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

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### 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

10 mm

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

Particle

foods

Milk, mayonnaise, iced cream

manufacture

Paintings, cosmetics, concrete

Nature

Fog, smoke, polluted water, blood

solvent

Innovate new materials,

Analysis on flows in micro devices

Interactions

Particle - Particle

Particle - Molecule

fluctuate

Electrochemical, DLVO

Brownian motion

Dispersion stability

Internal structure

External field

induce fluid flows

and

affected by others

Multi-physics and Multi-scale

How to approach ?

Macro scale

Continuum dynamics

Navier-Stokes eq.

+

Visco-elastic model

Meso scale

solute + solvent

dynamics

Micro scale

Molecular dynamics

Numerical Models

Meso scale

solute + solvent

Navier-Stokes eq.

FDM, FVM

Boltzmann eq.

LBM, FDLBM

Newtonian eq.

Top down

SPH, MPS

LGA, RLG

Bottom up

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

Complex phenomena might be reproduced or mimicked from bottom-up

Colloid Particles

- Rigid Particle
- Deformable Particle

RLG particle

A rigid particle model- The solvent fluid is represented by RLG particles.

- Rigid objects are composed of solid cells.

For example . . .

Object

Solvent

τ time step interval

AlgorithmThe RLG streaming process

The RLG - Object interaction

Translations and rotations

The rigid objects’ motions

Collisions

Δt += τ; if ( Δt < 1 time step )

else

1 time step interval

The RLG collision process

Object rule 1

The reflection of RLG particles

- 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.

Calculate the RLG particles’ collision with the object,

Calculate the change of their momentum ΔP.

The momentum of rigid object is changed with -ΔP.

Object rule 1

The reflection of RLG particles

An assumption:

A rigid object is regarded as a heat bath.

: The normal direction of the solid surface

: The tangential direction

where

A new velocity vector is generated randomly

from the above probability density distributions.

n

n

Vrigid_suface

Vrigid_suface

vrlg

after

before

before

after

Objects Collision

before

Calculate the impulse

(white arrows)

after

Object rule 2Translational velocity vector

Angular velocity vector

A simpler model on spherical particles

Colloid particle

r

Colloid particle

An electrochemical potential energy

is defined between “center to center”

normal

RLG

The colliding point and its normal vector

DLVO particles

van der Waals attractions

Electrostatic repulsions

DLVO potential curve varied with h

a: Amplitude of van der Waals

h: Amplitude of a repulsive barrier

k: Screen length ratio

DLVO is the superposition of van der Waals and repulsions

Internal structures of a colloid

h=0

h=10

h=0，10 ： Attractive

h=20，30 ： Repulsive

h=20

h=30

The amplitude of the repulsive barrier could affect the internal structure

t = 5000

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 h are demonstrated

A deformable particle model

- Red blood cells
- Vesicles

Background on vesicles

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

5nm

Fundamental structure of a bio-cell

Drug delivery systems

- vesicles could deliver medicines to the target of tissues

Contrast agents

- improve the contrast of Doppler images

vesicle

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

Flow of vesicles

1 cm

Vesicles are regarded as a passive scalar

Artery

Re > 100

100

Arteriole

Re < 1

The correlation between vesicles and blood

could not be neglected

10

Capillary

Re << 1

1

A direct modeling of dynamics in this field is required

A vesicle model

5nm

Neglect membrane

Immiscible droplet

vesicle

Assuming that vesicles would be regarded as immiscible droplets,

Immiscible multi-component fluids

Existence of membrane prohibits vesicles from coalescing

Immiscible droplets

Immiscible multi-component fluid

Vesicle dispersion

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

Algorithm of immiscible multi-component rlg fluid

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

color

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

repulsive

attractive

Same color

Different color

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

Algorithm: color collision

The Color field

is the color gradient

The Color flux

is relative velocities to CM.

Color potential energy

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

An example of an immiscible multi-component fluid

6 vesicles + 1 suspending fluid = 7 fluids

1

3

1

2

3

2

7

7

5

6

4

5

6

4

Time evolution

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

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