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Direct Numerical Simulations of Non-Equilibrium Dynamics of Colloids

“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:

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Direct Numerical Simulations of Non-Equilibrium Dynamics of Colloids

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  1. “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)

  2. 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

  3. 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)

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

  5. 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

  6. 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)

  7. 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

  8. Smoothed Profile Method for HI: Phys. Rev. E. 71, 036707 (2005) Profile function No boundary condition, but “body force” appears Regular Cartesian mesh SPM

  9. 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 >>

  10. Numerical test of SPM:1. Drag force This choice can reproduce the collect Stokes drag force within 5% error.

  11. 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.

  12. Demonstration of SPM:3. Repulsive particles + Shear flow

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

  14. Demonstration of SPM:4. LJ attractive particles + Shear flow attraction shear clustering fragmentation ?

  15. 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)

  16. SPM for Charged colloids + Fluid + Ions: need Y(x)in F

  17. SPM for Electrophoresis (SingleParticle) E = 0.01 E = 0.1 E: small → double layer is almost isotropic. E: large → double layer becomes anisotropic.

  18. 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

  19. SPM for Electrophoresis (Single spherical particle) Simulation vs O’Brien-White Z= -500 Z= -100

  20. SPM for Electrophoresis (Dense dispersion) E = 0.1 E = 0.1

  21. Theory for dense dispersions Ohshima (1997) Cell model (mean field) E b k-1 a

  22. SPM for Electrophoresis (Dense dispersion)

  23. 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…).

  24. 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-)

  25. Smoothed Profile method (SPM) : Basic strategy Particle Field smoothening superposition Newton’s Eq. Navier-Stokes Eq. + body force

  26. 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

  27. Our strategy: Solid interface -> Smoothed Profile Smoothening Fluid (NS) Particle (MD) Full domain

  28. 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

  29. 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.

  30. Numerical test: 2. Interaction between a pair of charged rods (cf. LPB) Deviations from LPB become large for r - 2a <lD . For 0.01 < x / 2a < 0.1, deviations are within 5% even at contact position. lD r r-2a=lD contact

  31. Part 1. Charged colloids + ions: Working equations for charged colloid dispersions Free energy functional: Grand potential: for charge neutrality Hellmann-Feynman force:

  32. Smoothed Profile Method becomes almost exact for r -a > ξ Numerical test: 1. Electrostatic Potential around a Charged Rod (cf. PB) 1%

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

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