Order Parameters and Defects in Liquid Crystals

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1 Order Parameters and Defects in Liquid Crystals Ferroelectric phenomena in liquid crystals June, 2007

2 Liquid Crystal-substance with a degree of electronic-liquid Crystal Crystal Presentations crystalline order that remains in a liquid state Simple picture- long thin molecules

3 Energy to describe static stable configurations Made up of competing energies Molecular packing (molecular alignment) Thermal effects (randomize alignment) External stresses (mechanical, electrical..) Temperature dependent-different terms become dominant at different temperatures

4 T Isotropic Nematic Smectic Nematic phase-energy seeks local alignment of long axes. n(x) is the local average of long axes. n(x) =1 n(x) electronic-liquid Crystal Crystal Presentations

5 n(x) is a unit vector field Molecules do not have a head and tail as n(x) does. In problems where this does not lead to inconsistencies n(x) is the simplist way to describe a nematic liquid crystal. n(x)-director field

6 Oseen Frank Energy Energy records the cost of distortions away from n(x)=const. Written in terms of pure splay, twist, and bend. These have K 1,K 2, K 3 as coefficients respectively

7 Pure Distortions electronic-liquid Crystal Crystal Presentations Splay Bend Twist

8 Special case, K = K 2 = K 3 = K, K 4 1 = vector identity F F (n ) = 0, one constant approximation electronic-liquid Crystal Crystal Presentations

9 Boundary Conditions electronic-liquid Crystal Crystal Presentations 1) n = on Ω n 0 2) 2 F b = c Ω ( n e ( x, n 0 =1 )) 1) Strong anchoring problem 2) F=F F +F b weak anchoring If c<0 then n(x) tends to be parallel to e(x). e(x) is the easy axis. If n(x) or e(x) are parallel to the boundary normal we are promoting homeotropic boundary values.

10 If c>0 then n(x) tends to be e(x) on the boundary. If electronic-liquid Crystal Crystal Presentations e(x) is parallel to the boundary normal then n(x) tends Thm towards tangential boundary values. Assume is a smooth surface, n 0 is smooth, K 1,K 2, K 3 >0. Then there is a minimizer for F(n) in the class A Ω 2 = { u u : Ω S, u = n 0, Ω Ω u provided A is nonempty. 2 < }

11 n(x) may have singularities, defects electronic-liquid Crystal Crystal Presentations ex. n(x)=x/ x Hedgehog. n(x) has homeotropic b.v. and a defect at one point. Minimizers from the theorem can not be singular on a curve. Line singularities in liquid crystals do exist, disclination lines. Try n(r,,z)=e r and get θ n 2 = Ω = B R

12 Cladis and Kleman consider : electronic-liquid Crystal Crystal Presentations r e z with n ( r ) = cos ϕ ( r ) e + sin ϕ ( r ) ϕ ( R ) = 0 z n(r) z r r R

13 F 2 R 2 cos = B R n dxdy = 2 π 0 (( ϕ ') + ) 2 r 2 ϕ rdr electronic-liquid Crystal Crystal Presentations 1 They prove the minimizer is ϕ ( r ) 2 tan ( ) R Solution is not singular. It escapes to the third dimension. = π 2 r R

14 Introduce an order parameter that allows n to melt near r=0 f( ϕ ) probability density that the director of molecules near x are within an angle Of n(x). f 0 ( ϕ ) = s=1 strongly nematic, s=0 isotropic 1 4π s(x) is the the isotropic order parameter. Measures how concentrated f is near ϕ = s ( x ) = cos ( )( 2 ϕ f 2 S f 0 )

15 Ericksen considers a nematic described by the pair (s,n) s 2

16 If k 1 electronic-liquid Crystal Crystal Presentations Mizel, Roccato, and Virga prove that this has a minimizer ( ~ s, n ~ ) in the family { ( r ),cos ϕ ( r ) e + sin ϕ ( r ) e, ( R ) = 0, s ( R ) = s > 0 s r z ϕ of the form ~ ϕ 0, ~ s ( r ) > 0 for > 0, ~ s (0) = 0 The solution is allowed to melt into the isotropicn phase at r=0, has finite energy, and has a disclination line along the z axis. If k > 1 then the solution has ~ s (0) > 0 and 0} ~ ϕ (0) = π 2.

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18 uniaxial biaxial

19 ρ (x) 2 = π Smectics

20 Molecules locally align and in addition form layers ν

21 Layer normal Density modulation ρ ν 2 π x ) = ρ + ρ cos( ν d 2π d ( 0 1 x q =, d = layer thickness If ρ 1 > 0 then layers exist - smectic phase If ρ 1 =0 then no layers - nematic phase )

22 If ν n - smectic A If not - smectic C Complex order parameter ψ = ρ ( x i ) e ϕ * 2 ρ ϕ = ψ + ψ 1 cos ψ = 0 nematic, ψ 0 smectic ( x )

23 Energy electronic-liquid Crystal Crystal Presentations III II I A>0

24 I ρ =1 1 prefers smectic phase II locally the level sets of ϕ are the layers, ϕ n, ϕ q III penalizes phase transitions

25 0 R r electronic-liquid Crystal Crystal Presentations

26 R 0 frdr + f surf R

27 K e ii easy 2 = S k, = e r ii

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30 Cholesteric Phases = π τ electronic-liquid Crystal Crystal Presentations

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32 Chirality introduces a stress onto the liquid crystal that affects its phase transition. τ lowers the transition temperature

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34 Smectic C

35 Set Then i ψ = e ω ( x) Now let ω = k x and let n= const, then k k n n

36 This minimizes for k 2 = and cos θ = = k k n 2 π k 2 π k θ

37 F = ( F + F F Sm ) dx F = ( F + F F Sm ) dx electronic-liquid Crystal Crystal Presentations

38 Boundary value problem for F. M. Calderer and J.Park electronic-liquid Crystal Crystal Presentations

39 N. Clark et al Jakli et al Bauman et al