the rheology of dilute solutions of flexible polymers

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The rheology of dilute solutions of flexible polymers: Progress and

problems

Ronald G. Larson

Citation: J. Rheol. 49, 1 (2005); doi: 10.1122/1.1835336

View online: http://dx.doi.org/10.1122/1.1835336

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Published by the The Society of Rheology

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The rheology of dilute solutions of flexible polymers:Progress and problems

Ronald G. Larsona)

Department of Chemical Engineering, University of Michigan, Ann Arbor,

Michigan 48109-2136

(Received 13 May 2004; final revision received 22 October 2004)

Synopsis

Recent progress toward understanding the rheology of dilute solutions of flexible polymers is

reviewed, emphasizing experimental results from flows imaging single deoxyribonucleic acid

DNA molecules and filament-stretching rheometry of dilute polystyrene Boger fluids, as well as

Brownian dynamics BD simulations of these flows. The bead-spring and bead-rod models arepresented, the range of their applicability discussed, and methods presented for inclusion of

hydrodynamics interactions, excluded volume, and other physical effects within BD simulations.

After reviewing and updating work in the linear viscoelastic regime, the primary focus shifts to the

more complex nonlinear regime. While BD predictions of the conformations of 20 to 100 micron

long DNA molecules in strong shear and extensional flows has been in good to excellent agreement

with the corresponding experiments, predictions of the polystyrene dilute solution rheometry data

have been hit or miss, with poorer results obtained for the higher molecular weights. This may be

due, in part, to the more important roles of hydrodynamic interactions and excluded volume

interactions in the more flexible, and therefore more condensed, polystyrene coils. Inclusion of these

effects in BD simulations has led to improved predictions, but does not lead to the accurate

prediction of the plateau Trouton viscosity for higher molecular weight samples, nor alleviate the

complete failure of simulations to predict measurements of coil distortion by light scattering. Thus,

despite enormous progress in the past decade, some significant gaps in understanding remain.

2005 The Society of Rheology. DOI: 10.1122/1.1835336

I. INTRODUCTION

Since the late 1980s, there has been enormous progress in molecular-level under-

standing of the rheological properties of dilute solutions of flexible polymers. This

progress is due primarily to four advances: 1 The application of methods of imaging the

conformations of isolated polymer deoxyribonucleic acid DNA molecules by optical

microscopy in well-defined flows Perkinset al.1995; 1997;2 the development of the

filament-stretching rheometer, which can impose a high-quality steady extensional flow

history on dilute solutions of polymers in viscous solvents Tirtaatmadja and Sridhar

1993; McKinley and Sridhar 2002; 3 the development of model dilute polymer solu-

tions, in which the polymer is nearly monodisperse and of high molecular weight and the

solvent very viscous Mackay and Boger 1987; Magda and Larson 1988; and 4 theincrease in computer speed and the development of methods that permit simulation of

ensembles of bead-spring or bead-rod chains containing enough beads to describe real-

istically the conformations of real polymers Liu 1989; Lopez Cascales and Garca de la

aElectronic mail: [email protected]

2005 by The Society of Rheology, Inc.J. Rheol. 491, 1-70 January/February 2005 0148-6055/2005/491/1/70/$25.00 1



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Torre 1991; Grassia and Hinch 1996; Somasi et al. 2002. A review of the status of

molecular rheology of dilute solutions of flexible polymers at the end of the 1980s can

be found in Chapter 8 of Larson 1988 as well as in Bird and O

ttinger 1992. Thisarticle updates those reviews, emphasizing results arising from the four advances listed

above. A complementary recent review can be found in Prakash 1999.

Since the polymer solutions used commercially are almost invariably nondilute a

notable exception being the dilute solutions used in turbulent drag-reducing flows Virk

1975, interest indilutepolymer solutions derives primarily from their importance in the

characterizationof a polymers molecular weight, stiffness, long-chain branching, and

interaction with solvent, and from a fundamental interestin understanding macromolecu-

lar response to hydrodynamic forces, free from the complications of intermolecular en-

tanglements. However, as we shall see, diluteness paradoxically creates added compli-

cations not present in concentrated entangled solutions. For purposes of molecular

characterization and fundamental understanding, the most useful dilute solutions are

those for which the polymer molecular weight is as monodisperse as possible. By defi-

nition of diluteness, the rheology of any polydisperse dilute solution can always bededuced by linearly combining the rheological properties of a series of monodisperse

dilute solutions representing the molecular weight distribution of the polydisperse

sample. Hence, nothing fundamental is learned by studying the effects of polydispersity

on dilute solution rheology. For this reason, only nearly monodisperse dilute solutions

will be considered in this review. We will also limit this review to uniform velocity

gradients, especially those of steady shear, extensional, and mixed shear/extensional

flows, since these provide the most clear-cut insight into rheological properties.

In the following, we assume that the reader has some familiarity with polymers and

polymer solutions, such as the random-flight configuration of ideal chains, the polymer

end-to-end vector, etc., and knows basic rheological concepts, such as cone-and-plate

rheometry, and the definitions of the storage and loss moduli, G andG . If needed, the

reader can find these concepts thoroughly described in numerous texts, including Bird

et al. 1987, Larson 1988; 1999, and Tanner 1985. This review is organized as fol-lows. We will present the criteria governing the term dilute solution Sec. II, and

describe the forces and interactions that control the dilute-solution rheology of flexible

polymers Sec. III. Next, we will present the conventional bead-spring and bead-rod

models of polymer rheology and limitations on their validity as well as moment equations

and Brownian dynamics equations for obtaining predictions from them Sec. IV. Their

predictions are then compared to experimental dilute-solution rheology data in the linearSec. V and nonlinear Sec. VI regimes. Finally, the results are summarized and the

gaps between predictions and the data are discussed Sec. VII.

II. CRITERION FOR DILUTENESS

Here we take the term dilute to have its usual meaning, that the polymer concen-tration is low enough that the polymers do not interact with each other either topologi-

cally i.e., through entanglements or hydrodynamically, and hence the effect of the

polymers on the rheological properties of the fluid in a fixed flow field is linear in the

concentration of polymer in solution. A different criterion for diluteness arises if one

wishes to measure rheological properties of dilute solutions in a flow without the polymer

molecules disturbing that flow. This criterion can in some cases be even stricter than the

normal criterion for diluteness Feng and Leal 1987, and would, for example, make all

polymer solutions that reduce drag in turbulent flows nondilute. Thus, the stipulation of a

linear effect of the polymer in a fixed flow fieldis a significant one.

2 LARSON



4/71

In the usual definition of a dilute solution, then, a rheological property such as the

viscosity is linear in the mass concentration c of polymer. One can, therefore, define

intrinsic properties such as the intrinsic viscosity of the polymer in dilute solution as

0 limc 0

s

cs, 1

where is the solution viscosity and s is the viscosity of the solvent. The intrinsic

viscosity can be obtained by measuring the viscosity of a series of polymer solutions

under increasingly more dilute conditions and extrapolating the ratio (s)/cs to

zero concentration. The subscript 0 on 0 implies that it is measured at low shear

rates, where it reaches an asymptotic plateau value that is independent of shear rate. From

its definition, it is evident that the intrinsic viscosity has units of volume per unit mass,

and can be thought of as the hydrodynamic volume occupied by a unit mass of the

polymer in dilute solution. Interpreted this way, one deduces that polymer molecules

begin to overlap with each other whenever c0 exceeds unity or so. Thus, a simplecriterion for diluteness of a polymer solution is that

c 1

0. 2

More precise expressions, and a thorough discussion of the criteria for diluteness of

flexible, semiflexible, and rigid polymers in both good and theta solvents defined in

Sec. III can be found in Bercea et al. 1999.

In a theta solvent defined below, the hydrodynamic volume of a polymer coil scales

with the 3/2 power of the molecular weight. Since the intrinsic viscosity is proportional

to the hydrodynamic volume of a polymer molecule divided by its mass, it follows that in

a theta solvent,

0 KM1/2, 3

whereKis a constant that depends on the equilibrium mean-square end-to-end distance

of a polymer,R20 , per unit molecular weight M, as

K R20/M

3/2 C

2

m0

3/2

. 4

Here, the brackets denote an ensemble average over a large number of polymer

molecules, and the subscript 0 on R20 denotes an equilibrium average, i.e., in theno-flow limit. C is the characteristic ratio of the polymer, defined as the ratio of the

mean-square end-to-end distance of the polymer to that for an ideal freely jointed chain

with the same number of backbone bonds n and same bond length as the real polymer;

that is Ferry 1980:

R20 Cn2, 5

whereC has a value of around 9.6 for polystyrene Fetterset al. 1994. In Eq. 4, m 0is the molecular weight per backbone bond of the polymer, which is 52 Daltons for

polystyrene, and 1.54 A for a carboncarbon bond.

The FloryFox parameter should have a universal value in a theta solvent; the

experimental value is around 2.51023 mol 1 Bercea et al. 1999. For polystyrene,

withC 9.6, this yields K 7.210 2 cm3 g 1 (g/mol) 1/2, close to the value of

K 8 10 2 cm3 g 1 (g/mol)1/2 reported for polystyrene by Flory 1969, who took

3DILUTE SOLUTIONS OF FLEXIBLE POLYMERS



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C 10.0. Thus, using Florys value for K, we find that for polystyrene of molecular

weight 10 million Daltons, 0 253 cm3/gm in a theta solvent. Hence, a polystyrene

solution of this molecular weight is dilute only if its concentration is well below about1/0 0.004 gm/cc, or about 0.4% polystyrene by mass or by volume for solvents of

typical density. In general, this criterion for diluteness is not very strict, as it implies that

at the cross-over condition, the viscosity of the solution is more than doubled by the

polymer. Sridhar and co-workers found for polystyrene of molecular weight 10.2 million

in a viscous theta-like solvent that the transition to a non-dilute regime begins at a

concentration of around 777 ppm, as measured by onset of a concentration dependence of

the longest relaxation time and by a failure of the extensional viscosity to scale linearly

with concentrationGuptaet al.2000. This concentration is about 20% of the cross-over

value estimated from 1/0 . Thus, one might use 1/50 as a stricter criterion for

diluteness, and 1/0 as a lenient one. When flexible polymers are highly deformed

under flow, the requirements for diluteness appear to become even more stringent than

under quiescent conditions, with diluteness apparently only being achieved in some cases

at concentrations more than a decade lower than 1/0 Keller and Odell 1985; Fengand Leal 1997; Clasen et al. 2004.

We define for future reference the radius of gyration R g of a polymer coil as the radius

of a sphere of equal moment of inertia to that of the polymer, if we were to put all of the

polymers mass on the surface of that sphere. Mathematically,

Rg2

12

n2

i 1

n

j 1

n

Rij2, 6

wheren is the number of monomers and R i j is the spatial distance between monomers i

and j. For a random-walk polymer, Rg2

R20/6.

III. PHYSICAL FORCES AND PHENOMENAOver the ranges of length scales a few nanometers to a few micronsand time scales

s to s of primary importance for rheological properties, the following polymer phe-

nomena are of real or potential importance:

1 Viscous drag,2 Entropic elasticity,3 Brownian forces,

4 Hydrodynamic interaction HI,5 Excluded-volume EV interactions,6 Internal viscosity IV, and7 Self-entanglement SE.

We have ordered these phenomena according to their importance, as currently under-stood. Viscous drag, which is the frictional force that the flowing solvent exerts on the

polymer, is always important, even under flows too weak to excite an elastic response

from the polymer. Entropic elasticity becomes important as soon as the flow is strong

enough to deform the chains away from their equilibrium distribution of conformations,

and is shown by viscoelastic i.e., time dependent rheological effects. In addition,

Brownian motion, due to the bombardment of the polymer by solvent molecules, also

influences the chains distribution of conformations under flow.

HI, which is the disturbance to the flow field produced by one part of the chain that

influences the drag on another part, is always important for high molecular weight flex-

4 LARSON



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ible polymers, but its effect can, in some cases as discussed below, be subsumed into

the effective drag properties of the chain. EV interactions are the repulsive forces be-

tween monomers that prevent their overlap, leading to a tendency of the chains to expandbeyond the ideal random-walk conformations they have in the melt. EV interactions can

be effectively cancelled out in some solvents at their theta point, which is the tem-

perature at which the weak repulsion the polymer feels toward the solvent is just strong

enough to cancel out the tendency of the coil to expand due to EV. Values for theta

temperatures for many polymer/solvent pairs can be found in Sundararajan 1996. Be-

cause the theta point is a single temperature, EV is therefore usually, but not always, of

at least some importance in the dynamics of dilute polymer solutions. Interestingly, in

concentrated solutions or in the melt, both HI and EV interactions are screened out and

so do not usually need to be considered, except perhaps at small length scales, much less

than the chains radius of gyration. In this sense, at least, concentrated solutions are less

complex than dilute solutions !, although in the former one must, of course, cope with

the great complicating factor of entanglements between chains, which are absent in truly

dilute solutions. HI and EV are at least partially to blame for the slowness of develop-ment of a detailed understanding of the rheology of dilute polymer solutions.

The last two phenomena on our list, namely IV and SE, are of uncertain significance.

They have been invoked by various authors to explain puzzling rheological data, but to

date their importance or even reality have not been clearly demonstrated. Recent de-

velopments, discussed below, may soon shed light on these phenomena.

IV. BEAD-SPRING AND BEAD-ROD MODELS

For years, meaningful comparisons between theories for the rheology of dilute poly-

mer solutions and corresponding experiments were frustrated not only by the lack of

high-quality experimental data especially data that directly measure polymer molecular

conformations, but also by the statistical nature of polymer theories, which give rise to

equations that in the past required some sort of linearization or preaveraging to reduce

them to forms manageable for the computers of the day. Until fairly recently, this meant

that rheological theories had to be simplified to closed-form constitutive equations relat-

ing stress or birefringence to flow history before comparisons to experiment were pos-

sible. As a result, disagreement between theory and experiment could be blamed, at least

in part, to the mathematical closure approximations of uncertain accuracy that were

needed to bring the molecular theory into a closed form, rather than arising entirely from

limitations of the physical model. Now, however, high-speed computers make possible

the direct solution of stochastic equations containing accurate expressions for the physi-

cal forces and interactions, for ensembles of polymer chains large enough to yield accu-

rate averages. Hence, nowadays, physical theories for polymer dynamics and stress can

be tested much more directly than before, without the uncertainties produced by math-

ematical closure approximations.However, even using the most advanced computers, atomic-level simulations of long-

chain polymers are still not close to being feasible since the longest time scales acces-

sible in such simulations are a fraction of a microsecond and, therefore, outside of the

rheologically most interesting range. Hence, there is still a need for coarse-graining

approximations in which only slow variables, representing coarse-grain features, are

tracked, while small-scale, fast, dynamics are assumed to remain at local equilibrium,

slaved to the slow variables. Thus, even with the very fast computers that are now readily

available, it is still important to choose the proper level of coarse graining for a given

flow, so that the simulations track all dynamics that are sluggish enough to be out of




7/71

equilibrium during the flow, while allowing the faster dynamics to remain at equilibrium,

where no computer time need be wasted simulating them.

A. Model definitions

The most commonly used coarse-grained models of polymer chains are the freely

jointed bead-rodand the bead-springmodels, depicted in Fig. 1. The physical bases for

these models have been explained in detail elsewhere Doi and Edwards 1986; Bird et al.1987; Larson 1988. In these models, the beads represent frictional drag centers. A single

rod in the bead-rod model represents a single link, or Kuhn step, in a freely jointed chain

model of a flexible polymer molecule, whose coarse-grained conformation is a random

walk in the absence of excluded volume forces, discussed shortly. A model related to

the bead-rod chain is a chain of freely jointed ellipsoids, where the aspect ratio of each

ellipsoid is chosen to capture the aspect ratio of a piece of the real chain that corresponds

to one Kuhn step of the chain Stigter and Bustamante 1998.

Real polymers are not freely jointed chains, but a freely jointed chain will have the

same equilibrium mean-square end-to-end length R20 and fully extended length L, asany real polymer in a theta solvent if the freely jointed step length b Kand the number of

steps NK of the freely jointed chain are chosen appropriately; i.e, so that b K2

NK

R2

0 and b

KN

K L . The fully extended length of a chain whose n backbone

bonds, each of length , are tetrahedrally bonded as is the case for carboncarbon

bonds, is given by

L 0.82n , 7

where the factor 0.82 enters because the backbone has a zig-zag configuration in the most

extended state that still preserves the tetrahedral backbone bond angle restrictions see

Fig. 1. From this, and the definition of C given in Eq. 5, we obtain the following

formulas for the equivalent freely jointed step length bKand the number of steps NK:

FIG. 1. Illustration of coarse-grain mapping of real polymer chain, with a carboncarbon backbone containing

fixed dihedral bond angles, onto a bead-rod chain whose configuration is that of a random walk, and furthercoarse-grain mapping of the bead-rod chain onto a bead-spring chain.

6 LARSON



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bK C

0.82; NK

0.822

Cn. 8

For polystyrene molecules, for example, with C 9.6, 1.54 A, and n M/52,

Eq. 8 yields a Kuhn step length bK 1.8 nm, and NK M/742, where M is the

molecular weight in Daltons.

Since the fully extended length of the polymer is proportional to its molecular weight,

the number of rods in a bead-rod model needed to represent the chain grows in proportion

to its molecular weight. And since the computational time grows with the number of rods

raised to a power of 2 or higher, the number of rods one can simulate with current

computers is limited to no more than a few hundred. For polystyrene, this limit corre-

sponds to molecular weights in the range of a few hundreds of Daltons. To reach a

molecular weight in the millions using the bead-rod model would require more than a

hundred-fold greater computational power, and so will remain out of reach for some time

to come.The bead-rod model is computationally very expensive not only because of the large

number of dynamical variables the bead positions, but also because the motions of

individual rods are very rapid, requiring small simulation time steps. If, however, the

deformation of the polymer molecule is slow enough that a subchain containing, say,

NK,s connected rods fully samples its distribution of conformations in the time required

for the beads to be convected appreciably in the flow, then on this slower time scale of the

convection the rapidly fluctuating force produced by motions of individual rods averages

out to an equilibrium entropic force fs that pulls the two ends of the subchain toward

each other, and the subchain, on these slower time scales, acts as aneffective spring, with

spring force fs.

Thus, for slow flows, each spring in the bead-spring model can be chosen to represent

a large number NK,s of Kuhn steps. Within the caveats discussed below, if the molecular

weight of the polymer is increased, one can simply proportionately increase the numberof Kuhn steps represented by each spring, and so the number of springs Ns

NK/NK,s in the bead-spring model does not necessarily need to grow with the in-

creasing molecular weight of the chain. The constants in the force law for the spring force

fs(r) depend on the number of Kuhn steps represented by that spring, but rules for

choosing these constants are well established see below.

B. Coarse-graining principles

While coarse graining is necessary to simulate high molecular weight polymers, there

are obviously limits to how far it can be pushed, before physical realism is lost. Gener-

ally, finer-scaled models are able to track higher-frequency chain motions, and are valid

up to higher velocity gradients or frequencies, than are coarser-grained models Larson

2004a. It is also worth noting that, contrary to intuition, the bead-rod model might not bevalid at frequencies any higher than that at which the bead-spring model fails because, at

frequencies too high to be described by an entropic spring, a chain of freely jointed rods

might also be inaccurate since the free joints fail to capture the rotational energy barriers

present in the real chain. One can estimate from the energy barrier the frequency where

this might occur and how it compares to the frequency where the bead-spring model failsLarson 2004b.

It is also possible to incorporate bending and torsional bond rotational potentials into

the bead-rod model, thereby mimicking the flexibility of a real polymer molecule at the

level of individual backbone bonds for synthetic polymers Ryckaert and Bellemans




9/71

1975; Lyulin et al. 1999. A bead-rod model for DNA chains that includes both torsional

and bending stiffness has been developed by Vologodshii and co-workers Jian et al.

1998. To maintain smooth bending of the molecule, the rod length was kept to onlyaround 1/20 of a Kuhn step, i.e., around 5 nm, or 15 base pairs. Using this model,

Brownian dynamics simulations of supercoiled i.e., twisted like a telephone cord DNA

plasmids circles of length up to 3000 basepairs around 1 m length were possible

over times of around 10 ms, including the effects of HI. If one wishes to simulate at a still

finer level, one can incorporate solvent molecules explicitly, and their intermolecular

interactions with each other and with the polymer can be included in great detail in fully

atomistic molecular dynamics MD simulations; see, for example, Aust et al. 1999.

However, such simulations are at present limited to timescales of less than a microsec-

ond, too slow to capture the dynamics of long polymer molecules. Still, computer power

is now great enough that one could, in principle, capture the full range of dynamics of a

polymer molecule in solution by using multiple methods with overlapping time scales

For example, using: 1 Atomistic MD, 2 Brownian dynamics of rods with torsional

and bending potentials, and 3 Brownian dynamics of bead-spring chains, that could inprinciple encompass the entire range of dynamics, from femtoseconds to seconds. Use of

overlapping time and length scales would allow the parameters of the coarse-grained

methods to be assigned by matching their predictions to those of the finer-grained mod-

els. It is even possible in principle to refine the modeling all the way to the quantum

level, if one restricts oneself to small enough portions of the macromolecule.

The need for such multiple-method, multiple-scale, modeling efforts is especially

acute in the biological world where very large molecules DNA and proteins interact

with each other via dynamics that vary from slow and long range to very fast and short

range, as occurs, for example, in DNA transcription, which involves both fast local

proteinDNA binding events, and much slower diffusive motions of the bulky DNA and

protein molecules. Some examples of what simulations can tell us about the physicome-

chanical properties of DNA and protein molecules can be found in Lavery et al. 2002.

A more mundane example of the vast range of time and distance scales that can be

important in the dynamics of dilute polymer solutions is that of flow-induced polymer

scissionKeller and Odell 1985; Islam et al.2004, where a local event bond breakage

governed by quantum-level forces acting on the femtosecond time scale, is driven by

flow-induced unraveling of a long polymer molecule on a time scale of milliseconds or

seconds. Modeling accurately such a process across the full range of time scales remains

an important unsolved, but possibly solvable, problem. Yet another problem area where

multiscale models are probably needed is that of polymers interacting with surfaces,

where both fast locally controlled surface binding events and slow chain motions interact.

A simple example is provided by a polymer confined to a gap comparable to the persis-

tence length or Kuhn length of the chain. In that event, the normal bead-spring model

does not predict the correct entropic spring force owing to the steric restrictions prevent-

ing the Kuhn steps in the subchains composing the springs from sampling a full three-dimensional ensemble Woo and Shaqfeh 2003. The bead-rod model may be a more

appropriate model for this situation.

At the larger time and length scales of primary interest here, guidance in choosing the

most appropriate level of coarse graining is provided by considering the modes of relax-

ation captured by bead-spring chains. The inverse characteristic frequencies of these

modes are the discrete relaxation times i . The simplest bead-spring model of a poly-mer molecule, namely the Rouse model, neglects both HI and EV forces and assumes

Hookean springs; for the Rouse model the mode relaxation times are given by Doi and

Edwards 1986:

8 LARSON



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i

16kBTs2 sin2i

2N

N2

4

2

kBTs

2

i

2

coilR20

6

2

kBTi

2, 9

where is the bead drag coefficient, which for spherical beads is given by

6sa, with s as the viscosity of the solvent, a as the bead radius, coil N as

the total drag coefficient summed over all the beads, and s2

3/2NK,sbK2

describes the

elasticity of a single spring. The approximation for the sine function used in Eq. 9

becomes accurate when N/i 1 and, in this limit, the relaxation times of the higher

modes decrease as 1/i2 with increasing mode number i. Therelaxation strengthsG i areall equal to each other in the Rouseor Zimmmodel; i.e., G i G vkBT, where v is

the number of molecules per unit of volume of solution. For any model of polymer chain

dynamics, once thespectrumof relaxation times and corresponding strengths is given, the

linear viscoelastic storage and loss moduli can be computed as

G i

Gii2

1i2

; G i

Gii

1i2

. 10

At a given oscillation frequency, , the storage and loss moduli are dominated by the

relaxation times, i, that are of order of, or larger than, the inverse of that frequency .

Modes with relaxation time constants, i, that are much smaller than the inverse fre-

quency,1, are not excited by the flow and do not contribute appreciably to the stress.

Experimentally, the longest of these relaxation times, 1 , is related to theintrinsic

viscosity 0 of the polymer by

1 0Ms

S1RT, 11

where M is the polymer molecular weight, R NAkB is the gas constant, and NA is

Avogadros number. For the simplest models, such as those of Rouse and Zimm, where

the relaxation strengths are all equal, the coefficient S1 is related to the distribution of

relaxation times by

S1ii

1. 12

For the Rouse model, one can show that S1 2/6 1.645, while for realistic poly-

mers that are influenced by HI and EV effects, S1 varies from around 2.369 Zimm

19562.387 Ottinger 1996b in theta solvents to values closer to the Rouse value,

1.645, in good solvents. We note here that an oft-used characteristic relaxation time 0 is

defined by

0 0Ms

RT. 13

From Eq. 11, we find that the time constant 0 is larger by a factor of S1 than the

longest relaxation time; i.e.,

0 S11 . 14

In general, for a monodisperse flexible polymer molecule described by a spectrum

with equal relaxation strengths, the mode relaxation times are distributed roughly as a




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power law in the mode numberi: i ip, where by convention the modes are ordered

so that the higher the mode number, the shorter the relaxation time. The exponent p

equals 2 for the Rouse model, but for real polymers, which are affected by HIs and EV,p ranges between 1.5 and 1.8 as the solvent quality varies from theta to very good. In

general, in an oscillatory flow, if one wishes to capture modes with frequencies up to

some frequency , then the bead-spring model must be able to describe all modes with

relaxation times down to at least 1. However, a bead-spring chain with Ns springs has

only Nsmodes, at least in the simplest approximate solutions, such as those of Rouse and

Zimm. More accurate solutions to the bead-spring model can have many more modes,

for example as many as Ns2

modes, and the relaxation strengths are typically then not all

equal Zylka and Ottinger 1992. If the highest of these mode numbers must have a

relaxation time of 1 or less, then this sets a minimum number of springs that must be

included in the model. However, there is not only a minimum, but also maximum number

of springs that ought to be used to represent a particular polymer molecule. This is

because the spring force laws fs(r) are derived in the asymptotic limit of many random-

walk steps or Kuhn steps NKper spring. In practice, we cannot increase the number ofspringsNs beyond the point where NK,s NK/Ns drops below 10 or so Underhill and

Doyle 2004. Thus, if the rate of deformation is fast enough to significantly distort the

molecular configuration at a length scale of only a few Kuhn steps of the chain, then no

simple asymptotic spring law can capture the dynamics.

C. Equations of the bead-spring model

The equations of the bead-spring model have been presented in numerous books and

articles; to fix our notation, and bring readers up to speed with what is needed later, we

review them very briefly here. Since the inertial forces on a polymer molecule are almost

always negligible, a force balance on each bead i in a bead-spring chain includes only the

drag, elastic spring, and Brownian forces, yielding

Fidrag

Fisp,b

FiB

0. 15

1. Drag force

The drag force on a bead is given by

Fidrag

riv ri , 16

whereis the bead drag coefficient per bead, ri is the position vector, ri dri/dtis the

velocity of the bead, and v(ri) is the velocity of the solvent at the position of bead i. For

a uniform velocity gradient with zero velocity at the origin, v(ri) ri, where is the

transpose of the velocity gradient tensor; i.e., (v) T. Substituting these expressions

into Eq. 15 and rearranging gives

d

dtri ri

1

Fi

sp,bFi

B. 17

Because the Brownian force FiB

is a random force, the above is a stochastic differential

equation, otherwise known as a Langevin equation. While it is beyond the scope of this

review, we note here that when nonlinearities are present there are subtle issues associ-

ated with the rigorous formulation of stochastic differential equations, alternative inter-

pretations of which are those of Ito and of Stratonovich, as is explained by O ttinger

1996b.

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2. Spring force

The total spring force acting on any bead i, other than an end bead ( i 1,N), is

Fisp,b

Fisp

Fi 1sp , 18

where Fisp

is the force that spring i exerts on bead i. For the end beads,

Fisp,b

F1sp

, FNsp,b

FN 1sp

. 19

The spring force is a function Fisp

Fisp

(Ri) of the extension R i of the spring, where

Ri ri 1ri is the distance between bead i and bead i1; Ri ri 1ri is thespring vector. The simplest spring force law is that for a Hookean, or linear, spring, which

is

Fsp

R HR Hookean spring, 20

where the linear spring constant His given by

H 2kBTs2

; with s2

3

2NK,sbK2

. 21

A Hookean spring is infinitely extensible; i.e., the force remains bounded for any finite

extension of the spring. A more realistic spring law should produce an asymptotically

large force when the spring is stretched to its full extension Ls b KNK,s , where Ls is

the fully extended length of a piece of chain represented by a single spring; i.e., Ls L/Ns. For a freely jointed chain, a statistical mechanical calculation yields for the

spring force law the inverse Langevin function Bird et al. 1987, L 1( ) :

Fsp

R kBT

bK

L 1

R

Ls

. 22

Here, the Langevin function is given by

L coth1

, 23

and coth (ee)/(ee) is the hyperbolic cotangent. The inverse-Langevin

spring law is plotted in Fig. 2. Note that the force grows linearly with extension for small

and modest extensions; i.e., the spring law reduces to the Hookean form for fractional

extension R/Ls less than around one-third. However, for large extensions, the force

grows rapidly, approaching a singularity at full extension, R/Ls 1.

Since the inverse of the Langevin function is not an analytic function of the spring

end-to-end vectorR, analytic approximations to it are frequently used. One such approxi-

mation is the Warner spring law:

Fsp

R 2s

2kBT

1 R/Ls 2

R H

1R/Ls2

R . 24

A more accurate analytic force law is the Cohen 1991 Padeapproximation:

Fsp

R H 3 R/Ls

2

3 1R/Ls2

R, 25




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which much more accurately approximates the inverse Langevin force law over the

whole range of extensions; see Fig. 2.

DNA and many other biopolymers have helical backbones that are bendable but do not

undergo the large torsional bond rotations about individual bonds that many synthetic

polymers do. The DNA double helix can experience torsional strain, however, causing

overwinding or underwinding of the helix. The backbone distance over which a semi-

flexible rodlikepolymer is rigid is called its persistence length p

. If the molecular length

L of a semiflexible rodlike polymer is much longer than its persistence length, then the

gross conformation of the molecule at equilibrium is that of a freely jointed chain with

NK L/ (2 p) Kuhn steps each of length ofb K 2p. Locally, semiflexible rod mol-

ecules are not well described as links with free hinges, but are better described by the

wormlike chainYamakawa 1971, which is a flexible thin rod. The statistical mechanics

of the wormlike chain model lead to a force law that is well approximated by the

followingMarkoSiggia spring law Marko and Siggia 1995:

Fsp

kBT

p1

41 R

Ls

2

1

4

R

Ls 2

3HLs1

41 R

Ls

2

1

4

R

Ls , 26

whereH

3kBT/ (b K2

NK)

3kBT/( 2Lsp) is again the force coefficient in the linear,Hookean, regime. Although usage of the MarkoSiggia formula is now well established,

a somewhat more refined approximation can be found in Bouchiat et al. 1999. The

MarkoSiggia force law, like the inverse Langevin force law, has a singularity at full

extension; see Fig. 2. The singularity in force makes the Langevin equation 17 stiff, and

hence small time steps must be taken in fast flows if the simplest, explicit, integration

schemes are used. However, Hechen and Ottinger 1997 introduced a semi-implicit

predictorcorrector method for solving the stochastic differential equation for the bead-

spring model that allows much larger timesteps to be taken. This method was refined by

Somasiet al. 2002 and extended by Hsieh et al. 2003 to allow inclusion of HI.

FIG. 2. Elastic spring force versus molecular extension for the freely jointed chain which is given by theinverse Langevin function, Eqs. 22and 23, the Warner spring, Eq. 24, the Cohen Padeapproximation to

the inverse Langevin function, Eq. 25, and the MarkoSiggia force law, Eq. 26. Note that the inverse

Langevin function and the Cohen Pade function are almost identical. The normalized elongation is the

end-to-end distance R divided by the fully extended distance L.

12 LARSON



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The predictions of the bead-spring model are rather insensitive to the form of the

spring law for steady-state flows or start-up flows, Hur et al. 2000, but are sensitive to

the spring law for relaxation from a highly stretched state Schroederet al.2004; Shaqfehet al. 2004.

3. Brownian force

The random Brownian force fluctuates extremely rapidly, on the time scale of pico-

seconds, which is the rate of thermal bombardment of the polymer by the solvent. Over

a somewhat longer time scale of nanoseconds, these bombardments establish a Maxwell-

ian distribution of molecular velocities. Over a somewhat longer, but still very short, time

scale of perhaps microseconds, there are so many random bombardments that the random

force imparted by them averages toward zero, except for a small drift force whose

magnitude gets smaller with the inverse square root of the averaging time. This is the

usual statistical rule for averaging out errors in random processes. Thus, the Brownian

force averaged over a time scale, d t, can be represented by a random number with zero

mean, distributed according to

FB 6kBTdt

1/2

n, 27

wheren is a random three-dimensional vector, each component of which has a magnitude

uniformly distributed between 1,1, generated from random numbers Grassia and

Hinch 1996. The square root of the time step,d t, appears in the denominator of Eq. 27,

as expected from the statistical rule for averaging noise, and the factors of kBT and

appear because of thefluctuationdissipation theorem, which relates Brownian motion to

drag force. A relationship must exist between Brownian motion and drag because the rate

of Brownian motion determines the diffusion coefficient, which is also reflected in the

magnitude of the drag coefficient. A more general expression of the fluctuation

dissipation theorem is Kubo et al. 1985:

FBt 0,

28FBtFBt 2kBTtt,

whereFB( t) orFB( t) is a one-dimensional component of the random force at times tor

t, respectively, and (t t) is the delta function. The factor of 2 in Eq. 28 is carried

over into Eq. 27, as is an additional factor of 3, which is the inverse of the average of

the square of a random number that is distributed uniformly over the interval 1,1. The

product of these two factors is the factor of 6 in Eq. 27.

The above provides a stochastic form for the Brownian force. An alternative method

of obtaining the Brownian force is to work with a molecular configuration distribution

function,(ri ,t). This function is defined such that (r

i ,t)dr

iis the probability that

at time t a randomly chosen chain has bead coordinates that lie between ri and ri dri, where dri is a small interval in multidimensional coordinate space. Thus,

c(ri ,t)dri is the mass concentration of chains whose bead positions lie within theinterval dri at time t. Now the Brownian force, once it is averaged over all chainswhose bead coordinates lie within this interval, will be zero, except for a drift force

FiB which is left over because (ri, t) is not quite uniform over the interval. The

brackets denote an average over the Maxwell velocity distribution at fixed beadpositions. As a result of this drift force, Brownian motion will carry chains from con-

figurations that are relatively numerous compared to their equilibrium frequency, to those




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that are less numerous. A statistical analysis shows that this Brownian drift term is

exactly analogous to the flux, per molecule, produced by the ordinary diffusion of small

molecules; that is, the drift force is given by

FiB kBT

rin, 29

where the analogous term for small molecule diffusion is the gradient in chemical poten-

tial, as described in standard textbooks.

Inserting the stochastic form of the Brownian force, Eq. 27, into Eq.17, we obtain

the simplest form of the Langevin equation for the bead-spring model:

dri

dt ri

1

Fi

sp,b6 kBT

dt

1/2

ni. 30

Integrating this equation numerically, using a random number generator to produce the

sequence of random vectors n, gives the evolution of bead coordinates for a singlerealization of a polymer molecule. To obtain the collective behavior of a dilute solution of

many molecules in a transient flow, one must solve this equation many times, each time

with a different sequence of random numbers to represent a different sequence of random

Brownian forces exerted on a different molecule. Then, these multiple realizations of the

chain dynamics can be averaged together to yield the ensemble-averaged behavior. For

steady-state flows, one can instead average a single set of equations representing one

molecule for a long period of time.

Alternatively, substituting the velocity-averaged Brownian force, Eq. 29, into Eq.17 yields

ri ri1

Fi

sp,b

kBT

ri

n. 31

This equation cannot be solved by itself; we must combine it with the conservation

equation for probability:

t

ri ri

ri ri 1

F

sp,bkBT

ri . 32

This is the Smoluchowski equation for the probability distribution function (ri, t),a function of all coordinates of all beads, and time. It is impractical to solve for this

function numerically. However, one can take moments of the equation, such as the second

moment, which, after appropriate closure approximations, can be put into a closed form.

The derivation of these moment equations can be found in standard references Birdet al.

1987; Larson 1988; Ottinger 1996b. These closed-form equations are much cheaper to

solve computationally than the Langevin equation, which requires averaging over en-sembles of hundreds or thousands of molecules to obtain accurate results. However, the

Langevin equation readily admits nonlinear phenomena, which in moment equations

require closure approximations of sometimes dubious accuracy. Hence, recent advances

in computer power have spurred the use of the Langevin equation, especially for the

prediction of nonlinear phenomena, where the simplest and cheapest closure approxima-

tions are especially dubious. For the prediction of linear viscoelastic phenomena, the

moment equations are often quite accurate, however, especially when Gaussian approxi-

mations are used to include fluctuation effects Ottinger 1989; Wedgewood 1989; Ot-

tinger and Zylka 1992. In fact, in the linear viscoelastic regime, moment equations are

14 LARSON



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sometimes preferable to Langevin equations, because the latter require extensive averag-

ing to reduce noise, which in the linear viscoelastic regime is very large relative to the

weak signal. While variance reduction methods can be very effective in reducing noisein solutions to the Langevin equation Melchior and Ottinger 1996, nevertheless the

moment equations may, in many cases, still provide the most accurate predictions of

rheology in the linear viscoelastic regime.

4. Hydrodynamic interaction

As mentioned in Sec. II, HI is always important for long, flexible, polymer molecules.

To account for HI using the bead-spring model, the disturbance to the velocity field

produced by the forces that each bead exerts on the solvent must be used to correct the

velocity acting on every other bead. By Newtons second law, the net hydrodynamic

force exerted by bead j on the solvent is equal and opposite the hydrodynamic drag force

Fjd

, exerted by the solvent on bead j. But this hydrodynamic force must be balanced by

the nonhydrodynamic spring and Brownian forces exerted on that bead, so that Fj

d

(Fjs

FjB). This force produces a disturbancevi to the velocity field at the position

ri of another bead i; this disturbance is a linear function of the hydrodynamic force Fjd

exerted by that bead:

vi i jFjd

i j Fjs

FjB

, 33

where i j, the hydrodynamic interaction tensor, is a function of the separation rirjbetween the two beads. The direct measurement of the disturbance velocity created by

movement a bead was recently achieved by Meiners and Quake 2000. When the dis-

turbance velocity given by Eq. 33 is incorporated into the stochastic differential Eq.17, one can show that Ermak and McCammon 1978:

dri

dt rij 1

ND

i j

Fj

sp,b

kBT

6

t1/2

j 1

i

i jnj, 34

where Di j (kBT/)i jIi j is the diffusion tensor. The tensor i j is introduced

into the term for the Brownian motion in the above because of the fluctuationdissipation

theorem relating the diffusivity of a bead to the magnitude of the random forces that act

on it. The overall diffusion tensor D and weighting factor are fourth-order tensors,

which means that each componentDi j of the diffusion tensor and each component i jof the weighting-factor tensor is itself each a 33 tensor. The fluctuationdissipation

theorem implies a square root relationship between Di j and i j:

Di j l 1

iljl . 35

A formula relating the components of i j to those ofDi j that satisfies Eq. 35 can be

obtained by a Cholosky decomposition as follows Ermak and McCammon 1978:

D 1

1

2

1/2

,

D 1

1

, , 36




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0, .

Here,,,indicate the row and column positions of the element in the overall fourth-

rank tensorsD and which, for example, are 9 by 9 tensors for three-bead chains. The

calculation of the tensor i j by the Cholesky decomposition is computationally very

expensive. Jendrejacket al. 2000 and Kroger et al. 2000 have implemented a faster

scheme developed by Fixman 1986 that takes advantage of the fact that in Eq. 34 only

j 1i

i jnj , not i j itself, appears, and the former quantity can be computed more

quickly than the latter, unless the excluded volume interaction exceeds a critical strength,

in which case the Cholesky decomposition is faster; for details, see Jendrejacket al.2000.

There is also a Smoluchowski form for the force balance with HI, namely:

ri ri j

Di j 1kBT

Fjsp,b

rjn . 37

Because the diffusion tensor Di j depends nonlinearly on the separations between all the

bead coordinates, it is difficult to develop closed-form moment equations from the

Smoluchowski equation that are valid in the nonlinear flow regime, when HI is present.

For the linear viscoelastic regime, Zimm 1956 preaveraged the diffusion tensor over the

known equilibrium distribution of configurations, leading to closed-form moment equa-

tions. The solutions to these equations will be discussed shortly. In the nonlinear vis-

coelastic regime, Ottinger1987introducedconsistent averagingof the HI tensor, which

allows it to change as molecules are deformed, but these changes are averaged over the

entire ensemble of molecules, thereby ignoring fluctuations in HI produced by fluctua-

tions in chain deformation. If consistent averaging is combined with a simplified and

apparently very accurate decomposition into normal modes introduced by Fixman1966, simulations with hundreds of beads can easily be carried out Magda et al.

1988b; however, in these simulations, fluctuations in HI are neglected. Fluctuations canbe included within moment equations using a Gaussian approximation for the distribu-

tion functions Ottinger 1989; Wedgewood 1989; Prakash 2002. Thorough comparisons

of moment equations with the Gaussian approximation against Brownian dynamics simu-

lations of Langevin equations for multispring chains with multiple sources of nonlinearityi.e., HI, EV, and nonlinear springs should be carried out to assess the accuracy and

computational efficiency of the former, relative to the latter, in simulations of real poly-

mers. For shearing flows at least, the Gaussian approximation shows very good agree-

ment with results from Brownian dynamics simulations for Hookean bead-spring chains

with HI Zylka 1991.

Forms for the diffusion tensor. There are several suggested forms for the diffusion

tensor Di j , which are approximations to the HI mediated by the fluid. The simplest of

these, theOseenBurgers tensorOseen 1927; Burgers 1938; Bird et al.1987, is derivedby assuming that the beads are well separated enough from each other that they can be

regarded as point sources of drag on the solvent:

Dii kBT

6saI,

38

Di j kBT

8sR i jI Ri jRi j

R i j2, i j ;

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whereI is the unit tensor and Ri j is the separation vector between the ith and jth beads,

Ri j rj ri . The OseenBurgers tensor is not suitable for Brownian dynamics simula-

tions, because it becomes nonpositive definite when bead separations are comparable to,or less than, the bead radius.

A better approximation, that accounts in a crude way for the finite size of the beads, is

the RotnePrager tensor, in which one expression for bead-separation distances larger

than twice the radius of the bead, and another for separations less than this Rotne and

Prager 1969; Zylka and Ottinger 1989:

Dii kBT

6saI,

39

Di j kBT

8s

1

R i j 1

2a2

3R i j2I1 2a

2

Ri j2 Ri jRi j

Ri j2 , for Rij 2a

Rij

2a83 3Rij4aIR i j4a Ri jRi jR i j2 , for Rij 2a; i j;

Note that as the ratio of bead size to bead separation a/R i j becomes small, the Rotne

Prager tensor, Eq. 39, reduces to the OseenBurgers tensor, Eq. 38. The strength of

the HI between beads can be indexed by the following parameter

h*

s H

363kBT

1/2

123

1/2

Rss

3

a

Rs, 40

whereH 3kBT/Rs2

is the elastic constant of the spring, 6sa is the drag coef-

ficient of a sphere of radius a, and Rs b KNK,s is the root-mean-square end-to-endvector of a spring at equilibrium. The largest reasonable value for a/R

s is around 0.5,

since for higher values the bead radius would be one-half of the average spring length

and the beads would therefore, on average, overlap. Hence, values of h* larger than0.53/ 0.49 are not physically reasonable. As we shall see later, a value of h* 0.25 gives predictions that are insensitive to the number of beads used and in agree-

ment with experimental linear viscoelastic data for polymers in theta solvents. For ex-

ample, the experimental value of the FloryFox parameter for theta solvents, 2.5

1023, can be derived from the Zimm theory defined below by setting h* 0.267,which is close to the special value h* 0.25 Bird et al. 1987.

Note, from the definition ofh *, that one can rewrite the RotnePrager tensor as




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Dii kBT

6saI,

41

Di j 3

4h*kBT

kBTH

6sa

1

R i j

1

2a2

3R i j2I1 2a

2

Ri j2 Ri jRi j

R i j2 , for Rij 2a

Rij

2a83 3Rij4aIRi j4a Ri jRi jRi j2 , for Rij 2a; i j;

In this form, one can choose to reduce or turn off HIs simply by arbitrarily reducing thevalue ofh * below that given by Eq. 40, or even by setting h * to zero. If one does so,then the form for the HI included in the Brownian dynamics simulations is no longer

consistent with that expected for spherical beads. Since the assumption of spherical beads

is arbitrary, there might be pragmatic reasons for treating h* as a separately adjustableparameter. One cannot, however, increase h * above the value given by Eq. 40, for thiswould make the tensor Di j nonpositive definite, and its square root could then not be

obtained, thus precluding Brownian dynamics simulations.

5. Excluded volume

For good solvents, a repulsive force Fev between the beads must be added to Eq. 17.

The repulsive force can be given by the gradient of a potential Uev(R); i.e.,

Fiev

j

riU

ev ri rj . 42

For simulations of small molecules, one normally imposes a steep repulsive potential,

such as that of the famous Lennard-Jones potential, whose repulsive part is a twelfth

power in the separation of the centers of mass of the molecules. A Lennard-Jones poten-

tial has in fact been used to simulate excluded volume effects in bead-spring models of

polymers Lopez Cascales and Garca de la Torre 1991. However, the steepness of the

repulsive potential should have little effect on the coarse-grained configurations of the

polymer chains. Hence, for simulations of polymers, it suffices to impose softer poten-

tials, which then allow larger simulation time steps than do steep, hard, repulsive poten-

tials, such as that of Lennard-Jones. It is convenient, then, to use an excluded volume

force that remains bounded, so that forces never become singular. A potential that decaysexponentially with separation serves this purpose, namely Jendrejacket al. 2002:

Uev

R 1

2vkBTNK,s

2 92Rs

23/2

exp9R22Rs

2, 43where, again, R s

2 NK,sb K

2is the mean-square end-to-end length of a spring. A similar

potential had been proposed earlier for polymers by Ottinger1996a, b; see also Prakash

and Ottinger 1999; and Prakash 2002:

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Uev

R z*

d*

3kBTexp

HR

2

2kBTd*

2

. 44

The two potentials are identical if in the Ottinger form we make the identification that the

dimensionless range of the potential is d* 1/3, and its dimensionless strength isz* 1/2(3/2) 3/2NK,s

1/2v/b K

3. The interaction volume v is most appropriately defined as

a microscopic volume, proportional to the volume occupied by a single Kuhn step of the

chain, that remains invariant for a given polymer/solvent combination as one changes

either the molecular weight of the polymer and henceNK) or the number springs used to

represent that polymer and hence NK,s). The influence of the EV on rheological prop-

erties grows with the parameterz z*Ns1/2

v/b K3

NK1/2

, until a saturation is reached at

high values ofz Prakash 2002. For shorter chains below the saturation condition, both

parameters z* or, equivalently, v) and d* affect the bead-spring model predictionsPrakash 2002. The EV force can be incorporated into either the Langevin or the Smolu-

chowski form of the force-balance equation. As usual, incorporation into the Smolu-chowski form necessitates invoking closures to obtain solutions.

The antithesis of EV interactions is polymerpolymer attraction, or, equivalently,

polymersolvent repulsion. If polymerpolymer attraction is weak, then its effect is

typically assumed to be equivalent to a weakening of the EV interaction, and is usually

modeled only in this indirect way. Of course, if the polymerpolymer attraction or

polymersolvent repulsion becomes strong enough, one crosses the theta point, and the

polymer dimension will shrink below the theta size, leading to a collapsed coil. Sol-

vents that are so bad that the coils collapse below theta dimensions usually produce

precipitation of the polymer, unless the polymer molecular weight is low or its concen-

tration is very low, below the overlap concentration. To model such a situation, one must

include net attractive interaction between beads. Simulations with bead attraction yield

collapsed coils that, when stretched, unravel abruptly into stretched filaments under

constant-force conditions and into ball and chain or tadpole configurations under

constant-stretch conditionsHalperin and Zhulina 1991; Cooke and Williams 2003. This

behavior is akin to a first-order phase transition, with the ball and chain state resembling

a coexistence of coil and stretched states, but within the same molecule. Stiff polymers in

poor solvents avoid completely collapsed states because of the penalty for molecular

bending, and instead at equilibrium are predicted to form exotic shapes, such as equilib-

rium torii and nonequilibrium racquets Schnurr et al. 2000; 2002; Montesi et al.

2004.

Experimentally, polymerpolymer attraction can be produced not only by use of a

poor solvent, but also through electrostatic effects, for example, by use of multivalent

counterions in polyelectrolyte solutions. The multivalent ions bridge two like charges on

the polymer chain, causing the chain to attract itself. This phenomenon is used to con-

dense semiflexible DNA molecules into a compact torroidal shapes for transport into cellsfor gene therapy, for example. Because of the free energy of the counterions, polyelec-

trolyte systems show complex behavior even at equilibrium. Such phenomena are beyond

the scope of this review, but the interested reader can pursue this topic through the

literature citations in a recent article by Zherenkova et al. 2003.

Finally, it might also be worth noting that polymerpolymer attraction typically is

longer ranged than excluded volume repulsion, even for uncharged polymers. Hence,

even when the attraction is not strong enough to collapse the chain below theta dimen-

sions, it is in principle possible that under some situations it might not be accurately

modeled simply by dialing down the strength of the EV effect. As an example, one could




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in principle generate the theta state in a bead-spring model by including both short-

ranged beadbead repulsion and long-range beadbead attraction, with magnitudes bal-

anced so as to reproduce the theta state. While the equilibrium statistical properties ofsuch a chain would hardly differ from one in which both the repulsive and attractive

interactions are neglected, it is possible that this will cease to be the case in highly

nonequilibrium situations. In general, it is fair to say that both EV interactions, and,

especially, polymerpolymer attractions, in highly nonequilibrium states, have received

insufficient theoretical attention.

6. Internal viscosity

It has long been theorized that resistance to deformation of a polymer molecule might

arise not only from the elasticity of the molecule and from the friction between the

polymer and the surrounding solvent, but also from friction within the polymer itself.

That is, there might be a dissipative force generated by bond rotations and other motions

required for the polymer to change its configuration. Such friction is called IV. One

would expect the presence of such a frictional force to be manifested by a contribution tothe stress that is dissipative, that increases with increasing flow rate, and that is indepen-

dent of the viscosity of the solvent. For the bead-spring model, a simple form for the IV

force that meets these criteria was suggested long ago by Kuhn and Kuhn 1945:

Fiv RR

R2

R, 45

where is a phenomenological coefficient that in principle should be independent of

solvent viscosity and only depend on the polymer type and possibly its molecular weight.

This form, when linearly added to the spring force in Eq. 17, treats the connector

between two beads as a spring and dashpot in parallel, so that rapid changes in spring

length produce both a purely elastic and a purely dissipative response. An alternative

formula was proposed by Cerf1957.

The need for such a term in a bead-spring model has never been conclusively dem-

onstrated; indeed, attempts to find a clear rheological signature of such a term have so far

failed Fuller and Leal 1980; Larson 1988. However, recent efforts to model the dynam-

ics of chromosomes, which are DNA bundles held together by proteins, may revive

interest in internal viscosity models Poirier and Marko 2002. Evidence for internal

viscosity was once thought to be manifested in the high-frequency viscoelastic response

of dilute polymer solutions, where a difference between the experimental dynamic vis-

cosity and the high-frequency prediction of the bead-spring model was found for some

polymers Massa et al. 1971; Ferry 1980. However, further investigation revealed that

the sign of this additional discrepancy was occasionally negative Morris et al. 1988,

which obviously could not be caused by addition of a dissipative and hence positivecontribution to the stress. By examining spectroscopically the motion of the solvent

molecules themselves, it was finally realized that the discrepancy was caused by a

polymer-induced modification to the dynamics of the solvent, which can be thought of as

a modification of the solvent glass transition temperature Morris et al. 1988; Lodge

1993. A polymer such as polystyrene that has a higher glass transition temperature Tgthan the solvent in which it is dissolved raises the Tg of solvent in its vicinity, making it

more viscous. Polybutadiene, on the other hand, has a Tg that is sometimes lower than

that of the solvent it is dissolved in, and the measured high-frequency viscosity in this

case is found to be lower than predicted; i.e., the deviation is negative in sign. Thus,

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high-frequency deviations from the predictions of the ordinary bead-spring model should

not automatically be attributed to viscous contributions arising from motions of the poly-

mer.Although at this point, there is no good reason for considering internal viscosity to be

of fundamental significance in the dynamics of flexible polymers, one might attempt to

use an IV model to represent crudely the high-frequency effects left out of a coarse-

grained bead-spring model. An IV model has been used, for example, to represent the

retarding effect of internal folds or polymer kinks on the unraveling dynamics of

polymers in strong extensional flows Larson 1990.

7. Self-entanglements

Another physical phenomenon sometimes entertained is that of SEs or intrachain

entanglements, in which an isolated polymer forms a knot, which restricts its ability to

unravel in a fast flow. Interchain entanglement is a well-established phenomenon in

concentrated polymers, even though a clear definition of an entanglement between twoor more chains is still lacking. Brochard and de Gennes 1977 have estimated that in a

good solvent knots should almost never occur, while in a theta solvent typical polymer

molecular weights would need to be in the millions before even a single knot becomes

likely at equilibrium. For an isolated chain, entanglements, or knots, can be precisely

defined for ring polymers, and one can use simulations of freely jointed ring polymers to

give an idea about the frequency of knot formation in linear polymers. Michels and

Wiegel 1986 find that the fraction of unknotted random-walk rings drops off roughly

exponentially with the number of Kuhn steps, such that at 300 Kuhn steps less than 40%

of the chains are unknotted. Similarly, Ten Brinke and Hadziioannou 1987, in a lattice

simulation of random-walk chains, found 40% unknotted configurations for rings with

160 steps, and noted that this length corresponds to a molecular weight of 120 000 for

polystyrene. Self-avoiding walks, on the other hand, impose an EV constraint, and are

much less likely to be knotted than random-walk chains. For example, Yao et al. 2001found only a 0.4% knotting probability in self-avoiding closed walks with 1000 steps on

a cubic lattice, and estimated that knots would only become prevalent for chains contain-

ing 2.5105 steps, corresponding to a polymer molecular weight in the hundreds of

millions. The large difference in knotting probability for random-walk versus self-

avoiding walks confirms the early estimates of Brochard and de Gennes 1977.

In linear chains, knots are not permanent, and their effect on polymer dynamics is

unknown. Also unknown is the effect of shear or extensional flow on knot formation, and

whether or not tumbling motions in shear might generate many more SEs than occur

under no-flow conditions. In long DNA chains whose ends are attached to beads that can

be manipulated by optical tweezers, self-knots can deliberately be created, and the move-

ment of the knots along the chains have been studied and found to obey simple diffusive

rules Bao et al. 2003. Experiments have not yet provided any direct evidence for any

effect of internal knots on rheology, although the failure to achieve the theoretical high

plateau extensional viscosity in filament-stretching and other extensional flows has been

attributed to the arrest of polymer stretch in these flows due to putative SEs James and

Sridhar 1995.

D. Stress tensor

Once the dynamics of the chain have been solved, one can compute macroscopic

quantities, such as stress, birefringence, scattering, or other observables. The most im-




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portant of these is the stress tensor . The polymer contribution to the stress tensor is

given, in general, by the Kramers expression Bird et al. 1987; Larson 1988:

p vi 1

Ns

Fis

Fiev

Fiiv

RiNsvkBTI, 46

where I is the unit tensor and v is here. The number of polymer molecules per unit

volume. The terms Fev and Fiv can be dropped when excluded volume and internal

viscosity, respectively, are absent. Further discussion of the formulas for the stress tensor

can be found in Larson1988and Bird et al.1987. A method to calculate the birefrin-

gence from the bead-spring model can be found in Li and Larson 2000a.

E. The bead-rod model

We have so far concentrated mainly on the bead-spring model, which is the least

expensive computationally for long-chain polymers, and consequently the most fre-

quently used. Within the considerations discussed in Sec. IV B, however, the bead-rodmodel is sometimes a more accurate model. Early Brownian dynamics simulations for a

bead-rod polymer chain were presented in a pioneering study by Acierno et al. 1974

and by Liu 1989. Lius solution method, still prevalent Grassiaet al.1995; Grassia and

Hinch 1996; Doyle et al.1997, relies on the method of Langrange multipliers, which are

in fact tensions in the rods, required to keep their lengths fixed. The method was earlier

used to keep atomic bond lengths fixed in MD simulations ofn-alkanesRyckaert et al.

1977, and in the context of MD simulations is known as the SHAKE algorithm. In a

Langevin equation, the constraint forces replace the spring forces of a bead-spring model,

so that Eq. 17, for example, becomes

d

dtri ri

1

Fi

con,b Fi

B, 47

with

Ficon,b

Tiui Ti 1ui 1 , 48

where the vector u i is the unit vector connecting beads i and i 1; i.e., ui Ri/R i (ri 1ri)/R i . In Brownian dynamics simulations, the tensions Ti enter the formula

for stress, necessitating noise reduction schemes to smooth the stress, which would oth-

erwise fluctuate drastically to balance the equally drastically fluctuating Brownian forces

that pull on the rods. Time stepping must also be handled with care, where a midpoint

algorithm for calculating the rod orientation Fixman 1978; Liu 1989; Doyle et al. 1997

has been validated as a safe choice. A new method for maintaining bond length, called

LINCS, which is claimed to be three to four times faster than SHAKEHesset al.1997

relies on projection of the bead or atom motion into directions that respect the bond-length restrictions. Application of this method to polymer simulations might be worth

exploring.

The effects of HI on the behavior of bead-rod simulations has been explored by

Neelov et al. 2002. Because in a bead-rod chain, each bead corresponds to a single

Kuhn length of the polymer, and inclusion of HI becomes prohibitively expensive beyond

100 beads, the effective molecular weights that can be studied with the bead-rod chain

and full HI are still quite low. Nevertheless, with full HI, Neelov et al. 2002were able

to observe close to the expected scaling law c N 1.5 for the dependence on number

of beads Nof the critical extension rate c for a coil-stretch transition discussed more

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below. Both Fixman 1978 and Ottinger 1994 have proposed general methods of

including HI into bead-rod Brownian dynamics simulations. However, there are numer-

ous subtle issues involved in simulations of chains with rigid constraints, as discussed indetail by Morse 2004.

An initially puzzling observation is that the steady-state chain conformational distri-

bution that emerges from typical Brownian dynamics simulations for the bead-rod i.e.,

Kramers chain in the absence of flow is not quite that of a random walk. For a two-bead,

three-rod Kramers chain, for instance, there is a modest preference for a 90 angle

between the rods over an angle of 0 or 180 Gottlieb and Bird 1976. This strange result

comes about because in statistical mechanics the equilibrium distribution is in general a

distribution not only over configurations but over momenta as well. Although the mo-

mentum or velocitydistribution equilibrates very rapidly typically on ns time scales as

alluded to earlier in Sec. IV C 3, nevertheless, averaging over this distribution, in the

case of a bead-rod chain, leads to a nonuniform statistical weighting of the different

bead-rod configurations. Brownian dynamics simulations do not keep track of velocity

distributions, but collapse information on momentum into the white noise Brownian

force term. The bead-rod algorithm by Liu, perhaps coincidentally, achieves the correct

equilibrium weighting of chain configurations for the case of beads of equal mass, but

would fail to do so for beads of unequal mass Morse 2004.

While these subtleties may seem worrying, for chains of many rods, the overall con-

figuration retains random-walk statistics perhaps with an effective Kuhn step length that

is slightly different from the rod length and, in reality, no real polymer is a chain of

perfectly rigid links. In fact, one could replace each rod of the bead-rod chain by a very

stiff spring in which the force is zero at a nonzero length equal to the desired rod

length; this is a so-called Fraenkel spring Fraenkel 1952. Then, since the momentum

space for Fraenkel springs is Cartesian, the averaging over momentum variables becomes

trivial, and a random-flight configuration distribution is recovered, so that, surprisingly, a

chain of Fraenkel springs has a slightly different configuration distribution than its coun-terpart chain of rigid rods, and this difference remains no matter how stiff the Fraenkel

spring becomes and how small the fluctuations in spring length become. Whether a very

stiff Fraenkel spring or a rod better represents the behavior of a real polymer bond or

segment can only be settled by consideration of quantum mechanical effects controlling

bond vibrationsMorse 2004. At the coarse-grained level of many bonds, however, such

fine details wash out, and Kramers chain is usually a more sensible choice than is a chain

of Fraenkel springs, because stiff Fraenkel springs require very small time steps to re-

solve their vibrational modes, and are therefore computationally expensive. Remember

that Hookean or Fene springs differ from Fraenkel springs in that in the former, many

Kuhn steps are subsumed into a single spring, while a Fraenkel spring represents a single

Kuhn step length.

In principle, polymers of moderate molecular weight can be simulated equally well by

either the Fene bead-spring chain or the bead-rod or Fraenkel-springmodel. In practice,

for short polymers, with say less than 100 Kuhn steps, accurate predictions cannot be

obtained with the bead-spring model, since a Fene or Marko Siggia spring should

represent at least 1015 rods, so that the spring force law correctly represents the entropy

of the rod configurations within the spring. On the other hand, polymers containing many

more than 100 Kuhn steps cannot be simulated by the bead-rod model, because of the

computational load required in simulating a chain with so many beads and rods. Chains

with around 100150 Kuhn steps can be simulated by either method, and in these cases,

both the bead-spring model with 1020 springs and bead-rod model give roughly similar




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predictions Hur et al. 2000; Somasi et al. 2002. Comparisons of the predictions of

bead-spring and bead-rod models can be found in Somasi et al. 2002.

V. COMPARISON OF PREDICTIONS OF MODELS TO LINEARVISCOELASTIC DATA

The predictions of bead-spring models in the linear viscoelastic regime are usually

obtained by solving for the moments of the Smoluchowski equation. The cheapest meth-

ods computationally involve performing a normal mode analysis; see below. A normal

mode analysis is much cheaper computationally than Brownian dynamics simulations,

and, in the linear viscoelastic regime, is reasonably accurate. Also, in the linear viscoelas-

tic regime, where deformations and stresses are small, long periods of averaging are

required to obtain accurate results from solutions of stochastic Brownian dynamics simu-

lations, making BD simulations even less attractive. Using normal modes, moment solu-

tions of bead-spring polymer models were already obtained 50 years ago, and since then

thorough investigations of their predictions have been made, with both HI and EV effectsincluded. In recent years, the emergence of very fast computers has allowed better solu-

tions to be obtained, both through more accurate calculation of moments of the Smolu-

chowski equation, and through Brownian dynamics simulations of the Langevin equation.

Comparisons of solutions of moments of the Smoluchowski equation with predictions

from Brownian dynamics simulations of the Langevin equation for bead-spring chains in

the linear regime have generally yielded satisfactory agreement between the two Zylka

and Ottinger 1989

the rheology of dilute solutions of flexible polymers

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