Here we illustrate the classic Ginzburg-Landau-de Gennes theory of isotropic nematic phase transition and show how fluctuations as well as deterministic kinetics can lead to phase equilibria.

1. Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion
Kinetic pathways to the isotropic-nematic
phase transformation: a mean field approach
Amit Kumar Bhattacharjee
Institute of Materialphysics in Space
Köln, Germany
February 21, 2012
Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 1 / 31

2. 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

3. Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion
Liquid crystals
States of matter ⇒ solid, liquid, gas.
F = E − TS : hard matter (minimize E), soft
matter (maximize S).
Changes of Phase; order of transition (e.g. liquid
to 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, ideas
apply 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

4. Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion
Nematic mesophases1
Consist of anisotropic molecules (e.g. rods,
discs), having long range orientational order
without translational order.
Uniaxial phase have rotational symmetry about
the direction of order, described through a
headless vector n: the director.
Biaxial phase have two directions of order,
described through two headless vectors: the
director n and the co-director l.
Isotropic-Nematic transition is weakly first order.
isotropic
nematic
1
de 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

5. Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion
Nematic mesophases
LC sample in
crossed polarizer
Schlieren texture
Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 5 / 31

6. Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion
Motivation
Topological defect entanglement in a nematic liquid crystal film of
width 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 four
brushes exhibited by a uniaxial nematic
film at 114.8◦
centigrade.
[Chandrasekhar et al, Current
Science (’98)].
Nucleation of ellipsoidal nematic droplet with an aspect ratio of 1.7
and 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

7. 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 based
ansatz of F (instead of explicit DOF coarse
graining).
Temporal coarse graining retaining thermal
fluctuation effects.
Numerical techniques in mesoscale
◮ Brownian dynamics, Dissipative particle
dynamics, Time-dependent Ginzburg-Landau.
Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 7 / 31

8. Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion
Definition of Order
Quantified through a symmetric traceless tensor Qαβ having five degrees
of freedom.
Molecular frame :
Q(x, t) = du f(x, u, t) uu ≡ uu (quadrupolar symmetry).
Principal frame : Qαβ = 3
2S(nαnβ −1
3 δαβ)+T
2 (lαlβ −mαmβ)(α, β = x, y, z).
Principal values represent strength of uniaxial (S) and biaxial (T)
ordering.
Principal axes designate the director n and the codirector l and the joint
normal m.
◮ S = T = 0 correspond to isotropic phase.
◮ S = 2
3 , T = 0 correspond to uniaxial nematic phase.
◮ T = 0 correspond to biaxial phase.
Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 8 / 31

9. Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion
Statics : “Minimal” model
FGLdG = d3
x[
1
2
ATrQ2
+
1
3
BTrQ3
+
1
4
C(TrQ2
)2
+ E′
(TrQ3
)2
+
1
2
L1(∂αQβγ)(∂αQβγ) +
1
2
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
isotropic
phase
biaxial
nematic
phase
discotic
phase
uniaxial
nematic
phase
Phase diagram
Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 9 / 31

10. Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion
Kinetics
Landau-Ginzburg (model-A) dynamics for non-conserved order
parameter2.
◮ ∂tQαβ = −Γ[δαµδβν + δανδβµ − 2
d δαβδµν] δF
δQµν
+ ξµν.
Equation of motion
∂tQαβ(x, t) = −Γ [(A + CTrQ2
)Qαβ(x, t) + (B + 6E′
TrQ3
) Q2
αβ(x, t) −
L1∇2
Qαβ(x, t) − L2 ∇α(∇γQβγ(x, t)) ] + ξµν(x, t)
Route to equilibrium ⇒
1 nucleation kinetics above T∗
.
2 spinodal kinetics beneath T∗
.
2
Stratonovich: 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

11. Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion
Numerical recipe
Projection in orthonormal basis
◮ Qαβ(x, t) =
5
i=1 ai(x, t)Ti
αβ,
◮ ξαβ(x, t) =
5
i=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 the
eigen-vectors corresponding to them.
Developed method: (Stochastic) Method of lines, Spectral collocation
methods and HPC of them.
Sytematic benchmark with scalar problem (e.g. Allen-Cahn equation),
OU process, linear and non linear theory of nematics 3.
3
Bhattacharjee et al: PRE (’08), JCP (’10)
Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 11 / 31

12. Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion
Application I Nucleation kinetics of
nematic droplet
Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 12 / 31

13. Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion
Classical nucleation theory
F(R) = −VρN∆µ + Aσ
V = 4
3 πR3, 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 = 16πσ3/3ρ2
N(∆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) = 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

14. 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-Papoular
anchoring 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
4
Fs ∼ 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

15. 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 from
circular to ellipsoidal 5.
t = 0
L2 = 0, t = 3000τ L2 > 0, t = 900τ L2 < 0, t = 1500τ
5
Bhattacharjee: PRE (’08)
Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 15 / 31

16. 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
) Q2
αβ(x, t) −
L1∇2
Qαβ(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

17. 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

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

19. 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

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

21. 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

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

23. Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion
Application II Defect morphology in spinodal
kinetics
Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 23 / 31

24. Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion
Topological characterization of point defects
Uniaxial nematic defects are characterized by the group Z2, having
unstable integer and stable half integer charged defects 6.
Biaxial nematic defects are characterized by the group Q8, having a
stable integer (¯C0 class, 2π rotation of director) and three half-integer
(Cx, Cy, Cz, π rotation of director) charged defects 7.
Defects are visualized and classified through scalar order.
The half-integer defect locations are identified in S(x, t), T(x, t) while the
textures show a subset.
6
Mermin ’79
7
Goldenfeld et al. ’95
Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 24 / 31

25. 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

26. 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

27. Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion
Defect core structure
0 5 10 15 20 25 30 35 40 45
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Distance
OrderParameter
S
T
Uniaxial defect
110 120 130 140 150 160
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Distance
Orderparameter
S
T
Defect class Cx
145 150 155 160 165 170 175 180
0.4
0.5
0.6
0.7
0.8
0.9
1
Distance
Orderparameter
S
T
Defect class Cy
Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 27 / 31

28. Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion
Line defects
Points in two dimensions correspond to
lines in three dimensions.
Annihilation of point defect-antidefect
correspond to formation and
disappearance of loop.
Line defects pass through each other
through intercommutation i.e.
exchanging segmentsa.
Intercommutation of lines depend on the
underlying abelian nature of the group
elements of that particular homotopy
groupb.
a
Turok et al ’91
b
Poenaru et al ’77
Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 28 / 31

29. 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

30. Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion
Summary
Methods
Formulation of a fluctuating equation for nematics within GLdG
framework; novel visualization technique of defects.
Nucleation kinetics
Anisotropy in the droplet shape found within GLdG theory.
Breakdown of the CNT due to nontrivial defect conformation within
droplet.
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
Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 30 / 31

31. Outline Introduction Numerics and Benchmarks Nucleation Coarsening Conclusion
Thanks for your attention
Amit Kumar Bhattacharjee (Institute of Materialphysics in Space Köln, Germany)KKK Seminar February 21, 2012 31 / 31