Brownian Suspensions of Rigid Particles
Aleksandar Donev &Steven Delong, Bakytzhan Kallemov and Florencio Balboa
and others
Courant Institute, New York University
Hydrodynamics at Small Scales, SIAM CSE15Salt Lake City, Utah,
March 18th 2015
A. Donev (CIMS) R-BD 3/2015 1 / 22
Introduction
Non-Spherical Colloids near Boundaries
Figure: (Left) Cross-linked spheres; Kraft et al. [1]. (Right) Lithographedboomerangs; Chakrabarty et al. [2].
A. Donev (CIMS) R-BD 3/2015 2 / 22
Introduction
Bent Active Nanorods
Figure: From the Courant Applied Math Lab of Zhang and Shelley [3]
A. Donev (CIMS) R-BD 3/2015 3 / 22
Introduction
Thermal Fluctuation Flips
QuickTime
A. Donev (CIMS) R-BD 3/2015 4 / 22
Introduction
Steady Stokes Flow (Re→ 0)
Consider a suspension of Nb rigid bodies with positionsQ =
%1, . . . ,%Nb
and orientations Θ = θ1, . . . ,θNb
.We describe orientations using quaternions.
For viscous-dominated flows we can assume steady Stokes flow anddefine the body mobility matrix N (Q,Θ),
[U , Ω]T = N [F , T ]T ,
where the left-hand side collects the linear U = υ1, . . . ,υNb and
angular Ω = ω1, . . . ,ωNb velocities,
and the right hand side collects the applied forcesF (Q,Θ) = F1, . . . ,FNb
and torques T (Q,Θ) = τ 1, . . . , τNb.
A. Donev (CIMS) R-BD 3/2015 5 / 22
Introduction
Brownian Motion
The Brownian motion of the rigid bodies is described by theoverdamped Langevin equation, symbolically:[
dQ/dtdΘ/dt
]=
[UΩ
]= N
[FT
]+ (2kBTN )
12 W (t) .
How to represent orientations using normalized quaternions andhandle the constraint ‖Θk‖ = 1?
What is the correct thermal drift (i.e., what does mean)?
How to compute (the action of) N and N12 and simulate the
Brownian motion of the bodies?
A. Donev (CIMS) R-BD 3/2015 6 / 22
Introduction
Difficulties/Goals
Stochastic drift It is crucial to handle stochastic calculus issues carefullyfor overdamped Langevin dynamics. Since diffusion is slowwe also want to be able to take large time step sizes.
Complex shapes We want to stay away from analytical approximationsthat only work for spherical particles.
Boundary conditions Whenever observed experimentally there aremicroscope slips (glass plates) that modify thehydrodynamics strongly. It is preferred to use no Green’sfunctions but rather work in complex geometry.
Gravity Observe that in all of the examples above there is gravity andthe particles sediment toward the bottom wall, often veryclose to the wall (∼ 100nm). This is a general feature of allactive suspensions but this is almost always neglected intheoretical models.
Many-body Want to be able to scale the algorithms to suspensions ofmany particles–nontrivial numerical linear algebra.
A. Donev (CIMS) R-BD 3/2015 7 / 22
Blob models of complex particles
Blob/Bead Models
Figure: Blob or “raspberry”models of: a spherical colloid, and a lysozyme [4].
The rigid body is discretized through a number of “beads” or “blobs”with positions Q = q1, . . . ,qN.Describe the fluid-blob interaction using a localized smooth kernelδa(r) with compact support of size a giving the effectivehydrodynamic radius of the blob (diffuse sphere).
Standard in fluctuating/stochastic immersed boundary methods butwith stiff springs instead of truly rigid agglomerates.
A. Donev (CIMS) R-BD 3/2015 8 / 22
Blob models of complex particles
Rigidly-Constrained Blobs
∇π − η∇2v =N∑i=1
λiδa (qi − r) +√
2ηkBT ∇ ·W
∇ · v = 0 (Lagrange multiplier is π)
N∑i=1
λi = F (Lagrange multiplier is υ) (1)
N∑i=1
(qi − %0
)× λi = τ (Lagrange multiplier is ω),
∀i :
∫δa (qi − r) v (r, t) dr = υ + ω ×
(qi − %0
)+ slip (Multiplier is λi )
A. Donev (CIMS) R-BD 3/2015 9 / 22
Blob models of complex particles
Notation
Composite velocity U = u1, . . . ,uN andrigidity forces Λ = λ1, . . . ,λN.Define the composite local averaging linear operator J (Q) operator,and the composite spreading linear operator, S (Q) = J ? (Q),
ui = (J v)i =
∫δa (qi − r) v (r, t) dr
λ (r) = (SΛ) (r) =N∑i=1
λiδa (qi − r) .
Denote the (potentially discrete) operators scalar gradient G ≡∇,vector divergence D = −G? ≡∇·, tensor divergence Dv, and vectorLaplacian L = −DvD?
v ≡∇2.
A. Donev (CIMS) R-BD 3/2015 10 / 22
Blob models of complex particles
Saddle-Point Problem
Define the geometric matrix K that converts body kinematics to blobkinematics,
U = KY = K [U ,Ω]T = U + Ω×(Q−Q0
).
We get the symmetric constrained Stokes saddle-point problem,−ηL G −S 0−D 0 0 0−J 0 0 K
0 0 K? 0
vπΛY
=
∇ ·
(√2ηkBT W
)00R
,where Y = [U , Ω]T and R = [F , T ]T , and recall that S = J ?.
A. Donev (CIMS) R-BD 3/2015 11 / 22
Blob models of complex particles
Mobility Matrix
Eliminate velocity and pressure using the Schur complement[M −K−K? 0
] [ΛY
]=
[slip
−(R + R
) ] ,where R =
√2ηkBT K?M−1JL−1DvW are the random
(stochastic) forces and torques.
Here the all-important 3N × 3N blob mobility matrix M is
M = JL−1S,
where L−1 = −L−1 + L−1G(DL−1G
)−1DL−1 denotes the Stokes
solution operator.
A. Donev (CIMS) R-BD 3/2015 12 / 22
Blob models of complex particles
Rigidly-Constrained Blobs
The physical interpretation is simple:
MΛ = KY + slip
K?Λ = R + R,
where the unknown Y = [U , Ω]T are the body kinematics,R = [F , T ]T are the applied forces and torques and R are therandom (stochastic) forces and torques.
Here Λ are the unknown rigidity forces (Lagrange multipliers) actingon the blobs that needs to be solved for.
The 3N × 3N block mobility matrix M has a simple pairwisephysical interpretation:The 3× 3 block Mij maps a force on blob j to a velocity of blob i ,
Mij ≈ η−1
∫δa(qi − r)G(r, r′)δa(qj − r′) drdr′ (2)
where G is the Green’s function (Oseen tensor for unbounded).
A. Donev (CIMS) R-BD 3/2015 13 / 22
Blob models of complex particles
Suspensions of Rigid Bodies
Taking yet one more Schur complement we get[UΩ
]= N
[FT
]+ (2kBTN )
12 W .
The many-body mobility matrix N takes into account rigidity andhigher-order hydrodynamic interactions,
N =(K?M−1K
)−1.
If a fluctuating fluid solver is used it gives an explicit square root of
N12 =
√2kBT NK?M−1JL−1DvW .
Observe that discrete fluctuation-dissipation balance isguaranteed,
N12
(N
12
)?= NK?M−1
(JL−1LL−1S
)M−1KN =
NK?M−1MM−1KN = N(K?M−1K
)N = NN−1N = N .
A. Donev (CIMS) R-BD 3/2015 14 / 22
Blob models of complex particles
How to Approximate the Mobility
If we have a way to approximate the (action of) the mobility matrixM we can also do this without invoking a fluid solver.
We need to be able to solve
N−1Y =(K?M−1K
)Y = R + R,
which we can either do using direct or iterative solvers.
There are different ways to obtain M:
In unbounded domains we can just use the Rotne-Prager-Yamakawatensor (RPY) (always SPD!).In simple geometries such as a single wall we can use a generalizationof RPY [5].For periodic domains we can use Ewald-type summations ornon-uniform FFTs with a fluctuating spectral fluid solver.In more general cases we can use a fluctuating FEM/FVM fluidStokes solver [6, 7].
A. Donev (CIMS) R-BD 3/2015 15 / 22
Results
Brownian motion under gravity
We consider the Brownian motion of a single rigid body near a no-slipboundary.
Temporal integration of the overdamped equations is done using arandom finite different (RFD) approach as described by StevenDelong.
Number of blobs is small and we have a simple geometry so we useapproximate Blake-Rotne-Prager tensor (Brady & Swan [8])
For this test we use direct linear algebra to compute N and
Cholesky factorization to compute N12 .
We add gravity which makes the equilibrium Gibbs-Boltzmanndistribution be
PGB (Q,Θ) ∼ exp
[−mgh + Usteric
kBT
],
where h is the center-of-mass height and Usteric is a Yukawa-typerepulsion with the wall.
A. Donev (CIMS) R-BD 3/2015 16 / 22
Results
Quasi-2D Diffusion
Brownian motion is confined near the bottom wall so it quasi-twodimensional.
Without external forcing the Brownian motion along the wall shouldbe isotropic diffusive at long time scales.
A naive guess for the effective 2D diffusion coefficient would bethe Gibbs-Boltzmann average of the parallel translational mobility:
D‖ = kBT⟨µ‖⟩
GB.
This is in fact a theorem for a sphere because rotational Brownianmotion does not change the mobility.Is it true for non-spherical particles?
A. Donev (CIMS) R-BD 3/2015 17 / 22
Results
MSD for a sphere
0 50 100 150 200time
0
2
4
6
8
10
12
MSD
MSD(t) for Icosahedron with Hydrodynamic Radius = 0.5
RFD Icosahedron Parallel MSDRFD Blob Parallel MSD (a = 0.5)Blob Parallel MobilityRFD Icosahedron Perpendicular MSDRFD Blob perpendicular MSDIcosahedron Asymptotic Perp MSD
Figure: Mean square displacement (MSD) for a non-uniform spherical particle ofunit diameter discretized as an icosahedron of 12 blobs or just a single blob.
A. Donev (CIMS) R-BD 3/2015 18 / 22
Results
MSD for a tetrahedron
0 100 200 300 400 500 600 700time
0
5
10
15
MSD
MSD(t) for TetrahedronFixman Parallel MSDRFD Parallel MSDParallel MobilityFixman Perpendicular MSDRFD Perpendicular MSDAsymptotic Perpendicular MSDAverage Perpendicular Mobility
Figure: MSD for a non-spherical particle (tetrahedron/tetramer).
A. Donev (CIMS) R-BD 3/2015 19 / 22
Results
The choice of tracking point matters
0 50 100 150 200 250 300 350time
0
2
4
6
8
10
12
14
16
MSD
MSD(t) for Free TetrahedronRFD Parallel MSD VertexRFD Parallel MSD CoMMu Parallel VertexMu Parallel CoM
Figure: MSD for a non-spherical particle (tetrahedron/tetramer).
A. Donev (CIMS) R-BD 3/2015 20 / 22
Results
Resolving lubrication forces
10−2
10−1
100
100
102
104
106
ε
F µU
41−cyl221−cyl121−cyl834−cyl39−shellTheory(dense)
Figure: The drag coefficient for a periodic array of cylinders in steady Stokes flowfor close-packed arrays with inter-particle gap ε, showing the correct asymptoticε−
52 lubrication force divergence.
A. Donev (CIMS) R-BD 3/2015 21 / 22
Results
References
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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.
Daisuke Takagi, Adam B Braunschweig, Jun Zhang, and Michael J Shelley.
Dispersion of self-propelled rods undergoing fluctuation-driven flips.Phys. Rev. Lett., 110(3):038301, 2013.
Jose Garcıa de la Torre, Marıa L Huertas, and Beatriz Carrasco.
Calculation of hydrodynamic properties of globular proteins from their atomic-level structure.Biophysical Journal, 78(2):719–730, 2000.
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Generalization of the rotne–prager–yamakawa mobility and shear disturbance tensors.Journal of Fluid Mechanics, 731:R3, 2013.
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.
Pat Plunkett, Jonathan Hu, Christopher Siefert, and Paul J Atzberger.
Spatially adaptive stochastic methods for fluid–structure interactions subject to thermal fluctuations in domains withcomplex geometries.Journal of Computational Physics, 277:121–137, 2014.
James W. Swan and John F. Brady.
Simulation of hydrodynamically interacting particles near a no-slip boundary.Physics of Fluids, 19(11):113306, 2007.
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