[PPT] - Brownian dynamics simulations with hard-body interactions: Exact PowerPoint Presentation

SLIDE 1

Brownian dynamics simulations with hard-body interactions: Exact numerical treatment

Ralf Eichhorn

PHYSICAL REVIEW E 83, 065701(R) (2011)

Hard-wall interactions in soft matter systems: Exact numerical treatment

Hans Behringer1,* and Ralf Eichhorn2,†

1University of Mainz, Institute of Physics, Staudinger Weg 7, D-55128 Mainz, Germany 2NORDITA, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden

(Received 29 March 2011; published 20 June 2011)

THE JOURNAL OF CHEMICAL PHYSICS 137, 164108 (2012)

Brownian dynamics simulations with hard-body interactions: Spherical particles

Hans Behringer1,a) and Ralf Eichhorn2,b)

1Johannes Gutenberg-Universität Mainz, Institut für Physik, Staudinger Weg 7, D-55128 Mainz, Germany 2Nordic Institute for Theoretical Physics (NORDITA), Royal Institute of Technology and Stockholm University,

Roslagstullsbacken 23, 106 91 Stockholm, Sweden

SLIDE 2

Brownian motion

interacting particles in a suspension (e.g. colloids)
driven by external forces
in a structured environment
solvent: viscous friction and thermal fluctuations

microfluidics, biomolecules in the cell, self-assembly, polymers, ... ֒ → numerical simulation of particle interaction with hard walls

SLIDE 3

Langevin equation

˙

r(t) = 1

η

F(

r(t), t) + √ 2D ξ(t) two common idealizations/approximations:

overdamped limit (“m = 0”)
hard-body interactions (singular!)

to represent the extremely short-ranged and strong repulsive contact forces

not included in F( r(t), t)

SLIDE 4

Euler algorithm d rt = vt dt + √ 2D dt Gt ,

r(t + dt) =

r(t) + d rt

with vt = 1

η

F( r(t), t), and Gt ∈ N(0, 1)d p(d rt) = 1 √ 4πD dt d exp

−[d

rt − vt dt]2 4D dt

r(t)
r(t + dt)

d rt

SLIDE 5

Euler algorithm d rt = vt dt + √ 2D dt Gt ,

r(t + dt) =

r(t) + d rt

with vt = 1

η

F( r(t), t), and Gt ∈ N(0, 1)d algorithm:

1) detect unphysical configurations (“collisions”) 2) rule to generate physically valid configuration

p(d rt) = ???

r(t)
r(t + dt)

d rt

SLIDE 6

Euler algorithm d rt = vt dt + √ 2D dt Gt ,

r(t + dt) =

r(t) + d rt

with vt = 1

η

F( r(t), t), and Gt ∈ N(0, 1)d algorithm:

1) detect unphysical configurations (“collisions”) 2) rule to generate physically valid configuration

p(d rt) = ???

r(t)
r(t + dt)

d rt

SLIDE 7

Heuristic methods (prominent examples)

rejection scheme

discard unphysical configurations (advance time or not???)

[B. Cichocki and K. Hinsen, Physica A 166, 473 (1990)]

event-driven scheme

propagate fraction of time step ǫdt until “collision point” use rejection scheme for remaining time step (1 − ǫ)dt

[Y.-G. Tao et al., J. Chem. Phys. 124, 134906 (2006)]

lack thorough justification

SLIDE 8

Euler algorithm d rt = vt dt + √ 2D dt Gt ,

r(t + dt) =

r(t) + d rt

with vt = 1

η

F( r(t), t), and Gt ∈ N(0, 1)d p(d rt) = 1 √ 4πD dt d exp

−[d

rt − vt dt]2 4D dt

r(t)
r(t + dt)

d rt

SLIDE 9

Euler algorithm d rt = vt dt + √ 2D dt Gt ,

r(t + dt) =

r(t) + d rt

with vt = 1

η

F( r(t), t), and Gt ∈ N(0, 1)d algorithm:

1) detect unphysical configurations (“collisions”) 2) rule to generate physically valid configuration

p(d rt) = ???

r(t)
r(t + dt)

d rt

SLIDE 10

Euler algorithm d rt = vt dt + √ 2D dt Gt ,

r(t + dt) =

r(t) + d rt

with vt = 1

η

F( r(t), t), and Gt ∈ N(0, 1)d algorithm:

1) detect unphysical configurations (“collisions”) 2) rule to generate physically valid configuration

p(d rt) = ???

r(t)
r(t + dt)

d r⊥ d r

SLIDE 11

Smoluchowski solution

[M. V. Smoluchowski, Phys. Z. 17, 557 (1916)]

driven diffusion on a half-line q ∈ [0, ∞) ∂ ∂tp = Dq ∂2 ∂q2p − vq ∂ ∂qp reflecting boundary at q = 0 −

Dq

∂ ∂qp − vqp

q=0

= 0 initial position q0 at time t = 0 solution: p(q, t; q0) = p1(q, t; q0) + p2(q, t; q0) + p3(q, t; q0)

with p1(q, t; q0) = 1

4πDqt

exp

−(q − q0 − vqt)2

4Dqt

p2(q, t; q0) =

exp

−vqq0

Dq

4πDqt

exp

−(q + q0 − vqt)2

4Dqt

p3(q, t; q0) = − vq

2Dq exp

vqq

Dq

erfc
q + q0 + vqt
4Dqt

q0 = 0.05 q0 = 0.35 q0 = 0.6 vq = 1.0, Dq = 1.0, t = 0.05

SLIDE 14

Smoluchowski solution

[M. V. Smoluchowski, Phys. Z. 17, 557 (1916)]

driven diffusion on a half-line q ∈ [0, ∞) ∂ ∂tp = Dq ∂2 ∂q2p − vq ∂ ∂qp reflecting boundary at q = 0 −

Dq

∂ ∂qp − vqp

q=0

= 0 initial position q0 at time t = 0 solution: p(q, t; q0) = p1(q, t; q0) + p2(q, t; q0) + p3(q, t; q0)

with p1(q, t; q0) = 1

4πDqt

exp

−(q − q0 − vqt)2

4Dqt

p2(q, t; q0) =

exp

−vqq0

Dq

4πDqt

exp

−(q + q0 − vqt)2

4Dqt

p3(q, t; q0) = − vq

2Dq exp

vqq

Dq

erfc
q + q0 + vqt
4Dqt

SLIDE 15

The algorithm

1. Perform standard integration as long as the displacements d

rt do not lead to unphysical configurations (“collisions”)

2. If a suggested displacement d

rt results in a “collision”, replace its component along the “collision axis” by a new dq = q − q0 drawn from p2(q, dt; q0) + p3(q, dt; q0) w with w =

∞

dq [p2(q, dt; q0) + p3(q, dt; q0)] = 1 −

∞

dq p1(q, dt; q0) =

−∞ dq p1(q, dt; q0)

SLIDE 16

The algorithm

“collision axis” ˆ = half-line (with “collision point” at the origin) random number created on the “collision axis”: Q = QG Θ(QG) + QC Θ(−QG) with pG(q) = p1(q) (q ∈ R) and pC(q) = p2(q) + p3(q) w (q ∈ R>0) ⇒ palg(q) =

∞

−∞ dq1

∞

dq2 pG(q1)pC(q2) δ(q − [q1Θ(q1) + q2Θ(−q1)]) =

∞

−∞ dq1

∞

dq2 pG(q1)pC(q2) [Θ(q1)δ(q − q1) + Θ(−q1)δ(q − q2)]) = Θ(q)pG(q) + pC(q)

−∞ dq1 pG(q1)

= p1(q) + p2(q) + p3(q)

SLIDE 17

Euler algorithm for hard wall d rt = vt dt + √ 2D dt Gt ,

r(t + dt) =

r(t) + d r ∗

t with vt = 1

η

F( r(t), t), and Gt ∈ N(0, 1)d algorithm:

1) “suggest” d rt using standard Euler scheme 2a) no “collision”: d r ∗

t = d

rt 2b) “collision”: d r ∗

t = d

rt + (q − q0 − n · d rt) n with n = −d r⊥/|d r⊥| note: d r⊥ and d r are uncorrelated

r(t)
r(t + dt)

d r⊥ d r

SLIDE 18

Euler algorithm for spherical particles d r1 = v1dt +

2D1dt

G1 , d r2 = v2 dt +

2D2 dt

G2

r1(t + dt) =

r1(t) + d r ∗

1 ,

r2(t + dt) =

r2(t) + d r ∗

2 algorithm:

1) “suggest” d r1, d r2 using standard Euler scheme 2a) no “collision”: d r ∗

1 = d

r1 and d r ∗

2 = d

r2 2b) “collision”: d r ∗

1 = d

r1 + η2 η1 + η2 [(d r2 − d r1) · e − (q − q0)] e d r ∗

2 = d

r2 − η1 η1 + η2 [(d r2 − d r1) · e − (q − q0)] e with e = ( r2(t) − r1(t))/| r2(t) − r1(t)| note: center of friction and relative motion are uncorrelated

r1(t)

d r1

r2(t)

d r2

SLIDE 19

Example: Mean first passage time

F

5 10 15

v [µm/s]

5 10 15 20

τ [s]

5 10 15

v [µm/s]

1 2

τcomp [a.u.]

F = (−f, f), v = f/η, particle radius 1 µm, dt = 0.01 s

SLIDE 20

In practice

detection of “collisions”

(any integration scheme)

integration time step dt is determined by

variations of F and curvature of structures/particles

generation of random number q according to distribution

pC(q) = [p2(q) + p3(q)]/w F(q) =

q

0 dq′ pC(q′) =

erfc

q0+vqdt

√

4Dqdt

− exp
vqq

Dq

erfc
q+q0+vqdt

√

4Dqdt

erfc
q0+vqdt

√

4Dqdt

.

Then: q = F −1(x) with x uniformly distributed on [0, 1] is distributed according to pC(q) numerical solution (Brent’s scheme from the GNU scientific library)

SLIDE 21

Brownian dynamics simulations with hard-body interactions: Exact numerical treatment

Ralf Eichhorn

extensions: non-spherical particles corners & wedges many particles (crowding)

PHYSICAL REVIEW E 83, 065701(R) (2011)

Hard-wall interactions in soft matter systems: Exact numerical treatment

Hans Behringer1,* and Ralf Eichhorn2,†

1University of Mainz, Institute of Physics, Staudinger Weg 7, D-55128 Mainz, Germany 2NORDITA, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden

(Received 29 March 2011; published 20 June 2011)

THE JOURNAL OF CHEMICAL PHYSICS 137, 164108 (2012)

Brownian dynamics simulations with hard-body interactions: Spherical particles

Hans Behringer1,a) and Ralf Eichhorn2,b)

1Johannes Gutenberg-Universität Mainz, Institut für Physik, Staudinger Weg 7, D-55128 Mainz, Germany 2Nordic Institute for Theoretical Physics (NORDITA), Royal Institute of Technology and Stockholm University,

Roslagstullsbacken 23, 106 91 Stockholm, Sweden