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 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

“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
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
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
Importance of HI: Sedimentation long-ranged, many-body

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
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: dispersion

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
Finite Element Method (NS+MD): dispersion

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
Smoothed Profile Method for HI: dispersion

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

Profile function

No boundary condition, but

“body force” appears

Regular Cartesian mesh

SPM


Definition of the body force
Definition of the body force: dispersion

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
Numerical test of SPM: dispersion1. Drag force

This choice can reproduce

the collect Stokes drag force

within 5% error.


Numerical test of spm 2 lubrication force
Numerical test of SPM: dispersion2. 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: dispersion3. Repulsive particles + Shear flow


Demonstration of spm 3 repulsive particles shear flow1
Demonstration of SPM: dispersion3. Repulsive particles + Shear flow

Dougherty-Kriger Eqs.

Einstein Eq.


Demonstration of spm 4 lj attractive particles shear flow
Demonstration of SPM: dispersion4. LJ attractive particles + Shear flow

attraction

shear

clustering fragmentation

?


DNS of colloid dispersions: dispersionCharged 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 dispersion Charged colloids + Fluid + Ions:

need Y(x)in F


SPM for dispersionElectrophoresis (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 h cke 1924 o brien white 1978
Theory for single spherical particle: dispersionSmoluchowski(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 (Single spherical particle) dispersion Simulation vs O’Brien-White

Z= -500

Z= -100


SPM for Electrophoresis (Dense dispersion) dispersion

E = 0.1

E = 0.1


Theory for dense dispersions ohshima 1997
Theory for dense dispersions dispersion Ohshima (1997)

Cell model

(mean field)

E

b

k-1

a


SPM for dispersionElectrophoresis (Dense dispersion)


SPM for dispersionElectrophoresis (Dense dispersion)Nonlinear regime

No theory for

E = 0.5

E = 0.1

E: small → regular motion.

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


Summary
Summary dispersion

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
Smoothed Profile method (SPM) : dispersion Basic strategy

Particle

Field

smoothening

superposition

Newton’s Eq.

Navier-Stokes Eq.

+ body force


Numerical implementation of the additional force in spm
Numerical implementation of the additional force in SPM: dispersion

“="

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


Our strategy solid interface smoothed profile
Our strategy: dispersion Solid interface -> Smoothed Profile

Smoothening

Fluid

(NS)

Particle

(MD)

Full domain


Demonstration of spm 1 aggregation of lj particles 2d
Demonstration of SPM: dispersion1. 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
Smoothed Profile Method for SCI: dispersioncharged 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
Numerical test: dispersion2. 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


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

Smoothed Profile Method becomes dispersion

almost exact for r -a > ξ

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

1%


Acknowledgements
Acknowledgements dispersion

1) Project members:

Dr. Yasuya Nakayama

(hydrodynamic effect)

Dr. Kang Kim

(charged colloids)

2) Financial support from JST


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