Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Kinetic pathways to the isotropic-nematicphase transformation: a mean field approach

Amit Kumar Bhattacharjee

Institute of Materialphysics in SpaceKöln, Germany

February 21, 2012

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 1 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Outline

1 Liquid crystals: another candidate in soft materials.2 Motivation for a theory of nematics.3 Computational approaches: complexity.4 Kinetic pathways in equilibrium phase transition.5 Invitation to a new direction in complex criterion.

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 2 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Liquid crystals

States of matter⇒ solid, liquid, gas.

F = E − TS : hard matter (minimize E), softmatter (maximize S).

Changes of Phase; order of transition (e.g. liquidto solid, paramagnet to ferromagnet).

Multistage transition process (e.g.Nematic,Smectic A-C, Cholesteric, Discotic, Coloumnar).Necessity to study:

◮ Technological applications : elctro-optic display,watches, temperature sensors etc.

◮ Physical interests : Statistical field theory, ideasapply from Biophysics to Cosmology!

[Spindle formation in mitosis]

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 3 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Nematic mesophases 1

Consist ofanisotropic molecules (e.g. rods,discs), having long range orientational orderwithout translational order.

Uniaxial phase have rotational symmetry aboutthe direction of order, described through aheadless vectorn: the director.

Biaxial phase have two directions of order,described through two headless vectors: thedirectorn and the co-directorl.

Isotropic-Nematic transition isweakly first order.isotropic

nematic1de Gennes & Prost, The physics of liquid crystals (’93)

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 4 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Nematic mesophases

LC sample incrossed polarizer

Schlieren texture

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 5 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Motivation

Topological defect entanglement in a nematic liquid crystal film ofwidth 790µm after a temperature quench, showing monopoles,boojums and various integer and half-integer defects[Turok et al,Science (’91)].

The schlieren textures with two and fourbrushes exhibited by a uniaxial nematicfilm at 114.8◦ centigrade.[Chandrasekhar et al, CurrentScience (’98)].

Nucleation of ellipsoidal nematic droplet with an aspect ratio of 1.7and homogeneous director field in a MC simulation(n = 20, L∗ = 15,∆P∗ = 0.052). [Cuetos et al, PRL (’07)].

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 6 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Mean field description

Landau (’62)

de Gennes (’91)

Theoretic treatment : broken symmetry variable,conservation laws, order of transition.

Coarse graining of space : Symmetry basedansatz ofF (instead of explicit DOF coarsegraining).

Temporal coarse graining retaining thermalfluctuation effects.Numerical techniques in mesoscale

◮ Brownian dynamics, Dissipative particledynamics, Time-dependent Ginzburg-Landau.

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 7 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Definition of Order

Quantified through a symmetric traceless tensorQαβ havingfive degreesof freedom.

Molecular frame :Q(x, t) =

∫du f (x, u, t) uu ≡ 〈 uu 〉 (quadrupolar symmetry).

Principal frame :Qαβ = 32S(nαnβ−

13δαβ)+

T2 (lαlβ−mαmβ)(α, β = x, y, z).

Principal values represent strength of uniaxial (S) and biaxial (T)ordering.Principal axes designate the directorn and the codirectorl and the jointnormalm.

◮ S = T = 0 correspond toisotropicphase.◮ S = 2

3,T = 0 correspond touniaxialnematic phase.◮ T 6= 0 correspond tobiaxial phase.

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 8 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Statics : “Minimal” model

FGLdG =

∫d3x[

12

ATrQ2 +13

BTrQ3 +14

C(TrQ2)2 + E′(TrQ3)2 +

12

L1(∂αQβγ)(∂αQβγ) +12

L2(∂αQαβ)(∂γQβγ)].

Free energy diagram−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

A

B

superheating spinodal line

I−N transition line

supercooling spinodal line

UN−BN transition line

isotropicphase

biaxialnematicphase

discoticphase

uniaxialnematicphase

Phase diagramAmit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 9 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Kinetics

Landau-Ginzburg (model-A) dynamics for non-conserved orderparameter2.

◮ ∂tQαβ = −Γ[δαµδβν + δανδβµ − 2d δαβδµν ]

δFδQµν

+ ξµν .

Equation of motion

∂tQαβ(x, t) = −Γ [(A + CTrQ2)Qαβ(x, t) + (B + 6E′TrQ3) Q2αβ(x, t) −

L1∇2Qαβ(x, t)− L2 ∇α(∇γQβγ(x, t)) ] + ξµν(x, t)

Route to equilibrium⇒1 nucleationkinetics aboveT∗.2 spinodalkinetics beneathT∗.

2Stratonovich:Zh.Eksp.Teor.Fiz. (’76), Bhattacharjee: PRE (’08)Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 10 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Numerical recipe

Projection in orthonormal basis◮ Qαβ(x, t) =

∑5i=1 ai(x, t)T i

αβ ,◮ ξαβ(x, t) =

∑5i=1 ai(x, t)ξi

αβ .

Integration of the equation of a’s.

Transformation back to the principle frame.

Extraction of the largest and second largest eigenvalue and theeigen-vectors corresponding to them.

Developed method: (Stochastic) Method of lines, Spectral collocationmethods and HPC of them.

Sytematic benchmark with scalar problem (e.g. Allen-Cahn equation),OU process, linear and non linear theory of nematics3.

3Bhattacharjee et al: PRE (’08), JCP (’10)Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 11 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Application I Nucleation kinetics of

nematic droplet

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Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 12 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Classical nucleation theory

F(R) = −VρN∆µ+ Aσ

V = 43πR3,A = 4πR2.

∆µ = L∆T/T∗.(L=latent heat,σ=surface tension)

Nucleated droplet grow (R > Rc) or shrink (R < Rc).

Rc = 2σ/ρN |∆µ|,Fc = 16πσ3/3ρ2N(∆µ)2.

Shape of nucleated phase⇒∫

dV = constant, minimum of∫

dA ?

Liquid-gas problem:ρN(x), σ(x) uniform⇒ spherical droplet.

Isotropic-nematic problem ?FV(R) 6= VρN∆µ; σ = σ[Q{S(x), n(x)}].

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 13 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Nematic droplet in isotropic background

Athermal system with handcrafted nematic droplet.

No approximation of surface free energy (e.g. Rapini-Papoularanchoring energy4) needed in our formulation.

Consequences : nucleation rate (∝ e−∆F/kBT ) can be calculated exactly,apart from the prefactors.

S(x, t); t = 0, θ = π/4

4Fs ∼ σ

∫[1+ ω(q⊥ · n)2], σ =interfacial tension,ω = anchoring strength

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 14 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Nematic droplet in isotropic background

Athermal system with handcrafted uniaxial nematic droplet.

No approximation of surface free energy needed in our formulation.

Consequences : nucleation rate (∝ e−∆F/kBT ) can be calculated exactly,apart from the prefactors.

Contribution from the anisotropic surface tension⇒ shape change fromcircular to ellipsoidal5.

t = 0

L2 = 0, t = 3000τ L2 > 0, t = 900τ L2 < 0, t = 1500τ

5Bhattacharjee: PRE (’08)Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 15 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Nematic droplet in isotropic background

∂tQαβ(x, t) = −Γ [(A + CTrQ2)Qαβ(x, t) + (B + 6E′TrQ3) Q2αβ(x, t) −

L1∇2Qαβ(x, t)− L2 ∇α(∇γQβγ(x, t)) ] + ξµν(x, t)

t = 0

L2 = 0, t = 3000τ L2 > 0, t = 900τ L2 < 0, t = 1500τ

Homogeneous director field throughout the nucleation process.

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 16 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Fluctuation induced nucleation

Temperature fluctuation drives spontaneously the nucleation event.

L2 = 0; t = 1200τ

S(x,t), n(x,t)L2 = 0; t = 3300τ

sin2[2θ(x, t)]

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 17 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

3D Nematic : L2 = 0

t = 600τ

t = 900τ

t = 690τ

t = 3000τ

droplet conformation

S(x, t) along the dropletintersection

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 18 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

2D Nematic : L2 > 0

t = 2040τ t = 2400τ

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 19 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

3D Nematic : L2 > 0

t = 2070τ

t = 6000τ

t = 2190τ

θ(x, t) at t = 2100τ

t = 2370τ

S(x, t) along dropletintersection

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 20 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

2D Nematic : L2 < 0

t = 900τ t = 2250τ

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 21 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

3D Nematic : L2 < 0

t = 990τ

t = 3900τ

t = 1380τ

t = 990τ

t = 1860τ

S(x, t) along dropletintersection

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 22 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Application II Defect morphology in spinodalkinetics

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Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Topological characterization of point defects

Uniaxial nematic defects are characterized by the groupZ2, havingunstable integer and stable half integer charged defects6.

Biaxial nematic defects are characterized by the groupQ8, having astable integer (̄C0 class, 2π rotation of director) and three half-integer(Cx,Cy,Cz, π rotation of director) charged defects7.

Defects are visualized and classified through scalar order.

The half-integer defect locations are identified inS(x, t), T(x, t) while thetextures show a subset.

6Mermin ’797Goldenfeldet al. ’95

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 24 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Uniaxial defect

S(x,t), n(x,t)

sin2[2θ(x, t)]

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 25 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Biaxial defect

S(x,t)

T(x,t)

Texture

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 26 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Defect core structure

0 5 10 15 20 25 30 35 40 450

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Distance

Ord

er

Pa

ram

ete

r

S

T

Uniaxial defect

110 120 130 140 150 1600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Distance

Order parameter

S

T

Defect classCx

145 150 155 160 165 170 175 180

0.4

0.5

0.6

0.7

0.8

0.9

1

Distance

Order parameter

S

T

Defect classCy

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 27 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Line defects

Points in two dimensions correspond tolines in three dimensions.

Annihilation of point defect-antidefectcorrespond to formation anddisappearance of loop.

Line defects pass through each otherthroughintercommutation i.e.exchanging segmentsa.

Intercommutation of lines depend on theunderlying abelian nature of the groupelements of that particular homotopygroupb.

aTurok et al ’91bPoenaruet al ’77

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 28 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Line defects

No topological rigidity found in biaxial nematics!

Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 29 / 31

Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Summary

Methods

Formulation of a fluctuating equation for nematics withinGLdGframework; novel visualization technique of defects.Nucleation kinetics

Anisotropy in the droplet shape found withinGLdG theory.

Breakdown of the CNT due to nontrivial defect conformation withindroplet.Coarsening kinetics

Classification and visualization of all defect classes in nematics.

No defect entanglement found in biaxial nematics within the “minimal”GLdG framework.

Animations :http://www.youtube.com/view_play_list?p=7F62606B554B63A6

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Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion

Thanks for your attention

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