LATTICE BOLTZMANN SIMULATIONS OF COMPLEX FLUIDS Julia Yeomans Rudolph Peierls Centre for Theoretical...
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Transcript of LATTICE BOLTZMANN SIMULATIONS OF COMPLEX FLUIDS Julia Yeomans Rudolph Peierls Centre for Theoretical...
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
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