LATTICE BOLTZMANN SIMULATIONS OF COMPLEX FLUIDS
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Transcript of LATTICE BOLTZMANN SIMULATIONS OF COMPLEX FLUIDS
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 modeldissipative particle dynamics
lattice Boltzmanncomputational fluid dynamics
experiment simulation
The lattice Boltzmann algorithm
Define a set of partial distribution functions, fi
ei=lattice velocity vectori=1,…,8 (i=0 rest) in 2d
i=1,…,14 (i=0 rest) in 3d
ieqii
fiii ftxftxftxftttexf ,,,1),(,
Streaming with velocity ei Collision operator
The distributions fi are related to physical quantities via the constraints
i
if i
ii uef
The equilibrium distribution function has to satisfy these constraints
i
eqif
ii
eqi uef
iii
eqi uueef
The constraints ensure that the NS equation is solved to second order
mass and momentum conservation
fieq can be developed as a polynomial expansion in the velocity
iisiississeqi eeEeeuuDuCeuBAf 2
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) NEMATICSLong axes (on average) aligned
n homogeneous
2) CHOLESTERICSNatural twist (on average) of axes
n helicoidal
Direction of the cholesteric helix
The director field model considersthe local orientation but not the local degree of ordering
This is done by introducing a tensor order parameter, Q
323 ij
jiij nnQ
ISOTROPIC PHASE
UNIAXIAL PHASE
BIAXIAL PHASE
yyxxyzxz
yzyyxy
xzxyxx
QQQQQQQQQQ
Q
21
2
1
000000
qqq
q1=q2=0q1=-2q2=q(T)
q1>q2-1/2q1(T)
3 deg. eig.
2 deg. eig.3 non-deg. eig.
220020
433/1
2
QA
QQQA
QA
fb
Free energy for Q tensor theory
bulk (NI transition)
distortion 2 220
1 22 2dK K
f Q Q Qq
surface term 200
2 QQW
f s
Beris-Edwards equations of liquid crystal hydrodynamics
uuuPuut 031
( , )t u Q S W Q H
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
0 ut
A rheological puzzle in cholesteric LC
Cholesteric viscosity versus temperature from experimentsPorter, 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?
How does the channel width affect the flow?
What happens if the flow is perpendicular to the helical axis?
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
Lattice Boltzmann simulations of spreading drops:chemically and topologically patterned substrates
2 21 2 3 2b c n n nf p
Free energy for droplets
bulk term
interface term 2
2df n
surface term 1s surfacef n
Wetting boundary conditions
1s surfacef n
1zn
An appropriate choice of the free energy leads to
Surface free energy
Boundary condition for a planar substrate
2/12121
3)(sincoscos1
3)(sincoscos22
cw1 p
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=0 t=20000t=10000 t=100000
Same point of impact in both simulations
Base radius as a function of time
tR
t0
*
Characteristic spreading velocityA. Wagner and A. Briant
c
2nn
UR
Superhydrophobic substrates
Bico et al., Euro. Phys. Lett., 47, 220, 1999.
Öner et al., Langmuir,16, 7777, 2000.
Two experimental droplets
He et al., Langmuir, 19, 4999, 2003.
Substrate geometry
eq=110o
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)