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