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A thermodynamically admissible reptation model 

for fast flows of entangled polymers. II. Model predictions 

for shear and extensional flows 

Jiannong Fang, Martin Kröger, and Hans Christian Öttinger a) 

Department of Materials, Institute of Polymers, ETH Zürich and Swiss F.I.T. 

Rheocenter, CH-8092 Zürich, Switzerland 

(Received 30 November 1999; final revision received 11 July 2000) 

Synopsis 

Numerical predictions of a previously proposed thermodynamically consistent reptation model for 

linear entangled polymers are presented for shear and extensional flows. Comparisons with 

experimental data and two alternative molecular-based models are given in detail. The model 

studied in this paper incorporates the essence of double reptation, convective constraint release, and 

chain stretching, and it avoids the independent alignment approximation. Here, no use is made of 

the ingredient of anisotropic tube cross sections of the previously proposed model. Simulation 

results reveal that the model at a highly simplified level with few structural variables, i.e., four 

degrees of freedom, is able to capture qualitatively all features of the available experimental 

observations and is highly competitive with recently proposed models in describing nonlinear 

rheological properties of linear entangled polymers. © 2000 The Society of Rheology. 

S0148-60550000406-5 

I. INTRODUCTION 

Stress–strain relationships for polymer melts are the main ingredient for the flow 

simulation of polymer processing such as injection molding, film blowing, and extrusion. 

The reliability and accuracy of these simulations depends crucially on the constitutive 

equations which describe the nonlinear viscoelastic properties of the underlying model 

polymer. Although the closed-form phenomenological models, such as K-BKZ equations, 

have been widely used in research and commercial codes, their degree of success is 

limited because of a lack of physical ingredient on the molecular level. For the purpose 

of realistic modeling, molecular-based models are uniquely suited. A molecular model 

was elaborated by Doi and Edwards 1978a, 1978b, 1978c, 1979 who extended the 

reptation idea introduced by de Gennes 1971 to a tube idea in order to describe the 

viscoelastic behavior of entangled polymers in the presence of ‘‘obstacles.’’ Within the 

tube and reptation pictures, the complex entanglement interaction between polymer 

chains has been treated in a rather direct approach, i.e., each chain in the polymer system 

is equivalent to a chain restricted to one dimensional motion so called ‘‘reptation’’ in a 

confining tube, except for its two ends which can move in any possible direction. In 

addition to the reptation mechanism, Doi and Edwards originally assumed instantaneous 

and complete chain retraction, affine tube deformation by the flow, and independent 

a Author to whom correspondence should be addressed. 

© 2000 by The Society of Rheology, Inc. 

J. Rheol. 446, November/December 2000 0148-6055/2000/446/1293/25/$20.00 

1293

1294 FANG, KRÖGER, AND ÖTTINGER 

alignment of tube segments. By doing so, they obtained a closed-form constitutive equation 

which only involves the second moment of the orientation vector for a tube segment. 

For highly entangled, linear polymers, the original Doi–Edwards DE model has been 

extended to incorporate chain contour length fluctuations Doi 1983; Ketzmerick and 

Öttinger 1989 and constraint release due to the motion of the surrounding chains so 

called ‘‘double reptation’’ Tsenoglou 1987; des Cloizeaux 1988. The combination 

of these two effects lead to a refined description of the linear viscoelastic properties 

O’Connor and Ball 1992, however, the model is much less successful for the nonlinear 

properties. The major experimental observations that the original DE theory fails to 

describe in the nonlinear regime are the following: 

1 There exist irreversible effects in double-step strain experiments with flow reversal 

Osaki and Kurata 1980; Osakia et al. 1981; Venerus and Kahvand 1994a, 1994b; 

Brown and Burghardt 1996. 

2 Over a wide range of shear rates ˙ above the inverse disentanglement time 1/ d , 

the steady shear stress is nearly constant for very highly entangled ones. The first normal 

stress difference N 1 increases more rapidly with shear rate than does the shear stress over 

the same range of shear rates. The slope of N 1 vs ˙ increases as the molecular weight 

decreases Bercea et al. 1993; Kahvand 1995. 

3 The steady-state shear viscosity of different molecular weights merge into a single 

curve in the high shear rate, power-law regime Stratton 1966. 

4 The shear stress xy shows transient overshoots in the startup of steady shear flow 

at low shear rates. The strain p at which the maximum in the overshoot occurs increases 

with shear rate at high rates Pearson et al. 1989; Menezes and Graessley 1982; 

Kahvand 1995. 

5 The first normal stress difference exhibits transient overshoots in the startup of 

steady shear flow at moderate shear rates Pearson et al. 1989; Menezes and Graessley 

1982; Kahvand 1995. 

6 The rate of stress relaxation following cessation of steady shear flow is shear rate 

dependent Attane et al. 1985; Menezes and Graessley 1982; Kahvand 1995. 

7 The steady-state extinction angle decreases more gradually with shear rate than 

predicted by the DE model Mead and Larson 1990; Kahvand 1995. 

8 The transient extinction angle shows an undershoot at the startup of steady shear at 

high shear rates; it also shows an immediate undershoot when the shear rate is suddenly 

decreased after a steady state has been reached, finally it reaches a higher steady-state 

value Mead 1996; Oberhauser et al. 1998. 

9 Steady-state values of the dimensionless uniaxial extensional viscosity are nonmonotonic 

functions of extension rate Munstedt and Laun 1981; Ferguson et al. 

1997. 

In order to improve the situation, many attempts of modifying the original DE model 

have been made during the last years. Several physical effects have been found to be 

important for more realistic modeling of nonlinear properties of entangled polymers. A 

short summary of the important effects found so far is given in Secs. IA–IE. 

A. Avoiding independent alignment „IA… 

Recognizing that the large discrepancy between model prediction and experimental 

data in double step strain with flow reversal is caused by the IA approximation. Doi 

1980a, 1980b, and Doi and Edwards 1986 made a detailed analysis for this situation 

and tried to derive a constitutive equation without using the IA approximation. Again, the 

instantaneous-chain-retraction assumption was employed in their derivation. It was 

shown that the model is able to correct the previous discrepancy whenever the time

THERMODYNAMICALLY ADMISSIBLE REPTATION MODEL 

1295 

interval between the two applied step strains is much larger than a characteristic time s , 

called the retraction time. Marrucci 1986 and Marrucci and Grizzuti 1986 also showed 

that the model without IA predicts the Weissenberg effect correctly. In an attempt to find 

a thermodynamically admissible formulation, the reptation model without IA has been 

reformulated by Öttinger and Beris 1999 in the general equation for the nonequilibrium 

reversible-irreversible coupling GENERIC formalism of nonequilibrium thermodynamics 

Grmela and Öttinger 1997; Öttinger and Grmela 1997. For a comprehensive 

introduction see Öttinger 1999a. The model has been shown to be thermodynamically 

consistent after modifying the production term and introducing an additional term in the 

extra stress tensor. A consistent model with a uniform monomer distribution along the 

chain was proposed by Öttinger 1999b. 

B. Double reptation 

Tsenoglou 1987 and des Cloizeaux 1988 derived a successful mixing rule for 

polydisperse melts based on the idea of ‘‘double reptation,’’ which represents the relaxation 

mechanism for the tube that does not arise from motion of the probe chain, but 

rather motion from the surrounding chains. Öttinger 1994 has considered a reptation 

model in which this additional constraint release mechanism is mimicked through a noise 

term in the time-evolution equation for the orientation of inner chain segments. By doing 

so, he recovered the correct mixing rule in the linear regime and extended the idea to the 

nonlinear regime. But, the prescribed relaxation spectrum is not fully recovered due to 

the lack of contour-length fluctuations in the model. The model predicts an improved 

power-law index of 4/3 for the steady-state viscosity in shear flow instead of 3/2 for 

the DE model. Accordingly, the shear stress versus shear rate curve still exhibits a 

maximum when only double reptation is considered. 

C. Chain stretching 

Marrucci and Grizzuti 1988 extended the DE model to allow for chain stretching and 

predicted steady-state properties. An effect is predicted to result in elongational flows, 

giving rise to an expected upturn of the elongational viscosity, but surprisingly, there is 

no effect for steady shear flows, in particular, the extended model does not improve the 

quality of the model prediction for the power-law index of the steady-state shear viscosity. 

Later, Pearson et al. 1991 studied the same type of model with chain stretching in 

transient situations. They found that both the shear stress and the first normal stress 

difference overshoot in the startup of steady shearing flows and the times required to 

reach these maxima are independent of the shear rate. Mead and Leal 1995 and Mead 

et al. 1995 presented a comprehensive study of the DE model by including chain 

stretching and a nonlinear finitely extensible spring law. Numerical predictions were 

given for steady two-dimensional flows with a continuously varying degree of extensional 

and shear character. The results revealed that significant steady state stretch is 

predicted as the flow becomes increasingly extensional in character. 

Although chain stretching is important for correcting some of the failings such as an 

overshoot in N 1 , it doesn’t solve a long-standing problem in the DE model, namely, the 

excessive shear thinning of the viscosity at high shear rates associated with a maximum 

in shear stress followed by a region in which shear stress decreases with shear rate 

asymptotically as ˙ 0.5 , which leads to constitutive instabilities in shear flow. The 

reason for this problem is that, in fast shear flow, the model predicts that the tube 

segments become highly oriented in the flow direction and hence present a very slim


profile to the flow. As a result, the flow ‘‘loses its grip’’ on the molecules, leading to 

anomalously low friction and hence low viscosity. 

D. Convective constraint release „CCR… 

Marrucci 1996 and Ianniruberto and Marrucci 1996 proposed a CCR mechanism 

which removes the problem just mentioned above. They proposed a model for which, 

under flow conditions, relaxation of chain orientation occurs by two mechanisms. One of 

them is ordinary diffusion reptation and double reptation due to thermal motion, which 

of course also takes place in the absence of flow. The second mechanism is CCR, i.e., the 

topological obstacles on a probe chain are renewed through the relative motion among 

chains due to chain retraction. In fast flow situations, this mechanism leaves the chain 

much freer to relax than is possible only by the previously described mechanism, and 

hence prevents the tube segments from becoming highly oriented in the flow direction. 

E. Anisotropic tube cross sections 

Ianniruberto and Marrucci 1998 introduced the idea that during deformation an 

initial circular tube cross section may become elliptical. They derived the corresponding 

expression for the stress tensor, but did not present a time-evolution equation for the tube 

cross section in flow. It was shown that the idea of anisotropic tube cross section has an 

important influence mainly on the ratio of normal-stress difference in shear flow. In view 

of the well know intimate relation between the time evolution of the structural variables 

and the stress tensor expression implied by various approaches to nonequilibrium thermodynamics, 

Öttinger 2000 developed a thermodynamically admissible reptation 

model with anisotropic tube cross section and the constraint release mechanisms associated 

with double reptation and CCR. For that model, he proposed relationships between 

the ratio of normal-stress differences and the mean-square curvature of the tube cross 

section in shear flow. 

Very recently, reptation models incorporating all the well-established phenomena except 

for anisotropic tube cross sections have been formulated by two groups, based on a 

full-chain stochastic approach suitable for computer simulations by Hua and Schieber 

1998 and Hua et al. 1998, 1999 and on rather complicated coupled integraldifferential 

equations by Mead et al. 1998. It is encouraging that these reptation models 

can quite successfully reproduce the experimentally observed rheological behavior in a 

large number of flow situations. In a previous work by Öttinger 1999b, which is referred 

to as part I of the present work, a new reptation model including anisotropic tube 

cross sections, chain stretching, double reptation, and CCR, while avoiding the IA approximation, 

has been developed under the guidance of nonequilibrium thermodynamics. 

Two versions of the model, referred to as ‘‘uniform’’ and ‘‘tuned,’’ have been proposed. 

The purpose of this paper is to give a detailed model evaluation of the recommended 

‘‘uniform’’ model in shear and extensional flows by numerical simulations. Because the 

experimentally observed features listed in the beginning of Sec. I can be explained 

without considering anisotropic tube cross sections their implementation seems to be 

nontrivial to us, and they weakly affect the ratio of predicted normal stress differences for 

which experimental data is rarely available, the variable Q representing a tube cross 

section in the previously proposed model has been omitted here. On this simplified level, 

the model has only four degrees of freedom, which will be recalled in Sec. II. Note that 

our model takes into account only a single relaxation time reflecting the linear viscoelastic 

response of the material. This limitation can be released by considering


1297 

a life span distribution for segments, being deduced from a measured relaxation modulus 

as described by Öttinger 2000. In this work, we focus entirely on the basic nonlinear 

behavior. 

II. MODEL DESCRIPTION 

The structural state variables chosen in Part I are the configurational distribution 

function f (u,s,r) and the stretching ratio of the chain contour length (r) L/L 0 , 

where u is a unit vector describing the orientation of a tube segment, s is the position 

label of a tube segment in the interval 0,1 the values s 0 and s 1 correspond to 

the chain ends, r is the space position vector, and L 0 denotes the equilibrium contour 

length of the chain. After determining all the state variables the hydrodynamic variables 

and the above structural state variables, the model has been formulated in the GENERIC 

formalism by constructing its ‘‘building blocks’’ step by step. Here we summarize the 

final time-evolution equations for the structural variables. 

The equation for the chain contour length stretching and relaxation reads 

D 

Dt 

˙ tot ˙ convect ˙ dissip , 

1 

where D/Dt is the material time derivative, and the total stretching rate is split into 

convective and dissipative contributions 

˙ convect :¯, 

2 

where is the transpose of the velocity gradient tensor, ¯ is the symmetric secondmoment 

orientation tensor defined by 

¯ 

0 

1 

ds d 3 ufu,s,ruu. 

3 

The dissipative contribution reads 

˙ dissip 1 s 

c 

3Z 1, 

4 

where Z the number of entanglement segments per chain which is given by M/M e M is 

the molecular weight of the chain and M e is the average molecular weight between 

entanglement points along one chain, s is the characteristic stretching time, and c() is 

the effective positive spring coefficient 

c 3Z 2 

max1 

2 

max 2 . 

5 

The spring coefficient is derivable from an entropy expression of the form 

3 2 Z ln 2 2 

max 1ln 2 

max 2 

2 , 

max 1 

6 

such that the spring force F() (1)c() vanishes at 1, and diverges for 

→ 0 and → max , respectively, while for small extensions Hookean spring behavior, 

and for large extensions FENE type spring behavior is recovered.


TABLE I. Values for the maximum possible stretching ratio max via Eq. 

6, the entanglement molecular weight M e , the average end-to-end distance 

of the entanglement segment at equilibrium d t , and the length of a 

Kuhn step l K for three important melts at 140 °C. 

Melts M e gmol 1 d t (Å) l K (Å) max 

PS 13000 76 16.2 4.7 

HDPE 860 33 12.9 2.6 

PP atactic 5400 61 10.7 5.7 

In Part I, the spring coefficient was deduced from an entropy expression containing a 

term squared in the relative stretch (1), together with the constraint of positive 

extension, thus motivating two additional parameters, called c 1 ,c 2 . Equation 6 is compatible 

with Part I, e.g., for a rather particular choice for c 1 together with c 2 0. The 

two approaches for the suggested setting c 1 1, c 2 0 possess measurable differences 

in their prediction of rheological quantities at startup of shear flow slightly increased 

stress amplitude at overshoot and slightly decreased strain at overshoot—for the 

model based on Eq. 6, steady shear flow weak decrease of steady stresses, and extensional 

flow critical strain rate postponed by a factor of 1.3–1.5, where the current 

setting has overall advantages when compared to experimental data for more details see 

Sec. IV. 

The parameter max clearly is the maximum possible stretching ratio of the chain 

contour length, which is equal to the square root of the number of Kuhn steps per 

entanglement N KE , 

max N KE d t 

l K 

, 

7 

where d t is the average end-to-end distance of the entanglement segment at equilibrium 

and l K is the length of the Kuhn step which is twice the persistence length. If we denote 

the molecular weight between entanglements for polymer melts as M e , then M e for 

polymer solutions can be estimated by using the relation M e M e / 1.2 , where is the 

volume fraction of polymer Ferry 1980. The values of d t and l K for some polymer 

melts can be calculated from the relevant experimental data tabulated in the literature 

Fetters et al. 1996, where the values of M e are also given. The value of l K is calculated 

from the characteristic ratio C by using the relationship 

C l K 

l 1, 

8 

where l is the bond length Flory 1988, p.111. Based on these available data, the 

values of max for several important melts are given in Table I. 

The diffusion equation for the configurational distribution function takes the form 

Df 

Dt 

u • 1 uu 

u 2 •–uf 

 

u •D 1 uu 

u 2 • 

u f , 

˙ s ṡ tot f dissip 

 

f 1 

2 d 

2 f 

2 s 

9


1299 

where 

ṡ tot 1 s 1 2˙ dissip . 

10 

This drift velocity for s means there is only a rescaling of the position label for the tube 

segment when the chain relaxes in the tube. The third term creation/destruction term on 

the right side of Eq. 9 compensates for configurations lost or gained at the boundaries. 

The terms involving second-order derivatives in Eq. 9 are of irreversible nature and 

express the erratic reptational motion along the chain contour second-order derivative 

with respect to s with the reptation time d and constraint release second-order derivative 

with respect to u with the orientational diffusion coefficient D, respectively. The 

form of D is 

D 1 6 1 

1 

˙ dissip 

 

2 

d 

H ˙ dissip 

 

, 11 

where H(x) is the Heaviside step function. The 1 term is interpreted as representing 

‘‘double reptation,’’ and the 2 term represents the CCR mechanism. The quantities 1/ d 

and ˙ dissip / determine the constraint release rate due to the loss of entanglements 

caused by reptation motion and chain retraction of side chains, respectively. The parameters 

1 and 2 determine the transfer rate from the constraint release rate to the relaxation 

rate of chain orientation, here we take 1 2 1/. This choice is motivated by 

the work of Mead et al. 1998 in connection with the appearance of their switch function. 

The argument is that the constraint release causes not only chain segments reorientation, 

but also contour length shortening this effect is not explicitly taken into account 

here. The role of the parameters 1 and 2 is to apportion the effects of constraint 

release between these two effects. When the chain is unstretched, the constraint release 

causes only chain segments reorientation; when the chain is highly stretched, the constraint 

release causes mainly chain contour length shortening. Hence, the parameters must 

be chosen in such a way that they approach unity when is near unity, and approach zero 

when is large. 

Conservation of the total probability implies the boundary conditions 

f˜ f˜ 

0, 

ss 0 

ss 1 

12 

where f˜(s,r) f (u,s,r)d 3 u 1. At the chain ends, we assume random orientation by 

specifying the distribution 

fu,s,r 1 

u1, s 0,1, 13 

4 

which is common practice, but has been opened to discussion by Kröger and Hess 1993. 

The extra stress tensor consists of two contributions, 1 2 , namely, the original 

Doi–Edwards contribution 

1 r 3Zn p k B T 

0 

1 

uuf u,s,rd 3 uds, 

14


which reflects a proportionality between stress and alignment for fixed Z, and a contribution 

associated with the chain stretching, 

2 r c1n p k B T 

0 

1 


15 

which may cause deviations from the stress-optic rule when stretching occurs. In these 

expressions, n p is the number density of polymers. The plateau modulus G N 

0 , to which 2 

does not contribute, is given by the Doi–Edwards result 

G N 

0 

3 

5 Zn p k B T. 16 

When max approaches infinity, we obtain the simple expression 

5G N 

0 

2 

0 

1 


17 

which may be regarded as a manifestation of the stress-optic rule, see also Fuller 1995. 

The presented model accounts for double reptation, CCR, and chain stretching and avoids 

the IA approximation by the drift term and the creation/destruction term. It has been 

verified to possess the full structure of GENERIC, in particular, the time–structure invariance 

of reversible dynamics. The model has only four structural degrees of freedom, 

one from the position label s, two from the unit orientation vector u, and one from the 

chain stretching . In this paper, we assume d / s 3Z for our model. In addition to 

0 

the plateau modulus G N and the reptation time d , the model has four parameters 

(Z, 1 , 2 , max ) with the preferable values 1 2 1/ for melts as motivated in 

0 

Part I and above. In this work we regard G N and d as the key adjustable parameters, 

while Z and max are fixed by the chemistry of a particular polymer. 

III. SIMULATION ALGORITHM 

In this section, we describe the numerical simulation of the model. According to the 

theory of stochastic differential equations SDEs, the diffusion equation 9, when ignoring 

the creation/loss term discussed separately below, is equivalent to the following 

set of Itô SDEs for the stochastic processes u t , and s t : 

du t 1 u t u t 

u t 2 •–u t 2Du tdt2D 1 u t u t 

u t 2 •dW t , 

ds t ṡ tot dt 1 2 d 

dW t , 

18 

19 

where W t and all the three components of the vector W t are independent Wiener processes. 

The mentioned equivalence means that the average of an arbitrary function 

X(u,s), evaluated as an integral with the solution f of the diffusion equation at the time 

t, can be obtained as the expectation of the stochastic process X(u t ,s t ) 

1 

 

0 Xu,sfu,s,td 3 uds Xu t ,s t . 

20


1301 

Here we assume that we consider the problem in a coordinate system moving with an 

arbitrary, given fluid particle, and we hence suppress the position argument. 

One should note that Eq. 18 is independent of Eq. 19. However, there exists a 

coupling between the processes u t and s t resulting from the boundary condition Eq. 13 

and the creation/destruction term in Eq. 9, which need to be discussed for obtaining a 

full equivalence between diffusion equations and SDEs. Whenever s t reaches the boundaries 

0 or 1, u t must be replaced by a random unit vector. Actually, there is a nonzero 

drift through the boundaries, which is exactly compensated by the creation/destruction 

term in the diffusion equation. Therefore, and in view of Eq. 12, the configurations 

diffusing through the boundaries are reflected back into the range of allowed s values. 

In order to calculate the statistical error of the results, we perform simulations for a 

number of independent blocks (N block ). In each simulation block, a number of trajectories 

(N sample ) of the stochastic processes u t and s t are propagated as follows. First, the 

drift in Eq. 19 is treated by a deterministic method. Given an s value at time t, the 

intermediate value after this treatment is 

s sṡ tot tt. 

21 

If there is a net flux of configurations out of the interval 0, 1, the lost configurations are 

randomly replaced by existing ensemble members; if there is a net flux of configurations 

into the interval 0, 1, the s values of the gained configurations are set in equal distance 

from the two outest existing points to the corresponding chain ends and the u vectors are 

randomly oriented according to the boundary condition. In order to keep the ensemble 

size constant in the latter case, the same number of configurations are randomly selected 

from the existing ensemble members and discarded. This first step takes care of both the 

drift in Eq. 19 and the creation/destruction term in Eq. 9. In the next step, we construct 

the new configurations at time tt 

s new s 1 2 d 

W, 

22 

u u–ut2DW, 

u new u 

u . 

23 

24 

If s new leaves the interval 0, 1, it will be reflected back into it. Upon any reflection, a 

new random unit vector u new is chosen according to the boundary condition. The accuracy 

of this straightforward Euler discretization of the stochastic part in Eq. 19 is of 

order t which is due to unobserved reflections Öttinger 1989. In order to obtain a 

higher-order (t) algorithm, we use the improved scheme proposed by Öttinger 1989, 

in which the effect of the unobserved reflections is taken into account through the conditional 

probability for their occurrence. Finally, the chain stretching at time tt is 

obtained as 

new ˙ convect ˙ dissip t. 

25 

For the results in this paper, we choose N sample 100 000, N block 10, and the time 

step size such that the strain is a maximum of 0.02 per time step and the maximum of the 

time step is 0.01 for small deformation rates. In each block, transient values are obtained 

by ensemble average, and steady values are obtained by average over time after reaching 

steady state in addition to ensemble average. The relative statistical errors of the simu-


lation data are within 2%, except for those with very small absolute values smaller 

than 10 3 at the very beginning of the startup of steady flows. There is no detectable 

difference in the results by increasing ensemble size or decreasing time step size. 

IV. RESULTS 

The experimental data used for comparison is from Kahvand 1995 which were also 

used by Hua et al. 1999. The test fluid is a solution in tricresyl phosphate of nearly 

monodisperse polystyrene with a molecular weight M w of 1.910 6 polydispersity index 

of 1.2 at a polymer density of 0.135 g/cm 3 . This fluid has a zero-shear-rate viscosity 0 

of 6.810 3 Pa s and a zero-shear-rate first normal stress coefficient 1,0 of 2.0 

10 5 Pa s 2 . The longest relaxation time and the average number of entanglement of the 

fluid are estimated to be 15 s and 7, respectively. Correspondingly, for our model we will 

0 

use the set of parameter values Z 7, 21, d 15 s, and G N 1160 Pa beside 

1 2 1/, as stated above to compare the simulation results with the experimental 

data. The value of the finite extensibility parameter max corresponds to 441 Kuhn steps 

per entanglement segment, or 6 monomers per Kuhn step. This value of monomers per 

Kuhn step is consistent with the value calculated from the data on polystyrene PS melts 

in Table I. The results given here except for extensional flows are not sensitive to finite 

extensibility. This set of parameter values is used for most of the cases, otherwise to be 

mentioned. In addition to the comparisons with the experimental data, comparisons with 

the two other models are also presented. These two models are the full chain simulation 

FCS model by Hua and Schieber 1998, Hua et al. 1999, and the simplified S form 

of the theory by Mead et al. 1998. Here, we shall denote them as ‘‘FCS’’ model and 

‘‘MLDS’’ model, respectively. For the FCS model, the simulations were done by Neergaard 

and co-workers, with the parameter b 3N K /(N1) 150, where N K denotes 

the number of Kuhn steps per chain and N is the number of beads for the FCS model 

compare Table II. The chosen value is more reasonable than the value b 25 used by 

Hua et al. 1999, while the other parameters remain the same. The former value of b 

corresponds to 150 Kuhn steps per entanglement, or 17 monomers per Kuhn step. This 

value of monomers per Kuhn step is larger than the value we estimated above. However, 

as for our model, the results given here for the FCS model are not affected significantly 

by choosing an even larger value of b. For the MLDS model in which double reptation is 

not taken into account, one may regard d and s both as adjustable parameters. The 

0 

values d 9s, d / s 10, and G N 800 Pa are obtained by a best overall fit to the 

available experimental data, where the exponential switch function is used. For d / s 

3Z 21 the fit would deteriorate considerably. Prior to further discussion of the 

different models, we refer to Table II for a characterization of the three models: FCS, 

MLDS, and the one under study. 

A. Start-up of steady shearing 

Figure 1 shows the growth of the dimensionless shear viscosity for several dimensionless 

shear rates predicted by our model. The linear viscoelastic limit forms an upper 

envelope for all the shear rates, the transient overshoots occur at medium and high shear 

rates, and the undershoots occur only at high shear rates. These undershoots are also 

predicted by the FCS model and reported by the experimental results. In Fig. 2 we 

compare our model predictions to the experimental data for the transient growth of the 

shear viscosity for several shear rates. Actually the model is able to capture the features 

of the data quantitatively. The predictions of different models for the same property are 

plotted in Fig. 3 where only the curves for the highest shear rate are shown because all


1303 

TABLE II. Characterization of the three models discussed in this work. Concerning the model predictions, the 

main discrepancies between FCS and our model are: larger stress overshoot predicted by FCS in the startup of 

steady shear flow Fig. 3 and larger second undershoot of the extinction angle predicted by FCS Fig. 12 due 

to segment connectivity only incorporated into FCS. The main discrepancies between predictions of MLDS and 

our model are: undershoots in transient stress and extinction angle only predicted by our model due to differences 

in the incorporation of CCR. For a detailed comparison see text. 

Parameters 

Model FCS MLDS This work 

Adjustable 

parameters 

The reptation time d 

and the number of 

beads per chain N fit 

from linear viscoelasticity 

alone 

The plateau modulus 

G N 

0 , the reptation 

time d , and the 

Rouse time s fit 

from nonlinear data 

The plateau modulus 

G N 

0 and the reptation 

time d fit from 

nonlinear data 

Physical 

effects 

incorporated 

and 

corresponding 

mathematical 

means 

Parameters 

fixed by the 

chemistry of 

the polymer 

or physical 

arguments 

Reptation 

The number of 

entanglements 

per chain Z, 

the Kuhn step length 

a K , and the number 

of Kuhn steps per 

chain N K 

None The number of 

entanglements 

and the chain 

Incorporated into a 

set of Langevin equations 

for the chain motion 

The second term in 

the equation for the 

tube survival probability, 

Eq. 10 of the 

reference for MLDS 

per chain Z, 

the constraint release 

parameters 1 , 2 , 

stretchability 

parameter max 

The stochastic term 

in Eq. 19 

Avoiding IA 

approximation 

As for ‘‘reptation’’ 

plus an equation for 

the tube motion 

Not considered 

The drift term in Eq. 

19 and the creation/ 

destruction term in 

Eq. 9 

Chain 

stretching 

As for ‘‘avoiding 

IA approximation’’ 

The equation for 

stretch, Eq. 12 of the 

reference 

The equation for 

stretch, Eq. 1 

Double 

reptation 

By a random, 

instantaneous constraint 

release algorithm 


The 1 -term in the 

orientational diffusion 

coefficient, Eq. 11 

Convective 

constraint 

release 

CCR 

As for ‘‘double 

reptation’’ 

The term with the 

‘‘switch’’ function in 

Eq. 10 and the last 

term in Eq. 12 of the 

reference 

The 2 -term in the 

orientational diffusion 

coefficient, Eq. 11 

Chain-length 

breathing 

and segment 

connectivity 

Incorporated into a 

set of Langevin equations 

for the chain 

motion 



the models give similar results for the two lower shear rates. It turns out that the FCS 

model overpredicts the overshoot, our model predicts the magnitude precisely, and the 

MLDS model slightly overpredicts the overshoot and does not predict an undershoot. 

Comparisons between our model predictions and experimental data are also made for the


FIG. 1. Dimensionless shear viscosity as function of dimensionless time after startup of steady shearing at 

many dimensionless shear rates predicted by our model. 

first-normal-stress difference coefficient 1 in Fig. 4. The overshoot in 1 occurs at a 

higher shear rate than the shear viscosity does. Here, our model underpredicts the magnitude 

of the overshoot at the highest shear rate. Not shown are the predictions by FCS 

and MLDS models: they are very comparable with our model for the two lower shear 

rates. For the highest rate, again the FCS model overpredicts the overshoot and the 

MLDS model gives a better but underpredictive fit. In Fig. 5 we plot the magnitude of the 

strain p at which the maximum in the overshoot of the stress occurs as functions of 

shear rate predicted by the models and the experiment. It appears that the values of p 

stay nearly constant at low shear rates and increase with the increasing shear rate at high 

rates. Here, our model predictions are in good agreement with the experimental data, 

while the MLDS and FCS models overpredict. Not shown are the model predictions and 

FIG. 2. Transient growth of normalized viscosity as function of time under startup of steady shear flow at 

several shear rates predicted by our model lines and experiment symbols.


1305 

FIG. 3. Transient growth of normalized viscosity as functions of time under startup of steady shear flow at 

˙ 10 s 1 predicted by the models and experiment. 

the experimental data of p for the first normal stress difference, which have a qualitative 

shape similar to that seen in Fig. 5 but differ by approximately a factor of 2. 

B. Cessation of steady shearing 

In Fig. 6 we plot the relaxation of the shear stress, normalized by its initial value, after 

cessation of steady shear flow for two shear rates. All the model predictions not shown 

are the predictions by MLDS which almost overlap with ours compare rather well with 

the experimental data. The stress relaxation rate increases with the increasing previous 

shear rate. Later in the relaxation process, at times much larger than s , the relaxation 

rate is governed only by the reptation process and constraint release due to double 

FIG. 4. Transient growth of normalized first-normal-stress difference coefficient as functions of time under 

startup of steady shear flow at several shear rates predicted by our model lines and experiment symbols.


FIG. 5. Magnitude of the strain p at which the maximum in the stress overshoot occurs as functions of shear 

rate for shear stress under startup of steady shear flow predicted by the models and experiment. 

reptation represented by a smaller value of d in the MLDS model. This behavior is 

also in accordance with the other experimental results Attane et al. 1985; Menezes and 

Graessley 1982. 

C. Steady shear flow 

In Fig. 7 the steady-state values of the shear stress xy and the first normal stress 

difference N 1 are plotted as functions of shear rate. Our model and the FCS model 

predict a slight decrease of the shear stress over a range of shear rates that extends from 

roughly ˙ 0.3 to ˙ 2s 1 , where the MLDS model predicts a slight increase which 

is consistent with the experimental data. This new observation of the MLDS model may 

be attributed to the fact that we use a relatively small ratio d / s 10 here. Over the 

same range of shear rates, all the model predictions for the first normal stress difference 

FIG. 6. Relaxation of the shear stress reduced by its value at steady state as a function of time after cessation 

of steady shear flow at two shear rates predicted by experiment and the models.


1307 

FIG. 7. Steady-state values of the shear stress and first normal stress difference as functions of shear rate 

predicted by the models and experiment. 

N 1 increase gradually with shear rate, which are consistent with the experimental data. 

For the shear stress at rates after the range, our model predicts an increase, the FCS 

model shows a lesser increase, and the MLDS model shows a decrease. This decreasing 

behavior of MLDS is caused by the exponential switch function used and the additional 

term accounting for the CCR effect in the equation for stretch. 

We examine the behavior of the steady-state shear stress and its dependence on choice 

of our model parameters in more detail in order to understand how to avoid the prediction 

of an instability region as mentioned earlier. In Fig. 8, we see that the inclusion of chain 

FIG. 8. Steady-state values of the dimensionless shear stress as functions of dimensionless shear rate predicted 

by the DE model, the model without any constraint release effects 1 0, 2 0, the model with double 

reptation effect 1 1/, 2 0, the model with double reptation and convective constraint release effects 

( 1 2 1/), and the model with double reptation and enhanced convective constraint release effects 

( 1 1/, 2 2/. The other parameters of the model are set as Z 20, and max 10.


FIG. 9. Steady-state shear viscosity vs dimensionless shear rate which here is proportional to a physical shear 

rate rather than expressed in terms of a relaxation time which depends on molecular weight, for Z 20 and 

Z 40 with the other parameters 1 1/, 2 1/, and max 10. 

stretching alone yields the onset of an upturn of the shear stress at high shear rates, but 

the prediction of the power-law index is worse. When double reptation is considered, the 

situation is improved. After CCR is added and adjusted to proper intensity, the maximum 

in the steady-state shear stress disappears and the model predicts an almost constant 

stress over a wide range of shear rates, which is also observed in experiments of a highly 

entangled polymer solution Bercea et al. 1993. We thus confirm the previous conclusion 

that the CCR effect is critical for predicting the right shape of the shear stress curve. 

In Fig. 9 we plot the steady-state shear viscosity for Z 20 and Z 40 in order to 

resolve the effect of molecular weight. As a basis for comparison, we assume that the 

reptation time scales like d M 3 . It turns out that the steady-state viscosity of two 

different molecular weights coincides in the power-law region, in agreement with the 

behavior observed in experiments Stratton 1966 and, e.g., molecular dynamics simulations 

Kröger et al. 1993. Similar results are found by the FCS and MLDS models. 

The effect of molecular weight on N 1 is reported in Fig. 10, the model prediction is in 

agreement with the second part of observation 2 in Sec. I. 

D. Steady and transient extinction angle 

The extinction angle is given by (1/2)arctan(2 xy /N 1 ). In Fig. 11 we show the 

steady-state extinction angle as a function of shear rate. Obviously, in our model, CCR 

and double reptation prevent the chain segments from aligning with flow dramatically at 

high steady shear rates, and thus solve the overestimated steady shear orientation problem 

of the original DE model. The predictions of our model compare very favorably with the 

experimental results, while the predictions of the FCS and MLDS models go back to the 

DE model predictions at very high shear rates in steady flow. Not shown here are the 

predictions of our model with the parameters 1 2 exp((1)), which also coincide 

with the predictions of the DE model at high rates. It can be concluded from the 

above observations that the exponential switch function tunes down the constraint release 

effect on orientation too strongly, and hence is not appropriate. The reorientation algorithm 

in the FCS model for constraint release is successful at low and intermediate rates,


1309 

FIG. 10. Steady-state first-normal-stress difference vs dimensionless shear rate for Z 20 and Z 40 with 

the other parameter values as for Fig. 9. 

but may profit from an adjustment if a plateau for the steady time-averaged extinction 

angle at high rates caused by unsteady rotation of chains would be experimentally confirmed. 

In Fig. 12 we show the transient extinction angle predicted in the startup of shear flow 

followed by a step down in shear rate. In both predictions by our model and the FCS 

model, the extinction angle shows an undershoot at the startup of the first shear rate and 

an immediate undershoot at the inception of the lower shear rate before reaching a 

steady-state value. These predictions are in agreement with the experimental data Mead 

1996; Oberhauser et al. 1998. However, the second undershoot predicted by our 

model is too small compared with both the experiment and the prediction of the FCS 

FIG. 11. Steady-state extinction angle as a function of shear rate predicted by the models and experiment.


FIG. 12. Time-dependent extinction angle as a function of dimensionless time after steady shearing and 

attainment of steady state followed by a step down in dimensionless shear rate predicted by our model and the 

FCS model. 

model. The MLDS model with the switch function either exp((1)) or 1/ does not 

possess these two observed undershoots. The undershoots predicted by our model is due 

to chain retraction and its effect on chain orientation through the noise term in Eq. 18. 

E. Extensional flows 

The steady-state uniaxial extensional viscosity ( 1 ( zz xx )/(3˙ )) as a function 

of extensional rate is considered next, see Fig. 13, where z is the direction of extension. 

One observes four regimes of extension rate: i a very low rate region, ˙ 0.1/ d , 

where the extensional viscosity is a constant equal to one third of the Trouton value; ii 

FIG. 13. Steady-state values of the dimensionless uniaxial extensional viscosity as function of dimensionless 

extension rate for 1 2 1/, Z 20, and max 10. The figure inside is the amplification of the region 

of extension rate range from 0.01 to 2.


1311 

FIG. 14. Dimensionless uniaxial extensional viscosity vs dimensionless time after startup of steady uniaxial 

elongation at many dimensionless elongational rates with the parameter values as for Fig. 13. 

a low rate region, 0.1/ d ˙ 1/ d , where the viscosity shows a weak extension 

hardening induced by orientation the maximum factor above the first constant region is 

around 1.05 here; iii an intermediate region, 1/ d ˙ 1/ s , in which the extensional 

viscosity is extension thinning, scaling as ˙ 1 ; and iv a high rate region, ˙ 

1/ s , where the extensional viscosity increases with increasing strain rate and 

reaches another constant value finally. The same conclusion for regimes i, iii, and iv 

has been drawn from the MLDS model by Mead et al. 1998. Regime ii is caused by 

double reptation, because at low strain rates the other effects have a minor influence. 

Accordingly, regime ii is not predicted by the MLDS model and DE theory. Because of 

the lack of experimental data on monodisperse, linear entangled polymers, these unusual 

predictions have not yet been fully established. Two of the few works showing a pronounced 

regime ii were performed by Munstedt and Laun 1981 and Ferguson et al. 

1997. The first of these papers stated that two of the four HDPE investigated in their 

study do not show a maximum of the steady-state elongational viscosity; for the others it 

is less pronounced than in the case of LDPE. 

In Fig. 14 we plot the transient elongational viscosity as a function of time in the 

startup of uniaxial extensional flow. After the extensional rate exceeds 1/ s , the viscosity 

shows strain-hardening behavior and finally saturates because of the finite chain stretchability. 

To study the strain-hardening property of our model in detail, a strain-hardening 

parameter versus Hencky strain at the extensional rate of 100 is plotted in Fig. 15. The 

strain-hardening parameter is defined as the ratio of the strain rate-dependent nonlinear 

elongational viscosity to the strain rate-independent linear elongational viscosity at the 

same time Koyama and Ishizuka 1983. The figure reveals that the first departure from 

linearity occurs at a Hencky strain value around 0.5, fully in accord with the experimentally 

observed value by Takahashi et al. 1999. Figure 16 shows the influence of the 

maximum stretch max on the transient uniaxial extensional viscosity. An increase of 

max does not affect the occurrence of strain hardening, but postpones the saturation and 

increases the saturation value. Not shown are the predictions of the steady-state and 

transient viscosity in equibiaxial extension, which have qualitative shapes similar to those


FIG. 15. The strain-hardening parameter as a function of the Hencky strain at the dimensionless elongational 

rate ˙ d 100 with the parameter values as for Fig. 13. 

seen in the uniaxial extension, except that i the second regime of steady-state viscosity 

is absent and ii it is ‘‘softer’’ and less strain hardening than uniaxial extension Nishioka 

et al. 2000. Simulations of planar extension of our model are also performed. The 

steady-state planar viscosity p1 ( zz xx )/(4˙ ) direction of extension z and 

p2 ( yy xx )/(2˙ ) neutral direction y are plotted in Fig. 17. The viscosity p2 

decays much faster than the p1 does in the intermediate region, 1/ d ˙ 1/ s . 

Again, there is no second regime for both of them. In Fig. 18 we show the transient 

FIG. 16. Dimensionless uniaxial extensional viscosity vs dimensionless time after startup of steady uniaxial 

elongation at the dimensionless elongational rate ˙ d 200 for 1 2 1/, Z 20, and three different 

values of max .


1313 

FIG. 17. Steady-state values of the dimensionless planar extensional viscosity p1 along the direction of 

extension z and p2 along the neutral direction y as functions of dimensionless extensional rate with the 

parameter values as for Fig. 13. 

 

viscosity p2 not shown is the 

p1 , which is similar to the transient viscosity in 

uniaxial extension. The model predicts overshoots at medium and high extensional rates. 

V. CONCLUSION 

A new thermodynamically admissible reptation model that includes chain stretching, 

double reptation, and convective constraint release, and that avoids IA approximation is 

numerically investigated for transient and steady properties in shear and extensional 

flows. Quantitative comparisons are made with experimental data of entangled PS solution 

in shear flows. We find that the model is able to capture qualitatively, or quantita- 

FIG. 18. Dimensionless planar extensional viscosity p2 along the neutral direction y versus dimensionless 

time after startup of steady planar elongation at many dimensionless elongational rates with the parameter 

values as for Fig. 13.


tively in most cases, all the nonlinear properties observed for shear flows summarized in 

Sec. I from item 2 to item 9 item 1 has already been checked in Part I. For extensional 

flows, the model exhibits unusual predictions. In order to confirm these, further measurements 

of the stress for ideal samples linear, monodisperse, entangled polymers in various 

extensional flows are required. There are difficulties in performing this kind of 

experiment: a monodisperse polymer is usually sold in powder form, which is often more 

difficult to shape into bubble free samples than granules; the polymer breaks at much 

lower Hencky strains and is less homogeneous than an industrial melt Schweizer 

1999. 

Comparisons with two recently formulated models, FCS Hua and Schieber 1998 

and MLDS Mead et al. 1998, respectively, in shear flows are also presented. It turns 

out that all three models are competitive with each other; in many cases, they show 

similar behaviors. Very detailed comparisons are given in the previous section, e.g., for 

the MLDS and our model it is shown that a weak switch function proportional to the 

inverse of relative stretch 1/ predicts more realistic results than obtained by applying 

a stronger one. 

In addition to the plateau modulus G N 

0 and the reptation time d , our model has two 

basic parameters, namely Z, the number of entanglements per chain which is proportional 

to the molecular weight and can be expressed in terms of the Rouse and reptation time, 

and the maximum stretch max , being equal to the square root of the number of Kuhn 

steps per entanglement segment. The remaining two model parameters 1 , 2 have been 

introduced but actually assigned the values 1 2 1/. 

A ‘‘stable’’ steady-state shear stress curve is obtained by setting 2 2/. Predictions 

of the steady shear stress at very high shear rates can be also improved by considering 

the stretching time s to be a decaying function in , to account for the effect of 

contour length shortening by constraint release in situations where highly stretched conformations 

occur. 

Notably, our model is a thermodynamically consistent single-segment theory which 

has only four degrees of freedom. In the sense of considering the time and memory 

requirements for the computation, it is very suitable to utilize the model to simulate 

complex flows by using the CONNFFESSIT idea Laso and Öttinger 1993. To our 

knowledge, variance reduction techniques Hulsen et al. 1997; Öttinger et al. 1997; 

van Heel et al. 1999; Gallez et al. 1999; and Bonvin and Picasso 1999 may not be 

directly applied to the model. The dynamics for the position label—and in particular its 

jump events upon touching the chain ends—depend on the local value for the macroscopic 

velocity gradient such that a synchronization of trajectories by cancellation of 

fluctuations is prevented at first glance. 

However, it should be a straightforward exercise to extend the presented model to 

account for polydispersity effects by a mean-field approach, such that the properties of 

surrounding chains enter into the dynamics of the test chain. The resulting model will 

necessarily locate beyond simple superposition models. 

ACKNOWLEDGMENTS 

The authors would like to thank Professor D. C. Venerus for providing the experimental 

data, Professor R. G. Larson for providing the simulation program of the MLDS 

model, J. Neergaard for providing calculations of the FCS model, and Professor J. D. 

Schieber for extremely helpful comments. J. Fang is grateful for the financial support 

from the European Community Program BRITE-EuRam.


1315 

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