gaussian approximation for finitely extensible bead-spring chains with hydrodynamic interaction

Gaussian approximation for finitely extensible bead-spring chainswith hydrodynamic interactionR. Prabhakar and J. Ravi Prakash

Citation: Journal of Rheology (1978-present) 50, 561 (2006); doi: 10.1122/1.2206715 View online: http://dx.doi.org/10.1122/1.2206715 View Table of Contents: http://scitation.aip.org/content/sor/journal/jor2/50/4?ver=pdfcov Published by the The Society of Rheology Articles you may be interested in Dynamics of individual molecules of linear polyethylene liquids under shear: Atomisticsimulation and comparison with a free-draining bead-rod chain J. Rheol. 54, 283 (2010); 10.1122/1.3314298 Brownian dynamics simulations with stiff finitely extensible nonlinear elastic-Fraenkelsprings as approximations to rods in bead-rod models J. Chem. Phys. 124, 044911 (2006); 10.1063/1.2161210 Iterated stretching and multiple beads-on-a-string phenomena in dilute solutions of highlyextensible flexible polymers Phys. Fluids 17, 071704 (2005); 10.1063/1.1949197 Variance reduced Brownian simulation of a bead-spring chain under steady shear flowconsidering hydrodynamic interaction effects J. Chem. Phys. 113, 4767 (2000); 10.1063/1.1288803 A Consideration of the Yamamoto Network Theory with NonGaussian Chain Segments J. Rheol. 31, 371 (1987); 10.1122/1.549928

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Gaussian approximation for finitely extensible bead-spring chains with hydrodynamic interaction

R. Prabhakara) and J. Ravi Prakashb)

Department of Chemical Engineering, Monash University, Clayton,Victoria-3800, Australia

(Received 10 February 2006; final revision received 5 April 2006�

Synopsis

he Gaussian Approximation, proposed originally by Öttinger �J. Chem. Phys., 90, 463–4731989�� to account for the influence of fluctuations in hydrodynamic interactions in Rouse chains,s adapted here to derive a new mean-field approximation for the FENE spring force. ThisENE-PG force law approximately accounts for spring-force fluctuations, which are neglected in

he widely used FENE-P approximation. The Gaussian Approximation for hydrodynamicnteractions is combined with the FENE-P and FENE-PG spring force approximations to obtainpproximate models for finitely extensible bead-spring chains with hydrodynamic interactions. Thelosed set of ordinary differential equations governing the evolution of the second moments of theonfigurational probability distribution in the approximate models is used to generate predictionsf rheological properties in steady and unsteady shear and uniaxial extensional flows, which areound to be in good agreement with the exact results obtained with Brownian dynamicsimulations. In particular, predictions of coil-stretch hysteresis are in quantitative agreement withimulations’ results. © 2006 The Society of Rheology. �DOI: 10.1122/1.2206715�

. INTRODUCTION

The importance of hydrodynamic interactions �HI� in determining the behavior ofilute polymer solutions has been widely recognized since the introduction of the Zimmheory �Zimm �1956��. The limitation, however, of the accuracy of Zimm theory to lineariscoelastic property predictions has spurred the development of more sophisticated treat-ents of HI. For instance, while the consistent averaging approximation �Öttinger

1987�� replaces the equilibrium averaged Oseen tensor in Zimm theory with a nonequi-ibrium average, the Gaussian approximation �Öttinger �1989�; Wedgewood �1989�� fur-her accounts for fluctuations in HI. The Gaussian approximation, which is the mostophisticated approximation to date, has been shown to be highly accurate in shear flowsy comparing its predictions of viscometric functions with exact Brownian dynamicsimulations �BDS� �Zylka �1991��.

The use of bead-spring chain models with linear Hookean springs in all these modelsas prevented the evaluation of their accuracy in extensional flows, in which the polymeroil undergoes a coil-stretch transition at a critical value Wic of the Weissenberg numberi=��, where � is a characteristic relaxation time, and � is the extension rate. As is well

�Present address: Research School of Chemistry, Australian National University, ACT-0200, Australia.�
Author to whom correspondence should be addressed; electronic mail: [email protected]
2006 by The Society of Rheology, Inc.561. Rheol. 50�4�, 561-593 July/August �2006� 0148-6055/2006/50�4�/561/33/$27.00

subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP:

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562 R. PRABHAKAR AND J. R. PRAKASH

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nown, the extensional viscosity becomes unbounded at Wic because of the infinite ex-ensibility of Hookean springs. The prediction of a bounded extensional viscosity waschieved in early theories by replacing Hookean springs with nonlinear finitely extensibleprings �Peterlin �1966��. However, these theories did not incorporate HI. The importantole played by HI in extensional flows has been established beyond doubt in a series ofecent publications, which have shown that the inclusion of HI leads to quantitativelyccurate predictions of both macroscopic and mesoscopic properties �Hsieh et al. �2003�;endrejack et al. �2002�; Prabhakar et al. �2004�; Sunthar et al. �2005�; Sunthar andrakash �2005��. All these theoretical predictions have been obtained by carrying outxact BDS incorporating fluctuating HI and nonlinear finitely extensible springs.

An additional aspect that has been shown in these works to be crucial to obtaininguantitative predictions is the need to use bead-spring chain models with a sufficientlyarge number of springs, NS �or equivalently, degrees of freedom�. This requirement isventuated by the need to accurately reflect both the changing nature of the drag expe-ienced by the polymer as it unravels in an extensional flow, and the self-similar characterf a high molecular weight polymer �which is responsible for the observation of universalehavior in dilute polymer solutions�. The requirement of large NS, coupled with the needo include HI �which leads to the CPU time scaling as NS

4.5�, makes the use of BDS foroutine calculations highly impractical. While the development of schemes for the accel-rated calculation of HI has alleviated this problem somewhat �Jendrejack et al. �2000��,here is a pressing need for the development of approximations capable of describingxtensional flows that are both accurate and computationally inexpensive. The aim of thisork is to introduce such an approximation by extending previous approximations for HI,hich were based on bead-spring chain models with Hookean springs, to models withonlinear finitely extensible springs.

Wedgewood and Öttinger �1988� have previously introduced a similar approximationy combining the consistent averaging scheme for HI with a FENE-P approximation forhe springs. While they carried out a detailed investigation of the predictions of the modeln shear flow, they did not evaluate its accuracy by comparison with exact BDS, nor didhey obtain any predictions in extensional flows. In this work, we introduce an approxi-

ation that extends the previously established Gaussian approximation by accounting foructuations in both HI and the spring forces. The accuracy of this model is then verifiedy comparison with BDS in a number of both transient and steady, shear and extensionalows.

The paper is organized as follows. In the following section, we recall the basic equa-ions for non-free-draining bead-spring models of dilute solutions of finitely extensibleolymer molecules. Subsequently, we develop in Sec. III the equations for the Gaussianpproximation. The symmetries in the flow patterns considered in this study lead toonsiderable simplifications in the implementation of the numerical methods for comput-ng the material functions, which are briefly discussed in Sec. IV. Predictions of thepproximations for the material functions in shear and extensional flows are presented inec. V, along with the exact results obtained with BDS. Section VI summarizes theentral conclusions of this study.

I. BASIC EQUATIONS

Under homogeneous conditions, the configurational state of a bead-spring chain, with

S springs, is completely specified by the set of connector vectors �Qi � i=1, . . . ,NS� �Birdt al. �1987b��. The Fokker-Planck equation governing the evolution of the configura-
ional probability distribution ��Q1 , . . . ,QNS
, t� is


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��

�t= − �

i=1

NS �

�Qi· � · Qi −

1

��j=1

NS

Aij · F jS,c� +

kBT

��i,j=1

NS �

�Qi· Aij ·

��

�Q j, �1�

here kB is Boltzmann’s constant, � is the transpose of the position-independent tracelesselocity gradient �v, T is the absolute temperature of the solution, and � is the Stokesianead-friction coefficient, which is related to the bead radius a through �=6�a�S, with �S

eing the viscosity of the Newtonian solvent. The dimensionless diffusion tensors Aij areiscussed in detail shortly. The spring connector force F j

S,c is given by

F jS,c =

�Sj

�Q j, �2�

ith Sj representing the potential energy of spring j. Here, the following FENE expres-ion for the spring force is used �Warner �1972��:

F jS,c = H� 1

1 − Q j2/Q0

2�Q j = H��Qj�Q j , �3�

here H is the spring constant, Qj = �Q j�, Q0 is the “fully stretched” length a single spring,nd ��Q j�= �1−Q j

2 /Q02�−1 is the nonlinearity in the FENE spring force law.

By considering a bead-spring chain of NS springs to represent a polymer molecule of

K Kuhn segments, each spring can be imagined to represent a “submolecule” of NK,S

NK /NS Kuhn segments. The parameters in the FENE spring-force law, H and Q0, canhen be related to the two parameters characterizing the equilibrium structure of a sub-

olecule, namely, bK �the length of a Kuhn step� and NK,S. Since the contour length ofhe submolecule is bKNK,S, it follows that Q0=bKNK,S. The equilibrium mean squarend-to-end distance of a submolecule, in the absence of excluded volume forces, is RS

2

bK2 NK,S. Using the equilibrium Boltzmann distribution for a single spring governed by

he FENE spring force law, �eqS exp�−Sj /kBT�, it can be shown that �Bird et al. �1987b��

H

kBT=

3

RS2 −

5

Q02 . �4�

ubstituting for RS2 and Q0

2 in terms of bK and NK,S leads to the following expression forhe spring contant H:

H

kBT=

1

bK2

3NK,S − 5

NK,S2 . �5�

The equations above indicate that taking the limit Q0→�, while keeping RS2 constant,

eads to �→1. In this limit, RS2=3kBT /H, and the FENE spring-force expression reduces

o the Hookean spring force law, FS,c=HQ. In this work, chains with Hookean springs areeferred to as “Rouse” chains, while finitely extensible bead-spring chains are denoted asFEBS” chains.

The diffusion tensors Aij in Eq. �1� are given by

Aij = Aij� + ��ij + �i+1,j+1 − �i,j+1 − �i+1,j� , �6�

here Aij =ij +i+1,j+1−i,j+1−i+1,j =2ij −�i−j�,1 is an element of the NSNS Rouse ma-rix, and �� ,� ,�=1, . . . ,N, are HI tensors, which are typically related to the interbead
isplacement r��=r�−r� between beads � and �, by expressions of the form

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�� =1

8��Sr��

C�r�� . �7�

ote that the interbead displacement vector r�� can be related to the spring connectorectors through

r�� = sgn�� − �� i=min��,��

max��,��−1

Qi. �8�

Models which neglect the influence of HI—such as the original bead-spring model ofouse �Rouse �1953��—can be interpreted as ones in which the tensor C is set to zero.arly models incorporating HI �Kirkwood and Riseman �1948�; Zimm �1956�� used theseen-Burgers form of the HI tensor, in which

C = � +r��r��

r��2 . �9�

lthough the Oseen-Burgers description of HI is inaccurate when the pairwise separationetween beads is comparable to the size of the beads, the hydrodynamic behavior of longolymer chains is dominated by the interactions between nonadjacent beads, and thehort-range inaccuracy of the Oseen-Burgers description has been shown to be relativelynimportant in the prediction of properties that depend on the polymer chain as a wholeÖttinger �1987�; Zimm �1980��. Further, its simple form is particularly useful for theevelopment of closure approximations. In numerical simulations, however, the Oseen-urgers expression becomes problematic when r�� a, since the 3N3N diffusion block

atrix comprising the Aij tensors as its constituent blocks can become nonpositive defi-ite �Rotne and Prager �1969�� when beads overlap. In its place, in all the Brownianynamics simulations performed here, the Rotne-Prager-Yamakawa �RPY� modificationPrabhakar et al. �2004�; Rotne and Prager �1969�; Yamakawa �1971�� which ensures thathe diffusion block-matrix always remains positive-definite, is used.

In order to express the Fokker-Planck equation in dimensionless form, the followingength scale and associated time scale are introduced �Prabhakar et al. �2004�; Suntharnd Prakash �2005��,

�S �RS

�3; �S �

��S2

4kBT. �10�

n the Hookean limit, both �S and �S reduce to their conventional form, �S=�H�kBT /H and �S=�H=� /4H=��H2 /4kBT, respectively. Defining t*� t /�S, Qi

*�Qi /�S,*��S�, the Fokker-Planck equation can be recast in dimensionless terms as

��

�t* = − �i=1

Ns �

�Qi* · �* · Qi

* −H*

4 �j=1

Ns

Aij · � jQ j*� +

1

4 �i,j=1

Ns �

�Qi* · Aij ·

��

�Q j* , �11�

here

H* �H�S

2

kBT=

�S2

�H2 �12�

s the dimensionless spring constant. When �H is used as the length scale, the parameter

0 is represented in dimensionless terms by the finite extensibility parameter, bQ0

2 /�H2 �Bird et al. �1987b��. The counterpart of this parameter in the nondimensional-

* 2 2 *
zation scheme adopted here is b �Q0 /�S. Since Q0=bKNK,S, it follows that b =3NK,S.

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ecause of the direct relation of b* to the degree of coarse-graining NK,S, the latter iseported instead of b* when presenting results.

On introducing the hydrodynamic interaction parameter, h*, defined by

h* =a

��S

, �13�

he dimensionless tensor �� occurring in the definitions of the diffusion tensors �Eq.6�� can be written as

�� =3��h*

4r��* C�r��

* � , �14�

here r��* = �1/�S�r��.

All the rheological properties of interest in the present work can be obtained fromramers’ expression for the polymer contribution to the stress tensor �p �Bird et al.

1987b�� which in nondimensional terms is

�p* =

1

npkBT�p = NS� − H*�

i=1

NS

��iQi*Qi

*� . �15�

ere, np is the number density of polymers, � is the unit tensor, and angular bracketsenote an average performed with the configurational distribution function �. Besidesheological properties, the mean square end-to-end distance Re

*2 is used here to providenformation on the size and shape of polymer molecules as they stretch and orient inesponse to the imposition of flow,

Re*2 = �

i,j=1

NS

tr�Qi*Q j

*� . �16�

urther, the prediction of the various models of the birefringence in extensional flows isvaluated using the following expression �Doi and Edwards �1986�; Wiest �1999��:

�n* �nxx

I − nyyI

A=

1

b*�i=1

NS

��Qi,xx*2 � − �Qi,yy

*2 �� , �17�

here ni is the intrinsic birefringence tensor, and A is a constant that, besides dependingn the chemical structure of the polymer and the solvent, is proportional to the polymeroncentration np. Recently, a more accurate expression for �n* has been derived by Lind Larson �2000�. Since we are only interested in a qualitative exploration of the stress-onformational phenomenon, we have adopted the simpler result in Eq. �17� above in thistudy.

Equations �15�–�17� above show that all the macroscopic properties of interest herean be described in terms of expectations of configuration-dependent functions. It isossible to obtain formally the equation for the time evolution of any configuration-ependent function by multiplying the Fokker-Planck equation with the function and thenntegrating over all possible configurations. However, as a direct consequence of theonlinearities in the model arising from FE and HI, the resulting equations are not closedith respect to the expectations whose evolution equations are desired. For instance, the

ollowing set of evolution equations for the second moments of the probability distribu-
ion � can be derived in this manner:

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�Qi*Q j

*��1� �d�Qi

*Q j*�

dt* − �* · �Qi*Q j

*� − �Qi*Q j

*� · �*T

= −H*

4 �m=1

NS

�Qi*Qm

* �m · Amj + Aim · �mQm* Q j

*� +1

2�Aij� , �18�

here the subscript “�1�” denotes the contravariant convected time derivative of a tensorBird et al. �1987a��. As a result of the nonlinearities in the functions � and � introducedy the spring force law and HI respectively, the equation above for �Qi

*Q j*� is not closed

ith respect to the second moments. It is this “closure problem” that has motivated bothhe introduction of various closure schemes and the development of exact Brownianynamics simulations for obtaining predictions of properties away from equilibrium. Asentioned previously, the Gaussian approximation for Rouse chains is the most accurate

losure approximation to date. In the section below, we develop the Gaussian approxi-ation for FEBS chains with HI.

II. GAUSSIAN APPROXIMATION

For Rouse chains, Öttinger �1989� and Wedgewood �1989� pointed out that the com-licated averages on the right-hand side of Eq. �18� above �with �m=1� can be expresseds functions of the second moments �Qi

*Q j*� by assuming a priori that the configurational

istribution function is a multivariate Gaussian,

� = N exp�−1

2 �i,j=1

NS

Qi* · �ij

* · Q j*� , �19�

here N is a normalization factor. The tensors �ij* are related to the second moments

ij* = �Qi

*Q j*� of the Gaussian distribution through the following set of NS

2 linear algebraicquations:

�m=1

NS

�im* · �mj

* = ij� . �20�

s is well known, this implies that the Gaussian distribution � is completely specifiednce its second moments �ij

* are known.The crucial property of Gaussian distributions that enables the simplification of com-

lex moments is described by Wick’s decomposition rule. In the present instance, becausehe right-hand side of Eq. �18� has complex averages of the form �xxf�x��, where x is theandom variable of interest, and f is a nonlinear function of x, Wick’s decomposition ruleeads to

�xxf� = �xx��f� + �xx� · � �f

�xx� . �21�

y applying Eq. �21� to Rouse chains with HI, Öttinger showed that Eq. �18� �with �m

1� reduces to a closed set of equations for the second moments �Qi*Q j

*�. In principle, theame approach can be used to obtain closure of Eq. �18� with both the nonlinear phe-omena of finite extensibility �FE� and hydrodynamic interactions included. However,irectly using the Gaussian assumption and Wick’s decomposition to simplify the expec-ations on the right-hand side of Eq. �18� leads to the incorrect prediction of a dependencef equilibrium static properties on h*. Similar behavior was earlier observed by Prabhakar
nd Prakash �2002� when the Gaussian approximation was used to achieve closure in a

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ookean dumbbell model simultaneously incorporating excluded volume and hydrody-amic interactions. The reason for this incorrect prediction is that, on direct application ofhe Gaussian approximation, the contributions from FE �and/or excluded volume� andhose from HI cannot be separated into independent factors at equilibrium �Prabhakar2005��, thus leading to an unphysical dependence of the predicted equilibrium secondoments on the HI parameter, h*. To avoid this problem, closure of Eq. �18� is achieved

ere in two steps. In the first step, a mean-field quadratic spring potential Sm* , which

pproximately accounts for fluctuations in the FENE spring force, is derived by using theaussian approximation for free-draining FEBS chains. A bead-spring chain model is

hen constructed which incorporates hydrodynamic interactions, and which has springs

escribed by Sm* . In the second step, the Gaussian approximation is again invoked to

btain a closed set of equations for the second moments of this model. In this manner, thencorrect prediction of a dependence of equilibrium static properties on h* is avoided.

Before discussing these two steps in detail in Secs. III A and III B below, it is appro-riate to note that if f had been replaced with its average �f� in Eq. �21�, thus neglectinghe fluctuations in f , one would obtain the first term on the right-hand side alone. Thus,or a Gaussian distribution, it is seen that fluctuations in f are exactly accounted for byhe second term on the right-hand side of the equation above. This convenient separationf the complicated expectation on the left-hand side into contributions from the mean anductuations will be used here to highlight the important role played by fluctuations in thepring force and in hydrodynamic interactions.

. Free-draining FEBS chains

In this section, the Gaussian approximation is used to achieve closure of the second-oment equations for chains with finitely extensible springs, but without HI. Prakash and

o-workers have previously used the Gaussian approximation for bead-spring chain mod-ls without HI, which are, however, still nonlinear because of the presence of conserva-ive nonhydrodynamic excluded volume forces between beads. The approximation wasound by them to lead to qualitatively accurate predictions �Kumar and Prakash �2004�;rakash, �2001a�, �2001b�; Prakash and Öttinger �1999��.

For free-draining chains, Aij =Aij�, and hence from Eq. �18�, one obtains for FEBShains

�Qi*Q j

*��1� = −H*

4 �m=1

NS

��Qi*Qm

* �m�Amj + Aim��mQm* Q j

*�� +1

2Aij� . �22�

he equation for the polymer stress is the same as in Eq. �15�. In this case, closure of thevolution equation above is not possible because of the FENE nonlinearity, �m, on theight-hand side. However, as in the case of the Gaussian approximation for HI, assuminghat the configurational distribution is a Gaussian, and using Wick’s decomposition rule,he problematic terms on the right-hand side of the equation above can be formallyimplified. Since �Sm

* /�Qm* =H*�mQm

* �where Sm* =Sm /kBT�, using Eq. �21� leads to

H*�Qi*Qm

* �m� = �Qi* �Sm

*

�Qm* � = �im

* · � �2Sm*

�Qm* �Qm

* � �23�

he expression on the right-hand side also could have been obtained by directly using a
uadratic mean-field spring potential,

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Sm* �

1

2� �2Sm*

�Qm* �Qm

* �:Qm* Qm

* , �24�

n place of the exact spring potential Sm* . In other words, applying the Gaussian approxi-

ation to simplify the average due to FENE springs in free-draining chains is equivalent

o using the quadratic mean-field spring potential Sm* , since this approximation leads to

�Qi* �Sm

*

�Qm* � =�Qi

* �Sm*

�Qm* � . �25�

rakash and Öttinger �1999� have previously shown that Fixman’s treatment of the ex-luded volume interaction potential �Fixman �1966�� corresponds to the use of a similarairwise mean-field quadratic excluded volume potential.

The expression for Sm* in Eq. �24� can be simplified by using �Sm

* /�Qm* =H*�mQm

* ,

Sm* = H*��m�� + � ��m

�Qm* Qm

*��:Qm* Qm

* . �26�

he first term within the brackets on the right-hand side of the equation above containing�m�, the average of the spring force nonlinearity, can be recognized as that arising fromconsistent-averaging treatment of that nonlinearity, while the remaining term accounts

or the influence of fluctuations in the spring force.Recursive application of Wick’s decomposition rule, using the result ��m /�Qm

*

2Qm* d�m /d�Qm

*2�, leads to a series expansion,

Sm* = H*��m�� + �

s=1

�

2s� ds�m

d�Qm*2�s��mm

*s �:Qm* Qm

* , �27�

here �mm*s denotes the matrix product of s copies of �mm

* . The expansion above is validrovided the Gaussian averages ��m� and �ds�m /d�Qm

*2�s� exist for all s=1, . . . ,�. ForENE springs, however,

�m =1

�1 − Qm*2/b*�

, �28�

nd

ds�m

d�Qm*2�s =

s!

b*s

1

�1 − Qm*2/b*�s+1 . �29�

he Gaussian averages of these functions do not exist since all of them have noninte-rable singularities at Qm

*2=b*, whereas the Gaussian distribution has an infinite range. Toake analytical progress, it is necessary to introduce further approximations. Here, analo-

ous to the Peterlin closure approximation �Peterlin �1966��,

��m� is replaced with1

��1 − Qm*2/b*��

, �30�

nd


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� ds�m

d�Qm*2�s� is replaced with

s!

b*s

1

��1 − Qm*2/b*�s+1�

. �31�

lthough each of these terms may remain bounded as long as �Qm*2�=tr �mm

* �b*, there isneed to carefully analyze the convergence properties of the infinite series in Eq. �27�

fter introducing these approximations. Here, only the first term in the infinite series inq. �27� is retained to represent spring force fluctuations. As a result, the followingean-field quadratic spring potential is used:

Sm* =

H*

2Lm:Qm

* Qm* , �32�

here

Lm �1

��1 − Qm*2/b*��

� +2

b*

1

��1 − Qm*2/b*�2�

�mm* ,

=1

1 − tr�mm* /b*� +

2/b*

1 − 2tr�mm* /b* + ��tr�mm

* �2 + 2�mm* :�mm

* �/b*2�mm* . �33�

he use of this potential leads to a reduction of the complex moments in Eq. �22�,

H*�Qi*Qm

* �m� = H*�im* · Lm, �34�

nd a mean-field spring force given by

FmS,c = H*Lm · Qm

* . �35�

As mentioned above, neglecting the spring force fluctuation term in Lm is equivalento a consistent-averaging treatment of the FENE nonlinearity, which leads to the well-nown FENE-P force law proposed originally by Bird et al. �Bird et al. �1980��,

FmS,c = H* 1

1 − tr�mm* /b*Qm

* = H*�mQm* , �36�

here �m= �1−tr �mm* /b*�−1. Incorporation of fluctuations through the Gaussian approxi-

ation leads to the spring force expression in Eq. �35�, denoted here as the “FENE-PG�pring force law.

The evolution equation for the second moments in the Gaussian approximation forree-draining FEBS chains is consequently

�ij,�1�* = − H*1

4 �m=1

NS

��im* · LmAmj + AimLm · �mj

* � +1

2Aij� , �37�

nd the polymer stress predicted by the approximation is given by

�p* = NS� − H*�

i=1

Ns

�ii* · Li. �38�

It is worth noting that for finite �mm* , as b*→�, the fluctuation contribution in Eq. �33�

ecomes vanishingly small, Lm→� and the spring becomes more Hookean-like. On the
ther hand, as the bead-spring chain approaches its full extension, and

2.235.136.167 On: Fri, 09 May 2014 13:47:33

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�mm* → �b* 0 0

0 0 0

0 0 0� , �39�

he scalar prefactor in the fluctuation term in Eq. �33� approaches �1/b*�, which howeverecomes negligible compared to the FENE-P contribution, which diverges. Thus, ashains become highly stretched, predictions obtained with the FENE-P or FENE-PGpproximations are expected to be nearly identical.

We note here that the method above for incorporating fluctuations in the spring forceor any other conservative nonlinear interbead force� leads to a noncentral force: the

ean-field force Fms,c is in general not directed along Qm. Such behavior is typical of

ean-field potentials �Kröger �2004��, the noncentral nature of the force arising as theesult of the anisotropy of the fluctuations in the spring’s configuration. Under isotropiconditions at equilibrium, Lm is isotropic and the mean-field force is directed in theirection of the corresponding connector vector.

Although the FENE-PG approximation is new and has been developed in this study,he FENE-P approximation has been studied in great detail by several workers �Ghosh etl. �2001�; Herrchen and Öttinger �1997�; Keunings �1997�; Öttinger �1989�; van denrule �1993�; Wedgewood and Bird �1988�; Wiest and Tanner �1989�; Wiest et al.

1989��. As pointed out in many of these studies, perhaps the single most importantifference between the exact FENE and the mean-field FENE-P and FENE-PG approxi-ations lies in the nature of the configurational probability distribution function. The

ingularity in the nonaveraged FENE spring force strictly restricts the lengths of theonnector vectors to lie in the domain �0,Q0�. However, with the Peterlin closure, theolution of the modified Fokker-Planck equation with FENE-P/PG springs is a Gaussian.herefore, there is a nonzero probability that a spring has a length greater than Q0. Thisas been clearly illustrated by Keunings �1997�, who used Brownian dynamics simula-ions of dumbbells with FENE-P springs to show that, in strong extensional flows, aignificant proportion of the dumbbells had length greater than the “maximum allowed”imensionless extension Q0. However, in all the literature on the FENE-P model, theean-squared lengths of the springs are observed to always be less than Q0

2. In otherords, the mean-field FENE-P model appears to satisfy the maximum extension con-

traint in an average sense.It is observed in the studies cited above that predictions with the FENE-P approxima-

ion compare well with the FENE model for steady-state properties, in both shear andniaxial extensional flows. An explanation for the good agreement at steady state introng flows is based on the observation that under such conditions most of the springs inhe chain are highly stretched. The exact steady-state distribution function for the lengthf the spring in a dumbbell model is sharply peaked around ��Q2�, and can be wellpproximated by a function �Lielens et al. �1998�; Wiest and Tanner �1989��. However,uring start-up of shear and extensional flows, predictions with the FENE-P model for theransient rheological properties deviate strongly from the exact results of BDS prior to thettainment of the steady state �Herrchen and Öttinger �1997�; van den Brule �1993��.

There have been several previous attempts to improve the FENE-P approximation. Inonditions where springs can be highly stretched, the true spring length distribution isetter approximated as the sum of a uniform distribution for smaller values of Q and a unction located at a large Q closer to �but less than� Q0. This FENE-L approximation,nd its three-dimensional counterpart, are shown to be more accurate than the FENE-P
odel, particularly in predicting the phenomenon of stress-conformational hysteresis

2.235.136.167 On: Fri, 09 May 2014 13:47:33

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Lielens et al. �1998�, �1999�; Sizaire et al. �1999��. Recently, the Lielens closure haseen extended to a FENE dumbbell with variable drag model as well �Nitsche et al.2006��. Gorban et al. �2001� show that an entropic argument can be used to improvepon the FENE-P approximation by first expanding the exact FENE potential in a Tay-or’s series expansion about �Q2� and then systematically regrouping terms in the seriesefore truncating it. This approach gives rise to a hierarchy of spring potentials of in-reasing complexity, the simplest of which is shown to be the quadratic potential corre-ponding to the FENE-P force law. By using the next higher order potential �which isenoted as the “FENE-P+1” potential by Gorban et al.� in the hierarchy, these authorsemonstrate with a one-dimensional dumbbell model that accuracy is significantly im-roved in an unsteady elongational flow. In Sec. V below, the accuracy of the FENE-PGodel proposed in this study is evaluated by comparison with exact Brownian dynamics

imulations.The choice of parameter values H* and b* for the approximate FENE-P and FENE-PG

odels is made here consistently with the procedure used in the case of the FENE forceaw; namely, the equilibrium mean square end-to-end distance of a spring predicted byhe model is equated to bK

2 NK,S, while the fully stretched length is equated to bKNK,S.urther, as before, �S

2=RS2 /3=bK

2 NK,S /3, and the corresponding time scale �S, are used ashe basic scales for obtaining equations and properties in dimensionless form. Thesessumptions lead to b*=Q0

2 /�S2=3NK,S, as in the earlier case. From Eq. �37�, it is clear that

he equilibrium second moments �ij,eq* , predicted by the Gaussian approximation for

ree-draining FEBS chains, are solutions to the following set of algebraic equations:

0 = −H*

4 �m=1

NS

��im,eq* · Lm,eqAmj + AimLm,eq · �mj,eq

* � +1

2Aij� . �40�

t equilibrium, since �ij,eq* =�ij,eq

* � and Lm,eq=Lm,eq�, the equation above can be rear-anged into the following set of equations:

0 = �m=1

NS

�UimAmj + AimUmj� , �41�

here

Uij � H*�ij,eq* Lj,eq − ij . �42�

ince the Rouse matrix is nonsingular, the solution to Eq. �41� is obtained by setting

ij =0. This implies �ij,eq* =ij�, and H*L j,eq=�. With all springs being identical at equi-

ibrium, L j,eq=Leq�, and therefore

H*Leq = 1. �43�

sing tr� j j,eq* =3, and b*=3NK,S, it can be shown using Eq. �33� that for FENE-PG

prings,

H* =3NK,S

3 − 9NK,S2 + 13NK,S − 5

3NK,S3 − 4NK,S

2 + 5NK,S

. �44�

or FENE-P springs, Leq= �m,eq �for all m�, where �m was introduced in Eq. �36� earlier.s a result,

H* =1

¯= 1 −

1

NK,S. �45�

�m,eq


2.235.136.167 On: Fri, 09 May 2014 13:47:33

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. FEBS chains with HI

The dimensionless evolution equation for the second moments of a bead-spring chainodel with springs obeying a mean-field potential given by Eq. �24�, and with fluctuatingI between the beads, is given by

�Qi*Q j

*��1� = −H*

4 �m=1

NS

�Qi*Qm

* · Lm · Amj + Aim · Lm · Qm* Q j

*� +1

2�Aij� , �46�

Since the tensors Lm are not direct functions of the connector vectors, expectations ofhe form �Qi

*Qm* ·Lm · ��mj�� appearing in the equation can be simplified by assuming

hat the configurational distribution function is Gaussian and using Wick’s decompositionule as in the case of free-draining FEBS chains above. The final equation thus obtainedor the approximate second-moments �ij

* is

�ij,�1�* = −

H*

4 �m=1

NS

��im* · �Lm · Amj + �mj� + �Aim · Lm + �mi

T � · �mj* � +

1

2Aij , �47�

here the tensor Aij describes contributions arising from averaging the Oseen-Burgersensor with the nonequilibrium distribution function, while the tensor �ij represents con-ributions due to fluctuations in HI. Since the expressions for these tensors are lengthynd not very illuminating in themselves, they are given by Eqs. �A1� and �A2� in the

ppendix. The important point to note here is that Aij and �ij are completely expressiblen terms of the second moments of the Gaussian distribution, and Eq. �47� is closed withespect to the tensors �ij

* .At equilibrium, all the tensorial quantities in the equations above are isotropic. Fur-

hermore, the HI fluctuation tensors �ij vanish under isotropic conditions �see the Ap-endix�. Consequently, one obtains an equation of the same form as Eq. �41�,

0 = �m=1

NS

�UimAmj + AimUmj� , �48�

n which the values of Uij are the same as in Eq. �42� for the free-draining model with

ENE-PG springs, and Aim are elements of the modified Rouse matrix �Eq. �49��,

�Aij�eq = Aij� = �Aij + �2h*� 2��i − j�

−1

��i − j + 1�−

1��i − j − 1�

�� , �49�

here, the convention 1/�0=0 has been used. Since the modified Rouse matrix is alsoonsingular, the equilibrium solution is once again obtained by setting Uij =0. Therefore,he equilibrium solution obtained for the Gaussian approximation for FENE-PG chainsith HI is identical to that obtained for free-draining FENE-PG chains. In other words,

he heuristic two-step procedure outlined above to achieve closure in the model combin-ng both FE and HI ensures that the equilibrium solution for the second moments isndependent of HI. This two-step procedure has been used to derive a closed set ofquations for the second moments in a FEBS chain model, which includes EV interac-ions �represented by the narrow-Gaussian excluded-volume potential�, and HI �Prab-akar �2005��. However, no calculations have been performed so far with this model,hich accounts for fluctuations in all the three nonlinear phenomena of FE, HI, and

xcluded volume.Nearly all the closure approximations introduced to date for the second moments* *
Qi Q j � can be derived by dropping appropriate terms in Eq. �47�. For instance, the Zimm

2.235.136.167 On: Fri, 09 May 2014 13:47:33

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odel is obtained by �i� using a Hookean spring force law, Lm=�; �ii� replacing the

onequilibrium averaged Oseen tensor contribution Aij with the modified Rouse matrix

ij �which arises due to equilibrium averaging the Oseen tensor�; and �iii� setting �ij

0. Ignoring fluctuations in hydrodynamic interactions in a model with Hookean springs�ij =0 and Lm=�� leads to the consistent-averaging �CA� approximation �Öttinger1987��, while retaining fluctuations in HI leads to the Gaussian approximation �GA�Öttinger �1989��. Setting �ij =0 and Lm= �m� simultaneously �which implies fluctuationsn both HI and the spring force are ignored� leads to a model combining the consistent-veraging treatment of HI with the FENE-P approximation for the spring force. Theredictions of this CA-P model for properties in shear flows have been studied in detaily Wedgewood and Öttinger �1988�. On the other hand, using only one of the twoonditions leads to a model in which fluctuations in the corresponding phenomenon alonere ignored, leading to the CA-PG model or the GA-P model. The model incorporatinguctuations in both nonlinear phenomena is referred to here as the GA-PG model. Note

hat in all these models, the polymer stress is given by Eq. �38�, with the appropriatehoice of Li.

In discussing predictions obtained using the various approximate models, it is conve-ient to use acronyms in which the first two letters refer to the approximation used for HI,nd the last letters refer to that used for FE. Table I lists the letter codes used for easyeference. Note that in this scheme, the Rouse model is denoted as the FD-H model,hile the Zimm model is denoted as EA-H. We next discuss briefly the numerical

chemes used in this work.

V. NUMERICAL METHODS

The total number of second moments in the bead-spring model equals NS2, and hence

he number of governing equations in the approximations described in this paper is equalo 9NS

2. However, �ij* =� ji

*T. Further, the second moments must not change if the beadumbering is reversed. Thus, �ij

* =�NS−i,NS−j

* . Therefore, the number of independent second

oments reduces by a factor of nearly 1/4 �Öttinger �1989��. The symmetry of themposed flow field can be used to further reduce the number of components for all theensorial quantities in the equations. In shear flows, the second moment tensors take the

TABLE I. Acronyms used for referring to approximate models.

Acronym

Treatment of HIFree-draining FDEquilibrium-averaging EAConsistent-averaging CAGaussian approximation GA

Treatment of FEHookean springs HFENE-P PFENE-PG PG

ollowing form:


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� = ��1 �4 0

�5 �2 0

0 0 �3� . �50�

n uniaxial extensional flows, in both stress growth and relaxation phases, all tensorialuantities in the approximations have the form

� = ��1 0 0

0 �2 0

0 0 �2� . �51�

The exploitation of the symmetries above to reduce the computation of the fourth-rankensorial function K�s� has been discussed thoroughly by Zylka �1991�. In general, forny symmetric tensor s, only 21 of the 81 components of the K function are nonzero intransformed coordinate system C in which the tensor s has a diagonal representation.ut of these 21 components, only nine require independent numerical evaluation. The

omponents K1111, K2222, and K3333 can be used to calculate the remaining 12 nonzeroomponents. Furthermore, the function K always appears in the equations as K: �. Inhear flows, this means that only 14 components of the K function in the laboratory fixedoordinate system are required to be evaluated. In uniaxial extensional flows, it can behown that only Kxxxx �where x is the direction of stretching� is required. As shown byylka, components of the H�s� and K�s� functions in the coordinate system C are effi-iently evaluated using elliptic integrals.

The initial condition for the time integration of the ordinary differential equationsODEs� starting from equilibrium is �ij,eq

* =ij� for the approximations. As pointed out in

he preceding sections, at equilibrium Aij,eq= Aij, �ij,eq=0, H*Li,eq=�. Since predictionsre obtained only for shear and uniaxial extensional flows in this study, separate computerodes were written for the two different flows to take advantage of the reduction in theomponents due to their individual flow symmetries. A variable-step Adams method �the02CJF routine in the NAG library� is used for the integration of the ODEs. Steady-state

olutions were obtained by setting the time derivatives in the ODEs to zero, and thensing a Wegstein successive substitution algorithm to solve the set of nonlinear equations.olutions are obtained for several values of the �shear or extensional� strain rate. Theonverged solution for the set of �ij

* at any strain rate is used as the initial guess for theext higher �or lower� strain rate for which a solution is desired. It is found that largeteps in the strain rates could lead to unphysical, non-positive-definite solutions for theecond moments �Wedgewood and Öttinger �1988��. Such behavior is avoided by manu-lly controlling the size of the steps in strain rate.

The Brownian dynamics simulations of Rouse and FENE chains with �and without�PY-HI performed in this study use a modification of the semi-implicit predictor-orrector scheme outlined recently by several workers �Hsieh et al. �2003�; Prabhakarnd Prakash �2004�; Somasi et al. �2002�� where the error between the predictor andorrector steps is used to adaptively control the time-step size in every simulation.

Before considering the results obtained in this study, we note that the computationaload per time step in the Gaussian approximation scales as NS

3 N3, since for every i andj, there is a summation over m=1, . . . ,NS on the right-hand side of Eq. �47� �Prakash and

ttinger �1997��. By appropriately storing results of calculations, the expense in the

eneration of the tensorial functions Aij, Li, and �ij does not scale faster than N3 in eachime step. The cost of a time step in the simulation of a single trajectory in a BD
imulation �with the numerical algorithm used in this study� on the other hand scales as

2.235.136.167 On: Fri, 09 May 2014 13:47:33

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2.25 �Fixman �1981� Jendrejack et al. �2000�; Kröger et al. �2000��. Although in prin-iple this means that for sufficiently long chains BDS should become more efficient thanhe Gaussian approximation, the number of trajectories required to bring down the errorn a BD simulation to an acceptable level is often of the order of 102 or more. We observehat the simulation time for a single trajectory for chains with about 20 to 25 beads isoughly of the same order of magnitude as a single run with the Gaussian approximation.herefore, a complete BD simulation takes O�102� times more CPU time than one runith the approximation, and if proven to be accurate, the Gaussian approximation is aiable alternative to BDS for the values of N 20 typically employed in the study ofilute polymer solutions, in situations where the conformations of individual polymersre themselves not the object of simulation interest. It is further possible to speed upalculations in the Gaussian approximation by using an additional diagonalization-of-ormal modes assumption �Prabhakar �2005�; Prabhakar and Prakash �2005�; Prakashnd Öttinger �1997�� An article on this faster version of the Gaussian approximation isurrently in preparation.

. RESULTS AND DISCUSSION

Attention is restricted here to rheological and conformational properties in some in-tances of simple shear and uniaxial extensional flows. Results for shear flows and ex-ensional flows are discussed separately in the respective sections below. In each case,esults are presented for �i� models with nonlinear force laws, but with no HI; �ii� modelsith HI incorporated, but with a linear Hookean force law; and �iii� models with nonlin-

ar force laws and HI incorporated. In particular, in case �i�, results obtained with the newENE-PG force law introduced here are compared with results obtained with the FENE-Porce law and with BDS; in case �ii� �which involves only models that have been intro-uced previously�, results for certain material functions that have not been predictedarlier are presented; and in case �iii�, the combined influence of both phenomena arexamined, aided by the results obtained in cases �i� and �ii�.

. Shear flow

For a simple steady homogeneous shear flow characterized by the dimensionless shearate �*, the rheological properties of interest are the polymer’s contribution to the dimen-ionless viscosity �p

* =�p,yx* / �*, the dimensionless first normal-stress difference coefficient

1*=−��p,xx

* −�p,yy* � / �*2, and the dimensionless second normal-stress difference coefficient

2*=−��p,yy

* −�p,zz* � / �*2, where x, y, and z are the flow, gradient, and vorticity directions,

espectively.We note here that, in the following discussion, the characterization of a shear rate as

low” or “high” depends on whether the Weissenberg number—defined as the product ofhe shear �or extension� rate and a time scale characterizing the large-scale dynamics ofsolated chains near equilibrium—is much lower, or greater than unity. As is well known,his characteristic time scale is proportional to N2 for free-draining chains, whereas inhains with HI, it is proportional to N3/2. The Weissenberg number in these two cases thuscales as �*N2 and �*N3/2, respectively. For N=20, a shear rate of �* 1 thereforeepresents a large strain rate, as will become evident in the results below. In extensionalows however, the critical strain rate for the coil-to-stretch transition �discussed in Sec.
B� marks the boundary between small and large extension rates.

2.235.136.167 On: Fri, 09 May 2014 13:47:33

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. Start-up of shear flow

The performance of the FENE-P and FENE-PG approximations for free-drainingEBS chains in the prediction of unsteady behavior in shear flows is first considered. At

ow shear rates close to equilibrium, finite chain extensibility has little influence on theynamic behavior predicted by the models. Therefore, the predictions of the free-drainingEBS models are close to those predicted with the Rouse model. At higher shear rates,arlier studies with free-draining FENE-P dumbbells �Herrchen and Öttinger �1997�� andhains with N=10 �van den Brule �1993�� show that this approximation leads to predic-ions for unsteady shear properties that agree only qualitatively with the exact resultsbtained using FENE springs. Figures 1�a� and 1�b� show that this trend is continued for0-bead chains. The incorporation of fluctuations in the spring force with the use of theENE-PG approximation leads to a clear improvement in the quality of the predictionsor all properties at moderate shear rates �represented by the curves for �*=0.23�. As theimensionless shear rate approaches unity, however, the FENE-PG approximation alsoegins to deviate from the exact BDS results. We observe in Fig. 1�b� that predictionsith the FENE-PG approximation for the second-normal stress difference coefficient are

n excellent agreement with the simulations’ results �within error bars�, in contrast to theonstant value of zero predicted with the FENE-P force law. It is interesting that springorce fluctuations lead to a transient nonzero positive �2

*, which however seems to vanisht steady state. In other words, the effect of spring force fluctuations on �2

* is only felt inhe transients, but not at steady state.

Although the qualitative behavior of the different approximations for HI in Rousehains has been well documented in the case of steady shear flows, their predictions ofheological and conformational properties in transient shear flows have not receiveduch attention. For Rouse chains with HI, use of equilibrium averaging in the Zimmodels results in a fixed time dependence of properties that is independent of the shear

ate �Bird et al. �1987b��. This behavior parallels the shear rate independence of theaterial functions predicted at steady state by the Zimm model. In contrast, the predic-

ions of the consistent-averaging and Gaussian approximations, and the exact results of

IG. 1. Growth of �a� viscosity and �b� second normal-stress difference coefficient, for free-draining FEBShains during start-up of steady shear flow.

he simulations for the growth of the shear material functions depend on the imposed


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hear rate, as demonstrated in Figs. 2�a� and 2�b� for the transient variation in �p* and �2

*,espectively, after the imposition of a steady shear rate. The importance of including aescription of the fluctuations in approximate models accounting for HI is clearly high-ighted by the superiority of the predictions of the Gaussian approximation. Interestingly,ig. 2�b� shows that fluctuations in HI lead to a negative second normal-stress differenceoefficient soon after the inception of flow. Zylka �1991� had earlier demonstrated theccurate prediction of the sign of �2

* with the Gaussian approximation at steady state.he results here show further that the approximation is excellent even in transient shearows.

For FEBS chains with HI, Fig. 3�a� shows that the GA-PG approximation is particu-arly useful in shear flows of moderate strength where fluctuations in the spring forces

IG. 2. Growth of �a� viscosity, and �b� second normal-stress difference coefficient, for Rouse chains with HIuring start-up of steady shear flow.

IG. 3. Growth of �a� first normal-stress difference coefficient, and �b� viscosity, for FEBS chains with HI
uring start-up of steady shear flow.

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nd HI are both important. The prediction of the GA-PG model is in remarkable agree-ent with simulations’ results for the unsteady growth in the first normal-stress differ-

nce coefficient following the imposition of a moderate shear rate of �*=0.23. Similaruantitative agreement is obtained for the unsteady variation in �p

* and Re*2 at this shear

ate. At lower shear rates FE is not important, and the behavior is the same as thatbtained with Rouse chains, for which it was shown above that the Gaussian approxima-ion does well in describing unsteady behavior.

At high shear rates, where FE becomes the dominant effect, however, predictions ofo single model are uniformly in agreement with the BDS results for all the propertiesxamined, as illustrated in Fig. 3�b�. To understand the source of the deviations in theredictions of the GA-PG model, we compare in Fig. 4 predictions of the GA-H, FD-PG,nd GA-PG models at �* 1. The transient evolution of the viscosity in the GA-PGodel can be broken into two phases: the early overshoot phase, and the later steady-state

pproach phase. At early times, we see that the GA-H model is in very good agreementith the BDS data �triangles� for Rouse chains with HI. In the same period, the FD-PGredictions deviate quite strongly from the corresponding BDS data for free-drainingEBS chains �squares�. The agreement of the composite GA-PG model with the BDSata for FEBS chains with HI �circles� is thus seen to be intermediate in the overshoothase. To understand the behavior at steady state, we note that fluctuations in HI at highhear rates aid in the penetration of the solvent’s velocity field into the deformed polymeroil, and the viscosity predicted is higher and closer to predictions with the correspondingree-draining model. This important aspect of HI’s influence on the behavior of diluteolymer solutions will be more clearly demonstrated shortly in predictions of steady-stateroperties. From the overprediction of the viscosity by the GA-H model at large times inig. 4, we see that at high shear rates, the Gaussian approximation for HI overestimates

he influence of fluctuations in HI. In Fig. 3�b�, we see that the CA-PG model’s predictions lower than the BDS results. The Gaussian approximation for HI thus pushes the

IG. 4. Comparison of the predictions of the GA-H, FD-PG, and GA-PG models for the growth of the sheariscosity with BDS results obtained with Rouse chains with HI �triangles�, free-draining FENE chainssquares�, and FENE chains with HI �circles�. The dimensionless shear rate is unity for Rouse chains and 0.91or FENE chains.

iscosity prediction of the CA-PG past the BDS results.


2.235.136.167 On: Fri, 09 May 2014 13:47:33

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. Steady shear flow

The steady-state shear viscosity predicted by the FENE-P approximation was observedarlier to be in quantitative agreement with the values obtained with BDS of smallree-draining FENE chains �Herrchen and Öttinger �1997�; van den Brule �1993��. Forhains with N=20, it is observed that the predictions of �p

* obtained with the FENE-P andENE-PG approximations are nearly identical. However, in Figs. 5�a� and 5�b�, it is seen

hat the FENE-PG approximation for the FENE force law does much better than theENE-P approximation in its prediction of the steady-state �1

* and Re*2, respectively, at

oderate shear rates. At �*�1, the predictions of Re*2 and �1

* with FENE-PG springslso begin to deviate from BDS results significantly, but are closer to the exact resultshan the predictions obtained with FENE-P springs. From the fact that the stresses and

e*2 values predicted with simulations and the FD-PG model are lesser than the predic-

ions of the FD-P model, we conclude that fluctuations in the spring forces lead to anncrease in the effective stiffness of the chains.

As mentioned before, the predictions of the Gaussian approximation for Rouse chainsn steady shear flows have been compared earlier with those of the consistent- andquilibrium-averaging �Zimm� approximations, and with the results of exact BDS �Öt-inger �1989�; Zylka �1991��. Figure 6 illustrates some of the well-known features of theredictions of the different approximations for Rouse chains with HI. It is important toote the contrasting roles played by fluctuations in HI at low and high shear rates. Theact that the predictions of the Gaussian approximation and BDS are lower in Fig. 6 thanhose obtained with the Zimm and consistent-averaging approximations near equilibriumnd at moderate shear rates indicates that, under these conditions, fluctuations in configu-ations at equilibrium result in a more effective screening of the velocity field within theoil. At high shear rates, where shear thickening is observed in Rouse chains with HI asconsequence of the decreasing influence of HI as the average separation between the

eads in the chains increases, the predictions of the GA-H model and the BDS results liebove the curve predicted by the CA-H model beyond a threshold shear rate, and areloser to the free-draining Rouse model’s prediction. This behavior is contrary to that

IG. 5. Variation of �a� the steady-state first normal-stress difference coefficient, and �b� the steady-stateean-squared end-to-end distance, with shear rate, for free-draining FEBS chains.

bserved near equilibrium, and shows that the role played by fluctuations in HI reverses


2.235.136.167 On: Fri, 09 May 2014 13:47:33

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t higher shear rates, and in fact aid the penetration of the solvent velocity field into theolymer coil as the coil deforms and becomes highly anisotropic. In general, it is ob-erved that at high strain rates in both shear and extensional flows, predictions withodels using the Gaussian approximation for HI are closer to their free-draining coun-

erparts than those using consistently averaged HI, indicating that the role reversal of HIccurs whenever polymer coils are highly anisotropic �Prabhakar �2005��.

Figure 7�a� shows the results for the steady-state �1* obtained with the different ap-

roximate models combining HI and FE. Predictions for the transient growth of �1* at

˙ *=0.23 were presented earlier in Fig. 3�a�. The qualitative features of the predictions forhe steady-state viscosity �p

* are the same, although the shear-thinning-thickening behav-or is more pronounced in the case of �1

*. As before, the predictions of these models are

IG. 6. The influence of HI on the variation of the steady-state polymer viscosity with dimensionless shear rate,or Rouse chains.

IG. 7. Variation of the steady-state �a� first normal-stress difference coefficient, and �b� second normal-stress
ifference coefficient with shear rate, for FEBS chains with HI.

2.235.136.167 On: Fri, 09 May 2014 13:47:33

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ompared with the results of BDS of FENE chains with fluctuating HI. It is clear fromig. 7�a� that among all the approximate models, only the predictions of the GA-PGodel are close to the simulations’ results across the whole range of shear rates exam-

ned. At moderate shear rates when FE begins to exert its influence, it is seen that theaussian approximation is not as accurate when used with FENE-P springs as it is withENE-PG springs. This suggests that fluctuations in the spring forces also become im-ortant for �p

* and �1* at moderately large shear rates. In contrast, as observed earlier in

ig. 1�b� for free-draining chains, Fig. 7�b� shows that spring force fluctuations have aegligible effect on the steady-state �2

*, although they are of greater significance inetermining the transients in �2

* �Prabhakar �2005��.

. Extensional flow

In a uniaxial extensional flow with a dimensionless extension rate �*, the sole rheo-ogical property of interest is the polymer contribution to the dimensionless extensionaliscosity, �p

* =−��p,xx* −�p,yy

* � / �*. In the case of the start-up of extensional flow followedy its cessation, we consider predictions for the magnitude of the polymer’s contributiono the dimensionless first normal-stress difference, N1,p

* = ��p,xx* −�p,yy

* �.

. Start-up and cessation of steady extensional flow

Models using Hookean springs predict unbounded extension in strong extensionalows. As a result of this unrealistic behavior, the predictions of approximations for HI inouse chains have not been examined in strong and unsteady extensional flows. How-ver, at any finite time following the imposition of the flow, the values of the stresses andther properties obtained with Rouse chains are still finite. Therefore, by comparing theredictions of the approximations and the results of BDS for Rouse chains, one can studyn isolation the influence of HI in strong extensional flows.

Figure 8�a� shows the growth of N1,p* upon the imposition of a steady extensional flow

˙*

IG. 8. �a� Growth and decay of the polymer contribution to the first normal-stress difference for Rouse chainsith HI, during start-up and following cessation of steady extensional flow, respectively. �b� Effect of fluctua-

ions in HI on the growth and decay of the polymer contribution to the first normal-stress difference for Rousehains with HI, during start-up and following cessation of steady extensional flow, respectively.

f � =0.04 �which is greater than the critical strain rate for the coil-stretch transition


2.235.136.167 On: Fri, 09 May 2014 13:47:33

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˙ c*=0.012 for N=20 and h*=0.25�. Also shown in Fig. 8�a� are the predictions for theelaxation of N1,p

* after the cessation of the extensional flow at different values of theencky strain. The results in Figs. 8�a� and 8�b� show several fascinating features. First,

he predictions of all the approximate models, including the Zimm model withquilibrium-averaged HI, appear to be in remarkably close agreement with the exactesults of BDS in the stress growth phase. The Zimm model only begins to deviateoticeably from the rest of the data during stress relaxation after flow stoppage at hightrains. Similar results are also obtained for Re

*2 �Prabhakar �2005��. Although the differ-nces between the different approximations appear to be small on the scale of the plot inig. 8�a�, plotting

�GA-CA =N1,pGA

* − N1,pCA

*

N1,pGA

* , �52�

he relative difference between the predictions of N1,p* of the two approximations in Fig.

�b� reveals that fluctuations in HI can account for as much as 30% of the polymer stressn extensional flows.

Predictions of free-draining FEBS models employing the FENE-P and FENE-PG ap-roximations are compared with exact results of the simulations for the growth andelaxation of N1,p

* and Re*2 in Figs. 9�a� and 9�b�, respectively. Also shown for comparison

re the predictions of the simple Rouse model. The strain rate for which the data arehown in Fig. 9 is larger than the critical strain rate �c

*=sin2�� /2N�=6.210−3 forree-draining chains with N=20. The influence of FE is first discerned in the growth ofhe stress when the curves for the FEBS chains separate from the prediction of the Rouse

odel. At higher strains, the deviations between the approximations and the exact resultsor N1,p

* become more marked and reach a maximum when the approximate results levelff sharply towards their eventual steady states. The approach of the BDS data to steadytate is more gradual. In the stress growth phase, the predictions of the FENE-PG ap-

IG. 9. Growth of �a� the polymer contribution to the first normal-stress difference, and �b� the mean-squarednd-to-end distance of free-draining FEBS chains, during start-up, and their relaxation following cessation, ofteady extensional flow. The horizontal dash-dotted line in �b� indicates the maximum possible value of Re

*2

3NK,SNS2.

roximation are closer to the exact results than those obtained with the FENE-P approxi-


2.235.136.167 On: Fri, 09 May 2014 13:47:33

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ation. Nevertheless, the sharp “knee” in the growth of N1,p* and Re

*2 is common to bothhe FENE-P and PG approximations, and appears to be the result of the Peterlin closure.

The relaxation behavior of free-draining FEBS chains following the cessation of ex-ensional flow has been previously studied �Doyle et al. �1998�; Wiest �1999��. Whileoyle et al. compared the predictions of the FENE-PM approximation with BDS ofENE dumbbells and also bead-rod chains, Wiest examined some interesting qualitativeeatures in the stress-relaxation phase using the FENE-P model. The FENE-PG approxi-ation is seen to lead to more accurate description of both the stress and extension in the

elaxation phase when the strain at flow stoppage is not much larger than the strain athich FE begins to exerts its influence, whereas the FENE-P approximation leads to anverprediction of both properties. When the Hencky strain at flow stoppage is larger, theENE-PG force law is seen to be in fact less trustworthy. This loss in accuracy of theENE-PG model when chains are highly stretched is consistent with the behavior ob-erved earlier in steady and unsteady shear flow.

The predictions of approximate models with both nonlinear phenomena incorporatedor the unsteady growth and relaxation of N1,p

* and Re*2 following start-up and cessation of

n extensional flow are shown in Figs. 10�a� and 10�b�. The nature of deviations betweenhe BDS results and the approximations’ predictions appears to be decided largely by thepproximation used for the FENE nonlinearity. On the whole, however, the GA-PGodel appears to be quite accurate in its description of the normal stresses and chain

eformation in unsteady extensional flows.The prediction of stress-conformation hysteresis is another test of the accuracy of

losure approximations. Stress-conformation hysteresis in dilute polymer solutions haseen studied in experiments �Orr and Sridhar �1999�; Spiegelberg and McKinley �1996��nd theoretically �Doyle et al. �1998�; Ghosh et al. �2001�; Li and Larson �2000�; Lielenst al. �1998�; Wiest �1999��. Figure 11 compares the N1,p

* -versus-�n* behavior predictedy BDS and the various approximations. It is immediately clear that closure approxima-ions fail to reproduce the size of the hysteresis loops observed in the simulations. Models

IG. 10. Growth and decay of the polymer contribution to �a� the dimensionless first normal-stress difference,nd �b� the mean-squared end-to-end distance, for FEBS chains with HI, during start-up and following cessa-ion, of steady extensional flow. The horizontal dash-dotted line indicates the maximum possible value of

e*2=3NK,SNS

2.

sing the FENE-P springs predict larger birefringence than the exact simulations at any


2.235.136.167 On: Fri, 09 May 2014 13:47:33

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Redistribution

evel of the stress. When fluctuations are accounted for in the spring forces through theENE-PG force law, the size of the hysteresis loop becomes even smaller. Nevertheless,

he curves predicted with FENE-PG springs now lie within the loops predicted by theDS.

Hydrodynamic interactions appear to have almost no influence on N1,p* −�n* hysteresis

n Fig. 11. Similar behavior is also observed on a plot of N1,p* -versus-Re

*2 �Prabhakar2005��. Although the maximum values of the stress and �n* are larger when free-raining models are used, the simulations’ and approximations’ predictions with andithout HI show significant overlap. This is remarkable because the variation of theroperties with Hencky strain for free-draining chains is clearly different from that ob-erved for chains with HI. However, when the data are plotted as in Fig. 11, the hysteresisoops for chains with and without HI are almost identical, showing that the stress-onfigurational paths taken by the chains are similar, although HI modifies the rates athich the changes occur.

. Steady extensional flow

Figure 12 compares the predictions of the steady-state polymer extensional viscosityy different approximations with the simulations’ results obtained for FEBS chains withnd without HI. The BDS results were obtained using step changes in the extension ratesnd integrating until a steady value is obtained for �p at each extension rate. The mostmportant characteristic of the curves for the steady-state extensional viscosity is theccurrence of a sudden rapid increase in its value within a narrow range of strain ratesround the critical elongation rate �c

* for the coil-to-stretch transition. For FEBS chainsith or without HI, �p

* remains bounded by the limit for fully stretched free-draininghains completely aligned in the direction of extension �Hassager �1974��,

�p* = NK,SNS�NS + 1��NS + 2� . �53�

Predictions with the FD-P approximation in the supercritical regime ��*��c*� are in

emarkable agreement with the BDS results, whereas the FD-PG approximation signifi-

FIG. 11. Stress-conformation hysteresis for FEBS chains with and without HI.

antly underpredicts the exact results. The accuracy of the FD-P approximation in its


2.235.136.167 On: Fri, 09 May 2014 13:47:33

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rediction of the steady-state extensional viscosity is well known �van der Brule �1993��.his behavior however is in contrast with that observed in steady shear flows �Fig. 5�,here the FD-PG approximation’s prediction are closer to the simulations’ results even atigh shear rates.

It is worthwhile to examine why the FENE-PG approximation does better at hightrain rates in shear flows than in extensional flows. As mentioned earlier, Lielens et al.1998� and Wiest and Tanner �1989� have argued that the accuracy of the FENE-Ppproximation at high extension rates stems from the fact that the exact distribution of

pring lengths is very narrow and close to being a function centered at some value Qesser than Q0. Under such conditions, the FENE nonlinearity � is well approximated by

he FENE-P function �. When Q is not very close to Q0, it is clear that the additional termn the FENE-PG function L in Eq. �33� leads to inaccuracies. We recall that higher ordererms in the series expansion in Eq. �27� have been dropped in obtaining L in Eq. �33�.ince every term �Eq. �29�� in the series in Eq. �27� is positive, it is evident that afterpplying the Peterlin approximation to every term in the series, the full summation �ifonvergent� will lead to an even greater overestimation of the influence of spring forceuctuations when the true spring length distribution is narrow. Since the series expansion

n Eq. �27� itself has been derived under the assumption of a Gaussian distribution, itppears that the extension of the Gaussian approximation to the FENE springs may not be

ppropriate in situations where the spring length distribution is narrow, but Q is not closeo Q0. The better agreement of predictions with FENE-PG springs with BDS data ob-erved earlier in shear flows not only indicates that spring length fluctuations are larger inhear flows, but also shows that the Gaussian approximation captures the influence of

hese fluctuations more accurately. As the strain rate increases and Q approaches Q0, it

IG. 12. Variation of the steady-state polymer extensional viscosity with extension rate, for FEBS chains withnd without HI. The circle and square symbols represent results obtained with BDS of FEBS chains with andithout HI, respectively. The horizontal dashed gray line is the dimensionless extensional viscosity of fully

tretched free-draining chains calculated using Eq. �53�, and marks the upper limit of the extensional viscosityor FEBS chains with parameter values shown.

as shown in Sec III A that the additional contribution in the FENE-PG spring force


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ecomes negligible in comparison with �, and the differences between the predictions ofhe two approximations become smaller.

For chains with HI, the superiority of the Gaussian approximation is clearly evident inig. 12 in the subcritical regime. At strain rates greater than �c

*, it is the GA-P model thaterforms best �also see Fig. 13�. This is somewhat surprising, since one might expectrom the better performance of the FD-P model for free-draining chains observed earlierhat fluctuations are small when chains are stretched out in the supercritical regime.herefore, if the influence of both spring force and HI fluctuations are unimportant, theredictions of the CA-P model should be close to the BDS data. Instead, however, theA-P model does much better than the CA-P model in Fig. 12. One plausible explanation

or this observation could be that, although fluctuations in spring lengths may be smallnd the FENE-P approximation for the spring force is accurate, the distribution of overallhain configurations may not be as sharply peaked at strain rates above �c

*. The resultantuctuations in HI may therefore still play an important role. The GA-PG model under-redicts the BDS results for the extensional viscosity when �*��c

*, and this is seen to behe result of the inaccuracy of the FENE-PG spring force at high extensions.

It is observed that the approximations for FEBS chains with HI can predict multipleteady-state values for the extensional viscosity for a range of values of the extension rate

˙* �Fig. 13�. The variation of the steady-state extensional viscosity with extension rateredicted by the approximations in Fig. 13 was obtained as follows. Starting with thequilibrium second moments ��ij,eq

* =ij�� as the initial guess, the set of coupled, non-inear equations for the steady-state second moments was first iteratively solved at amall value ��c

*� of the extension rate. After calculating the extensional viscosity withhe converged solution for the equilibrium second moments, this solution is used as theuess for obtaining the solution numerically at a slightly larger value of �*. This processs continued until �* is well beyond the value of �c

* for the coil-to-stretch transition. The

p*-versus-�* curve obtained thus is indicated as having been obtained with a “coiled

IG. 13. Prediction of coil-stretch hysteresis in the steady-state extensional viscosity by closure approximationsor FEBS chains with HI. The upright and inverted triangle symbols represent BDS data obtained by succes-ively stepping up, and stepping down, the extension rate, respectively.

nitial condition” in Fig. 13. The process is then reversed by successively lowering the


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xtension rate. In Fig. 13, it is seen that with the notable exception of the EA-P model,he predicted �p

*-versus-�* curve follows a different route as �* is decreased for all thether approximations for chains with HI. Since this steady-state solution branch is ob-ained starting from a highly “stretched initial condition,” it is indicated as such in Fig.3. A similar procedure described earlier by Schroeder et al. �2004� was used to obtainhe BDS data.

The existence of such “coil-stretch hysteresis” was first proposed in a landmark studyy De Gennes �1974�, who used a variable friction coefficient in a simple dumbbellodel to model the effect of the deformation of the polymer coil on its hydrodynamics.further closure approximation equivalent to the consistent-averaging treatment was

sed to obtain a solvable dumbbell model that predicted coil-stretch hysteresis in aniaxial extensional flow. Later, Magda et al. �1988� demonstrated that closure approxi-ations for bead-spring chain models with HI can predict coil-stretch hysteresis.It is interesting to note that the size of the window of strain rates where multiple

teady states are obtained depends strongly on the approximation used. The window sizeppears to depend on the difference between the extensional viscosities in the coiled statethe lower branch of the hysteresis curve� and the stretched state �the upper branch of theysteresis curve�. Fluctuations in HI lead to a widening of the window because theseuctuations decrease �p

* in the coiled state, but increase �p* when the chains are highly

tretched. As before, the BDS data are quite close to the predictions of the GA-P model.ignificantly, multiple steady states are never obtained with the EA-P model or in theree-draining models. As pointed out by De Gennes coil-stretch hysteresis occurs princi-ally because of the modification of the mobility matrix due to the deformation of theolymer coil. With equilibrium-averaged HI �and in all free-draining models�, the mobil-ty matrix is deformation independent, and hence no coil-stretch hysteresis is observed.

As mentioned above, De Gennes first obtained such multiple steady states using aean-field closure approximation in a simple dumbbell model with a variable

onfiguration-dependent drag coefficient. Later, Fan et al. �1985� obtained the solution tohe Fokker-Planck equation for the dumbbell model numerically without using any clo-ure approximations, and found that the resulting steady-state expectations are single-alued functions of the extension rate. They argued that De Gennes’ prediction of aoil-stretch hysteresis must therefore be an artifice of the mean-field approximations usedn his model. More recently, Schroeder et al. �2003� showed that the exact steady-stateonfigurational distribution in the variable drag dumbbell model is bimodal, and that aarge �effective� free-energy barrier between the coiled and stretched states leads to er-odicity breaking. In contrast, however, the use of closure approximations leads to twoistinct unimodal Gaussian solutions to the modified Fokker-Planck equations.

The BDS results obtained by Schroeder et al., and those in Fig. 13 suggest that thexact distribution of the steady-state extensional stress in FEBS chains with HI is alsoimodal. The linearity of the Fokker-Planck equation �Eq. �1�� in the configurationalrobability distribution function further guarantees that its exact solution is unique. Thentroduction of any mean-field approximation however leads to a modified Fokker-Planckquation that is nonlinear in the probability distribution function, since the coefficients inhe modified Fokker-Planck equation are functionals of the distribution function. Whilehe solution of this modified equation is a unimodal Gaussian distribution, the nonlinear-ty in the distribution function means that the solution may no longer be unique. Thus, thexchange of the nonlinearities due to FENE and HI for a nonlinearity in the distributionunction destroys the multimodal nature of the original solution, but allows the modifiedokker-Planck equation to have multiple solutions. In spite of this fundamental difference
etween the exact and approximate probability distributions, the results presented in Fig.

2.235.136.167 On: Fri, 09 May 2014 13:47:33

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3 above indicate that the multiple steady states obtained with the approximations canlosely follow the long-lived kinetically trapped states in the original model. Therefore,losure approximations for FEBS chains with HI can prove to be useful in a detailedxploration of hysteretic phenomena caused by ergodicity breaking in dilute polymerolutions.

I. SUMMARY AND CONCLUSIONS

The Gaussian approximation was originally suggested for Hookean dumbbells andouse chains with HI by Öttinger �1989� and Wedgewood �1989�, has been shown to

mprove predictions by approximately accounting for the influence of fluctuations in HI.n this work, we use the same idea in the context of a free-draining bead-FENE springodel to first obtain a new approximation for tackling the closure problem posed by theENE nonlinearity. We show that the resulting equations in the approximate model cane interpreted in terms of a mean-field spring potential, and that this “FENE-PG” springotential improves upon the well known FENE-P approximation in situations wherepring force fluctuations are important.

The direct application of the Gaussian approximation for bead-spring models simul-aneously incorporating the effects of both FE and HI leads to an undesirable dependencef the predictions of equilibrium static properties on the hydrodynamic interaction pa-ameter, h*. Instead, we use a two-step procedure to marry the FENE-PG mean-fieldpring potential with the Gaussian approximation for HI to arrive at a closed set ofquations for the second moments, which is consistent with the equilibrium Boltzmannrobability distribution. This approximate model accounts for the influence of fluctua-ions in both HI and the spring forces.

The two-step procedure for combining the Gaussian approximation for HI with theENE-PG approximation for HI suggested in this study is found to lead to predictions

hat compare well with the results of exact BDS for properties in steady and unsteadyhear flows. The better agreement of the predictions of this approximate model—dubbedere as the “GA-PG” model—than those obtained with the FENE-P and/or consistent-veraging approximations emphasizes the importance of fluctuations in HI, and alsoighlights the role spring force fluctuations play at moderate and high shear rates. Inxtensional flows, the GA-P model combining the Gaussian approximation for HI withENE-P springs is found to perform better when chains are highly stretched, perhaps

ndicating that under such conditions fluctuations in HI may still be important, evenhough local fluctuations in spring lengths are small. Although the improvementschieved with the FENE-PG approximation have not been dramatic, the results obtainedith this approximation have been useful in identifying situations where spring forceuctuations are important. The results in this study further demonstrate that it should beossible to obtain highly efficient closure approximations for FEBS chains with HI andxcluded volume interactions by combining accurate approximations for the FENE andxcluded volume forces with the Gaussian approximation for HI.

The predictions obtained with the Gaussian approximation for HI in steady and un-teady shear and extensional flows confirms previous observations on the role of fluctua-ions in HI on macroscopic behavior. When chains are close to their equilibrium isotropictates, fluctuations in HI enhance the screening of the solvent velocity gradient caused byI. The role of fluctuations in HI reverses in situations where polymer chains experience

ignificant stretching, and aid the penetration of the velocity field into the stretched
olymer coil. Consequently, the solvent is able to engage a firmer “grip” on the chains

2.235.136.167 On: Fri, 09 May 2014 13:47:33

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nd this tends to deform an already anisotropic chain further. Fluctuations in the FENEpring force are found to increase the chains’ resistance to stretching.

When the stress predicted in BDS of FEBS chains during the start-up and followinghe cessation of a strong extensional flow is plotted against predictions of the mean-quared end-to-end distance of the chains, or against the intrinsic birefringence, largeysteresis loops were observed. The hysteresis loops predicted with closure approxima-ions were much smaller than those predicted with the simulations. Surprisingly, HI wasound to have a negligible influence on the phenomenon of stress-conformational hyster-sis observed in extensional flows.

Closure approximations for FEBS chains with HI predict coil-stretch hysteresis. Inarticular, predictions of hysteresis in the extensional viscosity by the GA-P model are inxcellent agreement with the exact results obtained with BDS. It is observed that fluc-uations in HI enhance the hysteretic effect, increasing the width of the coil-stretchysteresis window on a plot of the steady-state extensional viscosity versus the strainate.

CKNOWLEDGMENTS

This work has been supported by a grant from the Australian Research Council, underhe Discovery-Projects program. We also thank the Australian Partnership for Advancedomputing for a CPU-time grant.

PPENDIX: TENSORS ASSOCIATED WITH HYDRODYNAMICNTERACTIONS

The tensor Aij in Eq. �47�, which describes contributions arising from averaging theseen-Burgers tensor with the nonequilibrium distribution function, is given by the ex-ression,

Aij � �Aij� = Aij� + �2h*�Hij + Hi+1,j+1 − Hi+1,j − Hi,j+1� , �A1�

hile the tensors �ij and �ijT, which represent contributions due to fluctuations in HI, are

iven by

�ij � �r,s=1

NS

�rjis:��sr

* · Lr�; �ijT = �

r,s=1

NS

�Lr · �rs* �:� jr

si , �A2�

ith

�rqps �

3�2h*

4��rq

psKrq + ��r+1,q+1ps Kr+1,q+1 − ��r+1,q

ps Kr+1,q − ��r,q+1ps Kr,q+1� . �A3�

he second-rank tensorial function H��=H�S��* � is related to the average of the HI tensor

hrough

�� = �2h*H�S��* � , �A4�

hereas the fourth-rank tensorial function K��=K�S��* � is related to the HI tensors

hrough


2.235.136.167 On: Fri, 09 May 2014 13:47:33

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F


Redistribution

� ��r��

r�� =3�2h*

4K�S��

* � · S��* . �A5�

hen the Oseen-Burgers form of the HI tensor is used, it can be shown that �Öttinger1987�; Zylka and Öttinger �1991��

H�� =3

2�2��3/2 � 1

k2�� −kk

k2 �exp�−1

2kk:S��

* �dk , �A6�

nd

K�� =− 2

�2��3/2 � 1

k2k�� −kk

k2 �k exp�−1

2kk:S��

* �dk . �A7�

n the equations above, S��* = �r��

* r��* �=�i,j=1

NS ��ij �ij

* , where ��ij =��

i ��j is the

wo-dimensional “box” function, ��i =�� , i�−�� , i�=−��

i is the one-dimensionalbox” function, with ��m ,n� �=1, if n�m, and =0 otherwise� denoting the Heavisidetep function. The dyad, r��r��=�i,j=1

NS ��ij QiQ j. The function ��

ij is symmetric withespect to an exchange of the indices � and �, or i and j. Thus, the functions H�� and

��, and therefore Aij and �ij, are completely expressible in terms of the second mo-ents of the Gaussian distribution.The inner product of K with a second-rank tensor a is defined in component form as

�K:a�� = ��,=1

3

K��a�, and �a:K�� = ��,=1

3

a�K��, for �,� = 1,2,3. �A8�

urther, since the inner product

1

k2�� −kk

k2 �k:� = 0 ,

or all k, we have K :�=0. A direct consequence of this result is that the � tensorsefined through Eqs. �A2� and �A3� vanish under isotropic equilibrium conditions.

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