Orientation and Flow in Liquid CrystalsMichael P Allen

H HWills Physics Laboratory, Royal Fort, Tyndall Avenue, BristolandDepartment of Physics, University of Warwick, Coventry

Outline

Introduction

Nematic Elasticity

Nematic Hydrodynamics

Liquid CrystalsTimeline

1880 1900 1920 1940 1960 1980 2000discoveryReinitzer, Lehmann characterizationFriedel

discotics, Chandrasekharroom-T nematics, Graytwist cells, Helfrich

1880 1900 1920 1940 1960 1980 2000

OnsagerOseen

Ericksen Leslie

Frankde Gennes

FrenkelElastic TheoryHydrodynamicsSimulations

TJ Sluckin, DA Dunmur, H Stegemeyer. Crystals that Flow: Classicpapers from the history of liquid crystals. Taylor and Francis (2004).DA Dunmur, TJ Sluckin. Soap, Science and Flat-Screen TVs: A history ofliquid crystals. OUP (2011).

Liquid Crystal SimulationsMolecular Models

Calamitic and Discotic Mesogens

OO

OO

OO

CN

Liquid Crystal SimulationsCoarse-Grained Models

Discs, Rods, Ellipsoids

Liquid Crystal SimulationsInteraction Potentials

The Gay-Berne Potential

0.0 1.0 2.0 3.0 4.0 5.0

−4.0

−2.0

0.0

2.0

4.0

r

φGB(r)

JG Gay, BJ Berne. J Chem Phys, 74, 3316 (1981).

Liquid CrystalsOrientational Order

I Short-ranged positional order.I Long-ranged orientational order.I The director n: a unit vector.I Second-rank order: n↔ −n.I Magnitude of ordering: S.

I Global reorientation of n costs zero free energy.I Quasi-conserved: influences hydrodynamics.I Molecular simulation:

I orientational elasticity;I effect on nematic flow.

Outline

Introduction

Nematic Elasticity

Nematic Hydrodynamics

Orientational ElasticitySpatial deformations of director n incur free energy penalty.Elastic Free Energy

∆F =

∫dr f

[n(r)

]f[n] = 1

2K1

(∇ · n

)2︸ ︷︷ ︸splay+ 12K2

(n · ∇ × n

)2︸ ︷︷ ︸twist+ 12K3

∣∣∣n× (∇× n

)∣∣∣2︸ ︷︷ ︸bend

I Description valid at long wavelengths.I Elastic constants K1, K2, K3.I Fourier transform: ∇ → ik.

Director Fluctuation ModesSplay-Bend and Twist-Bend

e3

e2e1

n1 n2

n

k

n = (0, 0, 1)

δn =(n1, n2, 0

)k = (k1, 0, k3)

bend k3splay

k1 bendk3twist

Director Fluctuation ModesFree Energy in Fourier Space

∆F =1

V

∑k

f(k)

f(k) = 12

(K1k

21 + K3k

23

)∣∣n1(k)∣∣2︸ ︷︷ ︸splay-bend+ 12

(K2k

21 + K3k

23

)∣∣n2(k)∣∣2︸ ︷︷ ︸twist-bend

Equipartition of (free) energy:Splay-bend: W13 ∝

⟨|n1(k)|

2⟩−1 ∝ K1k21 + K3k23

Twist-bend: W23 ∝⟨|n2(k)|

2⟩−1 ∝ K2k21 + K3k23

Inverse Orientational Fluctuations

N = 5× 105 Molecules

A Humpert, MP Allen. Elastic constants and dynamics in nematic liquidcrystals.Molec. Phys., DOI:10.1080/00268976.2015.1067730 (2015).

Colloidal PlateletsElastic Constant Measurements

D van der Beek, P Davidson, HH Wensink, GJ Vroege,HNW Lekkerkerker. Phys Rev E, 77, 031708 (2008).I-N Phase Separation

Reprinted with permission from Phys Rev E, 77, 031708 (2008).©(2008) American Physical Society.

Sterically stabilized gibbsiteAl(OH)3 platelets (hexagons).I DiameterD ≈ 230nm.I Thickness H ≈ 18nm.I H/D ≈ 1/13.I 20% polydispersity inboth dimensions.

I Alignment dictated by container walls.I Magnetic field: Freedericksz transition.I Gives bend elastic constant K3 = 6–8×10−14N.

Colloidal PlateletsElastic Constant Measurements

AA Verhoeff, RHJ Otten, P van der Schoot, HNW Lekkerkerker.J Phys Chem B, 113, 3704 (2009).

Tactoids in Magnetic Field

Reprinted with permission from J Phys Chem B, 113, 3704 (2009).©(2009) American Chemical Society.

Tactoids ≈ droplets.Low magnetic field:radial director structure,“hedgehog”, splay.High magnetic field:director structure deforms,tactoid elongates.

I Theory balances surface, elastic, and magnetic forces.I Gives splay elastic constant K1 = 9–26×10−14N.

Colloidal PlateletsElastic Constant Simulations

Experiment, Theory and Simulation

0.6 0.7 0.8 0.9 1.0

1

10

100

H/D = 0

bendsplay

twist

S

KD/kBT

0.6 0.7 0.8 0.9 1.0

H/D = 1/20

S

0.6 0.7 0.8 0.9 1.0

H/D = 1/15

S

0.6 0.7 0.8 0.9 1.0

H/D = 1/10

S

PA O’Brien, MP Allen, DL Cheung, M Dennison, AJ Masters.Phys Rev E, 78, 051705 (2008).PA O’Brien, MP Allen, DL Cheung, M Dennison, AJ Masters.Soft Matter, 7, 153 (2010).

Outline

Introduction

Nematic Elasticity

Nematic Hydrodynamics

NematodynamicsTheory

JL Ericksen. Arch. Rat. Mech. Anal., 4, 231 (1960).FM Leslie. Arch. Rat. Mech. Anal., 28, 265 (1968).O Parodi. J. de Physique, 31, 581 (1970).D Forster, TC Lubensky, PC Martin, J Swift, PS Pershan.Phys. Rev. Lett., 26, 1016 (1971).

I The director is effectively conserved on large length scales.I Coupled director n(k, t) and velocity v(k, t) fields.I Two Goldstone modes associated with director motion.

Nematic HydrodynamicsBalance Equations

ρvi = τji,j i, j = x, y, z

ρ ′ni = gi + πji,j f,j ≡∂f

∂rj

I We use the Einstein convention.I Incompressible fluid.I ρ =mass density, ρ ′ = director moment of inertia density.I vi(r, t) = velocity field, ni(r, t) = director field.I τji = stress tensor, πji = couple stress tensor.I gi = intrinsic director body force.

Nematic HydrodynamicsReactive Coefficients

Elastic EffectsτRji = −Pδji −

∂f

∂nk,jnk,i

gRi = γni − βjni,j −∂f

∂ni

πRji = βjni +∂f

∂ni,j

I f = Frank free energy density.I P, βj and γ are constants.I δij is the Kronecker delta.

Nematic HydrodynamicsDissipative Coefficients

Viscous EffectsτDji = α1 nknmdkmninj + α2 njNi + α3 niNj

+ α4 dji + α5 njnkdki + α6 ninkdkj

gDi = −γ1Ni − γ2 njdji

πDji = 0

I α1 . . . α6 = Leslie coefficients (viscosities)I γ1 = α3 − α2, γ2 = α6 − α5 = α3 + α2.I Ni = ni −wijnj = co-rotational time flux of the directorI dij =

12

(vi,j + vj,i

)= symmetric strain rate

I wij =12

(vi,j − vj,i

)= antisymmetric strain rate, vorticity

Nematic HydrodynamicsreactivecoefficientsτRji, πRji, gRi

dissipativecoefficientsτDji, gDi

balance equationsρvi = τji,j

ρ ′ni = gi + πji,j

hydrodynamicsI Drop ρ ′ni term; ni still appears throughNi = ni −wijnj.I Adopt same axis system as before.I Set n = (0, 0, 1) + δn; drop nonlinear terms in δn.I Use incompressibility: ∇ · v = 0, k · v = k1v1 + k3v3 = 0.

Define k = k(sinϕ, 0, cosϕ) and let v⊥ = v1 cosϕ− v3 sinϕ.

Nematic HydrodynamicsSplay-Bend Fluctuation Dynamics

(γ1∂

∂t+ Kk2

)n1 + ikα v⊥ = 0

−ikα∂

∂tn1 +

(ρ∂

∂t+ ηk2

)v⊥ = 0

K = K1 sin2ϕ+ K3 cos2ϕα = α2 cos2ϕ− α3 sin2ϕη = η1 sin2ϕ+ η2 cos2ϕ+ η12 sin2ϕ cos2ϕ

Nematic HydrodynamicsTwist-Bend Fluctuation Dynamics

(γ1∂

∂t+ Kk2

)n2 + ikα v2 = 0

−ikα∂

∂tn2 +

(ρ∂

∂t+ ηk2

)v2 = 0

K = K2 sin2ϕ+ K3 cos2ϕα = α2 cosϕη = η3 sin2ϕ+ η2 cos2ϕ

Linear Response TheoryI Consider perturbation coupling to A(r,p).I Measure response in B(r,p).I Define B such that ⟨B⟩ = 0.

The Response FunctionHamiltonian Perturbation: ∆H = −φ(t)A(r,p)

Response: ⟨B(t)⟩A=

∫ t−∞ dt

′ χBA(t− t′)φ(t ′)

Response function: χBA(t) = 0 for t < 0

Nonequilibrium response: ⟨B(t)

⟩A.

Equilibrium time correlation: CAB(t) ≡ ⟨AB(t)

⟩.

Nonequilibrium ResponseResponse and Correlation

χBA(t) = −1

kBT 〈AB(t)〉 = −1

kBT CAB(t) , for t > 0impulsive perturbation: φ(t) = constant× δ(t)⟨

B(t)⟩A∝ χBA(t) ∝ −CAB(t)

preparation and relaxation: φ(t) ={constant (t 6 0)

0 (t > 0)⟨B(t)

⟩A∝∫∞t

dt ′χBA(t′) ∝ CAB(t)

NematodynamicsExperiment

Director fluctuations, and time correlation functions or spectra,are investigated by light scattering experiments.PG de Gennes, J Prost. The Physics of Liquid Crystals. (Clarendon, 1995).For each mode the power spectrum (or frequencydistribution) has the form of one single Lorentzian,centred on the incident beam frequency. This meansthat the behaviour of the fluctuations is purelyviscous (no oscillations).

MJ Stephen, JP Straley. Rev. Mod. Phys, 46, 617 (1974).The orientation fluctuations of the director arecoupled to the fluid velocity by viscous effects, and infact are overdamped: the modes which the elastictheory . . . predicts do not propagate.

Twist DynamicsDirector and Velocity Fluctuations

TWISTn

nk

v

k

Hydrodynamic Equations(γ1

∂

∂t+ K2 k

2)n = 0(

ρ∂

∂t+ η3 k

2)v = 0

Twist DynamicsDirector and Velocity Fluctuations

TWISTn

nk

v

k

Hydrodynamic Equations(γ1

∂

∂t+ K2 k

2)n = 0(

ρ∂

∂t+ η3 k

2)v = 0

no couplingdirector rotationelastic constantn ∼ e−(K2k

2/γ1)t

Twist DynamicsDirector and Velocity Fluctuations

TWISTn

nk

v

k

Hydrodynamic Equations(γ1

∂

∂t+ K2 k

2)n = 0(

ρ∂

∂t+ η3 k

2)v = 0

no couplingdensityshear viscosityv ∼ e−(η3k

2/ρ)t

Twist DynamicsSimulation Results

Director and Velocity Correlation Functions

0 10 20 30

largerk

t

〈n(k,t)n(−k,0)〉

0 10 20 30

larger k

t

〈v(k,t)v(−k,0)〉

Twist DynamicsSimulation Results

Director and Velocity Correlation Functions

0 100 200 300 400

k2t

〈n(k,t)n(−k,0)〉

0 100 200 300 400

k2t

〈v(k,t)v(−k,0)〉

Splay DynamicsDirector and Velocity Fluctuations

SPLAYn

k

n

v

k

Hydrodynamic Equations(γ1

∂

∂t+ K1k

2)n+ ikα3 v = 0

−ikα3∂

∂tn+

(ρ∂

∂t+ η1k

2)v = 0

Splay DynamicsDirector and Velocity Fluctuations

SPLAYn

k

n

v

k

Hydrodynamic Equations(γ1

∂

∂t+ K1k

2)n+ ikα3 v = 0

−ikα3∂

∂tn+

(ρ∂

∂t+ η1k

2)v = 0

weak coupling

Splay DynamicsSimulation Results

Director and Velocity Correlation Functions

0 10 20 30

largerk

t

〈n(k,t)n(−k,0)〉

0 10 20 30

larger k

t

〈v(k,t)v(−k,0)〉

Splay DynamicsSimulation Results

Director and Velocity Correlation Functions

0 100 200 300 400

k2t

〈n(k,t)n(−k,0)〉

0 100 200 300 400

k2t

〈v(k,t)v(−k,0)〉

Bend DynamicsDirector and Velocity Fluctuations

BENDnn

k

v

k

Hydrodynamic Equations(γ1

∂

∂t+ K3k

2)n+ ikα2 v = 0

−ikα2∂

∂tn+

(ρ∂

∂t+ η2k

2)v = 0

Bend DynamicsDirector and Velocity Fluctuations

BENDnn

k

v

k

Hydrodynamic Equations(γ1

∂

∂t+ K3k

2)n+ ikα2 v = 0

−ikα2∂

∂tn+

(ρ∂

∂t+ η2k

2)v = 0

strong coupling

Bend DynamicsSimulation Results

Director and Velocity Correlation Functions

0 10 20 30

larger k

t

〈n(k,t)n(−k,0)〉

0 10 20 30

larger k

t

〈v(k,t)v(−k,0)〉

Bend DynamicsSimulation Results

Director and Velocity Correlation Functions

0 100 200 300 400

k2t

〈n(k,t)n(−k,0)〉

0 100 200 300 400

k2t

〈v(k,t)v(−k,0)〉

PG de Gennes’ ArgumentOrders of Magnitude

ρ ∼ 103 kgm−3 densitiesη ∼ α ∼ γ1 ∼ 10

−3–10−2 Pa s viscositiesK ∼ 10−11 N elastic constants

(K/γ1)k2 director decay rate

(η/ρ)k2 velocity decay rateµ =

ρK

γ1η∼ 10−4–10−2 time scale separation

Solving the HydrodynamicsSecular Equations for Bend Fluctuations

n(k, t)

v(k, t)

}∼ e−λk

2t ⇔ ∂

∂t→ −λk2

λ2ργ1 + λ(α22 − γ1η2︸ ︷︷ ︸small or large?

− ρK3︸︷︷︸small)+ K3η2 = 0

Define µ =ρK3

γ1η2, ξ = 1−

α22γ1η2

,

ξ2 . 4µ⇒ complex λ

What Did We Learn?I Simulations agree with hydrodynamics.I µ =

ρK3

γ1η2is small (de Gennes), typically µ ≈ 10−4–10−2.

I α22 ≈ γ1η2 (de Gennes again!).I ξ = 1−

α22γ1η2

also small, typically ξ ≈ 0.2.I Propagating modes if ξ2 . 4µ⇒

{µ ≈ 10−4, |ξ| . 0.02

µ ≈ 10−2, |ξ| . 0.2

I Experimentally observable for low viscosity liquid crystals?A Humpert, MP Allen. Propagating director bend fluctuations innematic liquid crystals. Phys. Rev. Lett., 114, 028301 (2015).A Humpert, MP Allen. Elastic constants and dynamics in nematic liquidcrystals.Molec. Phys., DOI:10.1080/00268976.2015.1067730 (2015).

Conclusionsliquid crystalline properties

↑↓molecular structureinsight fromstatistical mechanicaltheoriesoccasional surprisesrequire a rethink

stimulatenew experiments

simulationscan beuseful!

Conclusionsliquid crystalline properties

↑↓molecular structureinsight fromstatistical mechanicaltheoriesoccasional surprisesrequire a rethink

stimulatenew experiments

simulationscan beuseful!

Acknowledgements

PhD students:I Anja HumpertI Paul O’Brien

Collaborators:I David Cheung (Galway)I Andrew Masters (Manchester)

Funding:I EPSRC

AND THANK YOU FOR YOUR ATTENTION!

Acknowledgements

PhD students:I Anja HumpertI Paul O’Brien

Collaborators:I David Cheung (Galway)I Andrew Masters (Manchester)

Funding:I EPSRC

AND THANK YOU FOR YOUR ATTENTION!