Recent advances in glassy physics September 27-30, 2005, Paris Direct Numerical Simulations of Non-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: 1.

Introduction: colloid vs. molecular liquid Hydrodynamic Interaction (HI) Screened Columbic Interaction (SCI) 2. Numerical method: SPM to compute full many-body HI and SCI 3. Application 1: Neutral colloid dispersion External electric field: E 4.

Application 2: Charged colloid dispersion Double layer thickness: Mobility: Radius of colloid: 5. Summary and Future:

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

Brownian Dynamics with Oseen Tensor Hnm long-range HI Stokesian Dynamics (Brady), Lattice-Boltzman (Ladd)

Hnm Oseen tensor (good for low colloid density) no HI long-range HI + two-body short-range HI Direct Numerical Simulation of Navier-Stokes Eq. full many body HI

Importance of HI: Sedimentation 1) No HI 2) Full HI Gravity Color map Blue: u = 0 Red: u = large Gravity Gravity

Screened Columbic Interactions (SCI) in charged colloid dispersion -> long-ranged, many-body Models for simulation Effective pair potentials (Yukawa type, DLVO, ) anisotropic ionic profile

due to external field E linearized, neglect many-body effects no external field Direct Numerical Simulation of Ionic density by solving Poisson Eq. External force full many body SCI

(with external field) DNS of colloid dispersions: Density field of Ions Coulomb (Poisson) Colloid particles 2. DNS of charged colloid dispersions

(HI + SCI) Convection + Diffusion Hydro (NS) Velocity field of solvents 1. DNS of neutral colloid dispersions (HI)

Finite Element Method (NS+MD): Joseph et al. FEM V1 R1 R2 V2 Boundary condition (BC) (to be satisfied in NS Eq. !!) Irregular mesh

(to be re-constructed every time step!!) Smoothed Profile Method for HI: SPM Profile function Phys. Rev. E. 71, 036707 (2005) No boundary condition, but body force appears Regular Cartesian mesh

Definition of the body force: SPM (RY-Nakayama 2005) FPD (Tanaka-Araki 2000): Colloid: solid body Colloid: fluid with a large viscosity particle velocity intermediate fluid velocity (uniform f )

>> 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: 3. Repulsive particles + Shear flow Demonstration of SPM: 3. Repulsive particles + Shear flow Dougherty-Kriger Eqs. Einstein Eq.

Demonstration of SPM: 4. LJ attractive particles + Shear flow attraction clustering shear fragmentation ? DNS of colloid dispersions: Charged systems

Density field of Ions Coulomb (Poisson) Colloid particles 2. DNS of charged colloid dispersions (HI + SCI) Convection + Diffusion

Hydro (NS) Velocity field of solvents 1. DNS of colloid dispersions (HI) SPM for Charged colloids + Fluid + Ions: need (x) in F

SPM for Electrophoresis (Single Particle) 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), Hcke(1924), OBrien-White (1978) Dielectric constant: Fluid viscosity: External electric field: E

Drift velocity: V Colloid Radius: a Double layer thickness: Zeta potential: Electric potential at colloid surface SPM for Electrophoresis (Single spherical particle) Simulation vs OBrien-White

Z= -100 Z= -500 SPM for Electrophoresis (Dense dispersion) E = 0.1 E = 0.1 Theory for dense dispersions Ohshima (1997) Cell model (mean field)

E b a SPM for Electrophoresis (Dense dispersion) SPM for Electrophoresis (Dense dispersion) Nonlinear regime No theory for E = 0.1 E = 0.5

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 Newtons Eq. Navier-Stokes Eq. + body force Numerical implementation of the additional force in SPM:

=" Although the equations are not shown Usual boundary method ( ) here, rotational motions of colloids also taken into account correctly. Implicit method Explicitare method Our strategy: Solid interface -> Smoothed Profile Smoothening

Full domain Fluid (NS) Particle (MD) Demonstration of SPM: 1. Aggregation of LJ particles (2D) 1) Stokes friction Color mapp Blue: small p 2) Full Hydro Red: large p

Pressure heterogeneity -> Network Smoothed Profile Method for SCI: charged colloid dispersions Charge density of colloid along the line 0-L FEM 0 SPM L Present SPM

Numerical method to obtain (x) Iteration with BC vs. FFT without BC (much faster!) Numerical test: 2. Interaction between a pair of charged rods (cf. LPB)

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 r-2a=D contact Part 1. Charged colloids + ions: Working

equations for charged colloid dispersions Free energy functional: Grand potential: for charge neutrality Hellmann-Feynman force: Numerical test: 1. Electrostatic Potential around a Charged Rod (cf. PB) 1% Smoothed Profile Method becomes almost exact for r -a >

Acknowledgements 1) Project members: Dr. Kang Kim Dr. Yasuya Nakayama (charged colloids) (hydrodynamic effect) 2) Financial support from JST