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)

Cholesterics: Bistable (powerless) LCDs

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: homeotropic anchoring, R/p=0.50

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

BPIII: field response

t = 0.2, k = 2.0, e = 0.55

BPE candidate

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