Introduction to the real-coded lattice gas model of colloidal systems

Yasuhiro Inoue

Hirotada Ohashi, Yu Chen, Yasuhiro Hashimoto, Shinnosuke Masuda, Shingo Sato, Tasuku Otani

University of Tokyo, JAPAN

Background - Colloid -Colloid -> particles + a solvent fluid 　　

Particle

solvent

1 nm 10 m

foodsMilk, mayonnaise, iced cream

manufacturePaintings, cosmetics, concrete

Nature

Fog, smoke, polluted water, blood

Innovate new materials,

Analysis on flows in micro devices

Interactions

Particle - Particle Particle - Molecule

External field

Electrochemical, DLVO

induce fluid flows andaffected by others

Brownian motion

fluctuate

Dispersion stabilityInternal structure

Multi-physics and Multi-scale

How to approach ?

Navier-Stokes eq. +Visco-elastic model

Macro scaleContinuum dynamics

dynamics

Micro scaleMolecular dynamics

Meso scalesolute + solvent

Numerical Models

Meso scalesolute + solvent Navier-Stokes eq.

Boltzmann eq.

Newtonian eq.

FDM, FVM

LBM, FDLBM

SPH, MPS Top down

Bottom upLGA, RLG

A particle-model is free from the difficulty of mesh generations

Complex phenomena might be reproduced or mimicked from bottom-up

Algorithm of real-coded lattice gas

Streaming (inertia)

Multi-particle collision

beforeafter

Colloid Particles

Rigid Particle

Deformable Particle

A rigid particle model

For example . . .

• Rigid objects are composed of solid cells.

• The solvent fluid is represented by RLG particles.

Object Solvent

RLG particlesolid cell

Algorithm

The RLG streaming process

The RLG collision process

The RLG - Object interaction

The rigid objects’ motionsTranslations and rotations

Collisions

Δt += τ; if ( Δt < 1 time step )τ time step interval

else1 time step interval

A rigid particle model

Object rule 1• Solid Cell and RLG particles are exclusive to each other.

Solid Cell RLG particle

before after

• Forces exerted on the rigid object surface by bombardments of RLG particles.

The momentum of rigid object is changed with -ΔP.

Calculate the RLG particles’ collision with the object,Calculate the change of their momentum ΔP.

The reflection of RLG particles

Object rule 1

2

2

1exp)( nnnn mccmcP

2

2

1exp

2)( ttt mc

mcP

Vrigid_suface

The reflection of RLG particles

where

after

A new velocity vector is generated randomlyfrom the above probability density distributions.

n

vrlg

n

An assumption: A rigid object is regarded as a heat bath.

: The normal direction of the solid surface

: The tangential direction

Vrigid_suface

before

acerigid_surfrlg Vvc

Object rule 2

Translational velocity vector

Angular velocity vector

Object Motion

before after

before afterCalculate the impulse (white arrows)

Objects Collision

Application

A simpler model on spherical particles

r

The colliding point and its normal vector

An electrochemical potential energy is defined between “center to center”

RLG

normal

Colloid particle

Colloid particle

DLVO particles

2 2 2 2

2 2 2 2

2 2 4ln

4vdwU r R R r R

kT r R r r

*ln 1 exp 2EU r r

kT R

van der Waals attractions

Electrostatic repulsions

H 6A kT

202 R kT

* R

DLVO potential curve varied with

: Amplitude of van der Waals

: Amplitude of a repulsive barrier

: Screen length ratioDLVO is the superposition of van der Waals and repulsions

Internal structures of a colloid

The amplitude of the repulsive barrier could affect the internal structure

=0 ， 10 ： Attractive=20 ， 30 ： Repulsive

t = 5000

=0 =10

=20 =30

0 10

Aggregate forms varied with

20 30

Aggregate forms varied with

Summary: a rigid particle model

Any shape of rigid objects could be modeled by solid cells

Hydrodynamic and electrochemical interparticle interactions could be implemented

Various aggregate forms depending on are demonstrated

A deformable particle model

Red blood cells Vesicles

Background on vesicles

Drug delivery systems

Contrast agents

vesicle

• vesicles could deliver medicines to the target of tissues

• improve the contrast of Doppler images

The size of vesicle should be of the order of micro meter or smaller

5nm

Vesicles are closed thin membrane separating the internal fluid from the external solvent

Fundamental structure of a bio-cell

Flow of vesicles

m

m

1 cm

100

10

1

m

ArteryRe > 100

Arteriole Re < 1

CapillaryRe << 1

Vesicles are regarded as a passive scalar

The correlation between vesicles and blood could not be neglected

A direct modeling of dynamics in this field is required

A vesicle model

Assuming that vesicles would be regarded as immiscible droplets,

vesicle

5nm

m

Neglect membrane

Immiscible droplet

Immiscible multi-component fluids

Vesicle dispersion Immiscible multi-component fluid

Immiscible dropletsExistence of membrane prohibits vesicles from coalescing

A vesicle dispersion could be modeled as an immiscible multi-component fluid

repulsiveattractive

Algorithm of immiscible multi-component rlg fluid

• Color is for difference species• Define interparticle interactions based on color

A rlg particle is colored by either red, blue, green or so oncolor

Different color Same color

Interfaces of multi-component could be reproduced by the above rules

Algorithm: color collision

)(};{

Vvq redi

ired

The Color flux

The Color field

)(};{

Vvq bluei

iblue

is the color gradient

is relative velocities to CM.

Color potential energy

cc

ccccU,

Fq

cF

The color collision is done by a rotation matrix, where U takes the minimum

Phase segregation: 3 species

An example of an immiscible multi-component fluid

6 vesicles + 1 suspending fluid = 7 fluids

1 2 3

4 5 6

12

3

4 5 6

7 7

Time evolution

Brownian motion

Stable dispersion

Aggregate form

time

Micro bifurcation

time

Re ～ 2, Ca ～ 0.001

Zipper-like flow

Flows in a complex network

Summary: a deformable model

Vesicles are regarded as immiscible droplets.

The dispersion stability is able to be controlled by model parameters.

A preliminary example for the application of flows of a vesicle-dispersion in a micro-bifurcation was demonstrated