8 Fréedericksz-Like Positional Transition Triggered by An External Electric Field
331
With Dirichlet boundary conditions n μ (s) = 0 on the two walls, the solution to EulerLagrange equations can be written as [42]
n μ (x) =
V
d
3 x
G μ (x, x
)[−∂
μ P(x
) + ∂
μ ∂
z C(x
)],
(8.22)
where G μ is the Green’s function for n μ . Notice that here μ in the integral does not
follow Einstein summation notation.
8.4 Results and Discussions
8.4.1 Homeotropic Boundary Condition
8.4.1.1 External Field Perpendicular to the Two Plates
Here we choose the coordinate z axis along the normal direction of the two cell walls
where LC molecules are homeotropically anchored, as depicted in Fig. 8.2a). In
the first case, when an electric is applied perpendicular to the two plates, i.e., Ez in
Fig. 8.2a, the corresponding Euler-Lagrange equations are written as Eq. (8.19). With
Dirichlet boundary conditions n μ (z = 0) = n μ (z = L) = 0, the Green’s function
can be derived as [42]
G μ (x, x
) =
4
L
∞
n=1
∞
m=−∞
e
im(ϕ−ϕ
) sin
nπ z
L
sin
nπ z
L
I m (λ n ρ < )K m (λ n ρ > ). (8.23)
Here ϕ and ϕ
are the azimuthal angles, z and z
are the positional coordinates, I m
and K m are modified Bessel functions, ρ < is the smaller one between
x 2 + y 2 and
x
2 + y
2 , and λ n = [(nπ/L)
2
+ εE
2
/4π K ]
1/2 with L the thickness of the NLC
cell. Using the definition of self energy given in terms of Green’s function[42]
U
sel f
dd = −2π K p
2
∂ μ ∂
μ H μ (x, x
)| x=x
,
(8.24)
where H μ (x, x
) = G μ (x, x
) − 1/|x − x
|, we can obtain the elastic energy U
I
e for
an NLC cell with a microparticle suspended in the presence of an electric field.
Besides the elastic energy, the gravitational potential U g due to buoyant force should
be considered as well, leading to a total energy written as
U
I
total = U
I
e + U g
= − 2π K p
2
−
4
L
∞
n=1
λ
2
n sin
2 (
nπ z
L
)K 0 (λ n ρ) +
1
ρ 3
ρ→0
−
4
3
πr
3 (ρ LC − ρ mp )gz,
(8.25)
331
With Dirichlet boundary conditions n μ (s) = 0 on the two walls, the solution to EulerLagrange equations can be written as [42]
n μ (x) =
V
d
3 x
G μ (x, x
)[−∂
μ P(x
) + ∂
μ ∂
z C(x
)],
(8.22)
where G μ is the Green’s function for n μ . Notice that here μ in the integral does not
follow Einstein summation notation.
8.4 Results and Discussions
8.4.1 Homeotropic Boundary Condition
8.4.1.1 External Field Perpendicular to the Two Plates
Here we choose the coordinate z axis along the normal direction of the two cell walls
where LC molecules are homeotropically anchored, as depicted in Fig. 8.2a). In
the first case, when an electric is applied perpendicular to the two plates, i.e., Ez in
Fig. 8.2a, the corresponding Euler-Lagrange equations are written as Eq. (8.19). With
Dirichlet boundary conditions n μ (z = 0) = n μ (z = L) = 0, the Green’s function
can be derived as [42]
G μ (x, x
) =
4
L
∞
n=1
∞
m=−∞
e
im(ϕ−ϕ
) sin
nπ z
L
sin
nπ z
L
I m (λ n ρ < )K m (λ n ρ > ). (8.23)
Here ϕ and ϕ
are the azimuthal angles, z and z
are the positional coordinates, I m
and K m are modified Bessel functions, ρ < is the smaller one between
x 2 + y 2 and
x
2 + y
2 , and λ n = [(nπ/L)
2
+ εE
2
/4π K ]
1/2 with L the thickness of the NLC
cell. Using the definition of self energy given in terms of Green’s function[42]
U
sel f
dd = −2π K p
2
∂ μ ∂
μ H μ (x, x
)| x=x
,
(8.24)
where H μ (x, x
) = G μ (x, x
) − 1/|x − x
|, we can obtain the elastic energy U
I
e for
an NLC cell with a microparticle suspended in the presence of an electric field.
Besides the elastic energy, the gravitational potential U g due to buoyant force should
be considered as well, leading to a total energy written as
U
I
total = U
I
e + U g
= − 2π K p
2
−
4
L
∞
n=1
λ
2
n sin
2 (
nπ z
L
)K 0 (λ n ρ) +
1
ρ 3
ρ→0
−
4
3
πr
3 (ρ LC − ρ mp )gz,
(8.25)