Rigid Multiblob Methods for
Confined Brownian Rigid Particles
Aleksandar Donev, CIMS
Courant Institute, New York University
Soft Matter Seminar
Tufts University
Nov 4th 2015
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Introduction
Non-Spherical Colloids near Boundaries
Figure: (Left) Cross-linked spheres; Kraft et al. [1]. (Right) Lithographed
boomerangs; Chakrabarty et al. [2].
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Introduction
Bent Active Nanorods
Figure: From the Courant Applied Math Lab of Zhang and Shelley
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Introduction
Thermal Fluctuation Flips
QuickTime
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Brownian Motion in a Liquid
Fluctuating Hydrodynamics
We consider a rigid body Ω immersed in an unbounded fluctuating fluid.
In the fluid domain
1
−∇ · σ = ∇π − η∇2 v − (2kB T η) 2 ∇ · Z = 0
∇ · v = 0,
where the fluid stress tensor
1
σ = −πI + η ∇v + ∇T v + (2kB T η) 2 Z
(1)
consists of the usual viscous stress as well as a stochastic stress
modeled by a symmetric white-noise tensor Z (r, t), i.e., a Gaussian
random field with mean zero and covariance
Zij (r, t)Zkl (r0 , t 0 ) = (δik δjl + δil δjk ) δ(t − t 0 )δ(r − r0 ).
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Brownian Motion in a Liquid
Fluid-Body Coupling
At the fluid-body interface the no-slip boundary condition is assumed to
apply,
v (q) = u + q × ω for all q ∈ ∂Ω,
(2)
with the force and torque balance
Z
Z
λ (q) dq = F and
∂Ω
[q × λ (q)] dq = τ ,
(3)
∂Ω
where λ (q) is the normal component of the stress on the outside of the
surface of the body, i.e., the traction
λ (q) = σ · n (q) .
To model activity we can, for example, add active slip on the active parts
of the surface, or add an active stress.
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Brownian Motion in a Liquid
Steady Stokes Flow (Re → 0)
Consider
of Nb rigid bodies with positions
a suspension
Q = %1 , . . . , %Nb and orientations Θ = {θ 1 , . . . , θ Nb }.
We describe orientations using quaternions.
For viscous-dominated flows we can assume steady Stokes flow and
define the body mobility matrix N (Q, Θ),
[U , Ω]T = N [F , T ]T ,
where the left-hand side collects the linear U = {υ 1 , . . . , υ Nb } and
angular Ω = {ω 1 , . . . , ω Nb } velocities,
and the right hand side collects the applied forces
F (Q, Θ) = {F1 , . . . , FNb } and torques T (Q, Θ) = {τ 1 , . . . , τ Nb }.
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Brownian Motion in a Liquid
Brownian Dynamics
The Brownian motion of the rigid bodies is described by the
overdamped Langevin equation, symbolically:
1
dQ/dt
U
F
=
=N
+ (2kB T N ) 2 W (t) .
dΘ/dt
Ω
T
How to represent orientations using normalized quaternions and
handle the constraint kΘk k = 1?
What is the correct thermal drift (i.e., what does mean)?
1
How to compute (the action of) N and N 2 and simulate the
Brownian motion of the bodies?
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Brownian Motion in a Liquid
Difficulties/Goals
Stochastic drift It is crucial to handle stochastic calculus issues carefully
for overdamped Langevin dynamics. Since diffusion is slow
we also want to be able to take large time step sizes.
Complex shapes We want to stay away from analytical approximations
that only work for spherical particles.
Boundary conditions Whenever observed experimentally there are
microscope slips (glass plates) that modify the
hydrodynamics strongly. It is preferred to use no Green’s
functions but rather work in complex geometry.
Gravity Observe that in all of the examples above there is gravity and
the particles sediment toward the bottom wall, often very
close to the wall (∼ 100nm). This is a general feature of all
active suspensions but this is almost always neglected in
theoretical models.
Many-body Want to be able to scale the algorithms to suspensions of
many particles–nontrivial numerical linear algebra.
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Blob models of complex particles
Blob/Bead Models
Figure: Blob or “raspberry” models of a spherical colloid.
The rigid body is discretized through a number of “beads” or “blobs”
with hydrodynamic radius a.
Standard but usually with stiff springs instead of rigid multiblobs.
But first let’s consider blobs that are free to move relative to one
another.
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Blob models of complex particles
Rigidly-Constrained Blobs
The blob-blob mobility matrix M describes the hydrodynamic
relations between the blobs, accounting for the influence of the
boundaries:
U = MF
The 3 × 3 block Mij maps a force on blob j to a velocity of blob i.
For well-separated spheres of radius a we have the Faxen expressions
r=q
a2
a2
Mij ≈ η −1 I + ∇2r
I + ∇2r0 G(r − r0 )r0 =qj
i
6
6
where G is the Green’s function (Oseen tensor for unbounded).
This gives the well-known Rotne-Prager-Yamakawa tensor for the
mobility of pairs of blobs.
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Blob models of complex particles
Rigidly-Constrained Blobs
We add rigidity forces as Lagrange multipliers λ = {λ1 , . . . , λn } to
constrain a group of blobs to move rigidly,
X
Mij λj =u + ω × (ri − q), ∀i
(4)
j
X
λi =F
i
X
(ri − q) × λi =τ ,
i
where u is the velocity of the tracking point q, ω is the angular
velocity of the body around q, F is the total force applied on the
body, τ is the total torque applied to the body about point q, and ri
is the position of blob i.
This can be a very large linear system for suspensions of many bodies
discretized with many blobs: iterative solvers that require a good
preconditioner.
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Blob models of complex particles
Suspensions of Rigid Bodies
In matrix notation we have a linear system of equations for the
rigidity forces Λ and unknown motion Y,
MΛ = KY + slip
K? Λ = R,
where the unknown Y = [U , Ω]T are the body kinematics,
R = [F , T ]T are the applied forces and torques.
Taking the Schur complement of the linear system we get
U
F
= NR = N
+ slip terms.
Y=
Ω
T
The many-body mobility matrix N takes into account rigidity and
higher-order hydrodynamic interactions,
−1
N = K? M−1 K
.
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Blob models of complex particles
How to Approximate the Mobility
In order to make this method work we need a way to compute the
(action of the) blob-blob mobility M.
There are different ways to obtain M:
In unbounded domains we can just use the Rotne-Prager-Yamakawa
tensor (RPY) (always SPD!).
In simple geometries such as a single wall we can use a generalization
of RPY [3].
For periodic domains we can use Ewald-type summations or
non-uniform FFTs with a fluctuating spectral fluid solver [4].
In more general cases we can use a fluctuating FEM/FVM fluid
Stokes solver [5]:
Brownian Dynamics without Green’s functions! [6]
In the grid-based approach adding thermal fluctuations (Brownian
motion) can be done using fluctuating hydrodynamics.
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Rotational Diffusion
Bodies with rotation
We can extend our work to simulate bodies with rotational DOFs by
formulating the appropriate Langevin equation and using a RFD
approach to for temporal integration.
For simplicity, first we consider a single body with only rotational
degrees of freedom.
Orientation is an element of SO(3) so we need to parameterize it: we
use normalized quaternion (point on the unit 4-sphere)
θ ∈ R4 ,
kθk2 = θ · θ = 1.
This offers several advantages over several other common approaches,
such as rotation angles, rotation matrices, and Euler angles.
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Rotational Diffusion
Quaternions
Successive rotations can be accumulated by quaternion
multiplication.
In three dimensions, there exists a 4 × 3 matrix Ψ(θ) such that, given
a conservative potential U(θ),
θ̇ = Ψω,
τ = ΨT ∂θ U(θ).
Here τ is the torque applied to the body, and ω is the angular velocity.
One can also rotate a body by an oriented angle φ, denoted as
θ n+1 = Rotate (θ n , φ) .
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Rotational Diffusion
Equations for Rotation
We assume now that we know the mobility tensor Mωτ ,
ω =Mωτ τ .
Given Mωτ and a potential U(θ), the Overdamped Langevin
Equation for orientation is
1
p
2
∂t θ = − ΨMωτ ΨT ∂θ U + 2kB T ΨMωτ
W
+ kB T ∂θ · ΨMωτ ΨT .
This equation preserves the unit norm constraint and is time
reversible w.r.t. the Gibbs-Boltzmann distribution
Peq (θ) = Z −1 exp (−U (θ) /kB T ) δ θ T θ − 1 .
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Rotational Diffusion
Algorithm with Translation
To include translation, we introduce the matrix Ξ, letting u = q̇
where q is the location of the body,
h
iT
I 0
q̇, θ̇ = Ξ [u, ω]T
Ξ=
,
0 Ψ
The complete overdamped Langevin equations are
p
1
u
∂q U
?
= − (ΞN Ξ )
+ 2kB T ΞN 2 W + (kB T ) ∂x · (ΞNΞ? )
∂θ U
θ̇
We have developed specialized temporal integrators to solve these
equations efficiently for confined bodies [7].
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Rotational Diffusion
Random Finite Difference
To take a time step in a Brownian Dynamics algorithm with
rotational diffusion we do:
f
ṽ =W
q̃ =qn + δũ
θ̃ =Rotate (θ n , δ ω̃)
n
T
v = − NΞ ∂x U
n
r
+
2kB T 1 n n kB T e
f
N2 W +
N − Nn W
∆t
δ
qn+1 =qn + ∆tun
θ n+1 =Rotate (θ n , ∆tω n ) .
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Confined Brownian Motion
Brownian motion under gravity
We consider the Brownian motion of a single rigid body near a no-slip
boundary.
Temporal integration of the overdamped equations is done using a
random finite different (RFD).
Number of blobs is small and we have a simple geometry so we use
approximate Blake-Rotne-Prager tensor (Brady & Swan [3])
For this test we use direct linear algebra to compute N and
1
Cholesky factorization to compute N 2 .
We add gravity which makes the equilibrium Gibbs-Boltzmann
distribution be
mgh + Usteric
PGB (Q, Θ) ∼ exp −
,
kB T
where h is the center-of-mass height and Usteric is a Yukawa-type
repulsion with the wall.
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Confined Brownian Motion
Diffusion of a Confined Boomerang
Quasi-2D (g = 20)
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Confined Brownian Motion
Translational+Rotational Diffusion
We define the total mean square displacement (MSD) at time τ
D(τ ; x) =h∆X(τ ; x) (∆X(τ ; x))T i,
(5)
where ∆X (τ ; x) = (∆q(τ ; x), ∆û(τ ; x)), with orientation increment
∆û(τ ) [1]
3
1X
ui (0) × ui (∆t) .
(6)
∆û (∆t) ≡
2
i=1
The Stokes-Einstein relation gives the short-time mean square
displacement,
hDtrans (τ ; x)i
1
lim
= kB T hMuF (x)i .
(7)
2 τ →0
τ
In general, it is much harder to characterize the long-time diffusion
coefficient
1
hDtrans (τ ; x)i
χlt =
lim
(8)
2 τ →∞
τ
χst =
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Confined Brownian Motion
Quasi-2D Diffusion
Brownian motion is confined near the bottom wall so it quasi-two
dimensional.
Without external forcing the Brownian motion along the wall should
be isotropic diffusive at long time scales.
A naive guess for the effective 2D diffusion coefficient would be
the Gibbs-Boltzmann average of the parallel translational mobility:
Dk = kB T µk GB .
This is in fact a theorem for a sphere because rotational Brownian
motion does not change the mobility.
Is it true for non-spherical particles?
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Confined Brownian Motion
MSD for a sphere
8
Translational MSD
7
6
5
RFD ico par
RFD ico perp
RFD sphere par
RFD sphere perp
sphere par
sphere asymp perp
4
3
2
1
0
0
20
40
60
Time
80
100
120
Figure: Mean square displacement (MSD) for a non-uniform spherical particle of
unit diameter discretized as an icosahedron of 12 blobs or just a single blob.
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Confined Brownian Motion
The choice of tracking point matters
Parallel MSD
10
8
RFD par MSD vertex
RFD par MSD center
avg par mobility vertex
avg par mobility center
6
4
2
00
50
100 150 200 250 300 350 400
Time
Figure: MSD for a non-spherical particle (tetrahedron/tetramer).
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Confined Brownian Motion
Tracking Point
We want the translational MSD to be strictly linear in time so that
the long and short time diffusion coefficients are equal.
Does there exist a choice of tracking point that makes the MSD linear
in time? (No!)
But some candidates for a better choice of tracking point exist.
For any body shape and specific position relative to the boundary,
there exists a unique point in the body called the center of mobility
(CoM) that makes the coupling tensors symmetric,
T
MT
ωF = MωF = Muτ = Muτ .
This is the best tracking point for isotropic (bulk) diffusion.
For some bodies of sufficient symmetry, there exists a point called the
center of hydrodynamic stress (CoH), where the cross-coupling
vanishes,
MωF = Muτ = 0.
Track an approximate CoH for quasi-2D diffusion [2]?
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Confined Brownian Motion
Boomerangs: Translation
40
Translational MSD (um2 )
35
30
25
20
CP par MSD
tip par MSD
tip perp MSD
CoH par MSD
avg CoH par mobility
avg tip par mobility
avg CP par mobility
asymp tip perp MSD
15
10
5
00
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10
15
Time (s)
20
Figure: Translational
MSD for a boomerang
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Confined Brownian Motion
Rotational MSD (rad2 )
Boomerangs: Rotation
400
rotational MSD, g = 1.0
constant χθ = 1.42
350
rotational MSD, g = 10.0
constant χθ = 0.79
300
rotational MSD, g = 20.0
constant χθ = 0.22
250
200
150
100
50
0
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0
50
100
Time (s)
150
Figure: RotationalRigidBlobs
MSD for a boomerang
200
250
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Confined Brownian Motion
References
Daniela J. Kraft, Raphael Wittkowski, Borge ten Hagen, Kazem V. Edmond, David J. Pine, and Hartmut Löwen.
Brownian motion and the hydrodynamic friction tensor for colloidal particles of complex shape.
Phys. Rev. E, 88:050301, 2013.
Ayan Chakrabarty, Andrew Konya, Feng Wang, Jonathan V Selinger, Kai Sun, and Qi-Huo Wei.
Brownian motion of boomerang colloidal particles.
Physical review letters, 111(16):160603, 2013.
James W. Swan and John F. Brady.
Simulation of hydrodynamically interacting particles near a no-slip boundary.
Physics of Fluids, 19(11):113306, 2007.
Eric E. Keaveny.
Fluctuating force-coupling method for simulations of colloidal suspensions.
J. Comp. Phys., 269(0):61 – 79, 2014.
B. Kallemov, A. Pal Singh Bhalla, B. E. Griffith, and A. Donev.
An immersed boundary method for rigid bodies.
To appear in CAMCoS, ArXiv:1505.07865, software available at
https://github.com/stochasticHydroTools/RigidBodyIB, 2016.
S. Delong, F. Balboa Usabiaga, R. Delgado-Buscalioni, B. E. Griffith, and A. Donev.
Brownian Dynamics without Green’s Functions.
J. Chem. Phys., 140(13):134110, 2014.
Software available at https://github.com/stochasticHydroTools/FIB.
S. Delong, F. Balboa Usabiaga, and A. Donev.
Brownian dynamics of confined rigid bodies.
J. Chem. Phys., 143(14), 2015.
Software available at https://github.com/stochasticHydroTools/RotationalDiffusion.
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