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Lattice Boltzmann Simulations of Cholesteric LCs
• Nematics, cholesterics and blue phases
• Computational methods (hybrid LB)
• Free energy, phase diagrams
• Hydrodynamic equations
• What we can do with it:
• BP domain growth kinetics
• Particles in cholesterics
• The elusive BPIII
Collaborators: O. Henrich, J. Lintuvuori, K. Stratford, D. Marenduzzo
Liquid Crystals
s = molecular axis
sa = 0 (apolar)
order parameter
Qab = sasb dab/3
largest eigenvector = n
n
s
Cholesterics: Nematics with a twist
Chiral nematic molecules
nematic: n uniform
Qab = q(nanb dab/3)
in uniaxial case
cholesteric in B field
n cholesteric: n has helical pitch (~ 1mm)
Double twist cylinders
DTC: more stable than cholesteric at high chirality
Problem: Cannot fill space with these
Best solution: Interpenetrating DTCs
Requires defect lines (disclinations)
disclination
Blue Phase Structures (BPI+BPII)
Hierarchically ordered soft matter
• BPI,II now stable over 50 K temperature window • Strong field effect, fast switching
• Display devices, tunable lasers, etc..
H. Kikuchi et al, Nat Mat 1, 64 (2002)
H. J. Coles et al, Nature 436, 997 (2005)
W. Cao et al, Nat Mat 1, 111 (2002)
defect lattice (BPII)
defect lattice (BPI)
BP Devices
15-inch „Blue Phase Mode‟ LCD
240Hz frame rate
Low manufacturing cost
Exhibited by Samsung, 05/08
?? Competitive with OLED
http://www.samsung.com/us/aboutsamsung/news/
newsIrRead.do?news_ctgry=irnewsrelease&news_seq=8351
BP Devices
Tri-directional pumped laser
Mirrorless operation
Tunable colour
W. Cao et al, Nat Mat 1, 111 (2002)
Computational Methods
First principles: molecular dynamics
good, but very limited in length and time scales
one BP unit cell or CLCD pixel = 1010 molecules
supra-unit-cell structure (e.g. BPIII) not accessible
Coarse grained / hydrodynamic description using LB:
Navier Stokes equation for continuum fluid, coupled to:
liquid crystalline ordering (Q tensor)
discrete suspended objects (colloids, polymers)
random thermal forces (Brownian motion)
composition variable (binary fluids)
...
Lattice Boltzmann for Simple Fluids
= fast fluid mechanics on a lattice
• each site x has velocity set {ci}: ci t = lattice vector
• fi(x,t): population of fluid “particles” at x with velocity ci
(x,t) = ∑i fi fluid density v(x,t) = ∑i fi ci+Ft/2 fluid velocity
• local streaming, and relaxation:
fi(x+ci,t+1) – fi(x,t) = t(I+tL/2)ij
-1[∑j Lij (fj(x,t) – fj0(x,t))F.cfi]
• continuum limit: Navier Stokes equation
Lattice Boltzmann for Colloids
Discretizes momentum space
fast momentum transfer across lattice Fully local dynamics
excellent for parallel supercomputers
Typical runs: ~ 105 106 timesteps on 1283 – 10243 lattices
10243: one timestep (~110 ns) takes 4s on a 5 Tf machine (IBM BG/L)
one configuration = 0.2TB memory
Adding colloids (bounce-back of momentum):
K. Stratford et al., J Stat Phys 121, 163 (2005)
Adding thermal noise:
R. Adhikari et al., EPL 71, 473 (2005)
B. Duenweg et al., PRE 76, 036704 (2007)
Adding dipole-dipole interactions (Ewald summation):
E. Kim et al., J Phys Chem B 113, 3681 (2009)
Lattice Boltzmann for Complex Fluids
Either: Add more distribution functions gi(x,t)
Or: couple simple forced LB to FD equations for order parameter(s) φ Continuum limit recovers Navier Stokes coupled to φ dynamics:
e.g. binary fluids: conserved scalar φ=f
Conventional: M. Swift et al, PRE 54, 5041 (1996)
Hybrid: P. Sumesh, I. Pagonabarraga, R. Adhikari, to appear Liquid crystals: nonconserved tensor φ = Qab
Free energy functional (Landau–de Gennes)
f1 = bulk nematic free energy density
A0: energy scale
g ~ temperature
first order transition at g = 2.7, spinodal at g = 3
K = elastic constant (one-constant approximation)
p = 2p/q0 = pitch of cholesteric helix
field energy in external E or B
reduced temperature
t = 27/g – 9
k: reduced chirality
k2 = 108Kq02/A0g
e: reduced field
e2 = 27ea E2/32pA0g
Phase diagram of Landau–de Gennes model
zero field: e = 0
[Periodic phases only]
G. Alexander+ J. Yeomans,
PRE 74, 061706 (2006)
Yang + Crooker,
PRA 35, 4419 (1987)
Numerical phase diagram Classic experiments
t
k
Phase Diagram in E Field (e = 0.3)
Plotted: q-isosurfaces
(enclosing disclinations)
O. Henrich et al.,
PRE 81, 031706 (2010)
Beris-Edwards equations: order parameter f = Qab
Navier Stokes (h = viscosity)
Stress tensor (x = flow alignment parameter)
Order parameter relaxation (G = rotational mobility, S = rotational advector)
Molecular field (F = fdV)
HYBRID LATTICE BOLTZMANN
• single fi(x,t) to handle momentum transport only
• finite difference scheme for Q
• v output from LB gives advection terms for Q
• FD scheme for Q outputs forcing terms (.P) for LB
• exploits 2nd order accuracy of LB for forced simple fluid
D. Marenduzzo et al, PRE 76, 031921 (2007)
• can add thermal noise
R. Adhikari et al, EPL 71, 473 (2005)
• can couple to colloids
J. Lintuvuori et al, J. Mat. Chem., in press; PRL, in press
efficiency, stability simple fluid LB
discretized free energy f[Q] unambiguous
Blue Phase Domain Growth
BPI BPII Phase diagram + Quenches
O. Henrich, K. Stratford, D. Marenduzzo and MEC, PNAS 107, 13212 (2010)
Blue Phase Domain Growth
BPII nucleus in cholesteric
O. Henrich, K. Stratford, D. Marenduzzo and MEC, PNAS 107, 13212 (2010)
Blue Phase Domain Growth
BPII nucleus in cholesteric
O. Henrich, K. Stratford, D. Marenduzzo and MEC, PNAS 107, 13212 (2010)
Blue Phase Domain Growth
BPII nucleus in isotropic
O. Henrich, K. Stratford, D. Marenduzzo and MEC, PNAS 107, 13212 (2010)
Blue Phase Domain Growth
BPII nucleus in isotropic
O. Henrich, K. Stratford, D. Marenduzzo and MEC, PNAS 107, 13212 (2010)
Blue Phase Domain Growth
Hierarchical growth mechanism
Proliferation of topological defects
Initial nucleus disappears
Prolonged arrest in metastable amorphous network
Final transformation to optimal (BPI/II) network
Does not happen spontaneously in simulations
Requires second nucleation
Larger critical nucleus than first stage
O. Henrich, K. Stratford, D. Marenduzzo and MEC, PNAS 107, 13212 (2010)
Particles in Cholesterics
Statics
What is the equilibrium defect structure for planar or homeotropic anchoring?
How does this depend on colloid/pitch size ratio?
Dynamics
Active microrheology: what is force to drag a colloid through the system
(a) in the plane of cholesteric layers?
(b) along the helical axis?
Statics: homeotropic anchoring
R/p = 0
(nematic) R/p = 0.25
R/p = 0.5 R/p = 0.75
disclination line J. Lintuvuori et al, J. Mat. Chem.
in press
Statics: planar anchoring
R/p: 0.25 0.50 0.75
2D boojum
3D disclination line
J. Lintuvuori et al, PRL, in press
Active microrheology: planar anchoring
force is linear in velocity
for both and motion...
J. Lintuvuori et al, PRL, in press
Active microrheology: planar anchoring
Stokes law violation:
force law probes global state, not local material properties
effective viscosity for bulk permeation flow ~ sample size L
Leffective ~ R?
but force not linear in R
when v to helix
J. Lintuvuori et al, PRL, in press
Active microrheology: large v
Ericksen number Er = 2q2vR/GK = 0.7 (previously 0.1)
Disclination line forms a wake Preliminary experiment
A. Pawsey + P. Clegg,
work in progress J. Lintuvuori et al, PRL, in press
Phase diagram of Landau–de Gennes model
zero field: e = 0
[Periodic phases only]
G. Alexander+ J. Yeomans,
PRE 74, 061706 (2006)
Yang + Crooker,
PRA 35, 4419 (1987)
Numerical phase diagram Classic experiments
t
k
Phase diagram of Landau–de Gennes model
zero field: e = 0
[Periodic phases only]
G. Alexander+ J. Yeomans,
PRE 74, 061706 (2006)
Yang + Crooker,
PRA 35, 4419 (1987)
Numerical phase diagram Classic experiments
t
k
BPIII (aperiodic)
Theoretical proposals for BPIII Quasicrystal
Hornreich and Shtrikman PRL 1986
Rokhsar and Sethna PRL 1986
Spaghetti of double-twist cylinders
Hornreich et al, PRL 1982
Amorphous polycrystalline BPII
Belyakov et al, JETP 1986, Collings PRA 1984
A metastable phase
Finn and Cladis Mol Cryst Liq Cryst 1982
Another clue:
BPIII transforms to an ordered phase (BPE) at high field
BPE BPI, BPII or O5
O5 = ordered lattice of lowest known F at high k
O5
Numerical search for BPIII
Lowest F to date:
start from randomly oriented DTCs within C matrix
evolve with full hydrodynamics
visually similar to metastable amorphs in BPI/II quenches
Generic Problem:
finding aperiodic state of lowest F
Generic Answer:
take aperiodic initial states
evolve each dynamically
select that of lowest final F
O. Henrich et al, in review
BPIII structure
Amorphous state minimizes Landau – de Gennes free energy
Configurational entropy would stabilize BPIII further
Equilibrium glass?
S(q): isotropic
(modulo discretization)
O. Henrich et al, in review
BPIII free energy comparison
1 sim unit ~ 108 Pa
kT/unit cell ~ 1 Pa
F~ 100kT/cell
O. Henrich et al, in review
Lattice Boltzmann Simulations of Cholesteric LCs
• Nematics, cholesterics and blue phases
• Computational methods (hybrid LB)
• Free energy, phase diagrams
• Hydrodynamic equations
• What we can do with it:
• BP domain growth kinetics: metastable amorphs intervene
• Particles in cholesterics: Stokes law violations
• The elusive BPIII: equilibrium amorphous network
Collaborators: O. Henrich, J. Lintuvuori, K. Stratford, D. Marenduzzo
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