microscopic theory of convective constraint releasesms15c/papers/fullccr.pdf · molecular weight at...

Microscopic theory of convective constraint release

S. T. Milner

Exxon Research & Engineering, Route 22 East, Annandale, New Jersey 08801

T. C. B. McLeisha) and A. E. Likhtman

Department of Physics and Astronomy, University of Leeds, Leeds LS2 9JT,United Kingdom

(Received 22 August 2000; final revision received 15 December 2000)

Synopsis

We develop a microscopic description of the contribution to stress relaxation in entangled polymermelts of convective constraint release, which is the release of entanglement constraints due to theeffects of convective flow on chains surrounding a given chain. Our theory resolves three of themain shortcomings of the Doi–Edwards model in nonlinear rheology, in that it predicts~1! amonotonically increasing shear stress as a function of shear rate,~2! shear stress independent ofmolecular weight at sufficiently high shear rates, and~3! only modest anisotropies in the singlechain scattering function, in agreement with experiment. In addition, our approach predicts that astress maximum and resulting shear-banding instability would occur for living micelle solutions, asobserved. ©2001 The Society of Rheology.@DOI: 10.1122/1.1349122#

I. INTRODUCTION

The model of Doi and Edwards~DE! ~1986! of stress relaxation by reputation is aremarkably successful microscopic model, given its simplicity, in describing the flowproperties of entangled polymer melts. Its shortcomings in the regime of linear viscoelas-ticity have largely been addressed by a recent quantitative treatment of contour-lengthfluctuations@Milner and McLeish~1998!# inspired by a microscopic theory for stressrelaxation in star polymer melts@Milner and McLeish~1997!#.

Substantial discrepancies remain between the DE model and experiment in nonlinearrheology. The simplest of these is that in steady shear, the DE model predicts a viscosityh(g) for large shear rates shear thinning more strongly thang21, in fact, scaling ash; g23/2 @Doi and Edwards~1986!#. This behavior implies a maximum in the shear

stress as a function of shear rate, with the consequence that flows with shear ratesgslightly above an inverse reputation timetd

21 would be hydrodynamically unstable toshear banding. This striking behavior is not observed in experiment.

In contrast, entangled solutions of ‘‘living micelles,’’ while exhibiting many of thesame rheological signatures as conventional polymer melts, do, in fact, present a shear-banding instability, which is associated with a stress maximum@Grandet al. ~1997!#. Acredible microscopic theory of the nonlinear rheology of entangled polymers ought to be

a!Author to whom correspondence should be addressed; electronic mail: [email protected]

© 2001 by The Society of Rheology, Inc.J. Rheol. 45~2!, March/April 2001 5390148-6055/2001/45~2!/539/25/$20.00

able to distinguish between the case of unbreakable and living polymers, predicting astress maximum and shear banding for the latter case only.

Marrucci ~1996! has in a recent series of papers described a mechanism for stressrelaxation in entangled polymers under flow that would be absent from linear viscoelas-ticity, which he called convective constraint release~CCR!. The release of entanglementconstraints arises from the continual retraction of extra chain contour length generated bythe flow. An affinely deforming chain, initially an ensemble of random coils, will onaverage increase in length in a shear flow, the first correction to its length coming atsecond order in strain. Because entangled chains are nonetheless free to relax back totheir equilibrium length by retracting along their ‘‘tubes,’’ this means that in steady flowchains are continually being stretched and relaxing back.

As they do so, entanglements that the ends of a chain impose on other chains arecontinually being relaxed, at a rate equal to the rate at which chain contour length wouldincrease if chain retraction were suddenly turned off. This mechanism requires chainconformations to be anisotropic in steady state to operate, and so it first appears at secondorder in strain rate.

Following these ideas, Ianniruberto and Marrucci~1996! and Meadet al. ~1998! haveboth constructed nonlinear constitutive models in which it is assumed at the outset thatbecause of CCR, the renewal rate of the entire tube conformation of a chain is a functionof the shear rate, becoming of order the shear rate for highly aligned chains.

This strong assumption makes some progress towards avoiding the unphysical stressmaximum, but it does so by delaying it to higher strain rates and molecular weights andweakening its amplitude, not by eliminating it altogether. A maximum is still present, forexample, at a value ofN/Ne 5 50, which is well within the range of existing experi-ments. The assumption that CCR acts directly on the overall chain conformation via theeffective terminal time also has the drawback that no prediction can be made for thesingle-chain structure factor. This is because the extra relaxation is applied directly to therelaxation of stress, rather than via the local chain conformation. A proper simultaneoustreatment of chain conformation and stress is becoming increasingly important in thelight of the strong links between small angle neutron scattering and molecular theories ofrheology @McLeish et al. ~1999!#, and of a long-standing lack of explanation of therelatively mild anisotropy observed in sheared polymer melts@Muller et al. ~1993!# com-pared to predictions of the tube model of Doi and Edwards~1986!.

However, a careful microscopic consideration by Viovyet al. of constraint release byreputation of surrounding chains@Viovy et al. ~1991!#, strongly suggests an alternativemicroscopic view of the effect of CCR. Namely, constraint release motion may be re-garded as hopping motion of the tube itself, on a time scale set by the frequency ofconstraint release events, with a hopping distance of order the tube diameter. In effect,the tube itself undergoes Rouse motion mediated by constraint release.

According to this picture, the renewal rate for the entire tube conformation is the‘‘constraint release Rouse time,’’ which scales as the hopping time~inverse constraintrelease rate! times n2, wheren 5 Nb2/a2 5 5/4N/Ne is the number of steps of theprimitive path,N is a chain length,a and b are the tube diameter and the chain Khunsegment, andNe is defined by the plateau modulusNe 5 ckBT/G(0). This is evidentlymuch longer than the constraint release time for highly entangled chains, and the tubeconformation is not completely renewed until this much longer time passes. This picturehas been the subject of a recent series of stochastic chain simulations by Hua and Schie-ber ~1998! and Huaet al. ~1999!, that seek to combine the ingredients of reputation, pathlength fluctuation, and CCR. However, these calculations also fail to eliminate the shearstress maximum completely. We need a complementary analytic approach that allows

540 MILNER, MCLEISH, AND LIKHTMAN

incorporation of the missing physics. It should contain sufficient information to be able tocalculate both stress tensor and single chain structure factorS(q), at least on lengthscales greater than the tube diameter (q < 2pa21).

In this paper, we shall develop a microscopically based nonlinear constitutive equationincorporating CCR, that does not simply assume the tube conformation is renewed at arate of orderg. Instead, we shall describe the effects of CCR on the tube conformation interms of its Rouse motion. We shall work with the imposed constraint of fixed tubelength, corresponding to the DE assumption that the relaxation of the contour length of achain inside its tube is rapid.

We are thus limited to shear rates less than the inverse Rouse time of the chain insidethe tube,gtR , 1. BecausetR is smaller than the reputation time by a factor of 3n, forlong entangled chains there is an interesting regimetd

21 , g , tR21 in which the tube

conformations may be considerably anisotropic but the chains remain unstretched. This isthe regime to which our theory pertains.

Within our theory, the distinction between conventional unbreakable polymers andliving micelles appears naturally, because of the different mechanisms for stress relax-ation in the absence of CCR. Stress in conventional entangled polymers relaxes viareputation; in contrast, stress in living polymers relaxes primarily as breaks occur near agiven section of the chain. In living polymers, there are no permanent chain ends, and nopermanent arc-length labeling of the monomers along the chain. This ultimately leads toa nearly single-exponential stress relaxation function for living polymers, with the relax-ation time determined by the rate at which chain breaks occur sufficiently near a givenmonomer for it to reptate to the break point and relax its stress. When we modify ourtheory of CCR to apply to living micelles, a stress maximum appears without adjustableparameters.

The remainder of this paper is organized as follows. In Sec. II, we introduce a gener-alized stochastic model for the motion of the tube itself, which combines the effects ofreputation, constraint release Rouse motion, advection by the macroscopic flow, and theconstant-length constraint imposed by rapid chain retraction. In Sec. III, we introduce thequantities we shall compute, namely the stress tensor and single-chain structure factor,which can both be expressed in terms of the tube tangent correlation function. In Sec. IV,we develop an equation of motion for the tube tangent correlation function from thestochastic equation of motion for the tube, with an approximate treatment of the con-straint of fixed tube length and contour-length fluctuations. In Sec. V, we explore themodel predictions for chain conformations and stress for shear flows in startup and steadystate. In Sec. VI, we extend our theory to the case of living micelles. In Sec. VII, wediscuss the implications of our model for the so-called ‘‘damping function’’ describingthe stress amplitude immediately following a large-amplitude step strain. We present finalconclusions and discussion in Sec. VIII.

II. TUBE DYNAMICS

We seek a model for constraint release in flow that is compatible with the tube-confined processes of reputation, chain length fluctuation, and retraction@Doi and Ed-wards~1986!#, but which also describes the insight that chain ends passing through theentanglement volume of a chain segment may result in a local reorganization of itseffective tube. Like other work reviewed in the introduction above, we shall adopt theframework that the tube itself may be viewed as a Rouse object under the action of theseprocesses, with a local hopping rate determined self-consistently by the disentanglementdynamics of the real chains.

541CONVECTIVE CONSTRAINT RELEASE

Other recent work has pointed out possible differences between Rouse-tube and stan-dard Rouse dynamics. In particular, it is tempting to allow a strong biasing of the localhopping motion@Hua and Schieber~1998!#, so that removal of a constraint by convectionof a neighboring chain allows the chain to ‘‘straighten out’’ locally kinked configura-tions, but not to create them. This would also be the conclusion of a strictly ‘‘two-body’’theory of ‘‘entanglement points,’’ which would imply that constraint release by retractionof neighboring strands under flow would always annihilate entanglements, never creatingthem. Chains progressively align, and the tubes actually dilate in such a picture. Perhapsunsurprisingly, this asymmetry in the Rouse hops of the tube does not remove the stressmaximum, although it may weaken it@Hua and Schieber~1998!, Meadet al. ~1998!#.

In fact, such a picture is inconsistent with what we know about entanglements inmelts. The scaling of the plateau modulus with concentration is not consistent with atwo-body theory of entanglements@Colby and Rubinstein~1990!#. Theoretical ideas alsosuggest that entanglements arise via the mutual, delocalized topological interaction ofmany structures@Rubinstein and Panyukov~1997!#. Very recently Ianniruberto and Mar-rucci ~2000! have proposed an accounting for local chain reorientation in CCR. Further-more, empirical evidence on a wide range of melts@Fetterset al. ~1999!# implies that thetube diameter~or equivalently the value ofMe) is determined by universal criteria on thelocal melt structure~of order 10 distinct chains are required to be present in the volumeof one entanglement!. We will see that strong convective constraint release occurs fordeformation rates of the order ofg > td

21, yet the melt structure will not be strongly

perturbed at the level of an entanglement length untilg > te21. So at deformation rates

that would be expected to lead to CCR, the local criteria that govern the nature of theeffective confining tube are hardly perturbed from those of an equilibrium melt.

This leads to a model for CCR in which the local tube diameter is kept constant, butwhere the local tube contour is permitted a random hopping motion, coupled to theaverage rate of constraint release. TheO(1) constant that describes tube motion inducedby constraint release~how many constraint release events within a tube volume areneeded to provide one lateral hop of the tube of sizea! will be the one new parameter ofthis theory. We may, however, expect it to be related to the number of chains~ ; 10!defining the entanglement volume itself. So constraint release induces local Rouse-liketube motions that both straighten and buckle the primitive path.

In one respect the motion of the tube is not quite that of a Rouse polymer: the tube isactually defined by the chain it contains. Because of retraction, the primitive path retainsa constant contour length, in contrast to a real Rouse chain whose contour length canchange by stretching. We will need to include the constraint of constant length in theRouse-like description on the dynamics.

So we need to derive an equation of motion for the tube that includes four types ofmotion: ~1! reptation,~2! convection by the flow,~3! retraction, which maintains a fixedtube length, and~4! Rouse-like motion due to convective constraint release. Because ofthe very different nature of reptation and retraction motion on the one hand~motion alongthe tube! and Rouse-like motion on the other hand~isotropic motion!, it is impossible toreduce a description of this richer tube motion to a coordinate set that ‘‘diagonalizes’’ theequation of motion, analogous to Rouse coordinates for pure Rouse motion. In the re-mainder of this section we will derive a stochastic equation of motion for the tubetrajectory itself, which is described by coordinates of each pointR(s,t) as a function ofdimensionless curvilinear distances from the end along the tube (s 5 0 ands 5 n arethe ends! and timet. In Sec. III we will recast this stochastic dynamics forR(s,t) as an


ordinary~nonstochastic! equation of motion for convenient averages ofR(s,t), which aremore suitable for direct solution.

We want to express the position of the tube at timet1Dt via position of the tube attime t. Reptation and convection by the flow terms are well known@see Doi and Edwards~1986!#

R~s,t1Dt ! 5 R~s1Dj~ t !,t !1DtkR~s,t !1..., ~1!

wherek is the deformation rate tensor, which for simple shear has a single nonzero entrykxy 5 g. The random noiseDj(t) has zero average and white noise correlation function

^Dj~t!Dj~t8!& 5 2Dcd~t2t8! ~2!

with Dc the curvilinear diffusion coefficient

Dc 51

p2nte~3!

and is related to the reptation timetd as td 5 n2/(p2Dc). Note that throughout thispaper diffusion constants have the dimensions ofs21, since countour length variables(s,s8) are given in terms of the tube diametera. Our ‘‘microscopic’’ time scalete is theextrapolated reptation time of one tube segment consisting of (4/5)Ne monomers~equivalently, three times the Rouse time for such a chain!.

Constraint release motion results from a random release of entanglement constraintson the tube. Each constraint release event permits the tube to move locally a distance oforder the tube diameter, whereupon it finds itself once again constrained by entangle-ments. As constraints are released at the same rate on all portions of the tube, this impliesRouse motion for the tube. We therefore write the constraint release terms in an equationof motion for R(s,t) by using a Rouse model

R~s,t1Dt ! 5 ...1DtS 3n

2

]2R

]s2 1g~s,t !D 1..., ~4!

whereg(s,t) is a delta-correlated, zero-mean noise term.By the fluctuation-dissipation theorem, the noise amplitude is related to the coefficient

n by

^ga~s,t!gb~s8,t!& 5 na2d~t2t8!d~s2s8!dab ~5!

~Greek indices signify Cartesian coordinates$x, y, z% in this paper!. Thus we see thatn isthe frequency of constraint release events in the model.

Finally, we derive the term in the equation of motion responsible for retraction of thechain, which maintains the total length of the tube constant. Assume that at timet thechain is not stretched and at the next momentt1Dt convection stretches each segment ats along the tube by a factor of (11l(s)Dt). To get back to its previous position, eachsegment must move a distanceDs 5 *s

n/2(l(s)Dt)ds along the tube towards its centerbecause due to constraints the chain can move only along its own path.

The formalism and computation are eased by assuming that the stretching ratel(s)does not depend on the position along the chain; we getDs 5 (n/22s)lDt. This as-sumption was checked in~single-chain ensemble! Brownian simulations of our formalismand works well when constraint release is present. In fact, it is possible to lift thisassumption, deriving a set of equations withn Lagrange multipliersl i instead of onel.This was done and we show that the constant-l assumption does not affect the stress


behavior at all, so we procede with it here~but in some cases it does have minor effectson predictions of the single-chain scattering, so we have used this full formalism toproduce the relavant Figs. 7–9 below!.

Therefore the retraction part of the equation of motion could be written as

R~s,t1Dt ! → R~s1lDt~n/22s!,t ! ' R~s,t !1Dtl]R

]s~n/22s!. ~6!

Herel is a functional ofR(s), to be determined ultimately~see Sec. III A! from thecondition of the constant chain length

E0

ndsU]R

]sU 5 na. ~7!

From the definition ofl Eq. ~6! and the fact thatu]R/]su ' a everywhere it follows thatlan/2 is a speed of each chain end during retraction. We will thus calll the retractionrate.

We assemble these terms@Eqs.~1!, ~4!, ~6!# to obtain the stochastic dynamical equa-tion for the tube

R~s,t1Dt ! 5 R~s1Dj~ t !,t !1DtS kR13n

2

]2R

]s2 1g~s,t !1lS n

22sD ]R

]s D . ~8!

The four terms on the right-hand side correspond, respectively to reptation, convection,CCR~third and fourth terms!, and retraction. Equation~8! is a nonlinear stochastic partialdifferential equation with two noise termsDj(t) andg(s,t). Because we are only con-cerned with terms of first order inDt ~because equations of motion will be first order in]/]t), we can consider separately the different types of motion of the tube~since eachcontribute atO(Dt), cross terms are negligible!.

The constraint-release raten should be in fact related tol. Because constraints of agiven tube arise from other chains, we assume that the retraction of two ends of a givenchain by a distancea/2 should on average release a constraint somewhere on anotherchain. From Eq.~6! we therefore conclude that the frequency of convective constraintrelease should be proportional to the retraction rate.

Independent of convective effects, constraints on a given chain segment are releasedby reptation of the other chains. Therefore the total constraint release rate should be thesum of the rate of CCR and the rate of ordinary ‘‘reptative’’ constraint release, whichshould be inversely proportional to the mean lifetime (p2/12)n3te of chain segmentsfrom the theory of Doi and Edwards~1986!. Note that since CCR only becomes signifi-cant whengn3te . 1, the reptative constraint release term does not change our quali-tative predictions for large shear rates.

Summing the reptative constraint release and CCR contributions as described above,we have

n 5 cnSl112

p2n3teD . ~9!

Herecn is a phenomenological coefficient of order unity;cn 5 1 would mean that onetube retracting by one entanglement length releases one tube segment of some otherchain. We expectcn < 1. Together, Eqs.~7!, ~8!, and ~9! are sufficient to determineR(s,t), l andn. We note that the only new parameter in this development of the tubemodel is the constantcn itself.


III. STRESS TENSOR, STRUCTURE FACTOR

Our goal in this work is to compute the nonlinear, time-dependent flow behavior ofmonodisperse entangled polymer melts, including the effects of convective constraintrelease. First and foremost, we compute the stress tensorsab , developing methods thatapply for an arbitrary time-dependent flow history, although we will focus on startup ofshear flow.

But together with purely rheological investigations of stress relaxation in entangledpolymer melts, it is also useful to ask how the chains in the melt are situated in space, bymeans of scattering experiments. We therefore calculate the single-chain structure factorS(q), which can be observed by neutron scattering from a dilute admixture of labeledchains.

We imagine experiments in which scattering is performed after a given startup flow,and a subsequent rapid quench below the glass transition. This corresponds exactly to theexperiments of Mu¨ller et al. ~1993!. Thus the relevantS(q;t) is an equal-time density–density correlation function a timet after starting the flow. We restrict ourselves toscattering wave numbers satisfyingqa , 2p; that is, we do not look inside the tube.Therefore the chain structure factor and the tube structure factor are taken to be identical.

The stress tensorsab is given in terms of the chain-pathR(s) by

sab 5c

N

3T

a2 E0

ndsK]Ra

]s

]Rb

]s L. ~10!

Herec/N is the polymer concentration, and the arclength variables ranges from zero tothe total chain lengthn.

Later it will turn out to be convenient to introduce Rouse coordinates for the tube

Xp 5 ~1/n!E0

ndsR~s!cos~pps/n!, ~11!

R~s! 5 X012 (p 5 1

n

Xp cos~pps/n!. ~12!

The number of tube Kuhn segmentsn in the entire chain serves as an appropriate cutofffor the number of tube Rouse modes.

In terms of tube Rouse modes the stress tensor is

sab 5c

N

3T

a2

2p2

n(

p 5 1

n

p2^~Xp!a~Xp!b&, ~13!

where the angle brackets denote an average over the system.To compute the structure factor, we begin with its formal definition

S~q! 5 E0

ndsE

0

nds8^eiqaRa~s!e2iqbRb~s8!&. ~14!

With the separationR(s)2R(s8) between tube segments ats ands8 taken to be Gauss-ian random variables, we have

S~q! 5 E0

ndsE

0

nds8 exp@2~qaqb/2!^~Ra~s!2Ra~s8!!~Rb~s!2Rb~s8!!&#. ~15!

The structure factor may be expressed as


S~q! 5 E0

ndsE

0

nds8 expF2~qaqb/2!E

s

s8ds1E

s

s8ds2K ]Ra

]s~s1!

]Rb

]s~s2!L G . ~16!

Using the expansion of chain coordinates in Rouse modes, we may rewrite the struc-ture factor as

S~q! 5 E0

ndsE

0

nds8 expF22qaqb(

p,p8^~Xp!a~Xp8!b&

3FcosSpps

n D2cosSpps8

n DGFcosSp8ps

n D2cosSp8ps8

n DGG. ~17!

~Note that in the quiescent limit, the structure factor of Eq.~17! reduces to a Debyefunction as it should.!

IV. TUBE TANGENT CORRELATION FUNCTION

From Eqs.~10! and~16! for the stress tensor and structure factor, we see that knowl-edge of the tube tangent correlation function

fab~s,s8,t! [ ^~]Ra /]s!~s,t!~]Rb /]s!~s8,t!& ~18!

would be sufficient to compute bothsab and S(q). We now proceed to derive anequation of motion forf ab(s,s8,t).

First, we derive an equation of motion for the related quantityFab(s,s8,t)5 ^Ra(s,t)Rb(s8,t)&. This is obtained from Eq.~8! by ~1! expanding to first order in

Dt; ~2! expanding to second order inDj(t); ~3! averaging overDj(t) using Eq.~2! forthe correlations of the reptative noise~and the fact that the reptative noise is uncorrelatedwith the conformation!; ~4! averaging overg(s,t) using the Ito–Stratonovich relation

^ga~s,t!Rb~s8,t!& 5na2

2d~s2s8!dab ~19!

for the cross correlation betweeng(s,t) andR(s,t); and~5! assuming thatl is uncorre-lated with the chain conformation. The result after a few pages of algebra is

]F

]t5 ~3n/2!S ]2

]s2 1]2

]s82D F1k•F1F•kT1na2dabd~s2s8!1DcS ]

]s1

]

]s8D2

F

1lF ~n/22s!]

]s1~n/22s8!

]

]s8GF. ~20!

The sole assumption made in deriving Eq.~20!, that the retraction ratel(t) is uncor-related with the tube coordinateRa(s,t), or

^Ra~s,t!Rb~s8,t!l~t!& ' l~t!^Ra~s,t!Rb~s8,t!& ~21!

can be checked by direct simulation of Eq.~8!. A large ensemble of single chains weresubjected to the deterministic and Brownian noise terms, but were individually retractedto maintain fixed contour length. Preliminary results show that it turns out to be a goodapproximation when CCR is present. We leave more detailed comparison of stochasticand analytical approaches to future publications.


Now we apply the operator (]/]s)(]/]s8) to the equation to obtain an equation ofmotion for f ab

]f

]t5 k•f1f•kT1DcS ]

]s1

]

]s8D2

~ f2feq!1~3n/2!S ]2

]s2 1]2

]s82D ~ f2feq!

2lF21~s2n/2!S ]

]sD 1~s82n/2!S ]

]s8D G f. ~22!

Here the equilibrium valuef ab(eq) of the tube tangent correlation function is given by

fab~eq! 5 ~a2/3!d~s2s8!dab . ~23!

A. Discussion of the equation of motion

The equation of motion Eq.~22! for f ab(s,s8;t) is a partial differential equation inwhich s and s8 range over the interval$0,n%. The convective termsk•f1f•kT ~of theusual form for a second-rank covariant tensor! act onf ‘‘in place,’’ at a fixed$s,s8%. Thereptative diffusion acts only along diagonal liness2s8 5 const, while the constraintrelease Rouse diffusion acts isotropically.

The terms enforcing the fixed-length constraint give rise to a ‘‘current’’ flowing alongthe diagonal direction, with speed proportional to the distance from the center point$n/2,n/2%. This expresses the effect of relabeling the monomers along the chain followinga small retraction; the term 2l f ab arises from the effect of this relabeling on the tangentvectors themselves, i.e., from the effect on the derivatives]/]s and]/]s8 in the defini-tion of f ab .

We shall see below that typicallyDc @ DR [ 3n/2, so that the region wheref isnon-negligible remains confined to a narrow ridge close tos2s8 5 0.

Even with constraint release being a perturbation on reptation, these equations containa mechanism by which a maximum in the shear stress as a function of shear rate isavoided. Constraint release gives rise to a local relaxation towards the isotropicfeq allalong the diagonal spine, corresponding to the slight misalignment of tube segments thathave been able to make a local hop. These isotropic misaligned segments then give asource term~via the convective deformation,k•f1f•kT) for shear stress.

Without this mechanism~i.e., in the Doi–Edwards model!, the shear stress cata-strophically ‘‘aligns away,’’ resulting in a maximum in shear stress as a function of shearrate, and consequent flow instabilities. In the present model, because the hop rate fornearly aligned tubes becomes proportional tog ~see below!, the amount of tube misalign-ment becomes constant in the limit of high extension rates, leading ultimately to anasymptotically constant value of shear stress.

One may ask whether a more restricted set of degrees of freedom than the full functionf ab(s,s8) is sufficient to describe a tube undergoing both Rouse and reptative motion.However, this is not possible; we must retain the full function. To see this, consider thefollowing. In equilibrium, f ab(s,s8) is isotropic, local@proportional tod(s2s8)], andconstant, by Eq.~23!. After a rapid step strain,f ab(s,s8) becomes anisotropic~propor-tional to some tensorAab), but is still local and constant. Now the terms in Eq.~22!arising from reptation preserve the locality off ab(s,s8). This is because pure reptationmoves all parts of the chain along the tube together, so that the chain path at differentsands8 remains uncorrelated.


If only reptation was operative, we could writef ab(s,s8,t) 5 Aabd(s2s8)c((s1s8)/2,t) and thus describe stress relaxation with a simpler functionc(s,t)—whichwould be the tube survival probability of Doi and Edwards. Unfortunately, the terms inEq. ~22! describing Rouse motion are of the form of an isotropic diffusion operator in the$s,s8% plane, and spoil the locality off ab(s,s8), forcing us to abandonc(s,t) as adescription of the tube motion.

Likewise, if only Rouse motion were operative, the resulting diffusion operator in Eq.~22! would be diagonalized in Fourier space, i.e., by transforming to Rouse modes.Unfortunately, the reptation terms are not diagonal in Fourier space, so again we arestuck with the full equation.

B. Fourier-space equation of motion

It turns out to be numerically simpler to solve Eq.~22! in Fourier space, i.e., bytransforming to Rouse modes, than to solve directly in real space. This is because~1!transforming gives us a set of coupled ordinary differential equations to solve rather thana partial differential equation;~2! cutting off the number of Rouse modes atn is a naturalway to impose a short-distance cutoff in the theory, which would be problematic in realspace; and~3! for realistic polymer molecular weights, the number of Rouse modes istypically not very large (n , 50, say!.

Using Eq.~12! for the Rouse modes, we have a Fourier-transformed tangent correla-tion function (Cpq)ab , related tof ab(s,s8) by

~Cpq!ab [ ^~Xp!a~Xq!b& 51

p2pqE

0

nds ds8 sinSpps

n DsinSqps8

n Dfab~s,s8! ~24!

with an inverse relation

fab~s,s8! 54p2

n2 (p,q . 0

pqsinSpps

n DsinSqps8

n D~Cpq!ab . ~25!

The equilibrium value of (Cpq)ab is

~Cpq~eq!!ab 5

na2

6p2p2 dpqdab . ~26!

Now we Fourier transform Eq.~22! using Eq.~24!. After a few pages of algebra weobtain

]

]t~cpq! 5 ~k1kT!dpq1k•cpq1cpq•kT2S 1

n3D ~11chnl!~p21q2!cpq

18

p2n3 (p8,q8

M pp8Mqq8cp8q81lS (p8

M pp8cp8q1(q8

Mqq8cpq82dpq1D~27!

in which ch 5 3p2cn/2, and the1 in the last term denotes the Cartesian unit tensordab8 .

In Eq. ~27! we have made for convenience some rescalings and changes in notation.We measure time in units ofte 5 td /n3 and defined a scaled deviatoric mode amplitudecpq by


Cpq 5na2

6p2pq~dpq1cpq!. ~28!

Thuscpq 5 0 in equilibrium.

The coupling constant matricesM andM are given by

Mpp8 5 H2pp8@~21!p1p821#

p822p2 p Þ p8

0 p 5 p8

, ~29!

Mpp8 5 H2pp8@~21!p1p811#

p822p2 p Þ p8

21/2 p 5 p8

. ~30!

Note thatf ab(s,s8) is symmetric under the interchange of$a,s% and$b,s8%; indeed,only becomes nonlocal ins2s8 because of Rouse diffusionf; should be symmetric underseparate interchange ofa and b, and s and s8. This implies that (Cpq)ab must besymmetric under separate interchange ofa and b, andp and q. Also, because the twoends of a chain are the same,f ab(s,s8) is equal tof ab(n2s,n2s8), which implies thatCpq vanishes unless$p,q% are both odd or both even. These symmetries reduce thenumber of degrees of freedom by a factor of about 4. Forn modes andn even~odd!, wehave 3n(n12)/4@3(n11)2/4#° of freedom, or 330° of freedom forn 5 20. ~The factorof 3 corresponds to the three tensor componentsxx,xy, andyy in simple shear.!

In terms of the new variables (cpq)ab , the deviatoric part of the stress tensor takes thesimple form

dsab 5cT

Ne~1/n! (

p 5 1

n

~cpp!ab . ~31!

The front factorcT/Ne is in fact the plateau modulusG0 ~recall thatc is the monomerconcentration, and the plateau modulus is ‘‘kT per displaced volume of an entanglementsegment’’!. In what follows, we shall present stress values in units ofG0 .

C. Hopping rate

We turn now to determine the retraction ratel. Our procedure is approximate in thatwe do not choosel to fix the contour length Eq.~7!, but the more convenient quantity

E ds(a

faa~s,s! 5 na2, ~32!

which is proportional to the sum of the mean-square lengths of each tube segment. In theFourier-transform variables, the constraint reads

2p2

n(

p 5 1

n

(a 5 1

3

p2~Cpp!aa 5 na2. ~33!

Replacing the true constant-length constraint Eq.~7! by the approximate relation Eq.~32! can be checked for accuracy at any point in a calculation, i.e., at any moment,knowing cpq we calculate the segment lengthsR2(s,t). In the calculations presentedbelow this was done routinely. We found that no segment length deviates froma more


than 5%. This approximation is valid ultimately for the same reason that takingl inde-pendent ofs works: the local operation of CCR endows each segment with approximatelythe same stretchl and lengtha. It is perhaps worth noting that this is not itself a form ofthe ‘‘independent alignment approximation’’ of Doi and Edwards~1986!, a version ofwhich is instead implicit in the decoupling approximation of Eq.~21! when CCR isabsent or very small~see Sec. VII on damping functions below!.

We obtain an expression forl that precisely maintains Eq.~33! by applying the traceoperator(pqdpq(abdab to the equation of motion, to obtain

l 5

2k:cpp2~2/n3!S 1112ch

n2p2D p2trcpp18/~p2n3!M pp8M pq8trcp8q8

3n1~2ch /n2!q2trcqq~34!

with all indices$p,p8,q,q8% summed over.Consider how the hopping rate in shear flow depends on the shear rate. From Eq.~34!,

l evidently vanishes in the limit of low deformation rates, for whichf ab is isotropic.Sincel is an even analytic function ofg, for small shear ratesl is second order in shearrate.

At shear rates much higher than the inverse reptation time, if the average tube align-ment with respect to the velocity direction reaches an asymptotically constant small angle~as in experiments!, then*ds fab(s,s) approaches some tensor that is not quite alignedwith the flow ~and correspondingly, the shear stress approaches some constant value!. Inthis scenario, the hopping rate would asymptotically become linear in shear rate, with apossibly small coefficient because of the near alignment of the tubes.

Now we may ask whether convective constraint release is a big or a small effect withrespect to reptation. Recall that the reptation timetd scales astd ; ten3. We expectsignificant tube alignment, resulting in shear thinning and a retraction ratel of order g,when gtd exceeds unity. Note that the stretch relaxation timets of the chains, i.e., theRouse time corresponding to contour-length fluctuations inside the tube, scales asts; ten2. Therefore, the condition that we have shear thinning but not chain stretching

inside the tube isn @ gtd @ 1.Now the constraint release Rouse timetR scales astR ; n2/(cg), with c possibly

quite small. Thus we may havetR @ td , i.e., constraint release Rouse motion is aperturbation compared to reptation, ifn2/c @ gtd . That is, constraint release Rousemotion is never sufficient to completely relax the tube conformation, whenever the shearrate is low enough that chain stretching is negligible.

However, the highest Rouse modes of the tube will be relaxed by constraint releasebefore reptation renews the tube. Equating the constraint release Rouse time of thepthmodetR(p) ; n2/(cgp2) to the reptation time to find the lowestp* relaxed, we findp* 2 ; n2/(cgtd) . n/c. If c happened to be as small as 1/n, only the top few Rousemodes would relax (p* ; n) before reptation; ifc were of order unity,p* would be ofordern1/2, and all Rouse modes betweenn1/2 andn would be relaxed by CCR.

D. Contour-length fluctuations

For entangled polymers of practical molecular weights, the number of entanglementsis usually quite modest, ranging from 5 to 50 or so. For such smalln, rapid relaxation ofthe ends of the chain by contour-length fluctuations is quite significant. Indeed, thesefaster processes are responsible for the dynamic modulusG9(v) displaying a power law


greater than21/2 above the terminal frequency, and ultimately for the apparent 3.4power law dependence of the zero-shear viscosity on molecular weight@Doi ~1983!,Milner and McLeish~1998!#.

We may treat these fast processes approximately by including a local relaxation termon the right-hand side of Eq.~22! to make the ends of the chain relax more quickly. Thisapproach is quite similar to our previous work on contour-length fluctuations and lineardynamic rheology@Milner and McLeish~1998!#. The term we add takes the form

]f

]t5 ...2S 1

t~s!1

1

t~s8!D ~ f2feq!. ~35!

If either of the pointss or s8 is sufficiently near a free end of the chain, contour-lengthfluctuations will quickly relax the correlation functionf(s,s8) to its equilibrium isotropicvalue.

Fourier transforming this term leads to a final term in Eq.~27!

]cpq

]t5 ...2(

p8Lpp8cp8q2(

q8Lqq8cpq8 . ~36!

Here we have defined the matrixLpp8 as

Lpp8 5 ~1/2!(r

Kr~dp1r ,p81dp81r ,p2dp1p8,r ! ~37!

and we have written a Fourier series for 1/t(s) as

1

t~s!5 (

r > 0Kr cosSrps

n D ~38!

in which Kr vanishes by symmetry forr odd.For t(s), we used expressions developed in our previous work@Milner and McLeish

~1997!# on stress relaxation in star polymers, and effects of contour-length fluctuations inlinear polymer melts@Milner and McLeish~1998!#. Essentially, these expressions fort(s) interpolate between the fast contour-length fluctuation rate for monomers near thechain end, and exponentially slow activated retractions for monomers more distant fromthe chain end. The reader is referred to Milner and McLeish~1997! and Milner andMcLeish ~1998! for details.

V. RESULTS FOR STARTUP SHEAR FLOWS

We now explore the consequences of the model developed in the previous section, instartup shear flows.@We solved the equations of motion Eq.~27! numerically on aMacintosh G3 using Bulirsch–Stoer routines from Numerical Recipes in C, called fromMathematica via MathLink.#

First, the shear stress shows substantial overshoots, as shown in Fig. 1. The firstnormal stress shows no overshoot, because our treatment does not contain any chainstretching. The shear-stress overshoots for different representative values of shear rate areshown in Fig. 2. The overshoot appears atgtd > 1 and then grows with increasing shearrate, reaching about 20% at large shear rates. The shear strain at the maximum is ap-proximately constant,g* ' 1.6, as is clear from Fig. 2 where the shear stress is plottedversus strain.


The steady-state viscosityh(g) and first normal stress differenceN1(g) are found byintegrating the equations of motion to a sufficiently late time. The steady-state shearstress as a function of shear rate for different values of the hopping parameterch isshown in Fig. 3. The case ofch 5 0 corresponds to pure reptation without constraintrelease. In this case the shear stress decreases for largeg, giving rise to a shear stressmaximum, in contradiction with experiment. This behavior is very similar to the predic-tions of the Doi–Edwards theory, in which shear thinning becomes unstable~has slopeless than21! for a value of gtd just barely into the shear-thinning regime. Forch5 1.1 or larger (cn > 0.07) the shear stress monotonically approaches a plateau value,

corresponding to a shear-thinning exponent at high shear rates of21. Thus our theory ofconvective constraint release eliminates one of the main shortcomings of Doi–Edwardstheory.

We obtain this qualitative behavior whether or not we include the effects of contour-length fluctuations on the stress relaxation. Stress as a function of shear rate for differentvalues ofch with contour-length fluctuations included is shown in Fig. 4. Contour-lengthfluctuations alone are not sufficient to eliminate the stress maximum, although they do

FIG. 1. Shear and first normal stress vs time after startup of shear flow. In the absence of fluctuations, these

curves are essentially independent ofn and of shear rate in the plateau regiontd21

! , g ! ts21.

FIG. 2. Shear stress vs time after startup of shear flow, for shear ratesgtd 5 0.25,1,4,16,64,256,1024.


displace the stress maximum to higher shear rate. When contour-length fluctuations areadded to CCR, the approach to the asymptotic stress plateau is shifted to higher rates,with very little change in slope. A range of values forcn gives a large region over whichthe shear-thinning exponent is in the range 0.8–0.9, as observed experimentally fornearly monodisperse polymer melts. In principle, the value ofcn could be determined bycomparing the detailed shape of the shear-thinning curve to experiment, but in fact thisdoes not pin down the value too precisely once the stress maximum is eliminated. Hencewe takecn 5 0.1 unless otherwise specified. This corresponds to the case that ten chainsare needed to create one entanglement.

The chain-length dependence of the steady-state viscosity is displayed in Fig. 5, forn 5 10,20,40. Note then independence of the viscosity in the strong shear-thinningregion, in agreement with the experiment of Berceaet al. ~1993!. The zero-shear viscosi-ties for these three values ofn scale with an apparent exponent ofh(0) ; n3.4–3.5, inagreement with experiment@Berry and Fox~1968!#. This behavior is reproduced becauseof the contour-length fluctuations included via Eq.~35!; without them, the zero-shearviscosity in our theory scales asn3. Indeed, the shapes of the shear-thinning curves for

FIG. 3. Steady-state shear stress vs shear rate, forn 5 20 andcn 5 0,0.01,0.1,1.

FIG. 4. Steady-state shear stress vs shear rate, forn 5 20 andcn 5 0,0.01,0.1,1, including the effects ofcontour length fluctuations. Note that shear rate is made dimensionless with the ‘‘bare’’ reptation timetd ,calculated in the absence of fluctuations.


different n are such that they can be brought to a universal curve by scaling by thezero-shear viscosity and a characteristic time which scale the same way withn, i.e., by‘‘sliding’’ the curves along the common high-shear asymptote of slope21 ~inset to Fig.5!. The corresponding first normal-stress coefficients as a function of shear rate areshown in Fig. 6. They too can be similarly collapsed to a common universal curve~seeinset to Fig. 6!.

In our treatment we can also examine the single chain structure factorS(q), as ob-servable in neutron-scattering experiments from selectively deuterated chains. We havein mind an experiment in which a flow is suddenly stopped by quenching below the glasstransition, and neutron scattering performed afterwards, as has been performed by Mu¨lleret al. ~1993!. A typical structure factor computed from our theory is shown in Fig. 7.Note that constraint release drastically changes the overall picture: the degree of orien-tation decreases, and theq dependence of the orientation angle appears; contour lines ofS(q) at smallerq are more oriented than at largerq.

FIG. 5. Steady-state shear viscosity vs shear rate, forcn 5 0.1 andn 5 10,20,40. Inset: master shear-thinningcurve.

FIG. 6. First normal stress coefficient vs shear rate, forcn 5 0.1 andn 5 10,20,40. Inset: master shear-thinning curve.


To compare these results with neutron scattering experiments, we can extract variousquantities including the characteristic dimensionsRmax

2 (g) andRmin2 (g) in the major and

minor principal directions, as well as theq-dependent alignment angleb(q,g) betweenthe major principal axis contours ofS(q) and the flow~x! direction. These quantitiessummarize the changes in dimension and orientation of typical chain conformations as aresult of the flow, assuming that the contours ofS(q) are elliptical.

Our predictions forq-dependent alignment angle are in qualitative agreement withexisting experiments@Muller et al. ~1993!#. The typicalq-dependenceb(q) is shown inthe inset to Fig. 8 forn 5 20. The dependence on shear rate of the limiting valuesbmax 5 b(q 5 0) and bmin 5 b(q 5 2p/a) are shown in Fig. 8 for different chainlengths. Note the slow approach to an apparent limiting angle. This is a direct result of

FIG. 7. Single-chain structure factor, forn 5 20, cn 5 0.1 and high shear limit. Inset: the same structurefactor in the same scale without CCR (cn 5 0).

FIG. 8. Alignment anglesbmax andbmin vs shear rate, forn 5 20 andn 5 10, cn 5 0.1 andcn 5 0. Inset:q dependence of alignment angle.


CCR, and the value of this angle is in effect a measure of the hopping parametercn . Theoriginal Doi–Edwards theory (cn 5 0) predictsbmax 5 bmin 5 90. Increasingcn leadsto decreasing bothbmax andbmin and increasing the gap between them. The angleb alsodepends on chain length: increasingn leads to an increase of the gap betweenbmax andbmin. The arrows in Fig. 8 indicate experimental results of Mu¨ller et al. ~1993!. How-ever, in the paper, the authors did not specify the shear rate and did not reach steady-stateshear~total strain was 2.4, but to reach steady flow one needs more than ten strain units!.Therefore one needs more neutron scattering data for a detailed comparison to our pre-dictions.

The major and minor radii of gyration are shown in Fig. 9; note again the apparentlimiting anisotropy. In fact, neither Mu¨ller et al. nor any subsequent analysis have dis-cussed the strong disagreement between the single-chain structure factor measured in thiswork and the tube-based theory without CCR. The real chains are much less oriented thanthe simple theory would suggest. But on incorporating CCR into the model, we are ablesimultaneously to account for the chain anisotropy and the rotation of the principal axisof the scattering pattern withq. The latter is a striking feature of the data, and is a directconsequence of the local Rouse motion of the tube, and the convection of the perturbedstructures so formed to longer length scales by the flow.

Note that the deformations of the chains from isotropic random coils are never ex-tremely large. Specifically, in all we have discussed so far, the tube conformations remainfixed-length random walks, albeit anisotropic, described by the tube orientation tensor^(]Ra /]s)(]Rb /]s)&. Therefore, even if the tube orientation tensor becomes ‘‘com-pletely aligned’’ with the flow, the mean-square end-to-end radius in the flow directionRz

2 never gets more than three times larger than its equilibrium value ofna2/3.We might imagine that the tube would align completely with the flow, i.e., stretch out

to its fully extended length along the flow direction, if only the tube convection andlength truncation processes were active, such that eventually only the center portion ofthe tube is highly stretched to give rise to an aligned tube. However, with constraintrelease motion active, the center portion of the tube continually misaligns, and the smallloops so formed are continually stretched out to form the anisotropic random walk that isthe asymptotic state of the present model in the high shear rate limit.

FIG. 9. Mean-square gyration radii along principalS(q) axes vs shear rate, forn 5 20 andn 5 10 andcn

5 0.1 andcn 5 0.


VI. LIVING MICELLES

For living micelles, the stress relaxation term in the absence of CCR is not diffusivereptation along the tube, but is instead well approximated by local, single-exponentialrelaxation@Cates~1990!#. There is no distinguished place along the chain as the chainend for reptation, since a tube segment must wait for a break to occur sufficiently near itto relax.

Likewise, the fixed-length constraint may be regarded as killing off segments ran-domly, depending on whether or not they are currently near an end. Indeed, the fixed-length constraint itself collapses down from*dstrf(s,s) 5 const to trf(0) 5 (0)5 const.

The living micelle tube does undergo constraint release Rouse motion. Note that theCCR term in Eq.~22! in fact makes no reference to chain length, nor does the coefficientof isotropic diffusion in the (s,s8) plane depend on monomer position. Thus for livingmicelles, we expect a collapse of Eq.~22! to a one-dimensional partial differential equa-tion in which x 5 s2s8 is the dependent variable.

The resulting equation of motion is in fact much simpler than Eq.~22!

]f

]t5 k•f1f•kT2t21~ f2feq!1~3n/2!

]2

]x2 ~ f2feq!2lS 21x]

]xD f. ~39!

The first two terms of this equation are evidently the same convective terms as before;the third term is the simple exponential relaxation to equilibrium; the fourth term is CCRRouse diffusion in the difference coordinatex. The final term expresses the effects of theconstant-length constraint on the tube tangent correlator as a function of the differencevariablex 5 s2s8, in which small differences inx are continually advected to largerdifferences under the relabeling that follows retraction. The term 2lf arises from the‘‘stretching’’ of the tangent vectors, i.e., from the effect of relabeling after retraction onthe derivatives]/]s and]/]s8, just as in Eq.~22!.

We expand the deviatoric part off ab(x) in a Fourier series about the equilibriumstressf ab

~eq!(x) 5 (a2/3)d(x)dab

fab~x!2fab~eq!~x! 5 a2/~3n! (

p . 0~ f p!ab cos~ppx/n!

~40!

~fp!ab 5 3/~na2!E2n

ndx cos~ppx/n!@fab~x!2fab

~eq!~x!#

and Fourier transform the equation of motion to obtain

]fp

]t5 ~k1kT!1k•fp1fp•kT2t21fp2lS 2d12fp1

chp2

n2 fp1(p8

Jpp8~d1fp8!D .

~41!

Here the coupling matrixJp,q is given by

Jpp8 5 H2p82~21!p1p8

p822p2 p Þ p8

1/2 p 5 p8

. ~42!


The fixed-tube-length constraint becomes(trfp 5 0, since the equilibrium part offalready fulfills the constraint. By applying the operator(p(tr) to the equation of motion,we obtain an explicit expression forl

l 52k:(pfp

6n1ch /n2(p2trfp1(p,p8Jp,p8~31trfp8!. ~43!

Following the approach of Sec. II, we obtain an expression in this case for the devia-toric part of the stress tensor as

dsab 5cT

NeS1nD (

p 5 1

n

~fp!ab ~44!

and for the structure factor as

S~q! 5 E0

nds ds8 expF2a2qaqb /~6n!F~dab1~f0!ab!~s82s!2

14~n/p!2(p

~dab1~fp!ab!~12cos~pp~s82s!/n!/p2!GG. ~45!

Our model again takes the form of coupled ordinary differential equations for modeamplitudes, which we solve numerically. The results are qualitatively similar to theconventional unbreakable polymer case, with one important exception: the shear thinningis now strong enough to induce a maximum in the shear stress. Figure 10 shows thesteady-state shear stresssxy(g) for the living micelles withn 5 20 and various valuesof ch . For ch less than aboutch 5 4 ~or cn 5 0.27), the viscosity exhibits a region ofshear thinning with exponent less than21. Hence there is a range of values 1, ch, 4 of the phenomenological parameterch , which describes how often a tube segment

actually hops when a constraint is released, for which we predict the absence of a stressmaximum for conventional polymers and the presence of a stress maximum for livingmicelles.

FIG. 10. Steady-state stress vs shear rate for living micelles, forn 5 20 andch 5 0,0.3,1,3. A stress maxi-mum is evident untilch 5 4 or so.


VII. STEP STRAIN; DAMPING FUNCTION

Step strains are never instantaneous. We may consider ‘‘ultrafast’’ or ‘‘fast’’ stepstrains, at shear rates always large compared to the reptation time, but either large orsmall compared to the stretch relaxation timets . If the step strain is ultrafast, the chainsare initially stretched and then quickly retract in their tubes. There are issues of principleto be resolved in extending a tube-based picture of stress relaxation to this regime, yetthis is the regime typical of measurements of the damping function. If we confine our-selves to merely fast step strains, the chains do not stretch, but maintain fixed tube lengthadiabatically. It seems initially plausible that the state of the system reached a timetsafter an ultrafast step strain which must be the same as the state reached after a fast stepstrain of the same amplitude. However, we will find both evidence and arguments in thefollowing that CCR is effective during the step strain only in the fast case.

The present model implies that convective constraint release is operative during a faststep strain. Thus the damping functionh(g), which is the factors(g)/@gs8(0)# bywhich the stress after the step strain is lower than expected from linear response, mustdiffer from the Doi-Edwards result. Since the Doi–Edwards result forh(g) is in goodagreement with experiment~over a range of strains up to 10@Doi and Edwards~1986!#!,we might hope that the new damping function incorporating the effects of convectiveconstraint release is not too different.

To compute the damping function, we drop all terms in the equation of motion Eq.~22! resulting from reptation. Then, because the constraint release Rouse rate is explicitlyof order the shear rate~from k!, we may regard the equation of motion as describingevolution with strain rather than with time. We may then compute a new tube orientation‘‘ Q function’’ to replace that of Doi–Edwards, which incorporates not only the processesof affine tube convection~via the termsk•f1f•kT) and fixed tube length~via the con-straint terms!, but also constraint release Rouse motion. From this, we may compute thedamping functionh(g).

As a preliminary, if we drop not only the reptation terms but also the constraint releaseterms, Eq.~22! can be solved exactly, with the isotropic initial condition of Eq.~23!. Theresult is

f~s,s8,t ! 5E~ t !•ET~ t !

E~ t !:ET~ t !a2d~s2s8!, ~46!

whereE(t) is the deformation tensor (]E/]t 5 k•E), with initial conditionE(0) 5 1.~This approximate dynamical implementation of the constant tube length constraint is thesame as in the Larson equation for stress evolution@Larson~1988!#!.

The result Eq.~46! is equivalent to a factorization of the Doi–EdwardsQ functionsimilar to the independent alignment~IA ! approximation@Doi and Edwards~1986!#

Qab* ~E! 5^~E•u!a~E•u!b&

^uE•uu2&, ~47!

where the brackets denote averaging of the vectoru over the unit sphere.Recall that the Doi–Edwards expressions forQ with and without IA approximation

are @Doi and Edwards~1986!#

Qab~DE!~E! 5

^~E•u!a~E•u!b /uE•uu&

^uE•uu&, Qab

~ IA !~E! 5 K ~E•u!a~E•u!b

uE•uu2 L . ~48!


From eachQ function a corresponding approximation to the damping functionh(g) isobtained, by

h~g! 5Qxy~g!

gQxy8 ~0!. ~49!

From this and Eq.~46! we haveh* (g) 5 1/(11g3/3). Both h~DE!(g) and h~IA !(g)must be numerically evaluated. All three functions are quite close to each other, as shownin Fig. 11. All three approach a22 power law for large shear strains. Experimental dataof Osaki et al. ~1982! are in rather good agreement with this general trend, up to totalstrains of 10–20, as shown in Fig. 11.

Calculating the damping function with and without CCR is instructive~see Fig. 12!.Clearly, for the same reason that CCR remove the stress maximum in steady flow, thepredicted values for the stress in step strain are also larger, and the damping functionconsequently weaker, than in Doi–Edwards theory. At high strains the prediction seemsno longer to agree with data on model polymers. However, the calculation of the dampingfunction with CCR refers to a very particular condition: the forward step strain is done

FIG. 11. Doi–Edwards~DE, solid!, independent alignment~IA, dotted! and Larson-model~L, dashed! dampingfunctions vs shear rate, compared to data of Osakiet al. ~1982!.

FIG. 12. Early-time damping function vs shear rate forch 5 0 and ch 5 1, compared to Doi–Edwardsdamping function~dashed! and data of Osakiet al. ~1982!.


much faster than reptation, but also much slower than the Rouse time, since the contourlength has been kept constant here. In practice, experiments in step strain are not typicallydone at this fast rate, but at an ultrafast rate in which the chain stretches throughout thestrain almost affinely. In these circumstances there will be NO CCR during the forwardstrain, since there is no retraction. There is the possibility of CCR during the subsequentfast chain retraction, but in this phase of the dynamics there is no forward shear toconvect the new tube conformations and produce extra shear stress. So although CCRoccurs, the mechanism for translating it into extra stress is absent. Without further cal-culation at this stage, it is possible that a full theory incorporating stretch actually predictsa STRONGER damping function than Doi–Edwards, since the Rouse tube relaxation willonly add to the overall stress relaxation on retraction. As Meadet al. ~1998! have pointedout, in the case of strong stretch it is likely that the tube Rouse steps are suppressed bythe higher tension in the retracting chain. Experiments on step strain in the fast~but notultrafast! case would therefore be an interesting test of this work, but also point forwardto the need to incorporate stretch. This we leave to the future.

VIII. CONCLUSIONS

We have presented a microscopic theory of the mechanism introduced by Marruccicalled convective constraint release~CCR!, and its effect on the nonlinear rheology ofpolymer melts and concentrated solutions of living micelles. In CCR, entanglements arereleased when the free end of a chain retracts past an entangling chain. Flows of finiteamplitude would typically stretch chains were it not for the fast retraction of chainswithin their tubes; this continuing retraction gives rise to a finite rate of CCR.

Our theory applies to general flow types, so long as the deformation rates are notsufficient to stretch the chains, i.e., for deformation rates smaller than the inverse Rousetime 1/(ten2) @te is the relaxation time of one tube segment, andn is the number of tubesegments in the chain#. Since the reptation time scales asten3, this gives a range ofdeformation rates of widthn for which the flow behavior is non-Newtonian and ourtheory applies.

It is not correct to assume, as previous authors have done, that a single CCR eventcompletely relaxes stress on an entangled segment. This was made clear by Viovyet al.~1991! in another context, who pointed out the existence of the constraint release Rousetime. The time for a tube conformation to completely relax by constraint release motionis n2 times inverse hop rate.

Nonetheless, our more careful treatment gives very similar rheological behavior to theprevious work. This is because only a few CCR events per entanglement segment areneeded to stave off the catastrophic shear thinning of the Doi–Edwards model. Theoperative mechanism is that CCR events misalign tube segments, and keep them frombecoming completely aligned along the flow direction. Thus there can be some remainingshear stress held by the tube even at high shear rates.

The details of the stress relaxation mechanism that competes with the alignment areimportant. When we extend our theory to the case of living micelles, for which stressrelaxation is essentially a local process of waiting for a break in the chain to occursufficiently nearby, we find a stress maximum. This result is also in agreement withexperiments, which find shear banding and other phenomena in living micelles consistentwith a stress maximum, but no such behavior in conventional polymer melts.

The prediction of a maximum in the shear stress for living polymers holds for thehopping rate efficiencycn ~the only adjustable parameter in our theory! less than about0.27. In contrast, there is no stress maximum in the case of conventional polymers forcn


greater than about 0.07; thus for any choice ofcn in the range 0.07, cn , 0.27, wehave no stress maximum for conventional polymers and a stress maximum for livingpolymers, as observed.

In addition to rheological information, our calculations give the instantaneous struc-ture factor of chains in the flow. Neutron-scattering measurements have been made ofS(q) for deuterated chains in melts frozen in the act of flowing. These results giveadditional information about the way the chains move and relax in the flow that cannot bedetermined from the stress tensor alone. We can also compute scattering from chains withlabeled ends or centers, which could reveal even more detail, such as the greater extent towhich the middle of a chain is aligned by flow as compared to the ends.

Several authors have proposed in various contexts mechanisms by which the tubediameter or cross section becomes dependent on the applied strain in the entanglementnetwork @Mhetar and Archer~1999! and Rubinstein and Panyukov~1997!#. We haveexplicitly neglected any such mechanisms, in order to examine the consequences of CCRin isolation. According to these authors, effects on the tube cross section would set in atshear rates of order the inverse reptation time~i.e., when the tubes become significantlyaligned!. Therefore, if these effects are real, they would be active in the same regime ofdeformation rates as CCR. Combining these effects remains a subject for future work.

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