LATTICE BOLTZMANN SIMULATIONS OF

COMPLEX FLUIDS

Alexandre Dupuis Davide Marenduzzo Julia Yeomans

FROM LIQUID CRYSTALS TO SUPERHYDROPHOBIC SUBSTRATES

Rudolph Peierls Centre for Theoretical Physics University of Oxford

molecular dynamics

stochastic rotation modeldissipative particle dynamics

lattice Boltzmanncomputational fluid dynamics

experiment simulation

The lattice Boltzmann algorithm

Define a set of partial distribution functions, fi

ei=lattice velocity vector

i=1,…,8 (i=0 rest) in 2d

i=1,…,14 (i=0 rest) in 3d

ieqii

fiii ftxftxftxftttexf ,,,

1),(,

Streaming with velocity eiCollision operator

The distributions fi are related to physical quantities via the constraints

i

if i

ii uef

The equilibrium distribution function has to satisfy these constraints

i

eqif

ii

eqi uef

iii

eqi uueef

The constraints ensure that the NS equation is solved to second order

mass and momentum conservation

fieq can be developed as a polynomial expansion in the velocity

iisiississeqi eeEeeuuDuCeuBAf 2

The coefficients of the expansion are found via the constraints

Permeation in cholesteric liquid crystals

Davide Marenduzzo, Enzo Orlandini

Wetting and Spreading on Patterned Substrates

Alexandre Dupuis

Liquid crystals are fluids made up of long thin molecules

orientation of the long axis = director configuration n

1) NEMATICS

Long axes (on average) aligned

n homogeneous

2) CHOLESTERICS

Natural twist (on average) of axes

n helicoidal

Direction of the cholesteric helix

The director field model considers

the local orientation but not the local degree of ordering

This is done by introducing a tensor order parameter, Q

32

3 ijjiij nnQ

ISOTROPIC PHASE

UNIAXIAL PHASE

BIAXIAL PHASE

yyxxyzxz

yzyyxy

xzxyxx

QQQQ

QQQ

QQQ

Q

21

2

1

00

00

00

qq

q

q

Q

q1=q2=0

q1=-2q2=q(T)

q1>q2-1/2q1(T)

3 deg. eig.

2 deg. eig.

3 non-deg. eig.

220020

433/1

2 Q

AQQQ

AQ

Afb

Free energy for Q tensor theory

bulk (NI transition)

distortion 2 220

1 22 2d

K Kf Q Q Qq

surface term 200

2 QQW

f s

Beris-Edwards equations of liquid crystal hydrodynamics

uuuPuut 031

( , )t u Q S W Q H

coupling between director rotation & flow

molecular field ~ -dF/dQ

2. Order parameter evolution

3. Navier-Stokes equation

pressure tensor: gives back-flow (depends on Q)

1. Continuity equation

0 ut

A rheological puzzle in cholesteric LC

Cholesteric viscosity versus temperature from experiments

Porter, Barrall, Johnson, J. Chem Phys. 45 (1966) 1452

PERMEATIONW. Helfrich, PRL 23 (1969) 372

helix direction

flow direction

xy

z

Helfrich:

Energy from pressure gradient balances dissipation from director rotation

Poiseuille flow replaced by plug flow

Viscosity increased by a factor 2 2q h

BUT

What happens to the no-slip boundary conditions?

Must the director field be pinned at the boundaries to obtain a permeative flow?Do distortions in the director field, induced by the flow, alter the permeation?Does permeation persist beyond the regime of low forcing?

How does the channel width affect the flow?

What happens if the flow is perpendicular to the helical axis?

No Back Flowfixed boundaries free boundaries

Free Boundariesno back flow back flow

These effects become larger as the system size is increased

Fixed Boundariesno back flow back flow

Summary of numerics for slow forcing

•With fixed boundary conditions the viscosity increases by ~ 2 orders of magnitude due to back-flow

•This is NOT true for free boundary conditions: in this case one has a plug-like flow and a low (nematic-like) viscosity

•Up to which values of the forcing does permeation persist? What kind of flow supplants it ?

Above a velocity threshold ~5 m/s fixed BC, 0.05-0.1 mm/s free BC

chevrons are no longer stable, and one has a

doubly twisted texture (flow-induced along z + natural along y)

y

z

Permeation in cholesteric liquid crystals

•With fixed boundary conditions the viscosity increases by ~ 2 orders of magnitude due to back-flow

•This is NOT true for free boundary conditions: in this case one has a plug-like flow and a low (nematic-like) viscosity

•Up to which values of the forcing does permeation persist? What kind of flow supplants it ?

•Double twisted structure reminiscent of the blue phase

Lattice Boltzmann simulations of spreading drops:chemically and topologically patterned substrates

2 21 2 3 2b c n n nf p

Free energy for droplets

bulk term

interface term 2

2df n

surface term 1s surfacef n

Wetting boundary conditions

1s surfacef n

1zn

An appropriate choice of the free energy leads to

Surface free energy

Boundary condition for a planar substrate

2/12121

3

)(sincoscos1

3

)(sincoscos22

cw1 p

Spreading on a heterogeneous substrate

Some experiments (by J.Léopoldès)

LB simulations on substrate 4

Evolution of the contact line

Simulation vs experiments

• Two final (meta-)stable state observed depending on the point of impact.

• Dynamics of the drop formation traced.• Quantitative agreement with experiment.

Impact near the centre of the lyophobic stripe

Impact near a lyophilic stripe

LB simulations on substrate 4

Evolution of the contact line

Simulation vs experiments

• Two final (meta-)stable state observed depending on the point of impact.

• Dynamics of the drop formation traced.• Quantitative agreement with experiment.

Effect of the jetting velocity

With an impact velocity

With no impact velocity

t=0 t=20000t=10000t=10000

0

Same point of impact in both simulations

Base radius as a function of time

tR

t0

*

Characteristic spreading velocityA. Wagner and A. Briant

c

2n

nU

R

Superhydrophobic substrates

Bico et al., Euro. Phys. Lett., 47, 220, 1999.

Öner et al., Langmuir,16, 7777, 2000.

Two experimental droplets

He et al., Langmuir, 19, 4999, 2003.

Substrate geometry

eq=110o

A suspended superhydrophobic droplet

A collapsed superhydrophobic droplet

Drops on tilted substrates

A suspended drop on a tilted substrate

Droplet velocity

Water capture by a beetle

LATTICE BOLTZMANN SIMULATIONS OF

COMPLEX FLUIDS

Permeation in cholesteric liquid crystals•Plug flow and high viscosity for fixed boundaries•Plug flow and normal viscosity for free boundaries•Dynamic blue phases at higher forcing

Drop dynamics on patterned substrates•Lattice Boltzmann can give quantitative agreement with experiment•Drop shapes very sensitive to surface patterning•Superhydrophobic dynamics depends on interaction of contact line and substrate

Some experiments (by J.Léopoldès)