universal consequences of the presence of excluded volume interactions in dilute polymer solutions...

Universal consequences of the presence of excluded volume interactionsin dilute polymer solutions undergoing shear flowK. Satheesh Kumar and J. Ravi Prakash Citation: J. Chem. Phys. 121, 3886 (2004); doi: 10.1063/1.1775185 View online: http://dx.doi.org/10.1063/1.1775185 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v121/i8 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Universal consequences of the presence of excluded volume interactionsin dilute polymer solutions undergoing shear flow

K. Satheesh Kumara) and J. Ravi Prakashb)

Department of Chemical Engineering, Monash University, Clayton, Victoria-3168, Australia

~Received 28 April 2004; accepted 28 May 2004!

The role of solvent quality in determining the universal material properties of dilute polymersolutions undergoing steady simple shear flow is examined. A bead-spring chain representation ofthe polymer molecule is used, and the influence of solvent molecules on polymer conformations ismodelled by a narrow Gaussian excluded volume potential that acts pairwise between the beads ofthe chain. Brownian dynamics simulations data, acquired for chains of finite length, andextrapolated to the limit of infinite chain length, are shown to be model independent. This featureof the narrow Gaussian potential, which leads to results identical to ad-function repulsive potential,enables the prediction of both universal crossover scaling functions and asymptotic behavior in theexcluded volume limit. Universal viscometric functions, obtained by this procedure, are found toexhibit increased shear thinning with increasing solvent quality. In the excluded volume limit, theyare found to obey power law scaling with the characteristic shear rateb, in close agreement withpreviously obtained renormalization group results. The presence of excluded volume interactions isalso shown to lead to a weakening of the alignment of the polymer chain with the flowdirection. © 2004 American Institute of Physics.@DOI: 10.1063/1.1775185#

I. INTRODUCTION

The nature of the thermodynamic interactions betweenthe monomers in a polymer chain and the solvent moleculesin their neighborhood, determines the ensemble of spatialconfigurations adopted by the chain, and as a result, has asignificant influence on all conformation dependent proper-ties of polymer solutions. Extensive investigations of staticpolymer solutions has established that both the temperatureof the solution and the molecular weight of the dissolvedpolymer, control the strength of these interactions. Indeedtheir influence can be combined together in terms of a singlevariable, the so-called solvent quality parameter,z5v0(12Tu /T)AM , where,v0 is a polymer-solvent system depen-dent constant,T is the temperature of the solution,Tu is thetheta temperature, andM is the molecular weight. At largevalues ofz, usually termed the excluded volume limit, vari-ous properties of polymer solutions have been shown to obeypower law scalings with molecular weight. On the otherhand, at intermediate values ofz ~the crossover region!, thebehavior of a host of different polymer–solvent systems hasbeen found to be collapsible onto universal master plots witha suitable choice of the constantv0 .1–4

The entire range of observed behavior has been success-fully modelled by static theories of polymer solutions bymimicking the influence of the solvent molecules on the con-formations of a polymer coil, with the help of a potential thatacts pairwise between the various parts of a polymer chain.Typically, by representing the linear polymer molecule by abead-spring chain consisting ofN beads connected together

by (N21) Hookean springs, more or less repulsion by thesolvent molecules is accommodated by varying the magni-tude of the pairwise excluded volume interaction parameter,denoted here byz* , between the beads.5–8 In particular, bynoting thatz* ;(12Tu /T), andN;M , a quantitative pre-diction of both the excluded volume limit and the crossoverregime behavior of a number of equilibrium properties hasbeen achieved by analytical theories, with the help of themappingz5z* AN.

The growing recognition that polymer–solvent interac-tions also play a key role in determining the behavior ofpolymer solutions far from equilibrium is responsible for therecent appearance of a number of studies aimed at obtaininga better understanding of this phenomenon.9–22 Due to theanalytically intractable nature of the nonequilibrium theory, amajority of these investigations have been largely numericalin nature. As in the case of static theories, polymer–solventinteractions have been modelled with the help of an excludedvolume potential, such as the Lennard-Jones potential, actingpairwise between the different parts of the polymer chain.While a number of qualitative conclusions on the influenceof excluded volume interactions have been drawn by thesestudies, a methodical program aimed at cataloging the uni-versal behavior of polymer solutions in a variety of flowfields in terms of the solvent quality parameterz—as hasbeen done at equilibrium—has not yet been attempted. Nev-ertheless, some progress in this direction has been made inthe framework of approximate analytical theories. In the ex-cluded volume limit, O¨ ttinger and co-workers9,10,23have ex-amined, with the help of approximate renormalization groupmethods, the universal behavior of polymer solutions under-going shear flow. The solvent quality crossover scaling be-havior of viscometric functions was, however, not reported

a!Present address: Department of Physics, POSTECH San 31, Hyoja-dong,Nam-gu, Pohang, Kyungbuk, 790-784 Republic of Korea.

b!Electronic mail: [email protected]

JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 8 22 AUGUST 2004

38860021-9606/2004/121(8)/3886/12/$22.00 © 2004 American Institute of Physics


http://dx.doi.org/10.1063/1.1775185

by these authors. Recently, Prakash17,19 has obtained, withthe help of a Gaussian approximation, universal materialfunctions in shear flow, both in the crossover regime and inthe excluded volume limit.

A systematic methodology by which it is possible to gobeyond approximate results, and obtain exact results~albeitwithin numerical error bars! of the influence of solvent qual-ity, in terms of the parameterz, on both equilibrium andnonequilibrium properties, has been newly introduced byKumar and Prakash.24 Basically, they showed that model in-dependent predictions can be obtained by extrapolatingexactBrownian dynamics simulations data, acquired for chains offinite length, to the limit of infinite chain length.~Italics areused here throughout to qualify the term ‘‘exact,’’ with aview to indicating the fact that results are exact only towithin the numerical error bars of the simulations. Theseerrors can, of course, be made arbitrarily small by suitablyincreasing the number of simulation trajectories.! While theirmethodology is applicable to both regimes of behavior, theyrestricted their attention to the prediction of equilibrium andlinear viscoelastic properties. In this paper, we apply theirprocedure and obtain, for the first time,exactpredictions ofthe universal crossover dependence of viscometric functionson z, at various values of the characteristic nondimensionalshear rate,b5lpg, wherelp is a characteristic relaxationtime defined by

lp5@h#0Mhs

NAkBT. ~1!

HerekB is Boltzmann’s constant,hs is the solvent viscosity,@h#0 is the zero shear rate intrinsic viscosity, andNA isAvogadro’s number. The same methodology is also used hereto calculate the dependence of several properties onb, in theexcluded volume limit. This includes, for instance, thepower-law scaling of viscometric functions withb for largevalues ofb, and the dependence of the orientation of thepolymer coil with respect to the flow direction, on the mag-nitude ofb. The important question of whether the value ofthe critical exponentn ~which governs the scaling of themean size of a polymer chain with molecular weight in thelong chain limit!, is unaltered in shear flow, is also addressedhere within this framework.

In addition to excluded volume interactions, it is nowwell established that it is also essential to include solventmediated hydrodynamic interactions between different seg-ments of a polymer chain, in order to obtain an accuratedescription of the universal dynamics of polymer solutions.While several studies~based on bead–spring chain models!of the combined influence of excluded volume, hydrody-namic interaction, and even finite extensibility effects, onrheological properties have been reported so far, they havegenerally been restricted to finite chains.16,21,22,25,26However,Zylka and Ottinger10 have examined the universal conse-quences of the presence of both excluded volume and hydro-dynamic interaction effects on properties far from equilib-rium, with the help of approximate renormalization groupmethods. Several important advances in recent years, withinthe framework of Brownian dynamics simulations, have nowmade it feasible to treat hydrodynamic interaction effects,

free of any simplifying approximations, in bead-spring chainmodels with a large number of beads.27,28 In this work, how-ever, we confine attention to excluded volume interactionsalone, for two reasons. First, we wish to examine, in isola-tion, the influence of excluded volume interactions on longchain properties, before considering the more complex pos-sibilities that arise in the presence of other nonlinearities.The preliminary results of Prabhakar and Prakash,21 whoshow that in fact the various nonlinearities appear to be de-coupled from each other, suggests that there is some value inlooking at these effects in isolation from each other. Second,and more practically, the procedure adopted here of extrapo-lating finite chain simulation data to the long chain limitbecomes significantly more computationally intensive whenboth excluded volume and hydrodynamic interactions are in-cluded. The results of the present paper are therefore not yetdirectly comparable with experiments. Nevertheless, whilethe more ambitious task of a complete description of poly-mer solution dynamics is being simultaneously pursued, it isfelt that several important conclusions with regard to the roleof solvent quality, can still be made within the present ap-proach.

The plan of the paper is as follows. In Sec. II, we statethe governing equations, define the various properties calcu-lated in this work, and summarize the previously introducedprocedure for obtainingexactpredictions of properties in thecrossover regime and in the excluded volume limit. In Sec.III A, asymptotic results describing the dependence of thenormalized viscosity and the normalized first normal stressdifference onb andz, are presented. The excluded volumelimit dependence onb, of various properties, is discussed inSec. III B, along with a comparison with results obtainedpreviously with the Gaussian approximation. Finally, theprincipal conclusions of this work are summarized inSec. IV.

II. BEAD–SPRING CHAINS WITH EXCLUDEDVOLUME INTERACTIONS

A. Governing equations

Within the framework of a numerical solution strategy,the most appropriate model for the task of describing therheological behavior of dilute polymer solutions, in the limitof long chains, is a bead–spring chain model. The internalconfiguration of a bead–spring chain, suspended in a New-tonian solvent undergoing homogeneous flow, with the beadslocated at positionsr n ,n51,...,N with respect to an arbi-trarily chosen origin, can be specified by theN21 bead con-nector vectors, Qk5r k112r k ,k51,...,N21, connectingbeadsk and k11. For Hookean springs, and a pairwise in-tramolecular excluded volume force, the potential energy ofa bead-spring chain is given by the expression,

f51

2H (

i 51

N21

Qi•Qi11

2 (n,m51nÞm

N

E~r n2rm!, ~2!

where H is the spring constant, andE(r n2rm) is the ex-cluded volume potential between the beadsn and m of thechain. The key ingredient, in the procedure for obtaining

3887J. Chem. Phys., Vol. 121, No. 8, 22 August 2004 Universal consequences of excluded volume interactions


universal predictions in terms of the parameterz, is the useof the narrow Gaussian potential to represent excluded vol-ume interactions,

E~r n2rm!5S z*

d* 3D kBT expH 2H

2kBT

r nm2

d* 2J , ~3!

wherer nm5r n2rm , is the vector between beadsm andn, thenondimensional parameterz* , as mentioned previously,measures the strength of the excluded volume interaction,andd* is a measure of the range of excluded volume inter-action. Prakash and co-workers have shown, in their exami-nation of universal linear viscoelastic and steady shear flowproperties in the context of the Gaussian approximation,17,19

and in their calculation of universal equilibrium and linearviscoelastic properties with Brownian dynamicssimulations,24 that the usefulness of the narrow Gaussian po-tential stems from the fact that, asN→`, ~i! the parametersz* and N combine together to form the single variable,z5z* AN, and~ii ! the choice ofd* is inconsequential becausethe parameterd* always appears in the theory as the re-scaled variabled* /AN. These results form the basis of ourexploration, in this work, of the universal behavior of bead–spring chains subject to shear flow.

As will be discussed in greater detail shortly in Sec. II Bbelow, all properties of interest in the present work can becalculated from appropriately defined configurational aver-ages. Unfortunately, it is currently infeasible to solve thenonlinear diffusion equation that governs the time evolutionof the configurational distribution function in the presence ofexcluded volume interactions. However, configurational av-erages can be numerically evaluated directly from ensemblesof polymer configurations. Basically, stochastic trajectoriesthat describe the temporal evolution of an ensemble of chainconfigurations can be generated by solving the stochastic dif-ferential equation that governs the bead connector vectors,with the help of Brownian dynamics simulations.29 In termsof the nondimensional variables,

Qk* 51

,Qk ; t* 5

t

lH; f* 5

f

kBT; k* 5lHk, ~4!

where,5AkBT/H is a length scale that is proportional tothe root mean square extension of a single spring in theRouse model,lH5z/4H is a time constant~with z represent-ing the bead friction coefficient!, and k(t) is the tracelesstranspose of the velocity-gradient tensor for the homoge-neous flow field~which can be a function of time but isindependent of position!, the stochastic differential equationthat governs the trajectories of the bead connector vectors inthe presence of excluded volume interactions, is given by theexpression,19

dQj* 5Fk* •Qj* 21

4 (k51

N21

Ajk

]f*

]Qk*Gdt*

1A1

2 (n21

N

Bj n dWn* . ~5!

HereWn* is a Wiener process, whose 3N-dimensional com-ponents satisfy

^Wn, j* ~ t* !&50,~6!

^Wn, j* ~ t* !Wm,k~ t* 8!&5min~ t* ,t* 8!d jkdnm ,

for j ,k51,2,3 andn,m51,2,3,...,N. The quantityBkn is an(N21)3N matrix defined by,Bkn5dk11,n2dkn , with dkn

denoting the Kronecker delta, andAjk is the Rouse matrix,

Ajk5 (n51

N

Bj nBkn5H 2 for u j 2ku50,

21 for u j 2ku51,

0 otherwise.

~7!

Before discussing the two different simulation schemesused here for solving Eq.~5! to obtain predictions of theuniversal crossover and excluded volume limit behaviors, itis appropriate to first introduce all the properties, and theirdefining equations, that are calculated in the present work.

B. Material functions

We confine our attention in this work to steady simpleshear flows, which are specified by the following matrix rep-resentation for the tensork in a laboratory-fixed coordinatesystem,

k5gS 0 1 0

0 0 0

0 0 0D , ~8!

whereg is the constant shear rate.The principle material functions of interest here are the

polymer contribution to the viscosity,hp , and the first nor-mal stress difference,C1 . Since hydrodynamic interactionshave not been included in the present model, the secondnormal stress difference is identically zero.30 In terms of thecomponents of the polymer contribution to the stress tensor,txy

p ,txxp , etc.,hp , andC1 , can be obtained from the follow-

ing relations:31

txyp 52ghp , txx

p 2tyyp 52g2C1 . ~9!

Within the framework of polymer kinetic theory, the polymercontribution to the stress tensor,t p, can be calculated byeither the Kramers or the Giesekus expressions. While thelatter expression becomes invalid in the presence of hydro-dynamic interactions, it has been shown to remain valid inthe presence of excluded volume interactions.18 The relativemagnitude of the variance associated with the use of either ofthese expressions in Brownian dynamics simulations hasbeen examined previously by Prakash19 and it has beenshown that the Giesekus expression always leads, particu-larly at low shear rates, to a smaller value of variance. Forthis reason, we have used the Giesekus expression in ourwork, which at steady state has the form

tp52npz

2 (m,n51

N21

Cmn$k•^QmQn&1^QmQn&•kT%, ~10!

whereCmn is the Kramers matrix. The Kramers matrix is theinverse of the Rouse matrix, and is defined by

3888 J. Chem. Phys., Vol. 121, No. 8, 22 August 2004 K. S. Kumar and J. R. Prakash


Cmn5 (n51

N

BnmBnn5min~m,n!2mn

N. ~11!

Here theN3(N21) matrix Bnk is defined byBnk5k/N2Q(k2n), with Q(k2n) denoting a Heaviside step func-tion. As will be discussed in greater detail in Sec. II C below,we further reduce the variance in our simulations by imple-menting a variance reduction procedure based on the methodof control variates.29

In the limit of zero shear rate, the ratioUCh , which isdefined by the expression

UCh5npkBTC1

hp2 , ~12!

wherenp is the number density of polymers, has been shownto have a universal value, both in theta solutions,29 and ingood solvents.24 In the context of a dumbbell model, Prakashand Ottinger13 have evaluated its dependence on the nondi-mensional shear ratelHg. In this work, we seek to examinethe universal dependence ofUCh on the characteristic shearrateb, in the excluded volume limit.

In addition to rheological properties, it is also of interestto examine the anisotropy induced in any arbitrary tensorialproperty of the solution, due to the orientation and deforma-tion of polymer coils caused by the flow field. The precisedefinition of anisotropy is taken up shortly below. Of particu-lar relevance in this work, is the change in the degree ofanisotropy brought about by the presence of excluded vol-ume interactions.

In order to obtain an estimate of the anisotropy of anytensorial quantitys, we begin by noting that in simple shearflows s must reflect the symmetry of the flow field. In par-ticular, since shear flows are invariant when the direction ofthe z axis is reversed,31 the xz, yz, zx, andzy componentsof s must be zero. As a result,s must have the followingmatrix representation in the laboratory-fixed coordinate sys-tem:

s5S sxx sxy 0

syx syy 0

0 0 szz

D . ~13!

It is clear from Eq.~13! that thez axis is a principal direc-tion. It follows that the other two principal directions mustlie in thexy plane. We denote the principal axes in this planeby x and y, with the x axis making an anglexs with the xaxis. Thus by rotating the coordinate system byxs , the ma-trix that representss can be diagonalized. By exploiting therules for change of tensor components, the orientation of the(x,y,z) coordinate system relative to the (x,y,z) coordinatesystem can be calculated to be

tan 2xs52sxy

sxx2syy. ~14!

With the help of symmetry arguments, one can show that theorientation anglexs5p/4 close to equilibrium. As the shearrate increases,xs is expected to decrease to zero, reflecting

the increasing alignment of the polymer coils with the flowdirection. The orientation anglexs , therefore, serves as ameasure of the anisotropy ofs.15

It is straightforward to show that the orientation angleassociated with the polymer contribution to the stress tensor,xt , is given by15,32

tan 2xt52txy

p

txxp 2tyy

p 5mt

b, ~15!

where the quantitymt , defined by the above expression, isfrequently referred to as the orientation resistance.32 In theRouse model,mt is a constant, independent of the shear rate,with a value 2.5. In a similar manner, the orientation angleassociated with the radius of gyration tensor,G, is given bythe expression15,32

tan 2xG52Gxy

Gxx2Gyy5

mG

b, ~16!

where mG is the corresponding orientation resistance, andGxy , etc., represent components of the tensorG, defined by

G51

N (n51

N

^~r n2r c!~r n2r c!&. ~17!

Here r c denotes the position of the center of mass. In theRouse model,mG51.75. It is worthwhile noting that whilethe orientation of the macromolecule measured by flow bire-fringence coincides withxt ~according to the ‘‘stress opti-cal’’ law!, xG is the orientation angle measured by static lightscattering.32 In this work, we show that in the presence ofexcluded volume interactions, bothmt andmG are not con-stant, but rather are functions of the characteristic shear rateb, and we examine this dependence in the excluded volumelimit.

C. Simulation schemes

Two different simulation schemes have been introducedand discussed in detail by Prakash and co-workers in order tofind the universal properties of polymer solutions in thecrossover region, and in the excluded volume limit,respectively.19,24 Here, we summarize them briefly.

The universal dependence of various properties on theparameterz in the long chain limit is found by a two stepprocedure:~i! Simulations are carried out for increasing val-ues ofN, keeping the value ofz (5z* AN), constant, and~ii ! The accumulated finite chain data for any property is thenextrapolated to the limitN→`. The crossover behavior ofthe property is obtained by repeating steps~i! and ~ii ! for anumber of values ofz. In the Gaussian approximation, thevalue of the parameterd* was held constant during the pro-cess of accumulating finite chain data,17,19 and it was shownthat the particular choice of value ford* was inconsequentialin the limit N→`. On the other hand, while obtaining thecrossover behavior of equilibrium and linear viscoelasticproperties with Brownian dynamics simulations,24 the pa-rameterd* was chosen such that

d* 5k~z* !1/5, ~18!



wherek is an arbitrary constant. Note that, for a fixed valueof z, sincez* →0 ~or equivalently,T→Tu), asN→`, thischoice of d* implies that the asymptotic limit is reachedalong trajectories in the (z* ,d* ) parameter space that con-verge to the origin. It was shown by Kumar and Prakash24

that choosingd* values according to Eq.~18!, permits theuse of larger step sizes in the numerical integration scheme.The independence of the asymptotic results from the particu-lar choice of k @and consequently the trajectory in the(z* ,d* ) parameter space#, was also established in Ref. 24.Since Brownian dynamics simulations are used in the presentwork to obtain the crossover behavior,d* values will bechosen according to Eq.~18!.

Universal properties in the excluded volume limit (z→`) are obtained by accumulating finite chain data for con-stant values ofz* ~which is equivalent to the temperatureTbeing constant!, followed by extrapolation to the infinitechain length limit. Since for any constant value ofz* , z→`, as N→`, we expect that properties obtained in theexcluded volume limit will be independent of the particularchoice ofz* . Prakash and co-workers19,24 have shown thatthis is indeed the case, for the situations examined previouslyby them. As will be discussed in greater detail subsequently,this independence will be used in the present work to indi-cate the adequacy or otherwise of the accumulated finitechain data.

The various configurational averages required to calcu-late the properties listed in Sec. II B above were obtainedwith a second order predictor–corrector Brownian dynamicssimulation algorithm originally proposed by Iniesta and Gar-cia de la Torre.33 A detailed discussion of the algorithm,adapted for the problem at hand, can be found in Ref. 19. Ina typical simulation, roughly 80 000 trajectories, with Gauss-ian initial distributions, were generated, and the integrationwas carried out until a stationary state was attained. Messagepassing interface~MPI! clusters were used to distribute thetrajectories across several processors. Since the present algo-rithm belongs to the class of embarrassingly parallel algo-rithms, the wall time scales with the number of processorsused. Typically, three different time stepsDt* 51.0, 0.6, and0.5 were used, and the results were then extrapolated to zerotime step using the subroutineTEXTRA suggested byOttinger.29 This extrapolation procedure enabled the simula-tions to be carried out at relatively large time steps.

A key ingredient in the simulation strategy adopted here,as mentioned perviously, is the use of a variance reductionprocedure based on the method of control variates, in orderto reduce the statistical error in the simulations. This is cru-cial because the magnitude of the error in the finite chaindata significantly affects the accuracy of the extrapolation tothe infinite chain length limit. Since a fairly involved discus-sion of the procedure has been given previously in Ref. 24, itis not repeated here. However, a few salient features arehighlighted below.

The essence of the scheme~and the vital factor that de-cides its success or otherwise! is to find a control variablewhose mean value is known exactly, and whose fluctuationsare correlated with the variable whose variance we are inter-ested in reducing. It was found by Kumar and Prakash,24 that

the Rouse model serves as an excellent source of controlvariables for all the equilibrium and linear viscoelastic prop-erties that were evaluated by them. In this work, we haveagain used the Rouse model as the source of control vari-ables, and found that as before, this leads to a significantreduction in the variance of all properties. At first sight, thisresult is somewhat surprising. In the case of equilibrium andlinear viscoelastic properties, since only equilibrium aver-ages were evaluated, both the stochastic differential equa-tion, Eq. ~5!, and the stochastic differential equation in theRouse model, were solved with the term involvingk setequal to zero. In the present instance, since we are interestedin properties at finite shear rates, both equations were inte-grated with thek term in place. In spite of including thisterm in the Rouse model, as is well known, viscometric func-tions predicted by it are constant and independent of shearrate. This is not the case, however, in the presence of ex-cluded volume interactions, where, as we shall see shortly,viscometric functions decrease with an increase in shear rate.The success in the variance reduction strategy suggests,therefore, that even though the Rouse viscometric functionsare independent of shear rate, the fluctuations in the Rousecontrol variables remain correlated with their correspondingvariables in the presence of excluded volume interactions.The reason for this occurance might lie in the fact that withincreasing shear rate, the beads of the chain are drawn fur-ther apart, and the resultant weakening of the excluded vol-ume interactions actually brings the results of the two modelscloser together.

III. UNIVERSAL PROPERTIES

A. Crossover behavior

In this section, we are concerned with obtaining thecrossover behavior of the universal ratios,hp /hp,0 , andC1 /C1,0 ~where,hp,0 andC1,0 are the zero shear rate poly-mer contributions to the viscosity and first normal stress dif-ference, respectively!, as a function of the characteristicshear rateb, using the simulation scheme described in Sec.II C above. A strict pursuance of this procedure would re-quire the construction of each of these ratios at various val-ues ofN, using equilibrium simulations~as described previ-ously in Ref. 24! to find hp,0 and C1,0, and simulations atfinite shear rates~as described above! to find hp and C1 ,followed by extrapolation to the long chain limit. Unfortu-nately, this procedure suffers from a serious problem. Sinceboth the denominator and the numerator are found byBrownian dynamics simulations, the ratios have relativelylarge statistical error bars, which makes their extrapolation totheN→` limit highly inaccurate. Fortunately, this difficultycan be overcome by adopting the following procedure.

We first note that

limN→`

hp

hp,05

limN→`~hp /hpR!

limN→`~hp,0 /hpR!

, ~19!

limN→`

C1

C1,05

limN→`~C1 /C1R!

limN→`~C1,0/C1R!

, ~20!



wherehpR and C1

R are the viscosity and first normal stressdifference predicted by the Rouse model, respectively,30

hpR5nplHkBTF ~N221!

3 G , ~21!

C1R52nplH

2 kBTF ~N221!~2N217!

45 G . ~22!

Kumar and Prakash have previously evaluated the ratioshp,0 /hp

R andC1,0/C1R in the long chain limit.24 For the pur-

pose of evaluatinghp /hp,0 in the long chain limit, therefore,it suffices to evaluate the ratios,hp /hp

R and C1 /C1R , asN

→`. The simulation scheme described in Sec. II C abovehas, consequently, been used here to evaluate these ratios atvarious values ofz.

Figure 1 clearly indicates the independence of the ex-trapolated values ofhp /hp

R andC1 /C1R , in theN→` limit,

from the trajectories in the (z* ,d* ) parameter space used toobtain them. The parametersz andb have been held constantat a value of unity, while carrying out simulations at increas-ing values ofN, for three different values ofk. In eachsimulation,z* 5z/AN, while d* is found from Eq.~18!. Inorder to keepb constant in each simulation, the nondimen-sional shear ratelHg used in the simulation, at each value ofN, has been calculated from the expression,lH g5b (lHnpkBT/hp,0), which can be derived from Eq.~1! fora dilute polymer solution. As mentioned earlier,(hp,0 /lHnpkBT) can be evaluated by carrying out equilib-rium simulations for the same set of parameter values asthose used in the finite shear rate simulations. Extrapolationof the finite chain data has been carried out by plotting simu-lation results versus 1/AN, since Prakash18 has shown previ-

ously that leading order corrections to the infinite chainlength limit, of various material properties, are of order1/AN. Furthermore, as mentioned earlier, the parameterd*always occurs in the theory as the ratiod* /AN. Theasymptotic values obtained in this manner correspond to theviscosity and first normal stress difference ratios for ad-function excluded volume potential, atz51 andb51. Thesame procedure was then repeated for various values ofzandb.

The universal dependence ofhp /hp,0 , andC1 /C1,0 onb, at various values ofz, obtained by combining the infinitechain length results for the ratioshp /hp

R andC1 /C1R , with

the infinite chain length zero shear rate ratios evaluated pre-viously by Kumar and Prakash,24 are presented in Fig. 2. Ateach value ofz, extrapolated data for two different values ofk are displayed in order to clearly delineate the limit ofb upto which the current asymptotic results are valid. In essence,beyond some threshold value of the characteristic shear rate,say b†, the lack of coincidence of data for two differentvalues ofk, implies that the present finite chain data, accu-mulated for chains withN<36, is no longer sufficient toobtain an accurate extrapolation. Forb.b†, the flow fieldbegins to probe model dependent length scales, and scaleinvariance, which is responsible for the observed model in-dependence, no longer exists. The results in Fig. 2 suggestthatb† for C1 /C1,0 is smaller than that forhp /hp,0 , and inboth cases, it decreases with increasingz.

It is clear from Fig. 2 that the presence of excludedvolume interactions leads to shear thinning, and that the ex-tent of shear thinning increases with increasing solvent qual-ity. However, the incremental increase in shear thinning withincreasing values ofz, for a given value ofb, appears todecrease asz increases. This can be seen more clearly fromFig. 3, where the asymptotic viscometric ratios are plotted

FIG. 1. ~a! The ratio of the viscosity to the Rouse model viscosity and~b!the ratio of the first normal stress difference to the Rouse model first normalstress difference vs 1/AN at z51 andb51, for three different values ofk.The symbols, with error bars, represent the simulation results, while thelines are least-square curve fits to the data.

FIG. 2. ~a! Asymptotic nondimensional viscosity ratio (hp /hp,0) and ~b!asymptotic nondimensional first normal stress difference ratio (C1 /C1,0),vs the characteristic shear rateb, for various values ofz andk.



versus the solvent qualityz, at two representative values ofb. In each case, the ratios decrease, but with decreasing rate,as the solvent quality increases, before appearing to level offat a final solvent quality independent value, in the limit oflargez. In this excluded volume limit, consequently, univer-sal viscometric functions would depend only onb, and on noother parameter.

It is not possible here to explore further the approach ofthe crossover regime to the excluded volume limit, becausethe present finite chain data is not sufficient to obtain accu-rate extrapolations. Obtaining such data is, unfortunately,currently prohibitively expensive computationally. Neverthe-less, behavior in the excluded volume limit itself can be di-rectly obtained, as has been discussed earlier. The horizontallines in Fig. 3 correspond to excluded volume limit values ofthe viscometric ratios obtained by these means, the details ofwhich will be described in greater detail shortly. It is imme-diately apparent that the largez asymptote of the crossoverregime does not coincide with the excluded volume limit.This is in complete contrast to the universal results, in steadyshear flow, obtained earlier by Prakash19 with the Gaussianapproximation. While the general behavior in the crossoverregime predicted by the Gaussian approximation is identicalto that described above, values of viscometric ratios forz@1 were found to smoothly approach those obtained in theexcluded volume limit. This perplexing behavior, where pre-dictions of the Gaussian approximation differ qualitativelyfrom those of exact Brownian dynamics simulations, wasalso observed recently by Kumar and Prakash in their at-tempt to predict the behavior of universal equilibrium andlinear viscoelastic ratios.24 Before discussing the origins ofthis behavior in greater detail, however, it is necessary to

complete the presentation of a few further results obtained inthe excluded volume limit.

B. Excluded volume limit

Predictions in the excluded volume limit are obtained, asdescribed in Sec. II C above, by extrapolating finite chaindata acquired for various constant values ofz* , to the infi-nite chain length limit. Figure 4 is an illustrative example,where simulations have been carried out at several values ofN, for z* 50.24,0.28, and 0.35, respectively, and at a fixedvalue ofb56, in order to acquire data for the ratiohp /hp,0 .The common extrapolated value, in the limitN→`, for allthe three values ofz* , is the universal value, in the excludedvolume limit, ofhp /hp,0 , atb56. As in the case of extrapo-lated values in the crossover regime, insufficiency in the ac-cumulated finite chain data is revealed when curves for dif-ferent values ofz* fail to extrapolate to a common point. Inthe present instance, this typically occurs for values of char-acteristic shear rateb*10. Extrapolated data obtained in thismanner, at various values ofb ~up to b'10), for both theviscometric ratios, are displayed in Fig. 5. The horizontallines, displayed in Fig. 3, have also been obtained similarly.

Both the viscometric functions displayed in Fig. 5 ex-hibit a similar dependence onb—nearly constant values atsmall values ofb, followed by a crossover to a power lawdependence at large values ofb. Results obtained earlierwith the Gaussian approximation are also presented in Fig. 5for comparison. While the extent of shear thinning is under-predicted by the Gaussian approximation for the values ofbdisplayed in the figures, it predicts a larger slope in thepower law regime, which implies that the shear thinning pre-dicted by it will eventually be greater thanexactBrowniandynamics simulations.

The Carreau–Yasuda model, which proposes the follow-ing expressions for describing the variation of the viscomet-ric ratios withb,

hp

hp,05@11~ahb!nh#mh /nh,

~23!C1

C1, 05@11~ac b!nc#mc /nc

FIG. 3. ~a! Asymptotic nondimensional viscosity ratio (hp /hp,0) and ~b!asymptotic nondimensional first normal stress difference ratio (C1 /C1,0),plotted againstz for two different values ofb. Symbols denote values ob-tained in the crossover regime, while the horizontal lines correspond tovalues of the ratios in the excluded volume limit.

FIG. 4. Nondimensional viscosity ratiohp /hp,0 , plotted against 1/AN, forthree different values ofz* , and for b56. Symbols are predictions ofBrownian dynamics simulations, while lines are least-square curve fitsthrough the data.



has been shown previously by Prakash19 to provide a good fitto the results of the Gaussian approximation. As indicated inFig. 5, with a suitable choice of the fitting coefficientsah ,mh , nh , etc., it also leads to a good fit of Brownian dynam-ics simulation data. The values of the fitting coefficients usedin the two cases are displayed in Table I. The constantmrepresents the slope of the power law region at large valuesof b. In line with the expectation by visual inspection, thevalue ofm for the Gaussian approximation is larger than thatfor exactBrownian dynamics simulations. Using renormal-ization group methods, O¨ ttinger9,23 has previously found thatthe values,mh520.25 andmc520.5, describe the powerlaw shear rate dependence ofhp /hp,0 , and C1 /C1,0, re-spectively, in the excluded volume limit. As is evident fromthe values of the power law exponents displayed in Table I,the predictions of Brownian dynamics simulations are in re-markable agreement with renormalization group results.

The expression forUCh , Eq. ~12!, can be rewritten inthe form

UCh5UCh,0

~C1 /C1,0!

~hp /hp,0!2 ~24!

5UCh,0

@11~acb!nc# (mc /nc)

@11~ahb!nh# (2mh /nh) , ~25!

whereUCh,0 is the value ofUCh in the zero shear rate limit,and the Carreau–Yasuda model expression, Eq.~23!, hasbeen used for the viscometric ratios. Both renormalizationgroup results, and the present Brownian dynamics simula-tions suggest that,mc'2mh'20.5 ~see Table I!. For b@1, therefore, Eq.~25! can be simplified to

UCh5UCh,0Aah

ac. ~26!

Kumar and Prakash24 have estimated that in the excludedvolume limit, UCh,050.77160.009. Substituting this result,and the values forah andac from Table I into Eq.~26!, andusing standard methods for the estimation of error propaga-tion, leads toUCh50.56860.038. Thus, the universal ratioUCh decreases from a zero shear rate value of 0.77160.009, to a shear rate independent value of 0.56860.038in the limit of large shear rates.

Figure 6 displays the dependence ofUCh on b, obtainedby carrying out Brownian dynamics simulations~opencircles!, and with the Gaussian approximation~squares!. Ineach case, the symbols denote values that have been calcu-lated using Eq.~24!, with the excluded volume limit valuessubstituted for all the quantities on the right-hand side. Thedashed curves have been obtained from Eq.~25!, using meanvalues for the Carreau–Yasuda model parameters given inTable I. Unlike in the case ofexact Brownian dynamicssimulations, sincemcÞ2mh in the Gaussian approximation,UCh does not reach a constant value in the limit of largeb.

Perhaps the most celebrated result of the study of ex-cluded volume interactions in dilute polymer solutions is theobservation that at large molecular weights, the equilibriumroot-mean-square radius of gyration of the polymer coil,Rg ,scales with molecular weight as a power law,Rg;M n,wheren is a universal exponent, independent of the particu-lar polymer–solvent system. Pierleoni and Ryckaert15 have

FIG. 5. ~a! Asymptotic nondimensional ratiohp /hp,0 and ~b! asymptoticnondimensional ratioC1 /C1,0, for various values ofb, in the excludedvolume limit. Circles represent Brownian dynamics simulation data, whilesquares represent results of the Gaussian approximation. The dashed linesare fits to the data using the Carreau–Yasuda model, Eqs.~23!.

FIG. 6. The dependence of the universal ratioUCh on the characteristicshear rateb. The open circles are predictions of Brownian dynamics simu-lations, while the squares are predictions of the Gaussian approximation,calculated from data reported by Prakash~Ref. 18!. The dashed curves havebeen obtained from Eq.~25!, using values for the Carreau–Yasuda modelparameters given in Table I. The horizontal lines represent the zero-shearrate predictions in the excluded volume limit, reproduced from Ref. 24.

TABLE I. Values of Carreau-Yasuda model parameters forhp /hp,0 andC1 /C1,0. Comparison of exact Brownian dynamics simulations~BDS!,with the Gaussian approximation~GA!.

mh nh ah

BDS 20.25560.009 1.92560.100 0.71260.050GA 20.28860.006 1.67560.097 0.42960.025

mc nc ac

BDS 20.5260.02 2.39560.090 1.31260.020GA 20.75060.014 1.87860.795 0.50760.012



shown recently, by calculating the structure factor for bead–spring chains with up to 300 beads, that in the presence ofexcluded volume interactions, two distinct regimes can beidentified in the structure of a linear polymer chain undergo-ing shear flow. At large length scales, the polymer coil has aRouse-like behavior, while at small length scales, where theflow field has not yet distorted the isotropic coil structure, thepolymer coil exhibits typical self-avoiding statistics. Inthe light of these observations we expect, therefore, that inthe limit of infinite chain length, the scaling exponentn willremain unaltered from its equilibrium value, since, at anyparticular shear rate, more and more polymer length scaleswill be smaller than the smallest length scale probed by theflow field. Recently, Kumar and Prakash24 have obtained thevalue ofn at equilibrium, by exploiting a unique feature ofthe solution to the excluded volume problem. We show herethat the same procedure can also be applied to find the valueof n in the presence of shear flow. In order to do so, however,it is necessary to make a few introductory remarks.

While the power law scaling ofRg with chain lengthNoccurs only in the excluded volume limit, it is common todescribe the dependence ofRg on N, at all values of chainlength, with an apparent power law,Rg;Nneff, with neff rep-resenting an effective exponent that approaches its criticalvalue n, as N→`. Schafer and co-workers8,34 have shownthat the manner in whichneff approaches its asymptotic limitis strongly dependent on the magnitude of the parameterz*relative to its fixed point valuezf* . Discussions of the originof the fixed point, and its significance can be found in trea-tises on renormalization group methods.6–8 Basically, the ex-istence of power law behavior in the excluded volume limit,such as the one observed forRg , is intimately connected tothe existence of a fixed point value for the parameterz* .Schafer and co-workers have shown, using both renormaliza-tion group methods, and Monte Carlo simulations, thatneff

→n on two distinct branches,~i! the strong-coupling branchcorresponding toz* .zf* , and~ii ! the weak-coupling branchcorresponding toz* ,zf* . On the weak-coupling branch, forincreasing values ofN ~or equivalentlyz), neff approachesnfrom below, increasing rapidly at first before approaching theasymptotic value very gradually. On the other hand, on thestrong-coupling branch,neff approachesn from above, de-creasing rapidly before approaching the asymptotic valueslowly. Kumar and Prakash24 have shown that Brownian dy-namics simulations are able to capture the existence of thedual branched structure of the solution to the excluded vol-ume problem. Furthermore, they show that this distinctivefeature of the solution implies that at fixed values ofz andd* , a plot of neff versusz* , for a range of values spanningthe fixed point, has a characteristic shape exhibiting amarked inflection point, from which one can obtain both thefixed point zf* , and the critical exponentn ~see Fig. 11 inRef. 24!. In particular, they found that at equilibrium,n50.6, and 0.28<zf* <0.3.

By collecting data onRg as a function ofN, at variousconstant values of the parametersz* , and b, the value ofneff , at particular values ofz andd* , can be calculated withthe help of the expression,neff5] ln Rg /] ln N. The symbolsin Fig. 7 have been obtained in this manner, for two repre-

sentative values of the characteristic shear rateb. It is clearfrom the figure that the dual branched structure of the solu-tion persists into the nonequilibrium regime, and the charac-teristic shape of the dependence ofneff on z* , observed atequilibrium, is preserved at finite values of the shear rate.The point of inflection, for the data corresponding tob50.4, clearly indicates that bothzf* andn are unaltered fromtheir equilibrium values. Atb51.75, while zf* lies in thesame interval as at equilibrium, there is a small decrease inthe value ofn, arising perhaps due to the relatively smallvalues ofN used to acquire the data. The results displayed inFig. 7 are consistent with the argument that, for chains hav-ing sufficiently large number of beads, regardless of the char-acteristic shear rateb, there exists a local length scale belowwhich the beads do not experience flow, and are effectivelyat equilibrium.

It is appropriate now to discuss the earlier observation inFig. 3 that theexact crossover functions forhp /hp,0 andC1 /C1,0, in the limit of large z, do not approach theasymptotic predictions obtained in the excluded volumelimit. Kumar and Prakash24 have speculated, in connectionwith a similar observation made earlier for universal equilib-rium and linear viscoelastic ratios, that the origin of thisbehavior might lie in the fundamental difference betweensystems described by the crossover region, and those thatcorrespond to the excluded volume limit. As pointed out ear-lier, since the crossover behavior has been obtained by keep-ing z constant asN→` ~which implies thatz* →0, andT→Tu), points on a crossover curve correspond to systems,~i!that are infinitesimally close to theu temperature, and~ii !that satisfy the criteria for belonging to the weak-couplingbranch. On the other hand, since the excluded volume limitbehavior has been obtained for systems with a finite value ofz* , they correspond to systems,~i! with a nonzero differencebetweenT andTu , and~ii ! that belong to either the weak orthe strong-coupling branch. Interestingly, it was shown inRef. 24 that, at equilibrium, the Gaussian approximationdoes not possess the dual-branched structure of the exactsolution. We have not examined here whether such is thecase even at finite shear rates. Nevertheless, as pointed outpreviously, both at equilibrium and at steady state, the largezcrossover behavior of the Gaussian approximation coincides

FIG. 7. The effective exponentneff as a function of the strength of excludedvolume interactionsz* , predicted by Brownian dynamics simulations, attwo different values of the shear rateb.



with the behavior in the excluded volume limit, unlike in thecase of the exact solution.

A number of studies aimed at predicting the orientationangle and orientation resistance in shear flow, associatedwith various tensorial quantities, as a function of the shearrate, have been carried out previously.15,22,32,35,36The generalconsensus is that these quantities are universal properties of adilute polymer solution, in the sense that, at the same re-duced shear rateb, chains of different lengths have the sameorientation angles and resistances. The universal dependenceof xG andmG on b, predicted by the present Brownian dy-namics simulations in the excluded volume limit, will bepresented shortly below. However, the universal shear ratedependence of the orientation anglext , and the orientationresistancemt ~with the expected power law behavior at largevalues of b!, can be derived in a straightforward mannerfrom the results that have been obtained above for the shearrate dependence of the viscometric functions in the excludedvolume limit, namely, Eqs.~23!.

Substituting the expressions for the viscometric func-tions, Eqs.~9!, into the defining expressions forxt andmt ,Eq. ~15!, and using the Carreau–Yasuda model, and the re-lation betweeng andb, leads to the following expressions:

cot~2xt!5UCh,0

2

~C1 /C1,0!

~hp /hp,0!b ~27!

5UCh,0

2

@11~acb!nc# (mc /nc)

@11~ahb!nh# (mh /nh) b, ~28!

mt52

UCh,0

~hp /hp,0!

~C1 /C1,0!~29!

52

UCh,0

@11~ahb!nh# (mh /nh)

@11~acb!nc# (mc /nc) . ~30!

On using the relationmc52mh , Eqs.~28! and~30! simplifyto the following power law dependencies of cot(2xt) andmt

on b, in the limit of largeb,

cot~2xt!5UCh,0

2 S ac2

ahD mh

b11mh, ~31!

mt52

UCh,0S ah

ac2 D mh

b2mh. ~32!

Since both renormalization group theory,9,23 and the currentBrownian dynamics simulations suggestmh'20.25 ~seeTable I!, Eqs.~31! and ~32! indicate that cot(2xt) grows ap-proximately asb0.75, while mt has a weaker dependence,growing approximately asb0.25, in the limit of large b.These predictions are of course restricted to the influence ofexcluded volume interactions on these quantities, and a thor-ough comparison with experiment requires the incorporationof hydrodynamic interaction effects. Bossart and O¨ ttinger32

and Cifre and de la Torre22 have shown previously that thepresence of hydrodynamic interactions also leads to shearrate dependent orientation angles and resistances. Interest-ingly, however, in contrast to the prediction obtained here inthe presence of excluded volume interactions alone, they findthat at large values ofb, the orientation resistancemt de-

creases with increasing shear rate. As is well known, thepresence of hydrodynamic interactions leads to the predic-tion of shear thickening at large values ofb.37 It is clear fromEq. ~32!, that a positive value of the exponentmh , wouldexplain the observation of Bossart and O¨ ttinger, and Cifreand de la Torre.

Figure 8 displays the shear rate dependence of the ori-entation angles obtained here by Brownian dynamics simu-lations in the excluded volume limit. In the case of cot(2xt),values forhp /hp,0 andC1 /C1,0, obtained at various valuesof b, in the excluded volume limit, have been substituted onthe right-hand side of Eqs.~27!, with UCh,050.77160.009. On the other hand, cot(2xG) has been evaluated byaccumulating finite chain data using Eq.~16!, at a constantvalue of z* 50.28 ~the fixed point value!, followed by ex-trapolation to the long chain limit. Comparison with theRouse model~which predicts a linear dependence onb ineach case!, clearly indicates that the presence of excludedvolume interactions leads to a weakening of the alignment ofthe polymer coil in the flow direction. This has been ob-served previously both theoretically,15 and experimentally.38

Since, at the same shear rateb, cot(2xG).cot(2xt), the ten-sor G is more easily oriented thant p.

Figure 9 displays the universal dependence of the orien-tation resistancesmG andmt , in the excluded volume limit,on b. As in the case ofxt , the orientation resistancemt hasbeen obtained by substituting known values on the right-hand side of Eq.~29!. On the other hand,mG has been ob-tained from the Brownian dynamics simulation data forcot(2xG), displayed in Fig. 8~b!, by using the defining ex-pression,mG5b/cot(2xG). It is evident from the figure that

FIG. 8. Universal dependence of the cotangent of the orientation angles,~a!xt ~open circles! and ~b! xG ~filled circles!, on the characteristic shear rateb, in the excluded volume limit, obtained by Brownian dynamics simula-tions. The dashed curve through thext data has been drawn using Eq.~28!,with parameter values reported in Table I. The solid lines are the predictionsof the Rouse model.



both mG and mt increase withb, indicating that it gets in-creasingly difficult to orient the polymer coil with increasingshear rate. This has been previously predicted,39 and ob-served experimentally at relatively low values ofb.40 Thefact thatG is more easily oriented thant p can also be seenfrom the behavior of the orientation resistancesmG andmt ,sincemt.mG , for all values ofb. As pointed out previouslyby Bossart and O¨ ttinger,32 flow birefringence experiments~which measurext), reflect orientational ordering on lengthscales smaller than those probed by light scattering experi-ments~which measurexG). Since smaller length scales areless affected by flow, they exhibit greater resistance to orien-tational ordering, leading to the observed relative magnitudesof mG andmt .

It is of particular interest to examine the universal valuesof the orientation resistances in the limit of zero shear rate,namely,mG,0 andmt,0 , since a number of studies have beenconfined to the low shear rate regime. Bossart and O¨ ttinger,32

and Pierleoni and Ryckaert39 have argued previously that theeffect of excluded volume interactions on the orientation re-sistance is negligible. Bossart and O¨ ttinger,41 have found,from careful experiments, that in a good solvent~polystyrenein bromobenzene!, mt,053.7460.28, compared to the valueof 3.1360.42 in a theta solvent~polystyrene in 4-bromo-a-benzyl alcohol!. Given the magnitude of the error bars, it canonly be concluded thatmt,0 is probably larger in a goodsolvent than in a theta solvent. It must be borne in mind thatboth these values reflect the influence of hydrodynamic in-teractions. In the present instance, where only the influenceof excluded volume interactions is considered, the value ofmt,0 can be obtained from the following expression:

mt,052

UCh,0, ~33!

which is derivable from Eq.~29! in the limit of zero shearrate. Using the previously establishedexact Brownian dy-namics simulations value ofUCh,050.77160.009,24 leads tomt,052.59460.030, which is marginally larger than theRouse value of 2.5. Using nonequilibrium molecular dynam-ics simulations of bead–spring chains with FENE springs,Aust et al.36 have estimatedmt,052.460.3. Interestingly, inspite of the fact that the solvent is treated explicitly in these

simulations, and consequently hydrodynamic interactions aretaken into account, the predicted value is not significantlydifferent from the pure excluded volume prediction obtainedhere. It is worth noting that Bossart and O¨ ttinger,32 who usedthe Gaussian approximation, and de la Torre andco-workers,22,35 who used Brownian dynamics simulationswith hydrodynamic interactions, report the values:mt,0

'3.57, andmt,0'3.5, respectively—both of which are sub-stantially different from the prediction of the Rouse model.While an improvement in solvent quality seems to increasemt,0 fractionally, our results seem to suggest that it decreasesmG,0 fractionally. By fitting themG data in Fig. 9 with a leastsquares third order polynomial, and extrapolating tob→0,we find that in the excluded volume limit,mG,051.6960.03, which is slightly smaller than the Rouse predictionof mG,051.75. The value obtained here is in good agreementwith the explicit solvent nonequilibrium molecular dynamicssimulations of Austet al.36 and the results of renormalizationgroup calculations,32 both of which predictmG,051.7. Sincethese studies account for hydrodynamic interactions, it isclear that hydrodynamic interactions have no effect onmG,0 .Pierleoni and Ryckaert have also observed a similar patternwhen comparing their explicit solvent molecular dynamicssimulations39 with their Brownian dynamics simulations~inwhich only excluded volume interactions were taken intoaccount!.15 It is well known that hydrodynamic interactionshave a significant influence on dynamic properties, but haveno influence on static properties, since the equilibrium distri-bution function is unaltered in their presence. This suggests,perhaps not surprisingly, thatmG,0 is static property, whilemt,0 is a dynamic property.

IV. CONCLUSIONS

Results of a detailed Brownian dynamics simulationsstudy, of a polymer solution undergoing steady shear flow,have been presented. The polymer molecule has been mod-elled by a bead–spring chain, and a narrow Gaussian ex-cluded volume potential, that acts between pairs of beads,has been used to mimic the influence of solvent quality.

Material properties have been shown to become inde-pendent of the range of excluded volume interactionsd* ,and the number of beadsN, in the limit of largeN. Further-more, it has been found that master plots are obtained when,~i! the influence of the strength of excluded volume interac-tions is interpreted in terms of the solvent quality parameterz5z* AN, and ~ii ! the dependence on the shear rateg isinterpreted in terms of the characteristic nondimensionalshear rate,b. In this work, we have explored the universaldependence of the nondimensional viscosity ratio(hp /hp, 0), the nondimensional first normal stress differenceratio (C1 /C1, 0), the orientation anglesxt andxG , and theorientation resistancesmt and mG , on the characteristicshear rateb and the solvent qualityz.

The extent of shear thinning has been found to increaseasz increases~Fig. 2!. However, the incremental increase inshear thinning saturates asz increases~Fig. 3!. The existenceof universal viscometric ratio versus shear rate curves, inde-pendent of all model parameters, in the excluded volume

FIG. 9. Universal dependence of the orientation resistancesmt ~squares!andmG ~circles!, on the characteristic shear rateb, in the excluded volumelimit. The line through themt data has been drawn using Eq.~30!, withparameter values reported in Table I. The horizontal lines are the predictionsof the Rouse model.



limit, z→`, has been verified~Fig. 5!. The shear rate depen-dence of both the universal viscometric ratios has been foundto be well described by the Carreau–Yasuda model, withpower law decay exponents at large values ofb, that are inclose agreement with values predicted by renormalizationgroup theory~Table I!.

The asymptotic values of the exact crossover functionsfor (hp /hp,0) and (C1 /C1,0), in the limit of largez, arefound not to approach the universal predictions obtained inthe excluded volume limit, unlike previous predictions of theGaussian approximation~Fig. 3!.

The shear rate dependence of the universal ratioUCh

has been obtained in the excluded volume limit. The ratio isfound to decrease from the zero shear rate value of 0.77160.009, to a constant value of 0.56860.038, in the limit oflarge shear rates~Fig. 6!.

The dual branched structure of the solution, elucidatedrecently by Scha¨fer and co-workers,8,34 and shown to be cap-tured by Brownian dynamics simulations at equilibrium,24

has been found to persist into the nonequilibrium regime. Asin the equilibrium case, this distinctive structure has beenexploited here to obtain an estimate of both the fixed point ofthe strength of excluded volume interactions, and the criticalexponentn. Both quantities are found to remain unaltered inthe presence of shear flow~Fig. 7!.

The universal dependence ofxt , xG , mt andmG , onb,in the excluded volume limit, has been obtained. The pres-ence of excluded volume interactions has been shown toweaken the alignment of polymer coils in the flow direction~Fig. 8!. Of the two tensors, the radius of gyration tensorG isfound to be more easily oriented thant p ~Fig. 9!. The powerlaw shear rate dependence ofxt (;b0.75) and mt

(;b0.25), in the limit of large shear rate, has been obtainedby exploiting the Carreau–Yasuda model fit of the universalviscometric functions. The presence of excluded volume in-teractions leads to a marginal increase inmt,0 , while mar-ginally reducingmG,0 .

ACKNOWLEDGMENTS

The authors gratefully acknowledge the Victorian Part-nership for Advanced Computing~VPAC! for a grant underthe Expertise program, and both VPAC and the AustralianPartnership for Advanced Computing~APAC! for the use oftheir computational facilities.

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universal consequences of the presence of excluded volume interactions in dilute polymer solutions...

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