LATTICE BOLTZMANN SIMULATIONS OF COMPLEX FLUIDS

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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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Lattice Boltzmann simulations discovering new
physics
Binary fluid phase ordering and flow Wetting
and spreading chemically patterned
substrates superhydrophobic surfaces Liquid
crystal rheology permeation in cholesterics
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• Binary fluids
• The free energy lattice Boltzmann model
• The free energy and why it is a minimum in
  equilibrium
• A model for the free energy Landau theory
• The bulk terms and the phase diagram
• The chemical potential and pressure tensor
• The equations of motion
• The lattice Boltzmann algorithm
• The interface
• Phase ordering in a binary fluid

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The free energy is a minimum in equilibrium
Clausius theorem
Definition of entropy
B
A
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The free energy is a minimum in equilibrium
Clausius theorem
Definition of entropy
B
A
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isothermal
first law
The free energy is a minimum in equilibrium
constant T and V
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The order parameter for a binary fluid
nA is the number density of A nB is the number
density of B The order parameter is
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Models for the free energy
nA is the number density of A nB is the number
density of B The order parameter is
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F
Cahn theory a phenomenological equation for the
evolution of the order parameter
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Landau theory
bulk terms
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Phase diagram
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Gradient terms
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Navier-Stokes equations for a binary fluid
continuity
Navier-Stokes
convection-diffusion
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Getting from F to the pressure P and the chemical
potential
first law
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Homogeneous system
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Inhomogeneous system
Minimise F with the constraint of constant N,
Euler-Lagrange equations
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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

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Navier-Stokes equations for a binary fluid
continuity
Navier-Stokes
convection-diffusion
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The lattice Boltzmann algorithm
Lattice velocity vectors ei, i0,18
Define two sets of partial distribution functions
fi and gi
Evolution equations
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Conditions on the equilibrium distribution
functions
Conservation of NA and NB and of momentum
Pressure tensor
Velocity
Chemical potential
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The equilibrium distribution function
Selected coefficients
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Interfaces and surface tension lines analytic
result points numerical results
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Interfaces and surface tension
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N.B. factor of 2
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lines analytic result points numerical results
surface tension
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Phase ordering in a binary fluid
Alexander Wagner JMY
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Phase ordering in a binary fluid
Diffusive ordering
t -1
L-3
Hydrodynamic ordering
t -1 L t -1 L-1 L-1
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high viscosity diffusive ordering
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high viscosity diffusive ordering
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High viscosity time dependence of different
length scales
L(t)
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low viscosity hydrodynamic ordering
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low viscosity hydrodynamic ordering
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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 energy
3. Spreading on chemically patterned surfaces
4. Mapping to reality
5. Superhydrophobic substrates

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Lattice Boltzmann simulations of spreading
dropschemically and topologically patterned
substrates
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Surface terms in the free energy
Minimising the free energy gives a boundary
condition
The wetting angle is related to h by
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Variation of wetting angle with dimensionless
surface field linetheory pointssimulations
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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

• Two final (meta-)stable state observed depending
  on the point of impact.
• Dynamics of the drop formation traced.
• Quantitative agreement with experiment.

Simulation vs experiments
Evolution of the contact line
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Effect of the jetting velocity
Same point of impact in both simulations
With an impact velocity
t0
t20000
t10000
t100000
With no impact velocity
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Base radius as a function of time
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Characteristic spreading velocityA. Wagner and
A. Briant
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Superhydrophobic substrates
Bico et al., Euro. Phys. Lett., 47, 220, 1999.
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Two droplet states
A suspended droplet
q
A collapsed droplet
q
He et al., Langmuir, 19, 4999, 2003
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Substrate geometry
qeq110o
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Equilibrium droplets on superhydrophobic
substrates
Suspended, q160o
Collapsed, q140o
On a homogeneous substrate, qeq110o
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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
• What is a liquid crystal
• Elastic constants and topological defects
• The tensor order parameter
• Free energy
• Equations of motion
• The lattice Boltzmann algorithm
• Permeation in cholesteric liquid crystals

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An elastic liquid
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topological defects in a nematic liquid crystal
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The order parameter is a tensor Q
ISOTROPIC PHASE
q1q20
3 deg. eig.
q1-2q2q(T)
UNIAXIAL PHASE
2 deg. eig.
q1gtq2?-1/2q1(T)
3 non-deg. eig.
BIAXIAL PHASE
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Free energy for Q tensor theory

bulk (NI transition)
distortion
surface term
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Equations of motion for the order parameter
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The pressure tensor for a liquid crystal
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The lattice Boltzmann algorithm
Lattice velocity vectors ei, i0,18
Define two sets of partial distribution functions
fi and gi
Evolution equations
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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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PERMEATION W. Helfrich, PRL 23 (1969) 372
helix direction
flow direction
Helfrich Energy from pressure gradient balances
dissipation from director rotation Poiseuille
flow replaced by plug flow Viscosity increased
by a factor
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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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z
y
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)
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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 fluid phase ordering and
hydrodynamics two times scales are
important Wetting and spreading chemically
patterned substrates final drop shape determined
by its evolution superhydrophobic
surfaces ?? Liquid crystal rheology permeation
in cholesterics fixed boundaries huge
viscosity free boundaries normal viscosity,
but plug flow
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