1 / 19

Edge Dislocation in Smectic A Liquid Crystal (Part II)

Edge Dislocation in Smectic A Liquid Crystal (Part II). Lu Zou Sep. 19, ’06 For Group Meeting. Reference and outline . General expression

teagan
Télécharger la présentation

Edge Dislocation in Smectic A Liquid Crystal (Part II)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Edge Dislocation in Smectic A Liquid Crystal(Part II) Lu Zou Sep. 19, ’06 For Group Meeting

  2. 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) 931-937 • 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) 5923-5925

  3. z Surface Tension Burgers vectors A1, γ1 z = D z’ b x z = 0 A2, γ2

  4. 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

  5. The smectic A elastic energy WE (per unit-length of dislocation) • (1) • The surface energies W1 and W2 (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 = WE + W1 + W2

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

  7. In an Infinite medium z Burgers vectors z = 5D (A1A2)2b z’+4D z = 4D Surface Tension (A1A2)A1b -z’+4D z = 3D z’+2D (A1A2)b z = 2D A1b -z’+2D A1, γ1 z = D z’ b x z = 0 A2, γ2 A2b -z’ z = -D z’-2D (A1A2)b z = -2D -z’-2D (A1A2)A2b z = -3D z’-4D (A1A2)2b z = -4D

  8. (5)

  9. Error function :

  10. Interaction between two paralleledge dislocations • The interaction energy is equal to the work to create the first dislocation [b1, (x1, z1)] in the stress field of the second one [b2, (x2, z2)]. (6)

  11. (7)

  12. Interaction of a single dislocation with surfaces • Put b1 = b2 = b, x1= x2 and z1 = z2 = z0 Rewrite equ(7) as (8)

  13. In a symmetric case Polylogarithm function

  14. Minimize Equ. (8)

  15. In our case AIR thicker layers 8CB (n+1+1/2) BILAYER Trilayer (1+1/2) BILAYER H2O

  16. Calculation result with γ, λ, B, K for both AIR/8CB and 8CB/Water, t = 0.54 ≈ 0.5

  17. AIR EXAMPLE: If 10 bilayers on top of trilayer, (n = 10) Then, D = 375 Ǻ ξ= 173 Ǻ θ≈ 44o θ 8CB D H2O Obviously,θ with n

  18. Because of the symmetry, In our case, b = n d = ΔL d is the thickness of bilayer. } ΔL cutoff energy  γc = 0.87 mN/m

  19. worksheet AIR 8CB H2O

More Related