LATTICE BOLTZMANN SIMULATIONS OF COMPLEX FLUIDS Alexandre Dupuis Davide Marenduzzo Julia Yeomans FROM LIQUID CRYSTALS TO SUPERHYDROPHOBIC SUBSTRATES Rudolph Peierls Centre for Theoretical Physics University of Oxford

molecular dynamics stochastic rotation model dissipative particle dynamics

lattice Boltzmann computational fluid dynamics experiment simulation

The lattice Boltzmann algorithm Define a set of partial distribution functions, f i e i =lattice velocity vector i=1,…,8 (i=0 rest) in 2d i=1,…,14 (i=0 rest) in 3d Streaming with velocity e i Collision operator

The distributions f i are related to physical quantities via the constraints The equilibrium distribution function has to satisfy these constraints The constraints ensure that the NS equation is solved to second order mass and momentum conservation f i eq can be developed as a polynomial expansion in the velocity The coefficients of the expansion are found via the constraints

Permeation in cholesteric liquid crystals Davide Marenduzzo, Enzo Orlandini Wetting and Spreading on Patterned Substrates Alexandre Dupuis

Liquid crystals are fluids made up of long thin molecules orientation of the long axis = director configuration n 1) NEMATICS Long axes (on average) aligned n homogeneous 2) CHOLESTERICS Natural twist (on average) of axes n helicoidal Direction of the cholesteric helix

The director field model considers the local orientation but not the local degree of ordering This is done by introducing a tensor order parameter, 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

Beris-Edwards equations of liquid crystal hydrodynamics coupling between director rotation & flow molecular field ~ -dF/dQ 2. Order parameter evolution 3. Navier-Stokes equation pressure tensor: gives back-flow (depends on Q) 1. Continuity equation

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? How does the channel width affect the flow? What happens if the flow is perpendicular to the helical axis?

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

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

Free energy for droplets bulk term interface term surface term

Wetting boundary conditions An appropriate choice of the free energy leads to Surface free energy Boundary condition for a planar substrate

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.

Impact near the centre of the lyophobic stripe

Impact near a lyophilic stripe

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, Öner et al., Langmuir, 16, 7777, 2000.

Two experimental droplets He et al., Langmuir, 19, 4999, 2003.

Substrate geometry eq =110 o

A suspended superhydrophobic droplet

A collapsed superhydrophobic droplet

Drops on tilted substrates

A suspended drop on a tilted substrate

Droplet velocity

Water capture by a beetle

LATTICE BOLTZMANN SIMULATIONS OF COMPLEX FLUIDS Permeation in cholesteric liquid crystals Plug flow and high viscosity for fixed boundaries Plug flow and normal viscosity for free boundaries Dynamic blue phases at higher forcing Drop dynamics on patterned substrates Lattice Boltzmann can give quantitative agreement with experiment Drop shapes very sensitive to surface patterning Superhydrophobic dynamics depends on interaction of contact line and substrate

Some experiments (by J.Léopoldès)