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PHYSICAL REVIEW E MARCH 1997VOLUME 55, NUMBER 3
Theory of melt fracture instabilities in the capillary flow of polymer melts
Joel D. Shore,1,* David Ronis,1,2 Luc Piche,1,3 and Martin Grant11Department of Physics and Centre for the Physics of Materials, McGill University, 3600 University Street, Montre´al, Quebec,
Canada H3A 2T82Department of Chemistry, McGill University, 801 Sherbrooke Street West, Montre´al, Quebec, Canada H3A 2K6
3Industrial Materials Institute, National Research Council of Canada, 75 De Mortagne Boulevard, Boucherville, Que´bec,Canada J4B 6Y4
~Received 1 July 1996!
We present a model for the flow of a polymer melt through a capillary with nonlinear slip boundaryconditions at the wall of the capillary. The model consists of the linearized Navier-Stokes equations coupled toa Maxwell constitutive relation for the viscoelasticity and a phase-field model for a first-order transitionbetween stick and slip flow at the boundary. Specializing to the case of a two-dimensional capillary, weperform a linear stability analysis about the steady-state solutions and predict in which parameter regimes thesteady-state becomes unstable. A numerical study of the model shows regions of steady flow, as well asregimes with periodic oscillations, spatially uniform but temporally chaotic oscillations, and more complicatedspatiotemporal behavior. We show that the oscillations can account for the sharkskin texturing and defectstructures seen in the extrusion of polymer melts.@S1063-651X~97!10401-9#
PACS number~s!: 61.25.Hq, 83.20.Lr, 64.60.My, 47.50.1d
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I. INTRODUCTION
Understanding the instabilities which occur when a pomer melt is extruded through a capillary~or ‘‘die’’ ! is alongstanding problem in the plastics and chemical engining communities. The issue was of large technologicalportance for the plastics industry in the 1950s and 19since these instabilities lead to unwanted distortions infinal plastic product, which have been collectively term‘‘melt fracture.’’ Over time, engineers have found waysavoid or minimize the problem in practice, but little undestanding of the origin of these distortions has been reacand much discussion of the problem has continued inrheology and non-Newtonian fluid mechanics literature. Frecent reviews, see@1,2#.
The problem can best be described by reference tosketch of the ‘‘flow curve’’ presented in Fig. 1.~For actualexperimental curves, see, e.g., Refs.@1–4#.! The horizontalaxis specifies the apparent shear rate of the polymer methe wall of the capillary@5#, while the vertical axis specifiethe shear stress at the wall.~Shear rate and shear stress,though the more fundamental quantities, are not the mdirectly accessible; in terms of physically measurable or ctrolable parameters, the horizontal and vertical axes cospond to the flow rate and the applied pressure, respectiv!The inset in Fig. 1 is a cartoon of a typical experimensetup: a piston or screw feeds the polymer melt into a revoir and eventually through the die and out the other ewhere it cools and solidifies.
As the flow rate in the die is increased, one typically finseveral different regimes@2,3,6#: At the lowest flow rates, theextrudate is smooth and regular. Next, one encounters agime of surface distortion termed ‘‘sharkskin,’’ in which th
*Current address: Bldg. 83, 2nd floor, RL, Eastman Kodak Copany, Rochester, NY 14650-2216.
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extrudate develops a sawtooth texturing on its surface. Oa change in the slope of the flow curve is apparent nearonset of the sharkskin, as shown in Fig. 1. At still larger florates, one sees ‘‘stick-slip’’ or ‘‘spurt’’ flow, which is characterized by fairly long time oscillations in the flow ra~e.g., on the order of seconds!, with a noticeable effect on theextrudate. Sharkskin is frequently seen on the extrudateing all or part of the stick portion of the cycle. Finally, at thhighest flow rates, one sees ‘‘gross’’ or ‘‘wavy’’ melt fracture which is characterized by a highly irregular extrudaNot all these regimes are necessarily seen for all mateand, in particular, the ‘‘stick-slip’’ or ‘‘spurt’’ flow is seenonly in those experimental setups where it is the pisspeed, rather than the pressure, which is held fixed. Unpressure regulation, one instead sees hysteresis, wherincreasing and decreasing pressure the flow rate jumps wlimits of metastability are reached, as shown by the daslines and arrows in Fig. 1.
There has long been controversy about whether thesefects are due to instabilities inside the die, or are insteadto instabilities at the die entrance or exit, or some combitions of these. However, recent ultrasonic measurements@7#show that anomalous time-dependent behavior in the pmer flow occurs within the die, far from both the entranand exit~and even at flow velocities somewhat below thowhere sharkskin first becomes apparent!, thus suggesting thainstabilities inside the die itself are probably important.
Since the Reynolds number@8# is still very small whenthese instabilities occur, it is clear that they cannot be duturbulence in the die; hence, the instabilities are most likto be due to some of the special properties of polymer mewhich of course differ from classical fluids in a numberimportant ways@9#. First, the time scales for relaxation along and widely distributed. Therefore, the fluids can behaelastically on laboratory time scales and must be modewith viscoelasticconstitutive relations between the stress astrain. Second, the viscosity in the fluids is often found to-
2976 © 1997 The American Physical Society
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55 2977THEORY OF MELT FRACTURE INSTABILITIES IN . . .
FIG. 1. Sketch of a typical flow curve seen iexperiments on extrusion of polymer meltadapted from@2–4#. Shown is the shear stressthe wall, sxy
wall , as a function of the shear rateexywall , at the wall. Also shown are the approximaflow regimes where different imperfections aseen on the extrudate. For photographs of thetrudate produced in these regimes, see@1#. In thespurt regime, the flow oscillates in time; alterntively, if the experiment is performed at constapressure, this regime is not accessible and hysesis is seen instead, as shown by the horizondashed lines. The inset shows a typical expemental setup where the melt is pushed by a pisfrom a large reservoir through the capillary anout the other end where it solidifies.
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a decreasing function of the shear rate, a phenomenon te‘‘shear-thinning.’’ The proposed explanation of this is thunder larger shears the polymers tend to line up parallethe flow and can then more easily slide past one anothe
Third, there is increasing evidence that polymeric flumight not obey the ‘‘stick’’ boundary condition at the walof the container that is standardly assumed in hydrodynics. That slip behavior could occur in polymer melts wsuggested in some of the earliest literature on melt frac@10#; such ideas were introduced into the physics commuseveral years ago by de Gennes@11#. More recently, deGennes and co-workers have expanded on this idea, suging that a sharp transition should occur between stickslip behavior as the shear rate at the walls is increased@12#.Experimental evidence for the importance of slip has baccumulating in the study of polymer melts@1,2,13,14#, andalthough most of the evidence has been rather indirecrecent elegant experiment by Migleret al. @15# using a poly-meric fluid in the planar Couette geometry@8# actually mea-sured the velocity of the fluid within 100 nm of the waconfirming much of de Gennes’ picture. In particular, tauthors found that as the shear rate was increased therea sharp transition at which the slip velocity jumped by abfour orders of magnitude.
In order to understand the phenomenon of melt fractuwe present a hydrodynamic model that describes the flowa viscoelastic fluid in which the polymer near the surfaundergoes a first-order transition in conformation as a fution of the shear stress at the wall. This conformatiochange leads to a change in the frictional force betweenwall and the polymer in the bulk, producing stick-slip behaior, and leading in a natural way to an effectively multivaued flow curve@see Fig. 2~a!#, as has been much discussedthe recent literature@1,2,16,17#. When the die is coupled tolarge reservoir in which the polymer can be compressed,find that the multivalued flow curve gives rise to oscillatospurt flow, as has been discussed previously@2#. More sig-nificantly, and even in the absence of the effect of the revoir, we find that theelasticnature of the fluid can give riseto periodic oscillations, chaotic behavior, and large-sc
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spatial structures in the die, which we conjecture is respsible for sharkskin. A shorter communication of our wohas been presented in Ref.@18#. Here, we give a fuller de-scription of the model, discuss the determination of tsteady-state solution and the exact linear stability analabout this solution, and give much more detail of the numecal results and relation to experiment.
The plan of the paper is as follows: In Sec. II, we presthe basic elements of our model and discuss the Foutransform of the fluid flow equations to reexpress the moin a mixed spectral and real space representation. In Secwe examine the steady-state solutions and their linear sta
FIG. 2. ~a! Flow curve for our model produced using the paraeter values given in Table I. Shown is the shear stress at the wsxywall , as a function of the average flow velocity,vx in steady state.
Each point on this curve is associated with a uniform steady-ssolution of the model. The dotted part of the curve hasx21,0 andis thuspotentiallyunstable, while the solid part hasx21.0 and isstable. The inset shows the range ofvx , where the steady-statsolution is actually found to be linearly unstable as a functiontm /t, as obtained from an approximate result of the linear stabianalysis, Eq.~3.24!. ~b! Local free energy densityf (c) for threedifferent values ofsxy
wall . The solid, dashed, and dotted curves afor shear stresses ofsxy
wall 5 80, 160, and 200 kPa, respectively. Thdashed part of the flow curve in~a! corresponds to a maximum inf (c) while the solid part of the curve corresponds to minimaf (c).
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2978 55SHORE, RONIS, PICHE´ , AND GRANT
ity. In Sec. IV, we present numerical results, first briediscussing the numerical method, next considering osctory solutions which are spatially uniform in the flow diretion, and then considering spatially nonuniform solutioWe conclude Sec. IV with a discussion of two additionfeatures which must be added to the model in order to fureproduce features seen in most experiments: compressiof the melt in the reservoir and nonuniformity in the bounary conditions along the walls. Finally, in Sec. V, we coclude with a summary and a brief discussion of avenuesfuture research.
II. THE MODEL
A. General presentation
We consider Poiseuille~pipe! flow @8# in a circular orrectilinear geometry. Generalization to more complicatedometries or to other types of flows, such as planar or cydrical Couette flow@8# is straightforward. We assume perodic boundary conditions in the flow direction animpermeable walls on the sides of the capillary.
The bulk fluid flow is specified by the following equations:
r]v~r ,t !
]t52g~ t !x2¹dp~r ,t !1¹• s↔~r ,t !, ~2.1!
¹•v~r ,t !50, ~2.2!
and
tm] s↔~r ,t !
]t52@ s↔~r ,t !2h0 e
↔~r ,t !#. ~2.3!
Equation ~2.1! is the Navier-Stokes equation@8#, which issimply Newton’s law for the velocityv(r ,t) of a fluid ele-ment at positionr and timet with densityr. s↔(r ,t) is the~viscous! stress tensor. The pressure term has been divinto two parts:p(r ,t)5g(t)x1dp(r ,t), whereg(t) is a uni-form pressure gradient in the flow direction, anddp(r ,t) isthe excess part, which~like all the other variables! is periodicin the x direction. Equation~2.2! is the continuity equationwhich follows from the assumption that the fluid is incompressible. In solving these equations, the incompressibcondition and the boundary condition that the perpendicucomponent of the velocity must go to zero at the wallsused to determinedp(r ,t). Equation~2.3! is the Maxwellconstitutive relation for a linear viscoelastic fluid@9#, withh0 the viscosity ande↔ the rate of strain tensor with componentsei j[]v i /]xj1]v j /]xi . By varying the Maxwell re-laxation time, tm , we can tune between a viscous flu(tm→0) and an elastic solid (tm ,h0→`).
These three equations together constitute a standarddrodynamic model for an incompressible viscoelastic fl@9#. Note, however, that we have linearized the equationsreplacing the total or convective derivatives that would nmally appear on the left-hand side of Eqs.~2.1! and ~2.3!with a simple partial derivative. This is an excellent appromation for Eq.~2.1! because the instabilities we will studare found to occur at very low Reynolds number@8#, wherethese terms are very small. For Eq.~2.3!, the corresponding
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dimensionless ratio to consider is the Weisenberg numgiven by the product of the shear rate andtm . This is esti-mated to be of order 0.1 to 1 in the range where sharksand spurt flow are seen. Because the convective terms itically vanish for flows that are uniform in the flow directionwe expect that, in analogy with the Reynolds number,Weisenberg number generally has to be considerably lathan one before the convective terms become importThese terms do play a role in certain cases, such as inducing higher stresses near the ends of the die~as will bediscussed in Sec. IV C! and for the behavior at higher flowrates, i.e., in the gross melt fracture regime not studied inpaper. In this regard, we note that a recent paper by Bland Graham@19# argues that there exists alarge-wave-number instability when terms in their model, analogousthose proportional to the Weisenberg number discusabove, are large~that is, for high shear rates!. However, herewe believe that the most important nonlinearities in the prlem come in through the boundary conditions at the wallsthe die.~We could also add additional nonlinearities by icluding shear thinning@9# in our model; however, prelimi-nary investigations showed that considering such a comcation was not necessary to produce the physics of intehere.!
The boundary condition for the fluid velocity at the walof the die is coupled to the kinetics of a first-order phatransition@20# describing the conformation of the polymerthe wall by the following equations:
vi~r5r i,t !5l c~r i,t !~ n•¹vi!ur5r i, ~2.4!
F@c#5Ewalls
dr iF f ~c!1j02
2~¹r i
c!2G , ~2.5!
and
t]c~r i,t !
]t52
dF
dc~r i,t !. ~2.6!
Equation~2.4! specifies the slip boundary condition, witthe slip velocity assumed to depend on the shear rate awall @21#. Here, r i denotes a coordinate which ranges onover the walls,n(r i) is defined as the inward unit normal athe wall, andvi(r i,t)[v i(r i,t) vi(r i,t) is the component ofthe fluid velocity parallel to the wall.l is a constant with thedimensions of length andc is a dimensionless quantity defined locally at each pointx along the walls at timet. Equa-tion ~2.4! defines a slipping length,l c, which has the geo-metric interpretation as the distance between the die wallthe point outside the die at which the tangential velocitylinearly extrapolated based on its value and the value ofgradient at the wall, would go to zero. In particulal c→0 corresponds to ‘‘stick’’~tangential velocity zero athe wall! while l c→` corresponds to perfect slip. Equatio~2.4! is an equation commonly used to describe slip bouary conditions@1,12–15#; our one generalization here is tassume that the slipping length is a local quantity which cbe specified at each point along the wall.
Experiments show that the slipping length jumps sharat a critical value of the shear@13,15#, while hysteresis isobserved on the multivalued flow curves seen in extrus
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55 2979THEORY OF MELT FRACTURE INSTABILITIES IN . . .
@1,2#. Together, these two features suggest the changslipping length is the nonequilibrium analogue of a firorder phase transition, presumably the result of a transiin the local conformational state of the polymer at the waEquation ~2.5! presents the free energy functional ofGinzburg-Landau phase-field model@22# for c(x,t). Weconsider a standardc4 free energy model with local freeenergy density
f ~c!5c
4c42
b
3c31
a
2c22~H vi• s↔•n!c. ~2.7!
Note the inclusion of a linear term couplingc(x,t) to thevalue of the shear stress,vi• s↔•n, at the wall. The secondterm in Eq. ~2.5! is a square-gradient term which resischanges inc(x,t) in the direction along the wall. The constantsa, b, c, H, andj0 are all taken to be positive, with thlatter being a bare correlation length. With this choicesigns, and with the parameter values appropriately cho@23#, the equilibrium value ofc is zero whensxy50, with afirst-order transition to a large value ofc as the stress increases. The microscopic picture of de Gennes andworkers@12# suggests associating largerc with the stretch-ing and uncoiling of some polymers attached to the wunder the action of the shear stress, although it has bsuggested that other mechanisms such as desorption opolymer at the wall may be relevant@14#, and in fact that thespecific mechanism for the stick-slip transition might evvary from one system to another@13,14#. Our model is notdependent on a specific microscopic mechanism. Rathethe spirit of keeping our model simple and general, wesume only that the change in conformation can be specby its effect on the slipping lengthl c(r i,t) @24#. Note that,although our mesoscopic description is motivated by acroscopic picture, the collective gradient term proportionaj0 is a feature particular to our treatment. Since such a tis the most simple allowed by symmetry, and naturally ocurs in mesoscopic descriptions similar to ours, we expthat coarse graining or renormalizing the microscopicscription will give rise to it. At any rate, we will find that thmost complex spatiotemporal behavior occurs whenj0 ismuch smaller than the other length scales in the system
Finally, Eq.~2.6! is an equation for dissipative relaxatioof the polymer conformation toward states that are lominima in the free energy. The local rate of change ofc isproportional to the functional derivative of the free enerwith respect toc. The constant of proportionality is given b1/t wheret is the characteristic time scale for relaxationthe polymer conformation@25#. In general, one would alsoinclude noise terms in these equations, with their amplitudetermined by fluctuation-dissipation relations@20,22#. How-ever, for our purposes, these terms should be of minorportance and are neglected. It is through Eqs.~2.4!–~2.6! thatnonlinearities enter our description; the equations forbulk fluid, though viscoelastic, are linear. It is possible thamore complicated description of the dynamics of the pomer near the wall would be required to describe some pcesses of interest, but in the absence of any substantivetification, we have chosen as simple a model as the phyof the problem requires.
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Having defined the model, it is now useful to summarisome of its basic features. The most important is the induction of a first-order stress-induced phase transition forconformation of the polymer at the wall. The conformationchange couples back to the fluid flow by changing the sping length at the wall. Since there is a range of valuesusxyu where the local free energy densityf (c) has a double-welled structure, this gives rise to hysteresis associatedthe spinodal part of the van der Waals loop in] f /]c, wherethe local susceptibilityx[(]2f /]c2)21 is negative, whileretaining consistency with a sharp first-order phase transiin the infinite-time limit. This hysteresis in the conformatioof the polymer at the wall leads directly to hysteresis in tfluid flow because of the coupling of the polymer conformtion to the slip lengthl c. Such hysteresis has indeed beseen experimentally, and much recent theoretical workincorporated it in anad hocfashion@1,2,16,17#, i.e., by pos-iting a nonmonotonic relation between slip velocity and wstress, and then assuming a history dependence in detering which slip velocity to use in the case where the curvemultivalued. Here, such hysteresis arises naturally fromwell-accepted description of the dynamics of a first-ordphase transition.
It is not obvious that the instability associated with tpolymer conformation at the wall will necessarily lead toinstability of the entire fluid, and in fact, it might seem rathsurprising that an instability at the wall could be so effectiin driving an instability in a viscous fluid. We will see, ifact, in Sec. III B that theelasticityof the polymer melt playsa crucial role in determining whether or not the systemunstable: Only if the fluid is sufficiently elastic that it actsa solid on the time scales over which the polymer conformtion changes at the walls~i.e., if t&tm) will the system beable to sustain oscillations@26#. However, before discussinthe origin of the oscillations in the polymer melt in furthedetail, it is useful to rewrite the equations in a form mouseful for the study of the steady-state solutions and thstability.
B. Spectral representation
Since, in the description of the bulk fluid hydrodynamicEqs. ~2.1!–~2.3!, we have neglected all nonlinear termthese equations can be Fourier transformed analyticallyreduced to a simpler form. Before doing that, however,specialize to a specific geometry, namely, a two-dimensiocapillary of sizeLx3Ly with periodic boundary conditions inthe x ~flow! direction and impermeable walls aty50 andLy in the direction perpendicular to the flow. Since our sytem is finite in both directions, Fourier series are used awe have a discrete set of wave numbers. In they directionwe use a Fourier sine series forvy , reflecting the boundaryconditionvy(r ,t)50 at the walls. Once this choice has bemade, the structure of Eqs.~2.1!–~2.3! determines the sim-plest choice for the other variables, namely, sine seriessxy and cosine series forvx , sxx , syy , anddp. The trans-formation of the equations is then performed by multiplyieach equation by sin(kyy) or cos(kyy) and integrating withrespect toy, where
ky[np/Ly , ~2.8!
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2980 55SHORE, RONIS, PICHE´ , AND GRANT
with n50,1,2, . . . . For terms containing derivatives withrespect toy, integration by parts is performed; care musttaken in considering the surface terms which turn out tozero in all cases except in the equation for the propagatiovx .
Straightforward manipulation of the resulting equationsFourier space is then used to eliminate the variabdp(k,t), s↔(k,t), andvy(k,t), finally obtaining the follow-ing equation forvx(k,t):
tm]2vx~k,t !
]t21
]vx~k,t !]t
1n0k2vx~k,t !
52n0Ly
ky2
k2@~21!nTL~kx ,t !2T0~kx ,t !# ~2.9!
for kyÞ0. For the special caseky50, vx(kx ,ky50,t)50 forkxÞ0 and
r]vx~k50,t !
]t52g~ t !1
1
Lysxy~kx50,y,t !uy50
Ly . ~2.10!
In Eq. ~2.9!, n0[h0 /r is the kinematic viscosity@8# and wehave defined a shear rate at the wall as
Tj~x,t ![]vx~x,t !/]yuy5 j , ~2.11!
where j takes on the value ofy at the two walls:j5$0,Ly%. When we use a subscript onc or T to indicate atwhich wall the corresponding quantity is evaluated, we wreplaceLy simply by L for notational simplicity. Note thatthe last term in Eqs.~2.9! and ~2.10! arises from the aforementioned surface term that appears in the Fourier traformed version of Eq.~2.1!.
vx(kx50, ky50,t) in Eq. ~2.10! is simply the spatiallyaveraged flow velocity in the die, which we will hencefordenote asvx . Experiments are typically run under conditionof either constant pressure or constant flow rate. The forcondition is implemented by setting the pressure gradg(t) equal to a constant in Eq.~2.10!. A constant flow rate isimplemented simply by takingvx equal to a constant, i.e., brequiring
] vx /]t50, ~2.12!
which from Eq.~2.10! implies that
g~ t !51
Lysxy~y,t !uy50
Ly , ~2.13!
where sxy(y,t) is the shear stress spatially-averaged alothe flow direction. Henceforth, we will assume that E~2.12! holds until we return to discuss constant pressure fland a more complicated experimental flow conditionSec. IV C.
We will considerc only in real space. Hence, Eqs.~2.4!–~2.7! remain essentially unchanged. Here, we simply rewthem combining Eqs.~2.5!–~2.7! into one, specializing toour two-dimensional geometry, and incorporating the nnotation we have introduced
vx~x,y5 j ,t !5~21! j /Lyl c j~x,t !Tj~x,t ! ~2.14!
eof
s
l
s-
ernt
g.w
e
and
t]c j~x,t !
]t52cc j
3~x,t !1bc j2~x,t !2ac j~x,t !
1~21! j /LyHsxy~x,y5 j ,t !1j02 ]2c j~x,t !
]x2.
~2.15!
Finally, in order to obtainsxy appearing in Eq.~2.15!, it isnecessary to include Eq.~2.3! specialized to the shear component of the stress tensor at the walls
tm]sxy~x,y5 j ,t !
]t52@sxy~x,y5 j ,t !2h0Tj~x,t !#.
~2.16!
Equations~2.9! and~2.14!–~2.16! constitute a rewriting ofEqs. ~2.1!–~2.7! in a mixed real space-Fourier space repsentation often more amenable for both analytic and numcal work. In what follows, we will make use of both thmixed and real space representations, according to whicmore convenient for our purposes.
III. STEADY-STATE SOLUTIONS AND LINEARSTABILITY ANALYSIS
A. Steady-state solutions
The steady-state solutions are obtained from Eqs.~2.9!and ~2.14!–~2.16! by setting all time derivatives equal tzero. We are interested in steady-state solutions that areform in thex direction, and thus have only thekx50 com-ponents ofvx(k) andTj (kx) nonzero. These components asimply the uniform-in-x real space values which we writsuccinctly asvx(ky) andTj . Explicitly summing the Fourierseries for they component to obtain the velocities in respace at the two walls, we can write Eq.~2.14! as
l c0T05 vx1 (n51
`
vx~ky! ~3.1!
and
2l cLTL5 vx1 (n51
`
~21!nvx~ky!, ~3.2!
where we recall Eq.~2.8! relating ky and n. ~It is worthnoting that these equations also hold for a time-dependstate provided that it is uniform in thex direction.!
In the steady state, Eq.~2.9! reduces to
vx~ky!52Ly@~21!nTL2T0#
~np!2. ~3.3!
If we substitute Eq.~3.3! into Eqs.~3.1! and~3.2!, noting that(n51
` 1/n25p2/6 and (n51` (21)n/n252p2/12 @27#, and
solve forT0 andTL , we find that
T05~ l cL1Ly/2!vx
~ l c01Ly/3!~ l cL1Ly/3!2Ly2/36
~3.4!
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55 2981THEORY OF MELT FRACTURE INSTABILITIES IN . . .
and
TL52~ l c01Ly/2!vx
~ l c01Ly/3!~ l cL1Ly/3!2Ly2/36
. ~3.5!
Considering Eqs.~2.15! and~2.16! in the steady state ansubstituting in Eq.~3.4! and ~3.5!, the possible steady-statvalues ofc0 are given by the roots of a sixteenth-order pomonial. This polynomial factors into a fourth-order polynmial which gives those roots that describe steady states smetric under reflection about the centerline of the capillaand a twelfth-order polynomial describing the asymmesteady states. Moreover, only six of the 12 asymmetric stions are independent because the solutions must occupairs related by reflection about the midplane of the calary.
The roots of the full sixteenth-order polynomial candetermined using standard numerical techniques@28#. Forlarge enough slip length coefficientl , the only physicalroots ~those with bothc0 andcL real and positive! are thesymmetric ones. To investigate these analytically, we nthat whenc05cL , Eqs.~3.4! and ~3.5! reduce to
T052TL5vx
l c01Ly/6. ~3.6!
One can easily verify that this result agrees with the velocprofile obtained by a direct real-space calculation ofsteady-state solution of the Navier-Stokes equation for pflow with the prescribed boundary conditions.
Considering Eqs.~2.15! and~2.16! in the steady state ansubstituting in Eq.~3.6!, we obtain
cc032bc0
21ac02Hh0vx
l c01Ly/650. ~3.7!
Upon multiplying Eq.~3.7! by the denominator of the lasterm, we have a fourth-order polynomial equation. The sof the zeroth-order coefficient implies that the product offour roots is negative, and therefore for given values ofparameters~and, in particular, fixedvx), we have the follow-ing three possibilities:~1! two complex-conjugate roots antwo real ones, one positive and one negative;~2! three realpositive roots and one real negative one;~3! three real nega-tive roots and one positive one. Since only real positive roare physical, we see the possibility of either one or thphysical roots. The most relevant case of a flow curve whalways has only one physical root is shown in Fig. 2~a!.
For completeness, we show in Fig. 3 two unusual pathogies in the flow curve which can occur in our model fsmall values of the slip length coefficent but do not corspond, to the best of our knowledge, to any flow curvwhich have been observed experimentally. One patholoccurs when there is a region invx where Eq.~3.7! has threephysical roots while the other occurs when asymmesteady-state solutions appear in the region where thecurve for symmetric solutions is nonmonotonic@29#. In allthat follows, we will be interested in the case where tparameters of the model are such that the flow curve isthat shown in Fig. 2~a!, i.e., with neither of these two pathologies.
m-,c-inl-
te
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nee
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ee
B. Linear stability analysis
Before studying the linear stability of the steady-statelutions, it is useful to consider qualitatively how the firsorder transition inc, and the concomitant flow [email protected]. 2~a!#, can lead to oscillations in the system. Considersystem with the polymer being pushed into the capillary aflow rate where the slope of the flow curve is negative. Tsteady-state solution then has a value ofc which is a maxi-mum rather than a minimum off (c). Thus, we might na-ively expect that this solution is unstable. For the momewe assume the fluid is sufficiently elastic that this instabilstill persists when the equation of motion forc is coupled tothe equations of motion for the fluid. Then the kinetics of tfirst-order transition from smallc ~‘‘stick’’ ! to large c~‘‘slip’’ ! are the origin of oscillations in this regime: Consider the fluid initially in the stuck state. The stress alongwalls will increase. At a critical stress, the stick state bcomes metastable; at some stress beyond that@the spinodal,which is well-defined here because we have neglected nin Eq. ~2.6!#, the stick state becomes absolutely unstablethe system jumps to the slip state. However, once in the nslip state, the stress at the walls decreases, the slip stateeventually becomes metastable, and a transition to seventually ensues. Hence, the system can repeatedly cbetween stick and slip.
The range ofvx for which oscillations are expected toccur for any given parameter values can be determinedlytically using linear stability analysis about the steady staWe consider linear perturbations to the steady-state valuethe variables having the form
X~r ,t !5ei ~vt2kxx!X~kx ,y,v!, ~3.8!
whereX has been used to denote a generic variable sucvelocity, pressure, orc. First, substituting this form into Eq~2.3! and defining
FIG. 3. Pathological flow curves which can be produceddifferent sets of parameters. The solid curve shows~the symmetricbranch of! the flow curve whenl is reduced by a factor of 100 tol 50.0004 m. Note that this curve is multivalued not only asfunction of sxy
wall but also as a function ofvx . The dashed anddotted curve is forl 50.002 m. The new branches of the curv~dotted! are ‘‘asymmetric’’ steady-state solutions, i.e., with diffeent slip velocities at the two walls.@Such additional branches alsoccur for l 50.0004 m, but have not been shown in order to illutrate the two types of pathologies separately.#
ci
r
ar
ar-al-r-
er-
. We
lu-s
2982 55SHORE, RONIS, PICHE´ , AND GRANT
nv[n0
11 ivtm, ~3.9!
we find that
d s↔5rnvd e↔ . ~3.10!
Here, and in the equations that follow, we omit the explifunctional dependences onkx , y, andv for brevity.
Next, substituting the form~3.8! into Eqs.~2.1! and~2.2!,and using Eq.~3.10! and the fact that Eq.~2.2! allows usto write ¹• e↔5¹2v, we obtain the following equations fothe magnitude of the perturbed valuesdvx(kx ,y,v),dvy(kx ,y,v), anddp(kx ,y,v):
r@ iv1nv~kx22]y
2!#dvx5 ikxdp, ~3.11!
r@ iv1nv~kx22]y
2!#dvy52]ydp, ~3.12!
and
2 ikxdvx1]ydvy50. ~3.13!
Eliminatingdp anddvx from these three simultaneous lineequations then yields
@ iv1nv~kx2]y2!#@kx
22]y2#dvy50, ~3.14!
which is valid for allkxÞ0. The solutions to Eq.~3.14! arelinear combinations ofeqjy with (q1 ,q2 ,q3 ,q4)5(kx ,2kx ,k,2k) and
d
-
t
k[S kx21 iv
nvD 1/2. ~3.15!
However, because of the boundary conditions, Eq.~3.14! isan eigenvalue equation which can be satisfied only for pticular linear combinations and accompanying discrete vues ofv. Two conditions on the four coefficients are detemined by the requirement thatdvy vanish at the two walls,which is easily satisfied by writingdvy in terms of hyper-bolic sine functions. The remaining two conditions are detmined by using Eqs.~3.13! and ~2.11! to obtain dvx anddT from dvy and then requiring Eq.~2.14! be satisfied at thetwo walls, i.e.,
dvx~y5 j !5~21! j /Lyl ~ Tjdc j1c jdTj !, ~3.16!
where overbars are used to denote steady-state valuescan rewritedc j in Eq. ~3.16! in terms ofdTj by using thefollowing expression derived from Eq.~2.15! with the helpof Eq. ~3.10!:
@ ivt1x j211~kxj0!
2#dc j5~21! j /LyHrnvdTj . ~3.17!
Here, we have defined the inverse susceptibility by
x j21[
]2f
]c j2 53cc j
222bc j1a. ~3.18!
After some algebra, we find that the coefficients in the sotion to Eq. ~3.14! with the prescribed boundary conditionare nonzero only whenv satisfies the following condition:
F c01HrnvT0
ivt1x0211~kxj0!
2 2nv
ivl Lyh~kxLy ,kLy!G
3F cL2HrnvTL
ivt1xL211~kxj0!
2 2nv
ivl Lyh~kxLy ,kLy!G52F nv
vl G2F kxsinh~kxLy!
2k
sinh~kLy!G2, ~3.19!
re-m-he
me
.
whereh(x1 ,x2)[x1coth(x1)2x2coth(x2).Finally, specializing to the case of a symmetric stea
state (c05cL and T052TL), Eq. ~3.19! simplifies and wefind, making use of Eq.~3.6!, that the complex eigenfrequenciesv are given by the roots of
@ l c1Ly/6#@ l c1LyC6~kxLy ,ivLy2/nv!#
52H vxrnvl
ivt1x211~kxj0!2 , ~3.20!
where we have defined
C6~z,z
e
sswulsa
eop
et ooferas
d-de
ill
rictobulyth
es
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by
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uid
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55 2983THEORY OF MELT FRACTURE INSTABILITIES IN . . .
v51
2tmF i H 11
tmt
@x211~kxj0!2#J
6S 4H vxrn0l tm /t
@ l c1Ly/6#@ l c1LyC6~kxLy ,0!#
2H 12tmt
@x211~kxj0!2#J D 1/2G . ~3.23!
@In Eq. ~3.23!, unlike in Eq.~3.22! or previous equations, thsigns indicated by the two6 symbols are allowed to varyindependently.# In what follows, we assume the correctneof Eq. ~3.23!, and then return, at the end, to discuss howexpect the use of the exact expression to modify our res
Referring back to Eq.~3.8!, we see that instability occurwhen any of the eigenfrequencies acquires a negative imnary part. One can show from Eq.~3.23! that a necessarycondition for instability is that the inverse susceptibility bnegative, or in other words, that we be on the negative slpart of the flow curve in Fig. 2~a! @30#. Interestingly, how-ever, this is not a sufficient condition. There are two casdepending on the sign of the quantity in the square rooEq. ~3.23!, or equivalently, whether or not the real partv is zero at the point where the imaginary part crosses zFor the latter case, which seems to occur more gener~and, in particular, holds for the parameter values discusin this paper, for all values oft, tm , j0, or kx) @31# we cansee from Eq.~3.23! that the condition for linear instability issimply given by the criterion
x211~kxj0!2,
21
~tm /t!. ~3.24!
Note that the dependence onvx in Eq. ~3.24! enters implic-itly through x21. In a recent paper, an instability was founby Georgiou@17# for a viscoelastic fluid in the Couette geometry. Georgiou assumed perfect stick at one wall anhighly nonlinear slip-stick boundary condition at the othsuch that the resulting flow curve was nonmonotonic, asFig. 2. This is related to our present treatment of Poiseuflow, in the limits t[0 and j0[0. However, within ourmodel we find that the Couette flow case~with stick-sliptransitions at both surfaces! will generically lead to addi-tional branches of the flow curve involving asymmetsteady-state solutions analogous to those shown bydashed and dotted curve in Fig. 3. This suggests a mcomplicated scenario where instabilities occur in somenot all regimes; in particular, we find evidence of linearstable, but nonlinearly unstable, states. Further details ofwork will be presented in a future communication.
Equation~3.24! shows that whenx21 become sufficientlynegative, there will be an unstable band of wave numbcentered aboutkx50. The criterion for instability dependon the relative magnitudes of the elastic time scale,tm , andthe time scale for relaxation of the conformation of the pomer at the surface,t. The curves in Fig. 4 show the stabilitboundaries in thevx—kxj0 plane for various values otm /t as given by Eq.~3.24!, while in the inset to Fig. 2 wehave plotted the range ofvx for which the steady-state solution is unstable as a function oftm /t ~using the fact thatkx50 is the most unstable wave number!. We see that if
ets.
gi-
e
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o.llyed
arine
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rs
-
tm /t is too small there is no instability for anyvx . Instead,c at the walls relaxes to the steady-state value givensetting the left-hand side of Eq.~2.15! equal to zero. This istrue even forx21,0, when this steady-state solution of E~2.15! alone@with sxy (x,y5 i ,t) held constant at the steadystate value# is unstable, i.e., when the value ofc at the wallscorresponds to a maximum rather than a minimum off (c).Thus, as we would expect, the coupling to the viscous flproduces a stabilizing effect on Eq.~2.15!, but this stabiliz-ing effect becomes weaker as the fluid becomes more elaThis can be understood intuitively as follows: If the chanin the polymer conformation occurs on a time scalet whichis short compared totm then the fluid responds in an elast~solidlike! manner and, in particular, has the ability‘‘spring back.’’ However, if the change in polymer confomation occurs on a time scale long compared totm then thefluid responds in a viscous~fluidlike! manner and the oscillations are damped.
Note that, to the extent that criterion~3.24! holds, thestability is independent of any of the bulk hydrodynamproperties of the liquid besidestm @32#. Numerical investi-gations of our model to be presented in Sec. IV confirmprediction of Eq.~3.24! that it is the ratio of the two relax-ation times,tm /t, which primarily determines the behaviothere is only very weak dependence of the behavior ofsystem on the value oftm for fixed tm /t, provided that timeis measured in units oftm .
We conclude this section with a few comments on taccuracy of our approximate stability criterion~3.24!, givenour use of Eq.~3.23! for the case where the real part ofv isnonzero at the stability boundary@31#. To test it against theexact analysis, we have used a nonlinear root-finding rou@28# to numerically solve for the roots of Eq.~3.20! using
FIG. 4. Results of linear stability analysis. Solid curves shstability boundaries in thevx—kxj0 plane fortm /t52, 5, 10, 100,and 1000~from inside curve to outside curve!, computed usingapproximate criterion~3.24!. For a given value ofvx , the steady-state solution is predicted to be linearly unstable to all perturbatiwith dimensionless wave numberskxj0 having values inside theboundaries shown. Points shown fortm /t52 and 10 give the exacstability boundaries obtained from the numerical solution of E~3.20! and~3.21! for a few different values oftm andj0, includingthose values for which the deviations from the curves are mpronounced.
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2984 55SHORE, RONIS, PICHE´ , AND GRANT
Eq. ~3.21! for various values oftm and j0 at tm /t52 and10, and a range ofkxj0. The resulting stability boundarieare shown by the symbols in Fig. 4. We find that the symbapproach the curves in the limitsLy
2/(n0tm)→0,Ly2/(n0tm)→`, and kxLy→`. That they seem to agree i
these latter two limits in addition to the first one may at fiseem surprising. It is because the arguments leading to~3.24! still hold to a good approximation even if we substute C6(z,z) for C6(z,0) in Eq. ~3.23!, provided that anyimaginary part or any negative real part ofC6 is small@andthat the real part of the entire term inside the square rooEq. ~3.23! remains positive#. In fact, one can show that thz→6` limiting behavior of C6(z,z) is the same asC6(z,0) provided the additional requirementz2@uzu is sat-isfied @33#. Conversely, for uzu→` with z2!uzu,C6(z,z)→1/Az and thus uC6(z,z)u becomes small, soLyC6(z,z) is only a small perturbation on thel c term inEq. ~3.23!.
Because the exact stability criterion agrees with Eq.~3.24!in all these limits, the actual stability boundaries never seto deviate very far from those determined by Eq.~3.24!. Thiscan be seen in Fig. 4, where we have included the lardeviations that were found, which seem to occur roughlyLy2/(n0t)'20. We expect that other sets of paramet
would show similarly small deviations.Finally, we note that if we had assumed that the slip
locity is proportional to the shear stress rather than the srate of strain in Eq.~2.4! @21#, then the effect would havebeen to multiplyl c in the secondbracketed term of Eq~3.20! by a factor ofnv /n0. This, in turn, makes the analyssomewhat more complicated, but the most general consions concerning the possibility of an instability whex0
21,0 remain unaltered. The approximate~3.24! is re-placed by a more complicated criterion; however, in the limtm /t@1, criterion~3.24! is again recovered.
IV. NUMERICAL RESULTS
Further investigation of our model is carried out numecally. We do this using the mixed real space spectral repsentation introduced in Sec. II. The infinite set of equatiofor vx(k,t) are reduced to a finite number by neglectithose beyond a maximum wave number in bothkx andky ;for example, we restrict ourselves ton<Ny in Eqs.~2.8! and~2.9!. These equations, along with the real-space equat~2.14!–~2.16! discretized in thex coordinate, can then bpropagated forward in time using Bulirsch-Stoer or RunKutta routines for integrating ordinary differential equatio@28#. We handle the second derivative in Eq.~2.15! by thestandard discretization
d2c/dx2→c~xi11!1c~xi21!22c~xi !
~Dx!2. ~4.1!
Note that in Eqs.~2.14!–~2.16!, the values ofvx and]vx /]y in real space are needed only at the walls and thuis only there that numerical Fourier transforms need toperformed at each integration step. Since we use discFourier transforms, the discretization in thex direction isrelated in a natural way to the cutoff inkx : if the number of
ls
tq.
in
m
strs
-ar
u-
t
-e-s
ns
-
itete
points used in real space isNx , then the set of discrete wavnumbers considered iskx5$0,62p/Lx ,64p/Lx , . . . ,6Nxp/Lx%, where the values of the function at the positiand negative wave numbers are complex conjugates of eother. Typical values used for the discretization weNy515–100; forNx , we useNx532–2048, depending onthe size of the structure present in the solution. For the ‘‘odimensional’’ case, where the system is forced to be unifoin the flow direction, we have checked that there is vegood agreement between this mixed real-space-spemethod and a method employing real-space findifferencing of the original partial differential equations.
The model has a large number of parameters and ineither feasible nor useful to systematically investigateeffects of variations in all of them. Rather, the paramevalues that we study are guided by experiment wherever psible. Except when otherwise noted, the values usedthose given in Table I. For the material parametersr andh0, we have chosen values typical of commodity polymesuch as polyethylene. The parametersl , a, b, c, andH, inEqs. ~2.14! and ~2.15! are chosen so that steady-state sotions of the equations give a multivalued flow [email protected]~a!# typical of ones seen experimentally@1,2,23#. Since wehave neglected shear thinning and ourc4 model is the mostsimplistic, we have only attempted to roughly reproduceperimental flow curves, rather than actually fitting our prameters to experimental data. We will find it instructiveinvestigate the model over various ranges of the phenomelogical parameterst, tm , andj0, and the length of the capillary, Lx , but will also discuss what values we would expethese parameters to have in a typical experiment.
A. Spatially uniform solutions
We will first consider the limitj0*Lx , where the systemis uniform in the flow direction.~For most experiments, thisis probably not the relevant limit, since typical die lengtare on the order of 1022–1021 m, while j0 is expected to bea length on the order of polymer dimensions, i.e., onorder of 1028 m, to at most 1026 m. However the limit isstill useful because it is simpler to study and serves as atoward understanding the more complicated nonuniformgime.! In such a limit, we find that any initial nonuniformityof c j (x,t) along the walls decays away. Indeed, in orderstudy this limit most efficiently, we have often used a ‘‘ondimensional’’ version of the code in which the flow directiois ignored altogether. This eliminates the need to perfoFourier transforms in thex direction; it also makes it unnecessary to specify the actual values ofLx andj0.
TABLE I. Standard values chosen for various parameters.last four parameters appear in thec4 free energy, Eq.~2.7!.
density r 740. kg/m3
viscosity h0 1. 3 104 Pa sdie width Ly 0.004 mslip length coefficient l 0.04 m
a 5b 13c 10H 3.23 1026 Pa21
eorso
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55 2985THEORY OF MELT FRACTURE INSTABILITIES IN . . .
The results of the linear stability analysis are confirmby the numerics. In particular, the stability criteria fkx50 are in excellent agreement with where oscillatinglutions first occur in our numerical simulations@34#. Somenumerical results are shown in Fig. 5. The period ofoscillations isO(tm), increasing from within about a factoof 2 of tm in the limit tm /t→` to several timestm whentm /t becomesO(1). There is also some dependence of tperiod onvx ; except very near the stability boundaries, ttendency is for the period to become longer asvx increases.
In order to make contact with experiment, it is necessto estimate reasonable values fortm and t. Eq. ~3.24! ex-presses the approximate criterion for the ratiotm /t in orderthat oscillations occur. It is generally believed thattm ,which in a melt involves the dynamics of polymer entangments, is proportional to the reptation time@35#. Since thestick-slip mechanism proposed by Brochardet al. @12,15# in-volves a large change in polymer conformation, the limitifactor is again likely to be polymer entanglements, and het should also be proportional to the reptation time@25#.However, if a different physical mechanism is responsi
FIG. 5. Behavior ofc j at the two walls forj0*Lx , at variousflow rates all within the dashed region of the flow curve in Fig.On the left, we present both time traces~insets! and power spectra~main graphs! of c j (t) for ~a! period 1,~b! period 3,~c! period 6,and ~d! chaotic oscillations. Walls aty50 andLy are indicated bysolid and dashed lines, respectively. The graphs on the far rshow phase space portraits, i.e.,c0(t) vs cL(t) with time t treatedas a parametric variable; the range for both axes is the same afor the ordinate axis in the time traces. Parameters used artm5 1023 s, tm /t 5 10; the exact values ofj0 andLx are not rel-evant.
d
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e
for the stick-slip transition in some systems~for example, asurface adsorption-desorption mechanism, as has recebeen suggested@14#!, then the dependence oft could bedifferent. Clearly, the value oftm /t relevant for experimen-tal systems warrants further study. In most of our simutions, we have chosentm /t510, which gives oscillationsover most of the velocity regime wherex21 is negative withan oscillation period ranging from;2.1 to;3.2tm over theunstable regime ofvx using the parameters in Table I.
A definite value to use for the elastic timetm itself islikewise not easy to determine, since there is a large disbution of elastic time scales in polymeric materials@9#. How-ever, considering the values obtained by Joseph, Riccius,Arney for high molecular weight silicone oils using theshear wave speed meter@36#, we believe thattm'1023 sshould be a good order-of-magnitude estimate for long-chpolymer melts@37#. Such an estimate yields an oscillatioperiod which corresponds reasonably well, given the flrate, to the wavelength of experimentally observed sharks@3#. We thus believe that the viscoelastic oscillations thatobserve correspond to the sharkskin texturing seen in expments. This connection will be further elucidated below.
Over much of the unstable regime invx , the oscillationsare simple, with the two walls approximately 180° outphase. However, for values ofvx which are close to theminimal and maximal ranges of the unstable [email protected].,near the maximum and minimum in the flow curve of Fi2~a!#, we find some of the more complex oscillatory behavstandardly seen in driven, damped nonlinear systems@38#.This includes chaotic behavior as well as oscillations wperiodicities that are integer multiples of the fundamenoscillation period. For some values ofvx , more than onestable attractor exists in the phase space and thus onedifferent behavior depending upon the initial conditionSometimes the initial conditions can even determine wheor not the system oscillates at all@34#. Examples of the be-havior seen are presented in Fig. 5. Note the period doubin going from 5~b! to 5~c!. Also, note that the phase spacattractor need not be symmetric about the licL(t)5c0(t) @e.g., Fig. 5~b!, 5~c!, and 5~d!#; however, whenit is not, the attractor related to this one by interchange oftwo walls @cL(t)↔c0(t)# is also a stable attractor.
As was noted above, the oscillations ofc j at the twowalls are most often out of phase. Of course, in a real thrdimensional physical system, the walls of the containernot independent because they are connected to each oTreating the two walls independently therefore describeslimit of a slit with an aspect ratio large compared to tlength over whichc is correlated perpendicular to the flodirection. In order to also simulate the other extreme, whthe walls are forced to remain completely in phase, we hstarted the system with initial conditions in whichc(x) isequal at the two walls. Because there is no noise in thetem to break this initial symmetry, this effectively enforcmirror symmetry in the system with respect to the centerlat y5Ly/2 for all times. Oscillations still occur in this casealthough we have not found any of the more complicabehavior~larger period motion or chaos! that is present whenc(x) at the two walls varies independently.
Finally, we have performed some limited studies withreal-space finite-difference code that allows us to inclu
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2986 55SHORE, RONIS, PICHE´ , AND GRANT
shear thinning in the model. We choose a local viscoswhich varies with the magnitude of the local rate of stratensor,e[( 12 e↔ : e↔)1/2, as
h~e!5h0
11~e/e0!12n , ~4.2!
with, e.g.,e051.4s21 and n50.3 @9#. The results suggesthat the qualitative features described here remainchanged, with the period of the oscillations decreasing asshear thinning becomes more pronounced.
B. Spatially nonuniform solutions
Spatially uniform behavior in the flow direction~with ei-ther periodic or temporally chaotic motion! is the rule whenLx is small andj0 is large; however, whenj0 is sufficientlysmall, the behavior can become quite complicated with mcomplicated structure, including what we believe to be stiotemporal chaos@39#, occurring. The behavior does not dpend only on the ratio ofLx /j0, since even forj050, wefind only uniform behavior for small enough aspect ratLx /Ly . This is because, as we will see below, the develment of complicated structure involves the formation of vlocity rolls and these rolls seem to have a minimum charteristic size on the order of the channel widthLy .
The behavior~at least over the times that we can measit! also depends on the initial conditions. If we start the wawith only a small amount of nonuniformity@for example,variations inc(x) of magnitude 1023 or 1022# then forLx&Ly or Lx&j0 we find that this nonuniformity decaysFor largerLx , the sort of behavior we get seems to depeon the behavior for the uniform case: If the value ofvx issuch that the uniform case produces simple periodic behior, then the variation along the walls remains small~butdoes not necessarily decay away completely as for smLx). If the uniform case produces chaos or complicatedriodic behavior, then we get spatiotemporal chaos.
The scenario is less clear when the initial condition halarger amount of nonuniformity@for example, variations inc(x) of magnitude 0.30#. In this case, the nonuniformity stildecays forLx&Ly or Lx&j0. However, for larger lengthcapillaries, spatiotemporal chaos seems to be the rule, ipendent of the value ofvx . This could be simply transienbehavior, but it seems to be true out to times correspondto several hundred oscillations with no indication that tnonuniformity along the walls is decaying.
The spatiotemporal behavior we see inc j (x,t) also mani-fests itself as complicated structures in the melt velocityside the die. In particular, we see rolls spanning the die,6~b!, when we look at the velocity relative to the mean flovelocity, v(x,t)2 vxx. If there is only one roll across thwidth of the die, then the velocity at the two walls will banticorrelated at any point along the die. Since the velocitproportional to the slip lengthl c j (x,t) at the wall, thiswould be expected to lead to out-of-phase oscillationsc j (x,t) at the two walls@40#. In fact, there does generallseem to be one velocity roll across the capillary andoscillations ofc j (x,t) are most often out of phase@41#. Thatc j (x,t) at the walls in the two-dimensional capillary is mooften out of phase correlates with the experimental fact
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sharkskin in a cylindrical die often has a spiral pattern wopposite sides of the extrudate being out of phase.
Finally, we point out that the complicated spatiotempopatterns at the two walls, shown in Figs. 6~a! and 6~c!, haveimportant experimental consequences: These, and analostructures in the third dimension not included in our calcution, should manifest themselves as defects on the surfacthe extrudate. Although no direct comparison can be mwithout including the third dimension in our model, we aencouraged by recent experiments@7# showing defect pat-terns on the surface of the extrudate~Fig. 7! similar to thepatterns seen in thex-t plane in Fig. 6~a! and 6~c!. ~For otherexperimental pictures of the extrudate surface during shskin, see@3,42#.!
C. Reservoir compressibility and nonuniform boundaryconditions
In order to make closer contact with the experiment, tmore features must be included in our model:~1! the com-pressibility of the polymer in the reservoir, and~2! the factthat the stresses are not constant along the die, but tendlarger near the exit~and may also vary due to randomneproduced, for example, by imperfections in the die itse!,thus leading to nonuniform boundary conditions alongwall.
The inclusion of the extrinsic effect of compressibilitythe reservoir is motivated by recent work@2,1# showing that
FIG. 6. Typical spatiotemporal patterns seen forj0!Lx . ~a! and~c! show space-time plots ofc j (x,t) along the top and bottomchannel walls, respectively. The value ofc j is indicated by thebrightness on this gray scale plot.~b! shows the flow pattern in thechannel at the latest time in~a! and ~c!. Shown is the velocityrelative to the mean flow velocity,v(x,y)2 vxx, with the magnitude~in arbitrary units! indicated by the size of the arrow. Note that thtime increases from top to bottom in~a! in order to facilitate acomparison of the latest time in~a! with the top wall in ~b!. Thedefect structures evident in~a! and~c! are associated with the rollin the channel shown in~b!. Parameters used are the same as inbottom panel of Fig. 5, except herej0 5 0 m andLx 5 0.1 m. Notethat only part of the channel has been shown in these figures.
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55 2987THEORY OF MELT FRACTURE INSTABILITIES IN . . .
the long time-scale oscillations seen in the ‘‘spurt’’ regimare controlled by the volume and compressibility of the pomer in the reservoir, as had been suspected for some@43#. This was demonstrated by showing that the flow rexiting the die oscillates significantly in time and that expementally the oscillation period for the spurt flow is propotional to the length of that part of the reservoir which cotains the melt. In the case of Ref.@2#, a model was alsointroduced in which the reservoir was considered roughlya spring which coupled the constant velocity motion of tmelt into the reservoir to the resulting varying rate at whthe melt enters the die. It was found that the numericalsults of this model well produced the experimental oscitions both in form and period.
The incorporation of reservoir compressiblity by consering the reservoir roughly as a spring@2# will give us oneadditional second-order differential equation for the averfluid velocity in the die; we will then see ‘‘reservoir compressibility oscillations’’ which have the expected linear bhavior on the volume of polymer in the reservoir. Note ththe inclusion of compressibility in the reservoir, while stignoring it in the hydrodynamic equations we solve in theitself, is reasonable: In most experiments, the volume ctained in the reservoir is very large compared to the voluin the die and therefore a small change in the density ofpolymer in the reservoir can lead to a change in volumethe reservoir which, though small on the scale of the resvoir, is large on the scale of the die.
The role of the reservoir can be explained qualitativelyfollows: Let us consider the piston in the reservoir movinga velocity vR such that the steady-state flow velocityvx
0 inthe die is in the unstable regime~in particular,
vRAR5 vx0Ad , ~4.3!
with AR andAd being the cross-sectional areas of the resvoir and die, respectively!. If the die is first in the stuck statethen the flow rate and pressure in the die will be at sopoint near the leftmost branch of the multivalued flow curshown by the bold solid curve in Fig. 8. Since materialbeing pushed by the piston into the reservoir faster than
FIG. 7. The surface of linear low-density polyethylene extrudat two different extrusion rates in the sharkskin regime fromexperiment of Ref.@7#. Shown are gray scale plots of the heigwith white representing peaks and black representing valleys.polymer melt was extruded in the upward direction, relative toorientation of these images, out a slit die;0.15 m long with arectangular cross section approximately 0.04 m3 0.002 m. Theimages show a region of the surface approximately 0.01 m acrNote that the patterns on the extrudate look qualitatively similathe defect structures seen in Figs. 6~a! and 6~c!.
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exiting the reservoir and entering the die, the density apressure of the melt in the reservoir will increase. When tpressure exceeds the critical pressure given by the lmaximum in the flow curve, the polymer at the wall in thdie will change conformation and enter the slip state, andflow rate in the die will jump to the right branch of the flowcurve~as shown by the upper horizontal solid line in Fig. 8!.However, at that point, the melt is now exiting the reservat a faster rate than the piston is forcing it, and thusdensity and pressure of the melt in the reservoir~and die!decreases, until the local minimum in the flow curvereached, at which point the polymer at the wall of the dreenters the stuck state, i.e., the system jumps back toleftmost branch of the flow curve and the cycle repeats. Nthat this scenario is similar in some respects to that whleads to the viscoelastic oscillations we discuss in the resthe paper. The primary difference is that here it is the copressibility of the fluid in the reservoir rather than the fluidshear elasticity that allows for the storage of energy. Albecause the volume of the polymer in the reservoir depeon details of the experimental design and on how much pomer remains in the reservoir, these oscillations are aintrinsic part of the experiment than those leading to shaskin. The final distinction is that this scenario leads to oslations in the flow rate in the die, whereas when we consered only viscoelastic oscillations the flow rate wconstrained to remain constant in the die~although the dis-tribution of the velocity field across the die varied!.
We now derive an approximate equation which govethis process. The approach is essentially the same as in@2# except that here we concentrate on the basic phyrather than including terms to get all the quantitative beh
ee
ee
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FIG. 8. The spurt flow regime. The bold solid curve shows ostandard flow curve withl 5 0.04 m. When reservoir compressibility is included and the piston speed is such that average flowis in the unstable part of the flow curve, the system cycles clowise around the multivalued portion of the flow curve. The leftmobranch is the stuck part of the cycle and the rightmost branch isslip part of the cycle; the horizontal solid lines show the rapjumps between the stuck and slipping portions. The dashed, andotted curves are flow curves forl 5 0.04, and 0.006 m, respectively. If we allow l to vary along the wall of the capillary, then wwill effectively have different flow curves in different regions of thcapillary, with the possibility of interesting behavior as discussedthe text.
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2988 55SHORE, RONIS, PICHE´ , AND GRANT
ior correct~e.g., shear-thinning effects, etc.!. If we denote themass of the polymer in the reservoir byMR and the volumeby VR then the change in rate of change in mass inreservoir can be written as
dMR /dt5rdVR /dt1VRdr/dt. ~4.4!
Using the fact thatdr5rbdpR , where
b[2~1/V!~dV/dp!
is the compressibility, and notingdVR /dt52ARvR anddMR /dt52rAdvx ~because the mass leaving the reservmust be flowing into the die!, we can rewrite Eq.~4.4! as
Ad~ vx2 vx0!52LRARb
dpRdt
, ~4.5!
where we have made use of Eq.~4.3!. Assuming that thepressure in the reservoir is given by the pressure at theentrance@44#, pR(t)52g(t)Lx , we can use conservation omomentum in the form@cf., Eq. ~2.13!#
rdvxdt
52g~ t !11
Lysxy~y,t !uy50
Ly ~4.6!
to eliminatepR(t) from Eq. ~4.5!. After rearranging termsthis gives
d2vxdt2
1Ad
ARLRLxrb~ vx2 vx
0!51
rLy
dsxy~y,t !
dt Uy50
Ly
. ~4.7!
Equation~4.7! is an additional equation which must bintegrated in time, replacing the trivial Eq.~2.12!. It incor-porates the effect of reservoir compressibility when anperiment is performed by regulating the speed of the pis
We see that the effect of the compressibility is thusmake the flow rate in the capillary vary, even if the pistonthe reservoir is pushed at a constant speed. Note that~4.7! is that for a driven harmonic oscillator. The drivinterm couples the oscillator to the equations we have alreinvestigated. The oscillator is in a strongly driven regimwhere the relevant frequency of the spurt oscillations willdetermined not simply from the natural frequency of osciltion in Eq. ~4.7!, but instead can be estimated very simpfrom the qualitative discussion of the origin of the oscilltions given above: Consider the point as which the syshas just jumped from the stuck to the sliding state. The tit that it takes for enough mass of materialDMR to drainthrough the die that the pressure is lowered sufficientlythe system to return to the stuck state is given by
t5DMR
rAduDvxu, ~4.8!
whereDvx is the average difference during the slipping pof the cycle between the~spatially averaged! velocity in thedie and the average of the flow velocity in the die overentireperiod,vx
0 . A similar equation can be written down fothe time it takes for mass to build up in the reservoir durthe stuck part of the cycle, and the periodT of oscillation isthen equal to the sum of the two times. The quantityDvx is
e
ir
ie
-n.
q.
dy
e-
me
r
t
n
not easy to determine very precisely for either part ofcycle, but from Fig. 2~a! it seems thatDvx' vx
0/2 is a rea-sonable estimate to use for each part of the cycle@45#. Fur-thermore,DMR5ARLRDr where the change in density othe fluid isDr5rbDp. The pressure changeDp is deter-mined from the difference in the wall stressDsxy betweenthe local minimum and maximum in the flow curveDp52DsxyLx /Ly . Combining all this together gives us thfollowing estimate for the period of the oscillations:
T'8ARLRLxbDsxy
Advx0Ly
. ~4.9!
Note that this formula has a very different dependencevarious quantities~such as the volume of polymer in threservoir! than would be derived by assuming that the perwas the inverse of the natural oscillation frequency in E~4.7!; it agrees with the experimentally observed factT}LR@2# for spurt oscillations.
To study the effect of reservoir compressibility numecally, we must use values for the various parameters descing the reservoir dimensions and the melt compressibilHowever, all these new parameters appear in Eq.~4.7! in onecoefficient and thus there is only one new independentrameter that enters the equation. Noting thatDsxy'3.853104 Pa for our flow curve in Fig. 2~a!, using a rea-sonable materials value for the bulk compressibilityb51.031029 Pa21, and choosing a reasonable value for tdie length ofLx 5 0.1 m, we can then consider the contrparameter to be the ratio of the volumes of the reservoirof the die,VR /Vd . Figure 9 shows the observed oscillatioperiod as a function of this control parameter forvx
0 5 0.30m/s,tm /t510 and three different values oftm . We see thatin the limit in which the volume of the reservoir is muclarger than the volume of the die, the period of the osciltions in the die is controlled by reservoir compressibility ais in good agreement with Eq.~4.9!. We find that these oscillations are in phase at the two walls, with large oscillatioin the flow rate in the die. In the opposite limit in which thvolume of the reservoir is much smaller than that of the dthe period of the oscillations in the die is controlled by telasticity of the fluid, the oscillations are generally outphase at the wall~and, can show chaotic behavior!, withalmost no oscillation in the flow rate in the die.
The two types of oscillations do not appear to coexist;the limit where the reservoir compressibility oscillationdominate, the flow rate in the die switches rapidly fromsmall value to a large value, spending very little time aflow rate in the unstable regime of the flow curve~see insetof Fig. 9!. As a result, there is very little time to excite thelastic oscillations which thus show up only weakly, astransient ‘‘ringing’’ effect in the velocity~and, to a muchlesser extent, inc itself! following the transition betweenstick and slip. Thus, reservoir compressibility oscillations bcome the controlling phenomenon when they are includBy contrast, experiments show that sharkskin texturingoccur during part of the cycle in the spurt regime; furthemore, it also occurs at velocities below that at which the flcurve for the die appears to be multivalued at all.
To reconcile our model with the experimental observtions, it is necessary to include one important feature of
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55 2989THEORY OF MELT FRACTURE INSTABILITIES IN . . .
system which has been neglected until now: the fact thatis not uniform and, in particular, that full solutions of thfluid flow equations with realistic constitutive laws show thhigher stresses are present near the exit of the die@1,46#. Weexpect that the critical stresses at which the polymer conmation changes will be unaltered but that the flow ratewhich such stresses are reached will be lower in theregion of the die. We can implement this most simplyassuming that a portion~‘‘end’’ ! of the die has a smaller sliplength coefficientl , cf. Eq. ~2.14!, and thus the flow curvefor that region is shifted to the left of the primary flow curv~see Fig. 8!.
For example, we assume that 20% of the die has anl halfthat in the remainder of the die~see the dashed curve in Fig8!. Of course, in the real system,l might be expected tovary smoothly along the wall. However, the overlappiflow curves which we choose here should reproduce the mimportant aspects of the physics and will allow us to repduce all the important features seen in experiments. Inticular, they allow for the possibility of choosing flow rateat which the system is in a stable regime with respect toprimary flow curve but an unstable regime of the flow cur
FIG. 9. Observed periodT of the oscillations when the compressibility of the polymer in the reservoir is included, as a functof the ratio of the reservoir and die volumes,VR /Vd . Parametersused arevx
0 5 0.30 m/s,tm /t510 and three different values otm ~top to bottom:tm 5 1022, 1023, and 1025 s!. In the limitwhere the reservoir is large, the oscillation period is in good agment with the approximate formula for the period of reservoir copressibility oscillations, Eq.~4.9!, shown by the solid line. In theopposite~less physical! limit, the oscillation period is determinedby the elasticity of the fluid and is given byT'2.37tm ~shown bydotted lines!. The inset showsvx(t) for about one complete oscillation for VR /Vd 5 10 andtm 5 1023 s. The flow rate in the diejumps very rapidly through the unstable regime and the only edence of any oscillations at a period on the order oftm is the‘‘ringing’’ effect following the jump. ~We show here an unusualldramatic case; asVR /Vd becomes larger, the time interval ovewhich such ringing is observable becomes a much smaller pathe total cycle.!
ie
t
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e
for the end of the die. It also allows for the possibility omore complicated behavior: at higher flow rates, the sysis in the unstable regime for the primary flow curve ashould execute reservoir compressibility oscillations whiduring the stuck part of the cycle, will put the flow rate inthe unstable regime of the flow curve for the end of the dThis allows the possibility of elastic oscillations superimposed on the reservoir compressibility oscillations durthis part of the cycle.
Figure 10 shows us the expectations discussed aboveindeed realized: Forvx
0 5 0.15 m/s, shown in 10~a!, we seethat the system does not execute compressibility oscillatisince we are in the stable part of the primary flow curve,that there are elastic oscillations in the end of the die whthe system is in the unstable part of the flow curve. This isagreement with the experimental fact that sharkskin isserved in a regime before spurt flow begins when the flcurve for the capillary as a whole is not yet multivalueFurthermore, we find that this is true whether or not tpolymer is pushed into the reservoir at a constant flow ratea constant pressure~although, of course, which stable brancof the flow curve the system settles upon at a given presdepends on the initial conditions!. This is understood as follows: for an incompressible fluid, the average flow rate mbe uniform along the length of the die. Hence, the stable flin the majority of the die effectively enforces a constant florate through the end region even when the polymer is pusinto the die at a constant pressure. This behavior is agaiagreement with experiment@2#: sharkskin is insensitive towhether the experiment is performed at constant pressurflow rate.
For vx0 5 0.30 m/s, shown in Fig. 10~b!, compressibility
oscillations occur as expected. Furthermore, during the spart of the cycle, the flow rate spends a large amount of t
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i-
of
FIG. 10. Grayscale plots showing the magnitude ofc(x,t) atone of the two capillary walls, as a function of the positionx alongthe wall and the timet, for the case wherel 5 0.004 m in 80% ofthe capillary andl 5 0.002 m in the last 20%. Here,tm /t510,tm51025 s, andVR /Vd51. The values ofvx
0 are~a! 0.15 m/s and~b! 0.30 m/s. Note that only the last 40% of the capillary is show
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2990 55SHORE, RONIS, PICHE´ , AND GRANT
in the unstable regime of the flow curve for the end of theand, as a result, elastic oscillations are set up in the enthe die. Indeed, experimentally, sharkskin is frequentlyserved on this part of the extrudate in the spurt flow regi@2#.
In order to gain insight into how robust the behavior dscribed above is, we have also considered several alterchoices for the variation in the value ofl along the die. Forexample, we have considered the case where the finalof the die hasl 5 0.006 m, producing the flow curve showby the dotted line in Fig. 8. In this case, the behaviorqualitatively the same except that in the spurt regime welonger see sharkskin superimposed. This is because, duthe spurt oscillations,vx never drops low enough to enter thunstable regime of the flow curve for the end of the die,can be predicted by referring to Fig. 8.
A more realistic case would havel varying continuouslyalong the wall near the die exit from its value in the restthe die to some smaller value; this continuous variationexpected to make the sharkskin behavior more robust sthere will be a continuous distribution of local flow curvesthe end region, with some overlapping the flow curve formain part of the die such that simultaneous spurt and shskin are produced. As an example, we have investigatesystem in whichl 5 0.04 m in 80% of the die and thedecreases linearly tol 5 0.02 m in 10% before increasinlinearly back tol 5 0.04 m in the final 10%. As we wouldexpect, forvx
0 5 0.30 m/s there are reservoir compressibiloscillations with viscoelastic oscillations at a given pointthe last part of the die occurring when the flow rate is suthat it is in the unstable regime for a flow curve withl at thegiven value. Finally, in real systems, we expect that the wwill be quite inhomogeneous. To account for this, we hainvestigated a case wherel is chosen to have a randomvalue between 0.02 and 0.04 m at each of the discretpoints along the wall. For a given flow ratevx
0 there areviscoelastic oscillations at those points along the diewhich this flow rate falls in the unstable region of the locflow curve. Our general conclusions from all of these invtigations are that the oscillatory behavior is quite robust athat the behavior at any point along the die can easilyunderstood in terms of simple arguments based on the naof the local flow curve.
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V. CONCLUSIONS
In summary, we have presented a model for the melt frture instabilities seen in the extrusion of polymer melts.includes a first-order transition between stick and slip behior of the polymer at the walls as a function of the shestress. The incorporation of the viscoelasticity of the fluleads to oscillations. The system can in fact display vcomplicated spatiotemporal behavior which we suggestexplain sharkskin texturing. The additional incorporationthe extrinsic effect of the compressibility of the fluid in threservoir allows us to obtain a unified description of bothsharkskin and spurt flow regimes seen in experiments.
The mechanism for the oscillations in our model is vesimple—when the stress at the walls becomes too large,polymers at the walls orient themselves so that slip occThis, in turn, results in a decrease in the stress at the wwhich eventually causes the polymers to reorient themseback to the stick condition. Provided that the fluid in the buresponds to the change in boundary conditions in a suciently elastic manner, this scenario leads to oscillationsthe fluid.
Finally, we should point out that we have not attempthere to model how the oscillations in the flow in the dtranslate into texturing on the surface of the extrudate. Hoever, it seems clear that such oscillations should in genlead to features on the extrudate, particularly once the effof variations in the normal stress@9# are considered. In futurework, we hope to explicitly consider the behavior of the fluonce it exits the die. Such a study might be particularly uful in order to understand the gross melt fracture regiwhere the extrudate becomes most severely distorted.
ACKNOWLEDGMENTS
We thank Ken Elder and Rejean Ducharme for their claboration in earlier stages of this work. We thank M.Graham for useful correspondence. This work was suppoby the Natural Sciences and Engineering Research Couof Canada, and les Fonds pour la Formation de Chercheul’Aide a la Recherche de Que´bec.
@1# M. M. Denn, Ann. Rev. Fluid Mech.22, 13 ~1990!.@2# S. G. Hatzikiriakos and J. M. Dealy, J. Rheol.36, 845 ~1992!.@3# D. S. Kalika and M. M. Denn, J. Rheol.31, 815 ~1987!.@4# E. B. Bagley, I. M. Cabott, and D. C. West, J. Appl. Phys.29,
109 ~1958!.@5# The apparent shear rate at the walls is derived directly from
flow rate under the assumption of the standard paraboliclocity profile of pipe flow. The actual shear rate at the wacan be very different due to shear thinning and wall slip.
@6# The phenomenology can be very complicated: not all thegimes are seen for all materials and there are dependencesuch factors as molecular weight and polydispersity. Thescription we give here is that applicable to high-density po
ee-
-on--
ethylene and some other long-chain linear~as opposed tobranched! polymers.
@7# J. Tatiboue¨t, R. Gendron, and L. Piche´ ~unpublished!.@8# L. D. Landau and E. M. Lifshitz,Fluid Mechanics, 2nd ed.
~Pergamon, Oxford, 1987!.@9# R. B. Bird, R. C. Armstrong, and O. Hassager,Dynamics of
Polymeric Liquids, 2nd ed.~Wiley, New York, 1987!, Vol. 1.@10# J. J. Benbow, R. V. Charley, and P. Lamb, Nature192, 223
~1961!; J. J. Benbow and P. Lamb, S.P.E. Trans.3, 7 ~1963!;F. N. Cogswell and P. Lamb, Trans. J. Plastics Inst.35, 809~1967!.
@11# P. G. de Gennes, C. R. Seances Acad. Sci. Ser. B288, 219~1979!.
o
r,
.
ss
the
f
ha
-telitonethi
t ong
-
ion
iainonheofeitn
t ofron
,
P.
wallsver-, it
ehee
yofemt of
e
the
fm-
n-thexci-
m-r a
onisthe
s
10,the
55 2991THEORY OF MELT FRACTURE INSTABILITIES IN . . .
@12# F. Brochard and P. G. de Gennes, Langmuir8, 3033~1992!; F.Brochard-Wyart, P. G. de Gennes, H. Hervert, and C. Redibid. 10, 1566 ~1994!; A. Ajdari, F. Brochard-Wyart, P. G.deGennes, L. Leibler, and M. Rubenstein, Physica A204, 17~1994!.
@13# P. P. Drda and S.-Q. Wang, Phys. Rev. Lett.75, 2698~1995!.@14# Y. W. Inn and S.-Q. Wang, Phys. Rev. Lett.76, 467 ~1996!.@15# K. B. Migler, H. Hervet, and L. Leger, Phys. Rev. Lett.70,
287~1993!; K. B. Migler, G. Massey, H. Hervet, and L. LegeJ. Phys.6, A301 ~1994!.
@16# G. C. Georgiou and M. J. Crochet, J. Rheol.38, 639 ~1994!;38, 1745~1994!.
@17# G. C. Georgiou, Rheol. Acta35, 39 ~1996!.@18# J. D. Shore, D. Ronis, L. Piche´, and M. Grant, Phys. Rev. Lett
77, 655 ~1996!.@19# W. B. Black and M. D. Graham, Phys. Rev. Lett.77, 956
~1996!, and references therein.@20# J. D. Gunton, M. San Miguel, and P. Sahni, inPhase Transi-
tions and Critical Phenomena, Vol. 8, edited by C. Domb andJ. L. Lebowitz~Academic, London, 1983!, p. 267.
@21# Alternatively, one could relate the slip velocity to the strerather than to the rate of strain, by replacingn•¹vi byvivi• s↔•n/h0. ~The division byh0 allows us to preserve theinterpretation ofl c as a slipping length.! We discuss the im-plications of using this latter relation when we discusslinear stability analysis in Sec. III B.
@22# P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys.49, 435~1977!.
@23# Specifically, we require that the parametersa, b, andc satisfy3,b2/(ac),4. For b2/(ac),3, no first-order transition oc-curs at [email protected]., f (c) has only one minimum for all values othe shear#, whereas forb2/(ac).4, f (c) has two minimaeven when there is no shear. Positive values forb andH insurethat the minima inf (c) occur atc.0 for a shear stress sucthat the force of the fluid on the wall is in the same directionthe velocity.
@24# A full description of local polymer conformation would incorporate the polymer’s random structure through a complicaorder parameter. Furthermore, such a model could be appthroughout the fluid, with its local expectation value relatedshear thinning. In fact, a scalar order parameter defithroughout the fluid has recently been used to study shearning in very small capillaries@M. Urbakh, L. Dalkhin, and J.Klafter, J. Chem. Phys.103, 10707 ~1995! and referencestherein#. However, here we are interested only in that aspecthe conformation which is relevant in determining the slippilength.
@25# Because we will choose units ofc such that the inverse susceptibility x21[]2f /]c2 is of order one,t is amacroscopictime scale for an entire polymer to change its conformatrather than a microscopic dynamical time for polymer reorietation.
@26# In contrast to our model, in Ref.@16#, only compressibility ofthe fluid is included, i.e., they consider a viscous Newtonfluid with no shear elasticity. Also, because they have notrinsic dynamics associated with their nonmonotonic slip cdition at the walls, their simulation corresponds to taking tlimit t→0 in our model and the negatively sloped regiontheir flow curve is always unstable. In a very recent paper, R@17#, their work has been extended to include shear elasticThe constitutive law used therein decomposes the stress te
n,
,
s
ded
dn-
f
n-
n--
f.y.sor
into purely viscous and elastic parts; the purely viscous parthe shear viscosity plays a role somewhat analogous to out,allowing restabilization of part of the negatively sloped regiof the flow curve.
@27# I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Seriesand Products, edited by A. Jeffrey~Academic, Orlando, 1980!.
@28# W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B.Flannery,Numerical Recipes in C, 2nd ed.~Cambridge Uni-versity Press, Cambridge, England, 1992!.
@29# For asymmetric steady states, the shear stress at the twois not equal and the vertical axis of Fig. 3 represents the aage of the magnitude of these two stresses; alternativelysimply denotes~up to a multiplicative factor! the pressure dropacross the die.
@30# In general,x is negative on that section of the flow curvbetween the two horizontal tangents. This follows from tfact thatdc/dsxy5Hx and that, since there is a term in thexpression forvx that is proportional toc, dvx /dsxy divergesat the same values ofsxy asdc/dsxy does.
@31# When the crossing of the imaginary part ofv occurs with thereal part zero, it will happen at a less negative value ofx21,i.e., the region of instability will be larger than predicted bEq. ~3.24!. This seems to only occur for rather small valuestm /t and is thus, in practice, often precluded by the systbecoming completely stable. However, we have found a separameters (Lx increased to 0.04 m andtm /t51.7) where thecrossing at highervx occurs with vanishing real part for somkx , includingkx50. Numerical solution of Eq.~3.20!, for thiscase, shows that the boundaries of the instability region invx—kxj0 plane~which is small at any rate! do deviate a fairbit from those predicted by Eq.~3.24! and, for some values oj0, the unstable region even becomes topologically more coplicated, consisting of disconnected regions.
@32# In this limit where the hydrodynamics is unimportant, the liear stability analysis for our system is the same as that forFitzHugh-Nagumo model, used to describe, e.g., neural etation. @See A. Karma, Phys. Rev. Lett.68, 397 ~1992!; A. T.Winfree, Chaos1, 303 ~1991!; and references therein.# How-ever, the full equations differ and, in particular, our more coplicated system allows for the possibility of chaos even fosystem which is spatially uniform~in the flow direction!.
@33# It can easily be shown thatC6(z) given by the approximate~3.22! is a strictly positive real function with the followinglimiting behaviors: limz→0C1(z)51/6, limz→0C2(z)51/2,and limuzu→`C6(z)51/u2zu.
@34# In fact, we sometimes see a subcritical bifurcation@38#,whereby the oscillatory solution and a nonoscillatory solutiare both locally stable in a narrow velocity regime. In thcase, the stability analysis correctly predicts wherenonoscillatory solution becomes absolutely unstable.
@35# P. G. de Gennes,Scaling Concepts in Polymer Physics~Cor-nell University Press, Ithaca, N.Y., 1979!.
@36# D. D. Joseph, O. Riccius, and M. Arney, J. Fluid Mech171,289 ~1986!; 171, 309~1986!; D. D. Joseph,Fluid Dynamics ofViscoelastic Fluids, Vol. 84 ofApplied Mathematical Science~Springer-Verlag, New York, 1990!; D. D. Joseph~privatecommunication!.
@37# In some of the numerical work discussed below, e.g., Fig.we take advantage of the previously noted insensitivity ofbehavior to the value oftm ~for a fixed ratiotm /t and withtime measured in units oftm) and usetm51025 s since this
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speeds up the numerical computations significantly.@38# See, e.g., P. Berge´, Y. Pomeau, and C. Vidal,Order Within
Chaos ~Wiley, New York, 1984!; H. D. I. Abarbanel, R.Brown, J. J. Sidorowich, and L. S. Tsimring, Rev. Mod. Ph65, 1331~1993!.
@39# M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys.65, 851~1993!; M. Caponeri and S. Ciliberto, Physica D58, 365~1992!; D. A. Egolf and H. S. Greenside, Nature369, 129~1994!.
@40# The proportionality between slip lengthl c j (x,t) and velocityat the walls also involves the shear rate at the wall@see Eq.~2.14!#, and thus the correlation between the two quantitneed not be direct. In fact, we find thatc j (x,t) can vary onshorter length scales thanvx(x,y5 j ,t) does, although thevariation invx(x,y5 j ,t) does seem to roughly follow that oc j (x,t) with the latter coarse grained over a somewhat larlength scale, as can be seen in Fig. 6.
@41# The walls can behave symmetrically also, but this requimore than one roll across the channel, and so seems mdifficult to create and sustain.
@42# M. T. Dennison, Trans. J. Plastics Inst.35, 803 ~1967!.@43# A. P. Metzger, C. W. Hamilton, and E. H. Merz, S.P.E. Tran
3, 21 ~1963!; J. M. Lupton and J. W. Regester, Polym. EnSci. 5, 235 ~1965!.
.
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.
@44# A better approximation would be to use Bernouilli’s equati~Ref. @8#! to relate the pressures in the reservoir and in the dthus incorporating the fact that pressure is lower in thebecause of the higher velocity of the fluid. To lowest ordthis introduces a nonlinear damping term of the for( vx /Lx)(dvx /dt) on the left-hand side of Eq.~4.7!. For thevelocities of interest here, however, the effect of this termexpected to be very small; indeed we have found that numcal runs including this term differ negligibly from those whicdo not.
@45# For the stuck regime, one has the strict upper bouuDvxu, vx
0 . Note that the assumption that the times for the twparts of the cycle are equal is not a very good one whenvx
0 isclose to one of the edges of the unstable regime. In that cEq. ~4.8! suggests, and numerical simulations confirm, that oor the other of the two times~and thus the total periodT) willbecome rather long.
@46# M. J. Crochet, A. R. Davies, and K. Walters,Numerical Simu-lations of Non-Newtonian Flow~Elsevier, Amsterdam, 1984!;R. Keunings, inFundamentals of Computer Modeling foPolymer Processing, edited by C. L. Tucker III~Carl HanserVerlag, Munich, 1989!, pp. 403–469; G. G. Lipscomb, R. Keunings, and M. M. Denn, J. Non-Newt. Fluid Mech.24, 85~1987!.
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