A Unified Model for I-N-S A -S C Phase Transitions for Liquid - - PowerPoint PPT Presentation

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A Unified Model for I-N-SA-SC Phase Transitions for Liquid Crystal

Song Mei

School of Mathematical Science, Peking University

November 1, 2013 Joint work with Jiequn Han, Wei Wang, Pingwen Zhang

Song Mei 1 / 29

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Contents

. .

1

Introduction . .

2

Classical models in three levels . .

3

A mechanistic Q Tensor Model . .

4

Vector Model . .

5

Numerical Results . .

6

Conclusion

Song Mei 2 / 29

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Contents

. .

1

Introduction . .

2

Classical models in three levels . .

3

A mechanistic Q Tensor Model . .

4

Vector Model . .

5

Numerical Results . .

6

Conclusion

Song Mei Introduction 3 / 29

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Molecules

Song Mei Introduction 4 / 29

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Phases

Song Mei Introduction 5 / 29

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Contents

. .

1

Introduction . .

2

Classical models in three levels . .

3

A mechanistic Q Tensor Model . .

4

Vector Model . .

5

Numerical Results . .

6

Conclusion

Song Mei Classical models in three levels 6 / 29

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Onsager’s Molecular Theory

The energy functional: βF[ρ] = ∫

Ω

∫

S2 ρ(x, m) ln ρ(x, m)dmdx

+ α 2 ∫

Ω

∫

S2

∫

Ω

∫

S2 ρ(x, m)G(m, m′, x − x′)ρ(x′, m′)dxdmdx′dm′.

where ∫

Ω

∫

S2 ρ(x, m)dxdm = N

and G represents the interaction potential. For example, G may represent volume exclusion potential:

G(m, m′, x − x′) = { 0, molecule (x, m) is disjoint with molecule(x′, m′) 1, joint with each other.

• r G may represent the Gay-Berne potential:

G(m, m′, r) = ε(m, m′, r) ( (r/r0 − σ(m, m′, r) + 1)−6 − (r/r0 − σ(m, m′, r) + 1)−12) .

Song Mei Classical models in three levels 7 / 29

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Q-Tensor Theory for Liquid Crystal

Define a symmetric traceless second order tensor Q(x) to describe the

• rientation of molecule.

Q(x) = ∫

S2(mm − 1

3I)f (x, m)dm,

where

∫

Ω

f (x, m)dm = 1.

Q = 0 :⇒ isotropic; If Q has two equal eigenvalues: Q = S(nn − 1

3I), n ∈ S2. ⇒ uniaxial;

If S = const ̸= 0: nematic phase; The Landau-de Gennes model:

F LG [Q] = ∫

Ω

( A(T − T ∗) 2 tr(Q2) − B 3 tr(Q3) + C 4 (tr(Q2))2 ) dx + ∫

Ω

( L1∂jQik∂kQij + L2∂jQij∂kQik + L3|∇Q|2 + L4Qlk∂kQij∂lQij ) dx

Song Mei Classical models in three levels 8 / 29

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Vector Model and Oseen-Frank Energy

Assume Q(x) = S(nn − 1

3I), define n(x) as the director.

Define c(x) as the relative density, where ∫

Ω c(x) = 1.

Vector Model describes N-SA-SC transitions, with order parameter c(x) and n(x). The energy functional: βF[c, n] = βF1 + βF2, where F1 contains up to second order derivative terms of c(x) βF1 = ∫

Ω

(ac2 + D||[(n · ∇)2c]2 − C||(n · ∇c)2 + C 2

||

4D|| c2 + C⊥δT

ij ∇ic∇jc + D⊥(∇2 ⊥c)2)dx.

F2 is the Oseen-Frank free energy for distortions in the nematic director: βF2 = β 2 ∫ (K1(∇ · n)2 + K2[n · (∇ × n)]2 + K3[n × (∇ × n)]2)dx.

Song Mei Classical models in three levels 9 / 29

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Contents

. .

1

Introduction . .

2

Classical models in three levels . .

3

A mechanistic Q Tensor Model . .

4

Vector Model . .

5

Numerical Results . .

6

Conclusion

Song Mei A mechanistic Q Tensor Model 10 / 29

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A mechanstic Q Tensor Model

With order parameter c(x), Q2(x), the energy functional is:

βF[c, Q] = ∫ c(x)(ln c(x) + BQ : Q − ln Z(x))dx + θ 2 ( ∫ (A1 + 3 2 A2|Q2|2 + 35 8 A3|Q4|2)c2dx − ∫ {G1|∇c|2 + G2|∇(cQ2)|2 + G3|∇(cQ4)|2 + G4∂i(cQ2ij)∂j(c) +G5∂i(cQik)∂j(cQjk) + G6∂i(cQ4iklm)∂j(cQ4jklm) + G7∂i(cQ4ijkl)∂j(cQ2kl)}dx + ∫ {H1∂ij(cQ4ijpq)∂kl(cQ4klpq) + H2∂ij(cQ2ij)∂kl(cQ2kl) + H3∂ik(cQ2ip)∂jk(cQ2jp) +H4∂ij(cQ2pq)∂ij(cQ2pq) + H5∂ij(c)∂ij(c) + H6∂ij(cQ4ijkp)∂kl(cQ2lp) +H7∂ij(cQ4ijpq)∂kk(cQ2pq) + H8∂ij(cQ4ijkl)∂kl(c) + H9∂ij(cQ2ij)∂kk(c)}dx)

Here Ai, Gi, Hi are coefficients depend on molecular parameters. θ: the average concentration; β = 1/(kBT). c(x): the normalized local concentration. Q2(x): second-order tensor; Q4 is a fourth-order tensor depending on Q2.

Song Mei A mechanistic Q Tensor Model 11 / 29

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Taylor expansion

Onsager’s molecular theory: F[ρ] = ∫ ρ(x, m) ln ρ(x, m)dmdx ⇒ Entropy term + α 2 ∫ ρ(x, m)G(m, m′, x − x′)ρ(x′, m′)dm′dx′dmdx ⇒ Interaction = Fentropy + α 2 ∫ ρ(x, m)G(m, m′, r)ρ(x + r, m′)dm′dmdxdr where r = x′ − x. Using Taylor expansion: ρ(x + r, m) = ρ(x, m) + r · ∇ρ(x, m) + 1 2rr : ∇2ρ(x, m) + · · · . The energy functional becomes:

Finteraction = α 2 ∫ ρ(x, m)G(m, m′, r)ρ(x, m′)drdm′dmdx ⇒ Bulk energy + α 4 ∫ ρ(x, m)G(m, m′, r)rr : ∇2ρ(x, m′)drdm′dmdx + α 48 ∫ ρ(x, m)G(m, m′, r)rrrr : ∇4ρ(x, m′)drdm′dmdx + · · · ⇒ Elastic energy

Song Mei A mechanistic Q Tensor Model 12 / 29

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Spherical Invariants Expansion

Expand the interaction potential in terms of spherical invariants:

G(m, m′, r) = ∑

lmk

Jlmk(|r|)T lmk(m, m′, ˆ r).

Thus, the moments of G can be write down as summation of interaction

• f Legendre polynomials:

∫

Ω

G(m, m, r)dr = a1 + a2P2(m) : P2(m′) + a3P4ijkl(m)P4ijkl(m′) + · · · , ∫

Ω

G(m, m, r)rrdr = g1I + g2(P2(m) + P2(m′)) + g3P2(m) : P2(m′)I + g4P2(m) · P2(m′) · · · , ∫

Ω

G(m, m, r)rrrrdr = h1I4 + h2(P2(m) + P2(m′))I2 + h3P2(m)P2(m′) + h4P2(m) · P2(m)I2 + h5P2(m) : P2(m)I4 + h6(P4(m) + P4(m′)) + · · · .

where Pn is the n-th order Legendre polynomial and thus Pn(m) is an n-th order tensor. For example P2(m) = 3

2mm − 1 2I.

Here ai, gi, hi can be calculated by integration, and are expressed by molecular parameters. We truncated to 4-th order Legendre polynomials terms.

Song Mei A mechanistic Q Tensor Model 13 / 29

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Bingham Closure Approximation

Let ρ(x, m) = c(x)f (x, m), where c(x) = ∫

S2 ρ(x, m)dm.

We assume f (x, m) conforms Bingham distribution: f (x, m) = 1 Z(x) exp(BQ(x) : mm) where BQ(x) is a 3 by 3 tensor, and Z(x) is the normalization constant Z(x) = ∫

|m|=1

exp(BQ : mm)dm. We denote Qn(x) = 2n C n

2n−1

∫

|m|=1

Pn(m)f (x, m)dm then Q2(x) is the Q tensor defined before.

Song Mei A mechanistic Q Tensor Model 14 / 29

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Calculation

Then

Fentropy = ∫ c(x)(ln c(x) + BQ : Q − ln Z(x))dx

and

F (2)

elastic = α

4 ∫ ρ(x, m)G(m, m′, r)rr : ∇2ρ(x, m′)drdm′dmdx = − α 4 ∫ (∫

Ω

(G(m, m′, r)rrdr ) : ∇ρ(x, m′)∇ρ(x, m′)dm′dmdx = − α 4 ∫ ( g1I + g2(P2(m) + P2(m′)) + g3P2(m) : P2(m′)I + g4P2(m) + · · · ) ∇ (c(x)f (x, m)) ∇ ( c(x)f (x, m′) ) dxdmdm′ = − α 4 ∫ { G1|∇c|2 + G2|∇(cQ2)|2 + G3|∇(cQ4)|2 + G4∂i(cQ2ij)∂j(c) + · · · } dx

Fbulk and F (4)

elastic can also be calculated.

Song Mei A mechanistic Q Tensor Model 15 / 29

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Models for Liquid Crystals: Q-Tensor Theory

We can finally obtain a Q-Tensor model:

βF[c, Q] = ∫ c(x)(ln c(x) + BQ : Q − ln Z(x))dx + θ 2 ( ∫ (A1 + 3 2 A2|Q2|2 + 35 8 A3|Q4|2)c2dx ⇒ Derived by J. Ball et. al., 2009 − ∫ {G1|∇c|2 + G2|∇(cQ2)|2 + G3|∇(cQ4)|2 + G4∂i(cQ2ij)∂j(c) +G5∂i(cQik)∂j(cQjk) + G6∂i(cQ4iklm)∂j(cQ4jklm) + G7∂i(cQ4ijkl)∂j(cQ2kl)}dx + ∫ {H1∂ij(cQ4ijpq)∂kl(cQ4klpq) + H2∂ij(cQ2ij)∂kl(cQ2kl) + H3∂ik(cQ2ip)∂jk(cQ2jp) +H4∂ij(cQ2pq)∂ij(cQ2pq) + H5∂ij(c)∂ij(c) + H6∂ij(cQ4ijkp)∂kl(cQ2lp) +H7∂ij(cQ4ijpq)∂kk(cQ2pq) + H8∂ij(cQ4ijkl)∂kl(c) + H9∂ij(cQ2ij)∂kk(c)}dx)

The coefficients Ai, Gi, Hi are functions of molecular parameter and

• temperature. It can be determined through integration.

In particular, when G(m, m′, r) represent the exclusion volume potential, these coefficients can be analytically expressed.

Song Mei A mechanistic Q Tensor Model 16 / 29

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Contents

. .

1

Introduction . .

2

Classical models in three levels . .

3

A mechanistic Q Tensor Model . .

4

Vector Model . .

5

Numerical Results . .

6

Conclusion

Song Mei Vector Model 17 / 29

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Reduction to Vector Model

Assumptions: . .

1 The space is T 3, with periodic boundary conditions.

. .

2 The Q Tensor is uniaxial. Thus

Q2ij = S2(ninj − 1 3I), Q4ijkl = S4(ninjnknl − 1 7(ninjδkl)symmetric + 1 35(δijδkl)symmetric). . .

3 n(x), S2(x), S4(x) are constant all over the space.

Deduced vector model:

βF = ∫

Ω

( ˆ A1c2 − ˆ G1|∇c|2 − ˆ G2(n · ∇c)2 + ˆ H1(∇2c : nn)2 + ˆ H2(∇2c : nn)△c + ˆ H3(△c)2 + ˆ H4|n · ∇2c|2 + ˆ H5|∇2c|2 ) dx

Song Mei Vector Model 18 / 29

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Layer Thickness and Tilt Angle in Smectic Phase

Trial function c(x) = 1 + c1cos(kn · x). Plug into the energy functional: βF(k2, cos2 θ, c2

1) =

∫ ˆ A1c2 − ( ˆ G1 cos2 θ + ˆ G2)k2)(c − c0)2 + (( ˆ H1 cos4 θ + ˆ H2 cos2 θ + ˆ H3 + ˆ H4 cos2 θ + ˆ H5)k4 + λc4 Optimizing over k, cos θ and c1: ˆ k2 = ˆ G1 + ˆ G2 cos2 ˆ θ 2( ˆ H1 cos4 ˆ θ + ˆ H2 cos2 θ + ˆ H3 + ˆ H4 cos2 ˆ θ + ˆ H5) . cos2 ˆ θ = 1 − ˆ k2(2 ˆ H1 + ˆ H2 + ˆ H4) − ˆ G2 2ˆ k2 ˆ H1 . These two equations can be solved iteratively.

Song Mei Vector Model 19 / 29

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Elastic constants in the Oseen-Frank Energy

Constant density (c(x) = c0 is constant). Uniaxial approximation: Q = S2(nn − 1

3I).

Neglect higher order derivatives energy. For rode-like molecules:

Felastic = 1 2 (K1(∇ · n)2 + K2(n · ∇ × n)2 + K3|n × (∇ × n)|2). K1 = 2KBTL4Dc2

0π

{ ( 15 224 − 1131η2 784 + 24 ln 2η2 49 )S2

2 + (− 221η2

64 · 49 + 115 ln 2η2 49 )S2

4

+(− 15 224 + 75η2 49 − 90 ln 2η2 49 )S2S4 } , K2 = 2KBTL4Dc2

0π

{ (− 95η2 784 + 5 7 · 32 + 15η2 ln 2η2 49 )S2

2 + ( 495η2

64 · 49 + 15 ln 2η2 49 )S2

4

+(− 5 224 + 25η2 49 − 30 ln 2η2 49 )S2S4 } , K3 = 2KBTL4Dc2

0π

{ ( 15 224 − 1131η2 784 + 24 ln 2η2 49 )S2

2 + (− 2155η2

64 · 49 + 150 ln 2η2 49 )S2

4

+( 20 224 − 100η2 49 + 120 ln 2η2 49 )S2S4 } ,

Song Mei Vector Model 20 / 29

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Contents

. .

1

Introduction . .

2

Classical models in three levels . .

3

A mechanistic Q Tensor Model . .

4

Vector Model . .

5

Numerical Results . .

6

Conclusion

Song Mei Numerical Results 21 / 29

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Numerical Result: INAC transition

I-N, N-SA: First order transition. SA-SC: Second order transition. The tilt angle: an increasing function. The layer thickness: h ∼ 1.5 − 2L.

13 14 14.6 15.8 18 1 2 3 4 5 6

I N S A S C

Average Density θ (Degree) I−N−SA−SC

G4 = 0.1289, H2 = 0.0078.

Song Mei Numerical Results 22 / 29

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Numerical Result: INA and INC transitions

13 14.2 15.9 18 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

I N S A

Average Density Free Energy I−N−SA

G4 = 0.0759, H2 = 0.0020

N-SA: First order transition.

13 14.2 15.9 18 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

I N S C

Average Density Free Energy I−N−SC

G4 = 0.1518, H2 = 0.0120

N-SC: First order transition.

Figure : Phase Diagram

Song Mei Numerical Results 23 / 29

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Numerical Results: Approximate Layer Thickness and Tilt Angle in Vector Model

Smectic A: G4 = 0.0759, H2 = 0.0020, S2 = 0.86, S4 = 0.61 Optimized h = 1.51, approximately calculated ˆ h = 1.47. Smectic C: G4 = 0.1518, H2 = 0.0120, S2 = 0.86, S4 = 0.61 Optimized h = 1.86, approximately calculated ˆ h = 1.81. Optimized θ = 23◦, approximately calculated ˆ θ = 28◦.

Song Mei Numerical Results 24 / 29

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Numerical Result: Elastic Constants

For rod-like molecules liquid crystal, when η = D/L is small, the relationship is K3 > K1 > K2.

Song Mei Numerical Results 25 / 29

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Contents

. .

1

Introduction . .

2

Classical models in three levels . .

3

A mechanistic Q Tensor Model . .

4

Vector Model . .

5

Numerical Results . .

6

Conclusion

Song Mei Conclusion 26 / 29

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Conclusion

Based on Onsager’s molecular theory, using Taylor expansion, Spherical invariants expansion and Bingham closure, we obtained a new Q-Tensor model. The model can deduce Chen-Lubensky model, and determine the macroscopic quantity such as K1, K2, K3, h and θ. For rod-like molecules liquid crystal, the model can characterize I-N-SA-SC phase transition uniformly. The procedure of modelling can be applied to molecules of other shapes(polar, disklike, banana shape, . . .).

Song Mei Conclusion 27 / 29

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Further Topics

Molecules of other shapes. The isotropic - nematic interface. Characterizing defects.

Song Mei Conclusion 28 / 29

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Main Citation

Chen, Jing-huei, and T. C. Lubensky. ”Landau-Ginzburg mean-field theory for the nematic to smectic-C and nematic to smectic-A phase transitions.” Physical Review A 14.3 (1976): 1202.

• J. Han, Y. Luo, W. Wang and P. Zhang, From microscopic theory to

macroscopic theory: systematic study on static modeling for liquid crystals, arXiv:1305.4889. P.G. de Gennes and J. Prost, The physics of liquid crystals, Oxford University Press, USA, (1995).

• J. Xu and P. Zhang, From Molecular Symmetry to Order Parameters,

preprint (2013). Bingham, C. An antipodally symmetric distribution on the sphere,

• Ann. Stat. 2, 1201-1225, (1974).

J.M. Ball and A. Majumdar, Nematic liquid crystals: from Maier-Saupe to a continuum theory, Mol. Cryst. Liq. Cryst., 525, 1-11(2010).

Song Mei Bibliography 29 / 29