Download pdf - A constitutive model with moderate chain stretch for linear polymer melts

J. Non-Newtonian Fluid Mech. 123 (2004) 185–199

A constitutive model with moderate chain stretch forlinear polymer melts

M.A. Tchesnokova,∗, J. Molenaara,b, J.J.M. Slotc,d, R. Stepanyanc

a Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlandsb Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

c Department of Applied Physics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlandsd Material Science Centre, DSM Research, P.O. Box 18, 6160 MD Geleen, The Netherlands

Received 26 February 2004; received in revised form 26 August 2004; accepted 30 August 2004

Abstract

In our previous publication, we presented a molecular model to describe the dynamics of the interfacial layer between a flowing polymermelt and a die wall. We showed that the ensemble-averaged behavior of polymer molecules adsorbed on the wall could be successfullyd tion andc introducedf ell-knownD to includec er chains asr linear flor n of the fullt agreementw©

K unct

1

tttsEtTpcE

. Theearation

tti,otchedvedhootstress.. Inaxi-tee)

astead,

0d

escribed in terms of the so-called bond vector probability distribution function (BVPDF). The BVPDF couples the chain orientahain stretch on the level of single segment, and thus is an extension of the orientation distribution function of Doi and Edwardsor inextensible chains. In this paper, the developed formalism is extended to molecules in the polymer bulk. We show how the woi and Edwards theory (DE) for inextensible chains based on the orientation distribution function can be naturally extendedhain stretch and (convective) constraint release (CCR). The final constitutive equation accounts for such mechanisms on polymeptation, retraction, convection, contour length fluctuations, and (convective) constraint release. It is valid for both linear and non-wegimes. The proposed theory is quantitative, and contains the same input parameters as the original DE model. As an applicatioheory, a simple equation of motion for the stress tensor is derived. Despite the simplicity, its predictions are found to be in goodith available experimental data over a wide range of flow regimes and histories.2004 Elsevier B.V. All rights reserved.

eywords:Reptation; Polymer extrusion; (Convective) constraint release; Bond vector; Constitutive equation; Bond vector probability distribution fion

. Introduction

The flow behavior of entangled polymer melts has at-racted a lot of attention and has been a recurring topic overhe past decades. A number of theories have been proposedo describe the dynamics of such systems. One of the mostuccessful microscopic models was developed by Doi anddwards[1] who applied the tube concept by de Gennes[2]

o the case of entangled, monodisperse, linear polymer melts.heir model combines reptation with instantaneous and com-lete chain retraction within the mesh of moving constraintsreated by surrounding chains. Predictions of the Doi anddwards theory (DE) for large-step shear strains are known

∗ Corresponding author. Fax: +31 53 4894833.E-mail address:[email protected] (M.A. Tchesnokov).

to be in an excellent agreement with experimental dataDE model, however, failed to predict other non-linear shproperties, such as the steady-state viscosity or the relaxof stress after cessation of steady shearing.

A refinement of the DE model by Marrucci and Grizureferred to as the DEMG model[3,4], allows retraction tbe gradual and incomplete so that chains can be streby the flow. The inclusion of chain stretch leads to impropredictions for transient startup behavior, such as oversin the first normal stress difference and in the shear sHowever, it did not remove all the flaws of the DE modelparticular, both the DE and DEMG models predict a mmum in the shear stressσxy as a function of the shear raγ (at γ approximately equal to the inverse reptation timfollowed by a region whereσxy decreases asymptoticallyγ−0.5. This has never been observed experimentally. Ins

377-0257/$ – see front matter © 2004 Elsevier B.V. All rights reserved.oi:10.1016/j.jnnfm.2004.08.007

186 M.A. Tchesnokov et al. / J. Non-Newtonian Fluid Mech. 123 (2004) 185–199

the shear stress was found to be a monotonic function of theshear rate.

The flaws of the DE model and the more sophisticatedDEMG model led to a conclusion that an important relax-ation mechanism was missing in the theory. This mechanismbecomes especially important if the shear rate exceeds theinverse reptation time, that is, where both the DE and DEMGmodels predict a significant chain alignment with the flow.Recently, Marrucci[5] and Ianniruberto and Marrucci[6]have described a mechanism for stress relaxation in entangledpolymers under flow which they called convective constraintrelease (CCR). It stems from the fact that in the presenceof flow, chains are continually being stretched and relaxingback. As they do so, entanglements in which the ends of achain are involved are also continually being relaxed. It wasrecognized that constraint release is a slow, and probably notimportant, process when a monodisperse melt is at rest or inslow flow. However, when there is a flow that is fast comparedto the inverse reptation time, constraints surrounding a chainmay be swept away rapidly. In this regime, CCR may play acrucial role in the dynamics of polymer chains.

Following these ideas, Mead et al.[7] and Iannirubertoand Marrucci[8,9] have developed non-linear constitutivemodels that account for both chain stretch and CCR. Thesetheories model the effect of CCR on stress by modifying theo thatC outt n thec n ofcp f thet ncyo ce oft n them here-f ofc

i-c thes tiono entb rma werep rationo t ofs sim-u er[

cor-p dic-t redt ec-o ) isi ularw eady-s ive

model should account for CCR, CLF, and chain stretch ina consistent way. In this paper, an attempt is taken to con-struct a new microscopic non-linear model for linear entan-gled polymer melts which incorporates state-of-the-art ideasof the existing models, such as treatment of CCR as the localRouse-like motion of the tube, CLF, gradual and incompletenon-uniform chain stretch. The resulting model is quantita-tive and contains no adjustable parameters, except for thosethat can be extracted from independent molecular measure-ments. Its numerical evaluation is quite simple, yet it stillprovides a reasonable accuracy over a wide range of flowregimes and histories.

2. The parameterized chain

In order to describe the behavior of a melt under a flow,we need a reliable mathematical description of the singlechain dynamics in a mesh of moving constraints imposed bysurrounding molecules. According to de Gennes[2], theseconstraints build a tube in which the chain is confined. Inthe absence of constraint release, motion of the chain in itstube can be presented as one-dimensional “reptative” motion,so that the description of the single chain dynamics can besignificantly “eased”. However, in the presence of constraintr ouse-l thec ainf

for-m itionvm im-i illb -e allc si-cgt i-t ior ofaa on ofa f itsp edt stic( hats

u

w trya ctionso e

verall chain relaxation time. However, the assumptionCR acts in a global manner “hides” any information ab

he local chain conformation on length scales less thahain length. An essential refinement of the descriptioonstraint release was made by Viovy et al.[10] who pro-osed to treat (C)CR as a hopping (Rouse-like) motion o

ube itself with the time scale set by the inverse frequef constraint release events, and with a hopping distan

he order of the tube diameter. This approach is based oicroscopic consideration of constraint release, and t

ore allows a detailed description of the local influenceonstraint release on the chain conformation.

Recently, Milner and coworkers[11,12] proposed a mroscopic non-linear constitutive model which combinedophisticated treatment of CCR through a Rouse-like mof the tube together with chain stretch. Its further refinemy Graham et al.[13] allows chain stretch to be non-unifolong the chain contour. In parallel, a number of modelsroposed which are based on the microscopic considef the melt dynamics, and formulated in terms of a setochastic differential equations suitable for numericallations, e.g.Ottinger and Beris[14] and Hua and Schieb

15].The general conclusion of all these models is that in

oration of CCR and chain stretch leads to improved preions for both linear and non-linear flow regimes compao those of the DE (DEMG) model. Moreover, it was rgnized that inclusion of chain length fluctuations (CLF

mportant to predict the correct scaling laws for the moleceight dependence of the longest relaxation time and sttate viscosity[1]. This implies that a successful constitut

elease, the tube itself undergoes a three-dimensional Rike motion. In this case, motion of the primitive path ofhain is similar to the Brownian motion of a Rouse chound in dilute solutions[1].

As known from the Rouse theory, every spatial conation of a chain can be described by a set of the pos

ectors{Rn}n=Nn=1 , whereRn is the position vector of thenthonomer, andN is the number of monomers per chain. S

larly, the configuration of the primitive path of a chain we specified by the space curveR(s0, t), where the paramter s0 runs over a certain fixed interval, the same forhains (seeFig. 1). This parameter ‘labels’ the same phyal segment of the primitive path at all times, so thatR(s0, t)ives the position vector of the segments0 at time t. Note

hat the curveR(s0, t) determines the position of the primive path, and thus only captures time averaged behav

physical chain, in contrast to the position vectorsRn ofRouse chain. Since in a tube-based model the motiphysical chain is only represented via the motion o

rimitive path, the primitive path will, for short, be referro as ‘chain’ throughout the paper. Moreover, all stochai.e. pertaining to a single chain) variables will carry aign.

Since we have freedom in selecting the interval fors0, lets choose it as

−L0

2≤ s0 ≤ L0

2, (1)

hereL0 is the equilibrium chain length. Due to symmerguments, the ensemble-averaged values are even funf s0. The pointss0 = 0 ands0 = ±L0/2 correspond to th

M.A. Tchesnokov et al. / J. Non-Newtonian Fluid Mech. 123 (2004) 185–199 187

Fig. 1. The parameterized bulk molecule:s(s0, t) is the physical position ofthe segments0 along the primitive path at timet;L0 is the equilibrium lengthof the chain;R(s0, t) is the position vector of the segments0; λ is the localstretching of the primitive path ats0 and timet; u is the unit tangent vector.

center and free ends of a chain, respectively. A similar de-scription of the single chain spatial conformation was used in[11–13]. The physical position of a segment along the prim-itive path is defined by the curvilinear coordinates(s0, t), thecorresponding arclength from the origin to the segment (seeFig. 1). Attention must be paid thats(s0, t) is a function ofs0 and time. Contrary tos0, s(s0, t) is chain dependent whichis indicated by its hat sign. However, in the absence of flow,when chains are not stretched,s(s0, t) ≡ s0. Taking into ac-count Eq.(1), the local stretchingλ(s0, t) of the primitivepath at positions0 and timet is given by

λ(s0, t) = ∂s(s0, t)

∂s0. (2)

Clearly, it describes local stretch of the primitive path by theflow. Note that in the absence of flow,λ(s0, t) ≡ 1 at any pointalong the chain.

Alternatively to the description in terms of the positionvectorsR(s0, t), every spatial conformation of a chain canalso be described by a set of the so-called bond vectors (seeFig. 1) defined as

b(s0, t) = ∂R(s0, t)

∂s0. (3)

I d lo-c ondv evelo siblec -s

c om-p ofa ore ish helpo asedo on,i w toc nter-e thatt has ac

3. Equation of motion for the bond vectors

Given an equation of motion for each bond vectorb(s0, t)of a chain in the melt, the time evolution of its primitive pathis defined. However, Eq.(3) makes it clear that, in order toderive this equation, we must first study the dynamics of theposition vectorsR(s0, t). In the absence of constraint releaseand reptation,R(s0, t) obeys the following equation of motion[16]:

R(s0, t +�t) = [ ¯I +�t ¯K (t)]R(s0 + υ0(s0, t)�t, t). (4)

It describes motion of a chain in a moving mesh of constraintsproduced by the flow. The motion of the mesh deforms theprimitive path (this is usually referred to as convection) andis characterized by the so-called velocity gradient tensor¯K[1] which specifies how fast the velocity of the mesh changesin space. In general,K is a function of position and time. Weassume, however, convection to be homogeneous on the scaleof a single chain, and hence drop the spatial dependence in¯K . In the case of shear flow,K has only one non-zero com-ponentKxy equal to the shear rate. In Eq.(4), the expressionin square brackets specifies convection of the primitive pathover the time interval betweent and t +�t (�t is small).Since the flow stretches the chain, convection is always ac-c ments ngtgg

υ

ξ

H n Eq.(t f aconH oni il-n sionfw ted asa left e,ar

a fm n

ts orientation and modulus coincide with the tangent anal stretching of the primitive path, respectively. The bector ‘couples’ the local orientation and stretch on the lf a single segment. Note that in the case of inextenhains or absence of flow,b(s0, t) coincides with the correponding unit tangent vectoru(s0, t) to the primitive path.

Given the parameterization functionR(s0, t) for everyhain in the melt, the dynamics of the whole system is cletely defined. Unfortunately, this requires a solutionn enormous system of coupled equations, and therefardly possible. The description of the system with thef the bond vectors contains less information than that bn the position vectors. However, as will be shown later

t includes all the necessary information we need to knoalculate different macroscopic parameters of practical ist, such as stress and viscosity. Moreover, we will show

he formalism based on the bond vectors is simple, andlear physical interpretation.

ompanied by chain retraction. Due to retraction, the seg0 “slides” along the primitive path. Its displacement alohe s0-axis over the time interval betweent and t +�t isiven byυ0(s0, t)�t, whereυ0 is the retraction velocity[16]iven by (for definiteness we assumes0 ≥ 0)

ˆ0(s0, t) = 1

λ(s0, t)

∫ s0

0dx ξ(x, t),

ˆ = ∂λ

∂t−K��u�u�λ. (5)

ere, summation is assumed over repeating indices. I5), ξ(x, t) is the retraction rate of the segmentx at timet, sohat υ0(s0, t) actually gives the retraction rate of a part ohain betweenx = 0 andx = s0. The local stretchingλ(s0, t)f the segments0 at timetwas defined earlier in Eq.(2). Mil-er et al.[11] derived equations similar to Eqs.(4) and (5).owever, the retraction rateξ was assumed to be positi

ndependent. Graham et al.[13] extended the results by Mer to account for non-uniform chain stretch. The expres

or υ0(s0, t) found in [13] can be obtained from Eq.(5) ife assume that a chain inside its tube can be represenbead-spring system. Then,ξ is proportional to the tensi

orce along the chain contour given byκ (R′′ · u) whereκ ishe spring constant,u is the unit tangent vector to the tubndR′′ is the second derivative of the position vectorR withespect tos0.

Differentiating both sides of Eq.(4) with respect tos0nd taking the limit�t → 0 yield the following equation ootion for the bond vectorb(s0, t) in the absence of reptatio


and constraint release:

∂b(s0, t)

∂t= ¯K (t) · b(s0, t) + ∂

∂s0(υ0(s0, t) b(s0, t)). (6)

The RHS of Eq.(6)contains two contributions. The first termcorresponds to convection, and therefore describes affine ‘ro-tation’ and ‘stretch’ of the bond vector by the flow. The sec-ond term arises from retraction, and describes ‘sliding’ and‘shrinking’ of the bond vector along the chain contour. Notethat υ0 also depends on the flow rate (viaλ) and vanishes atrest.

The next step towards a complete equation of motion forthe bond vectors is to incorporate reptation into Eq.(6). Ac-cording to Doi and Edwards[1], reptation, i.e. curvilineardiffusion of the entire chain along its own contour, can bemodelled as an one-dimensional Rouse motion. Let�ζ(s0, t)be the physical shift of the segments0 along the primitive path(over a small time interval�t) due to reptation. Due to theBrownian nature of reptation,�ζ(s0, t) is stochastic and canbe modelled by a Wiener process. Hence,

〈�ζ〉 = 0, 〈�ζ2〉 = 2Dc�t, (7)

whereDc is the diffusion coefficient of the Rouse motion[1]given by

D

H m-b effi-c vari-a rp f mo-t

R

w ts ec ia

�

S tion,c nw nf

b

w tion.A m tot -t ndo

This implies that Eq.(5) is non-local with respect tos0, andequations of motion for all the segments of a chain are cou-pled. In the absence of chain stretch, i.e.λ ≡ 1, the retractionvelocity υ0 (5) reads as

υ0(s0, t) = −∫ s0

0dxK��u�u�, (12)

where use was made of the fact that in this regime,b(s0, t)coincides with the corresponding unit tangent vectoru(s0, t)to the primitive path. One may ascertain that Eq.(11) thenboils down to the equation of motion foru derived by Doiand Edwards[1] for inextensible chains.

4. The bond vector probability distribution functionof bulk chains

In the previous section, we derived the equation of motionfor the bond vectors of a chain which accounts for repta-tion, convection, and retraction. In general, the motions ofneighboring chains in the melt are highly correlated via con-straint release. This implies that its solution in the presence ofconstraint release would require consideration of macroscop-ically large system of coupled equations. In reality, however,we are only interested in macroscopic parameters of the meltw embleog

σ

G

w dv pere tively.I n-s hee ndv ec

〈

w nf

f

C tora tt∫

c = kBT

Nς. (8)

ere,kB, N, ς, andt are the Boltzmann constant, the nuer of monomers per chain, the monomeric friction coient, and the absolute temperature, respectively. Thence of�ζ is proportional to

√�t, as is typical for Wiene

rocesses. In the case of pure reptation, the equation oion for R(s0, t) can be written as[1]

ˆ (s0, t +�t) = R(s0 +�ζ0, t), (9)

here�ζ0 is a displacement along thes0-axis of the segmen0 over the time interval fromt to t +�t. It is related to thorresponding physical shift�ζ along the chain contour v

ζ0(s0, t) = �ζ(s0, t)

λ(s0, t). (10)

ince reptation is independent of retraction and convecombination of Eqs.(4) and (10)together with differentiatioith respect tos0 leads to the following equation of motio

or the bond vectors:

ˆ (s0, t +�t) = [ ¯I +�t ¯K (t)]b(s0 + υ0�t +�ζ0, t)

+[�t

∂υ0

∂s0+ ∂�ζ0

∂s0

]b(s0, t), (11)

hich now accounts for convection, reptation, and retracs is seen, the contribution of reptation has a similar for

hat of retraction. However, according to Eq.(5), the retracion velocityυ0 of the segments0 depends on the stretch arientation of other segments with the coordinatesx ≤ s0.

hich are actually expressed via averages over the ensf chains. For example, the local stressσ�� in the melt isiven by[1]

��(t) = G0

L0

∫ L0

0ds0〈b�(s0, t)b�(s0, t)〉,

0 = 3kBTc

Ne, (13)

hereb�, G0, Ne, andc are the�-component of the bonectorb, the elastic modulus, the number of monomersntanglement segment, and the chain density, respec

n Eq. (13), the brackets〈· · ·〉 denote averaging over the eemble of chains. IfNall is the total number of chains in tnsemble andN(b, s0, t) is the number of chains whose boector at positions0 and timet is equal tob, then the averagan be written as

b�(s0, t)b�(s0, t)〉 =∫R3

d3b b�b�f (b, s0, t), (14)

heref (b, s0, t) is the bond vector probability distributiounction (BVPDF) defined as

(b, s0, t) = N(b, s0, t)Nall

. (15)

learly,f (b, s0, t) is the fraction of chains whose bond vect s0 and timet is equal tob. Eq. (15) makes it explicit tha

he BVPDF is normalized

R3d3b f (b, s0, t) = 1. (16)


Here, the integral is taken over all possible values of the bondvector. From Eq.(14), it follows that the BVPDF can be alsoformally written as

f (b, s0, t) = 〈δ[b − b(s0, t)]〉, −L0

2≤ s0 ≤ L0

2. (17)

Both definitions of the BVPDF (Eqs.(15) and (17)) are equiv-alent. Note that the latter (see Eq.(17)) shows explicitly thatthe BVPDF involves ensemble averaging, and consequentlyis an even function ofs0

f (b, s0, t) = f (b,−s0, t).

As follows from Eq.(17), given the equation of motion for thebond vectors (see Eq.(11)), one can derive that forf (b, s0, t).To this end, let us calculatef (b, s0, t +�t) for small�t.Substituting Eq.(11) into Eq.(17), we arrive at

f (b, s0, t +�t) = 〈δ[b − b(s0 + υ0�t +�ζ0, t)

− g(b(s0, t), s0, t)]〉, (18)

where

g(b(s0, t), s0, t) = �t ¯K (t) · b(s0, t) +�t∂υ0

∂s0b(s0, t)

+ ∂�ζ0 b(s0, t). (19)

A on,a pro-p(

uchs ceo eousa thatt t ontls dtd

f

T

〈

where the notationb = |b| is used and the third-order BVPDFf (3)(b′, x|b′′, y|b, s0, t) is the fraction of chains whose bondvectors atx, y, ands0 are equal tob′, b′′, andb, respectively.In Eq.(21), use was made of the fact thatξ(s0, t) only dependson the positions0 and timet via the bond vectorb(s0, t) (seeEq.(5)), so thatξ(s0, t) = ξ(b(s0, t)). Note that the last termon the RHS of Eq.(21) is proportional toξ(∂λ/∂s0), which isof the order of〈(λ− 1)2〉, and can be neglected if the chainstretch is small.

Next, expanding the first term on the RHS of Eq.20 inpowers of�t and neglecting second-order contributions, wehave

〈δ[b − b(s0 + υ0�t +�ζ0, t), t]〉

=⟨{

1 + υ0�t∂

∂s0+�ζ0

∂

∂s0+ 1

2(�ζ0)2

∂2

∂s20

}

× δ[b − b(s0, t)]

⟩, (22)

where use was made of the fact thatζ0 is proportional to√�t (see Eq.(7)). The second term on the RHS was already

studied in[16]. It describes relaxation of the BVPDF due toretraction and can be evaluated as⟨ ⟩

w r.-

a nn bondvv nismf hains ibu-t n ben pta-t winga⟨

T ue tor in-c(

∂s0

s seen,g contains contributions of convection, retractind reptation. The convection and retraction terms areortional to�t, whereas the last term is proportional to

√�t

see Eq.(7)).The characteristic time scale of retraction is usually m

maller than that of reptation[1]. Therefore, in the presenf retraction, reptation can be considered as a simultannd coordinated motion of all the segments of a chain, so

he chain moves in its tube as a whole. This implies thahe time scale of reptation,ζ0 is independent ofs0, and theast term on the RHS of Eq.(19) can be neglected. Then,g isimply proportional to�t. For small�t, g is small compareo b. So, expanding the RHS of Eq.(18) in powers ofg andiscarding second-order terms, we have

(b, s0, t +�t) = 〈δ[b − b(s0 + υ0�t +�ζ0, t), t]〉

− ∂

∂b· 〈g(b, s0, t)δ[b − b(s0, t)]〉. (20)

he last term on the RHS can be evaluated as follows:

g(b, s0, t) δ[b − b(s0, t)]〉

= �t

[¯K (t) · b − ξ(b)

bb

]f (b, s0, t)

−�tbb2

∫R3

d3b′ ξ(b′)∫R3

d3b′′ |b′′|∫ s0

0

× dx

{∂

∂yf (3)(b′, x|b′′, y|b, s0, t)

}y=s0

, (21)

υ0∂δ[b − b(s0, t)]

∂s0

= 1

b

∫R3

d3b′ ξ(b′)∫ s0

0dx

∂

∂s0f (2)(b′, x|b, s0, t)

+ 1

b2

∫R3

d3b′∫R3

d3b′′ ξ(b′) |b′′|∫ s0

0

× dx

{∂

∂yf (3)(b′, x|b′′, y|b, s0, t)

}y=s0

, (23)

heref (2)(b′, x|b, s0, t) is the BVPDF of the second ordeThe last two terms on the RHS of Eq.(22)describe relax

tion of the BVPDF due to reptation. Sinceζ0 is a zero-meaoise, and is thus uncorrelated with the correspondingector at the same point, the linear (with respect toζ0) termanishes. Reptation is the dominant relaxation mechaor stress in the absence of flow. In flow regimes where ctretch becomes important, it gives only a minor contrion compared to fast convection and retraction, and caeglected. On the other hand, for flow regimes in which re

ion is relevant, chains are hardly stretched, and the follopproximation holds

(�ζ0)2∂2δ[b − b(s0, t)]

∂s20

⟩≈ 2Dc

∂2f (b, s0, t)

∂s20�t. (24)

his approximation actually means that displacements deptation along thes0-axis and along the chain contour coide. Finally, summing up all the contributions inEqs. (21)–24), the full equation of motion for the BVPDFf (b, s0, t)


reads as

∂f (b, s0, t)∂t

= Dc∂2f (b, s0, t)

∂s20+ 1

b

∫ s0

0dx

∫R3

d3b′ξ(b′)

× ∂f (2)(b′, x|b, s0, t)∂s0

− ∂

∂b· [· · ·]

+ 1

b2

∫ s0

0dx

∫R3

d3b′ξ(b′)∫R3

d3b′′|b′′|

× ∂f (3)(b′, x|b′′, y|b, s0, t)∂y

∣∣y=s0 , (25)

where

[· · ·] =[

¯K (t) · b − ξ(b)bb

]f (b, s0, t)

− bb2

∫R3

d3b′∫R3

d3b′′ξ(b′)|b′′|

×∫ s0

0dx

∂

∂yf (3)(b′, x|b′′, y|b, s0, t)

∣∣y=s0 . (26)

Eq. (25) describes the ensemble-averaged dynamics of thespatial configuration of a chain in the melt in the presence offlow. It accounts for reptation, convection, and retraction. Thefirst term on the RHS pertains to relaxation of the BVPDFd witht ata alt sec-o -tf eens genec ions.H eenn ima-tc

f

Ac st longt -t sb Notet ow,a lk sotc e so-

lution of full Eq. (25) is required. However, even for a flowwhose rate is of the order ofT−1

R , i.e. the inverse Rouse time[1], chain stretch is small[7]. This implies that on the lengthscale of one segment, the flow can still be considered as aperturbation, andlc is small compared toL0. Therefore, itis reasonable to expect that the closure approximation in Eq.(27) is applicable even in the non-linear flow regime withflow rates larger thanT−1

R . Incorporation of CCR which de-stroys any correlations between different chain segments maypostpone it to even higher flow rates.

The third term on the RHS of Eq.(25) has the form ofa divergence and contains both a contribution of convectionand retraction. It depends on the direction of the vectorb,and then determines deviation of the BVPDF from that atrest which is isotropic.

Eq.(25)shows that in general, the BVPDF cannot be de-composed in a pure ‘orientational’ and ‘stretch’ part. Thismeans that the local orientation and local stretch are coupled,so the pre-averaged approximation

〈u�u�λ〉 ≈ 〈λ〉〈u�u�〉,

which is widely used in the literature for all flow regimes, canonly be applied for regimes of small chain stretch whereλ isclose to unity. Finally, in order to solve Eq.(25), one mustae ll thedi tionp sc

f

5

uan-t aintr e-t asedb Thef ), thel to int itsr sidei werep hicht ins.I und-i oves( ments isec

ue to reptation. It has the form of a diffusion processhe coefficientDc defined in Eq.(8). This makes it clear thfter the timeTD = L2

0/π2Dc, the chain will escape its initi

ube, thereby “renewing” its spatial configuration. Thend and last terms on the RHS of Eq.(25)stem from retrac

ion. As seen, they contain higher order BVPDFsf (2) and(3) which include information about correlations betweparate chain segments along the chain contour. So, inral, the equation of motion for the BVPDFf (b, s0, t) is notlosed, and requires consideration of higher order functowever, in a flow regime where only correlations betweighboring segments are important, a closure approx

ion can be used. For example, the second-order BVPDFf (2)

an be approximated as follows:

(2)(b′, s′0|b, s0, t)≈ (1 − e−(|s0−s′0|/lc))f (b, s0, t)f (b′, s′0, t)

+ e−(|s0−s′0|)/lcf(b,s0 + s′0

2, t

)δ(b − b′). (27)

similar approximation can be written forf (3). Here,lc is theharacteristic correlation length (lc � L0) which specifiehe range of interactions between different segments ahe chain contour. Eq.(27) makes it explicit that if the disance between two segments is larger thanlc, any correlationetween their orientations and stretch can be ignored.

hat lc is a function of the flow rate. In the absence of flchain in the melt can be considered as a random wa

hat lc = 0. If the flow is fast enough, thenlc ≈ L0, and thelosure approximation becomes poor. In this regime, th

-

lso specify the boundary conditions forf (b, s0, t). Since thends of a chain are free to ‘choose’ their direction and airections are equally probable,f (b, s0, t) at s0 = ±L0/2

s isotropic. Besides that, due to fast retraction equilibrarocesses active at the free ends[7], stretching at both endan be neglected, and so(

b,± L0

2, t

)= 1

4πδ(|b| − 1). (28)

. Convective constraint release

The formalism developed above can be made fully qitative by specifying the frequency of convective constreleaseν as a function of the molecular and flow paramers. First, note that constraints on a chain can be reley either reptation or retraction of surrounding chains.

ormer is referred to as thermal constraint release (TCRatter as convective constraint release. To findν, let us poinut a single molecule in the melt and follow its evolution

ime. The flow “stretches” the chain. This is followed byetraction which results in a movement of the test chain ints tube. This process can be imagined as if the chainulled by one of its ends through the melt. The tube in w

he test chain is moving is built out of neighboring chan turn, the test chain also imposes constraints on surrong molecules. So, if one of the ends of the test chain minside its tube) a distance equal to the mean entanglepacing ¯a, it will release one (if entanglements are pair-wontacts) or more entanglements.


Fig. 2. Time evolution of the test chain under imposed flow: (a) the test chainat timet = 0; (b) the test chain in the absence of retraction at timet = t∗;(c) the test chain in the presence of retraction at timet = t∗.

Let t∗ be the time necessary for an end of the test chainto move the distance ¯a. An expression fort∗ can be found bycarrying out the following thought experiment. Imagine thatat timet = 0, we “switch off” retraction so that chains canbe infinitely stretched. Then, at timet∗, we take a snapshot ofthe test chain. Next, we “switch on” retraction and repeat theexperiment. The obtained pictures are presented inFig. 2.

Comparison of thet = 0 andt = t∗ (no retraction) snap-shots yields that the position vectors of the test chain takenat timet = 0 andt = t∗ satisfy

R(s0, t∗) = ¯E(t∗,0) · R(s0,0),

−L0

2≤ s0 ≤ L0

2, (29)

where ¯E(t∗,0) is the deformation tensor[1] which specifieshow the tube of the test chain is deformed by the flow overthe time interval fromt = 0 to t = t∗. From Eq.(29), it fol-lows that at timet = t∗, the segments0 = L0/2 − a0 has theposition vector

¯E(t∗,0) · R(L0

2− a0,0

). (30)

Now we compare thet = 0 andt = t∗ (with retraction) snap-shots. Let∆(s0, t∗) be the distance (along thes0-axis) that ispassed by the segments0 due to retraction over the time in-terval fromt = 0 to t = t∗. Therefore,

R

Isw

∆

wt or

(in the presence of retraction)

¯E(t∗,0) · R(L0

2+ ∆

(L0

2, t∗

),0

). (33)

By definition, t∗ is the time necessary for the segments0 =L0/2 to move a distance equal to the mean entanglementspacing. In terms of the position vectors, this implies that aftertime t∗, this segment arrives at the same point at which theposition vector of the segments0 = L0/2 − a0 of the ‘non-retractable’ chain is located (seeFig. 2). So the followingrelation holds:

¯E(t∗,0) · R(L0

2− a0,0

)

= ¯E(t∗,0) · R(L0

2+ ∆

(L0

2, t∗

),0

), (34)

which is in fact an implicit equation fort∗. Substitution ofEq.(32) into Eq.(34)yields⟨∫ t∗

0dt∫ L0/2

0dx ξ(x, t)

⟩= −a0. (35)

Here the averaging over the ensemble has been introducedsince every chain has in general its ownt∗. In Eq. (35), usewas made of the boundary conditions for the local stretch-i(i con-sd ofc 1( s fortf ntsn flowre fromEe

t

w ntw thanτ

b e-q f ac melta move-me rentp imet n-t ntsp time

ˆ (s0, t∗) = ¯E(t∗,0) · R(s0 + ∆(s0, t

∗),0),

−L0

2≤ s0 ≤ L0

2, (31)

f υ0(x, t) is the retraction velocity (along thes0-axis) of theegments0 at timet, then the displacement∆(s0, t∗) can beritten as

ˆ (s0, t∗) =

∫ t∗

0dt υ0(s0, t)

=∫ t∗

0dt

1

λ(s0, t)

∫ s0

0dx ξ(s0, t), (32)

here use has been made of Eq.(5). According to Eq.(30), atime t = t∗, the chain ends0 = L0/2 has the position vect

ng, that is,λ(±L0/2, t) = 1 (see Eq.(28)). Givenξ(x, t), Eq.35)becomes an integral equation with respect tot∗. Sincea0s the equilibrium entanglement spacing and thereforetant,t∗ does not depend on time explicitly. However,t∗ mayepend on time viaξ(x, t). Note that the retraction ratehain segments close to the free ends is of the order of/τeτe = TR/Z

2B is the Rouse time of one segment) wherea

hose in the middle, the retraction rate is given 1/TR. There-ore, Eq.(35) shows that it is mainly retraction of segmeear chain ends that causes constraint release. For aegime in whicht∗ � τe, on the time scaleTR, the chainnds can be considered to be at local equilibrium. Then,qs.(44) and (35), we find an explicit expression fort∗, thensemble averagedt∗

∗ ≈ a0

{∫ L0/2

0dx |ξ(x, t)|

}−1

, (36)

here ξ(x, t) is the mean retraction rate of the segmexhich is also averaged over the time scale much larger

e, but smaller thanTR. Note that in the steady-state, Eq.(36)ecomes exact. Givent∗, one can readily find the CCR fruencyν which is equal to the inverse mean life-time oonstraint. Let us assume that all entanglements in there pair-wise contacts between separate chains, so thatent of a chain over the distance ¯a will destroy only onentanglement. Since entanglements are built out of diffearts of chains, not all of them will be released after t∗. Let us point out a unit volume in the melt which coains, say,N chains. IfZB is the mean number of constraier chain, then the total number of entanglements (at


t = 0) in the chosen volume isNent = ZBN/2. The numberof entanglements in this volume at timet = t∗ is

Nent(t = t∗) = ZBN

2−N. (37)

Here, since we are only interested in the mean life-time ofan entanglement, the entanglement creation mechanism wasignored. LetτCCR be the mean life-time of an entanglementdue to CCR. Therefore,

Nent(t = t∗) −Nent(t = 0)

Nent(t = 0)= − t∗

τCCR= − 2

ZB. (38)

From Eqs.(36) and (38), we finally have

ν = 1

τCCR≈ 2

2

L0

{∫ L0/2

0dx |ξ(x, t)|

}, (39)

where we used thatL0 = a0ZB [1]. The extra-prefactor 2 onthe RHS stems from the fact that both ends of a chain con-tribute to CCR. In the single relaxation time approximation,ξ(x, t) ∝ [λ(x, t) − 1] (see Eq.(44)). Therefore, in the lin-ear flow regime at flow ratesγ < T−1

D (TD is the reptationtime), where chains are hardly stretched,ν < T−1

D so thatCCR plays only a minor role in the chain dynamics and theDE (DEMG) model can be used. However, at higher flowr antr veryh xi-m essiof

ξ

w q.( sf

ν

C eiro ndst ragev l toK shv , thism idesa t (thes m[ cyowata ins ction

rate ξ in Eq. (39) is given by Eq.(40) whereas in ([18]) byEq.(44).

Until now, we have ignored the presence of thermal con-straint release. This is only correct if the polymer chains arelong enough. In a real experiment, however, the mean num-ber of entanglement segments per chain is sometimes rathersmall, and TCR may also play a role in the chain dynam-ics. An explicit form of the frequency of TCR can be readilyfound from Eq.(7). If t∗ is the mean time needed for a chainto pass the distancea0 due to reptation, then

t∗ = a20

2Dc. (42)

Here, chain stretch is neglected. Therefore, fromEqs. (38)and (42), the frequencyνTCR of TCR can be written as

νTCR ≈ 4Dc

ZBa20

[1 + C√

ZB

]2

, (43)

where we also took into account the effect of CLF on thefrequency of constraint release (see, for example, Doi andEdwards[1]). The coefficientC ≈ 1.69 was found by nu-merical simulations in[19]. According to Eq.(51), the dif-f ofr

6

(( tr riuml odei im-p rm ta-n

ξ

T icha xationt i-c elax-a nt.Nfls ullt( dsa

atesγ > T−1D , τCCR< TD, and CCR becomes an import

elaxation mechanism on polymer chains. Note that atigh flow ratesγ > T−1

R , the single relaxation time approation becomes poor, and one has to use the exact expr

or the retraction rateξ(x, t) [16]

¯= 3ZBDc∂2λ

∂x2 , (40)

hereDc is the reptation diffusion coefficient defined in E8). Taking into accountEq. (5), Eq. (39)can be rewritten aollows:

= 22

L0

∫ L0/2

0dx

[K�� < u�u�λ > −∂λ

∂t

]. (41)

learly,ν contains two contributions. In order to explain thrigin, let us imagine a chain which is pulled by one of its e

hrough a melt. If the chain is inextensible, then the aveelocity between the chain and the melt is proportiona�� < u�u� > (the first term), the projection of the me

elocity on the chain contour. In the presence of stretchotion is also accompanied with retraction which provn additional velocity between the chain end and the melecond term). Graham et al.[13] and Likhtman and Graha18] also derived an explicit expression for the frequenνf CCR in terms of the local retraction rate. Eq.(39) can beritten in the form similar to that found in[13] if we furtherssume thatλ2 ≈ λ2 and change the integral over thes0-axis

o that along the chain contour. Note, however, that Eq.(39)grees with that found in[18] only in the case of small chatretch. The difference is that at high shear rates, the retra

n

usion coefficient due to TCR isZB times less than thateptation.

. The equation of motion for the BVPDF

In general, the retraction rate per single segmentξ(s0, t)40) is represented by a set of relaxation timesTp = TR/p

2

p = 1,3,5, . . .), where the Rouse timeTR is the longeselaxation time needed for a chain to restore its equilibength. The characteristic relaxation time of the second ms about 10 times smaller than that of the first mode. Thislies that for the flow rates less than 10T−1

R , higher ordeodes withTp (p = 3,5, . . .) can be considered instaneous so thatξ(s0, t) can be approximated as

ˆ(s0, t) ≈ − λ(s0, t) − 1

TR. (44)

his is the single relaxation time approximation in whll the segments are assumed to have the same rela

ime equal to the Rouse timeTR. It is, however, not applable for segments near the chain ends, since their rtion time is given byτe, the Rouse time of single segmeote that a similar approximation was used in[7]. For aow regime in which the approximation(44) is valid, chaintretch is still small[7], and a remarkable reduction of the fheory comes out. Taking into account Eq.(44), fromEqs.25) and (26), the equation of motion for the BVPDF reas (terms of the order ofl2c, lc〈(λ− 1)〉, and〈(λ− 1)2〉 are


neglected)

∂f (b, s0, t)∂t

=[Dc + 3νa2

0

4

]∂2f (b, s0, t)

∂s20

− 1

TR

λ(s0, t) − b

bf (b, s0, t)

+ 1

b

{∫ s0

0dx ξ(x, t)

}∂f (b, s0, t)

∂s0

− ∂

∂b·[(

¯K (t) · b + 1

TR

b− 1

bb)f (b, s0, t)

],

(45)

where use was made of the closure approximation forf (2)

(see Eq.(27)). Moreover, as was argued before, the termswith f (3) are of the order of〈(λ− 1)2〉, and therefore canbe neglected (compared to terms of the order of〈(λ− 1)〉).In Eq. (45), ξ(s0, t) and λ(s0, t) are the ensemble-averagedretraction rate and local stretching of the segments0 at timet, respectively. Note thatλ(s0, t) can be in turn expressed viathe BVPDF as

λ(s0, t) =∫R3

d3b bf (b, s0, t), (46)

w theb ro-p e toCo thef i-ai

ainsi vec-t tua-t ad-m oft ing ah chaine y aref n oftm asese canb

τ

w ere,u nds issT

d ofm

equilibrium BVPDF)

− 1

τ(s0)

[f (b, s0, t) − feq

], feq(b) = 1

4πδ(b− 1). (48)

The final equation of motion for the BVPDF (see Eqs.(45)and (48)) is at the heart of the present model. It describesboth evolution of the local chain orientation and the localchain stretch in flow, and includes all major mechanismson bulk chains. It is a natural extension of the well-knowntheory of Doi and Edwards[1] developed for inextensiblechains. In contrast to the DE model, the present formalismself-consistently includes chain stretch and CCR, and is validfor both linear and non-linear flow regimes. Let us show nowthat in the limit of inextensible chains, Eq.(45) can be re-duced to that of Doi and Edwards. In the absence of chainstretch, the BVPDF can be presented as

f (b, s0, t) = δ(b− 1)φ(u, s0, t), (49)

whereφ(u, s0, t) is the orientation distribution of unit tangentvectors introduced in the DE theory. Noting that for inexten-sible chains,λ ≡ 1 along the chain contour, from Eq.(5), itfollows that the retraction rate per segmentξ(b) is equal to−K��u�u�, and the retraction velocity ˆυ0 is given by Eq.(12). Substituting Eq.(49)into Eq.45and integrating overb,one may ascertain that Eq.(45) boils down (without the CRtw

7

nsti-t , letud

S

A o-p e ofS

ft ctor,w

here the integral is taken over all possible values ofond vector. So, Eq.(45)is a second-order non-linear integartial differential equation. Note that a contribution duR was added in Eq.(45). As shown in[16], it has the formf a diffusion process with the coefficient proportional to

requency of constraint releaseν and the equilibrium tube dmetera0 squared. The explicit expression forν was derived

n the previous section (see Eq.(41)).Eq.(45)accounts for such mechanisms on polymer ch

n the melt as reptation, retraction, convection, and (conive) constraint release, but ignores contour length flucions (CLF)[1]. However, the present formalism readilyits inclusion of CLF. These fluctuations involve motion

he chain ends into the tube, thereby temporarily creatigher than the average density of monomers near thends. When the chain ends move outward again, the

ree to choose their direction so that the initial orientatiohe tube relaxes. According to Milner and McLeish[17], theean relaxation time of chain segments due to CLF incre

xponentially with the distance from the chain ends ande written as

(s0) ≈ τ0e0.75ZB(1−2s0/L0)2, (47)

hereZB is the mean number of constraints per chain. Hse was made of the fact that chain stretch near the emall so thats(s0, t) ≈ s0. For the time constantτ0, we useR/2. A more accurate prefactor, which depends ons0, waserived in[17]. The contribution of CLF to the equationotion for the BVPDF(45)can be then written as (feq is the

erm) to the well-known equation forφ(u, s0, t) [1], derivedithout independent alignment approximation.

. The bond vector correlation function

As an application of the developed theory, a simple coutive equation will be derived in this section. To this ends introduce the bond vector correlation functionS��(s0, t)efined as the second moment of the BVPDF

��(s0, t) = 〈b�(s0, t)b�(s0, t)〉 =∫R3

d3b b�b�f (b, s0, t).

(50)

ccording to Eq.(13), the local stress in the melt is prortional to the averaged along the chain contour valu��(s0, t). The equation of motion forS�� follows directlyrom Eq. (45), namely multiplying both sides byb�b� andhen integrating over all possible values of the bond vee find

∂S��(s0, t)

∂t

= K�γS�γ (s0, t) +K�γS�γ (s0, t)

+[Dc + 3νa2

0

2

]∂2S��(s0, t)

∂s20− 2

λ(s0, t) − 1

TRS��(s0, t)

+∫ s0

0dx ξ(x, t)

∂S��(s0, t)

∂s0− 1

τ(s0)(S�� − S

eq��), (51)


where terms of the order of〈(λ− 1)2〉 are neglected. HereS

eq�� = δ��/3 is the equilibrium (no-flow) value ofS��. Note

that the mean local stretchingλ ≈ √S��, so that Eq.(51)is a

non-linear integro-partial-differential equation. The first twoterms on the RHS correspond to convection, and describelocal rotation and stretch of a chain. They show that in thepresence of flowS�� may contain non-diagonal elements,proportional to the corresponding shear stresses. The thirdterm contains contributions due to reptation and constraintrelease. Both have the form of a diffusion process, and thuscan be treated as diffusion along and perpendicular the chaincontour, respectively. The boundary conditions forS�� canbe readily obtained from those for the BVPDF (see Eq.(28)).

Note thatS�� ‘couples’ both the local orientation andstretch. Therefore, Eq.(51) does not only capture the dy-namics of the local orientation in flow, but also the dynamicsof the local stretch. In fact, the equation of motion for themean local stretchingλ(s0, t) of the segments0 directly fol-lows from Eq.(51)by taking the trace of both sides. Eq.(51)shows that apart from retraction, a chain may also relax itsstretch via CR. This agrees with the proposal of Mead et al.[7], who showed that CR may also lead to relaxation of thelocal stretch via removal of constraints on a ‘tout’ piece ofchain. They argued that at high flow rates, when chains arestretched, removal of a constraint will more likely result inr calo ech-a ribedb CRf eT raceo alt wefi

w tour,s tchd odel[

tm fa-m nta nf ci[ sedo haincw ndM hingi thec s forC

The molecular model developed by Milner and cowork-ers [11,12] and Graham et al.[13] provides an alterna-tive approach to constitutive modelling which is basedon the so-called tangent correlation functionf (s, s′, t) =〈b�(s′, t)b�(s, t)〉. Note thatf (s, s′, t) can be expressed viathe second-order BVPDFf (2), and therefore can be consid-ered as an extension of the “one-point” bond vector correla-tion functionS�� (50) introduced in this paper. The tangentcorrelation functionf (s, s′, t) satisfies a very sophisticatednon-linear equation of motion which is also based on the as-sumption of small correlation length between separate chainsegments. Its solution demands intensive numerical calcula-tions. In contrast, the equation of motion forS��(s0, t) (51)is only two-dimensional and can be easily solved numeri-cally using conventional computation methods for non-linearequations. Later on, we will show that despite the simplestructure of the constitutive equation(51), the present modelprovides a good agreement with experimental data over awide range of flow regimes and histories.

Recently, Likhtman and Graham[18] proposed a simpli-fied version of the ‘full’ model[13]. They derived an equa-tion of motion (the Rolie–Poly equation) for the stress tensorwhich accounts for convection, retraction, and CCR. Notethat Eq.(51)can be written in the form similar to that of theRolie–Poly equation if we assume that the chain stretch is po-s odeso

8

i-c eare onlyft -t romi annotb es noti , oneh tersr

l arep tn es i-c ncew eD gimeaIst mi-n doesn r in

elaxation of the local stretch than in relaxation of the lorientation. However when chain stretch is small, both mnisms occur at the same frequency. This is also descy Eq.(51), namely given an explicit expression for the

requencyν, one may directly find the total relaxation timtot of chain stretch in the presence of CCR. Taking the tf both sides of Eq.(51) and regrouping terms proportion

o (λ− 1), in the single relaxation time approximation,nd that

1

Ttot≈ 1

TR+ 1

TCCR, TCCR = 2

3

Z2B

π2

1

ν, (52)

here uniform chain stretch was assumed along the cono thatTCCR is the overall relaxation time of chain streue to CCR, in accord with the Verdier–Stockmayer m

1]. A similar expression forTtot was found in[7].In the absence of chain stretch, the correlation functionS��

urns into the orientation tensor〈u�u�〉 of the DE (DEMG)odel which is usually calculated with the help of theous Doi–Edwards tensorQ�� [1]. Note that the presepproach does not includeQ��, and the equation of motio

or 〈u�u�〉 is written out directly. Ianniruberto and Marruc8,9] proposed a simple constitutive model which is ban the equation of motion for the averaged (along the contour) orientation tensor〈u�u�〉. Note that Eq.(51)can beritten in the form similar to that found by Ianniruberto aarrucci if we further assume that the mean local stretc

s independent of the local orientation and position alonghain contour, and neglect high-order relaxation modeR and reptation.

ition independent, and neglect CLF and high-order mf reptation and constraint release.

. Results and discussion

The constitutive equation (Eq.(51)) was solved numerally using the conventional Newton method for non-linquations. As the original DE model, the theory contains

our parameters: the reptation timeTD, the Rouse timeTR,he mean number of entanglement segmentsZB, and the elasic modulusG0. All these parameters may be extracted fndependent rheological measurements, and therefore ce considered as adjustable. So the present model do

nvolve fitting parameters. In the absence of fluctuationsasTD = 3ZBTR and the number of independent parameeduces to three.

In Figs. 3–6, the steady-shear predictions of the moderesented. InFig. 4, we plot the shear stressσxy and the firsormal stress differenceN1 = σxx − σyy as functions of thhear rateγ. As is seen, bothσxy andN1 increase monotonally with γ and have three different regimes, in accordaith experimental observations[20,21]. As expected from thE model, in each case, there is a linear viscoelastic ret low shear ratesγ ≤ T−1

D for whichσxy ∝ γ andN1 ∝ γ2.n the non-linear regime at flow ratesT−1

D < γ < T−1R , the

hear stress increases only slightly withγ, whereasN1 con-inues to grow rapidly. In this regime, CCR plays a doant role in the stress relaxation. Since the DE modelot account for CCR, it leads to the unrealistic behavio


Fig. 3. Shear stress and first normal stress difference vs. shear rate forZB =30. The dotted lines are predictions for inextensible chains.

which σxy ∝ γ−0.5 andN1 approaches a constant value. Inthe vicinity of the shear rateγ ≈ T−1

R , the slopes of bothcurves again become steep due to significant tube stretch.The system enters the third regime in which these stretch

F fs

Fig. 5. Mean chain lengthening vs. shear rate for differentZB.

effects are very important.Fig. 3 shows that in this regime,the theory of inextensible chains with CCR (Eq.(51)withoutstretch) substantially underestimates the actual values ofσxyandN1.

Fig. 4 shows the shear stressσxy versus the shear rateγfor different molecular weights of polymer chains (ZB is pro-portional to the molecular weight). As is seen, longer chainsreach “the plateau regime” earlier than shorter ones, as ex-pected from the DE model. At high shear rates, all the curvescorresponding to different molecular weights nearly mergeinto a single one, in accord with the behavior observed inexperiments[22,23].

Fig. 5shows the mean chain lengtheningλ = 〈L〉/L0 ver-sus the shear rateγ for different molecular weights of poly-mer chains. Clearly, in the presence of flow, chains are alwaysstretched. However, chain stretch becomes especially impor-tant asγ approachesT−1

R (at γ ≈ T−1R , chains are found to

have the same amount stretch around 15%). Written as afunction of the dimensionless shear rateγTR, all the curvescorresponding to different molecular weights nearly super-impose to a single one. AsTR ∝ Z2

B [1], this implies thatλ ∝ Z2

B, and for a givenγ, longer chains are more stretched,as expected. Note also thatλ increases linearly withγ for alltested molecular weights.

In Fig. 6, the steady-state extinction angleχ is shown as afw r toto delspw ear

ig. 4. Shear stress vs. shear rate for variousZB (τe is the Rouse time oingle segment).

unction of the shear rate. It is seen thatχ first drops rapidlyith increasing shear rate, and then more slowly, simila

he behavior predicted by the CV model of Mead et al.[7] andbserved experimentally. Both the DE and the DEMG moredict an excessively steep decrease inχ with increasingγhich means a significant tube alignment with flow at sh


Fig. 6. Extinction angle (in degrees) vs. shear rate.ZB = 50.

rates aboveT−1D .

Fig. 7 shows early time relaxation of the shear stress af-ter cessation of steady shear flow. As found, the curves cor-responding to shear ratesγ much smaller thanT−1

R nearlymerge. This implies that the stress relaxation is independentof the steady-state shear rate. Clearly, these curves show re-laxation of the initially aligned with the flow chain configu-rations due to reptation. At high shear rates, the rate of re-laxation increases withγ, similar to the behavior observedin experiments[24]. Apparently, this is a consequence of re-laxation of chain stretch and its effect on the rate of CCR.According to Eq.(41), the CCR frequency is proportional tothe mean chain lengthening, so that more stretched chains areexpected to relax faster. Note that in the case of inextensiblechains with CCR, one would expect that the stress relaxationrate is independent ofγ.

In Figs. 8 and 9, the model predictions are shown for thecase of step shear deformation.Fig. 8shows the relaxation ofthe shear modulusG(γ, t) after step shear of various magni-tudes. As is seen, after a small step strain, chains are hardlystretched, so that stress relaxes via reptation. In contrast, aftera large step strain, chains are stretched substantially, so thatfirst stress relaxes via relaxation of stretch. However, at largertimes, the stress relaxation is again governed by reptation, sothat all the curves can be superposed into a single one byd ingf byO

a witht wn

Fig. 7. Early time relaxation of shear stress, normalized by its initial valueat steady state vs. time after cessation of steady-state shear flow for differentprior shear rates.ZB = 30.

Fig. 8. Relaxation of dimensionless shear modulusG = 3σxy/(γG0) vs.time after step shear strains of different magnitudesγ. ZB = 30.

ividing G(γ, t) by the corresponding value of the dampunctionh(γ) [1], in accordance with experimental datasaki et al.[25].In Fig. 9, we plot the step shear damping functionh(γ)

s a function of the shear strain. The result is comparedhat of the DE model. Note that the DE model is kno


Fig. 9. Step shear damping function vs. shear strain. Dots represent the resultof the DE model without IAA[1].

to be in an excellent agreement with experimental data onthe step–strain response of monodisperse melts and solu-tions [1]. Therefore, it is important that the present theoryshould preserve the ability to predict accurately the responsefor step strain deformations. As is seen, despite the different

form of the constitutive equation, the agreement with the DEdamping function is very good over a wide range of strainamplitudes.

In the rest of this paper, the predictions of the model willbe compared with the available experimental data for twoflow histories: steady-state shear flow and transient startupof simple shear. InFig. 10, the shear stressσxy and the firstnormal stress differenceN1 as functions of the shear rateare compared with data by Mead et al.[7] for a 3 wt% so-lution of 8.4 × 106 molecular weight polystyrene in tricre-syl phosphate at room temperature. The Rouse timeTR, thedisentanglement timeTD, and the mean number of entangle-ment segmentsZB were reported to be around 0.3 s, 2.0 s,and 20, respectively. The elastic modulus was found to beG0 ≈ 5900 Pa by a best fit to the data. It is seen that themodel predictions are in good agreement with the experi-mental curves over a wide range of flow rates from the linearto the non-linear regime.

In Fig. 11, model predictions for the viscosity and firstnormal stress difference as functions of time after startupof simple shear flow are compared with the data by Osakiet al. [25] for the f128-10 solution of polystyrene in tricre-syl phosphate at 40◦C. The Rouse timeTR and mean num-ber of entanglement segmentsZB per chain were estimatedby Graham et al.[13] to be around 3 s and 8, respectively.T ab ed in[ ree-m flowr im-p modelp

F for a 3 e atr

ig. 10. Shear stress and first normal stress difference vs. shear rateoom temperature[7]. The solid lines are the model predictions.

he elastic modulus was found to beG0 ≈ 4000 Pa byest fit to the data, which agrees with the value estimat

13]. As is seen, the model predictions are in good agent with the experimental data over a wide range of

ates.Fig. 11also shows that inclusion of chain stretchroves agreement between the experimental data andredictions.

wt% solution of 8.4 × 106 molecular weight polystyrene in tricresyl phosphat


Fig. 11. Shear viscosity and first normal stress difference vs. time after startup of shear flow of various rates for f128-10 solution of polystyrene in tricresylphosphate at 40◦C [25]. The solid lines are the model predictions. The dotted lines are the model predictions for inextensible chains forγ = 1.74 s−1.

9. Conclusion and remarks

In this paper, a quantitative molecular model is proposed todescribe the dynamics of polymer chains in flow. The formal-ism is based on the so-called bond vector probability distribu-tion function, found to be a successful mathematical ‘tool’ inconstitutive modelling. It is shown that the original DE theorybased on the orientation distribution function can be naturallyextended to incorporate chain stretch and constraint release.Basing on the equation of motion for the BVPDF, a simpleconstitutive equation is derived for the stress tensor whichaccounts for chain stretch, reptation, convection, retraction,CLF, and (C)CR. It contains no adjustable parameters, exceptfor those that can be extracted from independent rheologicalmeasurements.

Acknowledgements

This research is supported by the Technology FoundationSTW, applied science division of NWO, and the technol-ogy programme of the Ministry of Economic Affairs of TheNetherlands.

R

don

hem.

pre-118

[4] D.S. Pearson, E. Herbolzheimer, N. Grizzuti, G. Marrucci, Transientbehavior of entangled polymers at high shear rates, J. Polym. Sci., PartB: Polym. Phys. Ed. 29 (1991) 1589.

[5] G. Marrucci, Dynamics of entanglements: a nonlinear model consis-tent with the Cox–Merz rule, J. Non-Newt. Fluid Mech. 62 (1996)279.

[6] G. Ianniruberto, G. Marrucci, Dynamics of entanglements: a nonlinearmodel consistent with the Cox–Merz rule, J. Non-Newt. Fluid. Mech.65 (1996) 241.

[7] D. Mead, R. Larson, M. Doi, A molecular theory for fast flows ofentangled polymers, Macromolecules 31 (1998) 7895.

[8] G. Ianniruberto, G. Marrucci, A simple constitutive equation for en-tangled polymers with chain stretch, J. Rheol. 45 (2001) 1305.

[9] G. Ianniruberto, G. Marrucci, A multi-mode CCR model for entangledpolymers with chain stretch, J. Non-Newt. Fluid Mech. 102 (2002)383–395.

[10] J.L. Viovy, M. Rubinstein, R.H. Colby, Constraint release in polymermelts—tube reorganisation versus tube dilation, Macromolecules 24(1991) 3587–3596.

[11] S.T. Milner, T.C.B. McLeish, A.E. Likhtman, Microscopic theory ofconvective constraint release, J. Rheol. 45 (2001) 539.

[12] A.E. Likhtman, S.T. Milner, T.C.B. McLeish, Microscopic theory forfast flow of polymer melts, Phys. Rev. Lett. 85 (2000) 4550–4553.

[13] R.S. Graham, A.E. Likhtman, T.C.B. McLeish, Microscopic theory oflinear, entangled polymer chains under rapid deformation includingchain strectch and convective constraint release, J. Rheol. 47 (2003)1171.

[14] H.C. Ottinger, A.N. Beris, Thermodynamically consistent reptationmodel without independent alignment, J. Chem. Phys. 110 (1999)6593–6596.

thing,. The-1998)

ulesewt.

lax-66.for

lie–

eferences

[1] M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, ClarenPress, Oxford, 1986.

[2] P.G. de Gennes, Concept of reptation for one polymer chain, J. CPhys. 55 (1971) 572.

[3] G. Marrucci, N. Grizzuti, Fast flows of concentrated polymers:dictions of the tube model on chain stretching, Gazz. Chim. Ital.(1988) 179–185.

[15] C.C. Hua, J.D. Schieber, Segment connectivity, chain-length breasegmental stretch, and constraint release in reptation models. Iory and single-step strain predictions, J. Chem. Phys. 109 (22) (10018–10027.

[16] M.A. Tchesnokov, J. Molenaar, J.J.M. Slot, Dynamics of molecadsorbed on a die wall during polymer melt extrusion, J. Non-NFluid Mech., in press.

[17] S.T. Milner, T.C.B. McLeish, Parameter-free theory for stress reation in star polymer melts, Macromolecules 30 (1997) 2159–21

[18] A.E. Likhtman, R.S. Graham, Simple constitutive equationlinear polymer melts derived from molecular theory: Ro


Poly equation, J. Non-Newt. Fluid Mech. 114 (2003) 1–12.

[19] A.E. Likhtman, T.C.B. McLeish, Quantitative theory for linear dynam-ics of linear entangled polymers, Macromolecules 35 (2002) 6332–6343.

[20] M. Bercea, C. Peiti, B. Simonescu, P. Navard, Shear rheology of semidi-lute poly(methylmethacrylate) solutions, Macromolecules 26 (1993)7095–7096.

[21] J.J. Magda, C.S. Lee, S.J. Muller, R.G. Larson, Rheology, flow insta-bilities, and shear induced diffusion in polystyrene solutions, Macro-molecules 26 (1993) 1696–1706.

[22] J.D. Ferry, Viscoelastic Properties of Polymers, third ed., John Wiley& Sons, New York, 1980.

[23] W.M. Kulicke, R. Kniewske, The shear viscosity dependence on con-centration, molecular weight and shear rate of polysterene solutions,Rheol. Acta 23 (1984) 75–83.

[24] E.V. Menezes, W.W. Graessley, Nonlinear rheological behavior of poly-mer systems for several shear-flow histories, J. Polym., Sci. Polym.Phys. Ed. 18 (1982) 295.

[25] K. Osaki, T. Inoue, T. Isomura, Stress overshoots of polymer solutionsat high rates of shear, J. Polym, Sci., Part B: Polym. Phys. 38 (2000)1917–1925.