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

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 4.Application 2: Charged colloid dispersion 5.Summary and Future: Outline: External electric field: E Mobility:  Double layer thickness:   Radius of colloid: a Charge of colloid: -Ze

H nm → Oseen tensor (good for low colloid density) Brownian Dynamics only with Drag Friction 1/H mm → no HI Brownian Dynamics with Oseen Tensor H nm → 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 Gravity 1) No HI2) 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) Colloid particles Density field of Ions Velocity field of solvents DNS of colloid dispersions: Convection + Diffusion Coulomb (Poisson)

Finite Element Method (NS+MD): R1R1 V1V1 Boundary condition (BC) (to be satisfied in NS Eq. !!) Irregular mesh (to be re-constructed every time step!!) FEM R2R2 V2V2 Joseph et al.

Smoothed Profile Method for HI: SPM Profile function No boundary condition, but “body force” appears Regular Cartesian mesh Phys. Rev. E. 71, (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 reproduce the collect Stokes drag force within 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 fragmentation attraction shear ?

Hydro (NS) 2. DNS of charged colloid dispersions (HI + SCI) 1. DNS of colloid dispersions (HI) Colloid particles Density field of 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 = 0.01E = 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 potential at 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

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 methodExplicit 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 friction2) Full Hydro Color mapp Blue: small p Red: large p Pressure heterogeneity -> Network Demonstration of SPM: 1. Aggregation of LJ particles (2D)

1) Stokes friction 2) Full Hydro Demonstration of SPM: 2. Aggregation of LJ particles (3D)

Smoothed Profile Method for SCI: charged colloid dispersions Charge density of colloid along the line 0-L FEM Present SPM Iteration with BC FFT without BC (much faster !) vs. Numerical method to obtain  (x) 0 L 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 becomes almost 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