fluid.

3. Constitutive theories from molecular models

We have already described the generic components ofconstitutive theories derived from molecular modelinThese typically consist of an equation that relates the mroscale or continuum-level stress to the macroscopic strate and statistical averages of the microstructural svariables as well as an equation (or equations) describhow the microstructural state of a material element evolin time (Hinch and Leal, 1975). The change in micro-structure will generally have both a deterministic (hydrdynamic) and chaotic part. As a result, the microstructustate of the material will most often be expressed in terof a statistical distribution (or probability density) functionand the equation governing this distribution function meither be written as a generalized diffusion (or FokkPlanck) equation or as a stochastic (Langevin) equatHence, in either case, we may write:

σσσσ = σσσσ (average of microstructural state) (1)

and:

, (2)

where describes the evolution of the microstructustate vector for individual polymer chains. For example,the fluid can be modeled as a dilute solution of rigid rolike molecules in a Newtonian solvent, then q would sim-ply represent an orientation vector for each rod and ψ(q) isthen the orientation distribution function which specifiethe probability of an individual rod being in some directio(Hinch and Leal, 1972; Hinch and Leal, 1976). The equa-tion for would describe the orientation of a rod in th

ψ· q( ) ∇ q·ψ q( )[ ]⋅=

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L. Gary Leal and James P. Oberhauser

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flow subject to a deterministic (hydrodynamic) torque andBrownian rotation. We shall return to this example shortly.

The primary modeling issue is to determine what fea-tures of the microstructure, which will generally be exceed-ingly complex, need to be predicted to estimate the stressaccurately. The primary issue in using the set of equationsgiven by Eqs. (1) and (2) to make fluid dynamics pre-dictions (which generally means numerical solutions) is thecomplexity of Eq. (2), whether it is interpreted as a gen-eralized convection-diffusion or stochastic equation. In theexample of a solution of rigid rods, Eq. (2) is two-dimen-sional in orientation space (assuming the rods are axi-symmetric) and must be solved at each material point inthe physical flow domain. Eq. (2) will generally be a multi-dimensional convection-diffusion (or stochastic) equationin the microstructural configuration space, and its solutionfor even a large but finite number of discrete materialpoints is a formidable task, especially when coupled withthe equations of motion in the physical space. Furthermore,the full statistical distribution function contains far moreinformation than we really need to know for purposes ofcalculating the macroscopic stress. Returning again to theexample of the solution of rigid rods, the macroscopicstress requires only the second and fourth moments of ψrather than ψ itself. This fact has tended to favor attemptsto develop approximations that allow for the direct deter-mination of the moments of ψ rather than first calculatingψ and using it to compute the required moments.

Like most areas of statistical physics, however, thisattempt to develop a procedure for direct calculation of themoments of ψ rather than calculating it explicitly leads toa so-called closure problem. Using the rigid rod problem asan illustration, if we multiply Eq. (2) by uu (where u is aunit vector specifying the orientation of a rod) and integrateover all possible orientations, we obtain an equation for thesecond moment of ψ, written as:

. (3)

However, such a procedure generates a new equation con-taining the fourth moment uuuu, and the same is trueat higher orders. Thus, a governing equation can be derivedfrom Eq. (2), as described above, but it always containsmoments two orders higher. Closing this system of equa-tions requires a so-called closure approximation that relatesthe unknown higher order moment explicitly to one ormore lower order moments. Depending on the model, largenumbers of closure approximations have been proposed.However, these approximations are typically ad hoc innature and may inadvertently delete or add important phys-ics to the model, which is no longer based strictly onassumptions and procedures that have a clear physicalmeaning (Chaubal et al., 1995). One obvious issue is thatthe truncation of the hierarchy of moments at some level

limits the geometry of the approximate description of ψ.For example, if we retain only the second moment, most complex form for ψ that could be described exactlis a distribution function that is ellipsoidal in shape. However, the situation can be much subtler. A number of yeago, Hinch and Leal (1976) published a paper proposinclosure relating the fourth moment to the second momof the orientation distribution function for a dilute suspension of rigid, spheroidal Brownian particles. The clsure was developed by incorporating an asymptoticacorrect form for both weak and strong flows, with an arbtrarily chosen smooth interpolation between the two limiSubsequent work (Chaubal et al., 1995) has shown that thisclosure approximation adds a spurious branch of solutifor the second moment tensor that have no relation toexact solutions for ψ.

These difficulties with closure approximations have leto recent developments aimed at efficient solution of floproblems using the exact unapproximated models, wherψis calculated either via the stochastic (Langevin) or dfusion-based (Fokker-Planck) versions of Eq. (2) (Chauand Leal, 1999; Chaubal et al., 1997; Gallez et al., 1999;Halin et al., 1998; Hulsen et al., 1997; Laso and Ottinger,1993; Ottinger et al., 1997; Suen et al., 1999). We will notdiscuss this branch of non-Newtonian fluid mechanhere. Although this formalism can play a very useful roas a testing ground for macroscale constitutive theorbased upon closure (or perhaps other) approximationsmolecular models, we believe that the macroscale/conuum constitutive theories will continue to play a dominarole in non-Newtonian fluid mechanics for the foreseeafuture. The present paper will be based on the premise the ultimate goal of the molecular approach is to deveconstitutive theories of rheological behavior at a macscopic/continuum level.

4. Current status−−−−Dilute solutions

The problems of greatest technological interest and ccern are those involving polymer melts or solutions at hconcentration where the polymer molecules are in entangled state. However, the microscale behavior of susystems is complex. The resulting molecular-based cstitutive theories are complicated, and limited progress been made in incorporating them into non-trivial floproblems. Hence, by far the greatest emphasis to datebeen on relatively dilute polymer solutions, especiathose known as Boger fluids. Although this emphasisclearly disproportionate to the relevance of this class of fids in “ real world” application, the dynamics of dilute solutions has proven surprisingly fertile from an intellectupoint of view. It is argued (for reasons outlined earlier) theven this simple class of non-Newtonian fluids can playuseful role in the development of solution techniques a

uu⟨ ⟩ du uuψ u( )∫=

4 Korea-Australia Rheology Journal

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Non-Newtonian fluid mechanics for polymeric liquids A status report

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understanding of the flow behavior of viscoelastic liquids.Again, our own view is that there are serious limitations inusing dilute solution models in this capacity. However,there is no doubt of the intellectual motivation, due to boththeoretical and experimental discoveries. Consequently, wewill briefly discuss both dilute and concentrated entangledpolymer systems. Our goal is not to reproduce detailedequations or formulae. These can be found in a much morecomprehensive environment in original research papers andtexts to which we will refer along the way. Rather, weattempt to provide an indication of the current status, bothsuccesses and outstanding problems or issues.

4.1. Ultradilute solutionsBeginning with ultradilute polymer solutions, we imme-

diately recognize the importance of defining the meaningof the term ultradilute. From a theoretical standpoint, theword dilute implies a solution in which polymer moleculesare sufficiently far apart that the dynamics of any onemolecule are completely independent of the presence ofthe others (apart from changes in the flow itself). We firstfocus our attention on linear, flexible chain polymers. Atequilibrium, these exist in a random coil configuration thatcan be characterized by a length scale. One choice for thislength scale is the radius of gyration. Hence, in the equi-librium state, a dilute solution is one in which the meanseparation of random coils is large (say 10 times) theirradius of gyration. This large distance is required to min-imize hydrodynamic interactions between chains. In a flow,however, the motion of the surrounding solvent perturbsthe “shape” of the polymer molecule. Depending upon theflow, the polymer may undergo a transition from randomcoil to a highly elongated threadlike structure of lengthcomparable to the end-to-end contour length of the chain.This transition, known as the coil-stretch transition, is oneof the principle goals of molecular modeling of dilute solu-tions, because it corresponds to a large increase in thestress of O(L2), where L is the extended chain length scaledby the equilibrium end-to-end dimension (de Gennes,1974; Fuller and Leal, 1981; Hinch, 1976; Hinch, 1994;Larson, 1990).

Hydrodynamic interactions between non-spherical objects(i.e., particles, polymer molecules, etc.) in a viscous fluidoccur over a length scale that is proportional to their largestlinear dimension. Since the contour length of a high molec-ular weight chain may be 50-100 times its equilibriumradius, it is evident that the condition for diluteness in aflow will generally be much more severe than that for thesame polymer and solvent at equilibrium. We have foundit useful to distinguish between solutions that remaindilute, with negligible interactions between polymer mol-ecules, in flow from those that satisfy dilute conditions atequilibrium but not in flow. We denote the former ultradi-lute (Harrison et al., 1998).

All molecular models of dilute solutions assume that thedynamics of individual polymer molecules are independeof the presence of any other chains in the solution, botequilibrium and in a fully deformed/stretched state. Thuthese theories apply strictly to the subclass of dilute sotions that we have called ultradilute. The vast majority dilute solution models treat the polymer as a set of po(usually called beads) which interact with the solvent viahydrodynamic and Brownian forces. These beads are cnected by either inextensible spacers or springs (both ofwhich are hydrodynamically invisible) (Bird et al., 1987;Doi and Edwards, 1986). The latter are intended to incorporate the entropic tendency of a deformed chain to retto its statistically preferred equilibrium state and thus reresent a separation of the influence of Brownian flucations into this deterministic effect and a purely randocontribution. When rigid-rod connectors replace the entpic springs, the effect of Brownian motion cannot be spin this way; however, for a large number of beads, tentropic tendency to return to an equilibrium state will sbe incorporated.

The vast majority of fluid dynamics studies of dilutsolution behavior are in fact based upon the simplest two-bead dumbbell model with a linear or nonlinear spring. the latter case, a pre-averaged closure approximation isrequired due to the nonlinearity of the spring law. Tresulting models are generally known as FENE dumbbmodels (FENE refers to finitely extensible). The transitionto a dumbbell model from a many-bead chain means the dynamics of the polymer molecule are being specifonly in terms of a single vector (e.g., the length and oentation of the end-to-end vector). Indeed, the descripname dumbbell is somewhat misleading. The model simply a vector whose orientation and length can chanand whose hydrodynamic interaction with the solvent assumed to exist at two points corresponding to the hand tail.

The model is characterized by a single relaxation timalthough with a nonlinear spring law the actual rate relaxation depends on the degree of dumbbell deformatChain dynamics on length scales shorter than the whchain cannot be captured, and thus dumbbell models not reproduce the linear viscoelastic spectrum of relaxattimes that have been such an important historical focuspolymer physicists and physical chemists. From a fludynamics point of view, however, the critical point is ththe only important contributions of the polymer to thstress in an ultradilute solution are a result of transitiofrom the coiled to stretched state. Hence, it is essentiala model to provide an accurate description of this trasition. On the other hand, since the important rheologiproperties are all associated with transitions in polymconfiguration at the overall chain level, it is generalaccepted that the extra complexity of a multi-bead mode

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unwarranted. One consequence of the dumbbell modelassumptions is that the resulting constitutive equations willnot be terribly accurate in representing polymer behavior inshear flow. Hydrodynamically, the dumbbell only interactswith the solvent via a point force at its two ends. If thedumbbell becomes aligned in the flow direction in simpleshear, the hydrodynamic force that tends to produce stretchand rotation in other flows disappears. The dumbbell hasno cross-sectional area (though this can be alleviated tosome degree by assuming that the beads interact hydro-dynamically, which is equivalent to saying that they areactually spherical beads with a radius that is not vanish-ingly small compared to the length of the dumbbell). It hasgenerally been agreed that the failure of the dumbbellmodel to predict linear viscoelastic and steady shear flowproperties is an acceptable price to pay, given its simplicityand its ability to provide accurate predictions in extensionalor other flows in which the polymer can achieve a highlystretched state. In fact, standard rheological measurementsfor dilute solutions in shear flow are extremely difficult,precisely because the polymer contribution to stress is sosmall. Much of our knowledge of such solutions has comefrom studying them in more complex flows, which includeregions of polymer stretch, using rheo-optical techniquesthat are more sensitive to the polymer configuration andcan yield very localized measurements (Carrington andOdell, 1996; Cressely et al., 1979; Cressely and Hocquart,1980; Cressely et al., 1978; Dunlap and Leal, 1986;Dunlap et al., 1987; Fuller and Leal, 1980; Odell andKeller, 1985).

A few words about the current status of fluid mechanicsstudies for ultradilute solutions are merited. First, asalready noted, the only macroscopically significant con-tributions to flow occur if polymer chains are highlystretched. In most flows, this occurs only in extremelylocalized regions of sufficiently high strain rate and totalstrain (i.e., a long residence time for fluid elements). Thelatter condition tends to be realized only in the local neigh-borhood of stagnation points, if at all. The zone of highstretch may then extend downstream of such points forsome distance as the polymer relaxes back toward a lessstretched state. These zones are known as birefringencestrands (Harlen et al., 1991), primarily because the easiestway to perceive them is via the birefringence associatedwith the anisotropic state of polymer alignment and stretch.Examples include the region downstream of the stagnationpoint in such devices as the four-roll mill (Fuller and Leal,1980), the co-rotating two-roll mill (Dunlap and Leal,1986), and the cross-slot (Miles and Keller, 1980) oropposing jet (Tatham et al., 1995) apparatuses. The case ofa four-roll mill is illustrated in Fig. 1. Another such regionis known to occur along the symmetry axis (or plane)downstream of the rear stagnation point of a sphere orcylinder (Solomon and Muller, 1996a). Recent measure-

ments of both steady and startup flow in the two-roll mshow that the average polymer configuration, as reflecby birefringence levels, is predicted quite accurately the FENE-Chilcott-Rallison (FENE-CR) or FENE-P models using model parameters that are consistent with scaestimates for dilute solutions (Harrison et al., 1998).Data from the original publication are reproduced in F2. However, it should be noted that the maximum Wesenberg number (the product of steady strain rate andlongest Rouse relaxation time of the polymer) in thecomparisons was only approximately 6.0. Although thislarge enough for the polymer to achieve large fractiostretch, recent experimental and theoretical studies of dynamics of single polymer molecules at higher Wesenberg number show that the coil-stretch transition press takes a variety of detailed forms, many of which are well-represented by the pre-averaged dynamics of FENE-CR or FENE-P dumbbell models (Doyle anShaqfeh, 1998; Sizaire et al., 1999; Smith and Chu, 1998)Although alternative closure approximations have recenbeen proposed that appear to capture much of this cplexity (Lielens et al., 1998; Lielens et al., 1999), it isfair to say that an accurate model of reasonable coplexity for ultradilute solutions of flexible chain poly-mers is still a matter of research interest. In fact, elegexperimental studies of single DNA molecules in flo(Perkins et al., 1994; Perkins et al., 1997; Perkins et al.,1995; Quake et al., 1997; Smith and Chu, 1998) motivatfurther work in this area (Larson et al., 1999; Larson etal., 1997).

The only two examples in which there may be importatechnological interest in the dynamics of ultradilute pol

Fig. 1.The birefringent strand. Photograph of the birefringenpattern for a 100 ppmw solution of 2106 MW poly-styrene in Arochlor. The birefringent strand is centeralong the outflow axis from the central stagnation point a four-roll mill with a strain rate of 17.7 sec-1 [reference:Fuller and Leal, 1980].

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Non-Newtonian fluid mechanics for polymeric liquids A status report

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mer solutions are turbulent drag reduction and flow inporous media. Of course, both of these flows are far morecomplex than those discussed previously. It is known thatexceedingly small amounts of high molecular weight poly-mer (a few ppm by weight) are sufficient to produce amajor modification in certain turbulent flows (associatedwith a reduction in the apparent frictional drag at bound-

aries) (Berman, 1977; Den Toonder et al., 1997; Leal,1985; Oliver and Bakhtiyarov, 1983; Conference Proceings, 1977; Toms, 1977).

Small amounts of polymer are also known to be capaof producing substantial reductions in flow rate or increaes in the required pressure drop for a given flow rate flow through a porous media (Durst et al., 1981; Elata etal., 1977; Haas and Durst, 1982; James and McLar1975; Rodriguez et al., 1993). The latter occurs at flowrates that are clearly related to the onset of the large exsional viscosity associated with the coil-stretch transitioHowever, it is not known whether this effect occurs asconsequence of local stagnation point regions or asaccumulative effect of the extensional flow experienced ba fluid element as it passes through a series of “ throats”and “pores” in the porous matrix (Nollert and Olbricht1985).

Turbulent drag reduction is much more complex. It hgenerally been assumed that the polymer becomes histretched and then inhibits the extensional features of bursting process that is responsible for momentum traport between the wall and core regions of the flow. Thpicture is consistent with recent direct DNS simulationsturbulent drag reduction for a dilute solution of lineadumbbells (Dimitropoulos et al., 1998; Sureshkumar et al.,1997). In view of the exceedingly small concentrations thhave been observed to produce drag reduction, howeveis conceivable that the direct effect of polymer on the flois subtler and propagated to a large amplitude effect duthe complex dynamics of a turbulent wall flow. The recenumerical studies cited above show promise in predictand even understanding the effect. Nevertheless, it musstated that it remains to establish the robustness of thresults to details of the constitutive modeling. In a vecomplex, time-dependent flow, it is not obvious thqualitative changes in such model assumptions as spring law are unimportant. It seems possible to achinumerical drag reduction for reasons other than thothat actually occur in a very dilute polymer solution.

Apart from the two examples listed above, there are fapplications for which the fluid dynamics of ultradilutpolymer solutions play a role. Hence, though the dynamof these fluids is quite interesting from an intellectual poof view, their relative importance from a pragmatic pespective is certainly far less than more concentratedentangled systems.

4.2. Unentangled solutions−−−−The Boger or purelyelastic fluid

As concentration is increased, we encounter a classfluids that may still be dilute at equilibrium (in the sensthat the ratio c/c* < 1, where c* is the so-called overlapconcentration) but are sufficiently concentrated that chainteraction becomes important when the polymer is in

Fig. 2.A comparison of experimental and theoretical values ofbirefringence measured at the stagnation point of a two-roll mill. Case (a) is for a Weissenberg number of 3.2, andcase (b) is for a Weissenberg number of 6.0. Theoreticalvalues were obtained by simulation of the flow using theChilcott-Rallison model, with extensibility parameter L =50. The various curves correspond to different nonlinearspring laws as described in the original publication. Thepolymer solution was a 40 ppmw solution of narrowMWD polystyrene with MW = 3.4106 in a mixed tri-cresyl phosphate/polystyrene oligomer solvent. The valueL = 50, is obtained from the molecular weight assumingthat a statistical subunit corresponds to 15 monomer units.Time is scaled with the local strain rate, while the bire-fringence is scaled by 2πnc(n2 + 2)/3n2 (NA/MW)(µ1 - µ2),where n is the refractive index, c is the concentration in g/cm3, NA is Avogadro’s number, MW is the molecular weight,and (µ1 - µ2) is the difference in subunit polarizability par-allel and normal to the chain axis. Theoretical values arecalculated directly from the in-plane components of thesecond-moment tensor of the distribution function of theend-to-end vector [reference: Harrison et al., 1998].

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L. Gary Leal and James P. Oberhauser

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extended state. When the solvent is quite viscous, thesesolutions are generally called Boger fluids or purely elasticliquids.(Boger, 1977) The latter name comes from the factthat these fluids can exhibit quite substantial normal stressdifferences in simple shear flow but a constant shear vis-cosity (the shear viscosity being dominated by the viscosityof the solvent). In fact, the concentration range for Bogerfluids can sometimes be comparable to or even exceed c*by a factor of 2 or 3. This class of fluids has recently beenthe object of extensive investigation (Baaijens et al., 1995;Feng and Leal, 1997; Grillet and Shaqfeh, 1996; McKinleyet al., 1991; Purnode and Crochet, 1996; Purnode and Cro-chet, 1998; Satrape and Crochet, 1994; Singh and Leal,1994; Solomon and Muller, 1996a; Solomon and Muller,1996b; Verhoef et al., 1999; Yang and Khomami, 1999;Yong Lak and Shaqfeh, 1994). Driving this interest is thebelief that their rheology can be modeled by the dilutesolution FENE dumbbell models discussed in the previoussubsection, as well as the fact that they are relatively easyto work with in room temperature experiments. Anothermotivation is the belief (false, in our view) that these fluidscan provide understanding of the elastic properties of ageneric viscoelastic fluid, lacking only the complication ofshear or strain rate thinning of the viscosity that appearsin entangled polymer systems. We shall return to this pointlater in this subsection.

From a fundamental point of view, it seems obvious thatthe FENE dumbbell models should not apply to Bogerfluids (or any dilute solution that does not qualify asultradilute). As soon as interactions between polymermolecules occur, the basic assumptions underlying thesemodels, principally that the molecules are hydrodynami-cally and physically independent, break down, and weshould expect some type of failure of predictions based onthe FENE-CR or FENE-P models.

On the other hand, the FENE-CR model is nothing but adynamical approximation in which a polymer chain isrepresented by a vector whose length and orientation aredetermined by the flow. At this level, it is possible that amodel that simply purports to represent a polymer in termsof a single vector could apply equally well to ultradilutesolutions and Boger fluids, an idea that we shall explorelater. Nevertheless, as polymer interactions come into play,we may expect detailed aspects of the FENE-CR model tofail in some fashion. One possibility is that the materialcoefficients in the model may change relative to those forthe same molecular weight polymer and solvent in anultradilute solution. Another is that the physics of eitherchain dynamics in the flow or relaxation processes from anon-equilibrium state may change. If these changes are tobe approximated within the framework of dumbbell theory,it must be done either by changing the spring law and/orincorporating changes into the frictional interaction be-tween the polymer beads and suspending fluid.

Once the nonlinear dumbbell theory is converted into FENE-CR constitutive model, it is not necessary to restoneself to ultradilute solutions if we are willing to accemodel parameters or other modifications as ad hoc inputsin order to apply the model to a more general class of ids. Although it has not always been acknowledged, thisessentially the point of view that has been adopted in application of FENE-CR (or other) dumbbell theory Boger fluids and other polymeric solutions that are nstrictly ultradilute.

We shall see in the next section that there are aspecthe rheological behavior of entangled polymer liquids thcannot be captured even qualitatively by dumbbell-basdilute solution models. An obvious question is whether can identify physical phenomena in existing experimenon Boger fluids that cannot be explained via FENE-Cmodels. One clear indicator that the ultradilute FENmodels cannot be applied directly to Boger fluids is ththe model parameters necessary to match experimedata seem to be much different than would be applicafor the same polymer and solvent in the ultradilute solutilimit. We recently carried out one study using a dilusolution of monodisperse polystyrene as the Boger fluiddirectly explore this issue (Harrison et al., 1999). It wasshown via comparisons with birefringence and velocgradient measurements at the stagnation point of a rotating two-roll mill that a significant reduction in theextensibility parameter L is required as the concentration increased even to c/c* ~ 0.1. This is illustrated in Fig. which is reproduced from the original publication (Harison et al., 1999). Other studies of different flows havpreviously indicated that extremely small values of textensibility parameter are necessary to provide a qutative match to experimental data (Chilcott and Ralliso1988; Satrape and Crochet, 1994; Szabo et al., 1997).What is not clear at this point is whether the extensibilparameter can be treated as a material constant for a gpolymer/solvent system at a given concentration. In pticular, it is not known whether the same value of L willprovide adequate predictions in a variety of different flowIt is also unclear how or whether the need for reduced small values of L can be interpreted in a physical sensOne suggestion might be that the onset of flow-inducentanglements reduces the dynamically active entity fromwhole chain to the segment of a chain between entanments. The reduction of molecular weight, from the whochain to a segment, would be expected to reduce the exsibility. However, the hypothesis is purely speculative awithout foundation in fact at this time.

A related observation is that two different classes Boger fluids, one made from polyacrylamide in glucosecorn syrup and the other from PIB in polybutene oil polystyrene in styrene oligomer, exhibit fundamentaldifferent values of L for a given molecular weight

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Non-Newtonian fluid mechanics for polymeric liquids A status report

artan-e-

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(Chmielewski et al., 1990). In particular, whereas the poly-acrylamide solutions typically require values of L2 ~ 10(compared to ultradilute values of order L2 ~ 103), thevalues appropriate to PIB or polystyrene are often one ortwo orders of magnitude larger, even though the twopolymers may be of comparable molecular weight in sol-vents of comparable quality.†

Another unresolved question is the suggestion that thebead friction should be conformation-dependent (de Gennes,

1974; Fuller and Leal, 1981; Hinch, 1976). This proposalwas originally directed towards very dilute solutions as pof the effort to understand turbulent drag reduction (PhThien and Tanner, 1978). From a fundamental fluid mchanics perspective, there is no question that the strengthe frictional interaction between a polymer molecule athe solvent must depend upon the “shape” or conformationof the polymer chain. In the dumbbell theory, this must incorporated into the bead-solvent friction coefficie(Phan-Thien et al., 1984). However, the various coiled conformations are extremely “porous,” and it is not obvioushow strong this effect should be even though the visualchanges in shape can be quite dramatic (Carrington et al.,1997). Recent studies of single polymer chain dynamsuggest that the conformation dependence of bead fricfor ultradilute solutions is extremely weak (Larson, 199Conversely, flow data for more concentrated Boger fluiseem to suggest a much stronger effect. For example,birefringence measurements downstream of the stagnapoint in a cross-slot flow device show an extremely slorecovery toward equilibrium−slower by at least an order omagnitude than suggested by the measured relaxation at equilibrium (Miles and Keller, 1980). Recent comptational studies suggest that this may be explainedadding a conformation-dependent, scalar friction coeffi-cient to the basic FENE-CR model (Remmelgas et al.,1999b). This can be seen in Fig. 4, which is reproducfrom the paper of Remmelgas et al. (1999b). Anotherobservation is the accumulative stretch (and stress) obsed in Boger fluids that pass down a wavy-wall tube channel (Chin et al., 1989; Magueur et al., 1985; Nollertand Olbricht, 1985). In the absence of quite strong cformation-dependent friction, it can be shown that a pomer chain in a periodic extensional flow will exhibit a limitcycle with only very modest stretch (Szeri and Leal, 1996A large degree of chain stretch will occur in a periodic floonly if the friction increases quickly enough to producerelaxation time that increases with increasing chain stretch(recall that the effect of the nonlinear spring is in thopposite direction, contributing an increased relaxatrate as the chain extends).

Thus, it appears that, at least for some flows, a strodose of conformation-dependent friction is important modeling the flow behavior of Boger fluids using dumbbemodels. It is not clear to us why the conformation-depeent friction (CDF) effect should be enhanced in Bogfluids relative to ultradilute polymer solutions or whetheits inclusion for a particular polymer/solvent combinatiowill enhance the quality of model predictions for all flowor just some specific flows (Singh and Leal, 1996). Givthat the FENE-CR model is being applied well beyond strict range of validity in the case of Boger fluids, featursuch as CDF remain as ad hoc additions. Unless its uni-versality is proven by extensive comparison of mod

Fig. 3.A comparison of experimental and theoretical values ofbirefringence measured at the stagnation point of a two-roll mill. Details are the same as in Figure 2 except thatthe polymer concentration is 520 ppmw, which corre-sponds approximately to c/c* = 0.1. Theoretical curves arefrom the Chilcott-Rallison model with the inverse Lan-gevin spring law (denoted CR-IL in Figure 2), and twovalues of L = 50 and 32.2. In the original paper, com-parisons are also made between theoretical and experi-mental values of the velocity gradient, which is signifi-cantly changed at this polymer concentration from itsvalue for a Newtonian fluid [reference: Harrison et al.,1999].

†Aseparate issue is that it appears to be difficult (or impossible) toobtain medel parameters entirely from shear flow rheological data.This is not surprising in view of limtations discussed earlier fordumbbell models in simple shear flow. Two indications of the trou-ble are: (1) the non-uniqueness ofL identified by Starape and Cro-chet (1994), and (2) the multi-mode fits to experimental data, whichproduce a first mode with a very small L and large relaxation time τand a second with large L and small τ (Oztekin et al., 1994).

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L. Gary Leal and James P. Oberhauser

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predictions with experimental observations, the applica-bility of CDF must be assumed to be flow-type dependent.

Finally, we may return to the question of technologicalrelevance in the extensive studies of models for Bogerfluids in non-Newtonian fluid mechanics. Since Bogerfluids find relatively few direct applications, the answermust rest on the extent to which the methodology ofanalysis and the understanding of flow behavior carry overto the more common entangled polymer solutions andmelts. Our personal view is that the FENE-CR and relatedmodels have provided a useful ground for the developmentof computational techniques for viscoelastic flow prob-lems. Indeed, the fact that very strong, direct effects of thepolymer on the flow tend to occur in localized regions (forreasons similar to the localization of chain configurationchanges in ultradilute solutions) means that these flowproblems are almost certainly more difficult to solvenumerically than entangled systems, where the effect of thepolymer on the flow is felt more or less smoothly through-out the flow domain. Of course, the constitutive modelswill tend to be more complicated in the latter case, but atthis point we are just discussing issues of spatial resolutionand numerical convergence.

However, as a basis to develop understanding of vis-

coelastic flow phenomena, it is our view that the FENE-Cand Boger fluid studies are only marginally useful if thultimate goal is fluid mechanics for entangled solutionsmelts. There is a fundamental difference in rheologiproperties for these two classes of fluids. From a modepoint of view, both exhibit flow-induced orientation anstretching of polymer molecules or segments of molecuYet the fundamental distinction is that Boger fluids another unentangled solutions possess relaxation time scfor chain stretch and chain orientation that are comparawhereas entangled solutions and melts typically hawidely separated time scales. Thus, for the entangled tems, there is a wide range of flow conditions where material is characterized by a high degree of chain oentation with little or no stretch. In most flows, such fluidexhibit shear thinning−a decrease in the viscosity withincrease of strain rate. Shear thinning is only curtaiwhen chain stretching commences, leading to strain hardening−an increase in the viscosity with increase strain rate. On the other hand, for Boger fluids and otunentangled fluids, chain stretch and chain orientattypically occur under the same flow conditions. In thcase, the material rheology is dominated by chain streing and is characterized by strain rate hardening for mflows. Indeed, a signature of the onset of a significant leof entanglement with increasing polymer concentrationmixed-type flows is the transition from a monotonicalincreasing viscosity with strain rate to one with a larregion of pronounced strain rate thinning prior to strain rhardening (Harrison, 1997). Because of these fundamedifferences in the behavior of entangled and unentangpolymer solutions, it is difficult to translate much of thunderstanding of Boger fluid behavior to the more comon and important class of entangled polymeric liquid

5. Current status−Entangled solutions and melts

Since the original work of de Gennes (1971) and escially Doi and Edwards (Doi and Edwards, 1978a; Doi aEdwards, 1978b; Doi and Edwards, 1978c; Doi aEdwards, 1979; Doi and Edwards, 1986) twenty to thiyears ago, there has been tremendous progress in unstanding the behavior of entangled polymer solutions amelts. The Doi-Edwards reptation theory applies to lineflexible polymers and has had some remarkable succesas we shall briefly summarize below. Furthermore, its mrecent extensions to branched polymers (either “star-like”or the so-called “H” or “pom-pom” architectures) have inmany ways been even more successful in describing nlinear behavior.

5.1. Reptation models for entangled polymersThe basic concept of visualizing the effects of enta

glements on the dynamics of a test chain in terms of a tube

Fig. 4.Predicted values of the fractional extension of the dumb-bell model with several different bead friction laws alongthe centerline of the exit channel for a cross-slot flowdevice. The velocity field is Newtonian. The dumbbellconfiguration is obtained by numerical solution of thegoverning equation for the configuration tensor, A, (i.e.the second moment of the distribution function for theend-to-end vector), with an extensibility parameter L =100. The data is from Miles and Keller (1980), obtainedusing a Boger fluid. FENE-CR is the Chilcott-Rallisonmodel with constant bead friction. FENE-ML is the samemodel with a nonlinear spring-law suggested by Magdaand Larson (1989). FENE-CD uses the conformationdependent friction law of de Gennes and Hinch (1976)[reference: Remmelgas et al., 1999b].

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Non-Newtonian fluid mechanics for polymeric liquids A status report

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of constraints is no doubt familiar. It provides an elegantyet simple picture that yields a number of important pre-dictions with only three free parameters. One is the plateaumodulus of linear viscoelasticity, which relates directly tothe tube radius. The other two are relaxation times−namelythe longest Rouse time, τR, and the disengagement orreptation time, τd. The physical significance of these timescales will be discussed shortly.

It is convenient to model the chain in the tube as con-sisting of a series of segments between entanglements,each of which has a Gaussian distribution of end-to-endlengths at equilibrium. Hence, changes in chain configu-ration are described in terms of the orientation and lengthof these chain segments, though both types of deformationare ultimately caused by reorientation of the chain at theshorter scale of individual Kuhn steps. According to theoriginal Doi-Edwards (DE) theory, relaxation of stretch ofchain segments occurs on the Rouse time scale and isunimpeded by the tube of constraints. On the other hand, anon-equilibrium distribution of chain segment orientationscan only return to equilibrium as the test chain escapes itstube of constraints via the longitudinal diffusion processknown as reptation. In the DE theory, this process isassumed to occur in a tube whose radius is the same asexisted in the undeformed material (i.e., the entanglementmatrix is unchanged by flow). As a result, the relaxationtime τd exceeds that for chain length relaxation by a factor3Ne (where Ne corresponds to the number of entanglementsper chain). Since Ne may be large, τdτR for highly entan-gled systems (i.e., systems with large molecular weight andhigh concentrations if the liquid is a solution rather than amelt). A second consequence of the assumption of a fixedentanglement structure is that the relaxation of chain seg-ment orientation occurs from the ends of the chaintowards the center.

A key feature of the original DE theory is the assumption(based on the fact that τdτR) that chains retract instan-taneously from any stretched configuration. Hence, allmeas- urable polymer contributions to stress derive fromnon-equilibrium chain segment orientation distributions,and the corresponding relaxation toward equilibrium iscompletely described by the reptation process. Put anotherway, the effective initial chain configuration followingdeformation is not that associated with affine motion of theentanglement points, which would be described by theFinger strain tensor, but only the deformation remainingafter the chain has retracted to its equilibrium length(described by the DE universal strain tensor Q).‡

The DE model in this original form leads to a number of

impressive predictions (Doi and Edwards, 1986), incluing: (1) proper linear viscoelastic behavior for frequencup to and including a molecular weight independent pteau value for the storage modulus G*(ω); (2) a nearlycorrect scaling dependence of the largest relaxation tand the zero shear viscosity on molecular weight (the ter prediction being η0 ~ M3); (3) a nonzero second normal stress difference in steady shear with the correct sand at least approximately correct magnitude; (4)molecular weight independent relaxation modulus G(and (5) a quantitative fit to nonlinear step strain relaation data.

Although conceptually simple, even this original Dmodel is formidable as a basis of calculating stress afunction of time at a large number of material points required for any nonhomogeneous flow calculation (vHeel et al., 1999). Unfortunately, there are still a number features in the DE model that require improvement, eitto repair predictions that are clearly spurious or to increthe range of flows and materials to which the theory mbe applied. Fundamentally, the dynamics of changespolymer chain configuration can be thought of as a copetition between flow-induced changes in the chain configuration (determining the appropriate strain tensor) arelaxation processes that tend to drive the configuratback to an equilibrium state. Considerable effort has bmade to understand and modify both of these compone

If we begin by considering the mechanisms for relaation, the most obvious deficiency is that the reptatimechanism does not apply to many branched systemsillustrate the point, let us focus our attention on polymewith a “star” architecture−namely, a number of branches oarms emanating from a single common point with tbranch molecular weight being large enough to be stronentangled. If reptation were the only mechanism frelaxation of non-equilibrium configurations, a system star molecules could not relax at all. Since this is obviouuntrue, additional relaxation mechanisms must be presRecent work has shown rather conclusively that one smechanism corresponds to spontaneous retraction andtension of the arms, often called chain length fluctuatio(Milner and McLeish, 1997; Milner and McLeish, 1998aPearson and Helfand, 1984). These allow the armsescape the tube of constraints; however, the relaxation tfor this process tends to be very large. An arm only copletely relaxes from a flow-induced deformation whenhas completely retracted, the probability of which is lo(and decreasing rapidly with increasing length, or moleular weight, of the arm). Thus, relaxation times are tyically very large compared to the reptation times for linechain polymers. Recent theoretical studies (Inkson et al.,1999; McLeish and Larson, 1998) have introduced a vpromising model (the “pom-pom” model) for branchedpolymers with an H-type architecture, following earlie

‡The original DE model used the independent alignment (IA) approx-imation for the Q tensor in order to simplify the mathematics. IA hasbeen shown to introduce errors in certain flow conditions. Unless oth-erwise noted, references to the Q tensor in this paper correspond to itsexact version.

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L. Gary Leal and James P. Oberhauser

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work on star polymers. The spectacular success of themodels for branched systems not only provides a frame-work from which simplified macroscale constitutive mod-els may be derived for this class of polymeric liquids (apoint to which we will return later), but it also lends nearlyincontrovertible evidence supporting the basic reptationpicture of entangled polymers.

Application of the same arm retraction ideas has alsobeen pursued recently for linear chains, viewed as “ two-arm stars,” by Milner and McLeish (1998b) and muchearlier by Doi (Doi and Edwards, 1986). The inclusion ofchain length fluctuations in the basic DE theory for lin-ear polymers produces a change in the scaling of the zeroshear rate viscosity from the basic DE result (η0 ~ M3.0) toa result consistent with experiments (η0 ~ M3.4) for poly-mers with a large, but not asymptotically large, number ofentanglement points per chain (Doi, 1983).

A second limitation of the original DE theory is theimplicit assumption of a monodisperse molecular weightdistribution. It is manifested in the model via the assump-tion that the constraints formed by other polymer chainsare fixed and can only be released via reptation of the testchain. In reality, however, all of the polymer molecules arediffusing simultaneously, permitting entanglements to belost either by reptation of the test chain or by reptation ofthe chain responsible for the constraint. The latter processis known as constraint release. Constraint release becomesmore important with increasing polydispersity of themolecular weight distribution. For example, a constraintassociated with a relatively short chain is more likely to bereleased by means of its own diffusion since the diffusivityscales as M-2. A generalization of the basic reptation theoryto account for these diffusion-based constraint releaseprocesses has been developed by several researchers (Doiet al., 1987; Rubinstein and Colby, 1988; Viovy et al.,1991). A much simpler model, known as double reptation,was proposed by des Cloizeaux (1988, 1990a, 1990b).Recent work Milner (1996) has shown that the latter is arational approximation of the more general (and compli-cated) model of Viovy, et al. (1991) and demonstrated theaccuracy of the double reptation model in describing theeffects of polydispersity on the linear viscoelastic spectrumfor linear polymers (Mead, 1994; Mead, 1996). To date, anextension of the theories of branched systems to a poly-disperse molecular weight distribution has not been pub-lished. It is also unclear how the double reptation conceptwould translate to the nonlinear regime where, as we shallsee shortly, additional mechanisms of relaxation are believ-ed to play an important role.

A third limitation of the original DE theory is that noexplicit account is given to the relaxation of chain stretch.By assuming that this process is always infinitely fastcompared to relaxation of segmental orientation by rep-tation, the DE theory effectively assumes that chain

deformation occurs without stretch. Consequently, the stmeasure that appears in the theory is the tensor Q ratherthan the Finger strain or other more familiar formAlthough the time scale for relaxation of chain stretchsmaller than that for reptation by a factor of 3Ne, it is nottruly zero. Thus, it is clear that the DE theory will fail foflows in which the dimensionless velocity gradient, τR, isof order unity or greater. A theory that explicitly accounfor polymer chain stretching has been proposed by Mrucci and Grizzuti (1988) (the DEMG model), and it habeen further developed and analyzed by other researc(Mead and Leal, 1995; Mead et al., 1995; Pearson et al.,1991). This theory improves the predicted response tostartup of simple shear flow by adding overshoots in first normal stress difference N1 (Pearson et al., 1991).However, it includes a pre-averaging approximation in tdescription of the friction between the tube and the tchain that does not allow stretching modes with lengscales shorter than the entire chain. As a result, the DEmodel does not improve the high frequency predictioin the linear viscoelastic regime (Mead and Leal, 199Furthermore, there is an indication in current studies tthe DEMG model overpredicts the degree of chain streting in extensional or strong flows (Yavich et al., 1998).This result is unsurprising when one considers that stretching portion of the model is ad hoc and assumes thathe frictional interaction, caused by relative motion btween the test chain and the tube wall, is independenany flow-induced changes in the chain configuration.

With all or even a subset of the model extensiodescribed above, the original DE model becomes extrely complex. Yet it still exhibits some serious shortcominwhen compared with experimental data. Perhaps the known and important problem is the fact that the viscosin simple shear flow (or other 2-D flows that are smaperturbations from simple shear flow) decreases with shrate faster than .# As a result, the shear stress is prdicted to have a local maximum at ~τd

-1 and to decreasewith increasing shear rate thereafter (Doi and Edwar1986). Experimental data on entangled polymeric liqu(Ferry, 1980), both direct and indirect (via birefringenusing the stress-optical relationship) and for both podisperse and nearly monodisperse polymers (i.e., Mw/Mn < 1.05), disagree with this result, instead yieldinη ~ . An example of the comparison between data amodel predictions is shown in Fig. 5. The existence of sucha stress maximum has been proven for entangled, wormlike micelle solutions, which also show macroscopic flomanifestations of the non-uniqueness of stress versus srate that are known as shear-banding (Britton and Cal-laghan, 1999; Britton et al., 1999; Callaghan et al., 1996;Mair and Callaghan, 1996; Mair and Callaghan, 199

γ·

γ·1–

γ·

γ· 0.8–

#In simple shear flow, the viscosity decreases as for .γ· 3– 2⁄ γ· τd1–>

12 Korea-Australia Rheology Journal

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Non-Newtonian fluid mechanics for polymeric liquids A status report

e.

ill

8;9).

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Makhloufi et al., 1995). No such flow effects are observedfor polymeric liquids, again confirming that η decreaseswith shear rate more slowly than for all .

A number of mechanisms have been proposed to explainwhy the DE or DEMG models, with or without chainlength fluctuations and polydispersity, exhibit shear thin-ning at an excessive rate. All of these mechanisms have incommon the fact that they involve changes in the con-figuration state directly caused by the flow. One set ofproposals involves changes in the strain measure from Q toone that assumes that the tube radius (and/or tube cross-sectional shape) is changed by the deformation process sothat the test chain retains some residual stretch from theinitial affine deformation, even for <τR

-1 (Marrucci andde Cindio, 1980; Wagner, 1994; Wagner, 1997; Wagnerand Schaeffer, 1992).* A second fundamentally differentset of proposals is that the flow contrib utes to modifiedrelaxation processes. There are reasons to believe that bothof these proposals should be incorporated in some form.

Let us first consider the effect of deformation of the tubIf we begin with a tube of length L0 and a circular cross-section of radius a0 (at equilibrium), a simple affine defor-mation of points on the tube walls in simple shear flow wconvert it in time t to a tube of length L0|E(t)·u| with anelliptical cross-section (Ianniruberto and Marrucci, 199Marrucci and Ianniruberto, 1999; Mhetar and Archer, 199Here, E(t) is the deformation gradient tensor and u is a unitvector that specifies the orientation of a tube segment (and Edwards, 1986). The result is a dependence of sstress on shear rate that is not only monotonic, but in increases at a rate in excess of that predicted (from linelasticity) by the Cox-Merz rule. Another major changoccurs in the low shear rate limit for the ratio of N2/N1 insimple shear flow, from -1/7 for the DE or DEMG modto -0.46 for the affine elliptical cross-section. By comparison, experimental studies indicate that N2/N1 shouldbe approximately -2/7 (Brown and Burghardt, 1996; Browet al., 1995), a value between the two extremes. A simproposal is for the tube cross-section to remain circular for the diameter to decrease as the tube is stretched, mtaining a constant tube volume (Marrucci and de Cind1980; Wagner, 1990; Wagner and Schaeffer, 1994). Thothis produces a dependence of shear stress on sheathat is closer to satisfying the Cox-Merz rule, the ratio N2/N1 decreases in magnitude to a value of only -1/14(Marrucci and Ianniruberto, 1999). Although the individuvalues of N2 and N1 will change as the assumptions aborelaxation processes are modified, the ratio N2/N1 can onlybe altered via modification of the tube deformation proess, suggesting that a change in the strain measure is etial. Hence, the experimental value of this ratio seemsimply that the tube must deform into an elliptic cross-setion but with a weaker than affine deformation proce(Marrucci and Ianniruberto, 1999). There are several pathat could lead to such a result. We believe that one candidate that has not yet been incorporated is the effetube dilation due to chain alignment.

Although the additional stress produced by tube defmation can potentially cure the problem of excessive shthinning, birefringence measurements suggest that this mat best, be only a partial remedy. These measuremshow that the DE and DEMG models strongly overpredthe rotation of the chain toward alignment with the flowdirection as the strain rate is increased (Yavich et al.,1998), and the inclusion of tube deformation only makthis deficiency worse. In steady shear flow, this patholocauses the DE and DEMG models to produce too lichain stretch and a smaller than realistic contribution of polymer to the shear stress (i.e., excessive shear thinnConsequently, though it is likely that some form of tubdeformation must occur, it is clear that the predicted tedency of polymer molecules to rotate too strongly into tflow direction will only be cured by an additional mech

γ·1–

γ·

γ·

Fig. 5.A comparison of the experimental steady-state dimen-sionless shear viscosity η/η0 versus dimensionless shearrate (where τR is the longest Rouse time for this fluid).The experimental fluid is a 3.0 wt% solution of 8.42106 MW polystyrene in TCP. The theoretical points arecalculated using the Doi-Edwards model incorporatingonly segmental stretch in the basic model via the DEMGapproximation (the latter does not actually change the pre-dicted shear viscosity, which is identical to values for theoriginal DE model). Model parameters for the DEMGmodel calculations are τd = 78.6 sec. (the reptation time),Ne = 10 (the number of entanglements per chain), andnt = 8420 (the number of Kuhn steps), estimated via linearviscoelastic and other rheological data [reference: Yavichet al., 1998].

*These proposals are distinct from the sustained chain stretch that theDEMG model attempts to describe. The latter is viewed as being pro-duced by frictional interactions between the test chain and the walls ofthe tube within a tube whose cross-sectional area (and shape) is un-changed by the deformaion process. Since chain stretch relaxes on therelatively fast time scale τR and the steady-state length of a relaxedchain within a tube with an equilibrium radius is just the equilibriumlength scale L0, stretch can only be sustained by this mechanism if

.γ· τR1–>

Korea-Australia Rheology Journal March 2000 Vol. 12, No. 1 13

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L. Gary Leal and James P. Oberhauser

is- oftalvery

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anism (or mechanisms) that enhances the orientationrelaxation process.

Such a mechanism has been proposed by Marrucci andco-workers (Ianniruberto and Marrucci, 1996; Marrucci,1996) dubbed convective constraint release (or CCR). Thebasic idea is that relative motion exists between the testchain and the tube (or more precisely, the chains thatproduce the entanglement constraints that define the tube)in a flow. This convective motion is due to the fact that thetest chain either does not deform affinely in the flow oreven retracts, causing entanglement constraints to be lost ata rate that increases monotonically with increasing strainrate. In a simplified version of this basic idea, Iannirubertoand Marrucci (1996) suggested that CCR could be incor-porated into the basic DE model via a modified reptationtime of the form:

, (4)

where:

(5)

is the relative velocity between the tube and chain eval-uated at the end of the chain (Doi and Edwards, 1986).Here, τd,o is the disengagement time in the absence of CCR,and β is an ad hoc parameter. A value of β2.8 eliminatesthe shear stress maximum from the basic DE (or DEMG)model. A smaller value would be required if some versionof tube deformation were included in the model since thiscan also reduce or eliminate the stress maximum on itsown. One feature of CCR given by Eq. (4) is that it has itsstrongest effects (in 2-D flows) for purely extensionalhyperbolic flow and its weakest in shear flow, where

as the tube and test chain become increas-ingly aligned in the flow direction. With CCR included,however, the chain never becomes fully aligned in simpleshear, even as . Rather, a delicate balance is main-tained between increasing (and thus decreasing τd) andthe increasingly strong tendency of the flow to promoteperfect alignment (Ianniruberto and Marrucci, 1996).

A more complex, and in some ways qualitatively dif-ferent, implementation of CCR was published recently byMead, et al. (1998) In this case, a version of CCR isexplicitly incorporated into the DEMG framework. Inaddition, the influence of CCR is split between relaxationof chain segment orientation and stretch in such a way thatat low strain rates it acts solely on the relaxation of chainorientation. As , a transition occurs by means of a“switch function,” and CCR is applied preferentially toenhanced relaxation, of chain stretch. Finally, at yet higherstrain rates, the effect of CCR on chain stretch is alsoreduced (it is inversely proportional to the contour lengthof the tube), leading ultimately to stretch at the same rate

as in the basic DEMG model but shifted to higher Wesenberg numbers. Although both of the implementationsCCR are largely untested by comparison with experimendata, the basic concept of CCR appears to be another important addition to the basic DE or DEMG model.

Although great progress has been made in termsidentifying the important mechanisms of polymer chadeformation and relaxation, it is our personal view that versions of the reptation model are too complex to rotinely serve as a basis for computations of complex vcoelastic flows. Consequently, we believe that constituttheories that retain the critical physical features of themodels but which are greatly simplified in detail will be necessary tool in the future of non-Newtonian flumechanics. It is essential to the development of thsimplified theories, however, that a comprehensive evuation is made of all of the various proposed modificatioof the basic reptation theory. We discuss some compariswith experimental data from our own lab in the next setion, and speculate briefly on their implications for thfuture development of both comprehensive and simplifimodels for entangled polymers.

5.2. Comparisons with experimental data A number of publications have discussed comparisons

reptation model predictions with rheological and rheoptical experimental data for entangled polymer solutioand melts (Doi and Edwards, 1986; Larson, 1988; Lars1999). Rather than repeating these assessments, wefocus on additional issues arising from studies in our oresearch group.

The experimental studies in our group have focusprimarily on rheo-optical measurements for monodispepolymer solutions that are relatively lightly entangled (125 entanglements per chain). A number of flow historiestwo flow geometries have been considered: (1) transishear flow−including startup from rest, relaxation, and stechanges in shear rate (increase, decrease, and reversin−ina narrow gap Couette device; and (2) steady state, stafrom rest, and relaxation in planar extensional and mixetype (“strong”) flows produced in both four-roll and corotating two-roll mills. In the Couette device, the flow not modified by the polymer, allowing one to directlpredict it from knowledge of the flow cell geometry anthe angular velocity history of the rotating cylinder. In thtwo- and four-roll mill, measurements were mainly takenthe stagnation point and include both birefringence assess the polymer chain configuration) and dynamic liscattering (to obtain information on the velocity gradienThe latter is essential since the flow is nonhomogeneand can be profoundly modified relative to its form forNewtonian fluid. To assure proper comparison with theothe measured velocity gradient history is used as inputhe particular version of the reptation model under co

1τd---- 1

τd 0,------- β v s t,( )⟨ ⟩

s L2---=

+=

v s t,( )⟨ ⟩ ∇V : uu⟨ ⟩ 13---l–

0

s

∫=

v s t,( )⟨ ⟩s L

2---=

0→

γ· ∞→γ·

γ· τR1–→

14 Korea-Australia Rheology Journal

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Non-Newtonian fluid mechanics for polymeric liquids A status report

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sideration. The model is then used to predict the bire-fringence and orientation angle for comparison withmeasured data. Of course, this does not constitute a com-plete test of the model, because it does not determinewhether the model would be able to predict the flow.Nevertheless, we believe that it is a useful initial step. Thedata and model comparisons that we will discuss areabstracted from the Ph.D. work of Yavich (Wang et al.,1994; Yavich, 1995; Yavich et al., 1998), Harrison (Har-rison, 1997), and Oberhauser (Oberhauser, 2000; Ober-hauser et al., 1998; Oberhauser et al., 2000a; Oberhauser etal., 2000b). Several other papers are in various stages ofpreparation from the latter two theses, and a more detaileddiscussion of the work will appear there.

From this collected work, we can reach a number ofinteresting conclusions. First, in the steady shear andnear-shear flows, the tendency of the flow to rotate thepolymer chain away from the principle axis of the rate of

strain tensor and toward the outflow axis (i.e., the orietation of the principle eigenvector of the velocity gradietensor) is overpredicted by both the DE and DEMG mod(Yavich et al., 1998). A comparison of data and predictionfor simple shear and a mixed-type flow, with flow typparameter α = 0.15, is shown in Fig. 6 (where α = (|E |-|ΩΩΩΩ|)/(|E| + |Ω|), and |E| and |Ω| are the magnitudes of therate of strain and vorticity respectively). This is a consquence of the fact that the relaxation rate of segmenorientation is underpredicted by the reptation time τd,o.Indeed, for moderate to large velocity gradients, our dfor simple shear flow is quantitatively consistent with thCCR concept in the form of Equation (4) with a value β ≈ 3. However, overprediction of segmental orientatioalso occurs at small velocity gradients, a regime in whCCR should have a negligible effect. For the lightentangled systems that we have studied, further analysthe data suggests that relaxation due to chain length ftuations plays an increasingly important role (in a relatsense) at low strain rates (Milner and McLeish, 1998b

A second feature of the experimental data is that magnitude of the difference in measured and predicteorientation angle is largest for simple shear flow and dcreases as the flow becomes increasingly extensio(Yavich et al., 1998) (compare the results for simple sheand for α = 0.15 in Figure 6). Of course, in a purely extesional flow, the principle axis of the rate of strain tensand the outflow axis are coincident. Thus, the orientatangle is fixed at the angle of the outflow axis, and tpossibility that experiments and model can deviate froone another is eliminated. However, our data indicate tthe orientation angle is already reasonably well predicby the DEMG model without any form of CCR for a 2-Dflow with a flow type parameter as small as α = 0.15.Furthermore, for both α = 0.15 (Yavich et al., 1998) andpurely extensional flow (α = 1) (Harrison, 1997), thebirefringence is quite accurately predicted by the DEMmodel without CCR, at least for WiRO(1).

If we compare the measured orientation angle for steshear flow with predictions from the DEMG model, wcan determine an ad hoc value for τd at each shear rate sucthat the model gives the experimental value of the anPlotting these ad hoc values of τd versus shear rate, we sea plateau region at low shear rate in which the ad hoc τd islower than its predicted linear viscoelastic value (presuably due to chain length fluctuations as indicated abovHowever, the ad hoc τd eventually decreases with increasing shear rate at a slope that is consistent with the splified formula for τd given by Eq. (4) with β ≈ 3. On theother hand, if we make a similar comparison for the mixetype flows (i.e., flows with 0 <α < 1), we find that β mustdecrease as the flow becomes more extensional. This isto the fact that CCR, as given by Eq. (4), becomes stronin more extensional flows for a given value of β, while the

Fig. 6.A comparison of measured and predicted orientation anglesfor: (a) a mixed-type flow produced in a two-roll mill withflow type parameter α = 0.15, and (b) simple shear flow.The fluid for the shear flow data is described in Figure 5.The fluid for α = 0.15 is 0.076 g/cm3 of 2.89106 MWpolystyrene in a mixed solvent (toluene and polystyreneoligomer). The theoretical predictions are obtained usingthe DEMG model with parameters obtained via independ-ent rheological measurement [reference: Yavich et al.,1998].

Korea-Australia Rheology Journal March 2000 Vol. 12, No. 1 15

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L. Gary Leal and James P. Oberhauser

in00;

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measured effect on the orientation angle gets weaker.Another comparison that we have made with model

calculations is the Weissenberg number dependence ofchain stretching in the steady mixed-type and purelyextensional flows. Measured and predicted values of thebirefringence are shown in Fig. 7 for a polystyrene solu-tion in the two-roll mill. If we compare the measuredbirefringence with that predicted by the DEMG model inthe range of strain rates where the birefringence exceeds itsplateau value, we see that the predicted values increasemuch more rapidly with strain rate than the measuredvalues. Applying the CCR model of Mead et al. (1998)improves the prediction of chain stretch in the sense that itdelays the onset to larger WiR. However, it seems likelythat the frictional interaction between the tube and the testchain is too strong in the model of Marrucci and Grizzuti(used in both the DEMG model and the CCR model ofMead et al.)

Finally, we have compared measured and predictedbirefringence and orientation angle in several transientflows. In startup of purely extensional flow, Harrison(1997) performed model calculations using the DEMGmodel with measured values of the velocity gradient asinput. An example of his data and model predictions isshown in Fig. 8. We have already noted that DEMG givesnearly quantitative agreement for steady birefringence inthe same flow for WiRO(1). Therefore, it is not a greatsurprise that the transient predictions also agree well exceptfor those regions where the predicted birefringence appar-ently reflects greater transient stretch than appears exper-imentally, further evidence of problems with the chain

stretching portion of the DEMG model. Transient experiments for startup and step changes

shear rate in the Couette flow cell (Oberhauser, 20Oberhauser et al., 1998; Oberhauser et al., 2000b) havealso been compared with model predictions (Oberhauet al., 2000a; Oberhauser, 2000) using the basic DEMmodel, the basic DEMG model with the ad hoc modifi-cation of τd (chosen to match the steady state orientatangle data for simple shear flow) and the CCR modelMead et al. (1998) . A typical set of results is shown in Fig9. It can be seen that the basic DEMG model with theadhoc τd gives excellent quantitative agreement with meaured orientation angle data for both startup and for schanges in shear rate that involve a reversal of shdirection. In the startup flow case, it correctly exhibits thundershoot in orientation angle for WiRO(1) as has beenreported elsewhere (Oberhauser et al., 1998; Zebrowskiand Fuller, 1985), though the magnitude of this undershois smaller than measured for the specific case shownFig. 9. In contrast, the DEMG model using the linear vcoelastic value of τd produced very poor results for both thtransient behavior and, as already noted, the steady orientation angles. The model of Mead et al. (1998) showssome improvement over the DEMG model but is stunsatisfactory for these particular flows, though it shoube noted that the comparison was made without includchain length fluctuations.

For step changes in shear rate without flow reversal,experiments show that the orientation angle undergoe

Fig. 7.A comparison of measured birefringence with values pre-dicted by the DEMG model and the Mead et. al. (1998)CCR model. The data is for flow in the two-roll mill withflow type parameter α = 0.15, and the fluid is the 2.89106 MW polystyrene solution described in Figure 6. It canbe seen that the predicted birefringence increases muchtoo rapidly with strain rate (i.e., with Weissenberg num-ber) for WiRO(1), where chain stretching becomesimportant.

Fig. 8.A comparison of measured and predicted birefringenvalues for startup flow in a four-roll mill. The data is measured at the central stagnation point for a Weissenbnumber of 0.7 (based on the longest Rouse time). Tfluid is 10 vol% polyisoprene with MW = 1.15106 andMW/Mn = 1.023 in squalene. There are approximately entanglements per chain at this concentration, and longest Rouse time is 0.076 sec. The calculations wmade using these parameters in the DEMG model andmeasured transient strain rate at the stagnation point offlow as input data. We estimate the number of Kuhn steas 1700.

16 Korea-Australia Rheology Journal

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Non-Newtonian fluid mechanics for polymeric liquids A status report

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short-time scale transient in which the angle moves inthe opposite direction of its ultimate steady state value(Oberhauser et al., 1998). For example, if the shear rate isincreased, the steady state orientation angle shifts towardthe flow direction, but the initial change is away from theflow direction. Our early studies based on the DE andDEMG models with τd obtained directly from linear vis-coelastic data, did not show this effect. However, if we usethe DEMG model with the ad hoc reptation times cor-responding to the shear rates before and after the step,we find an initial transient that is similar to the one seenexperimentally but with an overly large amplitude and timeperiod. Interestingly, if care is taken in performing the

numerical calculations, a very small transient is found toexist even in the most basic DE model without chainstretching, CCR, or chain length fluctuations. It wouappear that the effect is inherent in the universal strtensor Q and is simply amplified in the DEMG modeusing the ad hoc τd values. This result contradicts claimby other researchers that it is necessary to include chlength fluctuations (Mead et al., 1998) or use a stochastiDEMG model with CCR (Hua et al., 1999) in order tomanifest this characteristic. On the other hand, incorrating of CCR and/or chain length fluctuations into model instead of simply changing τd in an ad hoc way, aswell as adding other mechanisms of relaxation or tudeformation, is apparently necessary to achieve quantive prediction of this phenomenon.

Finally, we close this section with a few comments abopossible areas for further improvement of reptation modFirst, though CCR as currently formulated seems to capable of eliminating the problem of excessive thinningthe viscosity in simple shear flow, it seems clear to us tone should also account for tube deformation (i.e., somemodification of the universal strain tensor Q). In fact, it islikely that the shift from the DE prediction of tothe observed rate of shear thinning is due to a sof these two effects. Wagner and co-workers (Wagn1990; Wagner and Schaeffer, 1994) have suggested deformation constrained by the condition of constant tuvolume can explain some of the minor discrepancbetween the measured and predicted shape of the damfunction for a step-strain deformation. However, the cculations of Ianniruberto and Marrucci (Ianniruberto anMarrucci, 1998; Marrucci and Ianniruberto, 1999) dicussed earlier indicate that neither a fully affine defomation nor a constant tube volume assumption is sufficito satisfy the Cox-Merz rule or give a correct value of N2/N1 at low shear rate. We would suggest that tube dilateffects, which will become increasingly important as tdegree of flow-induced alignment increases, must moerate the affine deformation of the tube. Tube dilation ha role not only in the deformation of the chain, but althrough an effectively increased tube radius (i.e., a decreanumber of entanglements per chain). It is still uncleprecisely how tube dilation will appear in the model fomulation, especially when combined with other mechnisms like chain length fluctuations and some versionCCR. Nevertheless, it is likely to improve agreement wthe Cox-Merz rule at the very least. In summary, it is clethat the choice of a proper strain measure remains an oproblem in reptation theory.

Another issue that we consider unresolved is the precform that convective constraint release should take. In sof the existence of a very elaborate model for CCR (Meet al., 1998), we believe that both of the existing modeare predicated on strong, but ad hoc, assumptions, and tha

η γ·3 2⁄–

∼η γ·

0.8–∼

Fig. 9.A comparison of experimental measurements and modelpredictions of the orientation angle for startup and a stepreversal of steady shear flow. The experimental fluid is a4.8 wt% solution of polystyrene in tricresyl phosphate with20 entanglements per chain and a Rouse time of 4.95 sec.The steady shear rate is 0.2 sec-1 in both flows. Theoreticalcalculations are for the DEMG model with the linear vis-coelastic estimate of the reptation time; the DEMG modelwith the ad hoc reptation time chosen to give the correctsteady state orientation angle for this shear rate (the sameresult would be obtained via the Marrucci model Eq. (4)with β ≈ 3); and the Mead et al. CCR model withoutchain length fluctuations.

Korea-Australia Rheology Journal March 2000 Vol. 12, No. 1 17

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L. Gary Leal and James P. Oberhauser

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there are still some open questions. One is the fact that thedata discussed above suggests that CCR should have itsgreatest impact for simple shear flow, whereas the forms ofIanniruberto and Marrucci and Mead et al. have a strongereffect as the flow becomes more extensional. In thesemodels, the relative motion is envisioned as derivedentirely from the component of the velocity gradient alongthe primitive chain, because finite extensibility and thetendency of the chain to retract within the tube leads tonon-affine longitudinal motion of points on the test chain.In the simple model of Ianniruberto and Marrucci, this idealeads to a decreased effective reptation time, implying thatthe enhanced rate of entanglement release appears first atthe ends of the chain. In addition, this form of CCR onlyaffects the degree of chain alignment and has no directeffect on the relaxation of chain stretch. In the moreelaborate model of Mead et al., the rate of constraintrelease due to CCR is uniform along the chain, and itseffect is gradually shifted from relaxation of segmentalorientation to relaxation of segmental stretch as .It is not obvious to us how the relative convective motionof polymer chains can enhance the relaxation of chainstretch. It seems more likely that it is the chain stretchingpart of the DEMG model that is flawed, rather than anadditional mechanism for enhanced relaxation of chainstretch as assumed in the Mead et al. model. The assump-tion that CCR acts uniformly along the test chain is alsodifficult to understand. Since the source of relative motionis the velocity gradient in the direction of the primitivechain or tube axis coupled with the retraction of the testchain within the tube, it is difficult for us to envision howconstraints would be lost anywhere but off the ends of thetest chain.

Perhaps more significantly, we have found that modelcalculations using the DEMG model with a modifiedreptation time provides good qualitative agreement with allaspects of our experimental data for transient shear flows,other than the small and short-lived initial transient inorientation angle discussed earlier. If the modified rep-tation times are interpreted using Eq. (4), however, it isnecessary to change the parameter β to account for theweakened effect of CCR in more extensional flows, whichseems undesirable (though not necessarily unacceptable onthe basis of any general principle). We are in the process ofconducting a similar comparison of the transient shear flowdata using the model of Mead et al. It does not appear thatthe results are going to be superior to those for the simplerIanniruberto and Marrucci model. Of course, comparisonwith a limited set of data does not necessarily provideconclusive evidence for the success or failure of a modelof this complexity.

Another question is whether the mechanism for CCRdescribed by the current models is appropriate and com-plete for all flows, especially for shear-like flows. In sim-

ple shear, the velocity gradient is perpendicular to the fldirection, possessing no component in the direction flow. Since the finite tube radius is ignored in the descrtion of the relative motion of the tube wall and the techain, there would be no CCR effect in simple shear flfor a chain that is fully aligned in the flow direction. On thother hand, there is still relative motion between theconstraining chains and points on the primitive chain dto the velocity gradient in the perpendicular direction. Athe primitive chain approaches an aligned state, this sec-ondary source of relative motion, derived from the finitcross-section of the tube, should come into play. Of couits significance depends on the asymptotic orientation anfor large shear rates. Since the tube diameter is smallemore highly entangled solutions or melts, such an effwould also tend to have greater import in less entangsystems.

Lastly, an additional confusing point in the incorporatioof CCR into the DEMG model is the proper role of thconvective term in the equation for the tube survivprobability function in the DE and DEMG models:

. (6)

The physical rationale for this term is that the outwaconvection of points on the tube acts as a constant soof constraints and makes it more difficult for the primitivchain to escape its tube by reptation. Whenever v(s,t) isnonzero, the convective contribution to the tube surviprobability function can have an enormous impact [thouless on the stress for reasons explained by Mead and (1995)]. At large strain rates, it drastically inhibits thorientation relaxation process and causes a boundary lstructure for G at the chain ends (Mead and Leal, 199We find it difficult to distinguish between the physicaprocess ascribed to this convective term from that respsible for CCR, with the obvious exception that their effecare opposite. The convective term in Eq. (6) slows relaxation of chain segment orientation at a rate tincreases with increasing v(s,t) (or strain rate), while thesame relative motion between points on the tube wall athe primitive chain is viewed as a source of enhancrelaxation in the current CCR model. Although the covective term is less important in calculations for simpshear flow due to the strong rotation of the chain towathe flow direction (where v(s,t)0), the same is trueof the enhanced relaxation due to CCR (which is aproportional to v(s,t)). Indeed, we have found that caculations using the DEMG model with the convective terof Eq. (6) omitted give results that are very similar to tfull Mead et al. model (i.e., including both the convectivterm and CCR) up to the point, WiR ~ O(1) when chainstretching becomes important. In short, the convective termgiven by Eq. (6) seems to be part of the same proc

∇V τR1–→

v s t,( )⟨ ⟩– ∂G s t t', ,( )∂s

-----------------------

18 Korea-Australia Rheology Journal

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Non-Newtonian fluid mechanics for polymeric liquids A status report

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envisioned by the current models for CCR.

6. Non-Newtonian fluid Mechanics for entangledpolymer liquids

Despite the fact that a number of issues remain unre-solved and the molecular models of entangled, linear poly-mers continue to evolve, there has been remarkableprogress in recent years to the point that we may anticipatevery satisfactory and quite complete models in the nearfuture. As an aside, although the scope of this paper pre-cludes any detailed discussion, progress is equally appar-ent for certain types of branched polymers. Consequently,it is disappointing, though not surprising that the cost ofimproved description of the macromolecular physics comesat the expense of complexity. The original DE model hadan enormous impact on our ability to model and under-stand the dynamics of entangled polymers because it wasbased on a delightfully simple picture of a very complexsystem. In some respects, as it becomes necessary to incor-porate more features, we are gradually pushing the theorytoward a detailed molecular model and losing the sim-plicity of the fundamental concept. Nevertheless, as a sourceof understanding of the dynamical response of an entan-gled polymer system to a given deformation, the extensionsto reptation theory will continue to have an enormousimpact on polymer physics and rheology.

However, the goal of non-Newtonian fluid mechanics isthe prediction of flow as well as material behavior, and forthis purpose, even the original DE theory is too demandingfor complicated flows. The increasingly detailed extensionsof the model exacerbate the situation. In order to use thetheory for flow calculations, one must solve for the chainconfiguration at a sufficiently large number of materialpoints to allow an accurately differentiable description ofthe stress. This procedure is analogous to obtaining a directsolution to the Fokker-Planck equation for the configura-tion space distribution function in dilute solution theory. Tomake matters worse, however, the stress calculation in DEtheory requires knowledge of the polymer configurationover the entire deformation history, an onerous burden fornumerical calculations. In spite of this complexity there hasrecently been sufficient progress in the development ofalgorithms for direct solution of the Doi-Edwards modelthat the computation of simple flows is in sight (van Heelet al., 1999). Nevertheless, it seems likely to us that themain emphasis will continue to push in the direction ofsimplified approximations of the model as the basis forfluid dynamics predictions. There are two possible options.One is to develop rational approximations to the reptationmodels while retaining their basic mathematical structure.One might view the independent alignment approximationfor the universal strain tensor Q, which leads to the Kaye-BKZ constitutive model (Larson, 1988), as an effort in that

direction. To date, however, relatively little progress hbeen made using this approach beyond the K-BKZ mod

The second option is to abandon the full models algether and develop the highly simplified descriptions thare often referred to as toy models. These models usuallyassume that the polymer configuration can be descriwithin a simple configuration space, often merely thlength and orientation of a single vector. Then, dynamiequations are derived so that the toy model mimics dynamics of the complete theory to the maximum possiextent. Clearly, some aspects of the chain dynamics desed by the full model cannot be reproduced by a toy modhowever, it is hoped that a judicious choice of micrstructural variables will at least lead to qualitative predtions of flow and polymer behavior in flow systemrepresentative of processing applications.

For the class of linear, highly entangled polymers, theis a history of toy model development beginning soon afthe advent of the original DE theory. The text by Lars(1988) describes many of these models. Perhaps the known is the model of Pearson et al. (1989), which uses asingle time scale diffusive relaxation of orientation describe the order parameter tensor:

, (7)

where Q is the universal strain tensor; an equation forscalar stretch function λ,

; (8)

and an equation for stress,

σ = . (9)

With no stretch (λ = 1), the model becomes a single timconstant version of the DE theory. In the equation for λ, theterm driving stretch involves the velocity gradient tens

, and the relaxation of stretch incorporates a secotime scale τs, where . The model of Pearson et al. isthus similar to DEMG but with a single time scale assciated with relaxation of orientation.

Closely related is the simpler model of Remmelgas, et al.(1999a) (the RHL model), which was also derived in attempt to mimic the behavior of the full DEMG modeThe RHL model assumes that a single vector can descthe polymer configuration. This is motivated by the fathat calculations with the full DEMG model show that thvast majority of the chain is oriented in a single directi(the exception being the regions at the very ends of chains) even when the flow is time-dependent. A simiassumption was used in the partially extending vecmodel (PEC) of Larson (1984) In this case, the confuration of a polymer strand is also represented by a sin

S dt'1τd---- t(– t')–

τd-----------------

exp Q E t t,( )( )∞–

t

∫=

λ· t( ) λ∇V :S 1τs----– λ 1–( )=

5GN0 λ2S

∇Vτs τd≠

Korea-Australia Rheology Journal March 2000 Vol. 12, No. 1 19

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L. Gary Leal and James P. Oberhauser

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vector. The PEC model can range from a non-stretching,rigid-rod model to a model in which each polymer strandvector rotates and stretches affinely. However, a limitationof the PEC approach, when compared with DEMG, is thatorientation and stretch occur on a single time scale. Con-versely, the RHL model assumes that stretch and orien-tation relax on two very different time scales and that thedistributions of orientation and stretch are statisticallyindependent. This leads to one equation for the secondmoment of the orientation distribution uu, where the vectorR representing chain configuration is written as Ru:

(10)

and d/dt is a codeformational derivative, a second equationfor the length of the vector R:

, (11)

and a third equation for stress:

σ = (12)

In this case, τd and τR are assumed to be distinct relax-ation times with the ratio τd/τR = 3Ne as in traditional rep-tation theory. Clearly, the RHL model is much simpler thanthe full DEMG model. As such, it is eminently suitable forcomputation of full viscoelastic flows. Since it is struc-turally similar to the dumbbell models used for dilute/Boger solutions, the computational algorithms developedfor these fluids can be applied more or less directly. At thesame time, the model reproduces much of the DEMGrheological behavior for steady shear and extensionalflows, as well as transients like startup that involvedeformation and orientation of the whole polymer chain(Remmelgas et al., 1999a). Of course, polymer chaindynamics occurring on length scales less than the entirechain cannot be captured due to the single vector nature ofthe model. Numerical solutions have recently been obtainedusing this model for simple shear and mixed-type flows(Remmelgas and Leal, 1999b). Unfortunately, like the DEand DEMG models, the RHL model also exhibits a shearstress maximum at . Although the nonhomogeneousnature of the flow in a two-roll mill precludes shear-band-ing, the shear thinning is so strong that the flow is largelyconfined to the region nearest the two rollers. This pro-duces a number of global changes in the flow, as well asqualitative changes in the flow near the stagnation point,that bear little resemblance to experimental observations.

Much more realistic behavior is produced when CCR isincorporated into the RHL model by means of the rela-tionship given by Eq. (4) (Remmelgas and Leal, 1999a).With the parameter β = 3, the model predicts a monoton-ically increasing shear stress with increasing shear rate and

also displays behavior for startup of simple shear flow tis very similar to experimental results, including the initiundershoot in orientation angle alluded to earlier. Despthe inclusion of CCR, however, comparison with expeimental data indicates that the RHL model still overprdicts the tendency of the polymer to orient with the flodirection, indicating that shear thinning remains too strowith β = 3. Of course, this should not be too great a sprise since the viscosity still decreases as for β = 3,whereas experimental data for the test fluid shows It is also likely that the toy RHL-CCR model is still toosimplified if employed with a single fixed value of β for allflow conditions. We noted in the previous section thcomparison of data with predictions using the DEMmodel, with τd given by Eq (4), suggested the need for β tohave a value that depends on flow type. In any event, fcalculations with the RHL model or other toy modeapproximations of reptation theory are still in their infancand continued study is warranted. We note that there anumber of the features of modern reptation theory that not incorporated into the RHL-CCR model, and thsuggests one direction for improved prediction. It is alikely that the details of the toy models will change as ounderstanding of reptation models continues to evolve

Recently, McLeish and Larson (1998) developed tso-called “pom-pom” model to describe a certain class highly branched, entangled polymeric liquids. The matematical constitutive equations of the complete model a set of three integro-differential equations that describedynamics of the microstructural state, a model that is complex to consider exact solutions for highly complflows. As a result, McLeish and coworkers have developa toy model approximation to the full model that is quisimilar to the RHL model, possessing one extra micstructural variable and one extra relaxation time (Bishet al., 1997; Bishko et al., 1999). Again, an initial set ofcalculations was reported for flow through a 4:1 plancontraction. In spite of the fact that the model also exhibia shear stress maximum, well-behaved solutions wfound for the contraction flow up to a Weissenberg numbof 8. Although experimental data is not currently availabfor a monodisperse melt of the specific “H” branch poly-mer architecture appropriate to the original “pom-pom”model, numerical solutions show qualitative agreemewith observations of a commercial LDPE. This may contrasted with the RHL model for linear polymers, whiconly gave qualitatively reasonable solutions for flow in thtwo-roll mill when the model was modified to eliminate thexcessive shear thinning. One difference, of course, is the 4:1 contraction flow is driven by a pressure gradieand is not as sensitive to the rate of shear thinning, win the two-roll mill case the flow is driven by the shestress at the rotating rollers.

δ uu⟨ ⟩δt

--------------- 2∇V :– uuuu⟨ ⟩ 1τd R2⟨ ⟩--------------- uu⟨ ⟩ 1

3---l–

–=

D R2⟨ ⟩Dt

--------------- 2 R2⟨ ⟩∇V : uu⟨ ⟩ 1τR----- f R2( )R2⟨ ⟩ 1–( )–=

GN0 f R2⟨ ⟩( ) R2⟨ ⟩ uu⟨ ⟩

γ· τd1–∼

η γ· 1–∼η γ· 0.8–∼

20 Korea-Australia Rheology Journal

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Non-Newtonian fluid mechanics for polymeric liquids A status report

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7. Conclusions

Although the potential of toy models for viscoelasticflow simulations remains to be fully evaluated, it is ourbelief that this will be a primary direction for futuredevelopments in the quest for a predictive branch of fluidmechanics of polymeric liquids and other complex fluids.Full molecular level models based on the reptation conceptof de Gennes and Doi and Edwards are useful in devel-oping understanding of the dynamics at a molecular level;however, the full molecular models typically lead to acomplex set of integro-differential equations. Thoughmethods are being developed in an attempt to solve themexactly in a fluid mechanics context (van Heel et al.,1999), it is our view that these exact flow simulationswill remain almost exclusively in the research realm forthe foreseeable future. They will provide a benchmarkfor the approximate toy model predictions but will notbe utilized on a routine basis for fluid mechanics cal-culations.

We have denoted simplified models such as the RHLmodel as toy models in recognition of the great simpli-fications inherent in their derivation. Nevertheless, it isimportant to recognize that they are still much morecomplex than nearly all of the continuum mechanics-basedconstitutive models used in the past. The toy models areusually characterized by multiple, distinct time scales forrelaxation from a flow-induced deformation, whereas themost common phenomenological constitutive equations(e.g., Phan-Thien-Tanner or Oldroyd) assume that orien-tation and stretch have the same relaxation times. Evenmore importantly perhaps, the toy models are designedaround a specific, microstructural description. Hence, anysimplifications or generalizations have a clear physicalmeaning, and the equations to which they lead are guar-anteed to be mathematically and physically acceptable.In other words, there is no need for principles such asmaterial frame indifference, as they are automaticallysatisfied.

There is still much work to do in evaluating, modifying,and fine-tuning the class of toy models for entangledpolymeric liquids. Nevertheless, we believe that suchmodels will be the basis for the next generation of suc-cessful simulations of viscoelastic flows, much as thedumbbell theories have provided a useful framework forthe purely elastic Boger fluids.

Acknowledgement

The preparation of this paper was supported by grantsfrom the fluid mechanics and polymer physics programs ofthe National Science Foundation. We have also benefitedfrom discussions and collaboration with Scott Milner overthe past several years.

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