BROWNIAN DYNAMICS SIMULATION OF
DNA IN COMPLEX GEOMETRIES
by
Yu Zhang
A dissertation submitted in partial
fulfillment of the requirements for the degree of
Doctor of Philosophy
(Chemical Engineering)
at the
UNIVERSITY OF WISCONSIN – MADISON
2011
i
To my wife, Xi
For your love, your support,
and a wonderful child.
To my son, Alexander
For your laughters, which brighten my life.
To my parents, Liangzhu Zhang and Zhenglan Yu
For your unconditional love.
ii
Acknowledgments
Thanks to my advisors, Michael Graham and Juan de Pablo for their support, pa-
tience, and guidance.
Thanks to former colleague Hongbo Ma, Juan Hernandez-Ortiz, and Patrick Un-
derhill for many useful discussions on fluid mechanics and polymer physics.
Thanks to current colleague Amit Kumar and Pieter Janssen for their helps and
many discussions.
Thanks also to other former and current colleagues of the Graham group – Wei
Li, Li Xi, Samartha Anekal, Matthias Rink, Mauricio Lopez, Aslin Izmitli, Pratik
Pranay, Kushal Sinha, Rafael Henriquez Rivera, and Friedemann Hahn – for all your
helps and many discussions.
Research projects presented in this dissertation are financially supported by the
University of Wisconsin-Madison Nanoscale Science and Engineering Center funded
by National Science Foundation.
iii
Abstract
This dissertation is concerned with the dynamics of a long DNA molecule in complex
geometries, driven by either electrostatic field or flow field. This is accomplished pri-
marily through the use of Brownian dynamics simulation, which captures the essential
physics at mesoscopic length scale, and allows us to simulate events happening on
long time scale, such as DNA pore translocation and cyclic dynamics of a tethered
DNA molecule in shear flow. Our work is novel in that, accurate and efficient algo-
rithms have been designed and developed for both electric field-driven and flow-driven
systems in complex geometries. We have focused on three distinct problems, which
we describe below.
In the electric field-driven case, we propose a novel class of electric field-actuated
soft mechanical control element for microfluidics. This type of element employs the
idea that under confinement a single polymer molecule is essentially a nanoscale
porous media. The chain could block the passageway of relatively large analytes such
as cells. At the same time, polyelectrolyte molecules, such as DNA, could deform
and squeeze through narrow pores when a sufficiently strong electric field is applied.
Brownian dynamics (BD)/Finite Element Method (FEM) simulation efficiently ex-
plore the design space, and results demonstrate that the On/Off switching could be
iv
achieved within a proper parameter space.
To study the effects of a solid impenetrable wall on the dynamics of a nearby
DNA molecule, we examine the cyclic dynamics of a single DNA molecule tethered
to a hard wall in shear flow, using Brownian dynamics simulation. We focus on the
dynamics of the free end (last bead) of the tethered chain and we examined the cross-
correlation function and power spectral density of the chain extensions in the flow
and gradient directions as a function of chain length N and dimensionless shear rate
Wi. Extensive simulation results suggest a classical fluctuation-dissipation stochastic
process and question the existence of periodicity of the cyclic dynamics, as previously
claimed. We support our numerical findings with a simple analytical calculation for
a harmonic dimer in shear flow.
In the case of flow-driven DNA molecule in micro/nano-fluidics, one big challenge
is that an efficient algorithm is required to calculate fluctuating hydrodynamic in-
teractions (HI) in complex geometries. We have developed an accelerated immersed
boundary method that allows fast calculation of Brownian motion of polymer chains
and other particles in complex geometries with HI. With this new method, the first
detailed analysis of a recent set of interesting nanofluidic experiments involving DNA
dynamics in a complex flow geometry is performed. This analysis explains the ob-
served dynamics over a wide range of parameter values (flow rate, molecular wieght)
and illustrates the important quantitative effect of the hydrodynamic interactions on
the behavior of the system.
v
Contents
Acknowledgments ii
Abstract iii
List of Figures viii
List of Tables x
1 Introduction 1
2 Problem Statement 4
2.1 Bead-spring DNA Model . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Kinetic theory of single polymer molecule . . . . . . . . . . . . . . . . 6
2.3 The Langevin equation and Brownian dynamics simulation . . . . . . 9
3 Intramolecular interactions and external forces 12
3.1 Spring force law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Excluded volume interactions . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Hydrodynamic interactions . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4 Electrophoretic force . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
vi
3.5 Polymer-wall steric interaction . . . . . . . . . . . . . . . . . . . . . . 19
4 Bistability and field-driven dynamics of confined tethered DNA 21
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Polymer model and simulation approach . . . . . . . . . . . . . . . . 28
4.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3.1 Pore-crossing geometry . . . . . . . . . . . . . . . . . . . . . . 31
4.3.2 Pore-entry geometry . . . . . . . . . . . . . . . . . . . . . . . 37
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5 Tethered DNA dynamics in shear flow 43
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.2 Discussion of Methods for Hydrodynamics of Polymer Solutions . . . 49
5.2.1 Implicit Solvent: Brownian Dynamics . . . . . . . . . . . . . . 50
5.2.2 Explicit Solvent: Continuum Methods . . . . . . . . . . . . . 52
5.2.3 Explicit Solvent: Particle Methods . . . . . . . . . . . . . . . 55
5.2.4 Coupled Methods . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.3 Simulation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3.1 Brownian Dynamics . . . . . . . . . . . . . . . . . . . . . . . 57
5.3.2 Lattice-Boltzmann . . . . . . . . . . . . . . . . . . . . . . . . 60
5.3.3 Stochastic Event-Driven Molecular Dynamics . . . . . . . . . 62
5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6 Flow-driven DNA dynamics in complex geometries 80
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
vii
6.2 Methods for hydrodynamics of confined polymer solutions . . . . . . 86
6.3 Polymer model and simulation method . . . . . . . . . . . . . . . . . 91
6.3.1 Model and governing equations . . . . . . . . . . . . . . . . . 91
6.3.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . 92
6.3.3 Mobility tensor and Fixman’s midpoint algorithm . . . . . . . 94
6.3.4 Chebyshev approximation . . . . . . . . . . . . . . . . . . . . 96
6.3.5 Fast Stokes solver with complex boundary conditions . . . . . 97
6.4 DNA flowing across an array of nanopits . . . . . . . . . . . . . . . . 114
6.4.1 Dynamics at low Peclet number . . . . . . . . . . . . . . . . . 117
6.4.2 Dynamics at high Peclet number . . . . . . . . . . . . . . . . 123
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7 Conclusion and future work 126
A Lattice random walk model of a tethered polymer 129
B BD/FEM algorithm for simulating DNA electrophoresis 137
C Dimer in shear flow 141
D Fixman’s midpoint algorithm 146
viii
List of Figures
2.1 Schematic of a bead-spring chain in Brownian dynamics simulation. . 6
4.1 Schematic representations of soft nanomechanical bistable elements. . 26
4.2 Soft nanomechanical elements. . . . . . . . . . . . . . . . . . . . . . . 28
4.3 Top view of the simulation domains. . . . . . . . . . . . . . . . . . . 29
4.4 Time trajectories for a 3D pore-crossing system . . . . . . . . . . . . 32
4.5 Snapshots of a pore-crossing event. . . . . . . . . . . . . . . . . . . . 33
4.6 Simulation results for the quasi-2D and 3D pore crossing geometries. 35
4.7 Time trajectory and probability density for the pore entry geometry . 38
4.8 Simulation results for the quasi-2D pore-entry geometry . . . . . . . . 39
4.9 Phase transition of the “competitive bistability” system. . . . . . . . 40
5.1 Cyclic dynamics of a tethered DNA molecule . . . . . . . . . . . . . . 46
5.2 Probability distribution of the end bead of the tethered DNA molecule 68
5.3 Relaxation time of the tethered DNA . . . . . . . . . . . . . . . . . . 70
5.4 Spectral analysis of the end-bead position time series . . . . . . . . . 73
5.5 Cross correlation functions for different beads . . . . . . . . . . . . . 74
5.6 Comparison of the cross-correlation function among different methods 76
6.1 DNA flowing through nanopits . . . . . . . . . . . . . . . . . . . . . . 85
ix
6.2 Schematic of the immersed boundary method . . . . . . . . . . . . . 99
6.3 Properties of the mobility tensor for the boundary particles . . . . . . 103
6.4 Screening function for the GGEM algorithm . . . . . . . . . . . . . . 105
6.5 Comparison between delta and regularized delta point forces . . . . . 106
6.6 Comparison between Hasimoto’s solution and GGEM . . . . . . . . . 110
6.7 Error analysis of GGEM . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.8 Laminar channel flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.9 Uniform flow around a sphere . . . . . . . . . . . . . . . . . . . . . . 115
6.10 Schematic of the immersed boundary representation of the device . . 116
6.11 Streamlines and contour plot of the streamwise velocity in a pit . . . 117
6.12 Snapshots of a hopping event . . . . . . . . . . . . . . . . . . . . . . 118
6.13 DNA dynamics at low Peclet number . . . . . . . . . . . . . . . . . . 120
6.14 Mean resident time τ v.s. Peclet number Pe. . . . . . . . . . . . . . . 121
6.15 Mean residence time τ v.s. chain length N and Peclet number Pe . . 122
6.16 DNA dynamics at high Peclet number . . . . . . . . . . . . . . . . . 124
A.1 A tethered polymer as lattice random walk. . . . . . . . . . . . . . . 133
A.2 The repulsive wall-polymer potential . . . . . . . . . . . . . . . . . . 135
B.1 Schematic of the nearest neighbor search algorithm. . . . . . . . . . . 140
C.1 Spectral analysis of a dimer in shear flow . . . . . . . . . . . . . . . . 145
x
List of Tables
2.1 Properties of λ-DNA. . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.1 Spring models and the corresponding force laws. . . . . . . . . . . . 14
1
Chapter 1
Introduction
Transport of polymer solutions in constricted spaces [1] is a long-standing research
topic with many applications including polymer enhanced-oil-recovery, size exclusion
chromatography [153], gel-electrophoresis [151], and recently single DNA molecule
analysis using micro- and nano-fluidic devices [140, 125]. In many fluidic devices de-
signed for separation, manipulation, and sequencing of DNA, the critical dimension
of the constriction approaches the size of the polymer molecule or smaller, and un-
derstanding the effects of spatial restrictions is essential to the rational design and
applications of those devices. In particular, capturing the interplays between polymer
dynamics, spatial confinement, and electrostatic or flow field is essential to a proper
description of these devices. It is therefore of great interest to develop validated fast
modeling and simulation tools to understand the fundamental physics of polymer
solutions in complex geometries.
There are two major types of driving forces to transport DNA molecules through
microfabricated devices: electrokinetic forces (mainly electrophoretic force) and fluid
2
flow drag force. Many devices with well-defined microstructures have been proposed
for DNA manipulation, especially for separation and stretching purposes using elec-
trophoresis [45]. By contrast, much less attention has been paid on pressure (flow)
driven DNA dynamics in microfabricated devices [141, 36, 143]. There are significant
differences between the electrophoresis case and the pressure driven case. Consider
for example DNA through a slit geometry. In electrophoresis, in a uniform field, the
velocity of each DNA segment is the same everywhere in the channel; by contrast,
the unperturbed velocity profile is parabolic across the channel in the pressure driven
case at low Reynolds number, and this velocity gradient can give rise to interesting
transport phenomena, such as Taylor dispersion [141]. Furthermore, hydrodynamic
interactions (HI) between objects such as polymer segments in an unconfined domain
are long-ranged, leading to strong many-body effects, for example, Zimm scaling of the
self-diffusion coefficient of polymer. In confined geometries, the long-ranged nature
of hydrodynamic interactions changes substantially, leading to significant changes in
polymer dynamics. [1]. In a slit geometry, for example, the velocity field due to a
point force perpendicular to the walls decays exponentially. For a force parallel to
the walls, the velocity field has a parabolic form in the wall-normal direction and
decays as 1/r2. This is still a slow decay, but the symmetry of the flow leads to
cancellations upon averaging that result in screening [9, 144, 14]. Both experiments
and simulations of the diffusion of long flexible DNA molecules in slits [31, 78, 75]
are consistent with screening of hydrodynamic interactions on the scale of slit height.
Nevertheless, on smaller scales, HI are not screened and lead to changes in segment
mobilities and boundary effects such as cross-stream migration, which leads to de-
pletion layers much larger than the equilibrium chain size [77, 90, 30, 68, 29, 81]. A
3
significant portion of this thesis is concerned with developing general methodologies
for Brownian dynamic simulations of DNA in complex geometries, driven by either
electrostatic field (Chapter 4 ) or flow field (Chapter 5 and Chapter 6).
This dissertation is organized as follows: In Chapter 2, we present the governing
equations for the dynamics of a single DNA molecule. Intramolecular interactions
and external potentials are introduced in Chapter 3. In Chapter 4 we present the
algorithm for DNA electrophoresis in complex geometries as applied to study the
properties of a class of soft nanomechanical control elements for microfluidics. In
Chapter 5, we discuss the Brownian dynamics simulation results of the cyclic motion
of a DNA molecule which is tethered onto a solid wall and experiences a shear flow
field. In Chapter 6, we develop and validate a method for the efficient calculation of
fluctuating hydrodynamics in complex geometries. In that chapter we also present
simulation results for the dynamics of a flow-driven DNA through a nanofluidic slit
with an embedded array of nanopits. Conclusions and future work are given in
Chapter 7.
The different sections in Chapters 4 through 6 correspond to different publica-
tions and manuscript. Those sections are therefore fairly self-contained, and some
repetition should be expected.
4
Chapter 2
Problem Statement
In this dissertation, we are concerned with the dynamics of a single DNA molecule
which is confined in a fluid domain with complex boundary conditions. As we are
interested in long-time ( > 1 second) dynamics of the DNA molecule, a bead-spring
DNA model and Brownian Dynamics simulation are used in this work. A general
discussion of the wide range of techniques for modeling the hydrodynamics of polymer
chains in solution is given in Chapter 5, and a brief overview of various methodologies
for modeling hydrodynamics of polymer solutions in complex geometries is given in
Chapter 6. In the remainder of this chapter, we describe the DNA model and the
governing equations.
2.1 Bead-spring DNA Model
Double-stranded DNA is a semiflexible polymer whose physical and chemical proper-
ties have been studied extensively. The physical properties of λ-DNA, the bead-spring
model of which is used in this work, are summarized in Table 2.1. The choice of DNA
5
Table 2.1: Properties of λ-DNA.
Property Symbol Value References
Number of base pares Nbp 48502 [137]
Contour length Lc 22 µm [137]
Persistence length lp 53 nm [138]
Kuhn length bk 2lp
Radius of gyration Rg 730 nm [137]
Diffusion coefficient D 0.47 µm2/s [137]
Electrophoretic mobility µ0 4.2× 10−4 cm2 /Vs [142, 102]1
model depends upon the molecule properties (chemical, mechanical, etc.) one wants
to model and the level of molecular detail one needs to retain [2]. For long time
Brownian dynamics simulation, one of the most commonly used coarse-grained mod-
els for linear DNA molecules is the bead-spring model. The molecule is modeled as
a series of Nb beads connected by Ns = Nb − 1 springs, as shown in Figure 2.1. The
total number of degree of freedom for a freely moving bead-spring chain in therefore
3Nb. Bead-spring model is the most coarse grained version of a DNA model. The
beads act as sources of fluid drag friction and the springs, which obey certain type
of spring force law, represent a collection of persistence lengths. Carefully calibrated
bead-spring chain model greatly enlarges the accessible time and length scales in nu-
merical simulation. In this work, we use the model developed by Jendrejack et al.[76],
which will be discussed in Chapter 3.
1Measured in TBE buffer (0.01M TBE, pH = 8.3, at 23 Celsius).
6
O
ri
1 2
3 i
i+1
Nb-1
Nb
Figure 2.1: Schematic of a bead-spring chain in Brownian dynamics simulation.
There are two equivalent ways to describe the dynamics of a DNA or, more gen-
erally, a polymer molecule. One is based on the Fokker-Planck equation (“Diffusion
equation”) for the time evolution of the Nb beads phase space distribution function
Ψ(R, t). The other approach uses Langevin equation to describe the motion of Nb
beads. By solving the diffusion equation, one can obtain Ψ(R, t) directly. On the
other hand, with the set of coupled Langevin equations, one can directly calculate
trajectories of the beads and then obtain Ψ(R, t) by averaging. We will discuss first
the diffusion equation, and then the Langevin equation and Brownian dynamics sim-
ulation.
2.2 Kinetic theory of single polymer molecule
Kinetic theory of polymer molecule describes the dynamics of a distribution function,
i.e., evolution of the distribution function, under the action of forces. Using the
bead-spring model of polymer, the positions of the beads represent the configuration
of the polymer, and the configuration distribution function Ψ is defined so that ΨdR
7
is the probability to find the position vector of the first bead within dr1 around r1 ,
second bead at dr2 around r2, and so on. The general equation for the configuration
distribution function Ψ of a bead-spring polymer is called the Fokker-Planck equation
or “Diffusion equation”. It is the derivation of this equation that we devote the
remainder of this section.
When the bead inertial relaxation times are short compared to the timescale of
interest, it is often possible to ignore inertia, that is, we can write a force balance
about bead i. The relevant forces on the bead i include: hydrodynamic force fhi ,
Brownian force f bi , and other non-hydrodynamic and non-Brownian forces fni which
is the sum of spring force, excluded volume force, and steric force between the bead
and wall. Then the force balance is written as
fhi + f bi + fni = 0, (2.1)
in which
fhi = −ζ · [ui − (u∞i + up
i )], (2.2)
f bi = −kBT∂
∂rilnΨ, (2.3)
fni = − ∂
∂riφ. (2.4)
Equation 2.2 is the Stokes drag law: the hydrodynamic force on bead i is assumed
to be linear in the slip velocity between the bead velocity ui and the solvent velocity
at the bead center (u∞i + up
i ) with the coefficient ζ . Here u∞i
is the unperturbed
velocity when the polymer is absent, and upi is the perturbed velocity due resulting
from the motion of other beads in the system. For now, we simply state that the
8
perturbed velocity at ri is linearly depend on the non-hydrodynamic, non-Brownian
forces acting on all beads in the system fni as upi =
∑Nb
j=1Mij · fnj . Here Mij is a
tensor describing the hydrodynamic interaction between bead i and j.
Equation 2.3 gives the Brownian force. This equation has been derived by Bird et
al. (Ref [21]) for the case of a structureless mass point in which the chain has been
equilibrated in momentum space.
Equation 2.4 represents the force from all the non-hydrodynamic and non-Brownian
forces which stem from intra- and intermolecular potentials, and external potentials
such as electric potential and bead-wall steric repulsive potential. We will return to
a discussion of these potentials and forces in Chapter 3.
Now substitute the expressions for various forces into the force balance equation
and we obtain
− ζi · [ui − (u∞i + up
i )]− kBT∂
∂rilnΨ + fni = 0. (2.5)
Rearranging above expression, we obtain
ui = (u∞i + up
i ) +1
ζi(−kBT
∂
∂rilnΨ + fni ) (2.6)
= u∞i +
∑
j
(1
ζjδijI+Mij) · (−kBT
∂
∂rjlnΨ + fnj ). (2.7)
Here, I is an identity matrix. From the first to the second line in above expression,
we use the expression for upi . According to the kinetic theory, the configuration
9
distribution function satisfies the continuity equation
∂Ψ
∂t= −
∑
i
(∂
∂ri· uiΨ). (2.8)
Defining the diffusion tensor D as
Dij =kBT
ζjδijI+ kBTMij , (2.9)
and combining 2.7 and 2.8, we obtain the diffusion equation.
∂Ψ
∂t= −
∑
i
(∂
∂ri· (u∞
i +1
kBT
∑
j
[Dij · (−kBT∂
∂rjlnΨ + fnj )]Ψ)) (2.10)
2.3 The Langevin equation and Brownian dynam-
ics simulation
In this section, we discuss the alternative descriptions of the dynamics of a polymer
molecule in terms of random variable dW and its defining stochastic differential
equation. This description is used in the Brownian dynamics simulation of a polymer
molecule.
The diffusion equation (Eq. 2.10) can be recast in the form of a Langevin equation
(a stochastic differential equation)[105]
dR = (U∞ +M · F+ kBT∂
∂R·M)dt +
√2B · dW, (2.11)
B ·BT = kBTM. (2.12)
10
Here R is the vector containing all bead positions ri, and the vector containing
total non-Brownian, non-hydrodynamic forces acting on the beads is denoted F. The
tensor B gives the magnitude of the Brownian displacement of the polymer beads,
and is coupled to M by the Fluctuation-Dissipation theorem (Eq. 2.12). The vector
dW is a random vector composed of independent and identically distributed random
variables according to a real-valued Gaussian distribution with mean zero and variance
dt.
In Brownian dynamics simulation, applying a stochastic integration scheme to
above equation generates the trajectories of a polymer molecule under forces. For
example, using a simple forward Euler integration scheme we obtain
R(t+∆t) = R(t) + (U∞ +M · F+ kBT∂
∂R·M)∆t +
√2B ·∆W. (2.13)
In electrophoresis, DNA is a “free-draining” (FD) polymer. Therefore, we can ignore
the hydrodynamic interactions (HI) between different chain segments, and the mobil-
ity tensor M is reduced to the product of 1ζand an identity matrix. Hence, Eq. 2.13
is reduced to
R(t+∆t) = R(t) +1
ζF∆t +
√
2kBT
ζ·∆W. (2.14)
Above equation is relatively easy to solve compared with that in the flow-driven case.
An efficient algorithm to find the electric field at the bead position in a field with
steep gradient is essential. This issue is addressed in Appendix B.
In the flow-driven case, to capture the full fluctuating hydrodynamic in complex
geometries, we first need to turn Eq. 2.13 into a derivative-free form, as the expression
for M is lacking when the molecule is confined in complex geometries. Then, the
11
stochastic differential equation can be solved based on a fast solver for Stokes flow in
complex geometries. This will be discussed in Chapter 6.
12
Chapter 3
Intramolecular interactions and
external forces
The dynamics of the DNA molecule obeys Newton’s second law, and therefore, it is
determined by the total force exerted on the chain. As discussed in last Chapter,
the total force on a bead is composed of a viscous drag force fh, a Brownian force f b
due to the collisions of the solvent molecules, and all other non-hydrodynamic non-
Brownian forces fn, which includes spring forces, excluded volume forces, and any
external forces such as electrophoretic force and DNA-wall steric interaction. In this
chapter, we discuss those forces and potentials.
13
3.1 Spring force law
Each spring models a fraction of the full DNA molecule which is long enough such
that its force-extension behavior obeys the Marko-Siggia spring law [26, 94],
f s =kBT
2bk[(1− |re|
Nkbk)−2 − 1 +
4|re|Nkbk
]re|re|
, (3.1)
where f s is the average force needed to get an end-to-end vector of re for each DNA
segment. The Kuhn length bk is twice the value of the persistence length, and when
a DNA molecule is divided into Nk = Lc/bk segments, each Kuhn segments can
be thought of as if they are freely jointed with each other (no bending, rotational,
or torsional potentials). This spring law was derived based on a worm-like chain
(WLC) model in polymer physics which is commonly used to describe the behavior of
semi-flexible polymers. In this model, the molecule is treated as a flexible smooth rod.
The rod’s local direction (tangential vector) decorrelates at distance s along the curve
according to exp(−s/lp), where the decay length, lp, is called the persistence length
of the chain. The stiffer the chain, the larger the persistence length. For DNA, the
persistence length is approximately 50 nm. The WLC model fits DNA force-extension
data very well up to 10 pN, or < 95% stretching ratio [26]. As long as the number of
Kuhn lengths contained in each spring is larger than 10bk, the Marko-Siggia spring
law can be used [149]. In this work, each spring represents 20bk.
There are other spring force laws proposed for simulation of polymer molecules,
which are summarized in Table 3.1, together with the WLC model.
14
Table 3.1: Spring models and the corresponding force laws.
Model Force expression
Hookean f s = Hre =3kBTNkb
2k
re
FENE f s = Hre
1−(|re|/Nkbk)2
Inverse Langevin model f s = kBTbk
L−1( |re|Nkbk
) re
|re|
L(x) = cothx− 1x
WLC f s = kBT2bk
[(1− |re|Nkbk
)−2 − 1 + 4|re|Nkbk
] re
|re|
3.2 Excluded volume interactions
In polymer physics, excluded volume interaction is a type of long-ranged1 interaction
referring to the idea that in any real polymer molecule, two monomers cannot occupy
the same space. This effect plays a far more important role in polymer solution
than it does in solution of small molecules [35]. We often idealize the chain to allow
overlap of monomers, and call it an ideal chain. By contrast, a regular chain with
excluded volume interactions is called a real chain. Ideal chain does not exist in
reality, but it is used extensively because it allows rigorous mathematical treatment
of many questions. More importantly, the real chain behaves like an ideal chain in
some situations, such as when the molecule is in concentrated solutions, melts, and
in theta condition. In Appendix A, we derive an effective repulsive potential between
a polymer molecule and a hard wall using the ideal chain model.
In Brownian dynamics simulation for λ-DNA, we use an exponential form of the
1Note that excluded volume interaction is long-ranged along the chain backbone. The potentialis actually short-ranged in space (Eq. 3.2).
15
repulsive excluded volume potential between bead i and bead j, when we consider a
real chain. It is obtained by considering the energy penalty due to the overlap of two
Gaussian coil [41],
Uevij =
1
2νkBTN
2k,s(
3
4πS2s
)3/2 exp(−3|ri − rj|2
4S2s
), (3.2)
where ν is the excluded volume parameter, and S2s = Nk,sb
2k/6 is the mean square
radius of gyration of an ideal chain containing Nk,s Kuhn segments of length bk.
The resulting expression describing the force acting on bead i due to the the
presence of bead j within some cut-off distance is then
f evij = νkBTN2k,sπ(
3
4πS2s
)5/2 exp(−3|ri − rj|2
4S2s
)(ri − rj), (3.3)
Although this potential is not self-consistent (any deformation of the coil caused by
the overlap has been ignored), it does provide the correct scaling relationships [76].
3.3 Hydrodynamic interactions
Beads immersed in a fluid generate flows as they move due to various forces, and
similarly they move in response to fluid motion through the Stokes drag. In this
work, we treat bead i of the bead-spring chain as a sphere of hydrodynamic radius
a. Then the relationship between the bead velocity ui and the drag force it exerts on
the fluid is given by the Stokes law fi = ζ(ui − u(ri)), where ζ is the bead friction
coefficient ζ = 6πηa, u(ri) is the fluid velocity at the bead position, and η is the fluid
viscosity. (Note that the finite size of the bead only arises in the friction coefficient.)
16
Through these hydrodynamic interactions (HI), beads interact with each other and
with the walls of the confining geometry. In kinetic theory, the velocity perturbation
due to the polymer molecule, up, is generally taken to be due to a chain of point forces
acting on the fluid, and obtained by solving the incompressible Stokes flow problem,
−∇p(x) + η∇2up(x) = −∑
fnδ(x− xn), (3.4)
∇ · up(x) = 0, (3.5)
up(r) = 0, r ∈ ∂Ωb, (3.6)
where p is the pressure, up is the fluid velocity, η is the fluid viscosity, fn is the force
exerted on the fluid at point xn, δ(x) is the three-dimensional delta function, and ∂Ωb
is the boundary of the fluid domain. In the actual implementation, regularized point
forces are used, as further discussed flow. The velocity perturbation at the position
of bead i due to the bead j is represented in a Greens function form as
upi =
Nb∑
j=1
Mij · fnj (3.7)
where Mij is called hydrodynamic interaction tensor. To illustrate the components
of the mobility tensor, we consider a two-bead chain (a dumbbell) in an unbounded
domain, neglecting Brownian effects for the moment. Here the force balance shows
that the fluid velocities u(ri) experienced by bead i are given by
u1
u2
=
u∞1
u∞2
+
1ζI G∞(r1 − r2)
G∞(r2 − r1)1ζI
·
f1
f2
, (3.8)
17
where G∞(r) = 18πηr
(I + rr/r2) is the Oseen-Burgers tensor, the free-space Green’s
function for Stokes’ equations, r = |r|, and I is the identity matrix. The quantity
G∞(r1 − r2) · f2 gives the velocity at position r1 generated by the force f2 exerted by
the particle at r2 on the fluid. We can write above expression succinctly as,
U = U∞ +M · F. (3.9)
We note that ∂∂r
·M = 0 for G∞. In simulation of coarse-grained polymer chain, the
singularity of the Oseen tensor must be regularized. The most common choice is the
RPY tensor [120, 157].
In a confined geometry, a wall correctionGW (r1, r2)has to be added to the mobility
tensor M, and it becomes
M =
1ζI+GW (r1, r2) G∞(r1 − r2) +GW (r1, r2)
G∞(r2 − r1) +GW (r1, r2)1ζI+GW (r1, r2)
. (3.10)
In general geometries, an analytical expression for GW (r1, r2) is not available, but
has to be calculated numerically. We note that it is not M that is needed, but rather,
the product M · f which is the velocity generated by the total non-Brownian, non-
hydrodynamic forces acting on the beads. As a result, using an accelerated Immersed
Boundary Method, we can calculateM·f for polymer in complex geometries efficiently.
We will return to a discussion on it in Chapter 6.
In Green’s function-based representation of the hydrodynamic interactions, we
assume that the beads are coupled instantaneously. In other words, the characteristic
time for the hydrodynamic interactions to propagate through the system should be
18
much smaller than the time scale of other signals. These conditions are equivalent to
that the Reynolds number is small Re ≪ 1 and the March number is small Ma ≪ 1,
which are satisfied explicitly through the use of the Stokes equation.
3.4 Electrophoretic force
Electrophoretic force is commonly used to transport and manipulate DNA molecules.
In this section, we discuss DNA electrophoresis and the effective charge assigned to
each bead calculated using the electrophoretic mobility of a long DNA molecule.
The velocity of a DNA molecule (> 100 bp) in free electrophoresis has little depen-
dence on length [142]. When an electric field is applied across an electrolyte solution
containing DNA molecules, the counter-ion cloud around the DNA experiences a force
in the opposite direction to that of the DNA. This force generates ab electroosmotic
flow which balances the flow perturbation generated by the DNA molecule. Hence,
we can say that hydrodynamic interactions are screened and DNA is “free-draining”
during free electrophoresis. Every segment feels the same friction from the fluid, so
DNA molecules of different sizes will have the same mobility.
We consider a DNA molecule subject to an external electric force E. The free
electrophoretic mobility of DNA µ0 is defined as the ratio between the electrophoretic
velocity v and electric field strength E. For a “free-draining” DNA bead-spring chain,
the total friction coefficient of the chain is the sum of all the friction coefficients ζ of
the beads. Likewise, the total electric charge of the chain is Q = Nbqb where qb is
the effective charge per bead. Balancing the electric force Fe = QE = NbqbE and the
19
friction force Fh = Nbζv, results in the following center of mass velocity
v =NbqbE
Nbζ=
qbE
ζ. (3.11)
Hence, the electrophoretic mobility of DNA in free-solution (µ0) is given by
µ0 ≡v
E=
qbζ. (3.12)
This expression indicates that the electrophoretic mobility of DNA in free solution is
independent of the length or the molecular weight of the chain. For a “free-draining”
chain with diffusivity D, the friction coefficient is ζ = kBT/NbD. Therefore, given
the electrophoretic mobility, the effective charge per bead is
qb = ζµ0 =kBT
NbDµ0. (3.13)
Using the diffusion coefficient of λ-DNA during free solution electrophoresis D = 0.55
µm2/s, and the electrophoresis mobility obtained under the same condition µ0 =
4.2×10−4cm2/Vs [102], qb is determined to be 178 electrons per bead, for the Nb = 11
bead-spring model used in this work.
3.5 Polymer-wall steric interaction
A potential needs to be devised to prevent penetration of beads through solid walls.
However, if the potential is steep near a wall, such as that of a Lennard-Jones po-
tential, a very small time step should be used. On the other hand, in the case of
20
soft potential, a bead can sometimes penetrate the wall. Jendrejack et al. devised a
potential barrier for bead-wall interactions which is harder than the Gaussian soft po-
tential used for bead-bead interaction and softer than typical Lennard-Jones potential
[77],
Uwalli =
Awallb−1k δ−2
wall(hi − δwall)3, hi < δwall
0, hi > δwall.
(3.14)
Here hi represents the perpendicular distance of bead i from the wall, δwall is the
cut-off distance.
The choice of the repulsive potential is arbitrary as long as it can prevent the
bead from penetrating the wall with reasonably small time step. In Appendix A, we
derive an effective repulsive potential between a polymer molecule and a hard wall
using the ideal chain model.
21
Chapter 4
Bistability and field-driven
dynamics of confined tethered
DNA
DNA is “free-draining” during electrophoresis as discussed in Chapter 3. As a con-
sequence, it cannot be separated by free electrophoresis. A sieving medium, such
as polymer gel or microfabricated structure, that induces a size-dependent mobil-
ity is necessary for DNA separations. Many microfluidic devices with well-defined
structures have been proposed for DNA manipulation, especially for separation and
stretching purposes using electrophoresis [45]. These devices are typically composed of
periodic obstacle arrays. When an electrostatic field is introduced, a nonuniform field
with steep gradient is generated in the device. Thus, understanding DNA dynamics in
nonuniform electric fields is an important issue for the design of high performance mi-
22
crofluidic devices. The motivation of the work presented in this chapter1 is to build a
general framework to analyze DNA dynamics in nonhomogeneous electrostatic fields
in complex microfluidic geometries. A Brownian dynamics/Finite element method
algorithm is presented in Appendix B. In the remainder of this chapter, we focus on
its application to study the properties a novel class of soft nanomechanical control
elements we proposed for microfluidic devices.
4.1 Introduction
A fundamental unit in many engineering systems is an element with two distinct
states – on/off, open/closed, left/right etc. Of particular interest and utility in many
cases is the property of bistability: each state is robust in the sense that a finite
perturbation must be temporarily introduced to switch the system from one state to
the other. Many classical macroscopic mechanical systems display bistability, and ef-
forts have recently been made on many fronts to design, construct and analyze “soft”
bistable systems on smaller and smaller scales, down to the molecular level. Steen
and coworkers [70, 152] have constructed systems of pairs of droplets connected by
a flow channel. This system exhibits bistability via the existence of situations where
two configurations with minimal surface energy can arise, with one drop large and the
other small, or vice versa. Switching between states can be achieved, for example, via
pressure fluctuations or electroosmotic pumping of fluid between droplets. Groisman
et al. [63] and Arratia et al. [10] have demonstrated microfluidic systems in which
elastic liquids (solutions of long flexible polymer molecules) in complex geometries
1This chapter is based on the publication: Y. Zhang, J. J. de Pablo, and M. D. Graham, SoftMatter, 5, 3694.
23
can exhibit bistability via symmetry-breaking flow instabilities. At the level of single
flexible molecules in solution, bistability arises through coil-stretch hysteresis in suffi-
ciently long DNA molecules in solution during extensional flow [126]. (We will use the
term “bistable” loosely here, to denote situations where the energy barrier between
two metastable states is greater that about kBT , where kb is Boltzmann’s constant
and T is absolute temperature.) A similar phenomenon is predicted for a chain teth-
ered on a no-slip wall, in the case where there is a stagnation point flow and the chain
tether point is at the stagnation point [18]. Banavar et al. [15] used simulations of
model chain molecules to predict bistability in the conformations of the molecules.
In particular, they predict for a certain case the coexistence of a single helix and a
dual helix in which the chain folds in half and self-interacts to form a double helix.
Our interest here is in bistability of polymer chain conformations associated with the
combination of tethering and confinement within a micro- or nanoscale device.
Partially confined configurations of polymer chains, where different parts of a
chain experience different degrees of confinement, give rise to interesting phenomena of
entropic origin. The entropy difference between a portion of polymer chain in confined
space and in open space gives rise to an entropically induced recoil force, which tries to
pull the whole chain into the open space. Using a nanochannel/microchannel device,
Mannion et al. estimated this entropically induced force of double-stranded (ds-)
DNA to be about 102 − 103fN [92]. According to the force versus extension curve of
ds-DNA, a force of this magnitude will generate a fractional extension of the molecule
of about 0.5 [94]. In a related calculation, Bickel et al. determined the force required
to tether an end of an ideal chain onto a hard wall, which is also a force due to the
decrease of available number of configurations [20]. This tethering force converges to
24
a constant value kT/lk as chain length diverges. For DNA with a Kuhn length of
lk ≈ 100nm, this force is about 40 fN at room temperature. Partial confinement
has been exploited to manipulate a DNA molecule using a nanoslit device (50 - 100
nm across), on which one wall had square nanopits (100 nm deep and 100 - 500 nm
wide)[116]. Because the confinement is weaker in the regions of the slit where there
is a pit, DNA segments are preferentially found there. This entropic well tends to pin
sections of the chain in these regions. When multiple pits are present (and not too
far apart), stretched sections of DNA will span the more confined regions between
pairs of pits. Binder and coworkers have used Monte Carlo simulations and theory
to study a tethered polymer in good solvent compressed by a circular disk centered a
distance H above the tether point [95, 71]. For moderate compression, the polymer is
“imprisoned” underneath the disk adopting a quasi-2D random walk configuration by
forming blobs with diameter equivalent to the disk height H . When a certain height
Himp, part of the chain “escapes” this imprisonment and forms a “stem-and-flower”
configuration where the blobs underneath the disk are stretched into an extended
configuration by the entropic force exerted by the escaped parts. Using Monte Carlo
simulations, they found that there is a regime of heights Hesc ≤ H ≤ Himp where the
chain exhibits “two-phase coexistence” (bistability) of the escaped and the imprisoned
configurations. Both simulation results and analytical theory indicate that the escape
transition is a first-order phase transition. This provides an example of bistability
created by a combination of tethering and partial confinement, specifically by the
competition between the entropy gain of the escaped segments and the entropy loss
and energy penalty of extending the portion of the chain that remains trapped under
the disk.
25
Partially confined configurations also lead to interesting dynamics [103, 99]. Trapped
within a two-dimensional array of spherical cavities interconnected by circular holes,
short linear DNA strongly localize in cavities and only sporadically “jump” through
holes [103]. To jump out of a cavity, small DNA segments penetrate into the neighbor-
ing cavity first and form a partial confined configuration. The fluctuation of the prob-
ing segments beyond a threshold leads to an abrupt jump of the entire molecule. This
is in accordance with previous theoretical studies of entropic barriers by Muthukumar
[100, 99].
In the present work, we examine, using Brownian dynamics simulations of a coarse-
grained model for long flexible DNA molecules, the possibility of generating bistable
behavior using long flexible linear polymer molecules end-tethered in a confined ge-
ometry. Fig. 4.1(a) shows a very simple example of what we will call an “entropically
bistable” system. (The system studied by Binder and coworkers [95] is also in this
class.) Here a polymer molecule is tethered within a small pore of width 2d connect-
ing two open regions. For a sufficiently long chain (Rg ≫ d, where Rg is the radius
of gyration of the chain in the absence of confinement), one clearly expects that the
most entropically favorable situations will be those where the entire chain is in one
open region or the other. Thermal fluctuations or transient application of an electric
field (if the chain is a polyelectrolyte) or flow field might drive the chain from one
region to the other over an entropic barrier. One realization of this type of system
would be the fully three-dimensional situation where the pore is a hole connecting
two half-spaces, as shown in Fig. 4.1(d). A more experimentally tractable version,
fabricated using nanolithography techniques, is the “quasi-two-dimensional” (Q2D)
situation shown in Fig. 4.1(c), where top and bottom bounding walls exist. Both
26
z y
xh=2d
r
z
y x
(c) (d)
l
2d
(a)
thermalnoise
transientf ie ld
z
y
x
(b)
mediumfield
z
y
x 2d
nofield
highfield
or
Figure 4.1: Schematic representations of soft nanomechanical bistable elements. (a) Pore-crossing geometry:an entropically bistable system. (b) Pore-entry geometry: a competitively bistable system. (c) Quasi-2D pore-crossing geometry. (d) 3D pore-crossing geometry.
27
situations will be studied here.
We also consider the possibility of systems where the origin of bistability is some-
what more subtle. Consider the system shown in Fig. 4.1(b), comprising a polyelec-
trolyte end-tethered at the mouth of a long pore. In the absence of an electric field,
the most likely conformations of the chain lie outside the pore (left). If a sufficiently
large field is applied pulling the chain into the pore, entropy will be overcome by
electrostatic energy and the chain will reside entirely in the pore (right). We hypoth-
esize that at intermediate field strengths, a nontrivial competition between entropy
and electrostatic energy will allow the possibility of two stable states (center), one
where the chain is mostly outside the pore, in a high entropy state, and the other
where the chain is mostly inside the pore, in a low entropy state. This system will
be called “competitively bistable”. We will study the quasi-two-dimensional version
of this geometry.
Many fundamental issues arise when considering these systems. What is the
nature of the transition states in these systems? How high are the free energy barriers
between states? How do these depend on the length of the chain and, in the Q2D
case, the degree of lateral confinement? What are the dynamics of transitions between
the states? How will those dynamics differ in the electrostatically driven and flow
driven cases? To what extent do the dynamics of these systems follow Kramers-type
kinetics? This initial report will only begin to touch on some of these issues.
Potential applications of these systems can also be considered. A chain tethered in
a micro/nanochannel under sufficiently confining conditions would behave somewhat
like a porous medium, resisting fluid flow and blocking the transport of sufficiently
large solutes. This property in combination with entropic bistability might be ex-
28
(a) (b)
TransientFieldTransientField
TransientFieldTransientField
z
yx
z
yx
Figure 4.2: Possible applications of tethered polymer molecules as control elementsin micro/nanofluidic devices. (a) Switch. (b) Gate.
ploited as shown in Fig. 4.2. In the left image, a polyelectrolyte chain is configured so
that it blocks transport of the round solute particles in the left channel, while allowing
transport of the square ones, or vice versa. Switching would occur through transient
application of an electric field in the z direction. In the right image, an entropically
bistable geometry serves as a gate that can be opened or closed by a transient electric
field.
4.2 Polymer model and simulation approach
We use Brownian dynamics simulation of a model of long ds-DNA molecules to study
the statics and dynamics of a tethered chains in confined systems. Following previous
work [76, 74], a double-stranded DNA molecule is described by a bead-spring chain
model composed of Nb beads of hydrodynamic radius a = 77 nm connected by Ns =
Nb − 1 entropic springs. Each bead represents a DNA segment of 4850 base pairs,
i.e., Nb = 11 corresponds to a stained λ-DNA, which has a contour length of 22 µm
29
2d=14 a
l=2 a
(a) (b)
z
y
x
ze
ze
2d=14 a
Pore crossing Pore entry
Figure 4.3: Top view of the simulation domains. Black areas are impenetrable walls;the free end of the chain is a filled circle. In quasi-2D cases, the tethered bead-springchains are bounded in x direction by walls at x = −d and x = +d. (a) Pore-crossinggeometry. (b) Pore-entry geometry.
and radius of gyration of 730 nm. An effective charge of Q = 178 e is assigned to
each bead, which is calculated based on the free solution mobility of DNA [142, 102].
The length unit in this work is a = 77 nm. The springs connecting the beads obey a
worm-like chain force law [94]
Fsij =
kBT
2bk
[
(1− |rj − ri|Nk,sbk
)−2 − 1 +4|rj − ri|Nk,sbk
]
]
rj − ri|rj − ri|
. (4.1)
Here, bk is the Kuhn length for DNA and Nk,s = 20 is the number of Kuhn length
per spring. The physical confinement is taken into account through an empirical
bead-wall repulsive potential of the form
Uwalli =
Awallb−1k δ−2
wall(hi − δwall)3, hi < δwall
0 hi > δwall.
(4.2)
30
Here hi represents the perpendicular distance of bead i from the wall, δwall is the
cut-off distance. In this work, we choose Awall = 50kBT/3 and δwall = bkN1/2k,s /2 =
3.01a = 0.24 µm. In this work, we ignore hydrodynamic interactions (HI) between
chain segments and assume that the chain is ideal. The force balance on the beads
leads to a stochastic differential equation[105]:
dR =1
ζFdt+
√
2kBT
ζdW, (4.3)
where R is the vector containing bead positions ri, ζ is the friction coefficient of
the bead and F is the vector of non-hydrodynamic and non-Brownian forces. The
components of dW are obtained from a real-valued Gaussian distribution with mean
zero and variance dt. We use a standard (stochastic) Euler scheme[105] for time-
integration.
In the pore crossing system (Fig. 4.3 (a)), we study the dependence of the free
energy barrier on chain length and confinement in both a quasi-2D and 3D geometries.
The wall thickness is 2 a and the pore size is 14 a. For the quasi-2D case (Fig. 4.1 (c)),
the chain is confined in the x direction, and the distance between the two impenetrable
planes are h = 14 a. For the 3D case (Fig. 4.1 (d)), the chain is unbounded in both
x and y directions.
In the pore entry system (Fig. 4.3 (b)), we consider only the quasi-2D case. The
chain is tethered to the center of the entrance to a square channel which runs along
the +z axis. An electrostatic field is applied along the −z direction. In the present
model electrostatic interactions between beads are neglected, consistent with behavior
in a high ionic strength solvent, where electrostatic interactions are screened. Elec-
31
troosmosis of counterions screens hydrodynamic interactions, as does confinement,
justifying their neglect in the present simplified model. The electric field strength E
at z = −∞ is E = E∞ez. The relative field strength is denoted as ER = αE∞/E0,
where E0 = kBT/aQ = 18.74 V/cm is the field strength unit and α is a constant
to make ER order O(1). When ER = 1, the corresponding E∞ = 1.23 V/m, and
the electric force fE = E∞Q = 3.51 × 10−2 fN. Even inside the channel where the
electric field gradient is around ten times larger than E∞, the electric force is very
small compared with thermal agitation kBT/a = 53.4 fN. The electric field is obtained
by solving the governing Laplace equation with no flux boundary conditions on the
channel walls and Dirichlet boundary conditions at the left and the right edges of the
domain. The commercial PDE solver COMSOL, based on the finite element method
(FEM), is used. When d = 7a, the simulation boxes are y × z = 200a × 100a and
14a× 50a for the outer chamber and the channel, respectively. As the field gradient
becomes uniform and constant going along the channel, we can use a short simula-
tion box for the channel. We use the Lagrange-quadratic square elements (1a × 1a)
provided by COMSOL. As the polymer model is coarse-grained to a length scale of
a, this resolution of the field is adequate.
4.3 Results and discussion
4.3.1 Pore-crossing geometry
We consider first the pore-crossing geometry of Fig. 4.3(a). Only the equilibrium
behavior in this system will be studied. Fig. 4.4(a) is a typical time series plot of the
z component of the free end ze of a chain with Nb = 15 in the 3D case. This plot
32
0 500 1000 1500 2000-4
-2
0
2
4
t (s)
z e (m
)
tw
(a)
1070 1080 1090 1100-4
-2
0
2
4
t (s)
z e (m
)
(b)
0 1000 2000 3000 4000 50000.0
0.1
0.2
0.3
(t w)
tw(s)
tw / exp(-tw/
(c)
Figure 4.4: Simulation results for an ideal chain tethered to the center of the porein a 3D pore-crossing geometry (Fig. 4.3(a)). (a) Typical time series plot of the zcomponent of the free end of a tethered chain with Nb = 15 and d = 7a. (b) Blowupof a segment of the time series plotted in (a). (c) Waiting time distribution and bestfit to an exponential distribution.
33
(a) (b) (c)
(d) (e) (f)
z( m)m
y(
)m
m
−3 −2 −1 0 1 2 3−2
−1
0
1
2
−3 −2 −1 0 1 2 3−2
−1
0
1
2
−3 −2 −1 0 1 2 3−2
−1
0
1
2
−3 −2 −1 0 1 2 3−2
−1
0
1
2
−3 −2 −1 0 1 2 3−2
−1
0
1
2
−3 −2 −1 0 1 2 3−2
−1
0
1
2
Figure 4.5: Snapshots of a crossing event in the quasi-2D pore-crossing geometry.Time interval between frames is 0.5s.
clearly shows that ze fluctuates around two metastable states. Fig. 4.4(b) is a blowup
of the time interval 1070-1100 s. The chain end stays on one side of the wall for a
time interval tw, which we will call the waiting time, and then crosses to the other
side. Fig. 4.5 shows the dynamics of a typical crossing event. The free end of the
tethered chain first diffuses to the pore and then through to the other side of the wall.
We notice that this step of end bead crossing does not necessarily cause a crossing
of the whole chain. A successful crossing event occurs when the number of beads on
one side of the wall changes from Nb − 1 to 0 or vice versa. The time points tic when
a crossing completes are recorded. We define the waiting time as the time between
two adjacent crossing events tiw = ti+1c − tic. As seen in Fig. 4.4(a), the crossing time
is very small compared to the waiting time, which is well-fit to an exponential with
chain-length-dependent decay rate 1/τ , as shown in Fig. 4.4(c). This result indicates
34
that, to a good approximation, the crossing events are independent of one another –
crossing is a Poisson process.
To gain further information about the crossing process and the transition state,
we consider the joint behavior of the z-component of the chain end, ze and the z-
component of the center bead, zm (where m = Nb/2 and (Nb + 1)/2 when Nb is
even and odd respectively). The joint probability density for these two variables is
denoted ρ(ze, zm). Fig. 4.6(a) shows − ln ρ(ze, zm) for a chain with Nb = 13 chain in
a 3D pore-crossing geometry. It clearly demonstrates that there are two metastable
states (maxima in ρ(ze, zm)), in which both ze and zm are on the same side of the
wall, and two transition states when the end bead and the center bead are located
on different sides of the wall. The transition states are in accordance with what we
observed in the snapshots shown in Fig. 4.5.
A simpler representation of the free energy surface explored by this system is
given simply by the probability density for the z-component of the end bead ρ(ze).
Unlike the joint distribution ρ(ze, zm), this representation is too low-dimensional to
provide insight into the structure of the transition state. On the other hand, it is
simple to construct and visualize, and a simple potential of mean force (PMF) can
be constructed as
F (ze) = F ∗ − kBT ln
[
ρ(ze)
ρ∗
]
, (4.4)
where F ∗ and ρ∗ are arbitrary reference values. In Fig. 4.6(b) we show ρ(ze) (dashed
circle) and corresponding F (ze) (solid line) when F ∗ = Fwell = 0 (Nb = 20) in a
quasi-2D system . As expected, the PMF is composed of two energy wells which
are separated by a energy barrier – the difference in energy between these is denoted
∆Fmax. Fig. 4.6(c) shows a plot of the energy barrier ∆Fmax/kT as a function of
35
-0.50
-1.0
-1.5
-2.0
-2.5
-3.0
-3.0
-3.5
-3.5
-4.0
-4.0
-4 -2 0 2 4-3
-2
-1
0
1
2
3
ze( m)
zm(
m)
-5.0-4.5-4.0-3.5-3.0-2.5-2.0-1.5-1.0-0.500
(a)
-3 -2 -1 0 1 2 30
1
2
3(b)
ze( m)F(
z e)/kT
Fmax
10 15 20 25 300
1
2
3
4
5
6
7
Fmax kT a ln(Nb)
a=2.58 0.12
a=1.41 0.03
(c)
F max
/kT
Nb
3D Q2D
10 15 20 25 30
10
100
1000(d)
4.16 0.14
2.45 0.05
3D Q2D
Nb
(s)
b
Figure 4.6: Simulation results for an ideal chain tethered to the center of the pore inquasi-2D and 3D pore crossing geometries (Fig. 4.3(a)). (a) The negative of logarithmof the joint probability density function of the z-component of the end bead ze and thez-component of the center bead zm for a Nb = 13 chain in 3D. (b) Reduced potentialof mean force ∆F(ze)/kBT for a Nb = 20 chain in Q2D. (c) Semilog plot of reducedenergy barrier vs. chain length when d = 7a (d) Log-log plot of mean first passagetime τ v.s. chain length when d = 7a. The error bars are about the same size as thesymbols.
36
Nb for the 3D and quasi-2D cases. For the range of Nb considered it appears that
∆Fmax/kT = a lnNb.
Fig. 4.6(c) shows the dependence of the mean waiting time τ on molecular weight.
Over the limited range of molecular weights studied, the results appear to follow a
power law: τ = τmNαb . If the transition process follows Kramers kinetics [160], one
would expect that τ = τ0 exp(∆Fmax/kbT ), which in the present case would imply
that τ0 = τmNα−ab . If hydrodynamic interactions were included in the model, one
might expect α to decrease slightly, based on simulation results for translocation of
a free polymer chain through a pore ([8, 73, 56, 54]). Since a is determined from the
free energy landscape of the system, it is unaffected by the presence or absence of
hydrodynamic interactions in the model. On the other hand, the pore geometry (size,
shape) might affect both these parameters; in particular changing the pore size will
change the free energy landscape and thus the exponent a. The nature and origin of
the observed exponents will be the subject of future work; we note that extension of
Kramers’ theory of barrier crossing of a point particle to a polymer system is a topic
of some current research [106, 129, 128].
Finally, we observe that the simulation results indicate that confinement from 3D
to quasi-2D greatly reduces the energy barrier and leads to a larger transition rate
(Fig. 4.6(c) and (d)). As the crossing rate is inversely proportional to the free energy
barrier, which is closely related to the number of accessible chain configurations, this
is not surprising by considering that the number of accessible chain configurations in
the quasi-2D case is greatly reduced compared to that in the 3D case.
37
4.3.2 Pore-entry geometry
The previous section reported a system where geometry alone determines the free
energy landscape of the tethered polymer. The present section turns to a case where
entropy and electrostatic energy compete, and address the possibility that bistability
of a tethered chain can arise from this competition. The situation under consideration
is the pore-entry geometry of Fig. 4.3(b). Only the quasi-2D case will be considered.
In the absence of a field, a chain with Rg ≥ d would not be expected to sample the
pore, while if the field is sufficiently large it is expected that the entire chain will
reside in the pore. The regime of interest is chain lengths such that Rg ≥ d and
intermediate values of the field strength.
Fig. 4.7(a) shows a time series of ze for the case Nb = 10 and ER = 5. As in the
pore-crossing geometry, bistability is observed, which can be seen more clearly from
the joint probability density function ρ(ze, zm) illustrated in Fig. 4.7(b). The path
connecting the two metastable states indicates, again, the transition state. Fig. 4.8(a)
shows PMF curves based on ρ(ze) as functions of ER when Nb = 10. As the electric
force is small in the outer chamber, it does not significantly contribute to the energy
of the chain when ER . 10.0 and the energy profile outside the channel is insensitive
to the change of ER. The energy well inside the channel is created by a balance
between the entropic force and the electric force. We denote the critical field strength
above which this energy well appears as Elow (≈ 2.2 for Nb = 10). As ER increases
further, the free energy barrier for entry to the pore weakens, and eventually vanishes
once ER exceeds a field strength denoted Ehigh When Elow < ER < Ehigh, the system
shows competitive bistability that is created by the interplay between entropy and
electrostatic energy. Fig. 4.8(b) shows the positions of the minima in Fig. 4.8(a) as
38
0 20 40 60 80 100
-2
-1
0
1
2
3
4(a)
t(s)
ze(
m)
-3 -2 -1 0 1 2 3 4-3
-2
-1
0
1
2
3
4
zm(
m)
ze( m)
(b) -4.00-3.50-3.00-2.50-2.00-1.50-1.00-0.5000
Figure 4.7: Simulation results for a tethered ideal chain in the quasi-2D pore-entrygeometry (Fig. 4.3(b)) for the case Nb = 10, ER = 5, d = 7a. (a) A typical timeseries plot of the z component of the free end of a tethered chain. (b) The negativelogarithm of the joint probability density function ρ(ze, zm).
39(c) (d)
(e) (f)
(g) (h)
-2 0 2 4 6 8 10 12 14-1
0
1
2
3
g
h
f
e
d
c
(b)
Elow
z e0(
m)
ER
OUT IN
Ehigh
z( m)m
y(
m)
m
−2 −1 0 1 2 3−2
−1
0
1
2
−2 −1 0 1 2 3−2
−1
0
1
2
−2 −1 0 1 2 3−2
−1
0
1
2
−2 −1 0 1 2 3−2
−1
0
1
2−2 −1 0 1 2 3
−2
−1
0
1
2
−2 −1 0 1 2 3−2
−1
0
1
2
-3 -2 -1 0 1 2 3 4-6-5-4-3-2-101234
E0 E1 E2.2 E3 E4 E5 E7 E9
ze0
(a)
ze( m)
F(z e)/k
T
Figure 4.8: Simulation results of an ideal chain tethered to the center of the entrance in a quasi-2D pore entrygeometry, with Nb = 10. (a) Potential of mean force as function of last bead position ze and reduced electric fieldgradient ER. Top to bottom curves correspond to ER = 0, 1.0, 2.2, 3.0, 4.0, 5.0, 7.0, and 9.0, respectively. (b)Most probable end bead positions ze0 (i.e., maxima of ρ(ze) or minima of ∆F (ze)) as a function of ER for state”IN” and state ”OUT”. The error bars are almost the same size as the symbols. (c)(d),(e)(f),(g)(h) are snapshotsat ER = 2.3, 5, and 9, respectively.
40
0.00 0.02 0.04 0.06 0.08 0.10 0.120
2
4
6
8
10
12
14
State "IN"+"OUT"
State "OUT"
State "IN"
Elow
Ehigh
(a)
ER
1/Nb
0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.10.20.30.40.50.60.70.80.91.0(b)
Pin
ER/E1/2
Nb=10 Nb=15 Nb=20 Nb=25
Figure 4.9: (a) Phase diagram showing the region in parameter space where the chainshows bistability in the pore-entry geometry. (b) The probability to find the chaininside the channel Pin as function of ER normalized by E1/2.
functions of ER – the regime of bistability is clearly seen. Fig. 4.8(c) shows typical
conformations of the chain at various points on Fig. 4.8(b).
Finally, we address the phase transition behavior of the tethered polymer molecule
in the competitively bistable system. In Fig. 4.9(a), the shaded area shows the param-
eter regime in ER and 1/NB where bistability occurs. Fig. 4.9(b) shows the probability
Pin =∫∞0
dzeρ(ze) of the chain end residing inside of the channel, as a function of ER
for various Nb. We have normalized ER with E1/2, the (molecular-weight dependent)
field strength where Pin = 0.5. From these plots it appears that as N → ∞, the
slope diverges and the region of bistability vanishes, indicating a first-order phase
transition. Similar behavior was noted in the escape transition system studied by
Binder and coworkers [95].
41
4.4 Conclusions
Simulations predict that flexible polymers end-tethered in a pore within a confined
geometry can display multiple free energy minima separated by substantial barriers
– bistability. For a tether point in a pore between two large chambers, bistability is
expected based on simple considerations of entropy and geometry -we denote this as
“entropic bistability”. In this situation dynamic simulations suggest that the crossing
of the chain end through the pore is the dominant mode of transition between the two
states, and consideration of the joint probability density function of the position of the
chain end and the chain midpoint indicates the presence of a transition state where
the chain end is on one side of the pore, while the chain midpoint is on the other. For
a chain end-tethered at the mouth of a long pore and subjects to a field that tends to
draw the chain into the pore, we predict the existence of a second kind of bistability,
arising from the interplay of conformational entropy and electrostatic energy; we
call this “competitive bistability”. In particular, simulations show the presence of a
regime of field strength in which there are two free-energy minima, one corresponding
to a chain that is largely outside the pore in an entropically favorable state, and one
where the chain is largely inside the pore, in an electrostatically favorable state.
Both fundamental and technological question are raised by these results. The
systems under consideration here comprise model systems for activated rate processes
in systems with many degrees of freedom and provide an opportunity to compare
experiment, theory and simulations for this class of systems. Of particular interest
may be the difference in dynamics when transition from one state to another is driven
by fluid flow rather than an electrostatic field. These systems or variations of them
might also have applications as electrically or fluidically actuated nanomechanical or
42
nanoelectromechanical elements, such as valves or switches.
43
Chapter 5
Tethered DNA dynamics in shear
flow
In Chapter 5 and Chapter 6, we discuss flow-driven DNA dynamics. To study the
effects of solid impenetrable walls on the dynamics of a nearby DNA molecule, we
start by re-examining a problem with simple geometry: the cyclic dynamics of a
tethered DNA molecule in shear flow where the chain is grafted to a flat wall. We
compare three simulation methods: Brownian Dynamics (BD), the Lattice Boltzmann
Method (LBM), and a recent Stochastic Event-Driven Molecular Dynamics (SEDMD)
algorithm.1 We focus on the dynamics of the free end (last bead) of the tethered chain
and we examine the cross-correlation function (CCF) and power spectral density
(PSD) of the chain extensions in the flow and gradient directions as a function of
chain length N and dimensionless shear rate Wi. Extensive simulation results suggest
1This chapter is based on the publication: Y. Zhang, A. Donev, T. Weisgraber, B. J. Alder, M.D. Graham, and J. J. de Pablo, J. Chem. Phys. 130, 234902. LBM and SEDMD calculations wereperformed by our collaborators at the Lawrence Livermore National Lab.
44
a classical fluctuation-dissipation stochastic process and question the existence of
periodicity of the cyclic dynamics, as previously claimed. We support our numerical
findings with a simple analytical calculation for a harmonic dimer in shear flow given
in Appendix C.
5.1 Introduction
The interaction of polymer molecules with fluid flow has been studied both theoret-
ically [130, 21] and experimentally [109, 108, 136, 135, 57] for several decades. The
behavior of polymer chains in flow is determined by an intricate interplay between
the flow gradients, chain elasticity, thermal fluctuations, and the physical confinement
[78, 75]. The dynamics of tethered polymer molecules (“polymer brushes”) in shear
flow has received considerable attention due to its relevance to diverse important
applications, such as colloidal stabilization, surface adhesion, and lubrication [96].
In contrast to previous work on the collective motion of polymer brushes [96, 55],
Doyle et al. studied the dynamics of a single tethered DNA in uniform shear flow
using fluorescence videomicroscopy [46]. Enhanced temporal fluctuations in the chain
extension were observed, and were attributed to the coupling of advection in the
flow direction and diffusion in the gradient direction. A cyclic dynamics mechanism
(Fig.5.1), closely related to the tumbling dynamics of a free polymer molecule in
shear flow [115, 57, 127], was proposed based on results from Brownian dynamics
simulations. No peaks were observed in the calculated power spectral density (PSD)
of the DNA extension in the flow direction, and the authors therefore suggested that
the cyclic dynamics of a tethered chain in shear flow is aperiodic. An important
45
physical question is whether there is a characteristic timescale associated with the
cyclic motion that is distinct from the internal relaxation time of the chain.
Several computational studies revisited the problem of a tethered chain in shear
flow by looking at different variables relating to both the flow and gradient directions,
such as extensions along both flow and gradient directions [37], polymer orientation
angle defined through the gyration tensor [127], and angle between the wall and
the vector joining the tethering point to the center-of-mass of the chain [62]. The
cross-correlation functions for such variables exhibit signatures of the proposed cyclic
motion in the form of peaks at non-zero delay time. Because of the particular choice
of variables in Ref. [62], the lack of such peaks at small Weissenberg numbers was
attributed to the existence of a critical Weissenberg number; as demonstrated in Ref.
[42], choosing a different sets of variables shows that the signature peaks exist even at
small shear rates. In Ref. [62], the position of the peak in the studied cross-correlation
functions was interpreted as a characteristic cycling time, and it was found to be a
fraction of the relaxation time of the polymer chain. In Ref. [127] the tumbling motion
of a free polymer chain in shear flow was studied experimentally and computationally,
and wide peaks were found in the Fourier spectra of the time series of the angle of
the chain relative to the flow direction. These peaks were identified as evidence of
periodic motion of the tumbling molecule. The characteristic tumbling time (period)
was extracted from the position of the peak in the spectrum and was found to be
in good agreement with the experimentally-measured tumbling frequency. These
finding for a free chain in shear flow inspired similar studies of a tethered chain,
and similar observations of periodic motion with a characteristic period about an
order of magnitude larger than the relaxation time were reported [127, 37, 38]. In
46
12
11
1)
2)
3)
4)
z
x
y
Figure 5.1: Snapshots taken from a simulation run withWi = 5 to show tethered DNAdynamics. The beads are labeled from 1 to 11 as shown. Cyclic motion mechanismproposed by Doyle et al. is composed of four stages: 1) (Re)coiling; 2) initiating; 3)stretching; 4) rotating [46]
several later studies of the tumbling emotion of a free polymer chain in shear flow,
experimental results [57], numerical simulation [115], and theory [34, 156] all suggest
that the intervals between successive tumbling events are exponentially-distributed
with a decay constant equal to the relaxation time of the chain. Such an exponential
tail implies that the tumbling events occur as a Poisson-like process, which is aperiodic
despite the existence of a characteristic timescale (frequency of repetition). If the
tumbling events of a free polymer in shear flow is aperiodic, intuitively, adding of a
wall to break the symmetry of the motion should leave the dynamics of a tethered
chain aperiodic.
Different authors use the terms “cyclic” (repetitive) and “periodic” with differ-
47
ent meanings, and it is therefore important to give our definitions. Periodicity is
the quality of occurring in regular time intervals (periods). Periodic motion has
correlation functions that are (possibly damped) oscillatory functions, and spectra
that have sharp peaks. Noise (fluctuations) and the associated dissipation will al-
ways broaden any peaks that are related to underlying deterministic periodic motion
(and consequently, exponentially dampen the oscillations in the real-space correla-
tion functions). As an example, for a rigid spheroid in shear flow, there is indeed
periodic motion (Jeffery’s orbits) in pure shear flow. Adding fluctuations, when they
are small, is expected to preserve that but introduce some broadening of the spectral
peaks [88]. In contrast, a cycle usually means a process that eventually returns to its
beginning and then repeats itself in the same sequence. The end-to-end tumbling of
a single polymer molecule in shear flow provides a relevant example. In this paper
we analytically calculate the power spectrum for a tethered dimer in shear flow, and
find an exponentially-decaying cross-correlation function that has the relaxation time
as the only characteristic timescale. More importantly, this analytical example shows
that the power spectrum can exhibit a wide peak at small frequencies without any
underlying periodic motion, and that the location of a maximum in the PSD is not
necessarily an indication of a new timescale. The analytical results for a tethered
dimer are consistent with our numerical observations for longer tethered chains in
shear flow. Therefore, our investigations do not confirm the existence of periodic
motion with a period distinct from the relaxation time of the chain, as previously
suggested in the literature [127, 37, 38].
We apply three different solvent representations to the same problem of a teth-
ered chain in shear flow. The models are an implicit solvent (Brownian Dynamics
48
[74], BD), a continuum solvent (Lattice-Boltzmann [5], LB), and a particle solvent
(Direct Simulation Monte Carlo [42], DSMC). Such comparison between widely dif-
fering methods on the same problem is important as a validation of their range of
applicability. It is also important to compare the computational performance of the
different methods. In this work, different chain representations and boundary condi-
tions make a direct quantitative comparison impossible. Specifically, the BD polymer
is a worm-like chain representative of semi-flexible DNA, in the LB simulations it is
a flexible chain of repulsive spheres, and in the DSMC solvent it is a flexible chain
of hard spheres. However, we can access the importance of the details of the chain
model, and in particular, of chain elasticity, and thus test the widely used assumption
that the dynamics scales with the Weissenberg number Wi = γτ , where γ is the shear
rate and τ the chain relaxation time, independent of the details of the model. For this
particular problem of a tethered chain in shear flow, we find good agreement between
the different methods.
A general discussion of the wide range of techniques for modeling the hydrody-
namics of polymer chains in solution is given in Section 5.2. Further details about the
three specific techniques we use in this paper to study the tethered polymer problem
are given in Section 5.3. In Section 5.4 we present our results, and finally, in Section
5.5 we give some concluding remarks.
49
5.2 Discussion of Methods for Hydrodynamics of
Polymer Solutions
In this Section we give a brief overview of various methodologies for modeling hy-
drodynamics of soft matter systems, notably, polymer solutions (see also review by
Duenweg and Ladd [47]). The various methods for computational hydrodynamics
of polymer solutions can be divided in two major categories. The first are purely
continuum methods that use constitutive equations for the polymer solution. These
models only apply at macroscopic scales, when the number of polymer chains in an
elementary fluid flow volume is large, so that statistical averages of the chain con-
formations can be used as parameters in constitutive models of the time-dependent
stress as a function of the strain rate history. The construction of such constitu-
tive models is ad hoc and rather difficult in situations where conformations of the
chains couple to an unsteady flow, as, for example, in the problem of turbulent drag
reduction. Additionally, such continuum methods do not apply to situations where
the dynamics of individual chains are of interest, such as a DNA molecule flowing
through a micro-channel or DNA translocation through a pore. The second major
category of methods explicitly simulates the motion of each polymer chain using some
form of molecular dynamics. The simplest chain model is a dumbbell. Multi-bead
representations of the chains are capable of complex chain conformations but require
models for the bead-bead interactions. Such details of the polymer model are impor-
tant for both physical fidelity and computational efficiency. For example, preventing
chain-chain crossing can require stiff interactions for excluded volume terms, which
in turn can lead to small time steps. There are also two major types of algorithms
50
for dealing with the solvent. One represents the solvent implicitly, and the others use
an explicit solvent. An implicit solvent is most efficient computationally, however, it
can only be used if the fluid flow in the absence of the polymer is known analytically
or can easily be pre-computed numerically (e.g., stationary flow), and if the polymer
chains themselves do not alter the background flow.
5.2.1 Implicit Solvent: Brownian Dynamics
The most widely used implicit-solvent algorithm is Brownian dynamics [74], described
in more detail in Section 5.3.1. The method involves solving first-order differential
equations of motion for the positions of the beads with additional forces due to the
presence of the solvent. These solvent forces can be separated into a deterministic
portion, for which a (linear) analytical approximation is used, and a stochastic por-
tion, which is assumed to be white noise. The fluctuation-dissipation theorem is used
to set the magnitude of the stochastic forcing. Brownian dynamics relies on several
assumptions usually valid in microfluidic applications. The first assumption is that
of small Reynolds number laminar (usually stationary) flow adequately described by
a linearized Navier-Stokes equation. The second assumption is that hydrodynamic
fields develop infinitely quickly relative to the rate at which the polymer conforma-
tion changes, so that a quasi-stationary approximation can be used to describe the
perturbation of the flow field induced by the motion of the beads. This approxi-
mation leads to Stokes friction on single beads, as well as hydrodynamic interaction
pairwise terms approximated with a long-range Oseen tensor as derived through an
asymptotic (t → ∞) analysis for point particles. The free-draining approximation
of Brownian dynamics neglects these pairwise hydrodynamic interactions. The in-
51
clusion of pairwise hydrodynamic interactions leads to a matrix formulation of the
fluctuation-dissipation theorem and therefore factorization of a matrix of the size of
the number of beads is required at every time-step. Various numerical tools have
been devised to avoid dense factorization [76, 132, 74, 67], thereby reducing the cost
of a single time step in Brownian dynamics with hydrodynamic interactions.
Brownian Dynamics should be distinguished from Langevin dynamics, in which
second-order (Newton’s) equations of motion are used for the beads, that is, both
the bead velocities and positions are included as explicit degrees of freedom (but the
solvent is still implicit) [122]. This assumes that there is a large separation of time-
scales between the fluid degrees of freedom and the velocities of the beads, which
is in fact only true if the beads are much denser than the solvent. Furthermore, a
much smaller timestep necessary to resolve the faster dynamics (relaxation) of the
bead velocities. Therefore, Langevin dynamics finds its use only when the solvent is
represented explicitly, so that calculating the friction and stochastic forces no longer
requires factorization of the mobility tensor.
An important advantage of Brownian dynamics is that it simulates the limit of
zero Reynolds number exactly. It can also often exactly account for simple boundary
conditions (e.g., flow in an infinite half plane) without resorting to approximations
that truncate the flow field to a finite domain, such as the commonly-used periodic
boundary conditions. Brownian dynamics is relatively easy to implement, however,
complex boundary conditions, such as indentations or bumps on walls, requires care
so that analytical approximations to the Oseen tensor that preserve the positive-
definiteness of the diffusion tensor [69]. While the computational cost can rise rapidly
as the number of beads is increased when direct implementations are used, novel
52
schemes can be used to truncate the long-range hydrodynamic interactions and yield a
linear dependence on system size, similarly to the handling of electrostatic interactions
in spectral [132] and multipole methods [67].
5.2.2 Explicit Solvent: Continuum Methods
In order to capture the bi-directional coupling between the motion of the polymer
and the flow around it, it is necessary to explicitly represent the solvent. The first
level of approximation is to use a continuum description of the solvent assuming the
applicability of the Navier-Stokes (NS) PDEs at small length scales. Typically an
incompressible assumption is made, which is appropriate at sufficiently low Mach
numbers if acoustic waves are not of interest. Additional approximations such as lin-
earization or an iso-thermal approximation may be appropriate. The time-dependent
(unsteady) NS equations can be solved by any of the numerous existing CFD al-
gorithms, including explicit, implicit, or semi-implicit algorithms of varying level of
complexity [131, 146, 12]. One of the advantages of the PDE formulation over par-
ticle methods is the ability to use powerful adaptive mesh resolution techniques that
allow coarsening of the mesh away from the region of interest, here polymer chains.
However, the case of complex boundary conditions such as needed, for example, in
the handling of moving beads or flow through porous media, presents difficulties. An
alternative to solving the Navier-Stokes PDEs is to use the Lattice-Boltzmann (LB)
method [150], as discussed in Section 5.3.2. It requires small time steps limited by
CFL-type conditions, however, each of the time steps is efficient. Recently, so-called
entropic LB schemes have been developed that posses a discrete H-function, resulting
in unconditional numerical stability even at high Reynolds numbers [23]. LB has been
53
found competitive with NS solvers in many situations and has the further advantage
that it is based on kinetic theory and allows a more detailed level of description than
NS. An important advantage of LB solvers is also their ability to handle complex
boundary conditions. Recently, Chen et al. have provided a detailed comparison
between BD and LB simulations on a DNA model that shows that the LB method
provides a reasonable description of the results of more precise BD simulations at low
Reynolds numbers [32].
Thermal Fluctuations
Most continuum fluid dynamics methods are deterministic and thus do not include
internal fluctuations of the hydrodynamic fields. Fluctuations become more important
the smaller the length scale of interest, and are crucial for polymer flows. Including
thermal fluctuations in a continuum formulation has been carried out for both CFD
and LB algorithms. The Landau-Lifshitz Navier Stokes (LLNS) equations include
thermal fluctuations in the stress tensor but numerical schemes to solve them are
not nearly as advanced as are the standard CFD solvers [131, 51, 19]. Fluctuations
have been included in LB and do not pose any particular numerical problems [3].
Fluctuations have also been included in incompressible solvers in conjunction with
the Immersed Boundary Method [12, 83]. The ability to turn fluctuations on or off
is an important advantage of continuum-based methods over particle methods.
Coupling with the Polymer Chains
Regardless of what continuum method is employed, it is necessary to couple that
method to the MD description of the polymer chains. The simplest and most com-
54
monly used coupling scheme is to approximate the beads as points and assume
for the solvent-induced force on the polymer beads the Stokes-Langevin form F =
−6πRHηvf+FS, where vf is an estimate of the local fluid velocity and FS is an uncor-
related stochastic force whose magnitude obeys the fluctuation-dissipation theorem
[146, 58]. This approximation is similar to that in Brownian dynamics, namely, Stokes
law is only valid in quasi-static continuum situations, relying on the separations of
time and length scales which are usually only marginally separated in realistic situa-
tions. Typically the strength of the coupling, RH , is empirically tuned to reproduce
experimental measurements. The coupling can also be dealt with when the beads
occupy an actual volume, free of fluid. Then stick or slip boundary condition at the
surface of the beads are employed, as in both NS [131] and LB [150] simulations of
colloidal dispersions. However, these methods are rarely used in polymer simulations
due to the complexity when many moving particles are involved, because, the grid
size needs to be smaller than the bead size and may need to be adaptively changed
when the bead moves.
A different alternative is provided by the Immersed Boundary method [12], where
the fluid occupies the whole space and the particles, represented as immersed struc-
tures, move together with the fluid with a velocity that is a localized average of the
fluid velocity. This eliminates the bead inertia from the problem and the need to ex-
plicitly enforce boundary conditions on the surface of the beads. The method can be
seen as an alternative to Brownian dynamics that correctly captures time-dependent
momentum transport in the fluid by explicitly representing the fluid flow, and also
includes thermodynamically-consistent thermal fluctuations [83].
55
5.2.3 Explicit Solvent: Particle Methods
An alternative to continuum methods is to use a particle representation of the fluid.
The most detailed description is a MD simulation of both the fluid and the solvent.
Unlike the classical NS equations, MD automatically and correctly includes fluctua-
tions, internal fluid structure, diffusion, and non-linear transport. Particle methods
are also typically simple to implement and can easily accommodate complex bound-
ary conditions.Typically a truncated repulsive Lenard-Jones potential is used for the
solvent-solvent interactions. However, even with massive parallelization such MD
simulations are limited to short total times and therefore efforts have been made to
coarse-grain the solvent to a mesoscopic representation. There, the fluid particles are
no longer representative of solvent molecules, but are larger having different dynamics
and interactions with each other. However, the viscosity and the stress fluctuations
in the solvent must be reproduced correctly. There are mesoscopic particle solvents
of progressively decreasing level of microscopic fidelity, and thus increasing efficiency.
The handling of the coupling between the solvent and the beads is a separate issue,
like for continuum solvents. A particle solvent may be coupled to a polymer chain
by including explicit short-ranged solvent-bead continuous [89] or hard-spheres [42]
interaction potentials. Efficiency can further be gained by coarse graining the bead-
solvent interactions as well, typically using the same ideas as used to coarse grain
the solvent-solvent interactions [82, 98]. Dissipative Particle Dynamics (DPD) [111]
further coarsens the solvent molecules to obtain a system of weakly-repulsive spheres
interacting with a mixture of conservative, stochastic, and dissipative forces. The
conservative forces can be used to reproduce the solvent equation of state, while the
dissipative forces model viscous friction. The stochastic forces act as a thermostat
56
that ensures detailed balance and correct thermal fluctuations in the DPD fluid. The
method has great flexibility and requires significantly less solvent particles and larger
time-steps than classical MD, however, it still requires costly integration of differen-
tial equations of motion for each of the solvent particles. Such integration of ODEs
can be avoided by using a kinetic Monte Carlo method, such as Direct Simulation
Monte Carlo (DSMC), to represent the solvent-solvent interactions. The idea is to
use stochastic conservative collisions between nearby solvent particles to represent
the exchange of momentum and energy. Both multi-particle collisions [82, 98] and bi-
nary collisions [42] have been used, as described in Section 5.3.3. The computational
efficiency comes at the cost of neglecting the structure of the solvent, as in contin-
uum methods. Recently a new Stochastic Hard-Sphere Dynamics method has been
proposed that also uses uncorrelated stochastic binary collisions but still produces a
non-trivial fluid structure and a thermodynamically-consistent non-ideal equation of
state, similar to those of a DPD fluid [43].
5.2.4 Coupled Methods
Methods that combine several of the techniques described above into a single con-
currently coupled simulation can take advantage of their region of validity. Such a
simulation may involve several levels each with a different level of microscopic de-
tail. For example, molecular dynamics with complete atomistic detail and realistic
potentials may be used for the polymer chain(s) and nearby solvent. The solvent can
then be coarse grained to a mesoscopic particle fluid sufficiently far from any chains.
The particle method can then be coupled to an explicit fluctuating hydrodynamic
description with a fine grid, for example, LB or a fluctuating NS solver. Finally, the
57
hydro grid can be adaptively coarsened in regions even farther from the chain, and a
non-fluctuating continuum solver used. This last macroscopic level can use a different
method from the fluctuating hydrodynamics level, for example, it could be an incom-
pressible NS solver. Much remains to be done to enable a truly multiscale simulation
capable of bridging from microscopic to macroscopic length and time-scales [155, 39].
5.3 Simulation Methods
In this Section we describe in further technical detail the three different techniques
we apply to the tethered polymer problem. The majority of the methodology has
been previously published so here we only summarize the essential points and cite
the relevant works.
5.3.1 Brownian Dynamics
Details of the DNA model and Brownian dynamics simulation method that we use
can be found in Refs. [76, 74]. We discretize a double-stranded DNA molecule into
a bead-spring chain composed of Nb beads of radius Rb = 77nm (the unit of length,
1 l.u. = 77nm) connected by Ns = Nb − 1 entropic springs. Each spring represents a
DNA segment of 4850 base pairs, so that Nb = 11 corresponds to a stained λ-DNA,
which has a contour length of 21 µm. In Brownian dynamics, a force balance on this
chain leads to a stochastic differential equation for the dynamics of the chain [105],
∆R = [U+D · FkBT
+∂
∂R·D]∆t +
√2B ·∆W (5.1)
58
where R is the vector containing bead positions, R = r1, ..., rN, U is the unper-
turbed velocity field at the bead centers, kB is Boltzmann constant, T is absolute
temperature, F is the non-hydrodynamic and non-Brownian forces, and D = B ·BT
is the diffusion tensor. The components of ∆W are obtained from a real-valued
Gaussian distribution with mean zero and variance dt. In a unbounded space, the
hydrodynamic interactions (HI) enter the chain dynamics through the diffusion ten-
sor,
Dij = kBT [(6πηa)−1Iδij +Ωij ] (5.2)
where η is the viscosity of the solvent, a is the bead hydrodynamic radius, I is the
unit tensor, δij is the Kronecker delta, and Ω is the HI (Stokeslet or Oseen) tensor.
Recent work has provided evidence of hydrodynamic coupling to the wall and ex-
perimental validation of the use of point-particle (Stokeslet) hydrodynamic interac-
tions (HI) to describe the motion of Brownian particles near a surface [48]. Therefore,
it is essential to have wall corrected HI in the simulation to capture the dynamics of
a tethered chain correctly. In a bounded space, like near a solid wall, the HI tensor
is modified to,
Ωij = (1− δij)ΩOB(ri − rj) +ΩW (ri − rj) (5.3)
where ΩOB is the free-space diffusion tensor, and ΩW is the correction which accounts
for the no-slip constraint on the wall. The solution for a Stokeslet above a flat plate
given by Blake allows us to calculate ΩW exactly [22]. In a square channel or complex
geometries, we need to solve this problem numerically with a finite element method
to determine ΩW at a grid of points [78]. Based on this description of near-wall HI,
Jendrejack et al. [77] predicted that the DNA molecules migrate away from the wall
59
in shear flow, leading to the formation of depletion layers in the near wall region. This
prediction has been verified in recent experiments of dilute DNA solutions undergoing
pressure-driven flow in microchannels [4, 30]. In different works, Delgado-Buscalioni
used a hybrid particle-continuum model method to describe HI [37] and Schroeder et
al. used unbounded space HI [127] to study the motion of a tethered chain.
We further assume that the chain is ideal (no self-excluded volume interactions
between different beads). The entropic springs connecting the beads obey a worm-like
chain law
Fsij =
kBT
2bk[(1− |rj − ri|
Nk,sbk)−2 − 1 +
4|rj − ri|Nk,sbk
]rj − ri|rj − ri|
, (5.4)
where bk is the Kuhn length for DNA and Nk,s is the number of Kuhn lengths per
spring. The physical confinement is taken into account through an empirical bead-
wall repulsive potential of the form
Uwalli = Awallb
−1k δ−2
wall(hi − δwall)3, (5.5)
when hi < δwall, where hi represents the perpendicular distance of bead i from the
wall, δwall is the cut-off distance. In this work, we choose Awall = 25kBT and δwall =
bkN1/2k,s /2 = 0.24 µm. All of the parameters a, bk, ν are the same as used in previous
work, where it has been shown to successfully reproduce the static and dynamic
properties of DNA with contour length 10µm− 126µm [76, 78]. For each parameter
set, the sample size is 30 chains unless otherwise specified. All results are presented
for DNA at room temperature in a solvent with a viscosity of 1 cP .
To study the dynamics of a tethered chain, beads are labeled from 1 to Nb + 1,
starting from the tethered point, as illustrated in Fig. 1. The fluid velocity in the
60
flow direction z is a linear function of distance from the wall in the gradient direction
x, vz = γx, where γ is the shear rate, and vx = 0 and vy = 0. Following common
experimental practice, the longest relaxation time is calculated by allowing a chain
that is initially stretched using a large shear rate to relax to equilibrium. Near
equilibrium, the relaxation time is determined by an exponential decay fit the chain
extension along the stretch direction,
〈X2〉 = (X2(0)− 〈X2〉eq) exp(−t
τ) + 〈X2〉eq. (5.6)
An exponential fit to the autocorrelation of the chain extension (relative to equilib-
rium) parallel to the wall gives similar results. The relaxation time for our λ-DNA
is estimated to be 0.59s at room temperature, which is in good agreement with the
experimental result of 0.51s [46] after extrapolating the viscosity to 1 cP .
5.3.2 Lattice-Boltzmann
In addition to Brownian Dynamics, we examine the short time correlations of a teth-
ered polymer in a uniform shear flow using a hybrid Lattice Boltzmann (LB) and
Molecular Dynamics (MD) code based on the method by Ahlrichs and Dunweg [5].
The Lattice Boltzmann method is a mesoscopic approach to fluid flow calculation
and is based on a discrete version of the Boltzmann equation with enough detail to
recover hydrodynamic behavior. The LB equation describes the evolution of a single-
particle distribution function, fi (x, t), which is the mass density of particles moving
61
with velocity ei at a time t and position x on a cubic lattice,
fi (x+ ei∆t, t +∆t) = fi (x, t) +∑
j
Aij
[
fj (x, t)− f eqj (x, t)
]
. (5.7)
The set of velocities ei is discrete and chosen such that x + ei∆t always remains a
lattice site. The last term describes the collision process in which the distribution
function relaxes to a local equilibrium, for which we utilize the BGK (Bhatnagar-
Gross-Krook) approximation to the collision operator, Aij = −τ−1δij , where τ is a
relaxation time. The macroscopic hydrodynamic quantities, density ρ, momentum j =
ρu, and momentum flux Π, are computed from moments of the particle distribution
function,
ρ =∑
i
fi, j =∑
i
fiei, and Π =∑
i
fiei ⊗ ei. (5.8)
The equilibrium distribution depends on the macroscopic variables and its form is
given by
f eqi (x, t) = wiρ
[
1 +ei · uc2s
+(ei · u)22c4s
− u2
2c2s
]
, (5.9)
where the weights wi depend on the particle velocity discretization and are determined
by mass and momentum conservation. The lattice sound speed is cs = ∆x/√3∆t,
where ∆x is the lattice spacing. In this work we solved the distribution function on
the standard D3Q19 lattice [150] where the 19 particle velocity components consist
of one rest particle, the 6 nearest neighbors in a simple cubic lattice, and the 12
next nearest neighbors in the [110] directions. The corresponding weights are 1/3,
1/18, and 1/36. The LB method avoids the additional mathematical complexities of
Navier-Stokes PDE solvers and is straightforward to parallelize efficiently. Using a
62
Chapman-Enskog expansion, the lattice-Boltzmann equation can recover the Navier-
Stokes equations for small Mach and Knudsen numbers, and, within these limits it
is second-order accurate in space and time. Compared to the other two methods we
apply to the tethered polymer problem, BD and SEDMD, LB is less efficient in this
case since it solves for the solvent in the entire domain, even relatively far from the
polymer chain.
In the LB calculations, the polymer is represented by 25 point particles joined by
finitely extendable nonlinear elastic (FENE) springs and interact through a repulsive
Lennard-Jones potential among each other and with the walls. Solvent fluctuations
are incorporated by adding a stochastic term to the right hand side of the LB equation.
This term introduces fluctuations into the momentum flux in a manner that satisfies
the fluctuation-dissipation theorem [150]. Coupling between the LB for the solvent
and the MD for the solute is achieved through Stokes drag forces and white-noise
stochastic forces acting on the monomers. The first monomer in the chain is tethered
to the stationary lower wall in a domain having 36, 22, and 24 lattice sites in the
streamwise, spanwise, and wall normal directions. The streamwise and spanwise
directions are periodic and the bounding upper wall moves with constant velocity,
providing the uniform shear.
5.3.3 Stochastic Event-Driven Molecular Dynamics
In addition to Brownian dynamics and Lattice-Boltzmann, we have also applied a
purely particle-based method to the tethered polymer problem. The Stochastic Event-
Driven Molecular Dynamics (SEDMD) algorithm introduced in Ref. [42] combines
Event-Driven Molecular Dynamics (EDMD) for the polymer particles with Direct
63
Simulation Monte Carlo (DSMC) [7] for the solvent particles. In SEDMD, the poly-
mers are represented as chains of hard spheres tethered by square wells. The solvent
particles are realistically smaller than the beads and are considered as hard spheres
that interact with the polymer beads with the usual hard-core repulsion. The al-
gorithm processes true (deterministic, exact) binary collisions between the solvent
particles and the beads, without any approximate coupling or stochastic forcing.
However, the solvent particles themselves do not directly interact with each other,
that is, they can freely pass through each other as for an ideal gas. Deterministic
collisions between the solvent particles are replaced with momentum- and energy-
conserving stochastic collisions between nearby solvent particles. This gives realistic
hydrodynamic behavior and fluctuations in the solvent, with tunable viscosity and
thermal conductivity, but without internal fluid structure. A recent modification of
the DSMC algorithm can be used to achieve a non-ideal equation of state for the
stochastic solvent that is thermodynamically-consistent with the density fluctuations
[43].
Hard-sphere models of polymer chains have been used in EDMD simulations for
some time [139, 107, 101]. These models typically involve, in addition to the usual
hard-core exclusion, additional square well interactions to model chain connectivity.
Recent studies have used square well attraction to model the effect of solvent quality
[104]. Even more complex square well models have been developed for polymers
with chemical structure and it has been demonstrated that such models, despite their
apparent simplicity, can successfully reproduce the complex packing structures found
in polymer aggregation [101, 107]. Here we use the simplest model of a polymer
chain, namely, a linear chain of Nb particles tethered by unbreakable bonds. This is
64
similar to the commonly-used freely jointed bead-spring FENE model model used in
time-driven MD. The length of the tethers has been chosen to be 1.1Db, where Db is
the diameter of the beads.
Several particle methods for hydrodynamics have been described in the literature,
such as MD [13], dissipative particle dynamics (DPD) [111], and multi-particle col-
lision dynamics (MPCD) [119, 89]. Molecular dynamics is the most accurate model
of the fluid structure and dynamics, however, it is very computationally demanding
due to the need to integrate equations of motion with small time steps ∆t and calcu-
late interparticle forces at every time step. The key idea behind DSMC is to replace
deterministic interactions between the particles with stochastic momentum exchange
(collisions) between nearby particles. The standard DSMC [7] algorithm starts with
a time step where particles are propagated advectively, r′
i = ri + vi∆t, and sorted
into a grid of cells. Then, a certain number Ncoll ∼ ΓcNc(Nc − 1)∆t of stochastic
conservative collisions are executed between pairs of particles randomly chosen from
the Nc particles inside the cell. For mean free paths comparable to the cell size, the
grid of cells should be shifted randomly before each collision step to ensure Galilean
invariance. The collision rate Γc and the pairwise probability distributions are chosen
based on kinetic theory.
In SEDMD the polymer chains and the bead-solvent interactions are handled us-
ing hard-sphere event-driven molecular dynamics (EDMD) [6, 44, 139, 104] instead
of the time-driven MD (TDMD) widely used for continuous potentials. The essential
difference between EDMD and TDMD is that EDMD is asynchronous and there is no
time step, instead, collisions between hard particles are explicitly predicted and pro-
cessed at their exact (to numerical precision) time of occurrence. Since particles move
65
along simple trajectories (straight lines) between collisions, the algorithm does not
waste any time simulating motion in between events (collisions). SEDMD combines
time-driven DSMC with EDMD by splitting the particles between ED particles and
TD particles. Roughly speaking, only the polymer beads and the DSMC particles
surrounding them are treated asynchronously as in EDMD. The rest of the DSMC
particles that are not even inserted into the event queue. Instead, they are handled
using a time-driven (TD) algorithm very similar to that used in traditional DSMC.
In three dimensions, a very large number of solvent particles is required to fill the
simulation domain. The majority of these particles are far from the polymer chain
and they are unlikely to significantly impact or be impacted by the motion of the
polymer chain. We therefore approximate the behavior of the solvent particles suffi-
ciently far away from any polymer beads with that of a quasi-equilibrium ensemble.
In this ensemble the positions of the particles are as in equilibrium and the velocities
follow a local Maxwellian distribution whose mean is the macroscopic local veloc-
ity. These particles are not simulated explicitly, rather, we can think of the polymer
chain and the surrounding DSMC fluid as being embedded into an infinite reservoir
of DSMC particles which enter and leave the simulation domain following the appro-
priate distributions. Using such open or stochastic boundary conditions dramatically
improves the speed, at the cost of small errors due to truncation of hydrodynamic
fields. This truncation can be avoided by coupling DSMC to a continuum fluctuating
hydrodynamic solver [155].
We have made several runs for different polymer lengths and also bead sizes. One
set of runs used either Nb = 25 or 50 large beads each about 10 times larger than
a solvent particle. Another set of runs used either Nb = 30 or 60 small beads each
66
identical to a solvent particle, with faster execution but nearly identical results. In the
simulations reported here we have used rough wall BCs for collisions between DSMC
and non-DSMC particles [42]. This emulates a non-stick boundary condition at the
surface of the polymer beads. Using specular (slip) conditions lowers the friction
coefficient, but does not qualitatively affect the behavior of tethered polymers. All of
the runs used open boundary conditions, where about 153 DSMC cells around each
bead were explicitly simulated. Note that for (partially) collapsed polymer chains the
total number of explicitly simulated cells is much smaller than 153Nb. The Nb = 30
runs were run for about 6000τ0 relaxation times, and such a run takes about 6 days
on a single 2.4GHz Dual-Core AMD Opteron processor. Even for such long runs
the statistical errors due to the strong fluctuations in the polymer conformations are
large, especially for correlation functions at long time lags t > τ .
5.4 Results
The main goal of our paper is to reinvestigate the tethered chain problem through
extensive long time simulations (thousands of longest relaxation time of the teth-
ered polymer, τ) involving different representations of polymer and solvent, including
Brownian dynamics [74] (BD), the Lattice Boltzmann method [5] (LBM), and a re-
cent Stochastic Event-Driven Molecular Dynamics [42] (SEDMD) algorithm. In this
section we present comparison results from our simulations. More extensive results
for the tethered polymer problem obtained using the SEDMD algorithm are presented
in Ref. [42]. Since the three different methods that we use give similar results and
Brownian Dynamics is the fastest methodology, the majority of the results we present
67
will be from BD simulations with Nb = 11 (Ns = 10), unless otherwise indicated. Of
the three methods used here, LB is the slowest and thus the LB results are of more
limited duration. We emphasize that direct computational comparison between the
methods is unfair. Most significantly, the LB runs use periodic boundary conditions
and have to fill the whole simulation domain with explicit solvent (lattice points).
By contrast, the SEDMD runs use open boundaries and thus use much less explicit
solvent, whereas the Brownian dynamics does not use an explicit solvent at all.
Doyle et al. proposed a cyclic dynamics mechanism for a tethered polymer chain in
shear flow (Fig. 5.1) based on Brownian dynamics simulation results [46]. According
to this scenario, when thermal fluctuations cause motion in the gradient direction
x (from state 1 to state 2), the chain is driven away from the wall and experiences
higher hydrodynamic drag. This leads to further stretching and an increase of the
extension in the flow direction z (state 3). Due to the finite extensibility of the chain,
the extension in the z direction is finite and depends on the shear rate and chain
properties. After stretching, the coupled torque of the hydrodynamic drag and spring
forces will rotate the chain towards the wall (state 4). As the chain get closer to the
wall, the flow velocity decreases and entropic recoiling becomes dominant, resulting in
a decrease of the z extension (state 1). The tethered chain could take other dynamical
paths than following the one described above, such as restretching or recoiling after
state 2 by random motion in −x and −z direction, respectively.
In Fig. 5.2 we show the probability distribution function (pdf) ρ(z, x) of the end
bead in the z − x plane at Wi = 0 and Wi = 2 for the three different methods. The
results are presented in dimensionless units by normalizing the unit of length by the
average radius of gyration in the x direction at Wi = 0. Here, again, we want to
68
-1 0 1 2 3 40.0
0.5
1.0
1.5
2.0
2.5
3.0
(a) Wi = 0
( )b Wi = 2
-3 -2 -1 0 1 2 30.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
-3 -2 -1 0 1 2 30.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
-3 -2 -1 0 1 2 30.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.000.0330.0650.0980.130.160.200.230.26
-1 0 1 2 3 40.0
0.5
1.0
1.5
2.0
2.5
3.0
-1 0 1 2 3 40.0
0.5
1.0
1.5
2.0
2.5
3.000.0810.160.240.330.410.490.570.65
Figure 5.2: Probability distribution of the end bead of the tethered DNA moleculein a dimensionless x − z plane at Wi = 0 and Wi = 2. The visible differences canlikely be attributed to the differences in the boundary conditions between the differentmethods, as well as the different elasticity of the chains. (Left) Brownian dynamics.(Middle) Stochastic Event-Driven Molecular Dynamics. (Right) Lattice BoltzmannMethod.
69
emphasize that we are not expecting perfect match between methods. In particular,
the different methods implement different effective boundary conditions at the wall
surface. In Brownian dynamics, an essentially reflective boundary condition appears,
while in the case of a hard-sphere chain a perfectly reflective boundary condition is
appropriate. For the LB runs an intermediate case appears, where the repulsion from
the wall is stronger than hard spheres but still finite-ranged. These boundary effects
are clearly visible in the results in Fig. 5.2, where the BD results show a depletion
layer near the wall where as the SEDMD and LB results show the bead spending
more time near the wall.
In Fig. 5.3 we compare the dependence of the relaxation times τx/y/z along the
three different axes on the flow rate among the three different methods. The figure
shows reasonable agreement between the different techniques, especially considering
the large errors inherent in determining relaxation times. We calculate the relaxation
times by fitting an exponential decay to the intermediate portion of the autocorre-
lation function 0.2 < C(t) < 0.8 of the position of the end bead along the three
coordinate axes. The LB calculations use periodic boundaries with a narrower box
in the spanwise (y) direction than in the streamwise (x) direction, which makes the
relaxation times τx(Wi = 0) and τy(Wi = 0) unequal, as they must be by symmetry.
We have scaled τy(Wi) (the shorter axes) in the LB results by a constant factor so as
to correct this strong boundary effect at Wi = 0. Among the three relaxation times,
the relaxation in the direction perpendicular to the wall τz is the shortest, even for
no flow. Note than in this work, following Ref. [42], the relaxation time along the
flow direction τx is used to define the internal relaxation time and thus Wi when
comparing among the different methods. Note also that it is τx that seems to get
70
0 1 2 3 4 5 6
Wi
0
0.5
1
1.5
2
τ /
τ 0
xzy
circles=BD, squares=SEDMD, diamonds=LB
0 1 2 3 4 5 60
0.5
1
1.5
2
τz / τ
x
τy / τ
x
Figure 5.3: Dependence of the dimensionless relaxation time τ(Wi)/τ(Wi = 0) of thetethered chain along the three coordinate axes as a function of dimensionless flowrate Wi. The inset shows the ratios of the different relaxation times as a function ofWi.
71
most strongly reduced as Wi increases.
To study the time scale associated with the fluctuating process (cycle) quantita-
tively and to find the correlation between different chain segments, we calculated the
cross-correlation functions (CCF) of beads’ positions. We also calculated the power
spectral density (PSD) in search of periodicity. The CCF and PSD are the natural
tools for examining the relationship between two time dependent random variables
in the time and frequency domain respectively. The mean-removed CCF of two time
series α(t) and β(t) is defined as
Cαβ(T ) =E[(α(t + T )− α)(β(t)− β)]
σασβ
(5.10)
where α = E(α) is the mean, σ2α = E(α2)−[E(α)]2 is the standard deviation, and T is
the time lag. A significant peak in the CCF at lag T indicates that α(t) is correlated
to β(t) when delayed by time T . In the frequency domain, the PSD is the norm of
the Fourier transform of the CCF,
Sαβ(ν) =
∥
∥
∥
∥
∫ ∞
−∞Cαβ(T )exp(−2iπνT )dT
∥
∥
∥
∥
(5.11)
Note that this is the standard definition used in the engineering literature, and here
the frequency ν = 1/T is actual frequency (inverse period) rather than angular fre-
quency ω = 2πν. To produce a PSD with accurate sampling around interesting
frequencies, long simulation times and a high sampling frequency are essential. We
examined various choices of variables to represent the motion of the chain and have
found little qualitative difference between them. We have chosen the position of the
end bead rNb= (x, y, z) as be the best option [42]. Extensive computational efforts
72
have been undertaken to determine the CCF and PSD of the end bead coordinates
as function of chain length N and shear flow parameter Wi.
The CCF Czx(t) of the end bead at various Wi is shown in Fig. 5.4(a). The shape
of the CCF is consistent with the cyclic dynamics mechanism proposed by Doyle.
Clearly, in the absence of flow, Wi = 0, the movements in the x and z directions are
uncorrelated on all time scales. When shear flow is introduced, the movements in flow
direction and gradient direction are coupled together due to the nature of the flow
and the finite extensibility of the chain, as reflected in the rise of a prominent peak in
the CCF. When thermal motion in +x direction occurs, the chain will be stretched
with an increase in +z, which leads to a positive correlation. Similarly, when motion
in −x direction is introduced, the chain will recoil in the −z direction as the drag
decreases, which also leads to a positive correlation. As expected, the larger the
shear rate, the greater the correlation. There’s only one significant peak in the long
time correlation function, shown in the inlet of Fig. 5.4(a), which suggests that all
correlations are short-lived and not periodic. Turning attention to the correlation
between different chain segments, Fig. 5.5 shows the CCFs of several beads along
the chain at Wi = 2. One striking feature is that for beads sufficiently far from the
tether all curves pass the time axis at the same time lag. The fact that all CCFs
have the same shape indicates that a common movement pattern exists for the whole
chain. The inset in Fig. 5.5(a) shows the CCFs of the x coordinates of different
beads. Although the correlation decays as the distance along the chain increases, it
confirms that all beads move in a cooperative manner, indicated by the fact that the
peak positions are all at zero time lag. The CCF for the end bead for various chain
lengths are compared in Fig. 5.5(b), to show that there is no fundamental difference
73
Figure 5.4: (a) Normalized cross-correlation functions (CCF) Czx(t) of the endbead’s coordinates in flow direction z and gradient direction x as a function of non-dimensional time, at various Wi for Ns = 10. The inset shows longer time lags.(b) Power spectral density (PSD) Szx(ν) of the end bead’s coordinates as a func-tion of non-dimensional frequency. The results are averaged over 30 runs for a totalsimulation time is 103τ , and 104τ for Wi = 5.
74
Figure 5.5: (a). CCFs of end beads’ coordinates at Wi = 2 for a chain with Ns = 10and simulation time is 1000 τ . The number in the legend is the bead label as shownin Fig. 5.1. The inset shows the CCFs for x coordinates of different beads to studythe correlation of the dynamics between different beads (similar results are obtainedfor the y axes). (b) CCFs of end bead as function of chain length at Wi = 5. Thenumber in the legend is the number of springs Ns in the chain. The inset shows longertime lags.
75
between different chain lengths, ranging from 20 µm (Ns = 10) to 80 µm (Ns = 40).
We have also established that these results are insensitive to the cut-off distance and
the magnitude of the repulsive potential between the wall and chain segments.
In Fig. 5.6 we compare the cross-correlation function Czx(t) at Wi = 2 among
the three different methods: Brownian Dynamics, Lattice-Boltzmann, and Stochastic
Event-Driven Molecular Dynamics. In particular, our goal is to verify the pervasive
assumption that the dynamics of polymer chains in shear flow is essentially universally
quantitatively determined by Wi for a wide range of flexible chains. Furthermore, it is
important to cross-validate the different methods against each other, given that each
of them makes certain assumptions and has somewhat different range of applicabil-
ity. The results in Fig. 5.6 indeed show reasonable agreement between the different
methods. Perfect agreement is not expected because the polymer models are different
among the different methods. The cross-correlations we measure are not consistent
with periodic motion. The PSD calculation does not show discernible peaks either,
as shown in Fig. 5.4(b). All that we can reliably extract from the results is that
the response of the chain to a large thermal fluctuation (the “cycle”) is reproducible
for short times, and we find no evidence of sustained correlations (oscillations) at
times longer than the internal relaxation time of the chain. For a free chain in shear
flow, where rotations of the chain are possible, one can count the number of tumbling
events per unit time and define that as a cycling time. The distribution of the delays
between successive tumbling events is itself important. If this distribution is sharply
peaked, that would be consistent with a periodic motion with a well-defined period.
If the distribution is exponential, this would indicate a Poisson-like tumbling process.
Several recent works have proposed an exponential distribution for the delay between
76
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
t / τ
-0.1
0
0.1
Czx
BD Wi=2.0 (N=20, error~0.005)
SEDMD Wi=1.8 (N=30, error~0.01)
LB Wi=2.0 (N=25, error~0.05)
Figure 5.6: Comparison of the cross-correlation function Czx(t) at Wi = 2 among thethree different methods: Brownian Dynamics (30 runs about ∼ 1000τ long), Lattice-Boltzmann (run is ∼ 600τ long), and Stochastic Event-Driven Molecular Dynamics(10 runs about ∼ 1000τ long).
77
successive tumblings [57, 34, 156]. Furthermore, the tumbling time was found to be
related to the internal relaxation time of the chain [34, 156]. For tethered chains, we
cannot even identify and count a unique event such as tumbling and thus we cannot
extract a repetition frequency for the “cycle”.
In Appendix C, we analytically calculate the CCF for a Brownian particle tethered
to the origin with a harmonic spring and subjected to shear flow. This simple dimer
model qualitatively reproduces the features we see in the CCF for the tethered chains,
namely, a single peak at t ∼ τ of width ∼ τ and height ∼ Wi. Better quantitative
agreement is obtained when a nonlinear spring and a hard wall surface are also in-
cluded (without hydrodynamics). The PSD for the dimer model shows no peaks and
there is only a single time-scale in the dynamics, namely, the intrinsic relaxation time
τ . Furthermore, the analytical form of the CCF shows that by a slight modification of
a tunable parameter one can obtain a CCF fully-consistent with our numerical results
for longer chains. This analytical CCF has an analytical PSD that does show a broad
peak at small frequencies ντ ∼ 0.1, very similar to the previously reported peaks
used to justify the claims to periodicity in the chain motion [127, 37, 38]. This peak
is weak and broad even when plotted on a logarithmic axes and its exact shape and
maximum will vary depending on the particular model, variables used in calculating
the PSD, Wi, the definition used for calculating τ and Wi, etc. We therefore believe
that its interpretation as evidence of periodic motion is not justified.
The calculations in Appendix C for a dimer in shear flow also demonstrate that
a qualitatively similar behavior is observed even without hydrodynamic interactions.
Our results from Brownian Dynamics simulations in the free-draining limit confirm
this and show that the HI do not affect the results significantly, so long as the relax-
78
ation time is recalculated when computing Wi. In the tethered case, we believe that
the competition between frictional and elastic restoring forcing dominates and the
hydrodynamic interactions are a weak perturbation. Therefore, it is not surprising
that the proper inclusion of hydrodynamic interactions is not essential for the teth-
ered polymer problem, as reasoned theoretically for a free chain in shear flow in Ref.
[156].
5.5 Conclusions
We studied the dynamics of a polymer molecule tethered to a hard wall and subjected
to a shear flow. We found consistent results among three methods utilizing different
representations of the solvent, Brownian Dynamics (BD), Lattice-Boltzmann (LB),
and Stochastic Event-Driven Molecular Dynamics (SEDMD). Specifically, BD im-
plicitly represents the solvent, LB explicitly represents the solvent flow on a discrete
lattice, and SEDMD utilizes a particle-based solvent. The three methods also utilized
different polymer chains, namely, the BD simulations used a worm-like chain, the LB
simulations used a FENE-LJ chain, and for SEDMD we used a tethered chain of hard
spheres.
The correlation functions of the position of the end bead question the existence of
periodic motion, as previously suggested. The cross-correlation function between the
bead positions along the flow and gradient directions shows a single peak indicative
of a fluctuation-dissipation cycle of duration comparable to the relaxation time of
the polymer. The corresponding Fourier representation, the power-spectral density,
shows no peaks. We find that neither the chain length of the polymer N , nor the
79
dimensionless shear rate Wi, qualitatively alter the results, and in the Appendix C we
give some calculations for a very simple model of a dimer in shear flow that reproduces
the essential features of the observed peak in the cross-correlation function.
While our conclusions are rather different from other authors, our results are sta-
tistically consistent with those presented in the literature. Specifically, the shape and
position of the peaks in the cross-correlation functions are very similar to reported re-
sults, however, we did not observe large oscillations in the CCFs previously identified
as signatures of periodic motion [37]. We believe that this is due to the requirement of
very long simulation times to obtain good statistics for the time-correlation functions
at long time lags, as necessary to establish periodicity. Not all previous studies have
been able to reach sufficiently long simulation times. Another important point we
clarified is that maxima in the power-spectral density does not necessarily indicates a
periodic motion, which we demonstrate in Section C using an analytic dimer model.
Namely, an analytical shape is suggested by the dimer calculations that can exhibit
peaks very similar to those reported in the literature through small adjustments of
a tunable parameter, whose appropriate value likely depends on details of the model
used and the exact variables used in the calculations of the power spectrum. Fur-
thermore, different if not conflicting ways have been used to define and calculate the
“cycling time”, without properly distinguishing between the duration of a cycle and
the interval between cycles. Even more importantly, the very concept of a cycle in the
chain motion as a well-defined countable event, analogous to the case of a free chain
in shear flow, should be questioned. Our results are consistent with a simple tradi-
tional picture of continuous thermal fluctuations dissipated by deterministic friction,
leading to exponentially-decaying correlation functions.
80
Chapter 6
Flow-driven DNA dynamics in
complex geometries
In this chapter, we consider the dynamics of a flow-driven DNAmolecule in micro/nano-
fluidic devices. In this case, the configuration-dependence of the mobility tensor can-
not be ignored and the solvent velocity field is in general non-linear on the length scale
of the molecule. We present an immersed boundary method that allows fast Brown-
ian dynamics simulation of polymer chains and other particles in complex geometries
with fluctuating hydrodynamics.1 This approach is applied to study the dynamics
of a flow-driven DNA molecule through a nanofluidic slit with an embedded array
of nanopits. We investigate the dynamics of the DNA molecule as a function of the
Peclet number and chain length, as well as the influence of hydrodynamic interactions
by comparing with free draining simulation results.
1This chapter is based on the manuscript: Y. Zhang, J. J. de Pablo, and M. D. Graham, submittedto J. Chem. Phys. (April 12, 2011)
81
6.1 Introduction
Transport of polymer solutions in constricted spaces [1] is a long-standing research
topic with many applications including polymer enhanced-oil-recovery, size exclusion
chromatography [153] and gel-electrophoresis [151] and, recently, single DNAmolecule
analysis using micro- and nano-fluidic devices [140, 125]. In many fluidic devices
designed for separation, manipulation, and sequencing of DNA, the critical dimension
of the constriction approaches the radius of gyration of the polymer molecule or
smaller. In particular, capturing the interaction between polymer dynamics and fluid
motion in a confined geometry is essential to a proper description of these devices.[1]
The purpose of the present work is to introduce a computational approach to the
dynamics of polymers in complex confined geometries, and to apply the approach to
an interesting recent set of experiments on the flow of DNA solutions over arrays of
nanopits [36].
Many devices with well-defined microstructure have been proposed for DNA ma-
nipulation, especially for separation and stretching purposes using electrophoresis[45].
By contrast, much less attention has been paid on pressure (flow) driven DNA dy-
namics in microfabricated devices [141, 36, 143]. There are significant differences
between the electrophoresis case and the pressure driven case. Consider for example
DNA through a slit geometry. In electrophoresis, in a uniform field, the velocity of
each DNA segment is the same everywhere in the channel; by contrast, the unper-
turbed velocity profile is parabolic across the channel in the pressure driven case at
low Reynolds number, and this velocity gradient can give rise to interesting transport
phenomena, such as Taylor dispersion [141]. Furthermore, hydrodynamic interactions
(HI) between objects such as polymer segments in an unconfined domain are long-
82
ranged, leading to strong many-body effects, for example, Zimm scaling of the self-
diffusion coefficient of a polymer chain. In confined geometries, the long-ranged nature
of hydrodynamic interactions changes substantially, leading to significant changes in
polymer dynamics. [1]. In a slit geometry, for example, the velocity field due to a
point force perpendicular to the walls decays exponentially. For a force parallel to the
walls, the velocity field has a parabolic form in the wall-normal direction and decays
as 1/r2. This is still a slow decay, but the symmetry of the flow leads to cancellations
upon averaging that result in screening [9, 144, 14]. Both experiments and simula-
tions of the diffusion of long flexible DNA molecules in slits [31, 78, 75] are consistent
with screening of hydrodynamic interactions on the scale of slit height. Nevertheless,
on smaller scales, HI are not screened and lead to changes in segment mobilities and
boundary effects such as cross-stream migration, which leads to depletion layers much
larger than the equilibrium chain size [77, 90, 30, 68, 29, 81].
Several recent computational studies investigated the effect of HI on dynamics
of polymer molecules in complex geometries.[54, 73, 93, 66, 154, 29, 56] Among the
problems studied in these works, forced translocation of a polymer through a nanopore
has received substantial attention because of its potential applications to rapid DNA
sequencing. In this problem of translocation, one end of a polymer molecule is placed
at the entrance to a nanopore embedded in a membrane, and then the polymer
molecule is pulled through the pore by a constant force exerted on that leading end
segment or on the chain segments in the pore. Using different simulation methods
and/or different polymer models, Izmitli et al. [73], Hernandez-Ortiz et al.[66], and
Fyta et al.[54] all found that HI shortens the translocation time and alters the scaling
exponents for the power-law dependence of the translocation time on polymer length
83
[54, 73]. Using the general geometry Ewald-like method (GGEM)[67], Hernandez-
Ortiz et al. studied the effect of HI on polymer solutions with finite concentration
flowing over a channel with grooves oriented perpendicular to the flow direction was
considered.[69] At low concentration and high Weissenberg number Wi, which is
defined as the polymer relaxation time times fluid deformation rate, the groove was
almost completely depleted of polymer chains. At concentration approaching overlap,
the concentration difference between the bulk and groove is substantially reduced, but
only if hydrodynamic interactions were included in the simulation. All these studies
suggest that HI is not only important but essential in the simulation of polymer
solutions in complex geometries.
The specific system of interest in the present work is a nanoslit device that exploits
partial confinement to manipulate a DNA molecule. [116, 36] In the experiments, the
height of the channel is H = 100 nm, and the bottom wall of the slit contains square
nanopits (100 nm deep and 500–1000 nm wide). When a solution of λ−DNA, which
has a radius of gyration Rg of 770 nm, is introduced into such a system, the DNA
is highly confined as reflected by the confinement ratio Rg/H = 7.7. Because the
confinement is weaker in the regions of the slit where there is a pit, DNA segments
are preferentially found there. This entropic well tends to pin sections of the chain in
these regions (Fig. 6.1(b)). When multiple pits are present (and not too far apart, 1-2
µm in experiments), stretched sections of DNA will span the more confined regions
between pairs of pits. When a pressure drop is applied across the channel, the DNA
molecule is transported downstream by the flow, hopping between neighboring pits
(Fig. 6.1(c)). When a molecule gets trapped in a pit, a free end of the molecule
first diffuses out of the pit, then is dragged downstream by the flow. When the free
84
end hits the next pit, the rest of the chain is dragged along into the second pit.
In Ref. [36], an equilibrium model was proposed, based on the hypothesis that the
dynamics of DNA at low Peclet number, which is defined as the ratio between the
diffusive and convective time scale, is thermally activated, and thus the Kramer’s
theory of barrier crossing could be applied. There are two important issues that
were not addressed in Ref. [36]. First, no direct evidence was shown to support the
hypothesis that the dynamics is thermally activated. Specifically, no analysis about
the distribution of the residence time in each pit as a function of Peclet number
(fluid velocity) was performed. Second, how the chain dynamics change with DNA
size was not systematically examined. We address these issues using an accelerated
Brownian dynamics method for modeling the hydrodynamics of polymer solutions
in complex geometries. This method is based on Green’s function solutions of the
Stokes equations, combined with a fast Stokes solver and a variant of the immersed
boundary method.
This paper is organized as follows: A general discussion of the wide range of tech-
niques for modeling the hydrodynamics of polymer solutions in complex geometries is
given in Sec. 6.2. In Sec. 6.3, we present the DNA model, governing equations, and
the new immersed boundary method that we have developed to calculate fluctuating
hydrodynamics in complex geometries. Sec. 6.4 gives results from the application of
the algorithm to the problem of flow-driven DNA dynamics in a nanofluidic device
with embedded nanopits array, and some concluding remarks are given in Sec. 6.5.
85
Figure 6.1: DNA flowing through a nanoslit with embedded nanopit arrays (repro-duced with permission from Ref. [36]. Copyright 2009 by the Institute of Physics) (a)Scanning electron micrograph of typical 1× 1 µm square pits embedded in a nanoslitat 1 µm intervals. (b) Epifluorescent images of stained λ-DNA molecules confined ina 107 nm deep slit embedded with 1× 1 µm pits spaced 2 µm apart. (c) Fluorescentimages of λ-DNA travelling across a nanopit array under an applied pressure of 40mbar in the same device as shown in (b).
86
6.2 Methods for hydrodynamics of confined poly-
mer solutions
In this section we give a brief overview of various methodologies for modeling hydro-
dynamics of polymer solutions in complex geometries. For theoretical analysis, the
reader is referred to Graham.[1]
The various methods for computational hydrodynamics of polymer solutions can
be divided into two major categories. The first category is purely continuum methods
that use constitutive equations for the polymer solution. These models only apply at
macroscopic scales, when the number of polymer chains in an elementary fluid flow
volume is large, so that statistical averages of the chain conformations can be used
as parameters in constitutive models of the time-dependent stress as a function of
the strain rate history. The second major category of methods explicitly simulates
the motion of each polymer chain using a coarse-grained (bead-spring) model of the
chains. Multibead representations of the chains are capable of complex chain confor-
mations but require models for the bead-bead interactions. There are also two major
types of approaches for dealing with the fluid dynamics of the solvent and thus the
hydrodynamic interactions between chain segments. One approach, exemplified by
Brownian dynamics (BD) and Lattice Boltzmann (LB) methods, uses a continuum
model (Stokes, Navier-Stokes or Boltzmann equation) to represent the solvent. The
other uses a discrete, particle-level, model to represent the solvent; examples include
full atomistic molecular dynamics (MD), Stochastic Rotational Dynamics (SRD), and
Dissipative Particle Dynamics (DPD). As we are interested in long-time (¿ 1 second)
dynamics of a single polymer molecule in complex geometries, this brief review focuses
87
on mesoscopic methods, namely, BD, LB, SRD, and DPD, especially their different
treatments of hydrodynamics, boundary conditions, and Brownian fluctuations.
In Brownian dynamics simulations of polymer molecules in unbounded or peri-
odic domain, the HI are usually incorporated using analytical Green’s functions for
Stokes’ equations: I.e. the beads exert point forces on the fluid and Green’s functions
are used to determine the fluid velocity at any other bead position. For a system with
N beads, these interactions are incorporated in theN×N mobility matrix[105]. While
for true point forces, the Green’s function for the Stokes equations – the Oseen tensor
– is appropriate, in Brownian dynamics simulation it is necessary to use regularized
point forces. The most common regularization is the Rotne-Prager-Yamakawa tensor
[120, 157], while in the present work a different, computationally convenient regu-
larization is used. Corresponding to the mobility matrix representation of the HI, a
matrix formulation of the fluctuation-dissipation theorem – a factorization of the mo-
bility tensor – must be implemented. Ermak and McCammon suggested a Cholesky
factorization which requires O(N3) operations[49], while Fixman [53] observed that
the square root matrix of the mobility matrix can be accurately approximated via
Chebyshev polynomials, and this algorithm brings down the computation complexity
to roughly O(N2) or O(N2.25).[76] Another advantage of Fixman’s algorithm, one
that is essential to this study, is that it works even when an analytical form of the
Green’s function is not available, as is the case for flow in complex geometries. For
periodic domains, the scaling can be dramatically improved, to O(N logN), by using
particle-mesh Ewald (PME) methods, which are based on Hasimoto’s solution for the
Green’s function for Stokes flow driven by a periodic array of point forces [65, 16].
BD can also be efficiently applied to nonperiodic geometries. In simple cases
88
like flow in a half-plane bounded by a no-slip wall, the exact Green’s function for
that geometry can be used. To handle more complex boundary conditions, such as
indentations or bumps on walls, it requires incorporating other methods such as finite
element methods [78, 69], which can be done in the framework of the general geometry
Ewald-like method (GGEM) [67], which also has O(N logN) or O(N) scaling. In
the present work, we combine GGEM with a variant of the Immersed Boundary
Method.[110, 97, 114]
The lattice Boltzmann method is a mesoscopic approach to fluid flow calculation
and is based on a discrete version of the Boltzmann equation with enough detail to
recover hydrodynamic behavior [47]. The LBM avoids the some of the computational
complexities of Navier-Stokes (NS) solvers and is straightforward to parallelize effi-
ciently. Using a Chapman–Enskog expansion, the LB equation can recover the NS
equations for small finite Mach and Knudsen numbers, and, within these limits it is
second-order accurate in space and time. Coupling between the LB for the solvent
and the molecular dynamics for the polymer is achieved through Stokes drag forces.
The Brownian fluctuation is introduced via random fluctuations added to the stress
tensor and the beads dynamics directly. [47] Usually, the no-slip boundary is intro-
duced by a bounce-back collision rule in which incoming fluid particles are reflected
back towards the nodes from which they originated, and this results in no-slip bound-
ary conditions. For complex boundaries, however, the boundary implementation may
not be trivial [85, 33, 28]. The Lattice Boltzmann-molecular dynamics scheme has
been successfully applied to the problems of polymer translocation through a pore
[54, 73, 93], polymer cyclic dynamics near a solid wall [158], and polymer migration
in a square channel [81].
89
Stochastic rotation dynamics is a relatively recent particle-based mesoscopic sim-
ulation method. In the SRD algorithm, the fluid is modelled as a system of point
particles. The algorithm consists of two steps, the streaming step and the collision
step.[80, 59] In the streaming step, the particles move ballistically during a certain
amount of time. In the collision step, the particles are sorted into cubic cells of a cer-
tain size. In each cell, the velocity vector of every particle is rotated an angle α about
a randomly chosen axis, relative to the center-of-mass velocity of all the particles
within that particular cell. In addition, a random shift is added in order to ensure
Galilean invariance. The polymer segments are coupled to the fluid by including
them in the collision step. The streaming and collision steps are designed to conserve
mass, momentum, and energy. In the long time limit, correct hydrodynamics are
recovered. SRD also naturally contains Brownian fluctuations. To implement no-slip
boundary conditions in SRD, there are two approaches: the bounce-back scheme and
the stochastic scheme. The bounce-back scheme was first proposed by Malevanets
and Kapral [91], and is a direct analog of the no-slip boundary condition commonly
applied in LBM. Lamura and Gompper found that the simple bounce-back rule is
not sufficient to guarantee no-slip without corrections. [87] In the stochastic scheme,
proposed by Inoue et al., the velocity of the SRD particle that comes into a wall
cell is changed to a random velocity obeying a Maxwell-Boltzmann distribution[72].
This boundary condition, however, allows flow to penetrate small objects.[60] Watari
et al.[154] and Chelakkot et al.[29] have used this method to study the dynamics
of polymer solutions between two parallel planes, and qualitatively reproduced the
hydrodynamic migration phenomenon.
Dissipative particle dynamics coarse-grains the solvent molecules to obtain a sys-
90
tem of weakly repulsive (soft) spheres interacting with a mixture of conservative,
stochastic, and dissipative forces.[50, 64, 118] The conservative forces can be used to
reproduce the solvent equation of state, while the dissipative forces model viscous
friction. The stochastic forces act as a thermostat that ensures detailed balance and
correct thermal fluctuations in the DPD fluid. The method has great flexibility and
requires significantly fewer solvent particles and larger time steps than classical MD.
The influence of solid boundaries is typically very strong in DPD simulations. The
modeling of no-slip boundaries in the context of DPD has been addressed in the past
by freezing the particles that represent the solid boundaries[117], and by modifying
the repulsive forces[79]. While these efforts introduce minimal additional algorithmic
complexity to the standard DPD formulation, care has to be taken to avoid numer-
ical artifacts such as depletion of particles near the wall or artificial ordering of the
near-wall particles leading to large density fluctuations[112].
To study the dynamics of polymer solutions in complex geometries, the ability to
treat no-slip boundary conditions efficiently and accurately becomes important. In all
three mesoscopic models (LBM, SRD, and DPD), it is not a trivial task to correctly
implement the no-slip boundary conditions. Another difference between BD and the
three mesoscopic models is that those methods describe compressible inertial fluids
in which hydrodynamics need time to propagate through the solution; the Reynolds
and Mach numbers Re and Ma are finite. In Green’s function-based methods, on the
other hand, the condition Re ≪ 1, Ma ≪ 1 are satisfied explicitly through the use
of the Stokes equation.
In the present study, we propose an immersed boundary method-based Green’s
function method, where the boundary is represented by many regularized points and
91
the Stokes equation is solved on a Cartesian grid. The validity and the accuracy of
the new method are scrutinized by simulating laminar flow in a slit, uniform flow
around a solid sphere, and flow generated by a Stokeslet near a single wall.
6.3 Polymer model and simulation method
6.3.1 Model and governing equations
In this section, we present the coarse-grained model used in this work to study the
dynamics of polymer in complex geometries. We focus attention on a model of λ-
DNA[158, 76, 74], described by a bead-spring chain model composed of Nb beads
of hydrodynamic radius a = 77 nm connected by Ns = Nb − 1 entropic springs. a
is also the length unit used through out this work for nondimensionlization. Each
bead represents a DNA segment of 4850 base pairs, i.e., Nb = 11 corresponds to
a fluorescently stained λ-DNA, which has a contour length of 21 µm and radius of
gyration of 770 nm.[136, 137] The length unit in this work is a = 77 nm. The spring
connecting the beads i, j obeys a worm-like chain force law [94]
f sij =kBT
2bk[(1− |rj − ri|
Nk,sbk)−2 − 1 +
4|rj − ri|Nk,sbk
]rj − ri|rj − ri|
. (6.1)
Here, bk is the Kuhn length for DNA and Nk,s is the number of Kuhn length per
spring. The physical confinement, or the steric interaction between DNA and wall, is
92
taken into account through an empirical bead-wall repulsive potential of the form
Uwalli =
Awallb−1k δ−2
wall(hi − δwall)3, hi < δwall
0, hi > δwall.
(6.2)
Here hi represents the perpendicular distance of bead i from the wall, δwall is the cut-
off distance. In this work, we choose Awall = 50kBT/3 and δwall = 1a. The excluded
volume potential between two distinct beads is
Uevij =
1
2νkBTN
2k,s(
3
4πS2s
)3/2 exp(−3|ri − rj|2
4S2s
), (6.3)
where ν is the excluded volume parameter, and S2s = Nk,sb
2k/6.
All of the parameters a, bk, ν are the same as used in previous works[76, 74],
where it has been shown to successfully reproduce the static (radius of gyration, Rg)
and dynamic properties (diffusivity D and longest relaxation time τ0) of DNA with
contour length of 10− 126µm.
6.3.2 Governing equations
We are concerned with the numerical simulation of the Brownian motion of a single
DNA molecule immersed in an isothermal incompressible Newtonian solvent in the
limit of zero Reynolds number. When the DNA molecule is represented by the bead-
spring model as described above, the equation of motion for each bead is determined
by the balance of Stokes drag on the bead due to the surrounding fluid, the Brownian
forces exerted by the fluid on the bead, and other non-hydrodynamic, non-Brownian
intra/inter molecular forces. The force balance on the beads leads to a stochastic
93
differential equation which governs the dynamics of the chain and can be solved
numerically[105]:
dR = (U∞ +M · F+ kBT∂
∂R·M)dt +
√2B · dW, (6.4)
B ·BT = kBTM. (6.5)
Here R is the vector containing all bead positions ri, kB is the Boltzmann constant
and T is the absolute temperature. The vector containing total non-Brownian, non-
hydrodynamic forces acting on the beads is denoted F. From Eq. 6.4 we can see that
there are four contributions to the change of chain configuration after a time step
dt: unperturbed flow velocity U∞ in complex geometries, perturbed velocity M · F
due to the polymer forces, drift due to the configuration dependence of the mobility
tensor ∂∂R
·M, and Brownian displacement√2B ·dW. The unperturbed flow velocity
U∞ at R is the velocity when DNA is absent in the system.
As noted, the second term in Eq. 6.4, M · F, is the velocity generated by the
motion of the polymer beads. It arises from the fact that beads immersed in a fluid
generate flows as they move due to various forces, and similarly they move in response
to fluid motion through the Stokes drag. As described in last section, we treat bead i
of the bead-spring chain as a sphere of hydrodynamic radius a. Then the relationship
between the bead velocity ui and the drag force it exerts on the fluid is given by
the Stokes law fi = ζ(ui − u(ri)), where ζ is the bead friction coefficient ζ = 6πηa,
u(ri) is the fluid velocity at the bead position, and η is the fluid viscosity. (Note
that the finite size of the bead only arises in the friction coefficient.) Through these
hydrodynamic interactions (HI), beads interact with each other and with the walls
94
of the confining geometry. The mobility tensor M, will be discussed in detail in Sec.
6.3.3.
For systems with configuration-dependent mobility tensors, the third term in Eq.
6.4 has to be included to obtain the correct dynamics. This is because the configura-
tion dependence of the mobility tensor results in a mean drift with a mean velocity
∂∂R
·M.[61] In complex geometries, the expression for M is lacking. Thus to solve Eq.
6.4, we need to turn it into a derivative-free form, and this will also be discussed in
Sec. 6.3.3.
The tensor B gives the magnitude of the Brownian displacement of the polymer
beads, and is coupled to M by the Fluctuation-Dissipation theorem (Eq. 6.5). The
vector dW is a random vector composed of independent and identically distributed
random variables according to a real-valued Gaussian distribution with mean zero
and variance dt. We will use the Chebyshev approximation given in Sec. 6.3.4 to
calculate B · dW.
6.3.3 Mobility tensor and Fixman’s midpoint algorithm
To illustrate the components of the mobility tensor, we consider a two-bead chain
(a dumbbell) in an unbounded domain, neglecting Brownian effects for the moment.
Here the force balance shows that the fluid velocities u(ri) experienced by bead i are
given by
u1
u2
=
u∞1
u∞2
+
1ζI G∞(r1 − r2)
G∞(r2 − r1)1ζI
·
f1
f2
, (6.6)
95
where G∞(r) = 18πηr
(I + rr/r2) is the Oseen-Burgers tensor, the free-space Green’s
function for Stokes’ equations, r = |r|, and I is the identity matrix. The quantity
G∞(r1 − r2) · f2 gives the velocity at position r1 generated by the force f2 exerted by
the particle at r2 on the fluid. We can write above expression succinctly as,
U = U∞ +M · F. (6.7)
In a confined geometry, a wall correction GW (r1, r2) has to be added to the mobility
tensor M, and it becomes
M =
1ζI+GW (r1, r2) G∞(r1 − r2) +GW (r1, r2)
G∞(r2 − r1) +GW (r1, r2)1ζI+GW (r1, r2)
. (6.8)
In general geometries, an analytical expression for GW (r1, r2) is not available, but
has to be calculated numerically.[78] In particular, we will see that it is not M that
is needed, but rather, the product M · F, which is the velocity generated by the
total non-Brownian, non-hydrodynamic forces acting on the beads. Using an accel-
erated Immersed Boundary Method, we can calculate M · F for polymer in complex
geometries efficiently, as discussed in Sec. 6.3.5.
Fixman proposed a method to numerically integrate Eq. 6.4 without calculating
the derivative term ∂∂R
·M.[52] The idea is to approximate the first-order derivative
96
with a finite difference approximation, and transform Eq. 6.4 into
R∗ = R+√2B(R) ·∆W,
∆R = [U∞ +M(R) · F]∆t +
√2
2kBTM(R∗) · [(B−1(R))
T ·∆W]
+
√2
2B(R) ·∆W, (6.9)
where ∆W is a vector of independent Gaussian random variables with variance ∆t.
A derivation of this algorithm is given in Appendix D.
6.3.4 Chebyshev approximation
In Eq. 6.9, there are three matrix-vector multiplications that need to be evaluated:
M · x, (B−1)T · x, and B · x, where x may be any real vector. The Brownian ten-
sor B is coupled to the mobility tensor M by the Fluctuation-Dissipation theorem
kBTM = B ·BT. A common choice for B is the square root matrix S, satisfying
kBTM = S · ST with S = ST. With that, we only need to evaluate M · x and S−1 · x,
because (S−1)T = (ST)−1 = S−1 and S · x = S · (S · S−1) · x = kBTM · (S−1) · x. Given
an algorithm to evaluate M · x, Fixman noted that one can evaluate S−1 · x using
Chebyshev polynomial approximation over the range [λmin, λmax], where λmin and
λmax are the minimum and maximum eigenvalues of M respectively [76, 53]. The
97
approximation yL to S−1 · x is a series of matrix-vector multiplications,
yL =L∑
0
alxl, (6.10)
x0 = dw, (6.11)
x1 = [daM+ dbI] · dw, (6.12)
xl+1 = 2[daM+ dbI] · xl − xl−1. (6.13)
Here al are the Chebyshev coefficients of the scalar function 1/√x over the domain
[λmin, λmax], and da = 2/(λmax−λmin)and db = −(λmax+λmin)/(λmax−λmin). From
the above equations, we see that rather than M, all that we need is the product
of M and a vector, which can be calculated efficiently by an Immersed Boundary
Method as will be described in the next section. To calculate the eigenvalues λ of
M, we use power iteration to calculate λmax and Arnold iteration (ARPACK) to
calculate λmin.[147] Both algorithms require only dot product calculations between
M and some vector x.
In summary, with a fast algorithm to evaluate M · x for arbitrary x, we can
simulate the Brownian movement of a bead-spring polymer in complex geometries.
This fast algorithm is discussed in next section.
6.3.5 Fast Stokes solver with complex boundary conditions
Note that M · F is the fluid velocity generated by the the vector of point forces, F.
Thus, to calculate M · F, we need to solve the Stokes flow equations with distributed
98
point forces with no-slip boundary conditions:
−∇p(x) + η∇2u(x) = −∑
fnδ(x− xn), (6.14)
∇ · u(x) = 0, (6.15)
u(r) = 0, r ∈ ∂Ωb, (6.16)
where p is the pressure, u is the fluid velocity, η is the fluid viscosity, fn is the force
exerted on the fluid at point xn, δ(x) is the three-dimensional delta function, and
∂Ωb is the boundary of the fluid domain. In the actual implementation, regularized
point forces are used, as further discussed below.
Provided a fast Stokes solver for distributed point forces within a periodic do-
main (GGEM), we developed an accelerated Immersed Boundary Method (IBM) for
the Stokes equations with complex boundary conditions by representing the no-slip
boundary as forcing term (momentum source) in the Stokes equation. To employ
the fast Fourier transform (FFT) technique to speed up the calculation, we chose a
periodic domain. We present first the IBM algorithm, then the GGEM algorithm.
Immersed boundary method
A generic three-dimensional domain is shown schematically in Fig. 6.2. Polymer
beads and solid boundaries are embedded in a rectangular periodic domain Ω =
[−Lx/2, Lx/2] × [−Ly/2, Lx/2 × [−Lz/2, Lz/2]. The beads of the polymer molecule
are represented by the filled symbols, and their position vector is R. The beads
exert forces obeying Stokes law on the surrounding fluid. A no-slip boundary ∂Ωb
is represented by regularized point forces located at boundary nodes Rb, which are
99
Figure 6.2: Schematic of the immersed boundary method. The system is a threedimensional periodic domain, in this study, a rectangular parallelepiped of Lx×Ly ×Lz. The beads of the polymer molecule is represented by the filled symbols, and theirposition vector is R. The no-slip boundary ∂Ωb is represented by regularized pointforces located at boundary nodes Rb, which are uniformly spaced with a boundarymesh size of h.
100
uniformly spaced with a boundary mesh size of h. These boundary nodes do not
move relative to the fluid, but rather exert forces on the fluid such that the velocity
of fluid at Rb is zero. Hence, the no-slip boundary condition is satisfied at each nodes.
The advantages of IBM include its ease of programming due to use of a Cartesian
grid, and its potential for parallel processing as the boundary nodes are treated in a
similar fashion as the polymer beads.
Denoting Mibm as the mobility tensor for all the regularized point forces in the
periodic domain, i.e., polymer beads and boundary nodes, the velocities generated by
those point forces are,
Mibm · Fibm = Uibm. (6.17)
Partitioning Mibm into smaller tensors for polymer-polymer, polymer-boundary, and
boundary-boundary interactions, the relationship between forces and velocities at the
polymer and boundary points is given by
Uibm =
Up
Ub
=
Uuni
Uuni
+
Mpp Mbp
Mpb Mbb
·
Fp
Fb
, (6.18)
where p and b denote polymer and boundary respectively, and Uuni is the uniform
background velocity which is generated by a constant pressure drop. Here Mpp and
Mbb are symmetric and positive definite (and thus invertible) matrices. The quantities
Mpb · Fp and Mbp · Fb are the velocities at boundary points generated by polymer
forces Fp and the velocities at polymer points generated by boundary forces Fb,
respectively, and we are able to calculate them using periodic GGEM, which will be
discussed in the next section. Note that the unperturbed Stokes flow velocity U∞ in
101
a complex geometry is the sum of the uniform background flow Uuni and a correcting
flow generated by the boundary points Ubr, U∞ = Uuni + Ubr. From Eq. 6.18 we
obtain:
Up = Uuni +Mpp · Fp +Mbp · Fb, (6.19)
Ub = Uuni +Mpb · Fp +Mbb · Fb = 0. (6.20)
Noting that Mbb is invertible, we obtain:
Fb = (Mbb)−1 · (−Uuni −Mpb · Fp), (6.21)
Up = Uuni +Mbp(Mbb)−1 · (−Uuni) +Mpp · Fp −Mbp(Mbb)−1Mpb · Fp
= Uuni +Mbp(Mbb)−1 · (−Uuni) + [Mpp −Mbp(Mbb)−1Mpb] · Fp. (6.22)
Comparing the above equation to U∞ = Uuni +Ubr, we obtain
Ubr = Mbp(Mbb)−1 · (−Uuni). (6.23)
By construction, Eq. 6.20 and Eq. 6.21 also suggest an algorithm to calculate Up:
(1) Given the polymer configuration R(tn), evaluate polymer forces Fp(tn) includ-
ing spring forces, excluded volume interactions, and repulsive wall-bead interactions;
(2) Calculate the perturbed velocities Upb = Mpb ·Fp generated by polymer forces
at the boundary points;
(3) Determine the boundary forces Fb(tn) that satisfy Mbb · Fb(tn) = −Uuni −
Upb(tn) (Eq. 6.21);
(4) Evaluate the polymer bead velocities Up = Uuni +Mpp · Fp +Mbp · Fb.
102
Step (3) in this algorithm requires an efficient iterative scheme to calculate the
boundary forces that specify the velocity field in the domain we are interested in. In
other words, our goal is to solve Mbb ·X = U, where U is given and X is unknown.
The number of iterations required to reduce the relative error to meet some specified
tolerance is a function of the condition number ofMbb (Figure 6.3(a)), which is defined
as the ratio of the largest over the smallest eigenvalues ofMbb. We found that a simple
conjugate gradient method [123] is sufficient. First, in the algorithm of conjugate
gradient method, only matrix-vector multiplications are calculated. Second, it can
reduce the relative residual ||Mbb · Fb − U||/||U|| (Eq. 6.21) to less than 10−4 in
tens of iterations as shown in Fig. 6.3(b), even when the condition number of Mbb is
very large. It should be noted that a similar approach, in a non-Brownian boundary
element context, has been proposed by Zhao et al. to study the flow of blood cells in
complex geometries.[159]
Periodic GGEM
The core of our algorithm is a fast solver for the Stokes equations with distributed
point forces. In this section, we present the general geometry Ewald-like method
(GGEM), which can serve as a “blackbox” to evaluate M · x for a given point force
distribution in periodic domain.[67, 68, 66, 114] For periodic boundary condition,
Ewald sum and particle-mesh Ewald (PME) methods are available, which are based
on the Hasimoto’s solution for Stokes flow driven by a periodic array of point forces.
These methods reduce the operation to O(N logN), compared with O(N2) opera-
tions of direct evaluation of all the pairwise interactions.[65, 132, 124] Fast multipole
method is another alternative, and the complexity is O(N).[145] GGEM is also an
103
Figure 6.3: Properties of the Mbb tensor for the case where a spherical boundary (R =3) is represented by uniformly distributed points on the surface and the parameters forthe calculation are Lx = Ly = Lz = 10, dx = dy = dz = 0.25, ξ = 4.0. (a) Conditionnumbers as a function of number of points on the sphere shell. (b) Relative residualas a function of number of iterations.
104
O(N logN) method when fast Fourier transforms are implemented.
The central idea that makes GGEM a fast algorithm is separating the point force
density ρ = Σfnδ(x− xn) into a local part ρl = Σfn(δ(x− xn) − g(x− xn)) and a
global part ρg = Σfng(x− xn) by introducing a screening function g(x − xn). In
this work, the screening function g(r) is a modified Gaussian with α as the screening
parameter
g(r) = (α3/π3/2)e(−α2r2)(5/2− α2r2), (6.24)
where r is the radial position vector and r = |r| (Fig. 6.4). This smooth function
satisfies
∫
g(r)dr = 1 for any value of α, and when α → ∞ it becomes a three-
dimensional Dirac delta function. Correspondingly, the velocity field generated by ρ is
separated to a local contribution and a global contribution u = ul+ug. We chose g(x−
xn) so that an analytical expression for ul is available and ul decays exponentially
fast. Thus, the interactions due to ρl are short ranged, and only interactions within
some cut-off distance Rc need to be considered in the calculation of ul. In the present
case, the global velocity ug is going to be calculated by solving the Stokes equations
on a structured mesh (grid) using three-dimensional fast Fourier transform. Note
in contrast to conventional Ewald-based methods, FFT is not necessary to achieve
computational efficiency in GGEM, so the GGEM approach is not limited to periodic
domains.[67, 69, 148, 114]
Local velocity field Consider first the local velocity at x, ul(x), which results
from the local force density ρl,
ul(x) =∑
n
Gl(x− xn) · fn, (6.25)
105
Figure 6.4: Screening function g(αr) (Eq. 6.24) for the GGEM algorithm.
where the “local Green’s function” is
Gl(x) =1
8πη(I+
xx
r2)erfc(αr)
r− 1
8πη(I− xx
r2)2α
π1/2e(−α2r2). (6.26)
Here, r = |x| and I is the identity matrix. In this work, to avoid singularity associated
with delta point forces, we use regularized point forces by replacing δ(r) with g(r)
(Eq. 6.24) with a regularizing parameter ξ and the corresponding regularized local
Green’s function is
GRl (x) =
1
8πη(I+
xx
r2)(erf(ξr)
r− erf(αr)
r)+
1
8πη(I− xx
r2)(
2ξ
π1/2e(−ξ2r2)− 2α
π1/2e(−α2r2)).
(6.27)
When x → 0,
GR0
l =1
8πη(4ξ√π− 4α√
π)I, (6.28)
and the velocity at x = 0 stays finite. From Eq. 6.27, we see that because of the
presence of the screening function, velocity decays exponentially over a distance pro-
106
Figure 6.5: (Free space) Comparison of the x component of the velocity field drivenby a delta point force (lines) acting along +x-axis and by a regularized delta pointforce (symbols) with a form of modified Gaussian (Eq. 6.24, α = 2.0). (Top) alongx-axis. (Bottom) along y-axis.
portional to 1/α, provided that α ≪ ξ. Calculation of the local velocity field begins
by identifying point forces located within the cut-off distance Rc = 4/α for a given
x. Any interactions beyond Rc is ignored. The value of α is empirically determined,
and is a constant for certain simulation. To make sure the local velocity satisfies the
periodic boundary conditions, we apply the minimum image convention which states
that, in the periodic system, the cutoff distance must be smaller than half the shortest
box length. In Fig. 6.5, we compare the velocity generated by a delta point force and
a regularized point force.
The regularizing parameter ξ is different for the polymer beads and the boundary
nodes. Consider first the boundary nodes. As with the conventional IBM [110, 97],
the boundary force density is distributed onto the grid, which is uniformly spaced
with a mesh size of h, through a regularized delta function. In this work, we choose
the modified Gaussian as in Eq. 6.24, and we choose ξ such that ξh = O(1). This
107
ensures that the force density associated with each grid node is spread over the length
scale of the associated elements, thereby preventing fluid from penetrating the solid
boundaries and also preventing unphysically large fluid. For the polymer beads, the
choice for ξ is straightforward and is inversely proportional to the hydrodynamic
radius of the polymer bead, ξa ≈ 1. In the present work, we choose ξ so that
ξa = 3/√π, with which the maximum fluid velocity is equal to that of a particle with
radius a and the pair mobility remains positive-definite.[69]
For any point x in the fluid that is not on the boundary or a polymer bead, the
local contribution to the velocity is given by
ul(x) =
n∑
i
GRl (x− xi) · f bi +
m∑
j
GRl (x− xj) · fpj . (6.29)
Here, n and m are the number of boundary points and polymer beads, respectively,
which lie within the cut-off distance centered at x. For a boundary point, the local
contribution is
ul(xk) =
n∑
i
GRl (x− xi) · f bi +
m∑
j
GRl (x− xj) · fpj . (6.30)
Note that in the first term, the i = k term requires evaluation of GRl (0), which is
given by Eq. 6.28.
For a polymer bead, the local contribution is
ul(xk) =n
∑
i
GRl (x− xi) · f bi +
m∑
j,j 6=k
GRl (x− xj) · fpj . (6.31)
The exclusion of the “self-term”, j = k, in the polymer local velocity calculation is
108
because in the Brownian dynamics simulation polymer beads do not move with the
fluid velocity. Thus the velocity “seen” by a polymer bead should not include the
velocity generated by the bead itself as it moves through the fluid.
Global velocity field The velocity field driven by the global part of force density
ρg is obtained by solving the Stokes equations with periodic boundary condition on
a grid using fast Fourier transform (FFT).
The continuous force density g(r) has to be replaced by a grid based force density.
In this work, we use the modified Gaussian function to distribute the force density
onto the grid points which are within the cut-off distance Rc. This is simply a Fourier
collocation approach to the global problem.[27] Again, we use periodic boundary
condition with minimum image convention to make sure the forcing term is periodic
in all three spatial directions.
The linearity of Stokes equation allows us to exploit the efficiency of FFT. We
assume the velocity, forcing term, and pressure gradient are periodic in all three
spatial directions. The velocity vector U(x) and pressure gradient ∇p(x) at the grid
point x = (x1, x2, x3) are
u(x) =∑
kuke
−ik·x,
∇p(x) =∑
kpke
−k·x. (6.32)
Here, k = (k1, k2, k3) is the wavenumber vector and k2 = k21 + k2
2 + k23. Now we apply
Fourier transform to the Stokes equations Eq. 6.15 and obtain
pk − ηk2uk = −f , ik · uk = 0. (6.33)
109
When k 6= 0, taking the scalar product of the Fourier transformed momentum equa-
tion with ik yields
pk = −(k · f)k2
k, (6.34)
uk =f
ηk2− (k · f)
ηk4k.
When k = 0, p0 = −f , which means that the force exerted on the fluid is balanced
by the mean pressure gradient of the field. Then the global velocity field is obtained
by inverse FFT. To get the velocity at the polymer points from the velocity on the
mesh, we perform back-interpolation using quadratic Lagrange polynomials. It is this
interpolation step that ultimately controls the order of accuracy of the solution.
Finally, note that in the global velocity calculation for polymer beads, we also
have to exclude the self-interaction term. This velocity field is determined by the
free-space regularized Stokeslet (Eq. 6.26). Therefore, the total velocity seen by a
polymer bead is
u(xpj ) = ul(x
pj ) + ug(x
pj )− lim
x→0
Gl(x) · fpj . (6.35)
Validation: GGEM We investigated the accuracy of GGEM as function of screen-
ing parameter α and mesh size ∆x, against Hasimoto’s solutions of spatially periodic
Stokes equation[65, 113]. The simulation box is a cube of side length L = 10. A
delta point force acting along x direction with unit strength is placed at the center
of the box. Fig. 6.6 shows Hasimoto’s solution and the numerical solution using
GGEM for the x component of velocity at line x = 0, y = 0 and line y = 0, z = 0.
Fig. 6.7(a) shows the error as a function of screening parameter α for three different
110
Figure 6.6: Comparison of Hasimoto’s result[65] (symbols) and numerical solution(lines) for the x component of velocity due to point force acting along +x directionin a periodic domain.
mesh sizes ∆x. Notice that, empirically, when 20/L < α < 0.8/∆x the error does
not depend on α. The lower bound is related to the restriction that cut-off radius
Rc = 4/α should be smaller than half of the box size, and the upper bound is about
the number of grid points which need to be included to resolve the length scale α−1.
Fig. 6.7(b) shows the error as a function of mesh size ∆x together with least squares
fitting to estimate the order of accuracy. The error is defined as the L2 norm of the
difference between GGEM results and Hasimoto’s ||E||2 = ||uGGEM − uHasimoto||2 at
200 sampling points which are randomly distributed within the simulation box. With
Lagrange back-interpolation, GGEM has order 3 of accuracy. This result does not
change with the positions of the sampling points. Unless otherwise noted, throughout
the rest of this study, we choose the screening parameter which satisfies α∆x = 0.8,
and ∆x is determined by the desired error tolerance.
111
Figure 6.7: Error ||E||2 as a function of (a) screening parameter α and (b) mesh size∆x for a point force at the center of a cubic periodic domain.
112
Validation: IBM We now show numerical results of three test cases using the
IBM algorithm proposed above: pressure driven flow through a slit geometry (Fig.
6.8), uniform flow around a sphere (Fig. 6.9(a)), and flow generated by a Stokeslet
near a solid plane (Fig. 6.9(b)). We use the slit case to investigate the numerical
error of IBM. For laminar flow in a slit geometry, the velocity only depends on the
perpendicular distance away from the wall, and it is parabolic Ux(z) = 3/2Uavg(1 −
(z/H)2). In the numerical calculations, the periodic domain is a cube with side of
L = 40, and the wall is represented by a square lattice with grid size of h. In all the
calculations, we used a mesh size of ∆x = 0.05L for the global velocity calculations.
Fig. 6.8(a) shows excellent agreement between the analytic solution and the numerical
solution obtained by IBM. The absolute error ||Unumerical −Uexact||2 is shown as a
function of h in Fig. 6.8(b); decays roughly as h1.2, consistent with the general result
that IBM is first-order accurate.[86] The local error is the largest near the boundary
and decays very rapidly into the flow (< 1× 10−5 on the center line).
The uniform flow around a sphere case demonstrates that our algorithm works well
for curved boundary as well. The exact solution to the flow field in an unbounded
domain is:
Ur = U∞[1− 3
2(a
r) +
1
2(a
r)3]cos(θ),
Uθ = −U∞[1− 3
4(a
r)− 1
4(a
r)3]sin(θ).
Here, θ is the angle away background flow direction, a = 1 is the radius of the sphere,
and r is the radial distance from the center of the sphere. We use a large simulation
box (L = 500) to minimize the effect of periodic boundary. When the background flow
113
Figure 6.8: (a) Velocity profile of laminar flow through a slit. The line represents theanalytic profile and the symbols represent the numerical result by IBM. (b) Error ofIBM as a function of boundary grid size h and the slope is 1.22.
114
is U∞ =(1, 0, 0), we use N = 200 regularized points, which are uniformly distributed
on a spherical shell of radius 1, to represent an unit sphere placed at the center of
the simulation box. The mesh size is ∆x = 1 and α∆x = 0.1 in this case. From Fig.
6.9(a) we see that IBM gives a satisfactory result.
In the case of a Stokeslet above a solid wall, numerical results are tested against
the exact results from the analytical expression of the mobility tensor for a point
force near a single solid wall, obtained by Blake[22]. The calculation is similar to that
in the slit case except that a point force is placed near a wall, and the separation
between the walls is large to minimize the effect from the other wall. The parameters
for this calculation are L = 40,∆x = 0.5, and h = 0.5. The wall is the x − y plane,
and the Stokeslet is situated at (0, 0, 2), pointing in the x direction. From Fig. 6.9(b)
we see that IBM gives excellent results for this case, which involves both force points
and boundary points.
6.4 DNA flowing across an array of nanopits
In this section, we apply the immersed boundary method described in last section to
study the dynamics of a DNA molecule flowing through a nanoslit with embedded
nanopit arrays (Fig. 6.10). The schematic of the problem is given in Fig. 6.10. The
device is periodic in the streamwise (x) and spanwise (y) directions. One periodic or
one unit cell is a rectangular parallelepiped of Lx ×Ly × 2H = 40a× 40a× 8a. Here,
a = 77 nm is the bead radius in the bead-spring model representing the DNA. One
pit is at the center of the unit cell and has a size of Ld × Ld ×H = 16a× 16a× 4a.
The pit-to-pit distance is 2Lc = 24a.
115
Figure 6.9: Validation of IBM algorithm. Open symbols are numerical results andlines are analytical calculations. (a) Velocity profile Ux around a unit sphere withno-slip boundary condition. Open circles represent points along the x-axis and opensquares are for points along y-axis. (b) Stokeslet near a single wall. Velocity profileof Ux along +z-direction for various (x, y) pair. Open squares are for (x, y) = (3, 3)and open circles are for (x, y) = (1, 1).
116
Figure 6.10: Schematic and the immersed boundary representation of the nanopitproblem. Gray spheres indicate points at which no-slip boundary conditions aresatisfied.
In our simulation, there are two unit cells in a three dimensional periodic simula-
tion box of 2Lx × Ly × 2H . The boundary is represented by boundary points which
are regularly spaced with a mesh size of 1a, as shown in Fig. 6.10. Correspondingly,
we chose α = 0.8a and ξ = 2.5a as the input parameters for the immersed boundary
method. A Peclet number, the ratio between the diffusive and convective time scale,
is defined as Pe = UpHD
, and the Weissenberg number is Wi = τ0γ = τ0Up
H, where Up
is the maximum x component of the unperturbed velocity in the y−z plane, D is the
chain diffusivity, τ0 is the longest relaxation time of the chain in bulk solution, and
γ is shear rate in the pit. Note that Pe and Wi are linked by the confinement ratio
Rg/H , as Wi ∼ Pe(Rg
H)2. For λ−DNA, which has a radius of gyration about 10a,
the effective confinement ratio is about 5, considering that the cut-off distance for
the excluded bead-wall interaction is 1a. For λ−DNA we use the τ0 value obtained
by Jendrejack et al.[74]. Figure 6.11 shows several streamlines of the Stokes flow and
the contour plot of the streamwise velocity in the absence of the DNA molecule.
117
Figure 6.11: (Top) Streamlines in the nanopit and (Bottom) contour plot of thestreamwise velocity on the x− z plane.
6.4.1 Dynamics at low Peclet number
In the introduction we described the experimental observation of the chain hopping
between pits, and the mechanism of a free end diffusing and being dragged down-
stream from one pit to the next. This behavior is reproduced in our simulations, as
shown by the sequence of snapshots in Fig. 6.12. To study the dynamics quantita-
tively, we tracked the center-of-mass of the DNA molecule and extracted dynamic
properties of DNA from its trajectories. Fig. 6.13(a) is a typical time series plot of
the x-component of the center-of-mass xc of a λ-DNA molecule. This figure clearly
demonstrates that DNA hops from pit to pit at low Peclet number: the molecule
stays in one pit for a time interval td, which we will call the residence time, and then
jumps to the next pit. As seen in Fig. 6.13(a) at low Peclet number, the time for the
118
Figure 6.12: Snapshots of a hopping event (from (a) to (f)) (top-down view, Pe = 3.5).
119
molecule to get to the next pit is very small compared to the residence time. The
probability distribution of td is well-fit to an exponential ρ(td) = 1/τ exp(−td/τ),
as shown in Fig. 6.13(b). This result indicates that, to a good approximation, the
hopping events are independent of one another—the hopping dynamics is a Poisson
process at low Pe, and the rate parameter which characterizes the Poisson process is
1/τ , where τ is the mean residence time.
Intuitively, one expects that the larger the applied pressure gradient, the faster the
chain hops, or the smaller τ becomes. Fig. 6.14 shows the Peclet-number dependence
of the mean residence time τ , nondimensionalized by the longest relaxation time τ0
of λ−DNA. Over the range of Peclet number studied, the results appear to follow
a exponential distribution τ ∼ τh exp(−αPe), where 1/τh is basically the hopping
frequency as Pe → 0. The physical mechanism of hopping becomes clear when we
write the Peclet number as Pe = (Upζc)HkBT
, where ζc is the friction coefficient of the
chain and we have written the chain diffusivity D as kBTζc
. The term Upζc is a measure
of the hydrodynamic drag force on a chain in the pit. Therefore, the product in the
numerator is the work done by the fluid to drag the molecule out of the pit, and Pe
is the ratio of that work to thermal energy. Hence, our simulation demonstrates that
at low Pe, the dynamics of the hopping is indeed thermally activated.
To examine the effect of HI on the dynamics, we also performed simulation for the
“free draining” (FD) case, where HI were ignored ( in Fig. 6.14). In FD simulation,
the beads are not coupled through hydrodynamical interactions. The mobility tensor
is reduced to the product of 1ζand the identity matrix. Comparing the results obtained
for HI and FD, we can see that the mean residence time is indeed affected by the
hydrodynamic interactions: it takes a longer time for the molecule to hop to the
120
Figure 6.13: (a) Time series plot of the x-component of the center of mass of a λ−DNA driven through a nanopit array for two low values of Pe. (b) Residence timedistribution and best fit to an exponential distribution for conditions shown in (a).
121
Figure 6.14: Mean resident time τ v.s. Peclet number Pe.
next pit in the FD case. This indicates that although HI is screened on a length
scale larger than the slit height, it plays an important role in dynamics happening
on smaller length scales. In this case, the “cooperative” motion of DNA segments
due to HI in the complex geometry gives rise to an increase of the hopping frequency.
We also noticed that the difference between HI and FD gets smaller as the Peclet
number increases. As Peclet number increases, the transport process is dominated by
DNA-wall steric interaction while hydrodynamic interactions play a relatively minor
role. This observation is in line with the observations of Hernandez-Ortiz et al. in
the problem of hydrodynamic effects on polymer translocation through a pore.[66]
We also studied the relationship between mean residence time and chain length. In
general, separation of a mixture of chains of different sizes can only occur if the mean
polymer velocity through the channel strongly depends on the chain length. Fig. 6.15
shows the dependence of the mean residence time on chain length for two different
122
Figure 6.15: Mean residence time τ v.s. chain length N and Peclet number Pe
Peclet numbers. First, as we increase the Peclet number from 3.5 to 5.4, τ becomes
smaller. Second, at a given Pe, we observe a strong molecular weight dependence of τ
at low Pe, while τ saturates in the long chain limit. The first observation is consistent
with the results shown in Fig. 6.14 that increasing Pe lowers the energy barrier
height. The mechanism responsible for the second observation is not clear. Based on
the barrier crossing theory, we have the following hypothesis. For a given pit size,
there is a corresponding DNA size Lc, which fills the pit. For a DNA molecule longer
than Lc, the energy barrier is determined only by the chain segments with the same
length as Lc, i.e. it is determined by the part of the chain residing in the pit. Since
the transport velocity of DNA in the slit is independent of chain length [141], we can
assume the reaction rate prefactor in the barrier crossing theory is also independent
of chain length, thus, in the long chain limit, the mean residence time saturates. The
effect of HI gets more pronounced as the chain length gets longer, as shown by the
123
empty square symbols in Fig. 6.15. This is consistent with the observation of Izmitli
et al.[73] and Hernandez-Ortiz et al.[66] in the problem of hydrodynamic effects on
polymer translocation through a pore.
6.4.2 Dynamics at high Peclet number
When we increase the pressure drop, the transport characteristics of DNA in the
device change, as indicated by the change of the shape of the probability density
function of residence time (Fig. 6.16(a)). In contrast with ρ(td) at low Peclet num-
ber, the distribution at high Peclet number can be fitted to a Gaussian ρ(td) =
1w√2π
exp(− (td−Tc)2
2w2 ), with mean Tc and variance w2. This shift from exponential to
Gaussian reflects the change of the dominant physics as Pe increases, from a pri-
marily stochastic process at low Peclet number to a primarily deterministic one at
high Peclet number. Qualitatively, our simulation results are consistent with exper-
imental observations [36]. We do not seek a quantitative comparison, as data about
the mean residence time were not reported in the experimental study. Fig. 6.16(b)
shows the mean residence time as function of Pe, which is also quite different from
that at low Pe. It appears to follow a power law at high Pe, Tc ∼ 1/Pe. This is
not surprising as at high Pe, as the deterministic drifting process dominates. Hence
the residence time is well-approximated by the time it takes to travel across one pit,
which is approximately Tc ≈ Lx/Up ∼ 1/Pe. The variance of the residence time,
w2, approaches a scaling of Pe−2 in the high Peclet number limit. This is consistent
with the macrotransport theory prediction of force-driven transport of a point-size
Brownian particle in a slowly varying periodic channel.[25, 84]
124
Figure 6.16: (a) Residence time distribution at high Peclet number and best fit to aGaussian distribution. (b) Fitting parameters for Gaussian distribution at differentPeclet numbers.
125
6.5 Conclusions
In this work, we presented an immersed boundary method (IBM) based on the gen-
eral geometry Ewald-like method (GGEM), aimed at simulating Brownian motion of
polymer in complex geometries with full description of hydrodynamic interactions.
The core idea is to replace the complex boundary with regularized point forces, cho-
sen to satisfy the appropriate boundary conditions. The IBM-GGEM methodology
correctly reproduces the velocity field in several test cases including cases involving
curved boundary and point momentum sources.
As an application of this methodology we have considered the simulation of a single
DNA molecule driven through a nanoslit with embedded nanopits array, by a pressure
gradient. We studied the dynamics of the DNA as a function of the Peclet number
and chain length, as well as the influence of hydrodynamic interactions by comparing
with free draining simulation results. We found that the transport characteristics of
the hopping dynamics in this device differ at low and high Peclet number, and for
long DNA, relative to the pit size, the dynamics is governed by the segments residing
in the pit. We also found that HI plays an important quantitative role in the hopping
dynamics even in such a highly confined system.
We expect that our algorithms will find many applications in micro- and nano-
fluidics.
126
Chapter 7
Conclusion and future work
In conclusion, we have developed general numerical frameworks to study electrophoretic-
driven and flow-driven DNA dynamics in complex geometries, and we presented a
systematic investigation of three different problems using Brownian dynamics simu-
lations.
In both the problem of soft nanomechanical elements and the problem of flowing
DNA through nanopits, the concepts of entropic trapping and barrier crossing give
satisfying explanations to the observed dynamics. It seems that we have a good
understanding of the basic physics in those microfluidic devices, where DNA dynamics
is determined by the combined effects of major driving forces and chain configuration
entropy. Now the goal is shifted to apply the known physics to the design of devices.
This echoes the motivation of this work, which is to build validated numerical tools
to help the design and optimization of novel processes and devices in a very broad
area.
Considering the future of both the simulation methods and their applications
127
to the problem of DNA dynamics in complex geometries, harnessing the increasing
computational power to conduct more detailed simulations of larger system for longer
time is an inevitable trend.
Future work may include:
• New mechanical model for DNA under nanometer scale confinement. When
a DNA chain is confined to a space smaller than its persistence length, which
is already achieved in experiments, the bead-spring model breaks down. We
thus need to build a finer-scale model of the DNA to study these systems, for
example, a bead-rod model with a rod length of 10 nm. For a λ-DNA molecule,
this means thousands of beads, which is out of the reach of traditional Brownian
dynamics simulation technique when long-ranged hydrodynamic interactions are
included. But it might be feasible with our new accelerated immersed boundary
method using graphic processing unit.
• Graphic processing unit (GPU) acceleration. The parallel architecture of GPU
makes it very effective for algorithms where processing of large blocks of data is
done in parallel. It has been applied to accelerate molecular dynamics simula-
tion and computational fluid dynamics calculation. As recently reported, GPU
calculations achieved tens of fold, even hundreds of fold in some case, speedup
over an optimized CPU implementation. The advantages of our accelerated
Brownian dynamics algorithm is that it divides the long-ranged interactions
into short-ranged pair-wise interactions, which can be calculated using analyti-
cal expressions, and long-ranged interactions, which change smoothly with small
gradient and can be calculated on a regular structured Cartesian grid. Both
calculations are straightforward to parallelize efficiently. Also, in our algorithm,
128
the boundary nodes are treated in a similar fashion as the polymer beads, which
gives its potential for parallel processing.
129
Appendix A
Lattice random walk model of a
tethered polymer
In this appendix, we use a random walk lattice model to calculate the thermodynamic
properties of an ideal polymer molecule tethered to a hard nonadsorbing wall.
For an ideal chain, the entropy S(r) associated with all chain conformations start-
ing from an origin and ending at r is simply related to the number of distinct walks
Z(r) connecting the origin and r in n steps.[35]
S(r) = kB ln[Zn(r)] (A.1)
Therefore, the free energy of an ideal chain is F(r)/kBT = − ln[Zn(r)]. It is useful to
know how does the free energy change as a function of a single reaction coordinate,
such as the perpendicular distance of the free end to the wall ze for a tethered molecule.
It could reduce the problem to one dimensional and allow us to apply Kramer’s theory
of barrier crossing to study the dynamic properties of the walk. Here we introduce
130
the potential of mean force of a polymer chain as a function of ze
F(ze) = F∗(ze)− kBT ln[〈ρ(ze)〉〈ρ∗(ze)〉
] (A.2)
where F∗(ze) and 〈ρ∗(ze)〉 are arbitrary reference values.[121] The average distribution
function 〈ρ(ze)〉 along the reaction coordinate is obtained from a Boltzmann weighted
average
〈ρ(ze)〉 =∫
drδ(z′
e(r)− ze)e−U(r)/kBT
∫
dre−U(r)/kBT, (A.3)
where U(R) represents the total energy of the chain as a function of the beads’
positions R and δ(z′
e(R)− ze) picks out the configurations with the same ze. For an
ideal chain, every path has the same weight, and above expression reduces to
〈ρ(ze)〉 =Zn(ze)
Ztotn
, (A.4)
and therefor, the potential of mean force is
F(ze) = F∗(ze)− kBT ln[Zn(ze)
Z∗n(ze)
]. (A.5)
Above expressions indicate one can calculate the free energy profile along the chosen
reaction coordinate (ze), by calculate the ratio of the number of configurations of a
specific state to that of the reference state.
The property of an ideal chain is often studied using a lattice random walk model
[35]. For a n-step random walk on an integer lattice, we can directly count the
number of walks starting origin and ending at (x, y, z). When the walk is near a wall,
absorbing boundary condition should be used, i.e., we should discard those walks
131
touching the wall at least once.[40, 134] Once the number of walks satisfying both
absorbing boundary and various constraints are obtained, the free energy is readily
obtainable using thermodynamic relationships presented above. Next, we present the
results of combinatorics calculations for counting the chain configurations.
According to Bertrand’s “ballot theorem”, the number of 1-D random walks on
the integers of n steps, from the origin to the point x > 0 and never return to the
origin, is
Z1dt (x,N) =
x
N
(
NN+x2
)
, (A.6)
assuming N and x have the same parity mod(N − x, 2) = 0. Thus, the total number
of walks is
Z1dt =
∑
x
W 1dt (x,N) =
∑
x
x
N
(
NN+x2
)
. (A.7)
Assuming N = 2k + 1(k = 0, 1, ...),
Z1dt =
∑
x=1
x
N
(
NN+x2
)
=1
N
∑
x=1
x
(
NN+x2
)
=1
N[
(
N
k + 1
)
+ 3
(
N
k + 2
)
+ ...+ (2k − 1)
(
N
k + k
)
+ (2k + 1)
(
N
k + k + 1
)
].
(A.8)
Note that (2i − 1) = (k + i) − (k − i − 1),(
Nk+i
)
=(
Nk+1−i
)
and (k + i)(
Nk+i
)
=
132
(N − k + 1)(
NN−k+1
)
. Last expression is
Z1dt =
1
N[
(
N
k + 1
)
+ 3
(
N
k + 2
)
+ ...+ (2k − 1)
(
N
k + k
)
+ (2k + 1)
(
N
k + k + 1
)
].
=1
N[(k + 1)
(
N
k + 1
)
− k
(
N
k
)
+ ...+ (2k)
(
N
2k
)
−(
N
1
)
+N ]
=1
N(k + 1)
(
N
k + 1
)
=
(
2k
k
)
. (A.9)
Similarly, we can prove that when N = 2k(k = 1, 2...), Z1dt =
(
2k−1k
)
. Therefore,
Z1dt =
(
N−1[N/2]
)
, where [x] is the largest integer smaller than x.
In 2-D case, assuming there are i steps in the wall normal direction and N − i
steps in the wall parallel direction, the total number of paths are
Z2dt =
N∑
i=1
(
N − 1
i− 1
)(
i− 1
[ i2]
)
2N−i =
(
2N − 1
N − 1
)
. (A.10)
Similarly, in 3-D, the total number of paths is
Z3dt =
N∑
i=1
(
N − 1
i− 1
)(
i− 1
[ i2]
)
4N−i. (A.11)
In summary, we obtain the analytical expressions of the number of configurations of
133
1
2N steps
x z
y
2
0
Figure A.1: A tethered polymer as lattice random walk.
a tethered polymer molecule, and the total numbers of paths are
Z1dt (N) =
N∑
x=1
x
N
(
NN+x2
)
=
(
N − 1
[N2]
)
≈ 2N√2πN
, (A.12)
Z2dt (N) =
N∑
m=1
(
N − 1
m− 1
)(
m− 1
[m2]
)
2N−m =
(
2N − 1
N − 1
)
≈ 4N√4πN
, (A.13)
Z3dt (N) =
N∑
m=1
(
N − 1
m− 1
)(
m− 1
[m2]
)
4N−m ≈ 6N√6πN
, (A.14)
where m is the number of steps in the confined(wall normal) direction. There is
no closed expression for the sum in 3-D. However, the approximation 6N/√6πN
converges to the exact result as N increases (data not shown), following the same
scaling as those for 1-D and 2-D cases. In this work, the total number of N -step tail
is called the fundamental solution of a tethered chain, and it is denote ZN0 .
When a polymer is grafted to a bilayer, it can induce gelation or other phase
134
changes in lamellar phases, which might be related to the interactions of locally
deformed membrane patches which are created by the entropic forces exerted by the
grafted polymer molecules[20]. When an ideal polymer chain is grafted to a hard wall,
the number of accessible configuration is greatly reduced which leads to an entropy
penalty. Phenomenologically, we observe a repulsive force which drives the polymer
away from the wall, i.e., we need to apply a tethering force to fix the end of polymer.
We are interested in the dependence of this tethering force on polymer chain length.
For a lattice polymer, the required force f0 is related to the number of configurations
Z as
f0 = −∂F(l)
∂l|l=0 ≈
F0 − F1
1a=
kBT
a(lnZN+1
0 − lnZN0 ), (A.15)
where l is the perpendicular distance of the tethered end from the wall, the lattice
grid size is a = 1 which has a physical meaning of persistence length of the polymer
molecule. Using the expression of the number of tail configuration (Eq. A.14), we
know that ZN0 ≈ 6N√
6πNand ZN+1
0 ≈ 6N+1√6π(N+1)
. Plug these into last equation and we
obtain
f0 =kBT
aln
ZN+10
ZN0
≈ kBT
aln 6
√
N
N + 1=
kBT
a(ln 6− 1
2ln
N + 1
N)
≈ kBT
a(ln 6− 1
2N). (A.16)
Clearly, the force is repulsive, and the longer the chain, the larger the force.
Interestingly, the force becomes independent of the chain length in the long chain
limit. The expression we obtained is similar to that of Bickel et al. [20] using the
135
0 1 2 310
−5
10−4
10−3
10−2
10−1
100
101
l/Rg
∆F(l/
Rg)/
k BT
N=6
N=8
N=10
N=12
Figure A.2: The wall-polymer potential calculated using the fundamental solution ofa tethered chain for various chain length (symbols). Dashed line is a fit to Eq. A.23
diffusion equation[20], f0 = kBTa
exp(− a2
4R2g) ≈ kBT
a(1 − a2
4R2g), and it has the same
dependence on persistence length a and on chain length n. The difference of the
constant ln 6 is due to the nature of the lattice. To our best knowledge, there is no
experimental results for us to compare with. For a semi-flexible fluorescently stained
λ-DNA with a = 50nm and Rg = 770 nm, the tethering force is about 0.82 pN. For a
flexible molecule of poly(methacrylic acid) (PMAA) with a=0.3nm and Rg = 700nm,
however, the tethering force is 11.2 pN.
Above analysis can be extended to calculate the effective repulsive potential be-
tween a free polymer molecule and a hard wall using the fundamental solution of a
tethered molecule. The total number of N -step paths in the half space starting l away
136
from the wall is denoted WNl , and it can be obtained recursively as,
WN0 = ZN
0 , (A.17)
WN1 = ZN+1
0 = 4WN−11 +WN−1
2 , (A.18)
WN2 = ZN+12
0 − 4ZN+10 = 4WN−1
2 +WN−11 +WN−1
3 (A.19)
WN3 = ZN+2
0 − 8ZN+10 + 15ZN+1
0 = 4WN−13 +WN−1
2 +WN−14 (A.20)
WNi+1 = WN+1
i − 4WNi +WN
i−1. (A.21)
We can see that WNl is a linear combination of the fundamental solutions ZN+i
0 , i =
1, 2, ..., l − 1, and the coefficients can be constructed recursively. Once all WNl are
calculated, taking the free chain as the reference state, the wall-polymer potential
can be calculated according to Eq. A.5 as
∆F(l, N)
kBT= − ln
WNl
6N. (A.22)
Results for various chain lengths are plotted in Fig. A.2. First, when l is normal-
ized by the size or the molecule Rg =√Na, where a is the lattice size, all results
collapse onto a master curve. Second, the interaction potential decreases exponen-
tially fast when l < Rg and it can be fitted to the following equation
∆F id(l, N)
kBT= − ln(erf(
l
Rg)), (A.23)
which is obtained from a continuous description of the random walk using the diffu-
sion equation[24]. This repulsive potential can be used in mesoscale simulation of a
polymer molecule near a single wall.
137
Appendix B
BD/FEM algorithm for simulating
DNA electrophoresis
The objective of this appendix is to present a numerical framework, Brownian dynam-
ics (BD)/Finite element method (FEM), to simulate DNA electrophoresis in complex
geometries. The proposed numerical scheme is composed of three parts:
• a bead-spring DNA model,
• a finite element scheme for the non-homogeneous electric field,
• and a coupling algorithm to find the values of the electric field at the DNA
beads location.
The bead-spring model has been described in Chapter 3. In the reminder of this
appendix, we discuss the finite element scheme and the coupling algorithm.
For a linear DNA molecule, the Debye length (κ−1), which is the length scale
over the charges on the DNA backbone are screened out by the counter-ions, is much
138
smaller than the persistence length of DNA in sufficiently concentrated salt solution
(typically κ−1 is approximately 2 nm in thickness under physiological conditions).
Thus, DNA globally behaves like a neutral polymer and the electric field is governed
by Laplaces equation, since local electric field disturbance due to a DNA molecule is
screened over the Debye length. Also, we consider the system where electroosmotic
flow is eliminated using a polymer layer grafted on the walls. In terms of geometry, we
consider a microfluidic device with a constant channel height, and the channel height
is usually much smaller than the other length scales of the device. Therefore, the
problem is reduced to two dimensional. Still, with increasing complexity of geometry,
a numerical method is required to calculate the electric field in a microfluidics with
complex geometries.
The electric field is computed from the electric potential with the relationship
E = −∇φ in an arbitrary domain, where φ denotes electric potential. The governing
equations for the electrostatic potential are,
∇φ2 = 0 in Ω, (B.1)
φ = φ0 or∂φ
∂n= 0 on ∂Ω, (B.2)
where n is the normal vector at the boundary. We consider two types of boundary
conditions. One is Dirichlet boundary condition which specifies the values of poten-
tial on the boundary of the domain, such as the electric potentials at the inlet and
outlet. The other is Neumann boundary condition which specifies the values that
the derivative of potential is to take at the insulating channel walls. In experiments,
the device is usually made of poly(dimethylsiloxane) (PDMS) which is an insulator.
139
Hence, a zero gradient boundary condition is assigned along the channel walls.
In this work, we use a commercial partial differential equation (PDE) solver,
COMSOL, which is based on the finite element method (FEM), to solve the governing
Laplace’s equation for the electrostatic potential (Eq. B.2). For detailed information
on FEM, the readers are referred to [17] and COMSOL manual. In a microfluidic
device, steep field gradient exists around obstacles or sharp corners, which means that
the space around an immersed structure like a post, and around a sharp corner like
near a contraction/expansion region, should be more finely discretized. Once electric
potential is obtained, E is simply computed from E = ∇φ and stored for lookup. To
calculate the field strength at the bead location, we need a fast algorithm to find the
triangular element surrounding the bead.
For an irregular triangular mesh, finding the electric field at a specified point in
the domain is not a trivial problem. As we just discussed above, to resolve the steep
field gradient around the obstacles and corners, very fine mesh needs to be used. This
means that we have a large number of elements to look through. We address this
task as follows. We first assume that the target point is closer to the centroid of the
element enclosing it, than to the centroids of all the other elements. Of course this is
not always true as shown in Fig. B.1. So we need to find the four nearest centroids
to the target, and then find the one enclosing it using a confirming algorithm. Now
the problem is reduced to find the 4-nearest centroids to the target. We can solve
it using the nearest neighbor search algorithm, which is a well-known algorithm in
computational geometry [133]. In this work, we use a variant of this algorithm called
Approximate Nearest Neighbor Searching (ANN) developed by David M. Mount and
Sunil Arya [11]. Finding the nearest point needs O(logN) operation in the case of
140
Figure B.1: Schematic of the nearest neighbor search algorithm (NNS). To find theelement enclosing the target point, we use NNS to find the k-nearest centroids of theelements.
randomly distributed points. The worst case search time, when a 2-dimensional KD-
tree containing N nodes is used, is O(N1/2). This allows us to use very fine mesh in
the calculation.
141
Appendix C
Dimer in shear flow
In this appendix we analytically and numerically consider the case of a Brownian
particle attached to the origin with a harmonic spring with stiffness k = κζ , where ζ
is the friction coefficient, subject to shear flow, vx = γy. If additionally we include a
hard wall at y = 0 such that the random walk is restricted to the upper half-space,
y > 0, we are essentially considering a tethered polymer chain composed of two beads.
Note that this problem is essentially two-dimensional.
The overdamped Langevin equations for the particle coordinates are
x = γy − κx+ Fx
y = −κy + Fy,
where F denotes the random forcing. After performing a Fourier transform in time,
142
we get the solution in Fourier space
x =(iν + κ)Fx + γFy
(iν + κ)2
y =Fy
(iν + κ),
from which we can obtain all cross-correlation functions using the identities⟨
Fx⋆Fx
⟩
=⟨
Fy⋆Fy
⟩
= α and⟨
Fx⋆Fy
⟩
= 0. In particular, we obtain the monotonically-
decreasing non-normalized PSD
Sxy(ν) =∥
∥
∥Cxy
∥
∥
∥= ‖〈x⋆y〉‖ =
αγ
(ν2 + κ2)3/2,
and, after an inverse Fourier transform, the non-normalized CCF
Cxy(t) =
αγe−κt(2κt + 1)/(4κ2) for t ≥ 0
αγeκt/(4κ2) for t < 0.
In the case of no shear flow, γ = 0, we obtain that Cxx(t) = αe−κ|t|/(2κ), showing that
the relaxation time is τ = κ−1 and thus Wi = γ/κ. For the harmonic spring dimer
the relaxation time does not depend on Wi. The cross-correlation function shows
a single peak at tmax = (2κ)−1 = τ/2, and after proper normalization, Cxy(t) =
Cxy(t)/√
Cxx(t = 0)Cyy(t = 0), the height of the peak in the CCF is found to be
Cmaxxy = Cxy(τ/2) =
√2e−1/2Wi
√
2 +Wi2. (C.1)
This analytically-solvable dimer model, even without a hard wall, reproduces the
143
characteristics of the CCF that we observe for tethered polymer chains in shear flow.
Specifically, Cxy(t) has an asymmetric peak of width ∼ τ centered at t = τ/2 and
height ∼ Wi. There is no periodicity in the motion of the dimer and no “cycling”
time-scale other than the intrinsic relaxation time τ .
The dimer problem can no longer be solved analytically if a hard wall is present
or if the spring is non-linear (e.g., FENE or worm-like). We can, however, study the
dimer with a non-linear spring and/or in the presence of a hard wall numerically using
Brownian Dynamics (without hydrodynamics). Some results for Wi = 2 are given in
Fig. C.1, where we also show the analytical solution for the harmonic dimer and the
results for longer tethered chains. When a hard wall is present, the numerical results
show that the position of the peak in the CCF shifts to smaller times and reduces in
height. For the non-linear springs, the position of the peak moves to smaller times as
Wi increases, exactly as we observe for the tethered chains. The height of the peak
is several times larger for a dimer than for a chain with N ≫ 1 beads, which is not
unexpected.
Even after including non-linearity and the hard wall, the dimer model fails to
reproduce the smaller but still substantial negative peak at t < 0 that we observe in
the CCFs for the longer tethered chains at small Wi. An analytical calculation for a
harmonic chain tethered to a point and subjected to shear flow might reproduce that
feature as well. We can mimic such a peak by constructing an artificial CCF,
Cxy(t) = Cxy(t)− αCxy(−t), (C.2)
where 0 < α < 1 controls the depth of the negative peak, and Cxy is the analytical
144
CCF for the harmonic dimer . As illustrated in Fig. C.1, such an empirical fit
matches the numerical results quite well. The Fourier transform of Eq. (C.2) gives
an empirical PSD of the form
S(2πν = Ω/τ) ∼ Wi
√
(1 + α2)(1 + Ω2)− 2α(1− Ω2)
1 + 2Ω2 + Ω4,
which for α > 1/3 exhibits a wide maximum at frequencies Ω = 2πτ/T ∼ 0.5, i.e.,
at a period T ∼ 10τ . As illustrated in Fig. C.1, the maximum in this PSD is very
reminiscent of the “peaks” in the PSD observed in Refs. [127, 37, 38], where they
were attributed to the existence of a periodic motion with period of about 10τ . The
analytical shape of the PSD only involves τ as a relevant timescale, and the cross-
correlation function has an exponential decay at large times ∼ exp(−t/τ), just like
the autocorrelation function for the end-to-end vector used to define relaxation times.
Such an exponential decay is inconsistent with periodic motion, but is consistent with
some recent theoretical models that suggest similar correlations for a free chain in
shear flow [34, 156]. In summary, as seen from this simple analytical example of a
dimer in a flow, a maximum in the PSD does not imply any periodic motion and the
claim of an existence of a new physical timescale other than the internal relaxation
time of the polymer is not justified.
145
0.01 0.1 1
Dimensionless frequency f=ντ
0.01
0.1
1
PS
D
α=1/2α=2/3α=3/4
-2 -1 0 1 2 3
t / τ
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Cxy
Rescaled theory Cxy
(2τ)/2
Harmonic dumbbellFENE dumbbellWormlike dumbbellRescaled chain 3C
xz(τ) (BD, N=20)
Empirical !t ( α=2/3)
PS
D
Cxy
0.3
0.2
0.4
0.1
0.0
-0.1
-0.2-2 -1 0 1 2 3
t/t
1
0.1
0.01
0.01 0.1 1
Dimensionless frequency f=nt
Figure C.1: The left panel shows the cross-correlation function for a dimer (dumbbell)tethered to a hard wall and subjected to shear flow, for a harmonic, FENE anda worm-like spring. We also show a rescaled form of the analytical solution for aharmonic dimer in shear flow (without a hard wall). The height of the peak diminishesby a factor of about 2 when a hard wall is present, so we have rescaled the analyticalsolution for the harmonic dumbbell accordingly. The position of the peak shiftsto smaller times when a hard wall is present as well, and we have thus rescaledthe time for the analytical solution. The CCF for a wormlike chain of N = 20beads, as obtained from Brownian Dynamics simulations, is also shown for qualitativecomparison after scaling by a factor of 3 to bring its height in agreement with thedimer case. We also show an empirical fit to the Brownian Dynamics simulations ofthe form proposed in Eq. (C.2), for which the PSD can be analytically calculatedand shows a maximum at period T ∼ 10τ , depending on the value of the tunableparameter α, as illustrated in the right panel.
146
Appendix D
Fixman’s midpoint algorithm
Fixman[52] proposed a method that avoids computing the derivative of diffusion
tensor, which is generally unknown for complex geometries. As a simple illustration
of how this method works, we start with the one dimensional case with only the drift
caused by the position-dependence of D, and a forward Euler step is simply
∆x = x(t+∆t)− x(t) = (d
dxD(x))∆t +
√2B(x(t))∆w. (D.1)
We can approximate the derivative using a finite difference, in which case this equation
becomes
∆x =D(x(t) + ∆x)−D(x(t))
∆x∆t +
√2B(x(t))∆w. (D.2)
This is an implicit method as information at t +∆t is needed. Fixman proposed an
intermediate step defined as
∆x∗ = x∗(t +∆t)− x(t) =√2B(x(t))∆w. (D.3)
147
This is a valid approximation, as in Eq. D.2 the first term is proportional to ∆t and
the second term is proportional to ∆w ∝ ∆t1/2. When ∆t → 0, the first term is van-
ishingly small compared to the second term. Then the finite difference approximation
of the derivative term becomes
D(x(t) + ∆x)−D(x(t))
∆x≈ D(x∗)−D(x(t))
∆x∗ =D(x∗)−D(x(t))√
2B(x(t))∆w. (D.4)
Inserting this into Eq. D.2 and notice that ∆t = (∆w)2 when ∆t → 0, we finally
have
∆x =D(x∗)−D(x(t))√
2B(x(t))∆w +
√2B(x(t))∆w. (D.5)
The vector version of this, with D = B ·B, is given by
∆r =√2/2(D(r∗)−D(r(t)))(B−1(r(t)))T ·∆w +
√2B(r(t)) ·∆w (D.6)
=√2/2D(r∗)(B−1(r(t)))T ·∆w +
√2/2B(r(t)) ·∆w. (D.7)
We now show that the first term on the RHS of Eq. D.6 equals (∂/∂r ·D)dt when
∆t → 0. The ith component of the first term is√2/2(Dij(r
∗)−Dij(r))(B−1)Tjk∆wk.
Expand Dij around r = r(t)
Dij(r∗) = Dij(r) +
∂Dij
∂rl∆r∗l +O(∆r∗2l )). (D.8)
Plug in Eq. D.3 and ignore the higher order terms:
Dij(r∗)−Dij(r) =
∂Dij
∂rl
√2Blm∆wm. (D.9)
148
Plug this into
√2/2(Dij(r
∗)−Dij(r))(B−1)Tjk∆wk =
√2/2
∂Dij
∂rl
√2Blm∆wm(B
−1)Tjk∆wk
=∂Dij
∂rlBlm(B
−1)Tjk∆tδmk
=∂Dij
∂rlBlkB
−1kj ∆t
=∂Dij
∂rl∆tδlj
=∂Dij
∂rj∆t. (D.10)
In the second step, we use the property of Wiener process dwidwj = δijdt.
149
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