LATTICE BOLTZMANN SIMULATIONS OF

COMPLEX FLUIDS

Julia Yeomans

Rudolph Peierls Centre for Theoretical Physics University of Oxford

Binary fluidphase ordering and flow

Wetting and spreadingchemically patterned substratessuperhydrophobic surfaces

Liquid crystal rheologypermeation in cholesterics

Lattice Boltzmann simulations: discovering new physics

Binary fluidsThe free energy lattice Boltzmann model

1. The free energy and why it is a minimum in equilibrium2. A model for the free energy: Landau theory3. The bulk terms and the phase diagram4. The chemical potential and pressure tensor5. The equations of motion6. The lattice Boltzmann algorithm 7. The interface8. Phase ordering in a binary fluid

The free energy is a minimum in equilibrium

dQ0

TÑ

F U TS

Clausius’ theorem

Definition of entropyreversible

dQS

T

B A

A B

dQ dQ0

T T A

B

The free energy is a minimum in equilibrium

dQ0

TÑ

F U TS

Clausius’ theorem

Definition of entropyreversible

dQS

T Ñ

A

B

dQS 0

T A

B

A

B

dQS 0

T

isothermalQ

S 0T

first lawU W

S 0T T

U T S 0 F 0

The free energy is a minimum in equilibrium constant T and V

W

nA is the number density of AnB is the number density of B

The order parameter is

A Bn n

The order parameter for a binary fluid

Models for the free energy

nA is the number density of AnB is the number density of B

The order parameter is

A Bn n

Cahn theory: a phenomenologicalequation for the evolution of the order parameter

d dF

dt d

F

Landau theory

2 24A B

2 4F dV nT ln n

bulk terms

Phase diagram

Gradient terms

22 4A BF dV nT ln n

2 4

Navier-Stokes equations for a binary fluid

t

t

t

n nu 0

nu nu u

1P u u u

3

u D

continuity

Navier-Stokes

convection-diffusion

Getting from F to the pressure P and the chemical potential

F U TS

dF dU TdS SdT

dU TdS PdV V d

dF PdV SdT V d

1 dF

V d

dFP

dV

first law

Homogeneous system

22 4A BF dV nT ln n

2 4

2 4A B

F V nT ln n2 4

31 dFA B

V d

2 4A BF V nT ln n

2 4

2 4

A B A B

3

2

2 4

A B A B

2 4

4

dFP

dV

N N N Nd A B N NV T ln

dV 2 V 4 V VV

V NNT

N V

nT

N N N NA 3B

2 4V VA 3B

2 4

Inhomogeneous system

22 4A BF dV nT ln n

2 4

Minimise F with the constraint of constant N, A BN N

A B

V V

L F N N NdV ndV

Euler-Lagrange equations

L L

0

3A B 0

The pressure tensor

Need to construct a tensor which

• reduces to P in a homogeneous system• has a divergence which vanishes in equilibrium

P 0

24P nTA 3B

2 4

Navier-Stokes equations for a binary fluid

t

t

t

n nu 0

nu nu u

1P u u u

3

u D

continuity

Navier-Stokes

convection-diffusion

The lattice Boltzmann algorithm

i i i ii i i

f n f e nu g Define two sets of partial distribution functions fi and gi

Lattice velocity vectors ei, i=0,1…8

i i i i i ,eq

f

i i i i i ,eq

g

1f t, t t f , t f f

1g t, t t g , t g g

x e x

x e x

Evolution equations

Conditions on the equilibrium distribution functions

eq eq eq

i i i ii i i

eq

i i ii

eq

i ii

eq

i i ii

f n f e nu g

f e e nu u P

g e u

g e e u u

Conservation of NA and NB and of momentum

Pressure tensor

Chemical potential

Velocity

The equilibrium distribution function

2 1 2 0 2

2 2

2 xx x 2 yy y

2 xy 2 yx x y 1 2

eq 2

i i i i i i

eq 2

i i i i

Tr PA A 4A A n 20A

24

Tr P Tr Pp pG G

8 16 8 16

G G G 4G

f A Be u Cu De e u u G e e

g H Ke u Ju Qe e u u

Selected coefficients

Interfaces and surface tension

lines: analytic resultpoints: numerical results

Interfaces and surface tension

3A B 0

3

3

x xB tanh B tanh

2 2

x xtanh tanh

2 2

xtanh

2A B

2d 8 B

dxdx 9

22 4A B d

F dx nT ln n2 4 dx

23

2

3

22 4

dA B 0

dxd d d

A B 0dx d dx

A B d0

2 4 2 dx

N.B. factor of 2

surface tension lines: analytic resultpoints: numerical results

Phase ordering in a binary fluid

Alexander Wagner +JMY

Phase ordering in a binary fluid

t u D

Diffusive ordering

t -1 L-3

1/ 3L t

Hydrodynamic ordering

t

1nu nu u P u u u

3

t -1 L t -1 L-1 L-1

2 / 3L t

high viscosity:diffusive ordering

high viscosity:diffusive ordering

L(t)

High viscosity: time dependence of different length scales

low viscosity:hydrodynamicordering

low viscosity:hydrodynamicordering

Low viscosity: time dependence of different length scales

R(t)

There are two competing growth mechanisms when binary fluids order:

hydrodynamics drives the domains circular

the domains grow by diffusion

Wetting and Spreading

1. What is a contact angle?2. The surface free energy3. Spreading on chemically patterned surfaces4. Mapping to reality5. Superhydrophobic substrates

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

22 4

s

A BF dV nT ln n

2 4

dS h

Surface terms in the free energy

Minimising the free energy gives a boundary condition

s

d h

dz

The wetting angle is related to h by1/ 2

wh 2 Bsign cos 1 cos2

2

warccos sin

Variation of wetting angle with dimensionless surface field

line:theory points:simulations

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.

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.

Two droplet states

A collapsed droplet

A suspended droplet

*

*

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

Substrate geometry

eq=110o

Equilibrium droplets on superhydrophobic substrates

On a homogeneous substrate, eq=110o

Suspended, ~160o

Collapsed, ~140o

Drops on tilted substrates

Droplet velocity

Dynamics of collapsed droplets

Drop dynamics on patterned substrates

•Lattice Boltzmann can give quantitative agreement with experiment•Drop shapes very sensitive to surface patterning•Superhydrophobic dynamics depends on the relative contact angles

Liquid crystals

1. What is a liquid crystal2. Elastic constants and topological defects3. The tensor order parameter4. Free energy5. Equations of motion6. The lattice Boltzmann algorithm7. Permeation in cholesteric liquid crystals

An ‘elastic liquid’

topological defectsin a nematic liquidcrystal

The order parameter is a tensor 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

t u ,

1 1, ( )( ) ( )( )

3 3

12 ( )Tr

3

F 1 FTr

3

S W Q H Q

S W Q D Ω Q I Q I D Ω

Q I QW

H IQ Q

Equations of motion for the order parameter

W u

( ) / 2

( ) / 2

T

T

D W W

Ω W W

1p 2 Q Q H

3

1 1H Q Q H

3 3

FQ Q H

QH Q

The pressure tensor for a liquid crystal

The lattice Boltzmann algorithm

i i i ii i i

f n f e nu G Q

Define two sets of partial distribution functions fi and gi

Lattice velocity vectors ei, i=0,1…8

i i i i i , eq

f

i i i i i , eq

i

i

1f t, t t f , t f f , t

1t, t t , t , t

p

G

x e x x

x e x G G xG G H

Evolution equations

i i i i i ii i i

ii

0 e e e 0p p p

h ,

H Q S W Q

Conditions on the additive terms in the evolution equations

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?

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

Binary fluidphase ordering and hydrodynamicstwo times scales are important

Wetting and spreadingchemically patterned substratesfinal drop shape determined by its evolutionsuperhydrophobic surfaces??

Liquid crystal rheologypermeation in cholestericsfixed boundaries – huge viscosityfree boundaries – normal viscosity, but plug flow