LATTICE BOLTZMANN SIMULATIONS OF COMPLEX FLUIDS Julia Yeomans Rudolph Peierls Centre for Theoretical Physics University of Oxford

Binary fluid phase ordering and flow Wetting and spreading chemically patterned substrates superhydrophobic surfaces Liquid crystal rheology permeation in cholesterics Lattice Boltzmann simulations: discovering new physics

Binary fluids The free energy lattice Boltzmann model 1.The free energy and why it is a minimum in equilibrium 2.A model for the free energy: Landau theory 3.The bulk terms and the phase diagram 4.The chemical potential and pressure tensor 5.The equations of motion 6.The lattice Boltzmann algorithm 7.The interface 8.Phase ordering in a binary fluid

The free energy is a minimum in equilibrium Clausius’ theorem Definition of entropy A B

The free energy is a minimum in equilibrium Clausius’ theorem Definition of entropy A B

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

n A is the number density of A n B is the number density of B The order parameter is The order parameter for a binary fluid

Models for the free energy n A is the number density of A n B is the number density of B The order parameter is

Cahn theory: a phenomenological equation for the evolution of the order parameter F

Landau theory bulk terms

Phase diagram

Gradient terms

Navier-Stokes equations for a binary fluid continuity Navier-Stokes convection-diffusion

Getting from F to the pressure P and the chemical potential first law

Homogeneous system

Inhomogeneous system Minimise F with the constraint of constant N, Euler-Lagrange equations

The pressure tensor Need to construct a tensor which reduces to P in a homogeneous system has a divergence which vanishes in equilibrium

Navier-Stokes equations for a binary fluid continuity Navier-Stokes convection-diffusion

The lattice Boltzmann algorithm Define two sets of partial distribution functions f i and g i Lattice velocity vectors e i, i=0,1…8 Evolution equations

Conditions on the equilibrium distribution functions Conservation of N A and N B and of momentum Pressure tensor Chemical potential Velocity

The equilibrium distribution function Selected coefficients

Interfaces and surface tension lines: analytic result points: numerical results

Interfaces and surface tension

N.B. factor of 2

surface tension lines: analytic result points: numerical results

Phase ordering in a binary fluid Alexander Wagner +JMY

Phase ordering in a binary fluid Diffusive ordering t -1 L -3 Hydrodynamic ordering t -1 L t -1 L -1 L -1

high viscosity: diffusive ordering

high viscosity: diffusive ordering

L(t) High viscosity: time dependence of different length scales

low viscosity: hydrodynamic ordering

low viscosity: hydrodynamic ordering

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 energy 3.Spreading on chemically patterned surfaces 4.Mapping to reality 5.Superhydrophobic substrates

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

Surface terms in the free energy Minimising the free energy gives a boundary condition The wetting angle is related to h by

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=0t=20000t=10000t= Same point of impact in both simulations

Base radius as a function of time

Characteristic spreading velocity A. Wagner and A. Briant

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 =110 o

Equilibrium droplets on superhydrophobic substrates On a homogeneous substrate,  eq =110 o Suspended,  ~160 o Collapsed,  ~140 o

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 crystal 2.Elastic constants and topological defects 3.The tensor order parameter 4.Free energy 5.Equations of motion 6.The lattice Boltzmann algorithm 7.Permeation in cholesteric liquid crystals

An ‘elastic liquid’

topological defects in a nematic liquid crystal

The order parameter is a tensor Q ISOTROPIC PHASE UNIAXIAL PHASE BIAXIAL PHASE q 1 =q 2 =0 q 1 =-2q 2 =q(T) q 1 >q 2  -1/2q 1 (T) 3 deg. eig. 2 deg. eig. 3 non-deg. eig.

Free energy for Q tensor theory bulk (NI transition) distortion surface term

Equations of motion for the order parameter

The pressure tensor for a liquid crystal

The lattice Boltzmann algorithm Define two sets of partial distribution functions f i and g i Lattice velocity vectors e i, i=0,1…8 Evolution equations

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

PERMEATION W. Helfrich, PRL 23 (1969) 372 helix direction flow direction x y z Helfrich: Energy from pressure gradient balances dissipation from director rotation Poiseuille flow replaced by plug flow Viscosity increased by a factor

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 Flow fixed boundaries free boundaries

Free Boundaries no back flow back flow

These effects become larger as the system size is increased

Fixed Boundaries no 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, 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 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