342
K. Xiao and C.-X. Wu
G x (x, x
) =
4
L
∞
n=1
∞
m=−∞
e
im(ϕ−ϕ
) sin
nπ x
L
sin
nπ x
L
I m (μ n ρ < )K m (μ n ρ > ),
G y (x, x
) =
4
L
∞
n=1
∞
m=−∞
e
im(ϕ−ϕ
) sin
nπ x
L
sin
nπ x
L
I m (ν n ρ < )K m (ν n ρ > ), (8.32)
with the same ν n and μ n as those in Eq. (8.29). In a similar way, the total energy
U
I V
total is given by
U
I V
total = U
I V
e + U g
= − 2π K p
2
4
L
∞
n=1
μ
2
n cos
2 (
nπ x
L
)K 0 (μ n ρ) −
2
L
∞
n=1
ν
2
n sin
2 (
nπ x
L
)
K 0 (ν n ρ) − K 2 (ν n ρ)
+
1
ρ 3
ρ→0
−
4
3
πr
3 (ρ LC − ρ mp )gx,
(8.33)
where U
I V
e is the elastic energy.
Similar to the previous cases, to probe the influence of electric field on the equilibrium position of microparticle, plots for the elastic energy and total energy against
the microparticle position for different electric field are presented in Fig. 8.12. When
ε > 0, the potential well of elastic energy and total energy in Fig. 8.12b are narrowed down as compared with that in Fig. 8.12a. When ε < 0, Fig. 8.12c and d
show the same trend. Those results suggest that no matter ε > 0 or ε < 0, the
microparticle is always trapped at the midplane of the NLC cell regardless of the
magnitude of the electric field applied, indicating that no positional transition occurs.
The reason lies in that the realignment of the liquid crystal molecules in the presence
of the external electric field does not flatten the interaction potential well substantially enough so as to decrease its corresponding equivalent restoring force on the
microparticle to a small magnitude, with which the asymmetric gravitational force
becomes competitive.
8.4.2.3 External Field Parallel to the Two Plates and the Anchoring
Direction
Finally, we consider an NLC cell in the presence of an electric field parallel to the
two plates and the anchoring direction as well, i.e., Ez in Fig. 8.2b. Given the
corresponding Euler-Lagrange equations Eq. (8.19), the Green’s functions are [42]
G μ (x, x
) =
4
L
∞
n=1
∞
m=−∞
e
im(ϕ−ϕ
) sin
nπ x
L
sin
nπ x
L
I m (λ n ρ < )K m (λ n ρ > ), (8.34)
with the same λ n as that in Eq. (8.23). Similarly, the total energy U
V
total can be derived
as
K. Xiao and C.-X. Wu
G x (x, x
) =
4
L
∞
n=1
∞
m=−∞
e
im(ϕ−ϕ
) sin
nπ x
L
sin
nπ x
L
I m (μ n ρ < )K m (μ n ρ > ),
G y (x, x
) =
4
L
∞
n=1
∞
m=−∞
e
im(ϕ−ϕ
) sin
nπ x
L
sin
nπ x
L
I m (ν n ρ < )K m (ν n ρ > ), (8.32)
with the same ν n and μ n as those in Eq. (8.29). In a similar way, the total energy
U
I V
total is given by
U
I V
total = U
I V
e + U g
= − 2π K p
2
4
L
∞
n=1
μ
2
n cos
2 (
nπ x
L
)K 0 (μ n ρ) −
2
L
∞
n=1
ν
2
n sin
2 (
nπ x
L
)
K 0 (ν n ρ) − K 2 (ν n ρ)
+
1
ρ 3
ρ→0
−
4
3
πr
3 (ρ LC − ρ mp )gx,
(8.33)
where U
I V
e is the elastic energy.
Similar to the previous cases, to probe the influence of electric field on the equilibrium position of microparticle, plots for the elastic energy and total energy against
the microparticle position for different electric field are presented in Fig. 8.12. When
ε > 0, the potential well of elastic energy and total energy in Fig. 8.12b are narrowed down as compared with that in Fig. 8.12a. When ε < 0, Fig. 8.12c and d
show the same trend. Those results suggest that no matter ε > 0 or ε < 0, the
microparticle is always trapped at the midplane of the NLC cell regardless of the
magnitude of the electric field applied, indicating that no positional transition occurs.
The reason lies in that the realignment of the liquid crystal molecules in the presence
of the external electric field does not flatten the interaction potential well substantially enough so as to decrease its corresponding equivalent restoring force on the
microparticle to a small magnitude, with which the asymmetric gravitational force
becomes competitive.
8.4.2.3 External Field Parallel to the Two Plates and the Anchoring
Direction
Finally, we consider an NLC cell in the presence of an electric field parallel to the
two plates and the anchoring direction as well, i.e., Ez in Fig. 8.2b. Given the
corresponding Euler-Lagrange equations Eq. (8.19), the Green’s functions are [42]
G μ (x, x
) =
4
L
∞
n=1
∞
m=−∞
e
im(ϕ−ϕ
) sin
nπ x
L
sin
nπ x
L
I m (λ n ρ < )K m (λ n ρ > ), (8.34)
with the same λ n as that in Eq. (8.23). Similarly, the total energy U
V
total can be derived
as