Edge Dislocation in Smectic A Liquid Crystal

(Part II)

Lu Zou

Sep. 19, ’06

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) 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

z’z = D

z = 0b

A1, γ1

A2, γ2

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 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

22

2

2

2

1

z

uB

x

uKdxdzWE

dxx

uWdx

x

uW

zDz

2

0

22

2

11 2

,2

Equilibrium Equation

(3)

Boundary Conditions at the sample surfaces

(Gibbs-Thomson equation)

(4)

Minimize W with respect to u,

2

2

4

4

z

uB

x

uK

0,,2

2

22

2

1

zx

u

z

uBandDz

x

u

z

uB

-z’

z’+2D

-z’+2D

z’-2D

-z’-2D

z’-4D

-z’+4D

z’+4Dz = 5D

z = 4D

z = 3D

z = 2D

z = D

z = 0

z = -D

z = -2D

z = -3D

z = -4D

z

A1b

b

A2b

(A1A2)b

(A1A2)A2b

(A1A2)2b

(A1A2)b

(A1A2)A1b

(A1A2)2b

A1, γ1

A2, γ2

Burgers vectors

x

Surface Tension

z’

In an Infinite medium

]}

2'212'

2'212'

2'212'

2'212'[

'21'

'21'{

4),(

2/11

21

2/12

21

2/121

12/121

2/12

2/1

Dmzz

xerfDmzzsg

A

AA

Dmzz

xerfDmzzsg

A

AA

Dmzz

xerfDmzzsgAA

Dmzz

xerfDmzzsgAA

zz

xerfzzsgA

zz

xerfzzsg

bzxu

m

m

m

m

m

(5)

Error function :

BwithA

BwithA

BK

22

2

22

11

1

11

2/1

1

1

1

1

)/(

x

dttxerf0

2 )exp(2

)(

Interaction between two parallel edge 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)

1

1

21 x

zzI dx

z

uBbW

]}

22

22[

{4

21

24/

1

21

21

24/

2

21

21

24/

211 21

24/

21

21

4/

2

21

4/21

212

21212

21

212

21212

21

212

21212

21

Dmzz

e

A

AA

Dmzz

e

A

AA

Dmzz

eAA

Dmzz

eAA

zz

eA

zz

ebbBW

DmzzxxmDmzzxxm

Dmzzxxm

m

Dmzzxxm

zzxxzzxx

I

(7)

Interaction of a single dislocation with surfaces

• Put b1 = b2 = b, x1= x2 and z1 = z2 = z0

Rewrite equ(7) as

1 0

221

02

2121

0

22 12

28 m

mmm

IDmz

AAA

DmzA

AA

Dm

AA

z

AbBW

(8)

In a symmetric case

Polylogarithm function

Minimize Equ. (8)

BwithA

BwithA

BK

22

2

22

11

1

11

2/1

1

1

1

1

)/(

In our case

AIR

H2O

8CB

Trilayer

thicker layers

(1+1/2) BILAYER

(n+1+1/2) BILAYER

Calculation result

with γ, λ, B, K for both AIR/8CB and 8CB/Water,

t = 0.54 ≈ 0.5

AIR

H2O

8CBθ

EXAMPLE:

If 10 bilayers on top of trilayer,

(n = 10)

Then,

D = 375 Ǻ

ξ= 173 Ǻ

θ≈ 44o

D

Obviously,θ with n

Because of the symmetry,

} ΔL

In our case,

b = n d = ΔL

d is the thickness of bilayer.

cutoff energy γc = 0.87 mN/m

worksheet

AIR

H2O

8CB