Edge Dislocation in Smectic A Liquid Crystal Lu Zou Nov. 21, ’05 For Group Meeting

Reference and outline General expression –“Influence of surface tension on the stability of edge dislocations in smectic A liquid crystals”, L. Lejcek and P. Oswald, J. Phys. II France, 1 (1991) Application in a vertical smectic A film –“Edge dislocation in a vertical smectic-A film: Line tension versus film thickness and Burgers vector”, J. C. Geminard and etc., Phys. Rev. E, Vol. 58 (1998)

z’ z = D z = 0 b A 1, γ 1 A 2, γ 2 x z Burgers vectors Surface Tension

Notations K  Curvature constant B  Elastic modulus of the layers γ  Surface tension b  Burgers vectors u(x,z)  layer displacement in z-direction λ  characteristic length of the order of the layer thickness λ= (K/B) 1/2

The smectic A elastic energy W E (per unit-length of dislocation) (1) The surface energies W 1 and W 2 (per unit-length of dislocation) (2) u = u (x, z)  the layer displacement in the z-direction The Total Energy W of the sample (per unit-length of dislocation) W = W E + W 1 + W 2

Equilibrium Equation (3) Boundary Conditions at the sample surfaces (Gibbs-Thomson equation) (4) Minimize W with respect to u,

-z’ z’+2D -z’+2D z’-2D -z’-2D z’-4D -z’+4D z’+4D z = 5D z = 4D z = 3D z = 2D z = D z = 0 z = -D z = -2D z = -3D z = -4D z A1bA1b b A2bA2b (A 1 A 2 )b (A 1 A 2 )A 2 b (A 1 A 2 ) 2 b (A 1 A 2 )b (A 1 A 2 )A 1 b (A 1 A 2 ) 2 b A 1, γ 1 A 2, γ 2 Burgers vectors x Surface Tension z’ In an Infinite medium

Error function :

(5)

Some discussions For free surfaces (i.e. γ 1 =γ 2 = 0) α 1 = α 2 = 0 and A 1 = A 2 = 0 Burgers vector equal to –b and b alternatively a crude approximation  wrong results A sample sandwiched between 2 plates K 1 = K 2 = 0 so α 1, α 2  ∞ and A 1 = A 2 = 1 all the image dislocations has Burgers vector b

Interaction between two parallel edge dislocations The interaction energy is equal to the work to create the first dislocation [b 1, (x 1, z 1 )] in the stress field …… of the second one [b 2, (x 2, z 2 )]. (6)

(7)

Discussion If α 1,α 2 > 1, then A 1, A 2 > 0, and W I > 0 Two dislocations of the same sign REPULSE each other Two dislocations of the opposite signs ATTRACT each other

Interaction of a single dislocation with surfaces Put b 1 = b 2 = b, x 1 = x 2 and z 1 = z 2 = z 0 Rewrite equ(7) as (8)