Direct Numerical Simulationsof

Non-Equilibrium Dynamics of Colloids

Ryoichi YamamotoDepartment of Chemical Engineering, Kyoto University

Project members:Dr. Kang Kim

Dr. Yasuya Nakayama

Financial support:Japan Science and Technology Agency (JST)

“Recent advances in glassy physics”September 27-30, 2005, Paris

1. Introduction: colloid vs. molecular liquidHydrodynamic Interaction (HI)Screened Columbic Interaction (SCI)

2. Numerical method: SPM to compute full many-body HI and SCI

3. Application 1: Neutral colloid dispersion

4. Application 2: Charged colloid dispersion

5. Summary and Future:

Outline:

External electric field: E

Mobility:

Double layer thickness:

Radius of colloid: aCharge of colloid: -Ze

Hnm 　 →　 Oseen tensor(good for low colloid density)

• 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

Models for simulation

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

Importance of HI: Sedimentation

GravityGravity

1) No HI 2) Full HI

Gravity

Color map Blue: u = 0 Red: u = large

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

External force

• 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)

Models for simulation

anisotropic ionic profile due to external field E

Hydro(NS)

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

1. DNS of neutral colloid dispersions

(HI)

Colloidparticles

Density fieldof Ions

Velocity field of solvents

DNS of colloid dispersions:

Convection+ Diffusion

Coulomb(Poisson)

Finite Element Method (NS+MD):

R1

V1Boundary condition (BC)(to be satisfied in NS Eq. !!)

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

FEM

R2

V2

Joseph et al.

Smoothed Profile Method for HI:

SPMProfile function

No boundary condition, but “body force” appears

Regular Cartesian mesh

Phys. Rev. E. 71,

036707 (2005)

Definition of the body force:

SPM (RY-Nakayama 2005)

Colloid: solid body

FPD (Tanaka-Araki 2000):

Colloid: fluid with a large viscosity

particle velocity intermediate fluid velocity (uniform f )

>>

This choice can reproducethe collect Stokes drag forcewithin 5% error.

Numerical test of SPM: 1. Drag force

Numerical test of SPM: 2. Lubrication force

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.

h

F

Demonstration of SPM: 3. Repulsive particles + Shear flow

Dougherty-Kriger Eqs.

Einstein Eq.

Demonstration of SPM: 3. Repulsive particles + Shear flow

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

clustering fragmentationattraction

shear

?

Hydro(NS)

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

1. DNS of colloid dispersions(HI)

Colloidparticles

Density fieldof Ions

Velocity field of solvents

DNS of colloid dispersions: Charged systems

Convection+ Diffusion

Coulomb(Poisson)

SPM for Charged colloids + Fluid + Ions:

need (x) in F

E: small → double layer is almost isotropic.E: large → double layer becomes anisotropic.

SPM for Electrophoresis (Single Particle)

E = 0.01 E = 0.1

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

External electric field: E

Drift velocity: V

Zeta potential: Electric potentialat colloid surface

Double layer thickness:

Colloid Radius: a

Fluid viscosity: Dielectric constant:

SPM for Electrophoresis (Single spherical particle) Simulation vs O’Brien-White

Z= -100 Z= -500

SPM for Electrophoresis (Dense dispersion)

E = 0.1 E = 0.1

a

b

Cell model (mean field)

E

Theory for dense dispersions Ohshima (1997)

SPM for Electrophoresis (Dense dispersion)

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

E = 0.1 E = 0.5

SPM for Electrophoresis (Dense dispersion)Nonlinear regime

No theory for

Summary

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

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

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

Smoothed Profile method (SPM) : Basic strategy

Particle

Navier-Stokes Eq.+ body force

Newton’s Eq.

smoothening

Field

superposition

“="

Numerical implementation of the additional force in SPM:

Implicit method Explicit method

Usual boundary method (ξ→ ０ )Although the equations are not shown here, rotational motions of colloids are also taken into account correctly.

Our strategy: Solid interface -> Smoothed Profile

Smoothening

Fluid(NS)

Particle(MD)

Full domain

1) Stokes friction 2) Full Hydro

Color mapp Blue: small p Red: large p

Pressure heterogeneity -> Network

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

Smoothed Profile Method for SCI:　 charged colloid dispersions

Charge density of colloid along the line 0-L

FEM

Present SPM

Iterationwith BC

FFT without BC

(much faster!) vs.

Numerical method to obtain (x)

0L

SPM

D

•Deviations from LPB become large for r - 2a < D .

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

r-2a=D

contact

r

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

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

Hellmann-Feynman force:

for charge neutrality

Grand potential:

Free energy functional:

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

Smoothed Profile Method becomesalmost exact for r -a > ξ

1%

Acknowledgements

1) Project members:

Dr. Kang Kim(charged colloids)

Dr. Yasuya Nakayama(hydrodynamic effect)

2) Financial support from JST