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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.
i
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
ii
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
iii
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
iv
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
v
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
vi
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
vii
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
viii
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
7.15 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 = 2.25, with
the plots showing (a) equilibrium, (b)Pe = 0.1, (c) Pe = 1.0, and (d)Pe = 10.0. . . . 154
7.16 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 white 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 = 10.0, with
the plots showing (a) equilibrium, (b)Pe = 0.1, (c) Pe = 1.0, and (d)Pe = 10.0. . . . 155
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
7.21 Depictions of the ion cloud surrounding an individual chain in dilute solution for various
values ofc/c∗. 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 (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
xiii
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
8.23 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
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
8.31 Sample trajectories for a pair of isolated swimmers initially moving in a “chasing”
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
8.32 Sample trajectories for a pair of isolated swimmers initially moving in a “chasing”
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
acc (o→ n) = min(1, e−β(U(rn)−U(ro))
). (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).
• The move is accepted with probabilityacc (o→ n) = min(1, e−β(U(rn)−U(ro))
).
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
acc (o→ n) = min(1, e−β(U(rn)−U(ro))
). (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).
• The move is accepted with probabilityacc (o→ n) = min(1, e−β(U(rn)−U(ro))
).
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
[(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)
+ 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
[(Di+1,µ(t)− Di,µ(t)) · (Fspr
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
(b) Hydrodynamically Interacting
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
(b) Hydrodynamically Interacting
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
(b) Hydrodynamically Interacting
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
(b) Hydrodynamically Interacting
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
(b) Hydrodynamically Interacting
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
(b) Hydrodynamically Interacting
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
(b) Hydrodynamically Interacting
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
(b) Hydrodynamically Interacting
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
(b) Hydrodynamically Interacting
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
(b) Hydrodynamically Interacting
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
(b) Hydrodynamically Interacting
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
(a) Equilibrium (b) Pe = 0.1
(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
(a) Equilibrium (b) Pe = 0.1
(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.
168
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
M (2,i),(µ,k) ·(Fspr
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
M (1,i),(µ,k) ·(Fspr
µ,k + Fexvµ,k
)(8.16)
r 2,i (t+ ∆t) = r 2,i (t) + ∆t∑
µ
∑k
M (2,i),(µ,k) ·(Fspr
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.
(a) Converging (b) Diverging
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
M (2,i),(µ,k) ·(Fspr
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
(a) Converging (b) Diverging
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
(a) Converging (b) Diverging
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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