Analysis of a Ginzburg-Landau Type Energy Model for Smectic C* - PowerPoint PPT Presentation

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References Analysis of a Ginzburg-Landau Type Energy Model for Smectic C* Liquid Crystals with Defects Sean Colbert-Kelly, joint work with Daniel Phillips and Geoffrey McFadden Applied and Computational Mathematics Division Seminar Series National Institute of Standards and Technology May 28, 2013 Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References Outline 3 Effects of Defects in Liquid Ginzburg-Landau (GL) 1 Crystals Functional The Generalized GL 4 Introduction to Liquid Functional 2 Crystals (LCs) References 5 Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References Outline 3 Effects of Defects in Liquid Ginzburg-Landau (GL) 1 Crystals Functional The Generalized GL 4 Introduction to Liquid Functional 2 Crystals (LCs) References 5 Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References GL functional is defined as E ε ( u ) = 1 Ω | ∇ u | 2 + 1 � 2 ε 2 ( 1 −| u | 2 ) 2 dx 2 Introduced in study of phase transition problems in superconductivity (also used in superfluids and mixture of fluid states) u - complex order parameter (condensate wave function/concentration/vector field orientation) ε - coherence length which can depend on temperature ( ξ ( T ) )/diffuse interface/core radius Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References When in equilibrium, the order parameter u minimizes E ε . Taking variations of u , the following must be satisfied Ω [ − ∆ u − 1 � ε 2 u ( 1 −| u | 2 )] δ u dx = 0 δ E ε = = ⇒ − ∆ u = 1 ε 2 u ( 1 −| u | 2 ) Ex: u t = u + tv , δ E ε = dE ε dt ( u + tv ) | t = 0 , δ u = v Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References Example in 1D The Euler-Lagrange (E-L) equation in 1D then becomes − u xx − 1 ε 2 u ( 1 − u 2 ) = 0 x Solution: u ε = tanh ( 2 ε ) given the boundary conditions √ u ( 0 ) = lim | x |→ ∞ u x ( x ) = 0. Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References The function y = ( 1 − t 2 ) 2 (Two-well potential in 1D) Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References Plot of solutions for various epsilons Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References Outline 3 Effects of Defects in Liquid Ginzburg-Landau (GL) 1 Crystals Functional The Generalized GL 4 Introduction to Liquid Functional 2 Crystals (LCs) References 5 Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References What are LCs Figure: The molecular orientation of different states of matter. Left - Solid, Middle - Liquid Crystal, Right - Isotropic Liquid Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References Types of LCs Figure: Arrangement of Molecules in particular LCs. Left - Nematic LCs, Middle - Cholesteric (Chiral Nematic) LCs, Right - Smectic LCs Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References Smectic C* Liquid Crystal Molecular Orientation Figure: Left Two Figures Source: http://barrett-group. mcgill.ca/teaching/liquid_crystal/LC03.htm Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References Director Projection onto Plane Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References Outline 3 Effects of Defects in Liquid Ginzburg-Landau (GL) 1 Crystals Functional The Generalized GL 4 Introduction to Liquid Functional 2 Crystals (LCs) References 5 Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References Introducing a dust particle Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References Figure: The effect of impurity ions on a thin film Smectic C* liquid crystal[LPM] Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References Energy Described over Smectic C* Liquid Crystals Consists of elastic energy, anchoring energy at domain boundary, and anchoring energy at boundary of defect core Energy from core boundary negligible. Anchoring energy at domain boundary results from polarization field. Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References Effect of polarization field p � n × v = ⇒ p ⊥ c The elastic energy contribution from the polarization field is described as � � ∂ Ω p · ν d σ Ω ∇ · p dx = where ν is the outer unit normal vector on ∂ Ω Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References � Want ∂ Ω p · ν d σ to be as negative as possible. = ⇒ p = − αν , α ∈ R + on ∂ Ω = ⇒ c � τ on ∂ Ω Introducing boundary values model effect of spontaneous polarization. Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References The resulting framework becomes minimizing � Ω k 1 ( div u ) 2 + k 2 ( curl u ) 2 dA u = ( u 1 , u 2 ) , | u | = 1 div u = ∂ x 1 u 1 + ∂ x 2 u 2 , curl u = ∂ x 1 u 2 − ∂ x 2 u 1 splay and bend constants k 1 , k 2 > 0, k 1 � = k 2 to incorporate electrostatic contribution from p . { u ∈ H 1 (Ω) : | u ( x ) | = 1 for x ∈ Ω and u = g on ∂ Ω } = / 0 for deg g := d > 0. Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References Outline 3 Effects of Defects in Liquid Ginzburg-Landau (GL) 1 Crystals Functional The Generalized GL 4 Introduction to Liquid Functional 2 Crystals (LCs) References 5 Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References We study instead J ε ( u ) = 1 Ω k 1 ( div u ) 2 + k 2 ( curl u ) 2 + 1 � 2 ε 2 ( 1 −| u | 2 ) 2 dx 2 (1) � = Ω j ε ( u , ∇ u ) dx u ∈ H 1 g (Ω) = { u ∈ H 1 (Ω; R 2 ) : u = g on ∂ Ω } where g is smooth on ∂ Ω , | g | = 1, and deg g = d > 0 Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References Set k = min ( k 1 , k 2 ) . k 1 ( div u ) 2 + k 2 ( curl u ) 2 = k 1 | ∇ u | 2 +( k 2 − k 1 )( curl u ) 2 + 2 k 1 det ∇ u = k 2 | ∇ u | 2 +( k 1 − k 2 )( div u ) 2 + 2 k 2 det ∇ u If k = k 1 , all constants in second line are positive and if k = k 2 , all constants in third line are positive. Colbert-Kelly Analysis of a G-L Energy

Ginzburg-Landau (GL) Functional Introduction to Liquid Crystals (LCs) Effects of Defects in Liquid Crystals The Generalized GL Functional References Splay Configuration u s = ± x | x | = ± ( x 1 , x 2 ) | x | 1 ⇒ ( div u s ) 2 = | ∇ u s | 2 = curl u s = 0 = | x | 2 for x � = 0 Colbert-Kelly Analysis of a G-L Energy