lattice boltzmann simulations of complex fluids julia yeomans rudolph peierls centre for theoretical...
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LATTICE BOLTZMANN SIMULATIONS OF
COMPLEX FLUIDS
Julia Yeomans
Rudolph Peierls Centre for Theoretical Physics University of Oxford
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Binary fluidphase ordering and flow
Wetting and spreadingchemically patterned substratessuperhydrophobic surfaces
Liquid crystal rheologypermeation in cholesterics
Lattice Boltzmann simulations: discovering new physics
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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
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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
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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
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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
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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
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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
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Cahn theory: a phenomenologicalequation for the evolution of the order parameter
d dF
dt d
F
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Landau theory
2 24A B
2 4F dV nT ln n
bulk terms
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Phase diagram
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Gradient terms
22 4A BF dV nT ln n
2 4
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Interfaces and surface tension
lines: analytic resultpoints: numerical results
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Interfaces and surface tension
3A B 0
3
3
x xB tanh B tanh
2 2
x xtanh tanh
2 2
xtanh
2A B
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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
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surface tension lines: analytic resultpoints: numerical results
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Phase ordering in a binary fluid
Alexander Wagner +JMY
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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
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high viscosity:diffusive ordering
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high viscosity:diffusive ordering
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L(t)
High viscosity: time dependence of different length scales
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low viscosity:hydrodynamicordering
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low viscosity:hydrodynamicordering
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Low viscosity: time dependence of different length scales
R(t)
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There are two competing growth mechanisms when binary fluids order:
hydrodynamics drives the domains circular
the domains grow by diffusion
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Wetting and Spreading
1. What is a contact angle?2. The surface free energy3. Spreading on chemically patterned surfaces4. Mapping to reality5. Superhydrophobic substrates
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Lattice Boltzmann simulations of spreading drops:chemically and topologically patterned substrates
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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
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Variation of wetting angle with dimensionless surface field
line:theory points:simulations
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Spreading on a heterogeneous substrate
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Some experiments (by J.Léopoldès)
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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.
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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
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Base radius as a function of time
tR
t0
*
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Characteristic spreading velocityA. Wagner and A. Briant
c
2n
nU
R
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Superhydrophobic substrates
Bico et al., Euro. Phys. Lett., 47, 220, 1999.
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Two droplet states
A collapsed droplet
A suspended droplet
*
*
He et al., Langmuir, 19, 4999, 2003
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Substrate geometry
eq=110o
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Equilibrium droplets on superhydrophobic substrates
On a homogeneous substrate, eq=110o
Suspended, ~160o
Collapsed, ~140o
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Drops on tilted substrates
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Droplet velocity
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Dynamics of collapsed droplets
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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
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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
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An ‘elastic liquid’
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topological defectsin a nematic liquidcrystal
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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.
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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
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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
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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
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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
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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
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A rheological puzzle in cholesteric LC
Cholesteric viscosity versus temperature from experiments
Porter, Barrall, Johnson, J. Chem Phys. 45 (1966) 1452
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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
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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?
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No Back Flowfixed boundaries free boundaries
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Free Boundariesno back flow back flow
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These effects become larger as the system size is increased
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Fixed Boundariesno back flow back flow
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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 ?
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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
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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
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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
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