Transcript COMPLEX FLUIDS - University of Warwick

LATTICE BOLTZMANN SIMULATIONS OF
COMPLEX FLUIDS
FROM LIQUID CRYSTALS TO SUPERHYDROPHOBIC SUBSTRATES
Alexandre Dupuis
Davide Marenduzzo
Julia Yeomans
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
ei=lattice velocity vector
i=1,…,8 (i=0 rest) in 2d
i=1,…,14 (i=0 rest) in 3d
Define a set of partial distribution functions,
fi
Streaming with velocity ei
Collision operator


 


1
eq 
f i x  ei t , t  t   f i ( x , t )  
f i x , t   f i x , t ,  f i 
f
The distributions fi are related to physical quantities via the constraints
f
i

i
 f e   u
i i
i
The equilibrium distribution function has to satisfy these constraints

i
f i eq  

f i eqei  u
i
f
eq
i
ei ei     u u 
i
mass and momentum conservation
The constraints ensure that the NS equation is solved to second order
fieq can be developed as a polynomial expansion in the velocity
f i eq  As  Bs u ei  Cs u 2  Ds u u ei ei  Es ei ei
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
 ij
3
Qij 
ni n j 
2
3
 Qxx Qxy
Qxz 


Q   Qxy Qyy
Qyz 


Q
Q

Q

Q
xx
yy 
 xz yz
ISOTROPIC PHASE
0 
 q1 0


Q   0 q2
0 
 0 0 q q 
1
2

q1=q2=0
3 deg. eig.
UNIAXIAL PHASE
q1=-2q2=q(T)
2 deg. eig.
BIAXIAL PHASE
q1>q2-1/2q1(T)
3 non-deg. eig.
Free energy for Q tensor theory
bulk (NI transition)
distortion
surface term
 
A0
A0
A0 2
2
fb  1   / 3Q 
Q Q Q 
Q
2
3
4
2
2
K1
K2
fd     Q     Q  2q0Q 
2
2

W0
0
fs 
Q  Q
2

2
2
Beris-Edwards equations of liquid crystal hydrodynamics
1. Continuity equation
 t    u  0
2. Order parameter evolution

t
 u   Q  S (W , Q )  H
coupling between director rotation & flow
molecular field ~ -dF/dQ
3. Navier-Stokes equation
 t  u   u            1  3  P0  u    u    u 
pressure tensor: gives back-flow (depends on Q)
A rheological puzzle in cholesteric LC
Cholesteric viscosity versus temperature from experiments
Porter, Barrall, Johnson, J. Chem Phys. 45 (1966) 1452
PERMEATION
z
W. Helfrich, PRL 23 (1969) 372
x
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
q2 h 2
y
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 ?
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)
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
fb  pc  n 1  n  2 n  3  2 
2
fd 

2
2
  n 
2
f s  1nsurface
Wetting boundary conditions
Surface free energy
f s  1nsurface
Boundary condition for a planar substrate
1
zn  

An appropriate choice of the free energy leads to
1  2  w

 cos1 (sin2  )  
 cos1 (sin2  )  
 1  cos 
 
2 pc cos 
3
3




 
1/ 2
Spreading on a heterogeneous substrate
Some experiments (by J.Léopoldès)
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
Impact near the centre of the lyophobic
stripe
Impact near a lyophilic stripe
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
Effect of the jetting velocity
Same point of impact in both simulations
With an impact velocity
t=0
With no impact velocity
t=10000
t=20000
t=100000
Base radius as a function of time

t 
t
 R0
*
Characteristic spreading velocity
A. Wagner and A. Briant
U
  nc
 n 
2
 R
Superhydrophobic
substrates
Öner et al., Langmuir,
16, 7777, 2000.
Bico et al., Euro. Phys. Lett.,
47, 220, 1999.
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)