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Direct Numerical Simulations of Non-Equilibrium Dynamics of Colloids

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“Recent advances in glassy physics”September 27-30, 2005, Paris### Direct Numerical SimulationsofNon-Equilibrium Dynamics of Colloids

Ryoichi Yamamoto

Department of Chemical Engineering, Kyoto University

Project members:

Dr. Kang Kim

Dr. Yasuya Nakayama

Financial support:

Japan Science and Technology Agency (JST)

Outline:

- Introduction: colloid vs. molecular liquidHydrodynamic Interaction (HI)Screened Columbic Interaction (SCI)
- Numerical method: SPM to compute full many-body HI and SCI
- Application 1: Neutral colloid dispersion
- Application 2: Charged colloid dispersion
- Summary and Future:

External electric field: E

Double layer

thickness:k-1

Mobility:m

Radius of colloid:a

Charge of colloid: -Ze

Hydrodynamic Interactions (HI) in colloid dispersions -> long-ranged, many-body

Models for simulation

Brownian Dynamics only with Drag Friction 1/Hmm→no HI

Brownian Dynamics with Oseen Tensor Hnm→long-range HI

Stokesian Dynamics (Brady), Lattice-Boltzman (Ladd)→long-range HI + two-body short-range HI

Direct Numerical Simulation of Navier-Stokes Eq.→full many body HI

Hnm→Oseen tensor

(good for low colloid density)

Importance of HI: Sedimentation

Color map

Blue: u = 0

Red: u = large

1) No HI

2) Full HI

Gravity

Gravity

Gravity

Screened Columbic Interactions (SCI) in charged colloid dispersion -> long-ranged, many-body

Models for simulation

- Effective pair potentials (Yukawa type, DLVO, …)→linearized, neglect many-body effects no external field
- Direct Numerical Simulation of Ionic density by solving Poisson Eq.→full many body SCI (with external field)

External force

anisotropic ionic profile due to external field E

DNS of colloid dispersions:

Density field

of Ions

2. DNS of charged colloid dispersions (HI + SCI)

Coulomb

(Poisson)

Convection

+ Diffusion

Colloid

particles

Hydro

(NS)

Velocity field of solvents

1. DNS of neutral colloid dispersions (HI)

Finite Element Method (NS+MD):

Joseph et al.

FEM

Boundary condition (BC)

(to be satisfied in NS Eq. !!)

Irregular mesh

(to be re-constructed every time step!!)

V1

R1

R2

V2

Smoothed Profile Method for HI:

Phys. Rev. E. 71, 036707 (2005)

Profile function

No boundary condition, but

“body force” appears

Regular Cartesian mesh

SPM

Definition of the body force:

FPD (Tanaka-Araki 2000):

Colloid: fluid with a large viscosity

SPM (RY-Nakayama 2005)

Colloid: solid body

intermediate fluid velocity (uniform hf )

particle velocity

>>

Numerical test of SPM:1. Drag force

This choice can reproduce

the collect Stokes drag force

within 5% error.

Numerical test of SPM:2. Lubrication force

h

F

Two particles are approaching with velocity V under a constant force F. V tends to decrease with decreasing the separation h due to the lubrication force.

Demonstration of SPM:4. LJ attractive particles + Shear flow

attraction

shear

clustering fragmentation

?

DNS of colloid dispersions:Charged systems

Density field

of Ions

2. DNS of charged colloid dispersions (HI + SCI)

Coulomb

(Poisson)

Convection

+ Diffusion

Colloid

particles

Hydro

(NS)

Velocity field of solvents

1. DNS of colloid dispersions

(HI)

SPM for Charged colloids + Fluid + Ions:

need Y(x)in F

SPM for Electrophoresis (SingleParticle)

E = 0.01

E = 0.1

E: small → double layer is almost isotropic.

E: large → double layer becomes anisotropic.

Theory for single spherical particle:Smoluchowski(1918), Hücke(1924), O’Brien-White (1978)

Dielectric constant: e

Fluid viscosity: h

External electric field: E

Double layer

thickness:k-1

Drift velocity: V

Colloid Radius: a

Zeta potential: z

Electric potential

at colloid surface

SPM for Electrophoresis (Dense dispersion)Nonlinear regime

No theory for

E = 0.5

E = 0.1

E: small → regular motion.

E: large → irregular motion (pairing etc…).

Summary

We have developed an efficient simulation method applicable for colloidal dispersions in complex fluids (Ionic solution, liquid crystal, etc).

So far:

- Applied to neutral colloid dispersions (HI):sedimentation, coagulation, rheology, etc
- Applied to charged colloid dispersions (HI+SCI):electrophoresis, crystallization, etc
- All the single simulations were done within a few days on PC

Future:

- Free ware program (2005/12)
- Big simulations on Earth Simulator (2005-)

Smoothed Profile method (SPM) : Basic strategy

Particle

Field

smoothening

superposition

Newton’s Eq.

Navier-Stokes Eq.

+ body force

Numerical implementation of the additional force in SPM:

“="

Although the equations are not shown here, rotational motions of colloids are also taken into account correctly.

Usual boundary method (ξ→０)

Implicit method

Explicit method

Demonstration of SPM:1. Aggregation of LJ particles (2D)

Color mapp

Blue: small p

Red: large p

1) Stokes friction

2) Full Hydro

Pressure heterogeneity -> Network

Smoothed Profile Method for SCI:charged colloid dispersions

Charge density of colloid along the line 0-L

FEM

SPM

0

L

Present SPM

Numerical method

to obtain Y(x)

Iteration

with BC

FFT without BC

(much faster!)

vs.

Numerical test: 2. Interaction between a pair of charged rods (cf. LPB)

Deviations from LPB become large for r - 2a

For 0.01 < x / 2a < 0.1, deviations are within 5% even at contact position.

lD

r

r-2a=lD

contact

Part 1. Charged colloids + ions: Working equations for charged colloid dispersions

Free energy functional:

Grand potential:

for charge

neutrality

Hellmann-Feynman force:

Smoothed Profile Method becomes

almost exact for r -a > ξ

Numerical test: 1. Electrostatic Potential around a Charged Rod (cf. PB)1%

Acknowledgements

1) Project members:

Dr. Yasuya Nakayama

(hydrodynamic effect)

Dr. Kang Kim

(charged colloids)

2) Financial support from JST

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