SIMULATION OF DILUTE POLYMER AND POLYELECTROLYTE SOLUTIONS

SIMULATION OF DILUTE POLYMER AND

POLYELECTROLYTE SOLUTIONS:

CONCENTRATION EFFECTS

by

Christopher Gerold Stoltz

A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF

Doctor of Philiosophy

(Chemical and Biological Engineering)

at the

UNIVERSITY OF WISCONSIN – MADISON

2006

To my wife, Amy

For your love, your support, your patience,

and occasionally talking me down from the ledge.

Thanks to my advisors, Michael Graham and

Juan de Pablo, for the opportunities given

me and for their guidance in this research...

Thanks to former colleagues Richard Jendrejack (3M)

and Philip Stone (NIST) for many helpful discussions

on mathematics, programming, and politics...

Thanks also to the other members of the MDG

group with whom I’ve shared an office at UW -

Juan Hernandez-Ortiz, Mauricio Lopez, Wei Li,

Hongbo Ma, and Li Xi - for all your help and for

putting up with me for so long...

And finally, thanks to the members of the UW

Condor team, especially James Drews, Jeff Ballard,

Colin Stolley, Todd Tannenbaum, and De-Wei Yin for

developing and maintaining the computational

resources essential for conducting this work.

This work was supported through the NSF Nanoscale Modeling and Simulation

Program and the Univeristy of Wisconsin Nanoscale Science and Engineering Center.

The author was personally supported by an NSF Graduate Research Fellowship and a

University of Wisconsin Grainger Fellowship.

SUMMARY

This dissertation focuses on the use of computer simulations to study the effects of alter-

ing concentration on the bulk rheological behavior of dilute polymer and polyelectrolyte

solutions. This is accomplished primarily through the use of Brownian dynamics sim-

ulations, in which we coarse-grain the polymer structure into a simple bead-spring rep-

resentation that captures the essential physics at the mesoscopic scale, but allows us to

eliminate many degrees of freedom as compared to an atomistic simulation and thereby

greatly improve the speed of the algorithm. We also employ Monte Carlo simulations

for the calculation of equilibrium properties as this enables us to more rapidly sample

the available configuration space. Our work is novel in that, to our knowledge, this work

is the first use of long-ranged hydrodynamic interactions in a simulation of flowing bulk

polymer solutions at nonzero concentrations. We have focused on three distinct problems,

which we describe below.

To study the effects of concentration on the structural and rheological properties of

dilute polymer solutions, we have used the model of Jendrejack et al. (2002b) forλ-

phage DNA under good solvent conditions, which incorporates excluded volume and

hydrodynamic interaction effects, and has been shown to quantitatively predict the non-

equilibrium behavior of the molecule in the dilute limit. Our work covers the entire dilute

regime, with selected investigations into the semi-dilute regime, as well as spanning mul-

tiple decades of both shear and extensional flow rates. In simple shear flow, as much as

a 20% increase in chain extension and 30% increase in the reduced polymer viscosity is

observed at the overlap concentration, as compared to the infinitely dilute case. Addi-

tionally, predicted relaxation times and shear viscosities are in very good agreement with

experimental observations. In elongational flow, we observe much stronger concentration

dependences than in shear, with a 110% increase in chain extension and 500% increase in

reduced viscosity when results are compared at equivalent extension rates. Finally, sig-

nificant concentration effects are observed in elongational flow at concentrations as low

as 10% of the overlap concentration and are largely the result of interchain hydrodynamic

interactions.

In our study of polyelectrolyte solutions, we use a simple coarse-grained kinetic the-

ory model incorporating explicit counterions to represent the polyelectrolyte. Brownian

dynamics simulations are used to conduct a systematic analysis of the behavior of poly-

electrolytes in simple shear flows, and to explore the relationships between flow rate,

Bjerrum length, and concentration. It is found that the polyelectrolyte chains exhibit a

shear thinning behavior at highPe that is independent of the electrostatic strength due to

the stripping of ions from close proximity to the chain caused by the flow. In contrast, at

low values ofPe, systems at different values of the Bjerrum length exhibit very different

viscosities owing to differences in the conformation of the chains and their surrounding

ion clouds. Furthermore, the presence of the ion cloud causes the viscosity to increase

monotonically with increasing Bjerrum length over the range studied here in contrast to

the nonmonotonic trend of chain size with increasing Bjerrum length. Concentration is

demonstrated to have a significant impact on the rheological behavior of polyelectrolyte

systems, despite its limited influence on the structure of chains when a simple shear flow

is imposed. These observations are explained by a previously unreported mechanism

based on the structure and orientation of the ion cloud enveloping an individual chain,

and its impact on the bead-ion electrostatic interactions. Finally, we have also considered

the role of hydrodynamic interactions in these simulations, finding that for low concen-

tration studies in shear flow, electrostatic effects dominate the hydrodynamic effects and

we are able to capture the correct qualitative behavior while ignoring the hydrodynamic

interactions.

Finally, we have considered the simulation of non-Brownian self-propelled particles

in bulk solution. We use a primitive model, treating the “swimming” molecule as a simple

bead-spring dumbbell, with an external force applied to one end along the director vector

of the dumbbell to represent the mode of propulsion. In addition, we account for the far-

field hydrodynamic interactions, allowing us to study the hydrodynamic coupling of the

motions of the individual swimmers. From these simulations, we find that hydrodynamic

coupling between the swimmers leads to large-scale coherent vortex motions in the flow

and regimes of anomalous diffusion that are consistent with experimental observations.

At low concentrations, we observe the existence of small-scale coherent motions. As

concentration increases, these coherent motions change in intensity depending on the type

of propulsion mechanism, and in turn, significantly alter the dynamics of such swimmers.

In addition, we find distinct differences in the types of collective motions evident in

solution owing to the type of propulsion mechanism, and discuss the dependence of our

results on the size of the system considered in a given simulation.

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Contents

1 INTRODUCTION 1

2 PROBLEM STATEMENT 5

3 INTRA- AND INTERMOLECULAR INTERACTIONS 13

3.1 Intramolecular Connectivity . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Excluded Volume Interactions . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Electrostatic Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Hydrodynamic Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 SOLUTION OF THE KINETIC THEORY EQUATIONS - STATICS 34

4.1 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2 Interior Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3 End Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4 Reptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 SOLUTION OF THE KINETIC THEORY EQUATIONS - DYNAMICS 45

5.1 Euler Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Implicit Integration Methods . . . . . . . . . . . . . . . . . . . . . . . . 47

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5.3 Decomposition of the Diffusion Tensor . . . . . . . . . . . . . . . . . . . 58

5.4 Nonequilibrium Simulations . . . . . . . . . . . . . . . . . . . . . . . . 61

6 CONCENTRATION DEPENDENCE OF SHEAR AND EXTENSIONAL RHE-

OLOGY OF POLYMER SOLUTIONS 71

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.4 Equilibrium Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.5 Dynamic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7 SIMULATION OF DILUTE SALT-FREE POLYELECTROLYTE SOLU-

TIONS IN SIMPLE SHEAR FLOWS 120

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.4 Equilibrium Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.5 Dynamic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

8 CONCENTRATION EFFECTS ON THE COLLECTIVE DYNAMIC BE-

HAVIOR OF SELF-PROPELLED PARTICLES 168

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

8.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

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8.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

8.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

9 ONGOING AND FUTURE RESEARCH DIRECTIONS 219

A BEAD-ROD SIMULATIONS 223

B STRESS TENSOR FOR MULTICOMPONENT SYSTEMS 231

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List of Tables

5.1 Comparison of the average time required (in seconds) for various implicit calculation

schemes to achieve1.0 ζσ2

kBT total units of simulation time. Euler time steps are 0.0002

for FD simulations, and 0.0005 for HI simulations. The time step for all semi-implicit

schemes was taken as 0.0025. Also, Newton’s method was evaluated using two different

equation solvers, one based on the conjugate gradient method, and the other being the

GMRES method.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 Strain periodic orientations for a square lattice in planar elongational flow.. . . . . . 66

6.1 Radius of gyration and overlap concentration,c∗, for chains of varying molecular weight.77

6.2 Minimum number of chains,NC , required to guaranteeL > 2L0 = 2NSq0 as a func-

tion of concentration and molecular weight.. . . . . . . . . . . . . . . . . . . . 80

6.3 Calculated longest relaxation times for 21µm DNA as a function of concentration both

with and without hydrodynamic interactions. Experimental values are those of Hur et al.

(2001), where the solvent viscosity has been normalized to match that of our simulated

system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.1 Radius of gyration and overlap concentration,c∗, for chains of varying molecular weight

at infinite dilution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

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List of Figures

4.1 Particle translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Interior rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 End rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.4 Reptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.1 Depiction of the sliding cell layers in simple shear flow illustrating the use of the Lees-

Edwards boundary conditions. Shown in the system is a 10-bead polyelectrolyte chain

straddling across the cell boundary along with surrounding counterions.. . . . . . . . 64

5.2 Depiction of the stretching cell layers over one period in planar elongational flow illus-

trating the use of the Kraynik-Reinelt boundary conditions. Shown in the system is a

10-bead chain with surrounding counterions. The dark lines give the original cell lattice

while the light lines represent the current lattice.. . . . . . . . . . . . . . . . . . 68

6.1 Reduced viscosity,ηr, as a function of Weissenberg number,Wi0, for systems subjected

to planar elongational flow. Comparison of results for systems atc/c∗ = 1.0 when

different numbers of chains per simulation cell are considered. With little difference in

the results for systems ofNC = 100 andNC = 200 chains, we useNC = 100 chains

for all other results presented in this work.. . . . . . . . . . . . . . . . . . . . . 79

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6.2 Mean square radius of gyration,⟨R2

g

⟩, plotted as a function of normalized concentra-

tion, c/c∗, for various chain lengths. . . . . . . . . . . . . . . . . . . . . . . . 81

6.3 Excluded volume energy contribution to the net system energy for 21µm DNA at vari-

ous concentrations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.4 Scaling of the static chain size as a function of normalized concentration. Solid lines

indicate fits following the scaling law⟨R2

g

⟩∝ (c/c∗)0 in the dilute regime (c/c∗ ≤ 1.0)

and⟨R2

g

⟩∝ (c/c∗)−0.25 in the semi-dilute regime (c/c∗ ≥ 1.0). . . . . . . . . . . . 83

6.5 Scaling of the static chain size as a function of molecular weight atc/c∗ = 0.1 and100.

Solid line indicates predicted dilute regime scaling (2ν = 1.2) while dashed line gives

the expected semi-dilute scaling (2ν = 1.0). . . . . . . . . . . . . . . . . . . . . 83

6.6 (a) Short-time and (b) long-time time diffusivity normalized against that of the infinitely

dilute case for 21µm λ-phage DNA systems as a function of normalized concentration

c/c∗ both with and without hydrodynamic interactions. At infinite dilution, the short-

time and long-time diffusivities match for each hydrodynamic case and areDHIS =

DHIL = 0.0115 µm2/s andDFD

S = DFDL = 0.0069 µm2/s. . . . . . . . . . . . . 87

6.7 Ratio of long-time to short-time diffusivity of 21µm DNA systems as a function of

normalized concentrationc/c∗ both with and without hydrodynamic interactions.. . . 88

6.8 Average number of chain crossings per chain during a given time step for systems sub-

jected to (a) simple shear flow and (b) planar elongational flow.. . . . . . . . . . . 95

6.9 Flow direction fractional extension as a function of shear rate for systems subjected to

simple shear flow both with and without hydrodynamic interactions.. . . . . . . . . 97

6.10 Flow direction fractional extension as a function of Weissenberg number for systems

subjected to simple shear flow both with and without hydrodynamic interactions.. . . . 100

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6.11 Reduced viscosity as a function of shear rate for systems subjected to simple shear flow

both with and without hydrodynamic interactions.. . . . . . . . . . . . . . . . . 102

6.12 Reduced viscosity as a function of Weissenberg number for systems subjected to simple

shear flow both with and without hydrodynamic interactions.. . . . . . . . . . . . 103

6.13 Comparison of polymer contribution to the viscosity in simple shear flow as calculated

from simulations including hydrodynamic interactions with experimental values of Hur

et al. (2001). The concentration isc/c∗ = 1.0. Simulation results have been rescaled to

account for differences in solvent viscosity.. . . . . . . . . . . . . . . . . . . . 104

6.14 Flow direction molecular extension as a function of extension rate for systems subjected

to planar elongational flow both with and without hydrodynamic interactions.. . . . . 106

6.15 Flow direction molecular stretch normalized against that of the infinitely dilute case as

a function of extension rate for systems subjected to planar elongational flow both with

and without hydrodynamic interactions.. . . . . . . . . . . . . . . . . . . . . . 107

6.16 Flow direction molecular extension as a function of Weissenberg number for systems

subjected to planar elongational flow both with and without hydrodynamic interactions.. 110

6.17 Flow direction molecular stretch normalized against that of the infinitely dilute case as a

function of Weissenberg number for systems subjected to planar elongational flow both

with and without hydrodynamic interactions.. . . . . . . . . . . . . . . . . . . 111

6.18 Reduced elongational viscosity as a function of extension rate for systems subjected to

planar elongational flow both with and without hydrodynamic interactions.. . . . . . 113

6.19 Reduced elongational viscosity normalized against that of the infinitely dilute case as a

function of extension rate for systems subjected to planar elongational flow both with

and without hydrodynamic interactions.. . . . . . . . . . . . . . . . . . . . . . 114

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6.20 Reduced elongational viscosity as a function of Weissenberg number for systems sub-

jected to planar elongational flow both with and without hydrodynamic interactions.. . 116

6.21 Reduced elongational viscosity normalized against that of the infinitely dilute case as a

function of Weissenberg number for systems subjected to planar elongational flow both

with and without hydrodynamic interactions.. . . . . . . . . . . . . . . . . . . 117

7.1 Mean square radius of gyration of the polyion chain,⟨R2

g

⟩, plotted as a function ofλB

for various molecular weight polyelectrolytes atc/c∗ = 10−4. Contour lengths of the

chains in increasing order are13.5σ, 28.5σ, and58.5σ. . . . . . . . . . . . . . . . 132

7.2 Molecular visualizations of an equilibrated 40-bead chain in three electrostatic regimes,

the neutral case (λB = 0), the peak extension case (λB = 1.5), and the condensed-ion

case (λB ≥ 10). Chain beads are shown as dark spheres, and counterions as light spheres.133

7.3 Illustration of one of the defects associated with the use of the Debye-Huckel theory

for electrostatic interactions. Shown is the mean-square radius of gyration for a 20-

bead chain atc/c∗ = 10−4 with the electrostatics calculated via the Debye-Huckel

approximation, and via explicit Coulombic interactions with monovalent counterions.. 134

7.4 Polyelectrolyte chain-chain radial distribution function,gC(r), in c/c∗ = 10−3 solution

at equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.5 Chain radius of gyration,⟨R2

g

⟩, plotted as a function ofλB for 10-bead chains at various

concentrations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.6 Effects ofλB on the size of the ion cloud at equilibrium, as determined by the calcula-

tion of PI(r). Systems shown atc/c∗ = 10−3. . . . . . . . . . . . . . . . . . . . 139

ix

7.7 Depiction of the equilibrium ion cloud surrounding an individual chain in dilute solution

(c/c∗ = 10−3) for various values ofλB . Pictures correspond to the plots ofPI(r) of

Figure 7.6. Shown is thex-y profile with data averaged through thez-direction. Scales

reflect the excess concentration of ions relative to the average concentration of ions in

the system, i.e.cI (r) = cI (r)−NI/V . . . . . . . . . . . . . . . . . . . . . . 140

7.8 Size of the counterion cloud surrounding a chain as a function ofλB for a 10-bead chain

at various concentrations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.9 Density of the counterion cloud surrounding a chain as a function ofλB for a 10-bead

chain at various concentrations.. . . . . . . . . . . . . . . . . . . . . . . . . 142

7.10 Degree of ionization as a function of1/λB . Also shown is the prediction from Manning

theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7.11 Depictions of the equilibrium ion clouds surrounding an individual chain in dilute so-

lution for various concentrations. Shown is thex-y profile with data averaged through

the z-direction. Scales reflect the excess concentration of ions relative to the average

concentration of ions in the system, i.e.cI (r) = cI (r)−NI/V . The system shown is

atλB = 2.25, with the panels showingc/c∗ = (a)10−4, (b) 10−3, (c) 10−2, and (d)10−1.145

7.12 Comparison of reduced viscosity results as a function ofλB for 10-bead systems both

with and without hydrodynamic interactions atPe = 1.0. . . . . . . . . . . . . . . 149

7.13 (a) Average chain stretch and (b) reduced viscosity for10 bead chains atc/c∗ = 10−4

as a function ofPe for various values ofλB . . . . . . . . . . . . . . . . . . . . . 151

x

7.14 Depictions of the ion cloud surrounding an individual chain in dilute solution for various

values ofPe. Shown is thex-y profile with data averaged through thez-direction;x

is the flow direction, whiley is the gradient direction. The average chain stretch and

orientation in flow is mapped by the solid black line in panels (c)-(d). Scales reflect the

excess concentration of ions relative to the average concentration of ions in the system,

i.e. cI (r) = cI (r)−NI/V . The system shown is atc/c∗ = 10−3 andλB = 1.5, with

the plots showing (a) equilibrium, (b)Pe = 0.1, (c) Pe = 1.0, and (d)Pe = 10.0. . . . 153




orientation in flow is mapped by the solid black line in panels (c)-(d). Scales reflect the







orientation in flow is mapped by the solid white line in panels (c)-(d). Scales reflect the




xi

7.17 Component contributions to the overall reduced viscosity of the system as a function of

λB for systems atc/c∗ = 10−4 andPe = 1.0. Component contributions are described

with subscripts according to the types of particles involved (BB for bead-bead interac-

tions, BI for bead-ion interactions, and II for ion-ion interactions) and with superscripts

for the type of interactions involved (EXV for excluded volume, and EL for electrostatic).156

7.18 Component contributions to the overall reduced viscosity of the system as a function of

λB for systems atc/c∗ = 10−4 andPe = 0.01 and0.1. Component contributions are

described with subscripts according to the types of particles involved (BB for bead-bead

interactions, BI for bead-ion interactions, and II for ion-ion interactions).. . . . . . . 157

7.19 (a) Cooperative and (b) competitive arrangements of a counterion and the chain center-

of-mass with regards to the attractive electrostatic interactions. For repulsive interac-

tions (e.g. excluded volume), the cooperative and competitive labels are reversed.. . . 159

7.20 Rheological behavior of10-bead polyelectrolyte chains plotted as a function ofλB for

various concentrations atPe = 1.0. Figure (a) depicts the flow direction chain stretch,

< X >, while (b) shows the reduced viscosity,ηr. . . . . . . . . . . . . . . . . . 162


values ofc/c∗. Shown is thex-y profile with data averaged through thez-direction;x


orientation in flow is mapped by the solid black line in panels (a) and (b) and by the

white line in panels (c) and (d). Scales reflect the excess concentration of ions relative

to the average concentration of ions in the system, i.e.cI (r) = cI (r) − NI/V . The

system shown is atλB = 1.5 andPe = 1.0, with the plots showingc/c∗ = (a) 10−4,

(b) 10−3, (c) 10−2, and (d)10−1. . . . . . . . . . . . . . . . . . . . . . . . . . 163

xii

7.22 Component contributions to the overall reduced viscosity of the system as a function

of concentration for a system withλB = 1.5 andPe = 1.0. Component contributions

are described with subscripts according to the types of particles involved (BB for bead-

bead interactions, BI for bead-ion interactions, and II for ion-ion interactions) and with

superscripts for the type of interactions involved (EXV for excluded volume, and EL

for electrostatic).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.23 Universal plot of the reduced viscosity as a function ofλB/Pe for systems at various

concentrations and values ofPe. . . . . . . . . . . . . . . . . . . . . . . . . 166

8.1 Bead-spring dumbbell model of a swimmer. The flagellum is represented by a force

exerted on one of the beads of the dumbbell, and a force in the opposite direction exerted

by the dumbbell on the fluid. The casep = +1 is shown. . . . . . . . . . . . . . . 174

8.2 The alga Chlamydomonas and its normal (p = -1, top right) and escape (p = +1, bottom

right) modes of flagellar motion (Bray, 2001).. . . . . . . . . . . . . . . . . . . 175

8.3 Mean-square displacement as a function of time for a swimming particle withp = +1

at a concentration ofc/c∗ = 0.02, illustrating the transition from ballistic to diffusive

motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

8.4 Mean-square displacement as a function of time for (a) swimmers and (b) tracer parti-

cles withp = +1 at various concentrations.. . . . . . . . . . . . . . . . . . . . 182

8.5 Trajectory traces for an individual swimmer in a collection of 100 swimmers atc/c∗ =

a) 0.01 and b) 1.00. Traces record100ts units of simulation time. . . . . . . . . . . 183

8.6 Trajectory traces for an individual tracer in a collection of 100 swimmers atc/c∗ = a)

0.01 and b) 1.00. Traces record100ts units of simulation time.. . . . . . . . . . . . 183

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8.7 Time scale,τC , over which the motion of the swimming particles changes from ballistic

to diffusive in nature as extracted from the intersection of the asymptotic fits to the

mean-square displacement vs. time.. . . . . . . . . . . . . . . . . . . . . . . 184

8.8 Time scale,τC , over which the motion of the swimming particles changes from ballistic

to diffusive in nature as extracted from the intersection of the asymptotic fits to the

mean-square displacement vs. time. Results are shown for various system sizes with

bothp = +1 andp = −1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

8.9 Diffusion coefficient as a function of concentration for both the swimmer and tracer

particles using different methods of propulsion.. . . . . . . . . . . . . . . . . . 186

8.10 Diffusion coefficient as a function of concentration for both the (a) swimmer and (b)

tracer particles using different methods of propulsion. Results are shown for various

system sizes and both types of propulsion.. . . . . . . . . . . . . . . . . . . . . 187

8.11 Velocities of both swimmer and tracer particles as a function of concentration for sys-

tems utilizing various forms of propulsion.. . . . . . . . . . . . . . . . . . . . 188

8.12 Velocities of the (a) swimmer and (b) tracer particles as a function of concentration for

systems of varyingNP . Results are shown for various system sizes and both types of

propulsion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

8.13 Swimmer diffusivity as a function ofv2τC . . . . . . . . . . . . . . . . . . . . . 190

8.14 Contour plot of the vertical component of the velocity perturbation field owing to the

presence of a force dipole in the dumbbell stemming from the application of (a) a push-

ing force (p = +1) or (b) a pulling force (p = −1). Dark regions indicate fluid moving

in the positive vertical direction, while dark regions indicate fluid moving in the negative

vertical direction. Streamlines illustrate the net velocity field. White circles indicate the

location of the dumbbell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

xiv

8.15 Concentration effects on the radial distribution of swimmers about a given swimmer

with p = +1. The dumbbell is represented by white circles at bottom of plot and

concentrations have been normalized against system concentration, as described in the

text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

8.16 Concentration effects on the orientation of swimmers about a given swimmer withp =

+1. The dumbbell is represented by white circles at bottom of plot and concentrations

have been normalized against system concentration, as described in the text.. . . . . . 194

8.17 Concentration effects on the radial distribution of swimmers about a given swimmer

with p = −1. The dumbbell is represented by white circles at bottom of plot and

concentrations have been normalized against system concentration, as described in the

text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

8.18 Concentration effects on the orientation of swimmers about a given swimmer withp =

−1. The dumbbell is represented by white circles at bottom of plot and concentrations

have been normalized against system concentration, as described in the text.. . . . . . 196

8.19 Contour plot of the vertical component of the velocity perturbation field owing to the

presence of force dipoles in a pair of dumbbells stemming from the application of (a)

a pushing force (p = +1) or (b) a pulling force (p = −1). Dark regions indicate fluid

moving in the positive vertical direction, while dark regions indicate fluid moving in the

negative vertical direction. Streamlines illustrate the net velocity field. White circles

indicate the location of the dumbbells.. . . . . . . . . . . . . . . . . . . . . . 198

8.20 Sample trajectories for a pair of isolated swimmers in the absence of excluded volume

illustrating the effects of pair hydrodynamic interactions. Trajectories shown for the

case ofp = +1. Dark circles refer to beads acted on directly by the flagellar force.. . . 199

xv

8.21 Sample trajectories for a pair of isolated swimmers in the absence of excluded volume

illustrating the effects of pair hydrodynamic interactions. Trajectories shown for the

case ofp = −1. Dark circles refer to beads acted on directly by the flagellar force.. . . 199

8.22 Sample trajectories for a pair of isolated swimmers initially moving in a “chasing”

configuration in the absence of excluded volume. Trajectories shown for the case of

p = +1. Dark circles refer to beads acted on directly by the flagellar force.. . . . . . 202


configuration in the absence of excluded volume. Trajectories shown for the case of

p = −1. Dark circles refer to beads acted on directly by the flagellar force.. . . . . . 202

8.24 Sample trajectories for a pair of isolated swimmers moving in opposite directions in the

absence of excluded volume illustrating the effects of pair hydrodynamic interactions.

Trajectories shown for the case ofp = +1. Dark circles refer to beads acted on directly

by the flagellar force.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

8.25 Sample trajectories for a pair of isolated swimmers moving in opposite directions in the

absence of excluded volume illustrating the effects of pair hydrodynamic interactions.

Trajectories shown for the case ofp = −1. Dark circles refer to beads acted on directly

by the flagellar force.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

8.26 Decay of orientation autocorrelation function with time for systems at various concen-

tration. Both propulsion mechanisms are included for comparison.. . . . . . . . . . 206

8.27 Swimmer orientation autocorrelation time as a function of concentration for systems

with different propulsion mechanisms.. . . . . . . . . . . . . . . . . . . . . . 207

8.28 Swimmer orientation as a function of concentration for various system sizes in the ab-

sence of excluded volume with a)p = +1 and b)p = −1. . . . . . . . . . . . . . . 208

xvi

8.29 Sample trajectories for a pair of isolated swimmers illustrating the combined effects of

pair hydrodynamic interactions and excluded volume repulsions. Trajectories shown for

the case ofp = +1. Dark circles refer to beads acted on directly by the flagellar force.. 209

8.30 Sample trajectories for a pair of isolated swimmers illustrating the combined effects of

pair hydrodynamic interactions and excluded volume repulsions. Trajectories shown for

the case ofp = −1. Dark circles refer to beads acted on directly by the flagellar force.. 210


configuration in the presence of excluded volume. Trajectories shown for the case of

p = +1. Dark circles refer to beads acted on directly by the flagellar force.. . . . . . 211


configuration in the presence of excluded volume. Trajectories shown for the case of

p = −1. Dark circles refer to beads acted on directly by the flagellar force.. . . . . . 211

8.33 Sample trajectories for a pair of isolated swimmers moving in opposite directions il-

lustrating the combined effects of pair hydrodynamic interactions and excluded volume

repulsions. Trajectories shown for the case ofp = +1. Dark circles refer to beads acted

on directly by the flagellar force.. . . . . . . . . . . . . . . . . . . . . . . . . 213

8.34 Sample trajectories for a pair of isolated swimmers moving in opposite directions il-

lustrating the combined effects of pair hydrodynamic interactions and excluded volume

repulsions. Trajectories shown for the case ofp = −1. Dark circles refer to beads acted

on directly by the flagellar force.. . . . . . . . . . . . . . . . . . . . . . . . . 214

8.35 Diffusion coefficient as a function of concentration for both the swimmer and tracer

particles in different solvent types. Panel (a) corresponds to the case ofp = +1 and

panel (b) top = −1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

xvii

8.36 Velocities of both the swimmer and tracer particles as a function of concentration for

different solvent types. Panel (a) corresponds to the case ofp = +1 and panel (b) to

p = −1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

8.37 Time scale,τC , over which the motion of the swimming particles in a good solvent

changes from ballistic to diffusive in nature as extracted from the intersection of the

asymptotic fits to the mean-square displacement vs. time.. . . . . . . . . . . . . . 217

1

Chapter 1

INTRODUCTION

Complex fluids - materials whose microstructure interacts in a nontrivial way with how

it is deformed by flow - are an important area of study because of both their intrinsically

interesting behavior and their widespread technological importance. Examples of such

fluids include liquid crystals, colloids, and the subject of this dissertation, polymer so-

lutions. While our understanding of such fluids continues to mature, there remain many

important unresolved issues. One such example is the so-called “polyelectrolyte effect”,

in which charged polymer molecules exhibit a nonmonotonic dependence of viscosity on

concentration in the transition from semi-dilute to dilute solutions. Understanding such

behavior has potentially significant consequences. Many biological molecules, including

DNA, are examples of polyelectrolytes. By understanding the physics underlying the

“polyelectrolyte effect”, it is foreseeable that we will be more readily able to manipulate

and process such molecules, potentially leading to improved methods of disease detec-

tion and therapy. In this work, we focus on three such unresolved issues, including the

“polyelectrolyte effect”, through the use of numerical simulations.

2

The use of computer simulations in the modeling of complex fluids has a long and

rich history. However, the issue of how to model a complex solution for use in computer

simulations is, for lack of a better word, complex. Unlike simple fluids, the presence of

a complex microstructure adds a wide range of length and time scales into the problem.

For example, a simple process involving a dilute solution of monodisperse linear polymer

contains time and length scales of the solvent, polymer, fluid deformation, and, of course,

the process. Complicating the issue further is the fact that the polymer molecule itself

contains a spectrum of length and time scales as well. As a result, it becomes a significant

challenge to model the behavior of such fluids, capturing the necessary physics of the

problem while maintaining a computationally feasible model.

One popular technique for modeling polymer solutions is molecular dynamics, in

which an atomistic model is formulated based on the physical system of interest, and

the system is allowed to evolve according to Newton’s equations of motion. However,

at present, molecular dynamics is only useful for the simulation of very short time and

length scales. In order to capture larger scale phenomena, it is necessary to turn to a less

descriptive model. To this end, we turn to the kinetic theory of macromolecules (Bird

et al., 1987), in which the solvent is treated as a viscous continuum which acts on the

microstructure through thermal fluctuations and viscous drag. The microstructure in turn

acts on the solvent through the microscopic contribution to the stress tensor. Together,

these interactions form the core of a kinetic theory model. With knowledge of the config-

urational probability distribution function, one can determine the exact interplay between

the fluid microstructure and the bulk flow behavior, and so can determine any configu-

rational property of the fluid. Unfortunately, in all but the simplest models, there is no

exact analytical solution for the probability distribution function. As a result, we turn to a

3

numerical solution in the form of Brownian dynamics simulations in order to estimate the

probability distribution function, and from this, calculate physical properties of interest.

As mentioned above, this dissertation focuses on three distinct problems dealing with

complex fluids. In the first, we consider the rheological behavior of dilute solutions ofλ-

phage DNA when simple flows are imposed. It has been observed by Owens et al. (2004)

and Clasen et al. (2004) that solutions of DNA in shear flow respond much differently

to changes in concentration than when an elongational flow is imposed. Using Brownian

dynamics, we explore these two types of flow for systems at various concentrations in

the dilute regime and explain the differences in the response of the fluids using hydrody-

namic arguments. This is presented in Chapter 6. The second topic of this work deals

with the aforementioned “polyelectrolyte effect”, in which we consider a simple model

of a polyelectrolyte molecule with explicit counterions for the calculation of electrostatic

interactions. Using this model, we again use Brownian dynamics simulations to study

the behavior of polyelectrolytes in shear flow and present a mechanism consistent with

the primary electroviscous effect in order to explain the increase in reduced viscosity ob-

served as concentration increases in the dilute regime. This work is presented in Chapter

7. Finally, we consider hydrodynamically induced collective motions in systems of non-

Brownian, self-propelled particles. Motivated by recent experimental investigations of

swimming microorganisms (Mendelson et al., 1999; Wu and Libchaber, 2000; Wooley,

2003; Kim and Powers, 2004; Dombrowski et al., 2004), we have used a primitive model

with hydrodynamic interactions to gauge the impact of concentration on the coordination

of swimmer motions. In Chapter 8, we discuss a variety of findings illustrating small

regions of both cooperative and competitive behavior. This initial investigation serves to

4

provide some guidance for future researches, especially for planned simulations involv-

ing swimming particles placed in a microchannel.

The remainder of this dissertation is organized as follows: In Chapter 2, we present

the conservation equations that govern the flow of a dilute solution of polymer molecules.

Chapter 3 focuses on the intra- and intermolecular potentials that describe the connectiv-

ity, solvent effects, and electrostatic interactions that constitute our models. In addition,

we consider hydrodynamic interactions in this chapter. In Chapter 4, we discuss the

Monte Carlo method as applied to this work for the simulation of static properties. The

calculation of dynamic properties via Brownian dynamics simulations is presented in

Chapter 5, where we consider a number of calculation schemes for solving the governing

equations of Chapter 2. Chapters 6-8 deal with the three main research topics described

above, and we conclude by briefly describing some ongoing and future avenues of re-

search in Chapter 9.

5

Chapter 2

PROBLEM STATEMENT

In this work, we are concerned with the numerical simulation of a dilute solution of

monodisperse linear polymers immersed in an incompressible Newtonian solvent. It is

the function of this chapter to describe the polymer model that we are using as well

as the development of the equations governing the behavior of the system. Throughout

this chapter and the rest of this work we observe the notation convention of Bird et al.

(1987) whenever possible in taking particle indices as lower case greek letters (ν, µ, ...)

and connector vector indices as lower case roman characters (i, j, ...).

Polymer molecules are extremely complex systems with an enormous number of de-

grees of freedom, and so it becomes prohibitively expensive to use numerical simulations

to study such molecules at the atomistic scale for chains longer than a few dozen repeat

units. As a result, we instead coarse-grain the molecule in order to construct a mechan-

ical model that significantly reduces the number of degrees of freedom in the problem

while retaining the proper physics governing the problem. The polymer is modeled as a

sequence ofNB “beads” connected byNS = NB − 1 “springs”. Each spring represents

6

a section of the overall molecule (asub-molecule) with contour lengthQ0, yielding an

overall chain contour length ofL0 = NSQ0. The3N Cartesian coordinates of the beads

in configurational space are represented by the vectorr , with the vectorr ν denoting the

position of theνth bead in physical space.

Given the model described above, we are primarily interested in two quantities - the

configurational probability distribution function,Ψ(r , t), and the polymer contribution to

the total stress tensor,τp. It is to the determination of these quantities that we devote the

remainder of this chapter.

The first quantity,Ψ(r , t), gives the probability of finding the system in a particular

state,r , at a particular time,t. All static structural properties may be determined from the

configurational distribution function. The evolution ofΨ is described by the diffusion

equation (Bird et al., 1987) which may be derived by combining the equations of motion

for each particle with the continuity equation that describes the conservation of system

points in the configuration space. Neglecting bead inertia, we may write a force balance

about each particleν in the system as:

F(h)ν + F(b)

ν + F(φ)ν = 0 (2.1)

in which

F(h)ν = −ζ · [[[r ν ]]− (vν + vν)] (2.2)

F(b)ν = −kBT

∂

∂r ν

ln Ψ (2.3)

F(φ)ν = − ∂

∂r ν

φ (2.4)

where[[·]] denotes an average with respect to the velocity distribution.

Equation 2.2 describes the hydrodynamic force acting on beadν stemming from the

difference between the bead velocityr ν and the local velocity of the solution about bead

7

ν, (vν + vν). Here, we make use of the fact that in the absence of external forces, the

hydrodynamic force on beadν exactly cancels the hydrodynamic force on the fluid about

beadν; that is,F(h),fν +F(h)

ν = 0. This is not the case for our simulations of self-propelled

particles, as discussed in Chapter 8. The local fluid velocity is in turn composed of two

contributions,vν = v0 + [κ · r ν ], the imposed homogeneous flow field at beadν, andvν ,

the perturbation of the flow field at beadν resulting from the motion of other particles

in the system. This perturbation is referred to as “hydrodynamic interaction” and will

be discussed in detail in Chapter 3. For now, we simply state that the hydrodynamic

interaction contribution is assumed to depend linearly on the hydrodynamic forces acting

on all of the other beads in the chain where the coefficients are given by the hydrodynamic

interaction tensorsΩνµ according to the relationshipvν = −∑N

µ=1 Ωνµ · F(h)µ . Finally,

the friction tensorζ is expressed as a diagonal tensor in which the diagonal elements

are given by the scalarζν , which, according to Stokes law, is directly proportional to the

radius of particleν.

It has long been known from microscopic observations that particles suspended in

a liquid are in a state of constant highly irregular motion. This motion stems from the

constant bombardment of the particles by the much smaller particles of the solvent. In-

stead of using a highly irregular functional form to capture this motion, we instead use a

statistically averaged force of the form in Equation 2.3. This expression has been derived

by Bird et al. (1987) for the case of a structureless mass point in which the force has been

equilibrated in momentum space.

Finally, Equation 2.4 represents the force stemming from the combined intramolec-

ular and intermolecular potentials. In the simulations considered in this work, these po-

tentials include the springs that comprise the polymer chains as well as both electrostatic

8

and excluded volume interactions. These interaction potentials will be considered further

in Chapter 3.

We obtain the equation of motion for each particle by first inserting these expressions

for the various forces into Equation 2.1 to obtain

−ζ · [r ν − (vν + vν)]− kBT∂

∂r ν

lnΨ + F(φ)ν = 0. (2.5)

Rearranging, we have

r ν = vν + vν +1

ζν

(−kBT

∂

∂r ν

lnΨ + F(φ)ν

), (2.6)

and substituting in the expressions forvν andvν , we have

[[r ν ]] = v0 + [κ · r ν ] +1

ζν

(−kBT

∂

∂r ν

lnΨ + F(φ)ν

)+∑µ 6=ν

[(1

ζµδνµδ + Ωνµ

)·(−kBT

∂

∂rµ

ln Ψ + F(φ)µ

)]= v0 + [κ · r ν ] +

∑µ

[(1

ζµδνµδ + Ωνµ

)·(−kBT

∂

∂rµ

ln Ψ + F(φ)µ

)]. (2.7)

By combining this equation with the equation of continuity

∂Ψ

∂t= −

∑ν

(∂

∂r ν

· [[r ν ]]Ψ

)(2.8)

and defining the diffusion tensor,D, according to

Dνµ =kBT

ζµδνµ + kBTΩνµ, (2.9)

we arrive at the so-called “diffusion” equation

∂Ψ

∂t= −

∑ν

(∂

∂r ν

·(

[κ · r ν ]

+1

kBT

∑µ

[Dνµ ·

(−kBT

∂

∂rµ

ln Ψ + F(φ)µ

)])Ψ

). (2.10)

9

This is the partial differential equation that describes the way in which the distribution of

configurations changes with time when the time-dependent homogeneous velocity field

is specified byκ(t). Finally, note that the constant solvent velocity has been arbitrarily

set to zero as it acts equally on all particles and hence does not contribute to the polymer

microstructure.

We now need to develop an expression for the stress tensor in order to connect the

configurational distribution function with the rheological behavior of our system. This

stress tensor expression accounts for the various mechanisms by which forces are trans-

mitted through the fluid. It is assumed that the overall stress tensor can be taken as the

sum of a solvent contribution,πs, and a non-solvent contribution,πp, stemming from the

polymer chain and, in the case of polyelectrolytes, from the surrounding counterions as

well. For a single, dilute polymer chain, the non-solvent contribution to the stress tensor

is in turn taken to be the sum of contributions from three sources - a kinetic contribu-

tion(π

(b)p

), a contribution from the pairwise conservative forces

(π

(φ)p

), and finally, one

stemming from any present external forces(π

(e)p

). For a complete derivation of these

quantities, we refer the reader to Bird et al. (1987) and present only the final expressions

here:

π(b)p =

∑ν

mν

∫[[(r ν − v) (r ν − v)]] Ψ (Q, t) dQ (2.11)

π(φ)p =

1

2

∑ν

∑µ

∫rµνF(φ)

νµ Ψ (Q, t) dQ (2.12)

π(e)p =

∑ν

∫RνF(e)

ν Ψ (Q, t) dQ. (2.13)

Here,mν is the mass of beadν located atRν , the position vector relative to the chain

center of mass.rµν is the vector from beadµ to beadν (i.e. r νµ = r ν − rµ) andQ

describes the internal coordinates for the molecule (Qi = r i+1 − r i). F(φ)νµ is the net

10

pairwise potential force acting on beadν by beadµ (i.e. F(φ)ν =

∑µ F(φ)

νµ ) and the external

force acting on beadν is given byF(e)ν . Following Bird et al., the resulting total stress

tensor for a system of chains at a number densityn, wheren is low enough such that the

chains do not interact, is given by

π = πs + n

NB∑ν=1

NS∑k=1

Bνk

⟨Qk

(F(φ)

ν + F(e)ν

)⟩+ n

NB∑ν=1

mν〈(rν − v) (rν − v)〉 (2.14)

where

Bνk =

k

NBk < ν

kNB− 1 k ≥ ν.

(2.15)

By applying the force balance of Equation 2.1 and assuming a Maxwellian velocity dis-

tribution for the kinetic term, we may rewrite the stress tensor as

π = πs − n

NB∑ν=1

⟨RνF(h)

ν

⟩+NSnkBTδ. (2.16)

We then subtract the equilibrium expression for the pressure component of the total stress

tensor,pδ = psδ + NSnkBTδ, and insert the explicit expression for the Newtonian

solvent contribution to arrive at the Kramers-Kirkwood form of the stress tensor,

τ = −ηsγ − n

NB∑ν=1

⟨RνF(h)

ν

⟩(2.17)

in which γ is the deformation rate. It should be noted that the kinetic contribution to

the stress tensor is isotropic, and as it thus does not contribute to material functions of

interest, we shall neglect it.

We now have a complete definition for the stress tensor for a dilute polymer solution.

However, for an undiluted polymer solution, the situation is somewhat more compli-

cated. Curtiss and Bird (1996) have derived an expression suitable for the calculation

11

of the stress tensor in a multicomponent mixture, such as those we presently consider.

For multicomponent polymer mixtures, the stress tensor contains four contributions - the

three previously discussed, as well as a fourth contribution stemming from intermolecular

interactions,π(d)p . These contributions are given for a particular sampling of the solution

by:

π(b)p =

∑α

∑ν

mν,α

∫[[(r ν,α − v) (r ν,α − v)]] Ψα (r ,Qα, t) dQα (2.18)

π(φ)p =

1

2

∑α

∑ν,µ

∫rµν,αF(φ)

νµ,αΨα (r ,Qα, t) dQα (2.19)

π(e)p =

∑α

∑ν

∫Rν,αF(e)

ν,αΨα (r ,Qα, t) dQα (2.20)

π(d)p =

1

2

∑α,β

∑ν,µ

∫rµν,αF(d)

νµ,αβΨαβ

(r ,Rαβ,Qα,Qβ, t

)dRαβdQαdQβ (2.21)

where the subscriptsα andβ describe the various molecular species,Rαβ is the vector

from moleculeα to moleculeβ, andΨαβ

(r ,Rαβ,Qα,Qβ, t

)is the configurational distri-

bution function for a pair of moleculesα andβ. Clearly, this intermolecular contribution

creates significant challenges in theoretical works involving the stress tensor due to the

required evaluation of the pair distribution function. However, in a computer simulation,

we may directly sample this distribution which allows us to greatly simplify the problem.

The simplification stems from the realization that the intermolecular and intramolecular

contributions to the stress tensor are essentially identical in form. Once the total pairwise

force is known for a given pair of particles, whether they are a part of the same molecule

or different ones, we need only compute the tensor contributionrµνFνµ. Then, using the

force balance of Equation 2.1 and following the development above, we have the result-

ing expression for the microstructrual contribution to the stress tensor in the undiluted

12

case,

τ = −ηsγ −1

2V

N∑ν=1

N∑µ=1

⟨r νµF(h)

νµ

⟩. (2.22)

This expression can be rewritten in a form identical to that of Equation 2.17 where the

summation limit has been changed fromNB toN , as illustrated in Appendix B. However,

the form of Equation 2.22 has the advantage of allowing us to analyze the component

contributions to the viscosity in our work of Chapter 7 in order to determine the dominant

contributions under a variety of conditions.

Simultaneous solution of Equations 2.10 and 2.22 yields the complete evolution of

the fluid flow and microstructural configuration for our system of interest. In this work,

we consider the solution of these equations for simple homogeneous flows to study the

structural and rheological behavior of our model polyelectrolyte systems.

13

Chapter 3

INTRA- AND INTERMOLECULAR

INTERACTIONS

The behavior of the polymer molecule within a process is defined by intramolecular and

intermolecular interactions, as well as interactions with other external fields. In this

chapter, we discuss molecular connectivity (spring forces), excluded volume interactions,

electrostatic interactions (for the simulation of polyelectrolytes), and the form of the hy-

drodynamic interaction tensor in unbounded domains. In addition, we briefly discuss the

effect of applying periodic boundary conditions to the system on the calculation of both

the long-range electrostatic and hydrodynamic interactions.

3.1 Intramolecular Connectivity

As noted in Chapter 2, we coarse-grain the microstructure of the polyelectrolyte chain

in order to obtain a more simplified model. In doing so, polymer chains are represented

by a freely-jointed bead-spring model in which we assume that we may replace a portion

14

of the chain representing multiple repeat units by an elastic “spring” and concentrate the

masses of these units into “beads”. The nature of the spring type plays a significant role

in determining the microstructure of the chain, and it is this term on which we focus in

this section.

To understand the origin of the spring representation, consider a simple description of

a polymer submolecule given by a freely jointed (i.e. no bending, rotational, or torsional

resistance) sequence ofNk,s rigid linear segments, each of lengthbk, where each segment

represents a Kuhn length of the polymer. The Kuhn length, equal to twice the persistence

length of the polymer, is a measure of the distance along the backbone of the chain over

which segments become statistically decorrelated. With this representation, the contour

length of the submolecule isQ0 = Nk,sbk. To a good approximation, the equilibrium

orientation of the chain may be taken as a simple random walk, and for a large number

of segments (Nk,s → ∞), the probability distribution of the end-to-end distance of the

chain approaches that of a Gaussian distribution. If we then assume that the change

in energy of the chain due to a small extension stems solely from the loss of entropy

involved (again, no bending, rotational, or torsional potentials), the effective “entropic”

potential between the ends of the molecule is given by a Hookean spring potential with

spring constantH = 3kBT/b2kNk,s. Extending this idea to our spring representation of a

polymer submolecule, we obtain an expression for the tension in theith spring:

Fspr,Hi = HQi. (3.1)

While this expression is satisfactory in the limit of small extension, it also implies that the

chain is infinitely extensible. This is, of course, physically unrealistic and stems from the

assumption that the polymer segment consists of an infinite number of segments, allowing

15

the use of Gaussian statistics. Treloar (1975) addressed this issue by proposing a non-

Gaussian statistical treatment of the chain which considers the case of finite extension

(i.e. a finite number of segments). The resulting connector force for theith spring is

known as the inverse Langevin model,

Fspr,Li = HL−1

(Qi

Q0

)Qi, (3.2)

whereQi ≡ |Qi|, and

L(x) = coth x− 1

x(3.3)

is known as the Langevin function. Note, however, that to obtain the force at a given

extension, one must solve a non-linear equation. As a result, this form of the connector

force is not particularly well-suited for use in numerical simulations. A popular alter-

native form of the connector force is the empirical expression known as the Finitely

Extensible Non-linear Elastic (FENE) model,

Fspr,FENEi =

HQi

1−(

Qi

Q0

)2 , (3.4)

whereQ0 is the maximum spring extension. Both the inverse Langevin and FENE models

accurately characterize the physical behavior of the molecules in that they both linearize

to the Hookean model in the limit of small extension and account for finite extensibility.

However, the singularity in the inverse Langevin model can not be expressed as a poly-

nomial while the FENE model has a simple singularity of(1− ( Qi

Q0)2)

. As a result, it

is much better suited for use in numerical simulations than the inverse Langevin model.

While having no special significance related to the physical problem, this form of the

connector force has been widely used in simulations of polyelectrolyte molecules, and it

is for this reason that we adopt it as well for our work in this area (Chapter 7).

16

Another popular model, useful for the simulation of very stiff polymers, is that of the

wormlike spring model of Marko and Siggia (1995), based on the Porod-Kratky wormlike

chain (Rubinstein and Colby, 2003). Unlike the freely-jointed model, the wormlike chain

model is a freely rotating model in which the bending angles are restricted to very small

values. Marko and Siggia applied this idea to describe the submolecule comprising an

individual spring forλ-phage DNA, yielding an expression that matches the asymptotics

of the wormlike chain in both the small and large force limits and fits the experimental

data of Bustamante et al. (1994). The resulting force is given by

Fspr,WLCi =

kBT

2bk

[(1− Qi

Q0

)−2

− 1 +Qi

Q0

.

]Qi

Qi

(3.5)

This expression has been successfully used in previous simulations ofλ-phage DNA

(Jendrejack et al., 2002b; Chen et al., 2004), and is the form that we adopt for our work

with this polymer in Chapter 6.

Another spring model worthy of note is the Fraenkel spring, which has Hookean

behavior in the limit of low extensions, but a nonzero equilibrium length. The force law

is given by

Fspr,FRAi = H(Qi −Qeq)

Qi

Qi

(3.6)

in whichQeq is the nonzero equilibrium spring length. We have employed a variation of

this model in Chapter 8 in which we incorporate finite extensibility to form the FENE-

Fraenkel spring,

Fspr,FFi = −

H(1− Qeq

Qi

)1−

(Qi−Qeq

Q0−Qeq

)2 Qi. (3.7)

In many cases, it is desired to simulate a polymer using a model that incorporates rigid

17

bond lengths; this is the so-called Kramers chain, in which the bead-spring representa-

tion is replaced by a bead-rod model. Based on the midpoint algorithm of Liu (1989) and

modifications byOttinger (1994), Petera and Muthukumar (1999) have successfully sim-

ulated bead-rod chains at infinite dilution in both shear and elongational flows. However,

to correctly simulate such molecules while including hydrodynamic interactions becomes

prohibitively expensive in a periodic domain, as illustrated in Appendix A. Hsieh et al.

(2006) have shown that a reasonable approximation to the Kramers chain can instead

be achieved by using the FENE-Fraenkel spring with a sufficiently large spring constant

at significant computational savings. As a result, it is this form that we use to simulate

collections of hydrodynamically interacting self-propelled particles in Chapter 8.

3.2 Excluded Volume Interactions

The choice of solvent plays a significant role in determining the conformational and rhe-

ological properties of a dilute polymer solution. Solvents are typically grouped into three

broad categories – good solvents, theta solvents, and poor solvents – based on the ener-

getic favorability of polymer-solvent interactions as compared to polymer-polymer inter-

actions. These categories can be qualitatively described in the following manner:

• In a good solvent, the polymer-solvent interactions are energetically more favorable

than the polymer-polymer interactions. As a result, the polymer molecules prefer to

be surrounded by solvent molecules rather than other polymer molecules, causing

the chain to swell. In kinetic theory, this class of solvents is realized through the

use of repulsive bead-bead potentials.

18

• In a theta solvent, the polymer-solvent and polymer-polymer interactions are ener-

getically indistinguishable. This causes the polymer to assume an “ideal” configu-

ration, in which, at large length scales, it does not “feel” itself. Theta solvents are

incorporated in kinetic theory by simply omitting solvation potentials.

• In a poor solvent, the polymer solvent interactions are energetically less favorable

than the polymer-polymer interactions. In other words, the polymer molecules pre-

fer to interact with one another rather than with the solvent, causing the chain to

contract. This can be accounted for in kinetic theorgy by using attractive interpar-

ticle potentials.

To date, the majority of work dealing with concentration effects in polymer solutions via

numerical simulation has focused on the use of good solvents as this condition is found

in a wide range of applications of interest. In this work, we follow suit and consider only

good solvents, and so will concentrate on such for the remainder of this section.

There are many different mathematical descriptions that are commonly used to sim-

ulate repulsive interactions. One common example is a power-law representation case in

the form of a Lennard-Jones potential,

ULJνµ = 4εLJ

[(σ

rνµ

)12

−(σ

rνµ

)6], (3.8)

whererνµ ≡ |r ν−rµ|, andεLJ andσ are model parameters with dimensions of energy and

distance, respectively. Often, the potential is shifted and truncated to give only repulsive

interactions. This form, also known as the Weeks-Chandler-Andersen (WCA) potential,

is the form we have incorporated into our simulations of polyelectrolytes (Chapter 7),

mimicing the model of Chang and Yethiraj (2002). The resulting pairwise interaction

19

energy and individual particle force terms are given by:

U exv,WCAνµ = 4εLJ

[(σ

rνµ

)12

−(σ

rνµ

)6

+1

4

](3.9)

Fexv,WCAνµ = 4εLJ

∑µ

[12

(σ

rνµ

)12

− 6

(σ

rνµ

)6]

r νµ

|r νµ|2(3.10)

whenrνµ <6√

2σ, and are equal to zero otherwise. These terms are applied to all parti-

cles, and we setεLJ = kBT for simplicity.

For simulations ofλ-phage DNA (Chapter 6), we instead use a form derived as the

energy penalty due to the overlap of two submolecules (i.e., beads) (Jendrejack et al.,

2002b). By considering each submolecule as an ideal chain with a Gaussian probability

distribution, the energy penalty due to the overlap of the two coils may be expressed for

two beads,ν andµ, as

U exv,GSNνµ =

1

2vkBTN

2k,s

(3

4πS2s

)3/2

exp

[−3 |r νµ|2

4S2s

], (3.11)

wherev is the excluded volume parameter andS2s = Nk,sb

2k/6 is the mean square radius

of gyration of an ideal chain consisting ofNk,s Kuhn segments of lengthbk. The resulting

expression describing the force acting on beadν due to the presence of beadµ is then

Fexv,GSNνµ = vkBTN

2k,sπ

(3

4πS2s

)5/2

exp

[−3 |r νµ|2

4S2s

](r νµ) . (3.12)

Although this potential is not self-consistent (any deformation of the coil due to the over-

lap has been ignored), it does provide the correct scaling relationships for good solvent

conditions. Furthermore, this potential is based solely on the discretization of the polymer

chains, and so does not require tuning to match simulations to actual physical systems.

20

3.3 Electrostatic Interactions

In Chapter 7, we present results concerning the case of polyelectrolytes in dilute solution

where the distribution function is determined by the competition of Coulombic electro-

static interactions and thermal motions. This competition results in a distribution that is

not random, even at considerable distances. Historically, simulations of polyelectrolytes

have concentrated on dealing solely with the structure of the polyelectrolyte rather than

considering any structure of the surrounding solvent. In large part, this has been due to

the expense of the calculations involved in accurately determining the electrostatic in-

teractions. In these simulations, the electrostatic effects of the solvent are accounted for

through the use of the Debye-Huckel approximation (Robinson and Stokes, 1955) of the

electrostatic potential,ψ. However, this approach has some severe limitations which we

seek to illustrate with this work.

In the Debye-Huckel theory, Poisson’s equation for charged bodies,

∇2ψ = − 4π

εε0ρ, (3.13)

where the charge density isρ, ε is the solvent permittivity, andε0 is the permittivity of

free space, is solved assuming a spherically symmetric Boltzmann distribution of charges

in the solvent about any particular ion and in the absence of external forces. Under the

conditions of spherical symmetry (valid based on time averaging), Poisson’s equation

reduces to

1

r2

d

dr

(r2dψ

dr

)= − 4π

εε0ρ. (3.14)

Now, selecting a particular ion,ν, with chargeqν , as the origin of coordinates, the con-

dition of electrical neutrality stipulates that the net charge in the solution outside the

21

selected ion must be−qν . Furthermore, the average charge density at any point in this

region must also be of opposite sign to the charge on the central ion. Debye and Huckel

then assumed the Boltzmann distribution law, according to which, since the electrical po-

tential energy of anµ-ion is qµψν , the average local concentration ofµ-ions at a location

is

nµ = nµ exp

(−qµψν

kBT

)(3.15)

wherenµ is the average concentration ofµ-ions in the system. The net charge density is

then

ρµ =∑

µ

nµqµ exp

(−qµψν

kBT

). (3.16)

According to Equation 3.16, the Boltzmann distribution thus leads to an exponential

relationship between the charge density and the potential. However, a theorem of elec-

trostatics known as the principle of linear superposition of fields, states that the potential

due to two systems of charges in specified positions is the sum of the potentials due to

each system individually. This discrepancy is integral to the failings of the Debye-Huckel

theory. Consider a linear expansion of the exponential term in Equation 3.16:

ρµ =∑

µ

nµqµ +∑

µ

nµqµ

(−qµψν

kBT

)+∑

µ

nµqµ2

(−qµψν

kBT

)2

+ . . . . (3.17)

The first term in the expansion vanishes under the conditions of electrical neutrality, and if

qµψν kBT , only the linear term is appreciable and we are left with a form of the charge

density that is consistent with the principle of linear superposition of fields. However, it

has been shown that in many solutions, the electrostatic interactions are generally not

weak compared with the thermal energy of the ions. We have performed simulations to

illustrate this situation and the resulting impact it has on the structure of the chain; the

22

results are presented in Section 7.4. We note that the linear approximation is less severe

when dealing solely with the case of a solution with a single electrolyte of symmetrical

valences. That is,n1 = n2 andq1 = −q2. In this situation, the quadratic term of the

expansion in Equation 3.17 drops out and the approximation is much improved. However,

this also serves to illustrate that the Debye-Huckel approximation may be expected to

perform less well in cases with multivalent ions.

Inserting Equation 3.17 into Equation 3.14, we may solve Poisson’s equation to obtain

the potential,

ψν =qµεε0

e−κr

r(3.18)

whereκ is the inverse Debye screening length defined by

κ2 =

∑µ nµq

2µ

εε0kBT. (3.19)

This is Debye and Huckel’s fundamental expression for the time-average potential at a

point in solution a distancer from an ion of chargeqν in the absence of external forces.

From this, the potential energy for a pair of particlesν andµ separated by a distancerνµ

is

Uνµ =qνqµεε0

e−κrνµ

rνµ

. (3.20)

In contrast, more recent simulations account for the electrostatic effects of the solvent

by an explicit description of the counterion cloud surrounding the polyelectrolyte. This

is the case we explore in Chapter 7. In this case, electrostatic interactions are treated

23

between all pairs of particles according to Coulomb’s law. The electrostatic energy stem-

ming from the interaction of two particlesν andµ is given by

Uνµ =1

2

qνqµ4πεε0rνµ

(3.21)

=kBT

2λB

zνzµ

rνµ

(3.22)

where thezν are the particle valences,λB = e2

4πεε0kBTis the Bjerrum length, ande is

the fundmental electron charge. The Bjerrum length represents the separation distance

at which the electrostatic energy arising from a pair of point charges will be equal to

the thermal energy. According to the theory of Manning condensation (Manning, 1969),

when the Bjerrum length is much less than1σ, counterions are uniformly distributed

throughout the simulation cell. WhenλB 1σ, however, counterions are expected to

be found closely bound to the chains in a phenomena known as counterion condensation.

For reference, the Bjerrum length of water at room temperature is 7.14A, which in the

present simulations is roughly equal to a distance of1σ. Most simulations to date have

concentrated onλB in ranges near1σ. We have extended this regime toλB ∈ (0, 20σ)

in our work with polyelectrolytes to account for the use of different temperatures or

alternative solvents with different permittivities.

The presence of the free-floating counterions necessitates the use of a special geomet-

ric framework to prevent entropy from causing the system to gradually expand without

bounds. While a confining potential could be used as a representation for a container,

it is not a satisfactory solution for the modeling of polyelectrolytes in bulk owing to the

influence of the walls on the solution. This problem can be overcome by applying peri-

odic boundary conditions (PBCs) to the system by which a particle leaving the simulation

cell through one face simply reenters the cell through the opposite face. While this solves

24

the problem of preventing entropy degradation of the system, it does present an additional

calculation hurdle - namely the fact that we must now properly account for the long-range

character of the electrostatic interactions.

Consider a system ofN particles with chargesqν and positionsr ν in an overall neutral,

cubic simulation cell of lengthL. If periodic boundary conditions are applied, the total

electrostatic interaction energy and the force on beadν are respectively given by:

U el =kBT

2λB

∑ν

∑µ

′∑n∈Z 3

zνzµ

|r νµn|(3.23)

Felν = kBTλB

∑µ

′∑n∈Z 3

zµ

|r νµn|2r νµn

|r νµn|(3.24)

wherer νµ ≡ r ν−rµ+nL, and the prime on the second sum indicates that the lattice vector

n = 0 is omitted forν = µ. The resulting summations converge very slowly at long

distances due to the reciprocal distance term and hence require tremendous computational

effort. We also note that since this sum is only conditionally convergent (i.e. the sum over

the absolute values diverges); its value is not well-defined unless one specifies the order

of summation over the lattice cells. Finally, we must also characterize the dielectric

conditions of the medium outside the cluster of periodic cells in order to determine any

potential dipole effects acting on the system stemming from the surroundings. These

difficulties have been partially overcome via the use of the well-known Ewald summation

technique (Allen and Tildesley, 1987) and further refined by the application of the Particle

Mesh Ewald (PME) (Hockney and Eastwood, 1981; Darden et al., 1993; Essmann et al.,

1995) and Particle-Particle-Particle-Mesh (P3M) (Hockney and Eastwood, 1981; Darden

et al., 1997; Deserno and Holm, 1998a,b) techniques for systems with a large number of

particles. Toukmaji and Board Jr. (1996) also provide an excellent review of Ewald-based

calculation methods; we present only the traditional Ewald method here.

25

In brief, the Ewald summation technique is based on choosing a function,f (r), that

splits the summation quantity1/r into two parts, one that incorporates the short distance

behavior and one that incorporates the long-range behavior:

1

r=f(r)

r+

1− f(r)

r. (3.25)

We see from Equation 3.25 that a suitable splitting function should causef (r) /r to be-

come negligible beyond a given cutoff distance and(1− f (r)) /r to be a slowly vary-

ing function for all r. The resulting sums converge exponentially and may be summed

straightforwardly, the former treated by imposing a simple cutoff and the latter summed

over only a few reciprocal vectors in Fourier space. The traditional selection forf (r) is

the complementary error function

erfc(r) :=2√π

∫ ∞

r

e−t2dt. (3.26)

This results in the well known Ewald formula for the electrostatic energy of the primary

simulation cell:

U el = kBTλB

(U (r) + U (k) + U (s) + U (d)

)(3.27)

whereU (r) is the contribution from real space,U (k) is the contribution from reciprocal

space,U (s) is the self-energy contribution, andU (d) is the dipole correction (Deserno,

2000). The latter two elements originate from the simplification of the real and reciprocal

space sums and will be discussed below. The four contributions are respectively given

26

by:

U (r) =1

2

∑ν

∑µ

′∑n∈Z3

zνzµerfc(α |r νµn|)

|r νµn|(3.28)

U (k) =1

2

∑k 6=0

4π

k2e−k2/4α2 |ρ (k)|2 (3.29)

U (s) = − α√π

∑ν

z2ν (3.30)

U (d) =2π

(1 + 2ε′)V

(∑ν

zνr ν

)2

(3.31)

wherek is a wavevector defined byk = 2πn/L, the Fourier transformed charge density

ρ(k) is defined as

ρ(k) =

∫ρ(r)e−ik·rd3r =

∑ν

zνe−ik·rν (3.32)

and the inverse length,α, often called the Ewald splitting parameter, tunes the relative

weights of the real space and the reciprocal space contributions. While the final result

is independent of the choice ofα, the calculation efficiency varies dramatically. Asα

increases, the calculation load of the reciprocal space sum increases while that of the

real space sum diminishes. Conversely, asα decreases, the opposite trend occurs. Typ-

ically, we chooseα large enough so as to set a real-space cutoff distance equal to half

the box length as the real-space calculations scale withN2 whereas the reciprocal space

calculations scale asN .

The dipole correction term assumes that the set of periodic replications of the simu-

lation box tends spherically towards an infinite cluster and that the medium outside this

sphere is homogeneous with dielectric constantε′. At any given instant, the cluster of

cells has a total dipole moment. This dipole moment induces a surface charge about

the cell cluster and a corresponding electric field. The surface charge then induces a

27

corresponding surface charge in the surrounding medium, which in turn imparts an ad-

ditional contribution to the net electrostatic energy of the system. In the extreme case

of a surrounding metal boundary condition, commonly called the “tinfoil condition”,

ε′ = ∞ and the electric field induced by the surrounding dielectric medium cancels out

the dipole moment of the cell cluster. In this case,U (d) = 0. At the other extreme,ε′ = 1

for a surrounding vacuum, and since there is no additional energy contribution from the

surrounding medium,U (d) = 2π3

(∑

ν zνr ν)2. The latter condition is identical to the sum

obtained from a naıve cell by cell summation over lattice vectors and it is this boundary

condition that we choose to use for this work.

The forceFelν on particleν is obtained by differentiating the electrostatic potential

energyU el with respect tor ν , i.e.,

Felν = − ∂

∂r ν

U el. (3.33)

Using Equations 3.27-3.32, we obtain the following Ewald formula for the forces:

Felν = kBTλB

(F(r)

ν + F(k)ν + F(d)

ν

)(3.34)

with the real space, Fourier space, and dipole contributions given by:

F(r)ν = zν

∑µ

zµ

′∑n∈Z3

(2α√πe−α2|rνµn|2 +

erfc(α |r νµn|)|r νµn|

)r νµn

|r νµn|2(3.35)

F(k)ν =

zν

V

∑µ

zµ

∑k 6=0

4πkk2

e−k2/4α2

sin (k · r νµ) (3.36)

F(d)ν =

−4πzν

(1 + 2ε′)V

∑µ

zµrµ. (3.37)

Since the self energy in Equation 3.30 is independent of particle positions, it does not

contribute to the force.

28

We have also investigated the use of the smoothed PME calculation scheme (Ess-

mann et al., 1995) for our simulations of polyelectrolytes. The fundamental idea behind

this scheme is to compute the reciprocal space contribution using a discretization of the

particle charges onto a mesh covering the physical space. In creating a regularly spaced

grid of charges, the Fourier transform of the traditional Ewald summation can be replaced

by a fast Fourier transform, allowing for a significant reduction in the computational ex-

pense for large systems. In the present set of simulations, however, we focus on systems

that are small enough so that the standard Ewald scheme is actually faster, and thus we

use the standard Ewald technique in this work.

3.4 Hydrodynamic Interactions

The motion of an object through a fluid perturbs the velocity field of that fluid, and

hence, affects the motion of all bodies in that fluid. This hydrodynamic coupling between

moving objects in a fluid is called hydrodynamic interaction. In Chapter 2, we remarked

that the hydrodynamic interaction contribution to the local fluid velocity about a particle

ν depends linearly on the hydrodynamic forces acting on all of the other particles in the

system according to the relationship

vν = −N∑

µ=1

Ωνµ · F(h)µ (3.38)

whereΩνµ is referred to as the hydrodynamic interaction tensor. The proper accounting

of these interactions plays a significant role in determining the dynamic properties of

dilute polymer solutions. The simplest models for use in describing the dynamics of

polymer solutions treat the particles asfree-draining, in which each bead of a polymer

chain contributes equally to the total viscous drag. This contrasts with the experimentally

29

observed behavior in which a polymer moves through the fluid in anon-drainingmanner.

That is, at equilibrium, a polymer coil diffuses through the fluid as though it were actually

a single large solid Brownian particle. Mathematically, this may be expressed via the

Stokes-Einstein relation (Doi and Edwards, 1986),

D =kBT

6πηsRH

(3.39)

in which ηs is the solvent viscosity andRH is the effective hydrodynamic radius of the

chain, which is proportional to the size of the polymer coil. Compare this to the free-

draining case,

D =kBT

6πηsaNB

(3.40)

wherea is the hydrodynamic radius of an individual bead. As the size of the polymer

scales withN0.588B in a good solvent, we can immediately see that the free-draining case

incorrectly predicts the diffusivity to scale withN−1B . Additional problems with the free-

draining model emerge when considering simple flow situations. As a result, we have

included explicit hydrodynamic interactions in our simulations of dynamic phenomena.

We now turn our attention to the specific form of the hydrodynamic interaction tensor

used in our numerical simulations. The traditional starting point for analytical analysis in

kinetic theory is the solution to the Stokes flow equations for a point force in an infinite

domain. This solution is given by the Oseen tensor,

ΩOBνµ =

0 ν = µ

18πηsrνµ

[δ + rνµrνµ

r2νµ

]ν 6= µ.

(3.41)

The Oseen tensor has the unfortunate drawback of being suitable only for far-field inter-

actions. As the interaction separation is decreased, the diffusion tensor stemming from

30

the Oseen hydrodynamic interaction tensor is not guaranteed to be positive-definite. This

may lead to a situation involving negative energy dissipation, which is clearly unphysical.

This difficulty has been addressed by Rotne and Prager (1969) and Yamakawa (1970), in

which the authors develop an expression for the interaction tensor by directly considering

the rate of energy dissipation by the motion of the surrounding fluid. The resulting hydro-

dynamic interaction tensor is guaranteed to be positive definite for all particle separations.

For identically sized particles, the Rotne-Prager-Yamakawa tensor has the form

ΩRPνµ =

1

ζ

0 ν = µ

3a4rνµ

[(1 + 2a2

3r2νµ

)δ +

(1− 2a2

r2νµ

)rνµrνµ

r2νµ

]ν 6= µ andrνµ ≥ 2a[

(1− 9rνµ

32a)δ + 3

32a

rνµrνµ

rνµ

]ν 6= µ andrνµ < 2a

(3.42)

where the Stokes Law relationζ = 6πηa has been assumed and the correction for

rνµ < 2a takes hydrodynamic overlap of the beads into account. This treatment of the

hydrodynamics essentially models the beads as point particles, ignoring the stresslet that

arises for a bead exposed to flow. This approximation greatly simplifies the computation

and is widely used in Brownian dynamics simulations of polymer dynamics (e.g. Hsieh

et al. (2003); Petera and Muthukumar (1999); Sunthar and Prakash (2005); Grassia and

Hinch (1996); Schroeder et al. (2004); Liu et al. (2004); Hernandez-Cifre and de la Torre

(1999); Neelov et al. (2002); Agarwal et al. (1998); Agarwal (2000)). Furthermore, one

can see two physical rationales for neglecting this effect. The first is based on intrachain

hydrodynamic interactions: the distance between adjacent beads is typically much larger

than the hydrodynamic radius of each bead, suggesting that the stokeslet associated with

each bead should dominate over the stresslet. The second is relevant to interchain be-

havior: one can estimate that the stress per chain associated with each bead’s stresslet

scales asNBa3 (the stresslet for a spherical bead scales asa3 and there areNB of them

31

per chain). Using the radius of gyration,Rg, as a measure of chain size, the scaling of the

stresslet associated with the whole chain due to its extension in space can be estimated as

R3g ∼ (N

3/5B a)3 ∼ N

9/5B a3, so forNB 1 the latter contribution is the dominant source

of interchain hydrodynamic interactions.

However, as in the case of the electrostatic interactions, the use of periodic boundary

conditions necessitates the proper accounting for the long-range character of the hydro-

dynamic interactions. Hasimoto (1959) addressed this problem for a periodic array of

point forces, and later Beenakker (1986) generalized this treatment to the RPY tensor.

Smith et al. (1987) showed that the direct application of the Poisson summation formula,

as applied by Beenakker, is incorrect as the summations do not converge by themselves.

Instead, one must assume the presence of some barrier surrounding the infinite lattice

that is responsible for creating a backflow contribution that cancels out the nonperiodic

contributions to the hydrodynamics. Nevertheless, Beenakker’s work does lead to correct

expressions for the periodic hydrodynamic interactions. For a complete derivation, we

direct the reader to the above works as well as the discussion by Brady et al. (1988); we

present only the resulting diffusion tensor here:

Dνµ =kBT

ζδ + kBTΩ

=

(1− 6√

παa+

40

3√πα3a3

)δ +

′∑n∈Z 3

M (1) (r νµ,n)

+1

V

∑k6=0

M (2) (k) cos (k · r νµ) (3.43)

whereδ is the3N x 3N identity tensor,r νµ,n = r ν − rµ + nL and the parameterα

determines the manner in which the computational burden is split between the two sums.

For our simulations, we have usedα = 6/L as this was found to provide a reasonable

division of the computational expense beween the two sums. The first summation in

32

Equation 3.43 is computed over all lattice pointsn = (nx, ny, nz) with nx, ny, nz integers

in the case of a cubic lattice, and the prime on the first sum indicates that the lattice vector

n = 0 is omitted forν = µ. The second summation is taken in reciprocal space over

reciprocal lattice vectorsk = 2πn/L. The tensorsM (1) andM (2) are given respectively

by:

M (1)(r) =

C1erfc(αr) + C2

exp (−α2r2)√π

δ

+

C3erfc(αr) + C4

exp (−α2r2)√π

rr, (3.44)

where

C1 =

(3

4ar−1 +

1

2a3r−3

),

C2 =

(4α7a3r4 + 3α3ar2 − 20α5a3r2 − 9

2αa+ 14α3a3 + αa3r−2

),

C3 =

(3

4ar−1 − 3

2a3r−3

),

C4 =

(−4α7a3r4 − 3α3ar2 + 16α5a3r2 +

3

2αa− 2α3a3 − 3αa3r−2

),

and

M (2)(k) =

(a− 1

3a3k2

)(1 +

1

4α−2k2 +

1

8α−4k4

)6π

k2

× exp

(−1

4α−2k2

)(δ − kk

), (3.45)

with r = r/ |r | and k = k/ |k|. For free-draining simulations, in which hydrodynamic

interactions are absent, we neglect the off-diagonal components ofD and simply take

D = δ. Finally, we note that∂∂r · D = 0.

It should be noted that we do not in fact require the explicit calculation of the diffusion

tensor at each time step, but rather, the product of the diffusion tensor with a force vector.

33

As a result, we use the addition formula for cosines to rewrite the lattice sum as

∑µ

DνµFµ =

(1− 6√

παa+

40

3√πα3a3

)Fν +

′∑n∈Z 3

M (1) (r νµ,n) Fµ

+1

V

∑k6=0

M (2) (k)

cos(k · r ν)

∑µ

cos(k · rµ) · Fµ (3.46)

−sin(k · r ν)∑

µ

sin(k · rµ) · Fµ

.

We may now calculate the sums over the bead indices in the final term as a function

of wavevector, thus requiring onlyO(N) operations as opposed to theO(N2) operations

required by calculating the full diffusion tensor prior to computing the dot product. This is

a significant improvement in computational efficiency as the operation count in reciprocal

space is multiplied by the number of reciprocal lattice vectors.

34

Chapter 4

SOLUTION OF THE KINETIC

THEORY EQUATIONS - STATICS

While the primary focus of this work centers on the dynamic behavior of dilute polymer

solutions, it is important to first understand the structure of such solutions at equilibrium

in order to better understand our observations when a flow is imposed. In Chapter 2, we

discussed the development of the Fokker-Planck equation,

∂Ψ

∂t= −

∑ν

(∂

∂r ν

·(

[κ · r ν ]

+1

kBT

∑µ

[Dνµ ·

(−kBT

∂

∂rµ

ln Ψ + F(φ)µ

)])Ψ

)(4.1)

which, together with the expression for the stress tensor (Equation 2.17), fully describes

the evolution of the microstructure of our system of interest and provides a means of

computing physical properties. However, at equilibrium, the situation is far simpler. In

the absence of flow, Equation 4.1 reduces to

0 =∑

ν

(∂

∂r ν

·

(1

ζ

∑µ

(−kBT

∂

∂rµ

ln Ψ + F(φ)µ

))Ψ

), (4.2)

35

and from equilibrium statistical mechanics, we can solve for the equilibrium configura-

tional distribution function as

Ψeq

(rN)

=e−φ/kBT∫e−φ/kBTdrN

. (4.3)

Here,φ represents the potential energy of the system, which stems from intra- and inter-

molecular interactions and depends only on the coordinates of theN beads in solution.

Using this quantity, we can then compute equilibrium properties via

〈B〉eq =1

nV

∫ ∫BΨeqdr . (4.4)

Unfortunately, due to the large number of independent coordinates present in the prob-

lem, direct numerical quadrature is not well suited to the solution of such integrals. In-

stead, we have employed the Monte Carlo method (Frenkel and Smit, 2002; Allen and

Tildesley, 1987) in the canonical ensemble for the calculation of equilibrium properties.

Rather than evolve the system according to deterministic forces and torques, Monte Carlo

simulations take an alternative approach and sample the available configuration space by

simply proposing various rearrangements (termed “moves”) of the existing system and

accepting new configurations according to specified probability criterion. As a result,

a well-designed Monte Carlo method can enjoy a significant computational advantage

in the calculation of equilibrium properties over techniques such as Brownian dynamics

as the clever selection of various “moves” allows for highly efficient exploration of the

available configuration space.

The basic principle underlying the Monte Carlo method is that of detailed balance. In

essence, this principle states that at equilibrium, the average number of accepted moves

from a state “o” to any other state “n” is exactly canceled by the number of reverse moves.

In other words, the “flow” from configuration “o” to configuration “n” must be equal to

36

the flow in the reverse direction. Mathematically speaking, this is equivalent to

N (o)α (o→ n) acc (o→ n) = N (n)α (n→ o) acc (n→ o) (4.5)

whereN (x) is the probability of finding a system in state “x”,α (o→ n) is the proba-

bility of performing a trial move from state “o” to “n”, andacc (o→ n) is the probability

of accepting the move. Furthermore, it is customary to assume thatN (r) is given by its

Boltzmann weight asN (r) = e−βU(r)

Z, where Z is the partition function of the ensemble

(Z =∑

ν e−βU(rν)). In this work, we have used four different types of moves: single

particle translations, interior rotations, end rotations, and reptation. The last three moves

have been adapted from lattice-based models to the present off-lattice simulations. Also,

while the translation move may be applied to any particle in the system, the other three

moves are only applied to polymer chains. In the remainder of this section, we describe

each move in detail and discuss appropriate modifications to the detailed balance for the

systems at hand.

4.1 Translation

The simplest move in any Monte Carlo simulation is the translation of a single particle

from its current location to some new location (Figure 4.1). To do so, we randomly

select one particle and move it to a new location within a cubic volume,V , centered

about the target particle. The probability of performing a trial translation move is equal

to the product of the probability of selecting one of a set ofN particles (1/N ) and the

probability of moving it to some new position withinV (1/V ), yielding a detailed balance

37

of

e−βU(ro)

Z× 1

N× 1

V× acc (o→ n) =

e−βU(rn)

Z× 1

N× 1

V× acc (n→ o) (4.6)

acc (o→ n)

acc (n→ o)=

e−βU(rn)

e−βU(ro)(4.7)

wherer o andrn are, respectively, the positions of the particles before and after the pro-

posed move. Taking the acceptance probabilities as

acc (o→ n) = min(1, χ) (4.8)

acc (n→ o) = min

(1,

1

χ

)(4.9)

whereχ is to be determined from the detailed balance, we have

χ =e−βU(rn)

e−βU(ro)(4.10)

= e−β(U(rn)−U(ro)), (4.11)

which gives an acceptance criterion of

acc (o→ n) = min(1, e−β(U(rn)−U(ro))

). (4.12)

We can thus summarize the translation move as follows:

• Select a particle at random and calculate the energy of the current configuration,

U (r o).

• Give the particle a random displacement to a location within a specified cubic vol-

ume,V , centered on the initial particle location. Calculate the energy of the new

configuration,U (rn).

38

Figure 4.1:Particle translation

• The move is accepted with probabilityacc (o→ n) = min(1, e−β(U(rn)−U(ro))

).

As a rule of thumb, the magnitude ofV is adjusted to give an acceptance rate of approx-

imately 30%.

4.2 Interior Rotation

An interior rotation (Figure 4.2) is a Monte Carlo move in which some random number

of segments along the backbone of the chain are rotated about an axis connecting the

beads to either end of the segments being rotated. For example, if we designate beadν

for rotation, we draw an imaginary axis connecting beadsν−1 andν+1, and then rotate

the designated bead(s) about this axis through some random angle. In doing so, we may

39

Figure 4.2:Interior rotation

cause drastic changes to the internal structure of a polymer chain in a very rapid manner.

The detailed balance for an interior rotation involving a single bead is quite simple.

For a chain ofNB beads, we designate a bead for rotation with probability1/ (NB − 2),

and rotate it through an arbitrary angle with probability1/ (2π). The probability of exist-

ing in the current state, as well as the acceptance probabilities are unchanged from above,

yielding a detailed balance equation of the form:

e−βU(ro)

Z× 1

NB − 2× 1

2π×min(1, χ) = (4.13)

e−βU(rn)

Z× 1

NB − 2× 1

2π×min

(1,

1

χ

)which, upon simplifying, gives an acceptance criterion of


). (4.14)

40

as above.

Thus, we summarize the interior rotation move for a single bead as follows:

• Select an interior chain bead (1 < ν < NB) at random and calculate the energy of

the current configuration,U (r o).

• Rotate the bead through a random angle about the axis created by connecting beads

ν − 1 andν + 1. Calculate the energy of the new configuration,U (rn).


).

For rotations involving multiple beads, the probabilities of selecting the number of beads

to be rotated and of choosing which set of beads are exactly balanced by the probabilities

of the reverse move (i.e.α (o→ n) = α (n→ o) ), and so the acceptance criterion is

exactly the same. Note that the spring lengths do not change in this move, and so the

calculation of the spring energy is unchanged as a result of this move.

4.3 End Rotation

The end rotation move is similar to the interior rotation in that a number of beads are

randomly selected and rotated through a random angle. However, as the name implies, the

beads are located at either end of the polymer chain as opposed to the interior. The axis

of rotation is taken as the bond vector joining the two beads preceding those designated

for rotation (Figure 4.3). To carry out an end rotation of a single bead for a chain ofNB

beads, we designate a target bead with probability1/2. As this probability is symmetric

with respect to forwards and backwards moves, and the additional probabilities involved

in the detailed balance are identical to those of the interior rotation, we have that the

41

acceptance criterion is once again given by


). (4.15)

The end rotation move for a single bead is then summarized as

• Select an end of the molecule at random and calculate the energy of the current

configuration,U (r o).

• Rotate the end bead through a random angle about the axis created by connecting

the two preceding beads of the chain. For example, if we select the end of the chain

corresponding to beadNB, the axis of rotation is formed by the connector vector

between beadsNB − 1 andNB − 2. Calculate the energy of the new configuration,

U (rn).


).

4.4 Reptation

The final move type used here is that of reptation (Wall and Mandel, 1975), in which a

polymer chain is thought to “slither”, moving along the path already described by the

chain contour in much the same manner as a snake. In a numerical simulation, this is

accomplished by removing the end of one chain and reattaching the bead(s) to the end

of the other chain (Figure 4.4). In doing so, we now have a more complicated detailed

balance as we must not only locate the position of the new bead, but that we must also

account for the probability of generating a new spring of a given length. As above, the

probability of selecting a particular chain end from which to remove a bead is1/2, the

42

Figure 4.3:End rotation

probability of exising in the current state isN (r) = e−βU(r)

Z, and the acceptance prob-

ability is (o→ n) = min(1, χ). In forming the trial state, the new particle location is

determined with probability p(rn)4π(rn)2

, wherep(rn) is the probability of selecting a spring

with length|rn| from the distribution of spring lengths for the system. The spring distri-

bution may be actively calculated during the simulation; while not absolutely rigorous, as

calculating the distribution in this way does not guarantee microscopic reversability, the

induced error is unlikely to be significant. As a result, the acceptance criterion is given

by

e−βU(ro)

Z× 1

2× p(rn)

4πr2n

×min(1, χ) =e−βU(rn)

Z× 1

2× p(ro)

4πr2o

×min

(1,

1

χ

)(4.16)

43

which, upon simplifying, gives

acc (o→ n) = min

(1,p(ro)

p(rn)

(rn

ro

)2

e−β(U(rn)−U(ro))

). (4.17)

Finally, we summarize the reptation move as

• Select an end of the molecule at random and calculate the energy of the current

configuration,U (r o).

• Remove the last bead from the selected end of the molecule and reattach the bead

to the other end of the molecule. The bead is to be placed at some distancern from

the previous bead, and at a random orientation. Calculate the energy of the new

configuration,U (rn).

• The move is accepted with probability

acc (o→ n) = min

(1,p(ro)

p(rn)

(rn

ro

)2

e−β(U(rn)−U(ro))

).

44

Figure 4.4:Reptation

45

Chapter 5

SOLUTION OF THE KINETIC

THEORY EQUATIONS - DYNAMICS

Returning once more to the general Fokker-Planck equation of Chapter 2,

∂Ψ

∂t= −

∑ν

(∂

∂r ν

·(

[κ · r ν ]

+1

kBT

∑µ

[Dνµ ·

(−kBT

∂

∂rµ

ln Ψ + F(φ)µ

)])Ψ

)(5.1)

we now focus our attention on various Brownian dynamics algorithms for computing the

numerical solution of the diffusion equation in simple homogeneous flows. In Section

5.1, we discuss the numerical solution of Equation 5.1 in a stochastic representation via

a straightforward Eulerian scheme. We consider more complex semi-implicit and fully

implicit solution schemes in Section 5.2 and discuss the utility of such schemes for the

problems at hand. Section 5.3 is devoted to a description of a rapid method of decom-

posing the diffusion tensor, a computationally challenging task that arises in each of the

46

methods of the first two sections. Finally, techniques for handling the boundary con-

ditions in the simulation of simple shear and planar elongational flows are presented in

Section 5.4.

5.1 Euler Integration

In order to solve Equation 5.1, we first take a stochastic representation ofΨ in which Ψ

is defined as

Ψ(t, r) ≈ 1

N

∑ν

δ (r(t)− r ν(t)) . (5.2)

Using this distribution function, we may then recast the Fokker-Planck equation in the

form of a stochastic differential equation (Ottinger, 1996)

dr ν =

([κ · r ν ] +

1

kBT

∑µ

Dνµ · F(φ)µ +∇r · D

)dt+

√2 B·dW, (5.3)

in whichD ≡ B·BT and where each component ofW(t) is a random Gaussian, or Wiener,

process with mean zero and variancedt. We may then integrate this equation to obtain

an expression suitable for numerical simulation,

r ν(t+ ∆t)− r ν(t) =

∫ t+δt

t

([κ · r ν ] +

1

kBT

∑µ

Dνµ · F(φ)µ

)dt′

+√

2

∫ w(t+∆t)

w(t)

B·dW′. (5.4)

In obtaining Equation 5.4, we have made use of the fact that the Rotne-Prager-Yamakawa

form of the hydrodynamic interactions satisfies the relationship∇r · D = 0, as noted in

Section 3.4.

The most common method for computing the integrals in Equation 5.4 is via an ex-

plicit Euler scheme in which the function values in the integral argument are assumed

47

to hold constant over the course of a finite time step at the value at the beginning of the

steps. That is,

r ν (t+ ∆t) = r ν (t) + [κ(t) · r ν(t)] ∆t +∆t

kBT

∑µ

[Dνµ(t) · F(φ)

µ (t)]

+√

2∑

µ

[Bνµ(t)·∆Wµ(t)] . (5.5)

This solution method has the benefit of being both simple to execute as well as scaling

asO (N2). However, due to the existence of a singularity in the spring potential, and

in the case of polyelectroltyes, the electrostatic and excluded volume potentials as well,

the Euler solution scheme requires the use of small time steps in order to guarantee both

accuracy and stability. This problem is further exacerbated when the system is subjected

to flow as the maximum time step must be reduced in accordance with the flow strength.

As a result, we consider various implicit schemes in the following section in order to

allow stable integration at larger time steps.

5.2 Implicit Integration Methods

As mentioned above, while the explicit Euler solution method is straightforward, it has

the unfortunate drawback of requiring small time steps to maintain both stability and

accuracy. Following the examples of Jendrejack et al. (2002a), Somasi et al. (2002),

and Hsieh et al. (2003), we have developed three alternative semi-implicit calculation

methods to carry out the stochastic integration. The underlying idea of each scheme is

to sacrifice some calculation expense at each time step to enable the use of a larger time

step, with the resulting product providing significant savings in total calculation time. We

briefly describe and compare the three methods in this section in order to illustrate the

48

computational difficulties associated with the use of semi-implicit schemes.

5.2.1 Newton’s Method

Newton’s method is perhaps the best known method of rapidly finding the roots of a

system of nonlinear equations. Following Jendrejack et al. (2002a), we have developed

a semi-implicit scheme utilizing Newton’s method for the stochastic integration. For the

given set of nonlinear equations,

f(r) = 0, (5.6)

we suppose that an approximate solution is given by the result at the previous time step,

r 0(t + ∆t) = r(t). We then improve on this trial solution by computing a correction

vector from the linear system of equations resulting from a two-term Taylor expansion of

the nonlinear system. That is, we improve on thekth iteration by

r k+1(t+ ∆t) = r k(t+ ∆t) + α∆r k (5.7)

whereα is the damping coefficient and the correction vector is determined from the

Jacobian system by

J(r k(t+ ∆t)) · (∆r)k = −f(r k(t+ ∆t)). (5.8)

The damping coefficientα ∈ (0, 1) is an adjustable parameter that allows one to improve

the stability of the algorithm at the expense of an increased number of iterations required

for convergence. In this work, we consider only the undamped case in whichα = 1.

For computational simplicity, we simplify the fully-implicit calculation scheme by

computing both the diffusion tensor and the Brownian term explicitly; all other terms are

49

computed implicitly. The resulting system of equations is given by:

fν = r ν (t+ ∆t)− r ν (t) − (v0 + [κ (t) · r ν (t+ ∆t)]) ∆t

− ∆t

kBT

∑µ

Dνµ (t) · F(φ)µ (t+ ∆t)

−√

2∑

µ

Bνµ (t) ·∆Wµ (t)

= 0 (5.9)

and the Jacobian given by:

Jνµ = δ − κ (t) δνµ∆t− ∆t

kBT

∑η

Dνη (t) ·∂F(φ)

η (t+ ∆t)

∂rµ (t+ ∆t). (5.10)

To improve the efficiency of the algorithm, the singularity in the spring force is removed

by linearizing the spring force when the spring length is greater than some predetermined

extension,Qm, taken to be99% of the maximum spring extension. AboveQm, the spring

force is then given by

Fsp,ext =

[F sp(Qm) +

(∂F sp

∂r

)Qm

(r −Qm)

]rr. (5.11)

Note that this spring force is used only during the iterative process and the final spring

lengths are checked so as to ensureQ < Q0.

There are two significant calculation expenses in this method - the calculation of the

Jacobian matrix,J = ∂f/∂r , and the solution of the linear system in Equation 5.8. At

first glance, the former problem does not appear as formidible as the latter; however,

this is not necessarily the case. As evidenced in Equation 5.10, the calculation of the

Jacobian requires anO(N2) operation for each periodic cell stemming from the multi-

plication of the diffusion tensor by the3N x 3N matrix ∂F(φ)(t+∆t)∂r(t+∆t)

, and in the case of

polyelectrolytes, the calculation of the electrostatic interactions as well. One simplifica-

tion we have utilized is to make the Jacobian term effectively free-draining, in which the

50

diffusion tensor is replaced by the identity matrix in Equation 5.10. This significantly

reduces the computational load per iteration at the expense of a few additional iterations

per time step without a noticeable loss in stability. The net result is a significant savings

in computational time.

In addition to the difficulties associated with the calculation of the Jacobian, we

must solve a system of3N linear equations at each time step. Classically, this requires

O(N3) operations, however, the computational load may be reduced toO(N2) operations

through the use of the GMRES solver (Saad and Schultz, 1986). For systems of interest

in this work, in whichN is typically on the order of 1000 or less, the solution of the

linear system of equations is not as expensive as the actual calculation of the Jacobian.

For larger systems, it is foreseeable that this trend may be reversed.

5.2.2 Broyden’s Method

A second, less well-known procedure for solving systems of nonlinear equations is a

quasi-Newton scheme known as Broyden’s method (Broyden, 1965, 1967). Rather than

use the true Jacobian matrix, Broyden proposed a modified form of the Newton’s method

algorithm in which a finite difference approximation to the Jacobian is used. This method

is essentially a generalization of the secant method to nonlinear systems. The primary

advantage in using Broyden’s method over Newton’s method is that with each iteration,

the Jacobian estimate may be updated without recalculating the entire Jacobian. For

systems in which the calculation of the Jacobian is burdensome, such as those of interest

here, this may prove highly advantageous. Updating the Jacobian may be done in varous

ways, but the most common method is to use a minimal modification of the Jacobian

estimate so that the change inf predicted byJk in a directionu orthogonal toxk − xk−1

51

is the same as predicted byJk−1. That is,

Jku = Jk−1u (5.12)(xk − xk−1

)· u = 0. (5.13)

This leads to a uniquely determined matrixJ. Broyden’s method for a system of equa-

tionsf(r) = 0 then is as follows:

Jksk = −f(r k)

(5.14)

r k+1 = r k + sk (5.15)

yk = f(r k+1

)− f(r k)

(5.16)

Jk+1 = Jk +

(yk − Jksk

) (sk)T

sk · sk. (5.17)

The iteration is initialized using the positions from the prior time step,r 0(t+∆t) = r(t),

as in the case of Newton’s method. An initial estimate for the Jacobian is also required,

which we take asJ (r(t)). As in Newton’s method, we simplify the iteration by taking

the Brownian term explicitly. However, in Broyden’s method, we may also implicitly

incorporate the diffusion tensor at each iteration with no added expense as there are no

additional evaluations of the functionf. The diffusion tensor is still excluded from the

calculation of the initial estimate of the Jacobian with little to no effect on the convergence

rate of the algorithm.

As noted above, the primary advantage to using Broyden’s method in lieu of New-

ton’s method is that the update of the Jacobian term may be performed at little expense.

However, while Broyden’s method may lessen the computational load per iteration, its

convergence rate is only superlinear and not quadratic. This implies an expected increase

in the number of iterations required for convergence as compared to Newton’s method.

52

In addition, Broyden’s updating technique does not maintain any symmetry or sparseness

in the Jacobian term, which reduces the speed at which Equation 5.14 may be solved.

5.2.3 Predictor-Corrector Method

The final semi-implicit method we have explored is a predictor-corrector method based

on the work of Somasi et al. (2002) and Hsieh et al. (2003). In these works, the authors

consider a single polymer chain in dilute solution with no intrachain interactions other

than the connector forces, which are taken as FENE springs. Hydrodynamic interactions

are included in Hsieh et al. (2003), but in an explicit fashion only. We have adapted this

method to the present system, incorporating both the additional interactions present and

the fact that we have multiple chains per simulation cell.

The predictor-corrector method differs from the other calculation methods presented

here in that it deals with the evolution of the springs, rather than the actual bead positions.

The translation between the two coordinate systems is straightforward:

Qi = ri+1 − ri

rc =1

NB

∑ν

rν

rν = rc +∑

i

BνiQi (5.18)

where

Bνi =

i

NBif i < ν

iNB− 1 if i ≥ ν.

(5.19)

In the original scheme, the use of the spring-based coordinates presents no difficulties and

is in fact more efficient than an equivalent scheme based on bead positions as it requires

53

the solution of onlyNB − 1 equations per chain. Unfortunately, in the present work, this

is not the case. We must also take into account the spatial evolution of each chain relative

to the rest of the system, and in the case of our polyelectrolyte simulations, the evolution

of the free counterions as well. This presents certain complications that will be addressed

presently.

The general structure of our predictor-corrector scheme is to first compute the updated

positions of the centers-of-mass of the chains via a standard Euler scheme, followed by a

semi-implicit determination of the actual chain structure about the chain center of mass.

The center-of-mass is updated via

r c (t+ ∆t) = r c (t) + [κ(t) · r c(t)] ∆t +∆t

NBkBT

NB∑ν=1

N∑µ=1

[Dνµ(t) · F(φ)

µ (t)]

+

√2

NB

NB∑ν=1

N∑µ=1

Bνµ(t) ·∆Wµ(t), (5.20)

whereF(φ)µ incorporates all inter- and intramolecular forces acting on beadµ. Following

this, we compute the microstructure of the chain in a three-part prediction-correction

algorithm. The prediction step is again a simple Euler step:

Q∗i = Qi (t) + [κ(t) ·Qi(t)] ∆t

+∆t

kBT

∑µ

[(Di+1,µ(t)− Di,µ(t)) · (Fspr

µ (t)− Fsprµ−1(t))

]+

∆t

kBT

∑µ

[(Di+1,µ(t)− Di,µ(t)) ·

(Fexv

µ (t) + Felµ (t)

)]+

√2∑

µ

(Bi+1,µ(t)− Bi,µ(t))·∆Wµ(t), (5.21)

whereFsprµ is the tension in springµ (note that this is not equivalent to the force on bead

µ due to the connecting springs as defined in Chapter 3). Next, we correct the Euler result

54

with two further steps. The first correction step is

Qi + 2∆t

kBTFspr

i = Qi (t) +1

2[κ(t) · (Qi(t) + Q∗

i (t))] ∆t

+∆t

kBT

∑µ


µ (t)− Fsprµ−1(t))

]+

∆t

kBT

∑µ

[(Di+1,µ(t)− Di,µ(t)) · (Fexv

µ (t) + Felµ (t))

]+

√2∑

µ

(Bi+1,µ(t)− Bi,µ(t))·∆Wµ(t)

+ 2∆tFspri (5.22)

where the spring force term,Fspr

µ , is taken implicitly as:

Fspr

µ =

Fsprµ if µ < i

Fsprµ if µ ≥ i.

(5.23)

Taking the right-hand side of Equation 5.22 asR and using the definition of the FENE

spring force from Equation 3.4, we compute the magnitude of each side of Equation 5.22

and rewrite it as a cubic equation:

Q3i −RiQ

2i −Q2

0(1 + 2∆tH)Qi +RiQ20 = 0 (5.24)

and solve for the resulting spring lengths. Note that this calculation can also be performed

for the wormlike spring chain, resulting in a cubic equation with different coefficients.

The second corrector step and cubic equation are essentially identical to the first set,

55

written only for notational simplicity:

Qi + 2∆t ˆFspri = Qi (t) +

1

2

[κ(t) ·

(Qi(t) + Qi(t)

)]∆t

+∆t

kBT

∑µ


µ (t)− Fspr

µ−1(t))]

+ ∆t∑

µ

[(Di+1,µ(t)− Di,µ(t)) ·

(Fexv

µ (t) + Felµ (t)

)]+

√(2)∑

µ

(Bi+1,µ (t)− Bi,µ (t)) ·∆Wµ (t)

+ 2∆tFspi (5.25)

Q3i −RiQ

2i −Q2

0(1 + 2∆tH)Qi +RiQ20 = 0 (5.26)

where the spring force term,Fsp

µ , is taken implicitly as:

Fsp

µ =

Fsp

µ if µ < i

Fspµ if µ ≥ i.

(5.27)

The second corrector step is then iterated until the difference betweenQi andQi is suf-

ficiently small, replacingQi by Qi at the start of each iteration. We note that the flow

term has been modified in the corrector steps in order to improve the convergence rate.

In addition, both the diffusion tensor and the nonbonded forces are computed explicitly.

Finally, rather than breaking the summation on the right-hand side of Equations 5.22 and

5.25, we have simply added the appropriate term outside the summation for simplicity.

While the predictor-corrector method works well for simple single-chain, dilute so-

lution simulations, it faces numerous difficulties in the simulation of more complicated

systems. The foremost problem has already been described above - namely that only the

spring forces are truly treated implicitly. The explicit treatment of the nonbonded inter-

actions severely weakens the stability of this method; in fact, for equilibrium simulations,

56

it has not proven to provide a significant increase in the suitable time step compared to

that used in the Eulerian simulations. A second difficulty lies in the explicit treatment

of the evolution of the chain centers-of-mass of the chain(s). There is no constraint pre-

venting elements of different chains from locating very near one another following a time

step, resulting in very large forces at the next time step and potentially causing the iter-

ation to become unstable. Such large forces may also be expected to lead to abnormally

large movements in the centers-of-mass of the chains leading to inaccuracies in calcu-

lated properties such as the diffusivity. Finally, in the calculation of the first summation

on the right-hand side of both Equations 5.22 and 5.25, the updating of the vectorF with

each successive spring requires an expensive recalculation of the productD · F. Due to

these difficulties, the performance of the predictor-corrector algorithm does not warrant

further study towards application to our polyelectrolyte system and will not be considered

further.

5.2.4 Comparison of Implicit Methods

We have examined the implicit calculation schemes of Sections 5.2.1-5.2.2 relative to

the standard Euler scheme for speed and stability as well as numerical accuracy. To test

our methods, we used statistical ensembles of 1000 equilibrated systems consisting of a

single 50-bead chain with 50 counterions at a monomer density of10−5. The static struc-

tural properties were identical to those of the Euler method within simulation error. In

Table 5.1, we present results describing the relative speeds of the various computational

methods.

As shown in Table 5.1, the standard Euler scheme is actually more efficient in equi-

librium simulations than either of the semi-implicit schemes for all free-draining cases as

57

Method FD,λB = 0 FD,λB = 1.5 HI, λB = 0 HI, λB = 1.5

Euler 4.4 24.4 46.6 55.0Broyden 65.1 71.6 80.1 87.9

Newton (CG) 26.6 34.2 43.6 56.9Newton (GMRES) 17.5 43.1 33.8 68.9

Table 5.1:Comparison of the average time required (in seconds) for various implicit calculation schemesto achieve1.0 ζσ2

kBT total units of simulation time. Euler time steps are 0.0002 for FD simulations, and0.0005 for HI simulations. The time step for all semi-implicit schemes was taken as 0.0025. Also, Newton’smethod was evaluated using two different equation solvers, one based on the conjugate gradient method,and the other being the GMRES method.

well as most hydrodynamically interacting cases. In addition, Newton’s method is com-

putationally faster than Broyden’s method. The difference in speed is not due to a large

difference in the number of iterations required per time step, but rather, due to the fact

that the Jacobian is symmetric, which leads to a more rapid solution of the corresponding

linear system of equations.

The time step utilized for the semi-implicit methods was chosen as 0.0025 regardless

of whether or not hydrodynamic interactions were included, compared to 0.0002 for the

free-draining Euler simulations and 0.0005 for the Euler simulations with hydrodynamic

interactions included. The semi-implicit time step was chosen as one that was able to

reproduce static structural data from the Euler scheme and suffer from no stability prob-

lems. It is likely that the time step could be chosen somewhat larger, which would reduce

the computational time of the semi-implicit methods. This has yet to be fully investigated,

though any additional increase is not expected to be significant. Ortega and Rheinboldt

(1970) discuss some means for improving the size of the convergence basin for these

iterative processes, such as implementing a steepest descent iteration for improvement

of the initial guess, although these methods further slow down the iterative process. As

a result, we continue to utilize the Euler scheme for the study of equilibrium systems.

58

The semi-implicit schemes may prove to be of more use for flowing systems in which

the time step must be reduced according to the strength of the flow. This has yet to be

investigated.

5.3 Decomposition of the Diffusion Tensor

The primary computational bottleneck in these stochastic simulations is the calculation

of the Brownian motion term. Recalling Equation 5.5, this term is of the form

F(b)ν ∆t =

√2∑

µ

Bνµ(t)·∆Wµ(t), (5.28)

in whichB is a nonunique operator satisfying the relationshipB ·BT = D. The two most

common choices forB are the square root matrix,S, satisfying

D = S · S, (5.29)

with

S = ST , (5.30)

and the triangular matrix resulting from a Cholesky decomposition ofD. While the

Cholesky decomposition is typically preferred, either choice is acceptable (Ottinger,

1996). Both decompositions scale asO(N3), making this step a highly expensive cal-

culation.

Fixman (1986) has circumvented this problem by noting that the actual term desired

is the vector productB · ∆W, and that the explicit calculation ofB is not required.

Using a Chebyshev polynomial expansion, Fixman was able to construct a vector ap-

proximation to the desired product that scales asO(N2.25). In this section, we present the

59

basic algorithm involved in computing this vector approximation as originally described

by Jendrejack et al. (2000) and discuss some of the key issues that arise in the use of

Fixman’s method.

Let s(d) be thepth order Chebyshev polynomial approximation (Canuto et al., 1988)

of the scalar function√d over the domain[λmin, λmax]. Thens(d) can be expressed as

s(d) =

p∑l=0

alCl, (5.31)

where the Chebyshev polynomials are given by

C0 = 1, (5.32)

C1 = dad+ db, (5.33)

Cl+1 = 2(dad+ db)Cl − Cl−1, (5.34)

and the translation from the desired domain to the Chebyshev domain[−1, 1] is

da =2

λmax − λmin

, (5.35)

db = −λmax + λmin

λmax − λmin

. (5.36)

For the of the Chebyshev coefficients,al, we refer the reader to Canuto et al. (1988).

Using the properties of functions of matrices (Wylie and Barrett, 1995), generalization

of the above scalar case givesS(D), the Chebyshev polynomial approximation of the

matrix functionD1/2 is:

S(D) =n∑

l=0

alCl, (5.37)

60

where

C0 = I , (5.38)

C1 = daD + dbI , (5.39)

Cl+1 = 2(daD + dbI)Cl − Cl−1. (5.40)

Theal are the same Chebychev coefficients as obtained in the scalar case and the eigen-

values ofD are bounded by[λmin, λmax]. As mentioned above, the explicit calculation of

S(D) is not necessary. Rather, we are interested in the quantityS · dw, whose polynomial

approximationy may be obtained by a series of matrix-vector multiplications

y = S(D) · dw =n∑

l=0

alxl, (5.41)

x0 = dw, (5.42)

x1 = [daD + dbI ] · dw, (5.43)

xl+1 = 2 [daD + dbI ] · xl − xl−1. (5.44)

Thus, assuming that we have knowledge of the bounds for the eigenvalues, we may di-

rectly compute the Brownian movement term to any desired accuracy without the need

for directly computing the diffusion tensor.

Hence, the only issue remaining is that of the calculation of the bounds for the eigen-

values. Even using a rapid solver capable of obtaining the upper and lower eigenvalues

in O(N2) operations, the recalculation of the eigenvalue limits at each time step signifi-

cantly reduces the computational savings of Fixman’s method. To avoid this, Jendrejack

et al. (2000) have proposed avoiding the calculation of the eigenvalue range at each step

and instead simply using the same range from step to step. In order to evaluate the error

stemming from the potential violation of the eigenvalue range, Jendrejack et al. (2000)

61

have developed a relative error term that may be used to judge convergence. Assuming

that the eigenvalue range is valid, we have from Equations 5.29, 5.30, and 5.41

limp→∞

[y · y] = dw · D · dw. (5.45)

A relative errorEf may then be defined as

Ef =

√|y · y− dw · D · dw|

dw · D · dw. (5.46)

If the error does not satisfy a given tolerance (0.05 in the present work) within a specified

number of iterations, the eigenvalue range is simply recalculated and used until another

violation occurs. In this manner, we are able to calculate the Brownian motion term at

reasonable computational expense.

5.4 Nonequilibrium Simulations

One of the difficulties in simulating a system at finite concentration is finding an appro-

priate means of treating the periodic geometry of the system when the system is not at

equilibrium. For infinitely dilute systems, there are no geometric constraints involved

in applying simple deformations. When systems at finite concentration are considered,

however, the geometry of the periodic cell lattice must be treated with care. There are two

essential considerations involved - lattice compatibility and lattice reproducibility. The

former issue deals with whether the deformation of the lattice by a specified flow causes

any aphysical behavior of the particles simulated, while the latter deals with whether or

not the lattice may be periodically reformed during the course of the simulation. Both

issues are critical to the simulation of infinitely long times under flow conditions. As

62

it turns out, however, compatibility follows from reproducibility, making reproducibility

the stronger condition to be satisfied.

These issues have been addressed for general lattice deformations by Adler and Bren-

ner (1985a,b), and more specifically by Lees and Edwards (1972) for simple shearing

flows and by Kraynik and Reinelt (1992) for a variety of extensional flows. To begin,

consider an arbitrary three-dimensional lattice consisting of all the pointsRn = n1b1 +

n2b2 + n3b3, whereb1, b2, b3, are linearly independent basis vectors andn = n1, n2, n3

is an integer triple. For a homogeneous, isochoric deformation, the time evolution of the

basis vectors satisfies

dbi

dt= D · bi (5.47)

whereD is a traceless, constant diagonal matrix. Thus, in terms of the initial basis vectors

boi , we get

bi = Λ · b0i (5.48)

whereΛ = exp(Dt). For a givenΛ, a lattice is reproducible if and only if there exist

integersNij, such that

Λb0i = Ni1b0

1 +Ni2b02 +Ni3b0

3; i = 1, 2, 3. (5.49)

As a result, we may rewrite these vector equations as an eigenvalue problem:

(N− λiI) · ci = 0 (5.50)

whereN is an integer matrix composed of elementsNij, λi = expDit, and the vectorci

contains theith components of the basis vectorsb0j .

63

The problem of reproducibility has now been reduced to finding a solution to the

eigenvalue problem, where the eigenvaluesλi represent potential strain periods and the

eigenvectorsci give a basis set for a reproducible lattice corresponding to the potential

strain periods. In the remainder of this section, we discuss specific solutions to this

problem for simple shear and elongational flows and their application to the simulations

at present. In addition, we also consider necessary adaptations to the Ewald summation

technique proposed by Wheeler et al. (1997) for noncubic lattices and to the calculation

of minimum image distances that are important to both classes of flow conditions.

5.4.1 Simple Shear Flows

In simple 2-D shearing flow, Adler and Brenner (1985a,b) have demonstrated that the

characteristic polynomial stemming from the eigenvalue problem in Equation 5.50 may

be solved for any strain period provided that the flow direction is parallel to one of the

basis directions. Lees and Edwards (1972) exploited this fact for the cubic simulation

cell to modify the standard periodic boundary conditions into a form that now bears their

names.

We begin with a system arranged in a perfect cubic lattice configuration with the

neighboring boxes aligned to the cell under consideration, as in an equilibrium study.

On imposing a simple shear flow with shear rateγ, we effectively impart an additional

velocity term in thex-direction to each particle, resulting in a linear velocity profile cen-

tered about the midpoint of the simulation cell. In order to maintain this velocity profile

across the periodic boundary, we could naıvely allow the periodic boundaries to deform

with the flow, and then reform the lattice following the simulation of each full strain pe-

riod. However, this type of boundary deformation leads to unneccesary computational

64

Figure 5.1: Depiction of the sliding cell layers in simple shear flow illustrating the use of the Lees-Edwards boundary conditions. Shown in the system is a 10-bead polyelectrolyte chain straddling acrossthe cell boundary along with surrounding counterions.

expense when computing periodic interactions, as will become clear shortly. Instead,

Lees and Edwards proposed a set of boundary conditions (LEBC’s) in which neighbor-

ing layers of cells in they-direction are allowed to “slide” past one another. A depiction

of this behavior is shown in Figure 5.1. As a result, a particle exiting the simulation cell

in they-direction will reenter the cell through the opposite face, but with some additional

displacement in thex-direction. For the case of simple shear flow, this displacement is

given by

∆x = (ntγ∆t− [ntγ∆t])L, (5.51)

wherent is the number of elapsed time steps,γ is the shear rate,∆t is the time step,L

is the box size, and[Y ] is the greatest integer which is less thanY . This description of

the boundary conditions is equivalent to that in which the periodic lattice is permitted to

65

deform with the flow, however, the standard cubic simulation cell is preserved.

5.4.2 Planar Elongational Flows

While simple shear flows may be handled relatively easily through the use of Lees-

Edwards boundary conditions, elongational flows are far more difficult to simulate. While

Adler and Brenner (1985a,b) have analyzed the reproducibility of periodic lattices in gen-

eral, their work concentrates primarily on simple shear flows. Further analysis by Kraynik

and Reinelt (1992) has produced a number of surprising results regarding specific elon-

gational flows, and it is primarily their work from which we draw upon here. To our

knowledge, the only other work using KRBCs in numerical simulation are the molecular

dynamics simulations of Daivis et al. (2003).

Using simple geometric arguments, Kraynik and Reinelt have demonstrated that in

elongational flow, the characteristic equation to this eigenvalue problem has three real

solutions only for a discrete set of strain periods. This contrasts sharply with simple

shear flow in which the characteristic equation has a continuum of solutions. Writing the

characteristic equation as

p(x) = x3 − kx2 +mx− 1 = 0 (5.52)

k = λi + λ2 + λ3 (5.53)

m =1

λ1

+1

λ2

+1

λ3

(5.54)

1 = λ1λ2λ3 (5.55)

there exist reproducible lattices for each of the elongational flows defined by the integer

pairs(k,m) that lie in the regionm ≤ k2/4 andk ≤ m2/4. When we restrict ourselves

to a specific geometry for the undeformed lattice, the set of permissible strain periods is

66

k(m) λ ε = lnλ θ

3 2.618 0.962 0.5546 5.828 1.763 0.39311 10.908 2.390 0.29415 14.933 2.704 0.61415 14.933 2.704 0.36918 17.944 2.887 0.232

Table 5.2:Strain periodic orientations for a square lattice in planar elongational flow.

further reduced. Finally, Kraynik and Reinelt have shown that it is impossible to find a

lattice that is reproducible in either biaxial or uniaxial extensional flow. We may, however,

find solutions for the case of planar elongational flow, in which the eigenvalues are given

by λ, 1/λ, and 1 (and thusk = m).

These conditions severely restrict the parameter space in which we can explore elon-

gational flows, and in this work, we will restrict ourselves to planar elongational flow on

square lattices. Table 5.2 describes some possible elongational flows and rotational ori-

entations (θ) for which reproducible square lattices may be found. The set of lattices that

may be coupled to an elongational flow are referred to as the Kraynik-Reinelt boundary

conditions (KRBCs). Note that there may exist more than one lattice orientation for a

particular strain period.

In simple shear flow, exact reproducibility is easily achieved via the use of LEBCs

by matching the time step of the simulation to the imposed flow rate as the deformation

occurs directly along one of the Cartesian directions. Unfortunately, this is not the case

for elongational flows. Instead, we allow the periodic lattice to deform during the course

of a strain period, and then reform the lattice at the end of the period to recover the initial

shape of the simulation cell. This is depicted in Figure 5.2 for a system consisting of

a single 10-bead polyelectrolyte chain with surrounding counterions. By deforming the

67

lattice in this way, the use of KRBCs causes the lattice points to move along curved

streamlines. Using an explicit Euler scheme, we approximate this curved streamline by

a series of small linear increments. This leads to a small approximation error in that the

deformed lattice does not exactly reproduce the original lattice at the end of a deformation

period. When a chain straddles the cell boundary at the end of a simulation period, the

bond straddling the boundary will be slightly altered due to the discrepancies between the

deformed lattice and original lattice. The number of chains straddling a cell boundary at

a given density scales asN−1/3C , indicating that the use of larger systems should decrease

the influence of this error. In addition, the difference between the coordinates of the

two dimensionless lattices at the end of a period is of the order of the time step used,

and so decreasing the time step will also reduce this source of error. We have carried

out simulations ofλ-phage DNA in Chapter 6 forNC = 100 with varying time steps in

order to estimate the effect of such an error and found that our results show negligible

dependence on the time step used. As such, we use identical time steps in shear and

elongational flows, as described in Section 6.3.

5.4.3 Additional Considerations

Modifications to the Ewald Sum

To this point, we have considered the use of Ewald summation based solely on cubic

periodic boundary conditions. As noted above, however, in a nonequilibrium simulation,

it becomes necessary to deform the periodic geometry in a manner consistent with the

imposed flow. As a result, we must update the set of lattice basis vectors,Γ, at each time

step for the accurate calculation of the Ewald-summed interactions. The resulting basis

68

Figure 5.2:Depiction of the stretching cell layers over one period in planar elongational flow illustratingthe use of the Kraynik-Reinelt boundary conditions. Shown in the system is a 10-bead chain with sur-rounding counterions. The dark lines give the original cell lattice while the light lines represent the currentlattice.

69

set in simple shear flow at time stept is

Γs (t) =

1 γt∆t 0

0 1 0

0 0 1

(5.56)

and in planar elongational flow is

Γp (t) =

(1 + εt∆t)b11 (1 + εt∆t)b12 0

(1 + εt∆t)b21 (1 + εt∆t)b22 0

0 0 1

(5.57)

with t = mod(

t+P/2P

)− P

2, whereP is the number of time steps for the lattice to

reproduce itself in flow. The vectors(b11, b21) and(b12, b22) describe the initial orientation

of the lattice in the flow plane as determined from the KRBC’s. Using these basis sets,

we may then select appropriate lattice vectorsn and wavevectorsk for the calculation

of D. Finally, as noted by Wheeler et al. (1997), care must be taken when computing

each summation to ensure that the selected lattice vectors result in a summation that

is in fact spherically symmetric regardless of the lattice deformation. In equilibrium

simulations, the reciprocal-space spherical cut-off value is typically established based on

the magnitude ofn rather thank for computational efficiency. This is permissible since

any permutation of a givenn will produce a wave-vector of the same magnitude as that

corresponding ton. For nonequilibrium simulations, however, this is no longer the case.

Instead, we must base the cut-off value on the magnitude of the wave-vectors themselves

to ensure that we maintain spherical symmetry.

70

Minimum image convention

While LEBC’s and KRBC’s are useful in properly coupling the flow with the periodic

boundary conditions in order to determine the proper particle trajectories, the calculation

of the long-ranged hydrodynamic interactions requires an accurate description of the de-

forming lattice conditions as well. This stems from the fact that the minimum distance

between a reference particle and a target particle, including periodic images of the target,

cannot be calculated for each basis direction independently as may be done for a sim-

ple Cartesian lattice (e.g. the undeformed cubic lattice). When the lattice is noncubic,

the situation is more difficult in that the minimum distance may require using a peri-

odic image that does not minimize one of the basis directions. This can be easily solved

through brute force by computing the distance between the reference particle and all of

the possible target images within some number of box sizes and selecting the minimum

pair. However, this is clearly computationally inefficient. Instead, we observe that the

minimum distance between a pair of particles in a planar periodic geometry must involve

a minimization in a direction orthogonal to at least one of the two basis directions. Thus,

we may find the nearest image particle to a given target particle by simply finding the

nearest images when minimizing against each basis direction independently, and then

selecting the one that provides the closer image overall.

71

Chapter 6

CONCENTRATION DEPENDENCE

OF SHEAR AND EXTENSIONAL

RHEOLOGY OF POLYMER

SOLUTIONS

6.1 Introduction

Considerable simulation work has been done regarding the single-chain behavior of poly-

mers in a solvent, representing the “infinitely dilute solution” case in which all inter-

molecular interactions are absent (Jendrejack et al., 2002b; Hernandez-Cifre and de la

Torre, 1999; Neelov et al., 2002; Schroeder et al., 2004, 2003; Kobe and Weist, 1993;

Agarwal et al., 1998; Agarwal, 2000; Fetsko and Cummings, 1995; Liu et al., 2004; Lar-

son et al., 1999; Sunthar and Prakash, 2005). Additional work by Ahlrichs et al. (2001)

72

has extended the single-chain model to the study of diffusion in semidilute solutions.

However, comparatively little study has been done regarding the dynamic behavior of

non-dilute, multiple chain polymer solutions at concentrations approachingc∗, wherec∗

is the overlap concentration (the concentration at which the combined pervaded volume

of the chains is equal to the volume of the system as a whole), especially when the so-

lutions are subjected to elongational flows. Despite the lack of study, these non-dilute

solutions are nevertheless of significant interest in many practical applications and dis-

play some highly interesting behaviors. For example, upon adding a small amount of

polymer to an otherwise Newtonian solvent, the flow resistance is far stronger when the

solution is subjected to an elongational flow than when a shear flow is imposed. This

has potential ramifications in many applications in which there is a strong elongational

component to the deformation of the solution, including fiber spinning, coating flows and

turbulent drag reduction.

Experimentally, far more work has been performed to study non-dilute, low concen-

tration solutions than has been done by simulation (Babcock et al., 2000; Hur et al., 2001;

Gupta et al., 2000; Ng and Leal, 1993; Link and Springer, 1993; Lee et al., 1997; Owens

et al., 2004; Dunlap and Leal, 1987). For example, Owens et al. (2004) have demonstrated

that the molecular relaxation time, as calculated by capillary thinning experiments, a tech-

nique utilizing extensional flows, exhibits a dependence on the polymer concentration for

concentrations as low as0.05c∗. When the relaxation time is calculated via shear exper-

iments using a cone-plate fixture, however, they do not see an appreciable concentration

dependence. This result corroborates the earlier work of Hur et al. (2001), in which the

authors investigated the behavior ofλ-phage DNA solutions in shear flows using fluo-

rescence microscopy with a parallel plate device and found no measurable change in the

73

distribution of chain extensions for individual molecules in solutions at concentrations

up to6c∗. It has been proposed by Owens et al. (2004) and Clasen et al. (2004) that the

concentration dependence exhibited by solutions in elongational flows may be rational-

ized by considering that chains under such flows stretch to a much larger degree than do

those in a shear flow, and so are more likely to interpenetrate at lower concentrations.

An interesting contrast, however, is exhibited in the work of Gupta et al. (2000) on di-

lute polystyrene solutions in which the the authors used a filament-stretching device to

reproduce uniaxial extensional flows. Their results indicate that the extensional viscosity

is simply proportional to concentration for concentrations in the rangec/c∗ ∈ (0.1, 1.0),

as expected for simple dilute solution (i.e. noninteracting molecules).

Previous computational studies of polyethylene solutions by Kairn et al. (2004a,b) in

shear flows have involved the use of nonequilibrium molecular dynamics (NEMD) and

have qualitatively confirmed some theoretical predictions, such as an expected increase in

shear viscosity as the concentration increases at low strain rates. However, their results do

not give quantitative agreement with scaling theories due to the fact that they are restricted

to the use of very short chains (≈ 12 Kuhn segments). The use of such short chains is

responsible for the absence of a semi-dilute regime in these solutions, and so attempting

to study the transition from dilute to semi-dilute solution becomes impossible. Thus, we

have undertaken a study of such non-dilute systems on much larger length scales in this

work to investigate the behavior of polymer solutions as the concentration approaches

the overlap concentration. To our knowledge, no work has yet been performed utilizing

Brownian dynamics for the simulation of flowing non-dilute, low concentration solutions

with fluctuating hydrodynamic interactions.

74

At present, we are interested in addressing the concentration dependence of the struc-

tural and rheological behavior of polymers in dilute solutions, including the effect of

hydrodynamic interactions at varying concentrations. To this end, we have carried out

Brownian dynamics simulations for a bead-spring model of 21µm λ-phage DNA both

at equilibrium and when subjected to simple shear and planar elongational flows. The

remainder of this work is organized as follows: In Section 6.2, we present the model

and governing equations, including a discussion of the system size and handling of the

periodic boundary conditions with respect to the hydrodynamic interactions. In Section

6.3, we discuss the simulation methods used as well as the applied boundary conditions.

We present the results of our simulations in Section 6.4, including descriptions of the

equilibrium structure, diffusivity, longest relaxation time, and response to simple shear

and planar elongational flows. We conclude in Section 6.5 with a summary of our results.

6.2 Model

In this work, we are concerned with the numerical simulation of a solution of monodis-

perse, linear polymer chains immersed in an incompressible Newtonian solvent. We

approach the problem at the mesoscale level and coarse-grain each polymer chain into a

sequence ofNB “beads” connected byNS = NB−1 “springs”. The maximum extension

of each spring is taken asq0, yielding an overall chain contour length ofL0 = NSq0. A

total ofNC chains are initially enclosed in a cubic cell of edge lengthL, giving a total of

N = NBNC beads per cell at a bulk monomer concentration ofc = NV

, whereV = L3 is

the volume of the simulation cell.

The model and parameterization used in this work are based on that of Jendrejack

75

et al. (2002b). This model has been used to successfully reproduce the transient and

steady state behavior of infinitely dilute 21µm YOYO-1 stainedλ-phage DNA in both

simple shear and planar elongational flows over a wide range of Weissenberg numbers.

It has also been used to successfully predict diffusivity results that are in quantitative

agreement with experimental data for infinitely dilute chains ranging from 21 to 126µm

(Jendrejack et al., 2002b), as well as for DNA in a slit (Chen et al., 2004). As a result,

we anticipate that with the modifications included here, this model should provide useful

predictive capabilities of both static and dynamic properties of bulk solutions of DNA at

nonzero concentration.

Adjacent beads of the polymer chain are connected via a worm-like spring model, in

which the force on beadν due to connectivity with beadµ is given by Equation 3.5 and

reproduced here

Fsprνµ =

kBT

2bk

[(1− rµν

q0

)−2

− 1 +4rµν

q0

]rµν

rµν

(6.1)

wherebk is the Kuhn length of the molecule. Good solvent conditions are incorporated

via Equation 3.12, which may be expressed for two particles,ν andµ, as

Fexvνµ = vkBTN

2k,sπ

(3

4πS2s

)5/2

exp

[−3 |r νµ|2

4S2s

]r νµ (6.2)

wherev is the excluded volume parameter andS2s =

Nk,sb2k6

is the mean square radius

of gyration of an ideal chain consisting ofNk,s Kuhn segments. Finally, hydrodynamic

interactions are accounted for through the periodic form of the Rotne-Prager-Yamakawa

tensor as in Equations 3.43-3.47.

Appropriate physical parameters for this model have been determined for this system

by direct comparison to bulk experimental data (Jendrejack et al., 2002b) for YOYO-1

76

stainedλ-phage DNA at room temperature in a 43.3 cP solvent, which has a contour

length ofL = 21 µm (Smith and Chu, 1998). We represent this molecule with a 10-

spring chain (i.e.NS = 10), and by comparing the model to available experimental values

of the relaxation time and equilibrium stretch (Smith and Chu, 1998) and an estimated

diffusivity (Smith et al., 1996; Smith and Chu, 1998; Sorlie and Pecora, 1990; Jendrejack

et al., 2002b), it was determined that suitable parameter values arebk = 0.106 µm,

a = 0.077 µm, andv = 0.0012 µm3. These parameter values set the number of Kuhn

segments per spring atNk,s = 19.8 and the bead diffusivity ofkBTζHI

= 0.065 µm2/s.

The free-draining model is then parameterized to give the same relaxation time as that of

the hydrodynamically interacting model at infinite dilution, yielding a free-draining bead

friction coefficient ofkBTζFD

= 0.076 µm2/s. This differs from the original value of 0.084

µm2/s of Jendrejack et al. (2002b) due to a difference in the calculation of the relaxation

time, as discussed in Section 6.4.3.

In this work, we have normalized the concentration with the overlap concentration,

c∗, to provide a common basis for comparing results of different molecular weights. The

overlap concentration is the concentration at which the combined pervaded volume of

the chains is equal to the volume of the system as a whole. We calculate the overlap

concentration on a monomer basis according to Doi and Edwards (1986) asc∗ = NB43πR3

g,

whereRg is the equilibrium radius of gyration of a polymer chain at zero concentration.

These values are tabulated for various molecular weights in Table 6.1.

Finally, we note that the model used here does not eliminate the possibility of chain

crossings, due to the high computational cost associated with their detection. However, as

we are dealing with solutions primarily in the dilute regime, crossings between separate

chains are expected to be few. We have tested this hypothesis by tracking the number of

77

L0 NS NB Rg c∗

(µm) (µm) (beads/µm3)

10.5 5 6 0.52 10.221 10 11 0.77 5.742 20 21 1.14 3.484 40 41 1.73 1.9

Table 6.1:Radius of gyration and overlap concentration,c∗, for chains of varying molecular weight.

times connecting springs of one or more chains cross one another for systems under flow

at various concentrations. The results are described in Section 6.5. While some chain

crossings are observed, they are infrequent, and we believe that the effects of preventing

such crossings in these systems would be quantitative in nature only, and not affect our

qualitative trends. Furthermore, in extensional flow at high Weissenberg number, perhaps

the most interesting regime described here, chain crossings are virtually absent.

6.3 Simulation

For the calculation of static equilibrium quantities, we used the Monte Carlo scheme

described in Chapter 4. For the calculation of dynamic properties, the main focus of this

work, we used Brownian dynamics simulations as detailed in Chapter 5.

Brownian dynamics simulations were carried out via an explicit Euler scheme for the

solution of Equation 5.1,

r ν (t+ ∆t) = r ν (t) + [κ(t) · r ν(t)] ∆t +∆t

kBT

∑µ

[Dνµ(t) · F(φ)

µ (t)]

+√

2∑

µ

Bνµ(t)·∆Wµ(t), (6.3)

whereF(φ)µ =

∑ω 6=µ Fexv

µω + Fsprµ,µ−1 + Fspr

µ,µ+1 and the Brownian term is calculated via

Fixman’s method as described by Jendrejack et al. (2000). The time step was chosen

78

based on the shortest time scale of the problem, either the bead diffusion time or the flow

rate (∆t = 0.1min

ζS2

s

kBT,[(∇v) : (∇v)T

]−1/2

), for all simulations except for some

planar elongational flows at high flowrates. Due to a high degree of chain extension in

these flows, we are forced to further reduce the time step to ensure computational stability.

Also, unless otherwise noted, simulations were run for sufficient time and ensemble sizes

to reduce the error bars to the order of the symbol size used here.

One of the primary difficulties that arises in the simulation of a bulk fluid at nonzero

concentration stems from the use of periodic boundary conditions (Allen and Tildes-

ley, 1987). Periodic boundary conditions are often employed in numerical simulations

to avoid spurious surface effects from artificially imposed containment. However, by

imposing periodic boundary conditions, we risk imposing artificial symmetries on the

system. In addition, we may introduce unphysical interactions in the system by allowing

a large molecule to interact directly with its own image through the periodic boundary.

Thus, we must take care in designing our systems so as to minimize such effects.

The calculation of dynamic properties, including the diffusivity and longest relax-

ation time, is difficult due to the long range nature of the hydrodynamic interactions.

Ideally, one should choose a system size at least twice the length of the contour length of

an individual chain so that a chain may not directly interact with its own image through

the periodic boundary. However, this becomes highly computationally demanding even

for low concentrations (Table 6.2). We have therefore adoptedNC = 100 as our basic

system, striking a compromise between accuracy and computational efficiency. In order

to estimate the effects of systems size, we have calculated select dynamic properties for

systems at various concentrations usingNC = 50,NC = 100, andNC = 200. Figure 6.1

is a representative example, in which we plot the reduced viscosity,ηr, as a function of

79

Figure 6.1:Reduced viscosity,ηr, as a function of Weissenberg number,Wi0, for systems subjectedto planar elongational flow. Comparison of results for systems atc/c∗ = 1.0 when different numbers ofchains per simulation cell are considered. With little difference in the results for systems ofNC = 100 andNC = 200 chains, we useNC = 100 chains for all other results presented in this work.

Weissenberg number,Wi0, for systems subjected to planar elongational flow (see Section

6.5 for definitions of these quantities). While our results show sensitivity toNC on com-

paring systems of 50 and 100 chains, the effect of system size is negligible on comparing

systems of 100 and 200 chains. Combining this observation with computational expense

considerations, we have thus chosen to useNC = 100 for all dynamic simulations pre-

sented here. Simulations of the equilibrium structural properties at higher concentrations

used larger system sizes, as appropriate. Note, however, that these simulations were per-

formed using Monte Carlo techniques for which the calculation expense is much lower

than that of Brownian dynamics.

80

c/c∗ NS = 5 NS = 10 NS = 20 NS = 40

0.001 19 43 101 2250.01 189 427 1003 22410.1 1887 4262 10029 224101 18869 42617 100283 2240922 37738 85234 200565 448183

Table 6.2:Minimum number of chains,NC , required to guaranteeL > 2L0 = 2NSq0 as a function ofconcentration and molecular weight.

6.4 Equilibrium Results

6.4.1 STATIC PROPERTIES

We begin our discussion of the behavior of DNA solutions at nonzero concentration with

an analysis of the equilibrium structure of systems at varying molecular weights and

concentration. To this end, we consider the static structure of the individual polymer

chains through the calculation of the mean square radius of gyration,⟨R2

g

⟩, defined for

an individual chain as⟨R2

g

⟩=

1

2N2B

NB∑ν=1

NB∑µ=1

⟨(r ν − rµ)2

⟩. (6.4)

Shown in Figure 6.2 is⟨R2

g

⟩as a function of concentration for chains of contour lengths

ranging between 10.5µm (Ns = 5) and 84µm (Ns = 40). As expected, the static size

is insensitive to changes in concentration for very low concentrations (i.e.c/c∗ < 0.1).

However, for concentrations greater than or equal to0.1c∗, we observe a decrease in chain

size as concentration increases. This identifies well with an expected gradual transition

from dilute to semi-dilute behavior, as also exhibited in the work of Paul et al. (1991).

The decrease in the size of the polymer chains with increasing concentration can be

explained by considering the excluded volume contribution to the total energy of the sys-

tem. At very low concentrations, there are effectively no energetic interactions between

81

Figure 6.2:Mean square radius of gyration,⟨R2

g

⟩, plotted as a function of normalized concentration,

c/c∗, for various chain lengths.

beads of different chains. However, as the concentration increases, chains are brought

into closer proximity to one another. Above a concentration of0.1c∗, the chains are close

enough to one another such that an interchain excluded volume potential develops (Figure

6.3). This results in each chain experiencing repulsions from its surrounding neighbors

which causes the chain to compact. The effect is enhanced as concentration continues to

increase.

From theoretical predictions (Rubinstein and Colby, 2003), we expect the polymer

chain size to scale with concentration and molecular weight as

R2g ∝ c0N2ν

S (6.5)

in dilute solution and as

R2g ∝ c

1−2ν3ν−1NS (6.6)

82

Figure 6.3:Excluded volume energy contribution to the net system energy for 21µm DNA at variousconcentrations.

in the semi-dilute regime. For a good solvent, the scaling exponentν is approximately

0.6. As demonstrated in Figure 6.4, we observe that our simulation results are in agree-

ment with theoretical predictions for the scaling of chain size with concentration in both

the dilute and semi-dilute regime, although the latter scaling applies only for sufficiently

large chain lengths, with a crossover region occuring forc/c∗ between 1.0 and 10.0. Our

results also agree well with the predicted scaling of polymer chain size with molecular

weight in both the dilute and semi-dilute regimes (Figure 6.5). As with concentration,

the crossover between predicted scaling behaviors lies at a concentration betweenc∗ and

10c∗. In both analyses, the semi-dilute regime is predicted to hold for concentrations as

large as100c∗.

83

Figure 6.4:Scaling of the static chain size as a function of normalized concentration. Solid lines indicatefits following the scaling law

⟨R2

g

⟩∝ (c/c∗)0 in the dilute regime (c/c∗ ≤ 1.0) and

⟨R2

g

⟩∝ (c/c∗)−0.25

in the semi-dilute regime (c/c∗ ≥ 1.0).

Figure 6.5:Scaling of the static chain size as a function of molecular weight atc/c∗ = 0.1 and100. Solidline indicates predicted dilute regime scaling (2ν = 1.2) while dashed line gives the expected semi-dilutescaling (2ν = 1.0).

84

6.4.2 DIFFUSIVITY

We begin our consideration of the dynamic behavior of the 21µm DNA with the calcu-

lation of the short- and long-time diffusive behavior as a function of concentration. The

short-time diffusivity of an individual polymer chain has been calculated via both the

center-of-mass definition and the Kirkwood formula, given respectively as

DCS = lim

∆t→0

kBT

6ζ

⟨|r c(t+ ∆t)− r c(t)|2

∆t

⟩, (6.7)

wherer c = 1NB

∑NB

ν=1 r ν is the center of mass of a chain, and

DKS =

kBT

3N2Bζ

NB∑ν=1

NB∑µ=1

(tr 〈Dνµ〉) . (6.8)

The latter expression is a simplified form of the true diffusivity found by preaveraging

the hydrodynamic interactions over a statistical ensemble at equilibrium and by assuming

that the external (non-connector based) forces acting on all beads are identical (Bird et al.,

1987; Liu and Dunweg, 2003). The Kirkwood formula produces results that match those

of the exact center-of-mass approach well within simulation error, and so we do not

distinguish between them, denoting the short-time diffusivity simply asDS. The long-

time diffusivity is calculated by tracking the mean-square displacement of the center of

mass of each chain,

DL =kBT

ζlimt→∞

⟨|r c(t)− r c(0)|2

6t

⟩. (6.9)

In this work, we calculated the short-time diffusivity for simulations at two different time-

steps,∆t = 0.1 ζS2s

kBTand∆t = 0.01 ζS2

s

kBT, with no discernable difference in the results.

As described in Section 8.2, the parameterization for our model of stainedλ-phage

DNA includes matching the short-time diffusivity of our full model with hydrodynamic

85

interactions included to an experimentally estimated value to obtain a bead diffusivity

of kBTζHI

= 0.065 µm2/s. The free-draining model was parameterized to match the re-

laxation time of the hydrodynamically interacting model, yielding a slightly higher free-

draining bead diffusivity,kBTζFD

= 0.076 µm2/s. Nevertheless, the presence of hydro-

dynamic interactions leads to higher chain diffusivities than in the free-draining case

(DHIS,c=0 = DHI

L,c=0 = 0.0115 µm2/s, DFDS,c=0 = DFD

L,c=0 = 0.0069 µm2/s). In the free-

draining case, each bead experiences identical frictional drag in the solvent, leading to

the Rouse value ofDFDS,c=0 = DFD

L,c=0 = kBT/NBζFD. When hydrodynamic interactions

are included, however, the motion of the solvent within the pervaded volume of a chain

is coupled to the motion of that chain, thereby screening the polymer segments at the

interior of the chain from frictional drag and causing the collective body to effectively

move as a single solid particle with less net drag.

We next consider the effects of concentration on both the short and long-time diffusiv-

ities. To avoid confusion when comparing free-draining and hydrodynamically interact-

ing results, we have normalized our diffusivity data for systems at varying concentrations

by the diffusivity of the infinitely dilute case in Figures 6.6(a) and 6.6(b). It is readily

apparent from Figure 6.6(a) that the inclusion of hydrodynamic interactions has a signif-

icant impact on the concentration dependence ofDS. In the absence of hydrodynamic

interactions, the short-time diffusivity is insensitive to changes in concentration in the di-

lute regime. When hydrodynamic interactions are accounted for, however, the short-time

diffusivity shows a notable decrease as concentration increases (≈ 15% at c/c∗ = 1.0)

throughout the dilute regime despite the fact that the chain structure is roughly unchanged

over much of this regime. As the polymer concentration increases, interchain hydro-

dynamics become significant, coupling the motion of multiple chains together with the

86

intermediate solvent to an increasing degree. Thus, each chain effectively sees a more vis-

cous solvent with increasing concentration, and the diffusivity decreases. These results

are consistent with the Stokesean Dynamics simulations of Sierou and Brady (2001) for

colloidal suspensions. Interestingly, forc/c∗ ∈ [0.1, 0.5], we observe a plateau region

in the calculated value ofDS before it resumes a decreasing trend as we enter the semi-

dilute regime. The origin of this plateau region is believed to be a competition between

the increased interchain hydrodynamic coupling (decreasingDS) and the onset of inter-

chain excluded volume repulsions, which serve to compact the chains, decreasing the

effective volume fraction and increasingDS. For concentrations higher thanc/c∗ = 0.5,

the interchain hydrodynamic coupling dominates and the short-time diffusivity decreases

further.

For low concentrations, the long-time diffusive behavior for each hydrodynamic case

(Figure 6.6(b)) is similar to the short-time diffusive behavior described above, with the

free-draining results independent of concentration while those with hydrodynamic inter-

actions included show a minor decrease. Above a concentration ofc/c∗ ≈ 0.1, however,

the solvent conditions become important as interchain excluded volume repulsions hin-

der the ability of chains to be able to diffuse about one another in solution and lead to a

decrease in chain diffusivity in both hydrodynamic cases.

Finally, we consider the ratio of the long-time diffusivity to the short-time diffusivity,

DL/DS, for each hydrodynamic case as a function of normalized concentration,c/c∗.

From Figure 6.7, we observe that this normalization tends to bring the free-draining and

hydrodynamically interacting results into agreement with one another, as observed for

87

(a) Short-Time Diffusivity

(b) Long-Time Diffusivity

Figure 6.6: (a) Short-time and (b) long-time time diffusivity normalized against that of the infinitelydilute case for 21µm λ-phage DNA systems as a function of normalized concentrationc/c∗ both with andwithout hydrodynamic interactions. At infinite dilution, the short-time and long-time diffusivities matchfor each hydrodynamic case and areDHI

S = DHIL = 0.0115 µm2/s andDFD

S = DFDL = 0.0069 µm2/s.

88

Figure 6.7:Ratio of long-time to short-time diffusivity of 21µm DNA systems as a function of normal-ized concentrationc/c∗ both with and without hydrodynamic interactions.

colloids (Moriguchi, 1997; Medina-Noyola, 1988). This indicates that the effects of in-

cluding hydrodynamic interactions are roughly identical in both the short-time and long-

time diffusivities, and so are offset in the calculation of this ratio. Thus, we may use the

hydrodynamic behavior of the short-time diffusivity to gain insight towards the behavior

of the long-time diffusivity. This is an important result as the short-time diffusivity is a

far easier quantity to determine.

6.4.3 RELAXATION

We next consider the calculation of the longest relaxation time for an individual polymer

chain as a function of concentration for the 21µm DNA model. We calculate the longest

relaxation time in a manner analogous to that employed in experiments (Smith and Chu,

1998), where one fits the decay of the mean square end-to-end distance for a suddenly

89

released extended chain to an exponentially decaying function. That is,

⟨R2⟩

= ae−t/τ + b (6.10)

whereτ is the relaxation time anda andb are constants determined by the boundary

conditions. Upon applying the boundary conditions〈R2〉 = 〈R20〉 at t = 0 and〈R2〉 =

〈R2∞〉 at t = ∞ and solving fora andb, we have,

⟨R2⟩

=(⟨R2

0

⟩−⟨R2∞⟩)e−t/τ +

⟨R2∞⟩. (6.11)

Hence, we plot log

(〈R2〉−〈R2

∞〉〈R2

0〉−〈R2∞〉

)against time, and obtain the longest relaxation time of

the system from the slope of the region spanning〈R2〉−〈R2

∞〉〈R2

∞〉 < 3.0. For the systems at

present, we initiate the calculation by applying a planar extensional flow to a previously

equilibrated system until the chains achieve a stretch that is many times the equilibrium

size. Each system is then allowed to relax to equilibrium in the absence of flow while

tracking the decay of the mean-square end-to-end distance.

The longest relaxation times extracted from these curves are summarized in Table

6.3, along with experimental results from the work of Hur et al. (2001) for flourescently

stainedλ-phage DNA. The inclusion of hydrodynamic interactions leads to a larger con-

centration dependence in the calculated relaxation time, though the differences are less

than 10%. More importantly, however, is the observation that our simulations, despite

being based on a model of an infinitely dilute chain, are able to accurately predict the

experimentally determined relaxation time of a chain atc/c∗ = 1.0 as determined by

Hur et al. (2001), once the difference in solvent viscosities between the simulations and

experiments has been accounted for. This provides confidence in the use of this model to

accurately predict dynamic behavior throughout the dilute regime. We note that the re-

laxation times calculated for the infinitely dilute cases are approximately 30% larger than

90

c/c∗ τFD(s) τHI(s) τexp(s)

0.0 5.4 5.4 5.40.001 5.4 5.40.01 5.4 5.40.1 5.6 5.80.5 5.9 6.31.0 6.4 6.8 6.82.0 7.0 7.3

Table 6.3:Calculated longest relaxation times for 21µm DNA as a function of concentration both withand without hydrodynamic interactions. Experimental values are those of Hur et al. (2001), where thesolvent viscosity has been normalized to match that of our simulated system.

the experimental value of4.1 sec from the work of Smith and Chu (1998) used in the orig-

inal parameterization of this model, though we have excellent agreement with the more

recent results. This discrepancy highlights the difficulty in obtaining a highly accurate

measure of the relaxation time due to the large degree of noise inherent in the tail region

of relaxation curve. The parameterization of Jendrejack et al. (2002b) was based in part

on attempting to match the simulation results to the experimental values. However, the

fitting of a single exponential function to the tail end of the decay of the mean-square end-

to-end distance is an inexact method requiring large ensembles to obtain good statistical

values. We have expanded the statistical ensemble used in this calculation to eliminate

much of the noise and improve the accuracy of this calculation from the original work

while using the same parameter values, leading to the observed difference. Finally, as

described in Section 6.2, we use the longest relaxation time of the free-draining chains

to establish a value for the bead friction coefficient by forcingτFD,c=0 = τHI,c=0. Using

τHI,c=0 = τFD,c=0 = 5.4 sec, this yields a bead diffusivity ofkBTζFD

= 0.076 µm2/s, which

is roughly10% smaller than the originally calculated value ofkBTζFD

= 0.084 µm2/s.

91

6.5 Dynamic Results

We next turn our attention to the behavior of our systems when subjected to an imposed

flow. In this work we consider two types of flow, simple shear and planar elongational.

In simple shear flow,∇v is given by

(∇v)s =

0 0 0

γ 0 0

0 0 0

(6.12)

while in planar elongation it is

(∇v)p =

ε 0 0

0 −ε 0

0 0 0

. (6.13)

We consider systems with concentrations up toc/c∗ ≈ 2 over a wide range of flow

rates for each flow type, and are primarily focused on the chain structure and rheology

in flow. The measure of chain size most easily obtained from fluorescence microscopy

experiments is the average flow-direction “stretch”,X, defined as the distance between

the upstream-most portion of the molecule and the downstream-most portion,

X = 〈max(r ν,x)−min(r ν,x)〉 , (6.14)

wherer ν,x is the x-component of the position vector of beadν. The rheological behavior

of the polymer solutions is investigated by considering the reduced viscosity,

ηr =ηpc

∗

ηsc, (6.15)

where the polymer contribution to the viscosity for simple shear flow is in turn given by

ηsp = −τp,12

γ(6.16)

92

and for planar elongational flow by

ηpp = −τp,11 − τp,22

ε. (6.17)

In calculating the viscosity in this way, we have normalized the viscosity against the

monomer concentration so as to eliminate the simple linear concentration dependence. In

both cases, the polymer contribution to the stress tensor for the system,τp, is calculated

as

τp =1

2V

∑µ

∑ν

r νµF(φ)νµ . (6.18)

The indicesµ andν are taken over all particle pairs andF(φ)νµ incorporates all nonhydro-

dynamic forces for a given particle pair.

In the following sections, we present our results in a number of ways to better clarify

certain trends of interest. Structural and rheological results are presented in terms of two

different Weissenberg numbers,Wi0 andWic. The Weissenberg number is defined as

the product of the solvent deformation rate,γ for shear flow andε for elongational flow,

and the longest molecular relaxation time. Given that the relaxation time depends on the

concentration of the system, we defineWi0 based on the relaxation time of the infinitely

dilute system,τ0, i.e. Wi0 = τ0γ in shear flow andWi0 = τ0ε in elongational flow,

and we defineWic based on the relaxation time at the concentration of interest,τc, i.e.

Wic = τcγ in shear flow andWic = τcε in elongational flow. We compare our results

based both on the actual strain rate as well as the concentration dependent Weissenberg

number in order to illustrate the effects of incorporating the concentration dependence

of the polymer relaxation times. In addition, we present both the actual property values

as well as the ratio of the actual property value at a given concentration to that of the

infinitely dilute case in selected studies. This allows us to more clearly express the effects

93

of altering the concentration of the system. Finally, we note that the simulations in this

work were carried out at a given set of strain rates, regardless of concentration. Thus,

in order to compare results from different concentrations at a common value ofWic, we

have used cubic splines as interpolating functions to find the desired values.

Finally, as discussed in Section 6.2, we do not include the effects of chain crossings

in our dynamic property calculations. In order to gauge the effect that the prevention of

chain crossings may have on our results, we have simulated polymer solutions at var-

ious concentrations in both simple shear and planar elongational flows strong enough

to deform the coils away from their equilibrium conformations, while incorporating the

bond-crossing detection algorithm of Padding and Briels (2001). This algorithm allows

us to track the number and location of events in which two connecting springs cross one

another, thus giving us an estimate of the frequency with which a bond may cross another

bond on either the same chain or a different one. The results for the two flow types are

given in Figures 6.8(a)-6.8(b), respectively, in which we tabulate the number of intra-

chain and inter-chain spring crossings individually. It should be noted that these results

are based on a simple count, so that a single pair of springs crossing one another repeat-

edly over a sequence of time steps may lead to a large number of tabulated crossings.

In simple shear flow, it is evident that at low concentrations, the only appreciable

source of chain crossings comes from a chain attempting to cross itself. Visual inspec-

tion indicates that this type of crossing typically occurs at the ends of the chain as the

chain begins each new tumbling cycle. As the flowrate increases, this effect diminishes

as the chains become increasingly stretched. With increasing concentration, we see an

expected increase in both intra-chain and inter-chain crossings, with the latter becoming

the dominant source of chain crossings forc/c∗ approaching 1. This would be expected

94

to lead to a lower polymer contribution to the viscosity as calculated by simulation than

would be expected from experiments, and is in fact shown to be the case in Figure 6.13

in which our simulations underpredict the experimentally determined viscosity of a so-

lution at c/c∗ = 1 by approximately 10% over a range of high Weissenberg numbers.

In addition, we are still able to capture the correct scaling behavior for such solutions

as a function of deformation rate. By preventing chain crossings, we may expect our

simulations to do an even better job of capturing the correct quantitave behavior of such

solutions. In considering the case of elongational flows, we see that in general, fewer

spring crossings are observed than in shear flow owing to the fact that the chains are

stretched and aligned with one another to a much greater degree. Hence, we do not ex-

pect chain crossing effects to substantially affect the majority of results presented in this

work. The notable exception is for the case of a system at the overlap concentration

and in a flow at moderate Weissenberg number. At moderate Weissenberg numbers (i.e.

Wi ≈ 1.0), the coils are not sufficiently deformed from their equilibrium state and so

are unable to align parallel to one another at a high packing density, as occurs at high

Weissenberg numbers. Instead, the coils will be found interpenetrating with one another,

leading to large numbers of chain crossings. In our work, we already observe signifi-

cant increases in viscosity with increasing concentration in these regimes, and so we do

not expect that including the effects of preventing chain crossings will not alter the basic

qualitative behavior described in this work.

95

(a) Shear Flow

(b) Planar Elongational Flow

Figure 6.8:Average number of chain crossings per chain during a given time step for systems subjectedto (a) simple shear flow and (b) planar elongational flow.

96

6.5.1 SHEAR

Stretch

We begin our discussion of the behavior of polymer solutions in simple shear flow by

considering the flow-direction stretch as a function ofWi0. In Figures 6.9(a) - 6.9(b),

we plot the flow-direction stretch (〈X〉) as a function ofWi0 for both free-draining and

hydrodynamically interacting systems, respectively. From these figures, we note the pres-

ence of two separate concentration-based behavior regimes, depending on the magnitude

of Wi0. For low shear rates,γ < 0.1 the flow is not strong enough to significantly

perturb the chains from an equilibrium coiled conformation, and the chain size is ob-

served to decrease with increasing concentration. For stronger flows, however, the chains

begin to deform and elongate in the flow direction. In the free-draining case (Figure

6.9(a)), the concentration dependence of the chain stretch is essentially eliminated for

Wi0 ≥ 1. However, when hydrodynamic interactions are included (Figure 6.9(b)), the

behavior is significantly different. At moderate to high shear rates (Wi0 ≥ 1), the stretch-

concentration trend reverses from that of the low-shear case, with our highest concentra-

tion systems exhibiting a stretch nearly20% larger than that of the infinitely dilute case

due to the intermolecular interactions. Finally, in both hydrodynamic cases, the onset of

chain extension occurs at a common value ofWi0 ≈ 1.0, regardless of concentration.

On directly comparing the actual molecular extension of free-draining and hydro-

dynamically interacting systems at a given concentration and at a Weissenberg number

sufficiently high to deform the chains from the equilibrium configuration (Wi0 ≥ 3), we

observe that including hydrodynamic interactions leads to a larger stretch forc/c∗ > 0.1.

At lower concentrations, we observe only a minor difference between the free-draining

97

(a) Free-Draining

(b) Hydrodynamically Interacting

Figure 6.9:Flow direction fractional extension as a function of shear rate for systems subjected to simpleshear flow both with and without hydrodynamic interactions.

98

and hydrodynamically interacting systems, with the free-draining systems slightly more

stretched than their hydrodynamically interacting counterparts. The low concentration

results are in agreement with those of Jendrejack et al. (2002b) and Petera and Muthuku-

mar (1999) for infinitely dilute chains, in which the authors argue that the hydrodynamic

interactions between beads of the same chain result in a reduction in the tumbling motion

of the chain, leading to a lower stretch in flow. However, for short chains such as those

examined here, the difference is slight. At higher concentrations, the onset of strong inter-

molecular hydrodynamic interactions leading to an increase in chain size with increasing

concentration causes the hydrodynamically interacting systems to stretch more than their

free-draining counterparts.

As illustrated in Section 6.4.3, however, the molecular relaxation time depends on

concentration for concentrations greater than or equal to 10% of the overlap concentra-

tion. As a result, simulations of systems above this concentration threshold are at lower

shear rates than a comparable low density system at the same true Weissenberg number.

Thus, we now consider the chain structure in flow as a function of the concentration de-

pendent Weissenberg number,Wic, in Figures 6.10(a) - 6.10(b). Previously, we observed

that the onset of chain extension takes place at a common value of the shear rate regard-

less of concentration. However, as a function ofWic, we observe a delay in the onset

of chain extension that increases with increasing concentration commensurate with the

increasing relaxation time. As a result, in the free-draining case (Figure 6.10(a)), higher

concentration systems exhibit smaller values of the flow-direction stretch than do those at

lower concentrations for allWic. When hydrodynamic interactions are included (Figure

6.10(b)), we observe a similar shift of the curves for higher concentrations, significantly

decreasing the effect of concentration on the chain stretch in flow. At highWic, the chain

99

stretch increases by a maximum of≈ 8% with increasing concentration for concentra-

tions up to2c∗ over the range of Weissenberg numbers considered here.

Reduced Viscosity

We now study the rheological behavior of our polymer solutions by considering the poly-

mer contribution to the solution viscosity. As above, we plot the reduced polymer contri-

bution to the viscosity (ηr) as a function ofWi0 for free-draining and hydrodynamically

interacting systems in Figures 6.11(a) - 6.11(b), respectively. Regardless of whether or

not hydrodynamic interactions are included, the low-Weissenberg regime displays results

that are rather unusual from the standpoint of our above structural descriptions. De-

spite the fact that the chains are increasingly compressed as the concentration increases,

the viscosity per chain exhibits a non-monotonic trend with respect to concentration for

c/c∗ ≥ 0.1, increasing as we raise the concentration toc/c∗ = 1.0, followed by a decrease

as we move into the semi-dilute regime (albeit a weak decrease when hydrodynamic in-

teractions are present). This may be explained through the use of steric arguments by

considering two competing effects, both stemming from the excluded volume potential.

As described in Section 6.4, the chain compression at high concentrations is a result of

interchain excluded volume repulsions. While the smaller chain profiles may be expected

to lead to a lower viscosity of the solution relative to the infinitely dilute case, as the con-

centration increases, the ability of the chains to tumble past one another in solution is

diminished due to the same repulsive interactions. This causes an increase in the overall

viscosity contribution to counter the decrease stemming from the chain compression. For

concentrations up toc/c∗ = 1.0, it is apparent that the latter effect dominates, leading

to a net increase in viscosity with increasing concentration. Above this point, the trend

100

(a) Free-Draining


Figure 6.10:Flow direction fractional extension as a function of Weissenberg number for systems sub-jected to simple shear flow both with and without hydrodynamic interactions.

101

reverses and the diminishing chain size effect dominates.

Unlike the lowWi0 case, when we consider the shear-thinning regime of moderate to

highWi0, the polymer contribution to the solution viscosity depends strongly on the hy-

drodynamic conditions. In the free-draining case (Figure 6.11(a)), the viscosity exhibits

a minimal dependence on concentration, consistent with our earlier finding that the chain

size is also roughly independent of concentration. When hydrodynamic interactions are

included (Figure 6.11(b)), however, increasing the concentration toc/c∗ = 2.0 raises the

reduced viscosity by as much as30% over the infinitely dilute case owing to the increase

in drag originating from the polymer perturbations to the solvent viscosity. Incorporat-

ing the concentration dependence of the relaxation time leads to a minor concentration

dependence of the viscosity in the free-draining case (Figure 6.12(a)), and an increased

dependence on concentration in the hydrodynamically interacting case (Figure 6.12(b)).

Finally, we note that these results are both qualitatively and quantitatively consistent with

the experimental findings of Hur et al. (2001) for solutions ofλ-phage DNA atc/c∗ = 1.0.

Taking into account the difference in solvent viscosities between our simulations (43.3

cP) and the experimental work (90-100 cP), for20 < Wic < 100 we achieve a rela-

tive error of approximately 10% (Figure 6.13). It must be noted, however, that we have

not accounted for the difference in contour lengths of the experimental polymer (bare

λ-phage DNA,L0 = 16 µm) and that used in our simulations (YOYO-1 stained DNA,

L0 = 21 µm), which would tend to increase the discrepancy. Nevertheless, we achieve

nearly identical scalings of the polymer contribution to the viscosity with Weissenberg

number (-0.53 (experiment) vs -0.51 (simulation)). As mentioned in Section 6.2, we have

neglected the effects of chain crossings in this model. Presumably, by including such in-

teractions, our calculated polymer contribution to the viscosity would increase, giving

102

(a) Free-Draining


Figure 6.11:Reduced viscosity as a function of shear rate for systems subjected to simple shear flowboth with and without hydrodynamic interactions.

103

(a) Free-Draining


Figure 6.12:Reduced viscosity as a function of Weissenberg number for systems subjected to simpleshear flow both with and without hydrodynamic interactions.

104

Figure 6.13:Comparison of polymer contribution to the viscosity in simple shear flow as calculatedfrom simulations including hydrodynamic interactions with experimental values of Hur et al. (2001). Theconcentration isc/c∗ = 1.0. Simulation results have been rescaled to account for differences in solventviscosity.

better quantitative agreement between our simulations and the experimental results.

6.5.2 EXTENSION

We have studied the behavior of our systems in planar elongational flows through the

use of Kraynik-Reinelt boundary conditions. As illustrated in the work of Adler and

Brenner (1985a), we are limited to the use of planar elongational flows for our systems at

nonzero concentration due to the need to maintain a periodic lattice that is reproducible

in the imposed flow, a condition that cannot be satisfied by axisymmetric flows. Within

this limitation, we consider the effects of imposing planar elongational flows on both

transient and steady-state structural and rheological properties.

105

Stretch

In Figures 6.14(a) - 6.15(a), we have plotted both the steady-state flow-direction exten-

sion and the extension normalized by the infinitely dilute case of free-draining systems

for various concentrations in the dilute regime. From these figures, we note the presence

of three distinct regimes corresponding to different extension rate ranges. For low exten-

sion rates, the chains do not significantly expand due to the imposed flow. However, at

moderate extension rates (0.3 < Wi0 < 3.0), the chains deform far from their equilib-

rium state, expanding and aligning in the flow direction. As a result, interchain excluded

volume repulsions are primarily directed normal to the extensional axis of the stretched

molecule. Rather than compressing the molecule into a coil by squeezing uniformly in all

directions as occurs in the absence of flow, these repulsions instead compress the chain

only in the directions orthogonal to the flow direction, and as a result, actually cause

the chain to stretch in the flow direction. The net result is a shift in the concentration

dependence of the chain size in elongational flow for moderate extension rates in which

the chain size increases with increasing concentration at a given extension rate due to the

larger interchain repulsions associated with higher concentrations (see Figure 6.15(a)).

This trend applies for concentrations as low as 10% of the overlap concentration. While

similar behavior is observed at high extension rates as well, the concentration dependence

becomes less pronounced as the chains become highly stretched. At high extension rates

(Wi0 > 3), the chain stretch approaches the molecular contour length and the transverse

chain size approaches zero. As a result, increasing the excluded volume repulsions via

an increase in concentration has little effect on the flow-direction stretch.

While the inclusion of hydrodynamic interactions (Figures 6.14(b) - 6.15(b)) does

not significantly alter the qualitative behavior of the three extension rate regimes, it does

106

(a) Free-Draining


Figure 6.14:Flow direction molecular extension as a function of extension rate for systems subjected toplanar elongational flow both with and without hydrodynamic interactions.

107

(a) Free-Draining


Figure 6.15:Flow direction molecular stretch normalized against that of the infinitely dilute case as afunction of extension rate for systems subjected to planar elongational flow both with and without hydro-dynamic interactions.

108

cause a significant increase in the magnitude of the concentration dependence of the

steady-state stretch vis-a-vis the similar free-draining system owing to the perturbation

to the flow of solvent as previously discussed. At low concentrations, we observe little

difference in the chain stretch for free-draining and hydrodynamically interacting sys-

tems. However, as the concentration increases at a givenWi0, the chain size increases

when hydrodynamic interactions are included, as in the case of an imposed shear flow.

For the free-draining system, raising the concentration toc/c∗ = 2.0 causes a maximal

increase of 20% in the average chain stretch over that of thec/c∗ = 0 case. When hy-

drodynamic interactions are included, however, the chains stretch to as much as 210% of

their c/c∗ = 0 size.

As in the analysis of shear flows, we now perform a similar analysis of the chain

stretch in elongational flow, but this time plotted as a function ofWic in order to in-

corporate the concentration-dependent relaxation time of the system. Using the longest

relaxation times of Table 6.3 for each concentration, we consider the chain structure in

flow as a function of the effective Weissenberg number in Figures 6.16(a) - 6.17(b). In

both the free-draining and hydrodynamically interacting cases, the stretching behavior

in the lowWic range remains unchanged from our earlier description, as the chains re-

main near their equilibrium configurations. However, the behavior in both the moderate

and highWic regimes is considerably different. Since the relaxation time increases with

increasing concentration, systems at higher concentration experience smaller smaller ex-

tension rates than do those at lower concentrations for a given sameWic. As a result,

for free-draining systems, we observe a significant decrease in the chain size as concen-

tration increases for moderate Weissenberg numbers (0.3 < Wic < 3.0). As illustrated

in Figure 6.17(a), the decrease in chain size relative to the infinitely dilute case is as

109

large as30% for c/c∗ = 2.0. In addition, it is notable that the stretching ratios achieve

a minimum atWic ≈ 0.5, regardless of concentration. The decrease in chain size with

increasing concentration continues into the highWic regime, with a measurable concen-

tration dependence persisting toWic = 10, despite a chain extension in excess of 90%

of the contour length.

For systems deformed far from equilibrium, the inclusion of hydrodynamic interac-

tions (Figures 6.16(b) - 6.17(b)) results in the virtual elimination of the concentration

dependence for the chain stretch. In the narrow range0.5 < Wic < 1.0, our systems

undergo a coil-stretch transition from the equilibrium state and some concentration de-

pendence is still evident. ForWic > 1.0, however, the chains adopt elongated configura-

tions that are roughly independent of concentration. While concentration independence at

high Weissenberg number may be attributed to chains approaching the molecular contour

length, as described previously (tail region of Figure 6.17(a)), the onset of concentration

independence atWic ≈ 1.0 when hydrodynamic interactions are present (Figure 6.17(b))

corresponds to chains that are stretched to only≈ 55% ofL0. This result indicates that the

decrease in extension rate associated with an increase in concentration for a givenWic

is essentially offset by the increase in solvent viscosity stemming from perturbations to

the flow field, and is consistent with our earlier description of shear flows in which the

chain size dependence on concentration was significantly weakened when systems were

considered at constantWic as opposed to constantWi0.

Reduced Viscosity

In Figures 6.18(a) - 6.19(b), we plot both the actual extensional viscosity and the exten-

sional viscosity normalized against that of the infinitely dilute case as a function of the

110

(a) Free-Draining


Figure 6.16: Flow direction molecular extension as a function of Weissenberg number for systemssubjected to planar elongational flow both with and without hydrodynamic interactions.

111

(a) Free-Draining


Figure 6.17:Flow direction molecular stretch normalized against that of the infinitely dilute case as afunction of Weissenberg number for systems subjected to planar elongational flow both with and withouthydrodynamic interactions.

112

extension rate for both the free-draining and hydrodynamically interacting cases. Un-

like the above trends described for the chain stretching, the extensional viscosity of both

systems increases with increasing concentrations at a given extension rate, regardless of

the actual magnitude of the extension rate. In the regime of moderate extension rates,

we observe an increase in viscosity over that of the infinitely dilute case correspond-

ing to increases in the chain size both when hydrodynamic interactions are included and

when ignored, although the former case shows a much sharper increase than is accounted

for by chain stretch alone. In the free-draining case (Figures 6.18(a), 6.19(a)), the vis-

cosity increases as much as 65%, while the hydrodynamically interacting cases (Figures

6.18(b), 6.19(b)) exhibit a viscosity up to 6 times larger than that corresponding to infinite

dilution. In the range of high Weissenberg numbers, however, the inclusion of hydrody-

namic interactions qualitatively affects the concentration dependence of the viscosity as

compared to the free-draining case. From Figure 6.19(a), we see that in the absence of

hydrodynamic interactions, the viscosity becomes roughly independent of concentration

as the chains approach full extension, in agreement with our earlier observation that the

polymer chain stretch is also independent of concentration. When hydrodynamic interac-

tions are present, however, the viscosity dependence on concentration persists throughout

the range ofWi0 investigated here. This contrasts with our earlier finding that the chain

stretch is independent of concentration at highWi0 for systems with hydrodynamic in-

teractions present.

When we instead consider the extensional viscosity as a function of the concentra-

tion dependent Weissenberg number, as was previously done for chain stretching, the

free draining simulations (Figures 6.20(a), 6.21(a)) correspond well with the physical de-

scription of the chain structure arrived at earlier; namely, the polymer contribution to the

113

(a) Free-Draining


Figure 6.18:Reduced elongational viscosity as a function of extension rate for systems subjected toplanar elongational flow both with and without hydrodynamic interactions.

114

(a) Free-Draining


Figure 6.19:Reduced elongational viscosity normalized against that of the infinitely dilute case as afunction of extension rate for systems subjected to planar elongational flow both with and without hydro-dynamic interactions.

115

viscosity decreases with increasing concentration at a given Weissenberg number. The

maximum decrease in chain size occurs at a common value ofWic = 0.5. This agrees

well with our earlier finding of a decrease in chain size with increasing concentration.

When hydrodynamic interactions are included (Figures 6.20(b), 6.21(b)), however, the

situation is different. Despite the insensitivity of the chain size to changes in concentra-

tion for sufficiently highWic, the viscosity shows a strong concentration dependence,

increasing steadily with increasing concentration. This is consistent with our earlier find-

ings based on comparing systems at equal extension rates (i.e. equalWi0).

To summarize, in free-draining systems, we observe a direct correlation between the

chain structure and viscosity with increasing concentration in both the moderate and high

Weissenberg number regimes, and we explain such behaviors by invoking steric argu-

ments. When hydrodynamic interactions are present, however, this correlation does not

hold, as the viscosity exhibits a much stronger dependence on concentration than does the

chain stretch. Changes in chain size and viscosity associated with increasing concentra-

tion in both shear and elongational flows result largely from hydrodynamic perturbations

to the solvent flow field.

6.6 Conclusions

This work is concerned with the numerical simulation of the effects of concentration on

both the static and dynamic properties ofλ-phage DNA in bulk solution. Using a simple

coarse-grained kinetic theory model, we have carried out a series of Brownian dynamics

simulations for systems at a variety of concentrations that span the entire dilute regime.

Simulations were performed on systems both at equilibrium as well as when subjected

116

(a) Free-Draining


Figure 6.20:Reduced elongational viscosity as a function of Weissenberg number for systems subjectedto planar elongational flow both with and without hydrodynamic interactions.

117

(a) Free-Draining


Figure 6.21:Reduced elongational viscosity normalized against that of the infinitely dilute case as afunction of Weissenberg number for systems subjected to planar elongational flow both with and withouthydrodynamic interactions.

118

to simple shear and planar elongational flows. Our results indicate that both the equi-

librium chain structure and the dynamic behavior of our polymer solutions are affected

by concentration at values as low as10% of c∗. At equilibrium, or at flow rates suffi-

ciently low so as to not significantly perturb the equilibrium coil size, our polymer chains

exhibit a decrease in chain size with increasing concentration, owing to a correspond-

ing increase in intermolecular excluded volume repulsions. However, when subjected

to sufficiently strong flows, significant increases in the chain extension in both simple

shear and planar elongational flow are observed as concentration increases, with the lat-

ter flow type exhibiting much larger concentration effects than the former. In the absence

of hydrodynamic interactions, the chain size and polymer contribution to the viscosity

display similar changes as concentration increases; we explain such trends using steric

arguments. When hydrodynamic interactions are present, however, the viscosity shows

a much stronger dependence on concentration than does the chain stretch. Changes in

chain size and reduced viscosity associated with increasing concentration in both shear

and elongational flows result largely from hydrodynamic perturbations to the solvent flow

field and the rheological effects are far more pronounced in elongational flows than in

shear. In simple shear flow and at moderate Weissenberg number, increasing the con-

centration toc/c∗ = 2.0 raises the reduced viscosity by as much as 30% over the value

for an infinitely dilute system. Over the same concentration and Weissenberg ranges,

an increase by a factor of six is observed in a planar extensional flow. Finally, we have

demonstrated excellent quantitative agreement between our simulations systems under

shear flow atc/c∗ = 1.0 and the experimental data of Hur et al. (2001), as well as quali-

tative agreement with the experimental findings of Owens et al. (2004) and Clasen et al.

(2004) regarding concentration effects on the dynamic behavior of polymer solutions in

119

extensional flows.

120

Chapter 7

SIMULATION OF DILUTE

SALT-FREE POLYELECTROLYTE

SOLUTIONS IN SIMPLE SHEAR

FLOWS

121

7.1 Introduction

Polyelectrolytes - polymers containing ionizable groups - have been extensively stud-

ied over the past few decades via experiments, theoretical investigations, and computer

simulations. This important class of polymers is used in a wide range of industrial appli-

cations, ranging from wastewater treatment to oil recovery operations. In addition, poly-

electrolytes in the form of DNA, RNA, and many proteins are central to many biological

processes. However, despite the importance of such polymers, the dynamic behavior of

polyelectrolytes continues to be poorly understood. One of the difficulties in the theoreti-

cal study of polyelectrolytes is the proper accounting of the balance between electrostatic

and hydrodynamic interactions. The presence of charged groups along the backbone of

the chain plays a significant role in both the static structure and the dynamic behavior

of such systems and gives rise to a number of perplexing properties. A notable example

is the non-monotonic viscosity relationship with polymer concentration provided by the

crossover from the dilute to semidilute regimes. In this work, we perform simulations

of polyelectrolyte solutions and use them to explore the relationship between viscos-

ity and Bjerrum length for dilute solutions, explaining an observed increase in viscosity

with increasing Bjerrum length via a novel mechanism. We then extend this mechanism

to account for the increase in viscosity observed as concentration increases toward the

semidilute regime.

A complete picture of the static structure and behavior of polyelectrolytes is devel-

oping from a combination of analytical theory, numerical simulations, and experimental

evidence. Based on the scaling theories of de Gennes et al. (1976), Pfeuty (1978), Ru-

binstein et al. (1994), and Dobrynin et al. (1995), a number of predictions are available

122

on the effects of polymer and salt concentration, charge density, and Bjerrum length (the

distance at which the energy of the Coulomb interaction between two elementary charges

is equal to the thermal energykBT ) on polyelectrolyte structure and dynamic behavior.

However, we note that these scaling theories have some significant shortcomings; Boris

and Colby (1998) have discussed the failings of some scaling predictions as do some

of the results presented in this work. Experiments on sodium poly(styrene sulfonate),

a common experimental polyelectrolyte, by Drifford and Dalbiez (1984), Krause et al.

(1989), and Johner et al. (1994) have studied the polyion structure in the dilute regime

via light scattering while the semi-dilute regime has been explored via small-angle X-

ray scattering by Kaji et al. (1984) and via small-angle neutron scattering by Nierlich

et al. (1979a,b, 1985a,b) and Takahashi et al. (1999). In addition, the effects of varying

salt concentrations have been investigated by Beer et al. (1997), Borochov and Eisenberg

(1994), and Wang and Yu (1988), in which it is demonstrated that the presence of added

salt serves to screen electrostatic interactions, and for low Bjerrum length systems, shields

the intrachain electrostatic repulsions and causes an overall decrease in chain size. These

experiments are supported by a wide range of computer simulations. Carnie et al. (1988);

Christos and Carnie (1989, 1990a,b); Christos et al. (1992) have performed Monte Carlo

simulations of polyelectrolytes using the Debye-Huckel approximation (Robinson and

Stokes, 1955) to account for electrostatic interactions. Stevens and Kremer performed

molecular dynamics simulations under similar conditions (Stevens and Kremer, 1996)

along with simulations using explicit counterions (Stevens and Kremer, 1995) to illus-

trate the defects of using the Debye-Huckel approximation and its failure to describe

ion condensation at large Bjerrum length. Further studies using Brownian dynamics by

Chang and Yethiraj (2002) and by Liu and Muthukumar (2002) extended the analysis

123

of the static structure of the system to include the structure of the counterion cloud that

surrounds the polyion. These simulations show that the size and shape of the counterion

cloud displays a nonmonatonic dependence on electrostatic strength (i.e. Bjerrum length)

and its collapse about the chain at high electrostatic strength plays a significant role in

determining the structure of the polymer chain itself. Winkler et al. (1998) and Chu and

Mak (1999) have further developed our understanding of the static structure of polyelec-

trolytes by considering the effects of altering the Bjerrum length over a wide range of val-

ues, providing further evidence of chain collapse at high Bjerrum length. Finally, Dubois

and Boue (2001) have performed experiments comparable to the simulations of Stevens

(2001) and Chang and Yethiraj (2003b) to describe the behavior of polyelectrolytes in the

presence of multivalent counterions. Generally, the simulation results presented to date

are in good agreement with one another as well as with experimental evidence. In this

paper, we present static results that are consistent with those already puslished and we

extend our analysis to dynamic properties.

Unfortunately, the case is not as clear in the analysis of polyelectrolyte dynamics.

Based on empirical studies of the viscosity of semidilute and moderately dilute salt-free

polyelectrolyte solutions, Fuoss (1948) proposed a relationship for the reduced viscosity,

ηr, as a function of monomer concentration,c, that now bears his name:

ηr =A

1 +Bc1/2(7.1)

where A and B are fitting parameters, and the reduced viscosity represents the polymer

contribution to the viscosity, normalized by both the solvent contribution and the net poly-

mer concentration (see Section 7.5). Rubinstein et al. (1994); Dobrynin et al. (1995) have

subsequently developed scaling theories for the dynamic behavior of polyelectrolytes that

124

confirm the Fuoss law scaling in the unentangled semi-dilute regime. These scaling theo-

ries represent the polyion by a chain of electrostatic blobs, with the statistics of the chain

inside each blob determined by the thermodynamic interactions between uncharged poly-

mer and solvent. On length scales greater than the blob size, it is assumed that electro-

static effects dominate and the blobs repel one another to form a fully extended chain.

However, while Fuoss’s law provides a means of accurately describing the reduced vis-

cosity in the semi-dilute regime, both the law and the scaling theories fail to accurately

predict the correct viscosity behavior for highly dilute systems as the reduced viscosity

is predicted to become constant at very low polyion concentrations. Rather, Eisenberg

and Pouyet (1954), and later Cohen et al. (1988) and Antonietti et al. (1996) showed via

experiment that the reduced viscosity in fact reaches a peak nearc∗, and then decreases

as the system becomes more dilute. One frequent explanation (Dobrynin et al., 1995;

Cohen et al., 1988; Rabin et al., 1988; Forster and Schmidt, 1995; Barrat and Joanny,

1996) of this phenomenon assumes that at a high degree of dilution, residual salt present

in the solvent eventually serves to screen the long-ranged electrostatic interactions, and at

this point, the viscosity may be expected to scale as that of neutral polymers. That is, for

c cs, wherecs is the residual salt concentration, we expectηr ∝ c. These assumptions

lead to the generalized Fuoss relationship developed by Cohen et al. (1988) and Rabin

et al. (1988), which predicts

ηr ≈ξ0c

κ3(7.2)

whereξ0 is the hydrodynamic friction coefficient of the polyion andκ =√c+ 2zscs is

the inverse Debye screening length. While this modified law describes some of the qual-

itative viscometric behavior adequately, it still fails to account for some effects such as

125

shear thinning (Boris and Colby, 1998) and the presence of a minimum reduced viscos-

ity at high concentrations (Fernandez Prini and Lagos, 1964). One potential reason for

these discrepancies is the fact that the generalized Fuoss relationship, along with other

descriptions of polyelectrolyte dynamics, relies on the Debye-Huckel theory (Robinson

and Stokes, 1955) to describe electrostatic interactions; as mentioned above, however,

simulations have shown that Debye-Huckel theory is unable to accurately predict the be-

havior of salt-free polyelectrolyte systems. As a result, we have taken an entirely differ-

ent path in this work, in which we seek to explain the relationship between viscosity and

concentration for salt-free dilute polymer solutions by means of analyzing the individual

contributions to the viscosity stemming from the interactions between the polyelectrolyte

chains and the dissociated counterions.

The primary focus of the present work is the dynamic behavior of dilute (c c∗)

salt-free polyelectrolyte solutions subjected to steady shear flows, addressing the connec-

tion between the rheological behavior of the solution and the structure of the chain and its

accompanying cloud of counterions. To this end, we use Brownian dynamics simulations

with a simple, coarse-grained model of a polyelectrolyte. We have incorporated elec-

trostatic interactions using explicit counterions and demonstrate in Section 7.5 that the

counterion contribution to the solution viscosity plays a significant role in determining

rheological trends with respect to both the concentration and electrostatic quality of the

solution. At present, the closest work to that presented here is that of Zhou et al. Zhou and

Chen (2006), in which the authors consider both the short- and long-time diffusive be-

havior of polyelectrolytes in dilute salt-free solutions. It was found that the incorporation

of hydrodynamic interactions supresses a coupling effect between the chain and its coun-

terions that is otherwise responsible for a noticeable increase in the long-time diffusivity.

126

We consider similar coupling effects here for polyelectrolytes subjected to flow in light of

the primary electroviscous effect, in which the retardation of the motion of the ion cloud

due to electrostatic attractions with the polyelectrolyte chain contributes significantly to

the reduced viscosity of the solution. This effect has both been observed via experiment

Jiang et al. (2001); Roure et al. (1996); Ganter et al. (1992) and studied both theoretically

Jiang and Chen (2001); Imai and Gekko (1991) and via numerical modeling Chen and

Allison (2001) using the Debye-Huckel approximation for polyelectrolyte systems. We

believe this to be the first use of numerical simulations incorporating explicit counterions

to demonstrate evidence of the primary electroviscous effect. Finally, in light of the work

by Zhou et al. Zhou and Chen (2006), we have incorporated fluctuating hydrodynamic

interactions into various flow simulations in order to evaluate the relative strengths of the

electrostatic and hydrodynamic effects.

7.2 Model

In this work, we simulate a dilute solution of monodisperse, linear polyelectrolytes with

explicit counterions immersed in an incompressible Newtonian solvent. We use the basic

model of Chang and Yethiraj (2002) and coarse-grain each polyelectrolyte chain into a

sequence ofNB “beads” connected byNS = NB − 1 “springs” of contour lengthq0

each, yielding an overall contour length ofL0 = NSQ0. The beads are assumed to be of

diameterσB and carry a chargeqB. A total ofNC chains are initially enclosed in a cubic

cell of edge lengthL, giving a total ofN = NCNB beads per cell at a bulk monomer

concentration ofc = NCNB

V, whereV = L3 is the volume of the simulation cell. We

also include a set ofNI free counterions, each of diameterσI and chargeqI , such that

127

NIqI = NCNBqB. For simplicity, we takeσB = σI = σ andqB = −qI = −e, wheree is

the electron charge. To map this model to an experimental system, we consider the case

of sodium poly(sytrene sulfonate), NaPSS. In room temperature water, a solvated sodium

ion is roughly 0.71 nm in diameter (Horvath, 1985), and the length of a monomeric

repeat unit along the chain is0.25 nm. Thus, we takeσI = σB = 0.71 nm, leading

to ≈ 3 monomers per bead and, correspondingly, a charge fraction of1/3. While this

is a much smaller “bead” than is customarily used in Brownian dynamics simulations

of bead-spring chains, it would be impossible at present to simulate systems in which

the chain scaling is more conventional due to the large number of ions that would need

to be included for charge neutrality. One alternative is to turn to a more detailed bead-

rod description of the polymer, where each rod represents one Kuhn length. However,

as we demonstrate in Appendix A, this also poses a significant calculation hurdle when

hydrodynamic interactions are included.

As stated in Section 3.2, adjacent beads of the polyelectrolyte chains are connected

via a Finitely Extensible Non-linear Elastic (FENE) spring model, in which the force on

beadν due to connectivity with beadµ is given by

Fsprνµ = − Hr νµ

1−(

Qi

Q0

)2 , (7.3)

whereQ0 is the maximum spring extension,r νµ = r ν − rµ, andH is the spring con-

stant. Following Chang and Yethiraj (2002), in this work, we useH = 30.0kBT/σ2 and

Q0 = 1.5σ. These parameter values, when combined with the repulsive excluded volume

potential (below), have been shown to prevent chain crossings. With this parameteri-

zation, the average spring length is approximately0.98σ in the absence of electrostatic

interactions.

128

Excluded volume interactions between beadsν andµ are accounted for through the

use of the Weeks-Chandler-Andersen (WCA) potential, where the resulting force acting

on beadν due to the presence of beadµ is given by

FWCAνµ =

4εLJ

[12(

σrνµ

)12

− 6(

σrνµ

)6]

rνµ

r2νµ

rνµ <6√

2σ

0 rνµ ≥ 6√

2σ.

(7.4)

Electrostatic interactions are described by pairwise Coulombic interactions, where the

Ewald summation technique of Equations 3.34-3.37 has been applied. The expressions

are repeated here:

Fν = kBTλB

(F(r)

ν + F(k)ν + F(d)

ν

)(7.5)

with the real space, Fourier space, and dipole contributions given by,

F(r)ν = zν

∑µ

zµ

′∑n∈Z3

(2α√πe−α2|rνµn|2 +

erfc(α |r νµn|)|r νµn|

)r νµn

|r νµn|2(7.6)

F(k)ν =

zν

V

∑µ

zµ

∑k 6=0

4πkk2

e−k2/4α2

sin (k · r νµ) (7.7)

F(d)ν =

−4πzν

(1 + 2ε′)V

∑µ

zµrµ (7.8)

wheren denotes the real-space lattice vector,k = 2πn/L is a wavevector, andα is

the splitting parameter that determines the relative computational loads between the real

and Fourier space summations. Finally, as was the case in Chapter 6, hydrodynamic

interactions are accounted for through the periodic form of the Rotne-Prager-Yamakawa

tensor as in Equations 3.43-3.47.

129

7.3 Simulation

The governing stochastic differential equation for this system is given by:

dr =

[κ · r(t)] +

1

kBT

[D · F(φ)

]+

∂

∂r· Ddt+

√2B·dW, (7.9)

in which kB is Boltzmann’s constant andT is the absolute temperature. The vectorr

contains the3N spatial coordinates of both the beads that constitute the polymer chain

and the counterions,D is a 3N × 3N diffusion tensor, andF(φ) is a force vector of

dimension3N . The 3N × 3N tensorκ is block diagonal with diagonal components

(∇v)T , with v being the unperturbed solvent velocity.B is a 3N × 3N tensor defined

by B · BT = D and the components of the3N dimension vectordW are obtained from a

real-valued Gaussian distribution with mean zero and variancedt. Note that in this work,

vectors and tensors listed without subscripts describe the full system and are of dimension

3N or 3N ×3N , respectively. Vectors and tensors with subscripts refer to specific beads.

For the solution of Eq. 5.1, we once again make use of an explicit Euler scheme:

r ν (t+ ∆t) = r ν (t) + [κ(t) · r ν(t)] ∆t +∆t

kBT

∑µ

[Dνµ(t) · F(φ)

µ (t)]

+√

2∑

µ

Bνµ(t)·∆Wµ(t), (7.10)

whereF(φ)µ =

∑ω 6=µ

(Fexv

µω + Felµω

)+ Fspr

µ,µ−1 + Fsprµ,µ+1, and where the Brownian term is

calculated via Fixman’s method as described by Jendrejack et al. (2000). Due to the ex-

istence of three separate singularities in the potentials of this system (FENE spring, elec-

trostatic, and excluded volume), the Euler scheme requires the use of much smaller time

steps here than for our work in Chapter 6. In the absence of flow, we use∆t = 0.0001 ζσ2

kBT

as this value was found to provide both accuracy and stability for the Euler method. Fi-

nally, the concentration is normalized by the overlap concentration,c∗, calulated on a

130

NB L0 Rg c∗

(σ) (σ) (beads/σ3)

10 15 1.67 0.5120 30 2.68 0.2540 60 4.18 0.13

Table 7.1:Radius of gyration and overlap concentration,c∗, for chains of varying molecular weight atinfinite dilution.

monomer basis according to Doi and Edwards (1986) asc∗ = NB

43π(R∗g)

3 . Here,R∗g is the

equilibrium radius of gyration of a neutral polymer chain at zero concentration. These

values are tabulated for various molecular weights in Table 7.1. Also, unless otherwise

noted, simulations were run for sufficient time and ensemble sizes to reduce the error bars

to the order of the symbol size used here.

7.4 Equilibrium Results

Polyelectrolyte chain structure

We begin our discussion of the behavior of polyelectrolyte solutions with an analysis of

the equilibrium structure as a function of Bjerrum length, molecular weight, and concen-

tration. While these topics have been previously discussed in the literature, including for

this model in particular, the results presented here both expand the scope of some previ-

ous works as well as provide some insight into interpreting the dynamic properties that

follow. We consider the static structure of the polyelectrolyte chains primarily through

the calculation of the mean-square radius of gyration, defined for an individual chain as

R2g =

1

2N2B

⟨NB∑ν=1

NB∑µ=1

(r ν − rµ)2

⟩. (7.11)

131

Shown in Figure 7.1 isR2g as a function ofλB for chains of various molecular weights at

a concentration ofc/c∗ = 10−4. For all systems considered here, the radius of gyration

displays a non-monotonic trend with respect to increasingλB. At small values ofλB,

the electrostatic attractions between the polyion and the surrounding counterions are not

sufficiently strong to overcome the effects of random thermal motion, and so few ions are

found in the immediate vicinity of the chain. As a result, the chain experiences little to no

screening of intrachain electrostatic repulsions and expands from the coiled state found

for λB = 0. As λB increases, so do the strength of these repulsions, and we observe

thatR2g increases correspondingly. AtλB ≈ 1.0, however, the electrostatic interactions

begin to overcome the thermal fluctuations and ions begin to condense about the chain.

This condensation, in turn, shields some of the electrostatic repulsions within the chain,

eventually causing the chain size to peak atλB ≈ 1.5. Continued increase in the Bjerrum

length causes the chain to contract until, at large values ofλB, the chain and ions have

effectively coalesced into a globular shape that is more tightly compacted than a random

walk with steps comparable to the size of the average spring length. This behavior may

be observed from the three individual bead-ion images of Figure 7.2; it has also been ob-

served via numerical simulation by both Chu and Mak (1999) and Winkler et al. (1998)

and is predicted by the scaling arguments of Schiessel and Pincus (1998). It should be

noted that while the average spring length within the chain monotonically increases with

respect to increasingλB due to the electrostatic repulsion of neighboring beads, the in-

crease is far too small to significantly impact the overall chain structure. These results

confirm that the equilibrium chain conformation is being governed by the long-range

electrostatic interactions, as opposed to simple spring forces. From the structure de-

scribed above, we identify three potential electrostatic regimes for further consideration:

132

Figure 7.1:Mean square radius of gyration of the polyion chain,⟨R2

g

⟩, plotted as a function ofλB for

various molecular weight polyelectrolytes atc/c∗ = 10−4. Contour lengths of the chains in increasingorder are13.5σ, 28.5σ, and58.5σ.

the neutral case (λB = 0), the peak extension case (λB ≈ 1.5), and the condensed-ion

case (λB & 10). In the remainder of this work, we use these electrostatic regimes as a

basis for describing the qualitative behavior of polyelectrolytes.

The behavior described above differs significantly from that exhibited in simula-

tions of polyelectrolytes in which the electrostatic interactions are treated via the use of

Debye-Huckel theory. In the Debye-Huckel theory (Robinson and Stokes, 1955), Pois-

son’s equation for charged bodies is solved assuming a spherically symmetric Boltzmann

distribution of charges in the solvent about any particular ion and in the absence of ex-

ternal forces. The smearing out of the ion cloud in this manner relies on the assump-

tion that the electrical interactions are generally weak compared with the thermal energy

of the ions and results in an expression for the electrostatic forces that is purely repul-

sive, and so is incapable of capturing the collapse of the polyelectrolyte chain at high

133

(a) λB = 0.0 – neutral case (b) λB = 1.5 – peak extension case

(c) λB ≥= 10.0 – condensed ion case

Figure 7.2:Molecular visualizations of an equilibrated 40-bead chain in three electrostatic regimes, theneutral case (λB = 0), the peak extension case (λB = 1.5), and the condensed-ion case (λB ≥ 10). Chainbeads are shown as dark spheres, and counterions as light spheres.

134

Figure 7.3: Illustration of one of the defects associated with the use of the Debye-Huckel theory forelectrostatic interactions. Shown is the mean-square radius of gyration for a 20-bead chain atc/c∗ =10−4 with the electrostatics calculated via the Debye-Huckel approximation, and via explicit Coulombicinteractions with monovalent counterions.

λB. We illustrate this effect in Figure 7.3, in which we have compared the equilibrium

size of 20-bead polyelectrolyte chains with explicit monovalent counterions to systems

using the Debye-Huckel approximation at the same ionic strength. Clearly, while the

Debye-Huckel approximation is suitable for low-λB simulations in which the ion cloud

is scattered throughout the simulation domain, it is insufficient for accurately describing

polyelectrolyte behavior at higher values ofλB. This deficiency stems from the counte-

rion condensation occurring forλB > 1.0. As the ions enter the vicinity of the chain, the

electrostatic interactions become large, causing the Debye-Huckel theory to break down

completely. Hence, in order to fully model the range ofλB, we have chosen to use an

explicit counterion model for a more complete exploration of polyelectrolye systems.

We next calculate the center-of-mass pair distribution function of the polyelectrolyte

135

chains in solution,

gC(r) =1

NCc

NC∑k=1

NC∑l=1

δ(r −

(r k

c + r lc

)), (7.12)

wherer kc = (1/NB)

∑NB

ν=1 r kν is the center of mass of thekth chain and the prime on the

second summation indicates that the termk = l is omitted. The results for a system at

c/c∗ = 10−3 are shown in Figure 7.4. We again see three regimes based onλB. For

neutral systems, the chains are found to be uniformly distributed in solution, leading to

a constant-valued distribution function. At moderate values ofλB, however, there is a

deviation from this behavior. With the counterion cloud surrounding a given chain be-

ing of low density, there is little shielding of electrostatic repulsions between different

chains. As a result, there is a correlation hole about each chain whose size corresponds

to that of the box size. AsλB becomes large, the condensed ions do provide significant

electrostatic shielding, and without the repulsive interchain interactions, the depletion

layer disappears. As a result, systems at highλB exhibit long-range behavior that is sim-

ilar to that of the neutral chains. At short distances, however, we observe a peak in the

chain-chain distribution function that is absent at lower values ofλB. The peak appears

to stem from an effect in which chains aggregate together due to attractive interactions

with shared counterions. Our initial investigations indicate that atλB = 10, the aggre-

gated state is actually slightly more energetically favorable than is the dispersed phase,

and asλB increases, the aggregated state becomes increasingly lower in energy relative

to the dispersed phase. As the primary focus of this work is the dynamic behavior of

polyelectrolytes, we have not considered this phenomenon in greater detail. We note,

however, that these observations are consistent with previous simulations (Chang and

Yethiraj, 2002, 2003a).

136

Figure 7.4:Polyelectrolyte chain-chain radial distribution function,gC(r), in c/c∗ = 10−3 solution atequilibrium.

Finally, we consider the effects of concentration on the static size of a polyelectrolyte

chain in Figure 7.5, in which we plot the polyion size for a 10-bead chain as a func-

tion of λB at various concentrations in the dilute regime, ranging fromc/c∗ = 10−4 to

10−1. At low λB, our results show that chain size is roughly independent of concentra-

tion in the dilute regime, in good agreement with the scaling theory of Dobrynin et al.

(1995). In their work, Dobrynin et al. (1995) describe a dilute salt-free polyelectrolyte as

an extended chain of electrostatic blobs, with the expectation that the chain size will be

independent of concentration. However, this description is predicated on the assumption

that counterions are homogeneously distributed throughout the system volume, which,

as shown above, is true only for low values ofλB. In the regime about the peak in the

static size, where we would expect the extended blob conformation to be most appli-

cable, we observe a weak decrease in the chain size as the concentration increases. In

this range ofλB, a significant fraction of the counterions remain dissociated in solution.

137

Figure 7.5:Chain radius of gyration,⟨R2

g

⟩, plotted as a function ofλB for 10-bead chains at various

concentrations.

As a result, each chain experiences unshielded electrostatic repulsions from the other

polyelectrolyte chains surrounding it in solution, causing it to compress. This effect is

enhanced as the concentration increases since neighboring chains are forced into closer

proximity, increasing the strength of the electrostatic repulsions. Finally, for high values

of λB, the condensed cloud of counterions shields the interchain electrostatic interac-

tions, and again, the chains do not significantly affect one another’s structure regardless

of concentration.

Ion cloud structure

As alluded to above, the nature of the cloud of ions enveloping a polyelectrolyte chain

plays an important role in determining the structure of that chain. As we shall see in

Section 7.5, the structure of the ion cloud also plays a significant role in determining the

rheological behavior of this model. Thus, we seek here to understand the nature of the ion

138

cloud at equilibrium, and later, to extend this description to systems in flow. To study the

structure of the equilibrium ion cloud, we use the counterion distribution function (Chang

and Yethiraj, 2002),PI(r), defined such thatPI(r)dr is the number of counterions in

the spherical shell surrounding a polyion at a distance from the chain center-of-mass

betweenr andr + dr. This definition of the distribution function incorporates a volume

element contribution, and thus should scale asr2 for a uniform distribution of ions. For

sufficient electrostatic strength, however, we expectPI(r) to display a peak at lowr

corresponding to the condensed ion cloud, followed by a tail region whose behavior at

largerr is governed by the density of the system. For dilute solutions, this tail is expected

to behave as the neutral case as the counterions become uniformly distributed at large

distances from the chain due to electrostatic shielding, regardless of electrostatic strength.

At higher densities, however, the tail may display additional peaks as the counterion

clouds corresponding to different chains are brought into close contact, possibly even

overlapping. These observations are illustrated in Figure 7.6, in which we plotPI(r) for

a 10-bead chain atλB = 1.5, 2.25, and10, respectively, andc/c∗ = 10−3. In addition,

we present histograms illustrating the radial density of ions surrounding a given chain for

each system in Figure 7.7, where the background ion concentration has been removed

to better illustrate the boundaries of the cloud. Note that at this low concentration, the

average distance between uniformly distributed chains is roughly29.0σ, which, as it turns

out, is much larger than the size of the ion cloud for any value ofλB. Thus, we can safely

assume that the clouds are non-overlapping at this density and we concentrate here solely

on the structure of an individual cloud surrounding a polyion.

From the plots ofPI(r), we define a counterion cloud as being composed of the

counterions that are within a specified cut-off distance and, for simplicity, we take this

139

Figure 7.6:Effects ofλB on the size of the ion cloud at equilibrium, as determined by the calculation ofPI(r). Systems shown atc/c∗ = 10−3.

cutoff distance to be equal to the distance corresponding to the local minimum ofPI(r)

separating the peak from the quadratic tail. Using this measure, we calculate both the

average cloud size, taken as equal to the local minimum ofPI(r), as a function ofλB

for various concentrations, as shown in Figure 7.8, and the bulk ion concentration for

the cloud, shown in Figure 7.9. While the number of ions associated with a given cloud

increases with increasingλB as one would expect, it is somewhat surprising that the

actual size of the counterion cloud exhibits a non-monotonic trend with increasingλB.

Echoing the same trend as the chain structure, the ion cloud first increases in size asλB

increases, and then decreases as nearly all of the ions condense very near the chain. It

deviates though in that the peak size of the counterion cloud occurs in the rangeλB ∈

(2.5, 5.0), as opposed toλB = 1.5 for the peak chain size. This may be explained by

considering the relative strengths of the thermal energy of each ion and the electrostatic

attraction to the polyelectrolyte. For low values ofλB, the electrostatic attractions are

140

(a) λB = 1.5 (b) λB = 2.25

(c) λB = 10.0

Figure 7.7:Depiction of the equilibrium ion cloud surrounding an individual chain in dilute solution(c/c∗ = 10−3) for various values ofλB . Pictures correspond to the plots ofPI(r) of Figure 7.6. Shown isthex-y profile with data averaged through thez-direction. Scales reflect the excess concentration of ionsrelative to the average concentration of ions in the system, i.e.cI (r) = cI (r)−NI/V .

141

Figure 7.8:Size of the counterion cloud surrounding a chain as a function ofλB for a 10-bead chain atvarious concentrations.

not sufficiently strong to bind the ions to a chain, and we expect the radius of the cloud

to grow roughly linearly with increasingλB. This is due to the fact that the electrostatic

potential scales withλB/r, and so doublingλB leads to a doubling of the range over

which the electrostatics act with equivalent strength. AsλB exceeds 1, however, the

electrostatic interactions are sufficiently strong such that the ions begin to condense about

the chain, substantially densifying the cloud (Figure 7.9), and eventually leading to the

collapse of the cloud that gives rise to the nonmonotonic size behavior observed in Figure

7.8.

We consider the degree of ionization, defined as the fraction of counterions considered

to be outside of the cumulative condensed ion clouds, for systems at various concentra-

tions as a function of reciprocalλB in Figure 7.10. Also shown is the prediction from the

Manning condensation theory, in which the polymer is modeled as an infinitely long line

charge and counterions are assumed to condense on the chain, effectively neutralizing

142

Figure 7.9:Density of the counterion cloud surrounding a chain as a function ofλB for a 10-bead chainat various concentrations.

some of the bead charges, to preserve a critical charge spacing in the case of strong elec-

trostatics. The ion adsorption process described by our simulations follows a sigmoidal

trend with respect to1/λB while the Manning theory predicts a linear process until sat-

uration is achieved atλB = 1.0. This trend stems from the fact that even under salt-free

conditions with no shielding, polyelectrolyte chains do not exist as rigid rods, but rather

as an expanded chain that may still explore multiple configurations. This discrepancy has

been previously noted in experiments (Beer et al., 1997), theory (Muthukumar, 2004),

and simulation (Winkler et al., 1998). Despite this discrepancy, however, the Manning

theory still provides an excellent guide as to the relationship between the degree of ion-

ization andλB.

Concentration also has a significant effect on the structure of the ion cloud, as evi-

denced by the above plots of the cloud size (Figure 7.8), density (Figure 7.9), and degree

of ionization (Figure 7.10), as well as the ion scatter plots presented in Figure 7.11 for

143

Figure 7.10:Degree of ionization as a function of1/λB . Also shown is the prediction from Manningtheory.

systems atλB = 2.25. Namely, increasing the concentration of the system leads to sub-

stantial decreases in both the cloud size and degree of ionization, in turn leading to a large

increase in cloud density. As described above, at low density, the chains are spaced far

enough apart that the ion clouds do not overlap. However, as the concentration of the sys-

tem increases, the ion clouds are increasingly packed, giving rise to the aforementioned

phenomena. These effects are largest for moderate values ofλB where there is little

electrostatic shielding between chains, and follow our earlier rationale for describing the

effects of concentration on chain size. We also note that while the cloud size decreases

with increasing concentration, the fraction of total system volume encompassed by the

counterion clouds increases substantially. For low values ofλB, changing concentration

has little effect on the ion cloud structure as the clouds are of very low density, with most

of the ions in the simulation cell dispersed uniformly throughout the cell. At the other

extreme, high values ofλB lead to the formation of a chain-ion cloud complex in which

144

the number of ions condensed in the vicinity of the chain is nearly equal to the number

of beads in the chain, creating a structure with a very low net charge. Thus, changing

concentration does not have a significant effect on the cloud structure at highλB until the

concentration is sufficiently high so that the clouds overlap one another.

7.5 Dynamic Results

Building on the static results presented above, we now turn our attention to the rheological

behavior of polyelectrolyte solutions when subjected to simple shear flows, where the

velocity gradient tensor,∇v, is given by

(∇v) =

0 0 0

γ 0 0

0 0 0

(7.13)

and whereγ is the shear rate. As described in the Introduction, the primary focus of this

work is to analyze the effects of bothλB and concentration on the polymer contribution

to the viscosity. In this section, we provide qualitative descriptions of these effects and

present a mechanism that attributes these effects largely to the electrostatic interactions

between a polymer chain and its enveloping cloud of counterions.

The measure of chain size most easily obtained from fluorescence microscopy ex-

periments is the average flow-direction “stretch”,X, defined as the distance between the

upstream-most portion of the molecule and the downstream-most portion,

X = 〈max(rν,x)−min(rν,x)〉 , (7.14)

wherer ν,x is thex-component of the position vector of beadν. The rheological behavior

of the polymer solutions is investigated by considering the polymer contribution to the

145

(a) c/c∗ = 10−4 (b) c/c∗ = 10−3

(c) c/c∗ = 10−2 (d) c/c∗ = 10−1

Figure 7.11:Depictions of the equilibrium ion clouds surrounding an individual chain in dilute solutionfor various concentrations. Shown is thex-y profile with data averaged through thez-direction. Scalesreflect the excess concentration of ions relative to the average concentration of ions in the system, i.e.cI (r) = cI (r) − NI/V . The system shown is atλB = 2.25, with the panels showingc/c∗ = (a) 10−4,(b) 10−3, (c) 10−2, and (d)10−1.

146

stress tensor for the system,τp, calculated as

τp =1

2V

∑µ

∑ν

r νµF(φ)νµ . (7.15)

The indicesµ and ν are taken over all particle pairs, including dissociated ions, and

F(φ)νµ incorporates all non-hydrodynamic forces for a given particle pair. By calculating

the stress tensor in this manner, we may isolate the contributions to the total stress tensor

stemming from the bead-bead, bead-ion, and ion-ion interactions individually. A reduced

viscosity is calculated according to

ηr =ηpc

∗

ηsc, (7.16)

whereηs is the solvent viscosity and the polymer contribution to the viscosity for simple

shear flow is given by

ηp = −τp,12

γ. (7.17)

By calculating the viscosity in this manner, we have normalized the viscosity against the

monomer concentration so as to eliminate the simple linear concentration dependence.

Finally, we note that our viscometric data is presented in terms of a bead Peclet number,

Pe, defined as the ratio of the convective time scale to the time scale of the diffusion of a

polymer bead over a distanceσ,

Pe =1/γ

ζσ2/kBT. (7.18)

Viscometric data for the study of polymer rheology is generally presented in terms of a

Weissenberg number,We = τ0γ, whereτ0 is the longest relaxation time of the polymer.

However, as the longest relaxation time of a polyelectrolyte is both difficult to accurately

determine and depends on many factors (e.g. molecular weight, concentration,λB), we

have instead chosen to present our results as a function ofPe.

147

Effect of Molecular Weight

For reasons of computational feasibility, we have restricted our study of polyelectrolyte

dynamics to the use of 10-bead chains. We have also calculated select dynamic properties

using both 20- and 40-bead chains with a variety of concentrations and values ofλB,

but find that the qualitative behavior is similar to that displayed by 10-bead chains. In

addition, we recognize that as we are dealing with shear flows in terms of a Peclet number

instead of a Weissenberg number, using longer chains actually leads to a problem in the

calculation of shear properties. For equivalent values ofPe, the longer chains will exhibit

a higher degree of shear-thinning than will shorter chains, leading us away from the zero-

shear plateau. At such flow rates, as we shall see below, there is little interplay between

the polyelectrolyte chains and the counterions in solution, leading to rheological behavior

that does not exhibit a dependence on either concentration orλB. Attempting to access

lower values ofPe is also problematic as the noise inherent in the calculation of material

functions such as the viscosity becomes unacceptable. As a result, we limit this work

to investigations of 10-bead chains alone in order to explain certain phenomenological

trends of interest. As we shall see below, even with such short chains, we may still draw

several general conclusions regarding the behavior of polyelectrolytes in dilute solution.

Effect of Hydrodynamic Interactions

Numerous previous works (Jendrejack et al., 2000; Hsieh et al., 2003; Petera and Muthuku-

mar, 1999; Sunthar and Prakash, 2005; Grassia and Hinch, 1996; Schroeder et al., 2004;

Liu et al., 2004; Hernandez-Cifre and de la Torre, 1999; Neelov et al., 2002; Agarwal

et al., 1998; Agarwal, 2000) have discussed the importance of including hydrodynamic

interactions in dynamic simulations of dilute polymer solutions in order to obtain an

148

accurate depiction of transport properties. We have considered the effects of including

hydrodynamic interactions in simulations of polyelectrolytes in simple shear flow, with

the calculated reduced shear viscosity for 10-bead chains at various concentrations and

Bjerrum lengths presented in Figure 7.12 forPe = 1.0. It is apparent that while the

inclusion of hydrodynamic interactions leads to a quantitative decrease in the reduced

viscosity when compared with free-draining results, the essential qualitative trends for

viscosity with respect to bothλB and concentration are unaffected. These results are not

surprising, however, as both experimental (Clasen et al., 2004; Owens et al., 2004) and

simulation (see Section 6.5.1) evidence have indicated that hydrodynamic concentration

effects are minimal for dilute systems in simple shear flows. As a result, in the remainder

of this work we consider only free-draining systems and focus on the electrostatic effects,

both as a function ofλB and concentration.

Effect of Bjerrum Length

We begin by considering the behavior of systems at low concentration (c/c∗ = 10−4),

where the system is sufficiently dilute so that there is little direct interaction between

different chains. In doing so, we may thus isolate the rheological effects stemming from

the interplay between an individual chain and the ions in proximity to that chain. In

Figure 7.13, we plot both the average chain stretch and the reduced viscosity as a function

of Pe for different values ofλB. From these plots, we identify a number of regimes of

interest on comparing the relative strengths of the imposed flow and the Bjerrum length.

Beginning with Figure 7.13(a), at lowPe, we observe that the imposed flow does not

significantly deform the chain-ion structure from its equilibrium conformation. We thus

see a nonmonotonic trend of〈X〉 with increasingλB, similar to that observed in Section

149

Figure 7.12:Comparison of reduced viscosity results as a function ofλB for 10-bead systems both withand without hydrodynamic interactions atPe = 1.0.

7.4. As the flow rate increases, however, ions are increasingly stripped away from the

condensed clouds by the flow, as illustrated in Figures 7.14 - 7.16. The actual deformation

of the ion clouds is discussed below; for now, we note that as a result of the ion cloud

deformation, the chains experience increased intra-chain electrostatic repulsions asPe

increases, leading to a shift in the trend of〈X〉 with λB. At sufficiently highPe (Pe ≈

10), enough ions have been stripped from the cloud so that the chains no longer collapse

over the range ofλB studied here, leading to a monotonic increase in〈X〉 with increasing

λB. For sufficiently highλB (i.e. whenλB/σPe

1), however, we should expect that the

electrostatic interactions would be strong enough to overcome the separating effects of

the flow, and we would once again observe a decrease in chain stretch with increasing

λB. As Pe increases further, we continue to observe the monotonic increase in〈X〉

with increasingλB for the values ofλB considered here, although the effect weakens

150

substantially. AtPe ≈ 1000, the flow rate is high enough to stretch and align the chains

such that electrostatic repulsions do not significantly affect the overall chain stretch.

On considering the effects of changingλB on the reduced viscosity of our dilute

systems (Figure 7.13(b)), we again note various behavioral regimes tied to the bead-ion

interactions. Specifically, at lowPe, we observe a slightly nonmonotonic trend ofηr

with λB, asηr increases with increasingλB for λB < 5, followed by a decrease asλB

approaches a value of10. As Pe increases, the deformation of the condensed cloud

causes this trend to shift to one in whichηr increases monotonically with increasingλB,

as we demonstrate below. Finally, at highPe, the ion cloud becomes highly dispersed,

leaving only intrachain repulsive interactions, and little dependence onλB. While there

are qualitative similarities to the trends described above for the chain stretch, we observe

effects onηr of much greater magnitude due to changingλB than may be accounted for

by analysis of the chain structure alone.

To better understand the origins of the strong dependence ofηr on λB, we decom-

pose the viscosity into contributions from various interactions. These contributions are

described according to the types of particles involved:ηr,BB for bead-bead interactions,

ηr,BI for bead-ion interactions, andηr,II for ion-ion interactions, and by the type of in-

teractions involved:ηEXVr for excluded volume andηEL

r for electrostatic interactions.

Figure 7.17 shows these contributions for a system atc/c∗ = 10−4 andPe = 1.0. This

particular system was chosen as it provides a clear illustration of the interplay between

Pe andλB. It is clear that asλB increases, the increase inηr is predominantly due to

the change in the total bead-ion contribution. As the ion cloud incorporates more ions

at a higher density with increasingλB, we observe a dramatic increase in the bead-ion

electrostatic contribution which overcomes all other contributions. These arguments hold

151

(a) Chain Stretch

(b) Viscosity

Figure 7.13:(a) Average chain stretch and (b) reduced viscosity for10 bead chains atc/c∗ = 10−4 as afunction ofPe for various values ofλB .

152

as we consider lower values ofPe as well. As we decreasePe, the deformation of the

cloud due to flow decreases, and from Figure 7.18, we see that the bead-ion contribution

becomes important at lower values ofλB. Thus, it is this contribution that we analyze in

greater detail.

At equilibrium, the cloud of ions attracted about a given chain is roughly spherical in

shape. When a shear flow is imposed, however, the cloud is deformed into an ellipsoidal

shape with the primary axis of the ellipsoid tilted at an angle from the direction of flow.

The actual size and orientation of the cloud are primarily determined for dilute systems

by the relative effects of the flow andλB. Representative examples of such ion clouds,

shown in shear profile (flow in thex-direction, gradient in they-direction) with the his-

tograms calculated by averaging in thez-direction, are given for chains atc/c∗ = 10−3

in Figures 7.14 and 7.16, withλB = 1.50 andλB = 10.0, respectively. We also include

the dominant principal axis of the chain for reference, depicted by the solid line across

the origin. Density plots are presented forPe = 0.0, 0.1, 1.0, and 10.0. From these

density plots, it is apparent that the primary axis of the ensemble averaged ion cloud lies

at an angle to that of the chain. This is due to the cumulative effects of the shear flow

and the electrostatic attractions between the counterions and the chain, and is crucial to

understanding the origins of theηr dependence onλB, as well as the effects of altering

concentration as discussed in Section 7.5.

Consider the cartoons of Figure 7.19, showing the two possible situations for the rel-

ative positions of the chain center-of-mass and a dissociated counterion (note that swap-

ping the positions of the two particles does not affect the orientations of the interactions

depicted). In Figure 7.19(a), both the flow and electrostatic forces serve to bring the

particles closer together (the “cooperative” case), while in Figure 7.19(b), the flow and

153

(a) Equilibrium (b) Pe = 0.1

(c) Pe = 1.0 (d) Pe = 10.0

Figure 7.14:Depictions of the ion cloud surrounding an individual chain in dilute solution for variousvalues ofPe. Shown is thex-y profile with data averaged through thez-direction;x is the flow direction,while y is the gradient direction. The average chain stretch and orientation in flow is mapped by thesolid black line in panels (c)-(d). Scales reflect the excess concentration of ions relative to the averageconcentration of ions in the system, i.e.cI (r) = cI (r)−NI/V . The system shown is atc/c∗ = 10−3 andλB = 1.5, with the plots showing (a) equilibrium, (b)Pe = 0.1, (c) Pe = 1.0, and (d)Pe = 10.0.

154


(c) Pe = 1.0 (d) Pe = 10.0

Figure 7.15:Depictions of the ion cloud surrounding an individual chain in dilute solution for variousvalues ofPe. Shown is thex-y profile with data averaged through thez-direction;x is the flow direction,while y is the gradient direction. The average chain stretch and orientation in flow is mapped by thesolid black line in panels (c)-(d). Scales reflect the excess concentration of ions relative to the averageconcentration of ions in the system, i.e.cI (r) = cI (r)−NI/V . The system shown is atc/c∗ = 10−3 andλB = 2.25, with the plots showing (a) equilibrium, (b)Pe = 0.1, (c) Pe = 1.0, and (d)Pe = 10.0.

155


(c) Pe = 1.0 (d) Pe = 10.0

Figure 7.16:Depictions of the ion cloud surrounding an individual chain in dilute solution for variousvalues ofPe. Shown is thex-y profile with data averaged through thez-direction;x is the flow direction,while y is the gradient direction. The average chain stretch and orientation in flow is mapped by thesolid white line in panels (c)-(d). Scales reflect the excess concentration of ions relative to the averageconcentration of ions in the system, i.e.cI (r) = cI (r)−NI/V . The system shown is atc/c∗ = 10−3 andλB = 10.0, with the plots showing (a) equilibrium, (b)Pe = 0.1, (c) Pe = 1.0, and (d)Pe = 10.0.

156

(a) Viscosity contributions by particle type

(b) Bead-ion viscosity contributions by interaction type

Figure 7.17:Component contributions to the overall reduced viscosity of the system as a function ofλB for systems atc/c∗ = 10−4 andPe = 1.0. Component contributions are described with subscriptsaccording to the types of particles involved (BB for bead-bead interactions, BI for bead-ion interactions,and II for ion-ion interactions) and with superscripts for the type of interactions involved (EXV for excludedvolume, and EL for electrostatic).

157

Figure 7.18:Component contributions to the overall reduced viscosity of the system as a function ofλB for systems atc/c∗ = 10−4 andPe = 0.01 and0.1. Component contributions are described withsubscripts according to the types of particles involved (BB for bead-bead interactions, BI for bead-ioninteractions, and II for ion-ion interactions).

electrostatic forces oppose one another (the “competitive” case). As a result, oppositely

charged particles tend to pass through configurations of type (a) more rapidly, and through

those of type (b) less rapidly, than would neutral particles. Thus, we see a higher con-

centration of ions in configurations of type (b) than those of type (a), leading to the tilted

ellipsoidal shapes of Figures 7.14 and 7.16. AsλB increases at a given value ofPe, the

elecrostatic forces act more strongly in competing or cooperating with the flow and thus,

in addition to simply causing a larger number of ions to condense about the chain, also

enhance the disparity between the ion-rich (quadrants I and III) and ion-poor (quadrants

II and IV) regions. An interesting, related effect occurs pertaining to the excluded volume

interactions. For those ions in the enveloping cloud that are in very close proximity to

the chain, there exist repulsive excluded volume interactions with the beads of the chain

that are stronger than the attractive electrostatic interactions. These interactions cause an

158

effect in opposition to that described above in the region immediately surrounding the

chain, in which ions tend to preferentially reside in the regions where the repulsive bead-

ion interactions compete with the actions of the flow, and deplete from the regions where

the flow and bead-ion repulsions cooperate. This two-tiered description of the ion cloud

can now be used to explain both the net positive bead-ion excluded volume contribu-

tion, ηEXVr,BI , and the net positive bead-ion electrostatic contributions,ηEL

r,BI , to the reduced

viscosity despite the fact that these forces act in opposition to one another. To see this,

we consider the calculation of the reduced viscosity of Equation 7.16. The key quantity

here isηr ∝ −∑

ν

∑µ r νµ,1Fνµ,2, so, for bead-ion arrangements such as those of Figure

7.19(a),sgn(r νµ) = sgn(Fνµ), giving a negative contribution toηr, while arrangements

of the type shown in Figure 7.19(b) havesgn(r νµ) = −sgn(Fνµ), and so yield a posi-

tive contribution toηr. Thus, as we increaseλB, and correspondingly increase both the

strength of the electrostatic forces and the difference in the number of interactions of

types (a) and (b), we observe a net increase inηr for dilute systems at moderate values of

Pe.

With this description, we may make a clear connection to the so-called electroviscous

effect, in which it is assumed that electrostatic effects cause three significant contribu-

tions to the reduced viscosity of a polyelectrolyte solution. Of particular interest here are

the primary and tertiary electroviscous effects, which describe the viscosity contributions

stemming from the retardation of the deformation of the ion cloud due to electrostatic

attractions and from the alteration of the electrostatic screening and resulting change in

conformation of the polyelectrolyte chain, respectively (the secondary effect, describing

the contribution from lubrication forces arising from the passage of ions past one another,

is not accounted for in this model owing to the point-charge description of the ions). At

159

(a) Cooperative

(b) Competitive

Figure 7.19:(a) Cooperative and (b) competitive arrangements of a counterion and the chain center-of-mass with regards to the attractive electrostatic interactions. For repulsive interactions (e.g. excludedvolume), the cooperative and competitive labels are reversed.

160

low λB, we observe from Figure 7.17(a), that the dominant contribution to the overall re-

duced viscosity stems from intrachain bead-bead interactions, indicating that the tertiary

effect is responsible for a significant contribution to the solution viscosity. However, as

we changeλB, the bead-ion effects are responsible for thechangein reduced viscosity,

indicating the influence of the primary electroviscous effect.

Effect of Concentration

We next turn our attention to the effect of varying concentration on the shear viscosity

of a dilute polyelectrolyte system. The rheological effects associated with changing the

concentration in the dilute regime have not been explored in great detail to this point from

an experimental standpoint as this regime is experimentally difficult to study. However, a

few studies (Eisenberg and Pouyet (1954); Cohen et al. (1988); Boris and Colby (1998))

have generally agreed that there is a significant increase in reduced viscosity associated

with an increase in concentration in the dilute regime. In this section, we seek to analyze

this trend for salt-free solutions, and in light of the above mechanism for the relationship

betweenλB andηr provide a new, alternative explanation for the observed rheological

behavior that does not require the presence of a salt to screen the electrostatics.

We again begin by considering the average chain stretch in flow in order to describe

the structural influence on the polymer contribution to the viscosity. Plotted in Figure

7.20(a) is the average chain stretch as a function ofλB for systems under an imposed shear

flow at various concentrations and different values ofPe, along with the equilibrium

case as a reference. For systems at equilibrium or sufficiently low flow rates such that

the systems are not significantly deformed from equilibrium, the chain stretch displays

only a weak dependence on concentration for moderate values ofλB (〈X〉 decreases

161

as concentration increases), and no appreciable dependence for either neutral chains or

for high λB. These trends continue for higher flow rates as well, as the concentrations

considered here are too low for the interchain electrostatic interactions to significantly

affect the chain structure.

As was the case above when comparing systems at differentλB, the effect of con-

centration on the reduced viscosity of the system cannot be explained by simple mild

changes in the chain structure. Plotted in Figure 7.20(b) is the reduced viscosity of10-

bead chains as a function ofλB for systems at various concentrations and flow rates.

From these figures, we observe that the viscosity increases substantially with increasing

concenctration for all values ofλB > 0 considered here, in direct contrast with both

the theoretical predictions of Cohen et al. (1988),Rabin et al. (1988) and Dobrynin et al.

(1995) and to the effect of concentration on chain stretch. As the flow rate increases,

the concentration dependence diminishes for a given value ofλB, owing to the fact that

more ions may be stripped from the cloud about a given chain, weakening the structure

of the ion cloud that is primarily responsible for changes inηr as a function ofλB. The

neutral case does not show any concentration dependence, regardless of whether or not

hydrodynamic interactions are included. As described in Section 7.5, the primary contri-

bution to the overall reduced viscosity stems from the electrostatic interactions between

beads along a given chain and the counterions in proximity of that chain. Thus, it is the

concentration dependence of this contribution that we consider in detail.

Plotted in Figure 7.22 are the contributions to the viscosity as a function ofλB for

systems at various concentrations and subjected to a flow ofPe = 1.0. At this flow rate,

we again observe that the bead-ion electrostatic contribution displays the greatest con-

centration dependence. To explain this effect, consider the effect of concentration on the

162

(a) Flow direction stretch

(b) Reduced viscosity

Figure 7.20:Rheological behavior of10-bead polyelectrolyte chains plotted as a function ofλB forvarious concentrations atPe = 1.0. Figure (a) depicts the flow direction chain stretch,< X >, while (b)shows the reduced viscosity,ηr.

163

(a) c/c∗ = 10−4 (b) c/c∗ = 10−3

(c) c/c∗ = 10−2 (d) c/c∗ = 10−1

Figure 7.21:Depictions of the ion cloud surrounding an individual chain in dilute solution for variousvalues ofc/c∗. Shown is thex-y profile with data averaged through thez-direction;x is the flow direction,while y is the gradient direction. The average chain stretch and orientation in flow is mapped by thesolid black line in panels (a) and (b) and by the white line in panels (c) and (d). Scales reflect the excessconcentration of ions relative to the average concentration of ions in the system, i.e.cI (r) = cI (r)−NI/V .The system shown is atλB = 1.5 andPe = 1.0, with the plots showingc/c∗ = (a) 10−4, (b) 10−3, (c)10−2, and (d)10−1.

164

structure of the condensed ion clouds. Recall that for systems sufficiently dilute so that

the clouds of ions associated with each chain do not overlap one another at equilibrium,

increasing the concentration results in denser ion clouds enveloping the polyelectrolyte

chains (see Figure 7.11). Similar trends are observed when a simple shear flow is applied,

although the clouds are increasingly stripped apart as the flow rate is increased for a con-

stant value ofλB, as demonstrated in Figures 7.14 and 7.16. This is a critical finding in

that the cooperative-competitive effect described above as being responsible for the bead-

ion contribution to the viscosity applies only for ions located within the cloud structure.

Beyond the boundaries of the cloud, the ion distribution is, on average, homogenous,

and so there is no net contribution to the viscosity stemming from electrostatic interac-

tions between the chain and ions in this region. Thus, as we increase concentration, the

increased number of ions per cloud leads to a greater absolute difference between the

ion-rich and ion-poor regions about the chain, leading to an increase in reduced viscosity.

As with changes inλB, this description is consistent with the primary electroviscous ef-

fect, in which a retardation in the deformation of the ion cloud due to electrostatic effects

is responsible for contributing to the change in reduced viscosity of the solution as we

change concentration. Hence, we are able to the “polyelectrolyte effect” by means of the

primary electroviscous effect, with direct evidence given by our simulation results.

Finally, we summarize the interplay between the electrostatic strength, imposed flow

rate, and concentration by plotting the reduced viscosity as a function ofλB/σPe

in Figure

7.23. This ratio gives the relative strengths of the ability of the imposed flow to deform the

system and the tendency to resist deformation owing to electrostatic interactions. When

λB/σPe

1, we observe that both the concentration andλB play a role in determining the

rheological behavior of our solutions. At the other extreme,λB/σPe

1, the electrostatic

165

(a) Viscosity contributions by particle type

(b) Bead-ion viscosity contributions by interaction type

Figure 7.22:Component contributions to the overall reduced viscosity of the system as a function ofconcentration for a system withλB = 1.5 andPe = 1.0. Component contributions are described withsubscripts according to the types of particles involved (BB for bead-bead interactions, BI for bead-ioninteractions, and II for ion-ion interactions) and with superscripts for the type of interactions involved(EXV for excluded volume, and EL for electrostatic).

166

Figure 7.23: Universal plot of the reduced viscosity as a function ofλB/Pe for systems at variousconcentrations and values ofPe.

interactions are not strong enough to prevent the cloud of ions from being stripped apart

by the imposed flow and the viscosity data collapses onto a common curve as the chains

become highly stretched and aligned.

7.6 Conclusions

We have conducted a systematic analysis of the behavior of dilute solution polyelec-

trolytes in simple shear flows, exploring the relationships between flow rate, Bjerrum

length, and concentration, for short chains of 10 beads. It was found that, due to the

stripping of ions from the vicinity of the chain caused by the flow, the polyelectrolyte

chains exhibit shear thinning behavior at highPe that is independent of the electrostatic

strength. In contrast, at low values ofPe, systems at different values ofλB exhibit very

different viscosities owing to differences in chain conformation and their surrounding ion

167

clouds. Furthermore, the presence of the ion cloud causes the viscosity to increase mono-

tonically with increasing Bjerrum length over the range studied here, in contrast to the

non-monotonic trend of chain size with increasing Bjerrum length. A specific mechanism

based on the structure and orientation of the ion cloud is presented to explain this effect.

In particular, the dominant contribution to the viscosity dependence on the Bjerrum

length stems from electrostatic attractions between beads of the polyion chain and coun-

terions in proximity to that chain. These attractive interactions, when combined with

a simple shear flow, result in an ion cloud that lies tilted from the primary axis of the

chain and the formation of both ion-rich and ion-depleted regions about the chain. The

ion density difference between these regions is directly related to the net bead-ion vis-

cosity contribution, in accordance with the primary electroviscous effect, and is highly

dependent on the value ofλB.

While concentration plays a weak role in determining the structure of a polymer chain

in dilute solution, both at equilibrium and when a simple shear flow is imposed, we have

demonstrated that changing concentration has a significant impact on the rheological be-

havior of such systems. We explain these effects with arguments similar to those used to

describeλB-based effects, as increasing concentration forces more ions into the vicinity

of the chains and enhances the disparity between the ion-rich and ion-poor regions about

each chain. Finally, we have also considered the role of hydrodynamic interactions in

these simulations; we find that for low-concentration studies in shear flow, the electro-

static effects thoroughly dominate the hydrodynamic effects and one may safely capture

the correct qualitative behavior without including hydrodynamic interactions.

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Chapter 8

CONCENTRATION EFFECTS ON

THE COLLECTIVE DYNAMIC

BEHAVIOR OF SELF-PROPELLED

PARTICLES

8.1 Introduction

The collective dynamics of swimming particles are interesting and important for a variety

of fundamental and technological reasons. For example, there is long-standing interest

in the theoretical biology and nonlinear physics communities in the collective motions

of groups of organisms such as flocks and herds. Central issues here include the mecha-

nisms by which autonomous agents interact to exhibit emergent collective behavior and

the properties of the resulting behavior. Another is the evolutionary significance of these

169

collective motions and whether different modes of collective swimming are more evolu-

tionarily favorable than others in various circumstances. Recently, researchers have be-

gun to experimentally study the fluid motions that directly arise in suspensions of swim-

ming microorganisms (Mendelson et al., 1999; Wu and Libchaber, 2000; Wooley, 2003;

Kim and Powers, 2004; Dombrowski et al., 2004), finding a fascinating variety of phe-

nomena including regimes of anomalous transport as well as spatiotemporally coherent

fluid motion on scales much larger than the organisms. Furthermore, it has recently been

experimentally demonstrated that mass transport in a microfluidic device can be enhanced

by the presence of swimming microorganisms (Kim and Powers, 2004).

The present work employs direct simulations to improve our understanding of these

experimental observations. We use a minimal model of the swimmers that captures the

dominant far-field hydrodynamics while keeping the structure of each swimmer very sim-

ple. This approach is taken for two reasons: first, it focuses attention on the “univer-

sal” long-range interactions without the complicating, computationally expensive, and

nonuniversal details of swimmer shape and detailed mechanism of propulsion, and sec-

ond, it allows for relatively rapid solution of the equations of motion, enabling simula-

tions of large populations. These simulations clearly illustrate that hydrodynamic inter-

actions alone are sufficient to yield complex collective dynamics in swimming particle

suspensions.

Wu and Libchaber (2000) have experimentally characterized correlated motions in

1% – 10% suspensions ofE. coli confined to a horizontally suspended soap film of thick-

ness 10µm. The fluid displayed intermittent flows in the form of swirls and occasionally

jets, with length scales of10 – 20 µm; this is of the same order of the film thickness, but

the film thickness was not varied so it is not known if that is what set the scale of the

170

motions. The authors studied the transport of tracer particles suspended in the film note

that these particles were4.5 - 10 µm in diameter, significantly larger than the bacteria.

The mean squared displacement〈∆r2 (t)〉 of the particles displayed two distinct regimes,

a short time regime with anomalous (superdiffusive) transport, where〈∆r2 (t)〉 ∝ t1.5,

and a longer time regime where the transport was diffusive. The crossover time between

the anomalous and classical diffusion regimes increased with increasing bacterial con-

centration, varying between 1 and 10 s as concentration increased from about1% - 10%.

In related work, Soni et al. (2003, 2004) studied the motion of a particle in an optical trap

contained within a suspension ofE. coli, at volume fractions up to 0.1. They found that

the correlation time for the position of the trapped particle increased monotonically with

cell concentration, reaching a value of 1.2 s for the most concentrated suspensions.

Goldstein, Kessler, and co-workers have experimentally studied cell-driven motions

in droplets of suspensions ofBacillus subtilis(Dombrowski et al., 2004). In sessile

drops, conventional bioconvection patterns form, driven by a Rayleigh-Taylor instabil-

ity induced as the denser cells swim upward toward the free surface, where the oxygen

concentration is high (Pedley and Kessler, 1992). In pendant drops, where the flow is

gravitationally stable, flow patterns are also observed, with a length scale of 100µm

and a correlation time of 12 s. Dramatically enhanced tracer diffusion is also found. The

authors conjecture that the origin of these patterns is hydrodynamic interactions. Mendel-

son et al. (1999) describe quite similar patterns in a slightly different situation. Colonies

of B. subtiliswere grown on agar surfaces. When a drop of water was placed on a colony,

cells immediately began to swim, forming “whirls and jets” that persisted until the water

soaked into the agar.

Modeling of the collective dynamics of moving organisms has been performed at a

171

number of levels. A number of researchers have studied active agent models of mov-

ing groups of self-propelled particles: here each particle moves and interacts with its

neighbors according to an ad hoc set of rules. A simple but rich model of this type was

proposed by Vicsek et al. (1995). In this model, at each time step every particle moves a

constant distance in the direction of its current orientation, and the orientation is updated

so that it is the average of the orientations of its neighbors, plus a bit of noise. As the

magnitude of this noise is changed, the systems behavior undergoes a transition from or-

dered to disordered motion. Gregoire and coworkers (Gregoire et al., 2001a,b; Gregoire

and Chate, 2004) found that this model was able to reproduce the main features of the

experiments of Wu and Libchaber. A related approach was taken by Toner and Tu (1995,

1998), who wrote down general field equations for a conserved quantity (number density

of particles) and a nonconserved one (flux of particles). This model can exhibit various

solutions, including ordered phases in which all particles move in the same direction and

disordered ones with large fluctuations in number density.

Another field-theoretic approach, this time with a more direct connection to the prob-

lem of interest here, was taken by Ramaswamy and coworkers (Simha and Ramaswamy,

2002; Hatwalne et al., 2004). In their theory, the number of particles is conserved, as is

fluid momentum. The effect of the particles on the fluid as they swim is accounted for by

including a dipole forcing term in the Navier-Stokes equations. (To leading order in the

far field, a neutrally buoyant swimming particle is a force dipole.) A third, nonconserved

field is the orientation field of the swimmers, which is treated in a way similar to phe-

nomenological treatments of the director field in nematic liquid crystals. With this model,

the authors predict that (1) oriented (“nematic”) suspensions of self-propelled particles

at low Reynolds number are always unstable to long wavelength disturbances and (2)

172

the number density fluctuations in this case are anomalously large: for a system with N

particles, the scaled variance⟨(δN)2⟩ /N of the number of particles in a given volume

diverges asN2/3. (This divergence is reminiscent of the controversial Caflisch-Luke di-

vergence prediction in sedimentation (Caflisch and Luke, 1985).) A similar model has

been developed by Liverpool and Marchetti (2005) in the context of solutions of filament-

motor-protein mixtures. Again, a uniform oriented state is predicted to be unstable.

The results obtained from the aforementioned studies are suggestive and intriguing.

They show that simple models obtained from general arguments predict nontrivial spa-

tiotemporal patterns in the dynamics of self-propelled particles. But even the models

described last, which do incorporate the Navier-Stokes equations, are limited. They do

not capture from first principles the details of the hydrodynamic interactions, they are

limited to very large length scales (as they treat the particle phase as a continuum field),

and there are too many free parameters for conclusive analyses beyond linear stability

to be performed. In the present work, we avoid these limitations, performing and ana-

lyzing the first direct simulations of suspensions of model self-propelled particles at low

Reynolds number.

8.2 Model

Simulations are performed with a minimal swimmer model, shown in Figure 8.1, that

captures the leading order far-field effect of hydrodynamic interactions between swim-

mers without specifying in detail the structure of the swimmer or its method of propul-

sion. Each swimmer is modelled by a nearly-rigid, neutrally buoyant dumbbell comprised

of two beads connected by a FENE-Fraenkel spring, which has been shown by Hsieh et al.

173

(2006) to provide a good approximation of a bead-rod (Kramers) chain for highly stiff

springs. The resulting force in connectori is given by

Fspri = −

H(1− Qeq

Qi

)1−

(Qi−Qeq

Q0−Qeq

)2 Qi, (8.1)

whereQ0 is the maximum spring extension,Qeq is the equilibrium spring length,Qi =

r i+1 − r i, andH is the spring constant. In this work, we useQ0 = 1.0σ, Qeq = 0.2σ

andH = 100.0kBT/σ2, whereσ is some arbitrary length scale. While not maintaining

a strictly rigid dumbbell, at low concentration, this parameterization allows the spring

length to vary by no more than3.5% from the equilibrium length and gives a reasonable

estimate of the rigid dumbbell model. The orientation of each swimmer is denoted by

a unit director vectorn. All the drag on the swimmer is concentrated on the two beads.

The propulsion is provided by a “phantom” flagellum, which exerts a constant force of

magnitudeFfl in the n direction on one of the beads (which we designate the “head”;

the other bead is correspondingly the “tail”) and an equal and opposite force on the fluid.

The phantom flagellum can either push or pull on the dumbbell; the “pushing” case cor-

responds to most spermatozoa and many other microorganisms, but the “pulling” case

is also commonly found in nature Bray (2001), such as in the case of the green alga

Chlamydomonas(8.2). In our model, the parameterp characterizes the “polarity” of the

flagellar force: ifp = +1, the flagellum pushes the swimmer; ifp = −1, it pulls. Some

organisms, such asE. coli, execute a complex “run-and-tumble” motion, during which

they change directions at random intervals. In the present model, we do not account

for such organism-specific effects; in isolation in an unbounded domain, each swimmer

would move in a straight line with constant speedvsw = F fl/2ζ + O (a/Qeq), whereζ

anda are the Stokes law friction coefficient and the hydrodynamic radius of each bead,

174

Figure 8.1:Bead-spring dumbbell model of a swimmer. The flagellum is represented by a force exertedon one of the beads of the dumbbell, and a force in the opposite direction exerted by the dumbbell on thefluid. The casep = +1 is shown.

respectively. Here, we takea = 0.05σ. Overall, the swimming motion exerts no net

force on the fluid, so in the far field the swimmer appears to be a moving symmetric force

dipole (stresslet) (Pedley and Kessler, 1992; Simha and Ramaswamy, 2002; Hatwalne

et al., 2004). In general, the torque balance on the swimmer also needs to be considered.

However, the leading order far-field flow due to the torques is weaker than the stresslet

contribution, so in the present minimal model, we neglect this effect.

We have performed direct numerical simulations of the particle motions in suspen-

sions of these simple swimmers in bulk solution, considering the situation where the

swimmers interactonly through the low-Reynolds-number hydrodynamics of the solvent.

The domain is a periodically replicated cube with edge lengthL. Within each cell, we

enclose a total ofNP swimmers at a number density ofc = NP/L3 as well asNT mass-

less tracer particles in order to gauge the hydrodynamic effects on the fluid. Positions

are nondimensionalized withQeq and time withQeq/vsw. Concentration is normalized

by the overlap concentration, defined on a swimmer basis asc∗ = 1/Q3eq as per Doi and

175

Figure 8.2:The alga Chlamydomonas and its normal (p = -1, top right) and escape (p = +1, bottom right)modes of flagellar motion (Bray, 2001).

Edwards (1986).

Hydrodynamic interactions between different particles are again introduced through

the off-diagonal components of the mobility tensor,M , which we compute using the

Rotne-Prager-Yamakawa (RPY) expression (Rotne and Prager, 1969; Yamakawa, 1970)

for the hydrodynamic interaction tensor,

M νµ =1

ζ

δ ν = µ

3a4rνµ

[(1 + 2a2

3r2νµ

)δ +

(1− 2a2

r2νµ

)rνµrνµ

r2νµ

]ν 6= µ andrνµ ≥ 2a[

(1− 9rνµ

32a)δ + 3

32a

rνµrνµ

rνµ

]ν 6= µ andrνµ < 2a

(8.2)

whereδ is the identity tensor and the Stokes Law relationζ = 6πηa has been assumed.

The hydrodynamic interactions are long-ranged and are calculated using the Ewald sum-

mation technique (Beenakker, 1986; Smith et al., 1987; Brady et al., 1988; Zhou and

Chen, 2006) to account for the periodic contributions.

176

8.3 Simulation

The basic algorithm for the simulation of self-propelled particles is slightly different from

that of the Brownian dynamics scheme used in Chapters 6 and 7, owing both to the lack

of a random Brownian term and the presence of a nonconservative force representing the

flagellar motion. As such, in this section, we derive the equations of motion for the self-

propelled particles. We begin by considering the force balance involving the fluid about

an individual swimmer:

Fhyd,f1,i + Fhyd,f

2,i + Ffl,fi = 0 i = 1, . . . , NP , (8.3)

whereNP is the number of swimmers,Fhyd,fν,i is the force acting on the fluid due to the

presence of theνth bead of theith swimmer andFfl,fi is the flagellar force imparted on

the fluid. Note that throughout this section, we use greek subscripts to denote individual

beads (ν, µ = 1, 2), while roman subscripts denote swimmers (i, k = 1, . . . , NP ). Hence,

r ν,i denotes theνth bead of theith swimmer. Now, we may also write the force balance

about each bead of theith swimmer as

Fhyd1,i + Fspr

1,i + Fexv1,i + Ffl

i = 0

Fhyd2,i + Fspr

2,i + Fexv2,i = 0 (8.4)

whereFsprν,i andFexv

ν,i are the connector (spring) and excluded volume forces, respectively,

acting on beadν of the ith swimmer, andFfli is the flagellar force acting on swimmeri.

Note that here we use the superscript “f” to denote forces acting on the fluid, while the

absence of such a superscript shall indicate that the force acts on a bead of a swimmer.

From Stokes Law, we express the hydrodynamic force about a given beadν as

Fhydν,i = −ζ

(vν,i − v′ν,i

). (8.5)

177

Here,ζ is the bead friction coefficient,vν,i contains the velocity components of theνth

bead of theith swimmer, andv′ν,i is the pertubation to the velocity field surrounding this

bead stemming from hydrodynamic interactions with other particles. Substituting this

into Equation 8.4, we have

−ζ(dr 1,i

dt− v′1,i

)+ Fspr

1,i + Fexv1,i + Ffl

i = 0

−ζ(dr 2,i

dt− v′2,i

)+ Fspr

2,i + Fexv2,i = 0. (8.6)

This may be recast to give the basic evolution equations as

dr 1,i

dt= v′1,i +

1

ζ

(Fspr

1,i + Fexv1,i + Ffl

i

)dr 2,i

dt= v′2,i +

1

ζ

(Fspr

2,i + Fexv2,i

). (8.7)

Next, we focus on the perturbation velocity, which depends linearly on the hydrody-

namic forces acting on all of the other beads in solution as

v′(r) =∑

µ

∑k

Ω (r − rµ,k) · Fhyd,fµ,k +

∑k

Ω (r − r 1,k) · Ffl,fk (8.8)

whereΩ is the hydrodynamic interaction tensor. Taking the perturbation at each bead

position and rearranging, we have,

v′1,i =∑k 6=i

Ω (r 1,i − r 1,k) ·(

Fhyd,f1,k + Ffl,f

k

)+∑

k

Ω (r 1,i − r 2,k) · Fhyd,f2,k

v′2,i =∑

k

Ω (r 2,i − r 1,k) ·(

Fhyd,f1,k + Ffl,f

k

)+∑k 6=i

Ω (r 2,i − r 2,k) · Fhyd,f2,k . (8.9)

Now, we may relate the forces acting on the fluid with those acting on the beads of the

swimmer according to

Fhyd,f1,i = Fspr

1,i + Fexv1,i + Ffl

1,i (8.10)

Fhyd,f2,i = Fspr

2,i + Fexv2,i (8.11)

178

and sinceFflµ,i = −Ffl,f

µ,i on inserting these into Equation 8.9, we obtain

v′1,i =∑k 6=i

Ω (r 1,i − r 1,k) ·(Fspr

1,k + Fexv1,k

)+∑

k

Ω (r 1,i − r 2,k) ·(Fspr

2,k + Fexv2,k

)v′2,i =

∑k

Ω (r 2,i − r 1,k) ·(Fspr

1,k + Fexv1,k

)+∑k 6=i

Ω (r 2,i − r 2,k) ·(Fspr

2,k + Fexv2,k

). (8.12)

We insert these expressions into Equation 8.7 to obtain

dr 1,i

dt=

∑k 6=i

Ω (r 1,i − r 1,k) ·(Fspr

1,k + Fexv1,k

)+

∑k

Ω (r 1,i − r 2,k) ·(Fspr

2,k + Fexv2,k

)+

1

ζ

(Fspr

1,i + Fexv1,i + Ffl

i

)dr 2,i

dt=

∑k

Ω (r 2,i − r 1,k) ·(Fspr

1,k + Fexv1,k

)+

∑k 6=i

Ω (r 2,i − r 2,k) ·(Fspr

2,k + Fexv2,k

)+

1

ζ

(Fspr

2,i + Fexv2,i

). (8.13)

Finally, by combining terms and taking the mobility tensor asM = Ω + 1ζδ, we get

dr 1,i

dt=

1

ζFfl

i +∑

µ

∑k

M (1,i),(µ,k) ·(Fspr

µ,k + Fexvµ,k

)(8.14)

dr 2,i

dt=

∑µ

∑k


µ,k + Fexvµ,k

)(8.15)

which can be easily solved using an Euler scheme:

r 1,i (t+ ∆t) = r 1,i (t) +∆t

ζFfl

i + ∆t∑

µ

∑k


µ,k + Fexvµ,k

)(8.16)

r 2,i (t+ ∆t) = r 2,i (t) + ∆t∑

µ

∑k


µ,k + Fexvµ,k

). (8.17)

The mobility tensor is calculated in a similar manner to the diffusion tensor of Section

3.4. The time step was chosen based on the relaxation time of a Hookean dumbbell

(∆t = 0.5/4H), and unless otherwise noted, simulations were run for sufficient ensemble

sizes to reduce the error bars to the order of the symbol size used here.

179

One of the primary difficulties that arises in the simulation of a bulk fluid at nonzero

concentration stems from the use of periodic boundary conditions (Allen and Tildes-

ley, 1987). Periodic boundary conditions are often employed in numerical simulations

to avoid spurious surface effects from artificially imposed containment. However, by

imposing periodic boundary conditions, we risk imposing artificial symmetries on the

system. Thus, we must take care in designing our systems so as to minimize such effects.

In this work, we have considered systems usingNP = 100, 200, and400 swimmers so as

to evaluate the effect of changing system size, with the results presented below.

8.4 Results

Our primary focus in this chapter is the study of the collective motions of self-propelled

particles induced by hydrodynamic interactions between different swimmers. In this sec-

tion, we consider the behavior of such particles in the absence of excluded volume in

order to determine the nature of effects that are purely of hydrodynamic origin. In addi-

tion, we consider the behavior of our particles when excluded volume is present in order

to address certain computational issues stemming from particle overlap that arise in the

case of no excluded volume.

8.4.1 No excluded volume

We begin by considering the mean-squared displacement (MSD) as a function of time for

both the swimmers and non-Brownian tracer particles in the absence of excluded volume

interactions. A representative curve describing the motion of swimmers at a concentra-

tion of c/c∗ = 0.02 is shown in Figure 8.3. We identify two regimes of interest: at low

180

concentrations, the transport is ballistic at short times, reflecting the straight-line swim-

ming of an isolated particle. At longer times, a crossover to diffusive behavior occurs,

with the crossover time (τC , taken as the intersection of the linear fits to the ballistic and

diffusive regimes) decreasing and the breadth of the crossover region increasing as con-

centration increases. As concentration increases, the ballistic region disappears almost

entirely, and the behavior can be characterized as diffusive on all appreciable time scales.

Conversely, at very low concentration, the transport is almost purely ballistic. We further

elucidate these two modes of transport in Figures 8.5(a) and 8.5(b), in which we plot

sample trajectories for an individual swimmer in systems ofc/c∗ = 0.01 andc/c∗ = 1.0

over 100ts units of time. From Figure 8.5(a), we observe the linear motion characteristic

of ballistic transport for periods of time on the order of 10ts, with large scale diffusive

motions occuring on longer time scales. At higher concentrations, however, the ballistic

time scale is much shorter owing to the closer proximity of the swimmers and the result-

ing increase in hydrodynamic perturbations to one anothers motions. From Figure 8.5(b),

we observe that particles move in a ballistic fashion for periods of less thants. The dif-

ference is even more evident when we compare the motions of fluid tracer particles at

the two different concentrations. At low concentration, there is only a weak perturbation

to the velocity field at at any given point not in close proximity to one of the swimmers.

As a result, the tracer particles do not move to any great degree, as illustrated in Figure

8.6(a). At higher concentrations (Figure 8.6(b)), however, there are strong perturbations

throughout the solution volume, leading to much larger tracer motions.

Using the crossover time,τC , we can characterize the relationship between concen-

tration and the time scales over which we observe a transition from ballistic to diffusive

181

Figure 8.3:Mean-square displacement as a function of time for a swimming particle withp = +1 at aconcentration ofc/c∗ = 0.02, illustrating the transition from ballistic to diffusive motion.

motion. This is plotted in Figure 8.7 for swimmers using various mechanisms of propul-

sion, and in Figure 8.8 for systems of varying size for both thep = +1 andp = −1

cases. From Figure 8.7, we observe that at low concentration, there is little difference in

the value ofτC based on the method of propulsion. However, as we consider higher con-

centrations, we observe that propulsion via the pulling mechanism leads to significantly

higher values of the crossover time than when the pushing mechanism is used, indicat-

ing a lower degree of hydrodynamic coupling between swimming molecules forp = −1

than forp = +1. Somewhat surprisingly, when we consider a system of 50% swimmers

with p = +1 and 50% withp = −1, we observe little deviation from the case in which

all swimmers move via the pushing mechanism. Finally, on comparing systems of 100,

200, and 400 swimmers per cell at equivalent concentrations, there appears to be a minor

decrease in the crossover time with increasingNp for p = +1. We find little difference

182

(a) Swimmers

(b) Tracers

Figure 8.4:Mean-square displacement as a function of time for (a) swimmers and (b) tracer particleswith p = +1 at various concentrations.

183

(a) c/c∗ = 0.01 (b) c/c∗ = 1.00

Figure 8.5:Trajectory traces for an individual swimmer in a collection of 100 swimmers atc/c∗ = a)0.01 and b) 1.00. Traces record100ts units of simulation time.

(a) c/c∗ = 0.01 (b) c/c∗ = 1.00

Figure 8.6:Trajectory traces for an individual tracer in a collection of 100 swimmers atc/c∗ = a) 0.01and b) 1.00. Traces record100ts units of simulation time.

184

Figure 8.7:Time scale,τC , over which the motion of the swimming particles changes from ballistic todiffusive in nature as extracted from the intersection of the asymptotic fits to the mean-square displacementvs. time.

in our results for the case ofp = −1 for systems of 100, 200, and 400 swimmers.

Figure 8.9 shows the effective long-time self-diffusion coefficient of both swimmers

and passive tracers as a function of concentration. At low concentrations, the effec-

tive diffusivity is high because the swimmers travel a long distance on a nearly straight

path before the weak hydrodynamic fluctuations signficantly alter their trajectories. The

flow is barely disturbed by the swimmers, so tracer particles diffuse very slowly. As

the concentration is increased, the diffusivity of the swimmers decreases as their natu-

rally ballistic trajectories are increasingly perturbed by hydrodynamic interactions with

other swimmers. Correspondingly, the naturally motionless tracers increasingly feel the

motion of the swimmers as concentration increases, leading to an increase in the tracer

diffusivity. For c/c∗ > 0.5, we observe that the diffusivity of both our swimming par-

ticles and the fluid tracers reaches a plateau value. This contrasts with the simulations

185

Figure 8.8:Time scale,τC , over which the motion of the swimming particles changes from ballistic todiffusive in nature as extracted from the intersection of the asymptotic fits to the mean-square displacementvs. time. Results are shown for various system sizes with bothp = +1 andp = −1.

of Hernandez-Ortiz et al. (2005) in which the authors considered the case of swimming

particles in a confined domain. In the confined domain, it was observed that a transition

occurs aroundc/c∗ ≈ 0.3 where the diffusion coefficient of both the swimmers and trac-

ers show a sharp increase as concentration increases. This type of behavior is indicative

of the emergence of strong large-scale coherent motion of the swimmers, and the lack of

such a transition indicates that in the present simulations, no such large scale motions are

evident.

At low concentration, we observe no appreciable difference in the diffusivity of the

swimmers regardless of the type of propulsion used; at infinite dilution, both types of

swimmers move in a straight line with velocities identical to that achieved in the absence

of hydrodynamic interactions. At higher concentrations, however, we see that the pulling

case leads to a higher swimmer diffusivity than does the pushing case. In contrast, the

186

Figure 8.9:Diffusion coefficient as a function of concentration for both the swimmer and tracer particlesusing different methods of propulsion.

tracer diffusivity is significantly higher in the pushing case than in the pulling case. In-

terestingly, the swimmer diffusivity is lowest for systems with 50% pushers and 50%

pullers. Furthermore, from Figure 8.10, we observe that the tracer diffusivity exhibits a

dependence on the number of swimmers per simulation cell, increasing with increasing

NP , for p = +1. No dependence is apparent forp = −1 or for the swimmers in either

case.

We next consider these findings in light of the velocities of both the swimming parti-

cles and the tracer particles. Plotted in Figure 8.11 is the particle velocity for both species

as a function of concentration for each type of propelling force considered above. At low

concentration, the swimmer velocities approach a uniform velocity as all intermolecu-

lar hydrodynamic interactions become increasingly weak. At higher concentrations, we

observe that thep = +1 case yields higher swimmer and tracer velocities than does the

187

(a) Swimmers

(b) Tracers

Figure 8.10:Diffusion coefficient as a function of concentration for both the (a) swimmer and (b) tracerparticles using different methods of propulsion. Results are shown for various system sizes and both typesof propulsion.

188

Figure 8.11:Velocities of both swimmer and tracer particles as a function of concentration for systemsutilizing various forms of propulsion.

p = −1 case. While this seems sensible for the tracer particles in light of our earlier re-

sults regarding the tracer diffusion, this finding appears to contrast with our earlier finding

that thep = +1 swimmers have a lower diffusivity than theirp = −1 counterparts. To

explain this phenomenon, we consider the crossover time for ballistic to diffusive behav-

ior from Figure 8.7. On plotting the swimmer diffusivity as a function of the product

v2τC (Figure 8.13), we observe that our data collapses well for all three propulsion cases

considered. Thus, the decrease in particle velocity is offset by longer correlation times,

with the result producing a consistent diffusivity across propulsion mechanisms. Finally,

as above, the velocities of both the swimmer and tracer particles exhibits much stronger

dependence onNP than do the pulling cases; both particle types exhibit an increase in

velocity with increasingNP .

189

(a) Swimmers

(b) Tracers

Figure 8.12:Velocities of the (a) swimmer and (b) tracer particles as a function of concentration forsystems of varyingNP . Results are shown for various system sizes and both types of propulsion.

190

Figure 8.13:Swimmer diffusivity as a function ofv2τC .

In order to better understand the exact nature of the hydrodynamic interactions be-

tween swimming molecules, consider the contour plots of Figure 8.14, in which we show

the orthogonal component of the velocity perturbation caused by the presence of a dumb-

bell moving in the lateral direction with panel 8.14(a) illustrating the case ofp = +1 and

panel 8.14(b) showingp = −1. In each case, the presence of the flagellar force, and en-

suing perturbation away from the equilibrium spring length, creates a force dipole in the

dumbbell that gives rise to the velocity perturbation in the fluid. For thep = +1 case, the

force dipole results in a net velocity perturbation where the fluid is drawn orthogonally

towards the body of the swimmer while being expelled axially from the ends of the swim-

mer. For thep = −1 case, the opposite trend occurs, with fluid expelled orthogonally

from the body of the swimmer and drawn to the swimmer axially. As a result, we observe

very different distributions of swimmers in solution based on the propulsion type.

191

(a) p = +1

(b) p = −1

Figure 8.14: Contour plot of the vertical component of the velocity perturbation field owing to thepresence of a force dipole in the dumbbell stemming from the application of (a) a pushing force (p = +1)or (b) a pulling force (p = −1). Dark regions indicate fluid moving in the positive vertical direction, whiledark regions indicate fluid moving in the negative vertical direction. Streamlines illustrate the net velocityfield. White circles indicate the location of the dumbbell.

192

Shown in Figures 8.15 and 8.16 are contour plots describing the distribution and ori-

entation of swimmers about a given central swimmer, respectively, withp = +1. Equiv-

alent plots for the casep = −1 are shown in Figures 8.17 and 8.18. In each of these

plots, we consider a given dumbbell of lengthQ = Qeq = 0.2 located at the bottom of

the panel. For Figures 8.15 and 8.17, we compute the spatial distribution of swimmers

at a point (r,z) relative to the center of mass of a given swimmer, where r corresponds to

the orthogonal direction and z to the axial direction. In Figures 8.16 and 8.18, we instead

consider the average orientation of swimmers at a point (r,z) relative to the center of mass

of a given swimmer. All distributions are normalized with the net swimmer concentra-

tion in the system. From the plots ofp = +1, we observe that at low concentrations,

there appears to be a depleted region immediately about each swimmer. As concen-

tration increases, we observe an increase in the number of swimmers locating about a

given swimmer, with a region of higher concentration about the swimmer apparent for

c/c∗ = 1.0. In thep = −1 case, nearly the opposite phenomena occurs, with an initially

concentrated region apparent at low system concentration disrupted as the system con-

centration increases. Atc/c∗ = 1.0, there is a slightly depleted region about the swimmer.

Furthermore, at low concentrations, swimmers tend to align with one another and move

in the same direction in thep = +1 case, as expected based on the flow perturbations

described in Figure 8.14(a). Forp = −1, however, the swimmers actually tend to align in

the opposite direction from the central swimmer, which is not at all expected from Figure

8.14(b).

To better interpret these phenomena, consider the contour plots of Figures 8.19(a) -

8.19(b), in which we plot the magnitude of the vertical component of the velocity pertur-

bation, as well as streamlines indicating the flow pattern, for a pair of swimmers oriented

193

(a) c/c∗ = 0.01 (b) c/c∗ = 0.1

(c) c/c∗ = 1.0

Figure 8.15:Concentration effects on the radial distribution of swimmers about a given swimmer withp = +1. The dumbbell is represented by white circles at bottom of plot and concentrations have beennormalized against system concentration, as described in the text.

194

(a) c/c∗ = 0.01 (b) c/c∗ = 0.1

(c) c/c∗ = 1.0

Figure 8.16:Concentration effects on the orientation of swimmers about a given swimmer withp = +1.The dumbbell is represented by white circles at bottom of plot and concentrations have been normalizedagainst system concentration, as described in the text.

195

(a) c/c∗ = 0.01 (b) c/c∗ = 0.1

(c) c/c∗ = 1.0

Figure 8.17:Concentration effects on the radial distribution of swimmers about a given swimmer withp = −1. The dumbbell is represented by white circles at bottom of plot and concentrations have beennormalized against system concentration, as described in the text.

196

(a) c/c∗ = 0.01 (b) c/c∗ = 0.1

(c) c/c∗ = 1.0

Figure 8.18:Concentration effects on the orientation of swimmers about a given swimmer withp = −1.The dumbbell is represented by white circles at bottom of plot and concentrations have been normalizedagainst system concentration, as described in the text.

197

at 90 degrees from one another. Using the streamlines, we observe that in thep = +1

case, the hydrodynamic interactions between molecules cause them to rotate and align

with one another while moving in the same direction while also drawing the molecules

towards one another. This behavior is displayed in Figure 8.20(a), where we plot sample

trajectories for a pair of swimmers moving towards one another withp = +1. Inter-

estingly, even for swimmers originally moving in diverging directions, if the separating

angle and distance are not too great, the swimmers may still rotate parallel to one an-

other and converge. This converging effect leads to an interesting numerical issue for

the p = +1 case stemming from the use of the Rotne-Prager-Yamakawa tensor as the

only form of intermolecular interaction. In the absence of excluded volume effects, there

is no mechanism to prevent swimmers from overlapping one another. As the RPY ten-

sor approaches the identity tensor for short separations, the two pushers shown in Figure

8.20(a) do not “feel” one another once they overlap and move as though in highly dilute

solution. One can also observe this effect by considering the nonmonotonic trend of the

velocities of the two swimmers as they approach one another; for separations greater than

2a, the swimmers increase in velocity with decreasing separation due to hydrodynamic

coupling. For separations less than2a, however, the use of the RPY tensor actually causes

the velocity to decrease and approach that of the infinitely dilute swimmer.

In the case ofp = −1, we observe that the hydrodynamic interactions between swim-

mers again rotate the bodies, but here the rotation is in the opposite direction to that of

the p = +1 case. As a result, two swimmers originally oriented towards one another

(such as in Figure 8.21(a)) rotate and approach one another in a colliding-type motion,

rather than in the slower, partnering approach of thep = +1 case. This again leads to

molecular overlap, and a second numerical issue regarding the use of the RPY tensor in

198

(a) p = +1

(b) p = −1

Figure 8.19: Contour plot of the vertical component of the velocity perturbation field owing to thepresence of force dipoles in a pair of dumbbells stemming from the application of (a) a pushing force(p = +1) or (b) a pulling force (p = −1). Dark regions indicate fluid moving in the positive verticaldirection, while dark regions indicate fluid moving in the negative vertical direction. Streamlines illustratethe net velocity field. White circles indicate the location of the dumbbells.

199

(a) Converging (b) Diverging

Figure 8.20:Sample trajectories for a pair of isolated swimmers in the absence of excluded volumeillustrating the effects of pair hydrodynamic interactions. Trajectories shown for the case ofp = +1. Darkcircles refer to beads acted on directly by the flagellar force.


Figure 8.21:Sample trajectories for a pair of isolated swimmers in the absence of excluded volumeillustrating the effects of pair hydrodynamic interactions. Trajectories shown for the case ofp = −1. Darkcircles refer to beads acted on directly by the flagellar force.

200

the absence of intermolecular repulsions. Once the two molecules have converged (right

hand side of panel 8.21(a)), the molecules become frozen in a configuration in which the

swimmers are oriented in opposite directions and the tail beads (beads without an applied

flagellar force) overlap. In this configuration, the swimmers are essentially frozen to one

another as the forces of each swimmer exactly balance one another. To see this, consider

the evolution equation for the tail bead of swimmer 2:

r 2,i (t+ ∆t) = r 2,i (t) + ∆t∑

µ

∑k


µ,k

)(8.18)

= r 2,i (t) + ∆t(

M (2,1),(1,1) · Fspr1,1

)+(M (2,1),(2,1) · Fspr

2,1

)+(

M (2,1),(1,2) · Fspr1,2

)+(M (2,1),(2,2) · Fspr

2,2

). (8.19)

Using the symmetry of the spring forces and the fact that the RPY form of the mobility

tensor approaches the identity tensor for overlapping particles, we have

r 2,i (t+ ∆t) = r 2,i (t) + ∆t(−M (2,1),(1,1) · Fspr

2,1

)+(Fspr

2,1

)+(Fspr

1,2

)+(−M (2,1),(1,1) · Fspr

1,2

)(8.20)

= r 2,i (t) + ∆t(−M (2,1),(1,1) · Fspr

2,1

)+(Fspr

2,1

)−(Fspr

2,1

)+(M (2,1),(1,1) · Fspr

2,1

)(8.21)

= r 2,i (t) . (8.22)

Thus, the swimmers become frozen in place barring outside interactions. This also ex-

plains the trend of swimmers aligning opposite one another for thep = −1 trend. At low

concentration, these type of opposing interactions are more likely to occur over long pe-

riods of time as the hydrodynamic perturbations that can break the symmetries involved

are weak. As concentration increases, however, the swimmers are less likely to remain

stagnant for any significant period of time, and so both the distributions of swimmer

201

locations and of swimmer orientations become uniform.

Next, consider the cases shown in Figures 8.22 - 8.23, in which we plot the trajecto-

ries of isolated swimmers initially in a “chasing” configuration, in which the swimmers

initially move in the same direction with one offset slightly in both the horizontal and ver-

tical directions from the other swimmer. Trajectories for the case ofp = +1 are shown

in Figure 8.22, while those forp = −1 are shown in Figure 8.23. In both cases, the

initial offset between the swimmers causes a significant curvature in the trajectories of

the two particles, and while the separation between the swimmers increases, their motion

remains coupled over long distances. The degree of curvature is greater in thep = −1

case than in thep = +1 case, and appears to increase with increasing initial separation.

The latter effect, however, is related to the phenomenon described above in which over-

lapping particles feel diminished hydrodynamic effects as compared to those separated

by a distance of2a. For larger separations, the degree of coupling between the swimmers

again decreases, and little curvature in the trajectories is observed.

Thus far, we have considered cases in which the swimmers initially move in concert

to some degree. In Figures 8.24-8.25, we consider the case of two swimmers initially

moving in opposing directions (here, in thex-direction) with varying degrees of offset

in the y-direction. For the case of zero offset, in the absence of excluded volume, the

swimmers come together into a stable conformation where the “tail” particles exactly

overlap, as was observed for the above case of swimmers converging at an angle. With a

small initial offset, however, we observe that the swimmer paths are significantly altered

as the hydrodynamic interactions between swimmers cause the swimmers to rotate from

their initial trajectories. The resulting trajectories cross one another and the swimmers

leave the interaction moving at some clockwise-measured angleθ ∈ (0, π/2) from their

202

(a) ∆y0 = 0.01 (b) ∆y0 = 0.1

Figure 8.22:Sample trajectories for a pair of isolated swimmers initially moving in a “chasing” config-uration in the absence of excluded volume. Trajectories shown for the case ofp = +1. Dark circles referto beads acted on directly by the flagellar force.

(a) ∆y0 = 0.01 (b) ∆y0 = 0.1

Figure 8.23:Sample trajectories for a pair of isolated swimmers initially moving in a “chasing” config-uration in the absence of excluded volume. Trajectories shown for the case ofp = −1. Dark circles referto beads acted on directly by the flagellar force.

203

initial path. The increased cooperative motion of thep = +1 case results in a larger

degree of rotation than does thep = −1 case for small offsets; for∆y0 = 0.2, the final

swimmer trajectories are nearly identical.

Returning to bulk measurements, we conclude by considering one final measure of

the influence of intermolecular hydrodynamic interactions - the decay of the swimmer

orientation autocorrelation function, as shown in Figure 8.26 for systems at various con-

centrations. At low concentration, it is apparent that the case ofp = −1 decays more

slowly than does the pushing case, while the reverse case holds for higher concentra-

tions. This is not surprising in light of the above discussion. We quantify the decay of the

swimmer orientation autocorrelation function by computing an autocorrelation time,τO,

which represents the time required for a given swimmer to move from one orientation to

a new, statistically independent orientation. This is accomplished by fitting a decaying

exponential function of the form

f(t) = A exp

(−t+B

τO

)(8.23)

to the temporal decay of the orientation,

C (t) =Qi (0) ·Qi (t)

|Qi (0)| |Qi (t)|(8.24)

and extracting the time constant from the functional form. In this work, we fit the ex-

ponential function to the region covering the final 20% of the orientation decay. The

results are summarized in Figure 8.27. As interchain hydrodynamic interactions are the

only mechanism through which chains may change orientation in this model, we observe

thatτO → ∞ asc → 0. Also, as concentration increases, the orientation autocorrelation

time decreases owing to the increased proximity of neighboring chains and the resulting

increase in the velocity perturbations caused by said chains. Little difference is observed

204

(a) ∆y0 = 0.001 (b) ∆y0 = 0.01

(c) ∆y0 = 0.1

Figure 8.24:Sample trajectories for a pair of isolated swimmers moving in opposite directions in theabsence of excluded volume illustrating the effects of pair hydrodynamic interactions. Trajectories shownfor the case ofp = +1. Dark circles refer to beads acted on directly by the flagellar force.

205

(a) ∆y0 = 0.001 (b) ∆y0 = 0.01

(c) ∆y0 = 0.1

Figure 8.25:Sample trajectories for a pair of isolated swimmers moving in opposite directions in theabsence of excluded volume illustrating the effects of pair hydrodynamic interactions. Trajectories shownfor the case ofp = −1. Dark circles refer to beads acted on directly by the flagellar force.

206

Figure 8.26:Decay of orientation autocorrelation function with time for systems at various concentra-tion. Both propulsion mechanisms are included for comparison.

by altering the number of chains per simulation cell, as shown in Figure 8.28. When both

pushing and pulling swimmers are included, we see behavior in which the orientation

autocorrelation decay tracks with that of the purely pushing case, as was the case for the

crossover times.

8.4.2 Excluded volume

To address the issue of swimmer overlap, we have also performed simulations of self-

propelled particles incorporating intermolecular repulsions through the use of an ex-

cluded volume potential. The excluded volume used here is a softened version of the

Weeks-Chandler-Andersen potential described in Chapter 3,

U exv,SPPνµ = εSPP

[(Qeq

rνµ

)3

−(Qeq

rνµ

)](8.25)

207

Figure 8.27:Swimmer orientation autocorrelation time as a function of concentration for systems withdifferent propulsion mechanisms.

with the resulting force given as,

Fexv,SPPνµ = εSPP

∑µ

[3

(Qeq

rνµ

)3

−(Qeq

rνµ

)]r νµ

|r νµ|2. (8.26)

whenrνµ <√

3Qeq, and are equal to zero otherwise. The excluded volume interaction

was tuned so that a pair of swimmers withp = +1 swim in parallel at a stable separation

of 2a, yieldingεSPP = 5.67× 10−3ζvsw.

We begin by comparing sample trajectories for a pair of swimming particles when ex-

cluded volume is included withp = +1 in Figures 8.29 - 8.33 and withp = −1 in Figures

8.30 - 8.34. It is apparent that swimmers moving via the pushing mechanism move and

couple with one another in similar fashion to those in the absence of excluded volume,

with one important distinction - the presence of the excluded volume interactions pre-

vents the overlap and eventual collapse of the swimmers atop one another. Interestingly,

the paired swimmers with excluded volume andp = +1 move with identical velocities

208

(a) p = +1

(b) p = −1

Figure 8.28:Swimmer orientation as a function of concentration for various system sizes in the absenceof excluded volume with a)p = +1 and b)p = −1.

209


Figure 8.29:Sample trajectories for a pair of isolated swimmers illustrating the combined effects of pairhydrodynamic interactions and excluded volume repulsions. Trajectories shown for the case ofp = +1.Dark circles refer to beads acted on directly by the flagellar force.

to those of the same motivation but without excluded volume. When the swimmers are

separated by distances slightly larger than2a, there is an observed increase in particle

velocities. However, when the pair is at equilibrium, the fluid perturbations in the flow

direction stemming from the excluded volume interactions that prevent particle overlap

exactly cancel the perturbations stemming from the paired force dipoles moving in paral-

lel. In the pulling case (p = −1), the initial behavior of converging swimmers is identical

to those in the absence of excluded volume. When the swimmers come into close contact,

however, they achieve a steady-state conformation in which the swimming trajectories are

frozen at an angle to one another owing to the excluded volume repulsions. As a result,

the swimmers continue to move laterally, frozen with respect to one another with the

swimmer heads abutting one another.

When particles are initially in the “chase” configuration, we observe that the presence

210


Figure 8.30:Sample trajectories for a pair of isolated swimmers illustrating the combined effects of pairhydrodynamic interactions and excluded volume repulsions. Trajectories shown for the case ofp = −1.Dark circles refer to beads acted on directly by the flagellar force.

of excluded volume repulsions causes an increase in the curvature of the swimmer tra-

jectories. This effect simply stems from the rapid repulsion of a pair of swimmers that

are initially close to one another. As the distance between swimmers increases to2a,

the hydrodynamic interactions become larger, leading to an increase in path curvature, as

described above.

Finally, we again consider the behavior of swimmers initially moving in opposite di-

rections with a small offset in their initial trajectories (Figures 8.33 - 8.34). As in the

case of the theta solvent, we observe a significant change in the swimmer paths follow-

ing a near-collision owing to the presence of intermolecular hydrodynamic interactions.

However, the paths resulting from such a near-collision are quite different than those pre-

viously observed when excluded volume was absent and the initial offset was small. For

large initial offsets, there are negligible intermolecular repulsions, and the resulting paths

are governed by the intermolecular hydrodynamic interactions. At small initial offsets,

211

(a) ∆y0 = 0.01 (b) ∆y0 = 0.1

Figure 8.31:Sample trajectories for a pair of isolated swimmers initially moving in a “chasing” config-uration in the presence of excluded volume. Trajectories shown for the case ofp = +1. Dark circles referto beads acted on directly by the flagellar force.

(a) ∆y0 = 0.01 (b) ∆y0 = 0.1

Figure 8.32:Sample trajectories for a pair of isolated swimmers initially moving in a “chasing” config-uration in the presence of excluded volume. Trajectories shown for the case ofp = −1. Dark circles referto beads acted on directly by the flagellar force.

212

however, the excluded volume repulsions during the near-collision strongly deflect the

two swimmers from one another, resulting in trajectories oriented at some clockwise-

measured angleθ ∈ (−π/2, 0) from the initial path.

With these pair-interactions in mind, we conclude this section by considering selected

bulk properties for the swimming particles with excluded volume. Shown in Figure 8.35

is the self-diffusion coefficient for both the swimmers and the massless fluid tracer par-

ticles both with and without excluded volume repulsions. Little difference is observed

between the two cases in either thep = +1 or p = −1 case for most concentrations.

The only notable difference is at our highest studied concentration, where the particles

are packed together at a high enough density so that many intermolecular excluded vol-

ume repulsions are present. This causes a small retardation in the diffusivity of both the

swimmers and tracers relative to the no-excluded volume case. A similar retardation of

the particle motion is observed in Figure 8.36, in which we plot the velocities of both

the swimmers and fluid tracers for each case. Finally, from Figure 8.37, we also ob-

serve that the presence of excluded volume causes an increase in the time for crossover

from ballistic to diffusive transport, with the increase larger in thep = −1 case than for

p = +1.

8.5 Conclusions

We have carried out numerical simulations of non-Brownian, self-propelled particles in

dilute bulk solution in order to gain insight into the role of intermolecular hydrodynamic

interactions in establishing cooperative motions between the swimming particles. We

have found that both concentration and the method of propulsion play a significant role

213

(a) ∆y0 = 0.001 (b) ∆y0 = 0.01

(c) ∆y0 = 0.1

Figure 8.33:Sample trajectories for a pair of isolated swimmers moving in opposite directions illustrat-ing the combined effects of pair hydrodynamic interactions and excluded volume repulsions. Trajectoriesshown for the case ofp = +1. Dark circles refer to beads acted on directly by the flagellar force.

214

(a) ∆y0 = 0.001 (b) ∆y0 = 0.01

(c) ∆y0 = 0.1

Figure 8.34:Sample trajectories for a pair of isolated swimmers moving in opposite directions illustrat-ing the combined effects of pair hydrodynamic interactions and excluded volume repulsions. Trajectoriesshown for the case ofp = −1. Dark circles refer to beads acted on directly by the flagellar force.

215

(a) p = +1

(b) p = −1

Figure 8.35:Diffusion coefficient as a function of concentration for both the swimmer and tracer parti-cles in different solvent types. Panel (a) corresponds to the case ofp = +1 and panel (b) top = −1.

216

(a) p = +1

(b) p = −1

Figure 8.36: Velocities of both the swimmer and tracer particles as a function of concentration fordifferent solvent types. Panel (a) corresponds to the case ofp = +1 and panel (b) top = −1.

217

Figure 8.37:Time scale,τC , over which the motion of the swimming particles in a good solvent changesfrom ballistic to diffusive in nature as extracted from the intersection of the asymptotic fits to the mean-square displacement vs. time.

in determining the behavior of the swimmers. When a pushing mechanism is used, the

resulting force dipole that develops in the dumbbell representation of the swimmer tends

to expel fluid away from the ends of the swimmer in an axial direction while drawing

fluid towards the body of the swimmer from the orthogonal directions. This leads to

the development of cooperative structures in which multiple swimmers positioned very

near one another align and move together through the fluid. Such a cooperative effect

is largely absent when the pulling mechanism is used, and in fact, pulling particles can

actually retard the motion of one another. As a result, the pushing mechanism leads to

larger particle velocities and shorter correlation times than are observed for the pulling

case. Interestingly however, the diffusivity is actually lower for the pulling case. On con-

sidering the swimmer diffusivity as a function ofv2τC , we observe that our data collapses

218

into a single curve independent of propulsion mechanism, illustrating the relationship be-

tween swimmer correlations and particle velocities. These results are shown to be a direct

consequence of the use of the Rotne-Prager-Yamakawa hydrodynamic interaction tensor

in the absence of excluded volume. When excluded volume repulsions are included,

particle overlap is prevented. However, there are few significant qualitative behavioral

changes aside from some retardation of the particle motions at high concentration owing

to intermolecular repulsions.

219

Chapter 9

ONGOING AND FUTURE

RESEARCH DIRECTIONS

In conclusion, we have presented a systematic analysis of three different complex fluid

flow problems using numerical simulation. However, many issues related to these prob-

lems remain to be addressed. Here, we describe some of the ongoing work from our

research group and discuss some avenues for potential future investigations.

At present, much of the work in our research group focuses on the behavior of di-

lute polymer solutions in microchannels, working in close collaboration with the group

of David Schwartz1. This important area of research has many practical applications,

including directed drug delivery and the sequencing of DNA molecules. To date, we

have considered only simulations of individual molecules in microchannels. Projects

are currently underway within our group aimed at bridging the gap between these single-

molecule simulations and the work presented in this dissertation, in which intermolecular

1Department of Chemistry and Laboratory of Genomics, University of Wisconsin - Madison

220

effects were considered in bulk solution. One of the difficulties in this type of analy-

sis is the high computational expense associated with correctly calculating the hydrody-

namic interactions with the walls of the microchannel. As a result, Juan Hernandez-Ortiz

is developing new, more efficient computational methods for the simulation of macro-

molecules in a highly confined domain (Hernandez-Ortiz et al., 2005).

Another current project in our group is focused on investigating the depletion layer

that forms near a solid surface when a shear flow is applied parallel to the wall. One

colleague, Hongbo Ma, has developed theoretical expressions to describe the migration of

polymer molecules away from a solid surface using a simple point-dipole representation

of the polymer molecule. Simulations of freely-jointed bead-spring chains show excellent

agreement with the theoretical expressions, both near a single wall and in microchannels.

Work is currently underway to extend this analysis both to the study of free surface flows

as well as to the study of semidilute polymer solutions.

Hongbo has also performed simulations to study the flow of DNA into a small pore

from a large reservoir. This problem is of great significance as a potential avenue for

DNA sequencing as the small pore restricts the DNA molecule to a highly stretched and

aligned conformation. Simulations are currently being used to investigate the influence

of flow into a pore on the chain conformation as a function of distance and orientation

from the pore entrance. In addition, this work focuses on the manner in which the chains

enter the pore by predicting which portion of the chain is most likely to enter the pore

first and the probability of entering a pore for a given flow strength.

The area of polyelectrolyte dynamics continues to be an active field of study as well.

221

While our work here rationalizes the viscometric behavior of salt-free, dilute polyelec-

trolyte solutions, there is still a rich parameter space to be explored. Many different vari-

ables can influence the behavior of polyelectrolytes; some effects that may considered in

the future include:

• the influence of salt concentration

• the use of multivalent ions

• altering the charge fraction of the polyelectrolyte

• the influence of macroions

• imposition of an electric field

The last topic is currently being pursued along one avenue in our research group by

Aslin Izmitli, who is using Lattice-Boltzmann simulations to study the transport of a

polyelectrolyte chain through a nanopore in a square channel. While closely related

to Hongbo’s project described above, Aslin’s current focus is the actual passage of the

polyelectrolyte through the nanopore and how this is related both to the structure of the

portion of the chain that has already passed through the pore and to the conformation of

the chain that has yet to enter the pore.

Finally, Juan has developed computational methods extending the study of concentra-

tion effects on self-propelled particles to the use of microchannel domains. The presence

of containing walls induces a number of interesting hydrodynamic effects observed in the

behavior of the swimming particles. For example, depending on the type of motivating

force used (pushing or pulling), the swimming particles will tend to migrate towards or

away from the channel surfaces as a result of the dipole moment induced by the flagellar

222

force. Preliminary results indicate that, at a given concentration, there is also a much

higher degree of coordination for particles in a microchannel than in bulk solution. At

present, this work has not considered concentrations much beyond the dilute regime;

however, with the new, more efficient method for calculating hydrodynamic interactions,

it is expected that this work will be able to extend to the semidilute regime as well.

223

Appendix A

BEAD-ROD SIMULATIONS

As mentioned in Chapters 7 and 8, an alternative to the bead-spring model in which

the chain bonds are made rigid is sometimes appropriate in the numerical simulation of

polymer solutions. In this appendix, we describe a simulation algorithm for carrying out

Brownian dynamics simulations using the Kramers bead-rod model. The scheme used

here is based on the midpoint scheme of Liu (1989) for free-draining systems and of

Ottinger (1994) for systems with hydrodynamic interactions, in which the authors im-

plemented constraint forces in each rigid bond via the use of Lagrange multiplers. We

illustrate here how to adapt this algorithm for the inclusion of both nonbonded poten-

tials and hydrodynamic interactions, and discuss the feasibility of implementing such an

algorithm from the standpoint of computational expense. For simplicity, we focus this

discussion on a single chain consisting ofNB beads andNS = NB − 1 bonds, but the

discussion is easily extended to the treatment of multiple chains.

The initial stages of the derivation are similar to that for the bead-spring chains, as

detailed in Chapter 2. Once again, we begin by considering the force balance about each

224

bead,

F(h)ν + F(φ)

ν + F(m)ν + F(b)

ν + F(c)ν = 0 (A.1)

whereF(h)ν is the hydrodynamic force acting on beadν. F(φ)

ν represents the combined

forces acting on beadν from all nonbonded potentials (e.g. excluded volume, electrostat-

ics). F(m)ν is a metric force (Ottinger, 1994; Hinch, 1994) that represents the influence of

the constraints on inertial and frictional effects and has the form

F(m)ν =

1

2kBT

∂

∂r ν

(ln(

detGjk

))(A.2)

where

Gjk =∑

ν

1

Mν

∂gj

∂r ν

∂gk

∂r ν

(A.3)

for beads of massMν and constraintsgj. The constraints are dealt with in more detail

below. The metric force is exactly the negative of the corrective force necessary to make

a truly rigid system behave like a system consisting of infinitely stiff springs, and so

its omission actually yields the simulation of a bead-spring system with an infinite spring

constant. The random Brownian force,F(b)ν , on beadν owing to the thermal motion of the

solvent molecules is taken as in Chapter 2. Finally, the constraint force,F(c)ν , represents

the force exerted on beadν so as to satisfy the constraint condition of a rigid bond length.

From Stokes Law, we express the hydrodynamic force about a given beadν as

F(h)ν = −ζ (rν − (vν + v′ν)) . (A.4)

Here,ζ is the bead friction coefficient,vν contains the velocity components of theνth

bead,vν is the local fluid velocity, andv′ν is the perturbation to the velocity field sur-

rounding this bead stemming from hydrodynamic interactions with other particles. The

225

local fluid velocity of theνth bead may be written as a combination of the system velocity,

v0 and an imposed flow field,[κ · r ν ],

vν = v0 + [κ · r ν ] . (A.5)

Now, substituting in the expressions of Equations A.4 and A.5 into Equation A.1, we

have

−ζ(dr ν

dt− (vν + v′ν)

)+ F(φ)

ν + F(m)ν + F(b)

ν + F(c)ν = 0, (A.6)

which can be recast to give the basic evolution equation as

dr ν

dt= v0 + [κ · r ν ] + v′ν +

1

ζ

(F(φ)

ν + F(m)ν + F(b)

ν + F(c)

ν

). (A.7)

Next, we focus on the perturbation velocity, which depends linearly on the hydrody-

namic forces acting on all of the other beads in solution as

v′(r) = −∑

µ

Ω (r − rµ) · F(h)µ (A.8)

whereΩ is the hydrodynamic interaction tensor. Taking the perturbation at each bead

position and rearranging, we have,

v′ν = −∑µ 6=ν

Ω (r ν − rµ) ·(Fhyd

µ

)(A.9)

and, on replacing the hydrodynamic force using the force balance of Equation A.1, we

obtain

v′ν =∑µ 6=ν

Ω (r ν − rµ) ·(F(φ)

µ + F(m)µ + F(b)

µ + F(c)µ

). (A.10)

We insert these expressions into Equation A.7 to obtain

dr ν

dt= v0 + [κ · r ν ] +

∑µ 6=ν

Ω (r ν − rµ) ·(F(φ)

µ + F(b)µ + F(c)

µ

)+

1

ζ

(F(φ)

ν + F(m)ν + F(b)

ν + F(c)

ν

). (A.11)

226

Finally, by combining terms and taking the diffusion tensor as1kBT

D = Ω + 1ζδ, we get

dr ν

dt= v0 + [κ · r ν ] +

1

kBT

∑µ

Dν,µ ·(F(φ)

µ + F(m)ν + F(b)

µ + F(c)µ

). (A.12)

To this point, we have considered the evolution of our systems in the absence of con-

straints. Here, however, we must solve Equation 8.14 subject to the rigid rod constraint.

That is,

gi = (r i+1 − r i)2 − a2 = 0 (A.13)

wherea is the length of the constrained bond. Using the method of Lagrange multipliers,

we express the total constraint force acting on beadν as

F(c)ν = −

∑i

γi∂gi

∂r ν

gi (A.14)

= −2∑

i

γiBiν (r i+1 − r i)

where

Biν = δi+1,ν − δi,ν (A.15)

and theγi are theNS undetermined Lagrange multipliers associated with theNS con-

straints. We accomplish this by means of a two-step Brownian dynamics algorithm, in

which we first calculate the displacements for each bead in the absence of the constraints,

and then use an iterative method to compute the Lagrange multipliers that restrict our

bonds to rigid lengths.

In the unconstrained step, we use a straightforward Euler scheme:

rν (t+ ∆t) = r ν (t) + [κ · r ν (t)] +∆t

kBT

∑µ

Dνµ ·(F(φ)

µ + F(m)ν

)+

√2kBT

∑µ

Bν,µ ·∆Wν (t) . (A.16)

227

Note that we have dropped thev0 term as the uniform fluid velocity does not affect the

microstructure of the chain. Following this, we use the definition of Equation A.13 to

correct the unconstrained positions,

r ν (t+ ∆t) = rν (t+ ∆t)− 2∆t

kBT

∑i

γi

[∑µ

BiµDν,µ · ui(t)

]c

(A.17)

= rν (t+ ∆t)− 2∆t

kBT

∑i

γi [(Dν,i+1 − Dν,i) · ui(t)]c

whereaui = r i+1 − r i and [ ]c denotes that the corresponding term is evaluated at the

positions(1− c)r ν (t) + crν (t+ ∆t). Typically,c is taken equal to1/2 (Ottinger, 1994).

The Lagrange multipliers,γi, are determined such that the constraint equations

gi = [ui (t+ ∆t)]2 − 12 = 0 (A.18)

are satisfied within a specified tolerance at each time step. Subtracting the expression

from Equation A.18 for the(ν)th from that of the(ν + 1)th bead, we obtain

uj (t+ ∆t) = uj (t+ ∆t)

− 2∆t

kBT

∑i

γi [(Dj+1,i+1 − Dj+1,i − Dj,i+1 + Dj,i) · ui]c , (A.19)

which can be inserted into the constraint equation to give

(uj (t+ ∆t))2 − 1

− 4∆t

kBTuj (t+ ∆t)

∑i

γi [(Dj+1,i+1 − Dj+1,i − Dj,i+1 + Dj,i) · ui]c

+

(2∆t

kBT

∑i

γi [(Dj+1,i+1 − Dj+1,i − Dj,i+1 + Dj,i) · ui]c

)2

= 0. (A.20)

Finally, this equation is rearranged to solve forγi via an iterative procedure where thenth

228

approximation toγi, γ(n)i , is given by

4∆t

kBTuj (t+ ∆t)

∑i

γ(n)i (Dj+1,i+1 − Dj+1,i − Dj,i+1 + Dj,i) · ui =(

(uj (t+ ∆t))2 − 1)

+ (A.21)(2∆t

kBT

∑i

γ(n−1)i [(Dj+1,i+1 − Dj+1,i − Dj,i+1 + Dj,i) · ui]c

)2

.

The procedure is then iterated until convergence is achieved for the set ofγi, and the

resulting values are inserted into Equation A.18 to calculate the positions of the beads at

the end of the time step.

While this method provides a self-consistent means of evolving a system of bead-rod

chains, it suffers a few obvious drawbacks regarding computational speed. To determine

the set of Lagrange multipliers, we are required to solve a linear system of equations

with each iteration at each time step. Contrasting this with the simple Euler solution for

a bead-spring chain of Section 5.1, this adds significant computational expense. More

distressing, however, is the treatment of the hydrodynamic interactions in this algorithm.

For the simulation of bead-spring chains, we have demonstrated that it is possible to

compute the product∑

µ DνµFµ directly while avoiding the expensive explicit calculation

of D. For the simulation of Kramers chains, however, we do require the calculation

of the explicit diffusion tensor in order to correctly isolate the terms of Equation A.22.

Furthermore, withc = 1/2, the calculation ofD must be updated with each iteration of

the constraints algorithm at each time step. Such calculations are currently prohibitely

expensive for any but very small systems.

One caveat may be made in the simulation of Kramers chains in the absence of hy-

drodynamic interactions. In this case, Equations A.16, A.18, and A.22 may be simplified

229

to give

rν (t+ ∆t) = r ν (t) + [κ · r ν ] ∆t+∆t

ζ

(F(φ)

µ + F(m)ν

)+√

2kBT∆Wν (t) (A.22)

r ν (t+ ∆t) = rν (t+ ∆t)− 2∆t

kBT

∑i

γiBiν (r i+1(t)− r i(t)) (A.23)

and

4∆t

ζuj (t+ ∆t)

∑i

γ(n)i Ajiui =

((uj (t+ ∆t))2 − 1

)+

(2∆t

ζ

∑i

γ(n−1)i Ajiui

)2

, (A.24)

whereAji is the Rouse matrix,

Aji =∑

ν

BjνBiν =

2 i = j

−1 i = j ± 1

0 otherwise.

(A.25)

In this case, the system of equations to be solved with each iteration through the proce-

dure for determining the Lagrange multipliers is tridiagonal, and so may be solved very

rapidly. Coupled with the computational savings stemming from not requiring the de-

termination of the diffusion tensor, this algorithm may be sufficiently fast to be used for

simulations in which the hydrodynamic interactions have negligible effect on the dynam-

ics of the system, such as when strong electrostatic interactions are present. We have

not yet considered this case for the simulation of polyelectrolytes as the main focus of

our work did not require consideration of the fine-grained structure of the chain. Also,

in our simulations of self-propelled particles, we have focused solely on systems where

hydrodynamic effects have a strong influence on the dynamic behavior, and so have used

230

a stiff spring in lieu of a truly rigid rod. Should some of the difficulties regarding the cal-

culation of the diffusion tensor be overcome, however, the Kramers chain model would

be the preferred choice for the simulation of self-propelled non-Brownian particles.

231

Appendix B

STRESS TENSOR FOR

MULTICOMPONENT SYSTEMS

In Chapter 2, we discussed the derivation of the stress tensor for a dilute polymer solution

at nonzero concentration. In this Appendix, we derive the form of the stress tensor used

in this work (Equation 2.22) beginning with the Kramers-Kirkwood expression (Equation

2.17).

The Kramers-Kirkwood expression for the non-solvent contribution to the stress ten-

sor for a single chain at infinite dilution was given by Equation 2.17, reproduced here:

τp = −nNB∑ν=1

RνF(h)ν . (B.1)

This can be rewritten as

τp = −nNB∑ν=1

(r ν − r c) F(h)ν (B.2)

232

wherer c = 1NB

∑NB

µ=1 rµ is the center of mass of the chain. Separating the summations

τp = n

(NB∑ν=1

r νF(h)ν −

NB∑ν=1

r cF(h)ν

)(B.3)

= n

(NB∑ν=1

r νF(h)ν − r c

NB∑ν=1

F(h)ν

), (B.4)

and as all interparticle forces are conservative, the second term on the right hand side of

Equation B.4 is equal to zero. Hence,

τp = n

NB∑ν=1

r νF(h)ν (B.5)

= −1

2n

(NB∑ν=1

r νF(h)ν +

NB∑µ=1

rµF(h)µ

). (B.6)

Using the relationshipFν =∑NB

µ=1 Fνµ, we obtain

τp = −1

2n

(NB∑ν=1

r ν

NB∑µ=1

F(h)νµ +

NB∑µ=1

rµ

NB∑ν=1

F(h)µν

)(B.7)

= −1

2n

(NB∑ν=1

r ν

NB∑µ=1

F(h)νµ −

NB∑µ=1

rµ

NB∑ν=1

F(h)νµ

)(B.8)

= −1

2n

(NB∑ν=1

NB∑µ=1

r νF(h)νµ −

NB∑ν=1

NB∑µ=1

rµF(h)νµ

)(B.9)

= −1

2n

NB∑ν=1

NB∑µ=1

(r ν − rµ) F(h)νµ (B.10)

= −1

2n

NB∑ν=1

NB∑µ=1

r νµF(h)νµ . (B.11)

From this expression, we can calculate the total non-solvent contribution to the stress

tensor by considering all pair interactions between particles on the chain. The general-

ization to a system at nonzero concentration is obvious, and by replacingNB by N , we

obtain the total stress contribution from the microstructure:

τp = − 1

2V

N∑ν=1

N∑µ=1

r νµF(h)νµ . (B.12)

233

This is the form of the stress tensor given in Equation 2.22 and used throughout this work.

234

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SIMULATION OF DILUTE POLYMER AND POLYELECTROLYTE SOLUTIONS