336
K. Xiao and C.-X. Wu
the midplane of the NLC cell, indicating that an application of external electric field
does not trigger a positional phase transition. Moreover, in these two figures we also
find that the electric filed has a crucial impact on the shape of potential well of elastic
energy and total energy. This is because when Ez and < 0, the realignment
of liquid crystal molecules with the increase of the electric field narrows down the
interaction potential well rather than flatten it (see Fig. 8.6), which creates a force
directing toward the midplane much larger than the gravitational contribution and
thus denies any positional transition.
8.4.1.2 External Field Parallel to the Two Plates
For the case of an electric field parallel to the two plates, i.e., Ex in Fig. 8.2a, the
Euler-Lagrange equations for n x and n y are written as Eq. (8.20). With Dirichlet
boundary conditions n μ (z = 0) = n μ (z = L) = 0, the related Green’s functions G x
and G y are given by
G x (x, x
) =
4
L
∞
n=1
∞
m=−∞
e
im(ϕ−ϕ
) sin
nπ z
L
sin
nπ z
L
I m (ν n ρ < )K m (ν n ρ > ),
G y (x, x
) =
4
L
∞
n=1
∞
m=−∞
e
im(ϕ−ϕ
) sin
nπ z
L
sin
nπ z
L
I m (μ n ρ < )K m (μ n ρ > ), (8.27)
where ν n = [(nπ/L)
2
− εE
2
/4π K ]
1/2 and μ n = nπ/L. In analogue to the previous case, we can obtain the elastic energy U
I I
e and thereby the total energy U
I I
total is
written as
U I I
total = U I I
e + U g
= − 2π K p 2
−
2
L
∞
n=1
sin 2 (
nπ z
L
)(α n + β n ) +
1
ρ 3
ρ→0
−
4
3
πr 3 (ρ LC − ρ mp )gz,
(8.28)
where α n = ν
2
n K 0 (ν n ρ) + μ
2
n K 0 (μ n ρ), and β n = ν
2
n K 2 (ν n ρ) − μ
2
n K 2 (μ n ρ).
In order to seek for the effect of electric field on the equilibrium position of
microparticle, plots for the elastic energy and total energy against the microparticle
position for different electric fields are presented in Fig. 8.7. When ε > 0, the narrows down of the potential well of elastic energy and total energy with the increase
of electric field in Fig. 8.7a indicates that the microparticle tends to be trapped in
the midplane of the NLC cell. When ε < 0, the increases of electric field tends
to deepen and narrows down (which can hardly tell) the potential well of elastic
energy and total energy, as illustrated in Fig. 8.7b. Due to the deep potential well,
the microparticle is trapped in the midplane of the NLC cell regardless of the sign of
the molecular dielectric anisotropy, indicating that an application of external elec-
K. Xiao and C.-X. Wu
the midplane of the NLC cell, indicating that an application of external electric field
does not trigger a positional phase transition. Moreover, in these two figures we also
find that the electric filed has a crucial impact on the shape of potential well of elastic
energy and total energy. This is because when Ez and < 0, the realignment
of liquid crystal molecules with the increase of the electric field narrows down the
interaction potential well rather than flatten it (see Fig. 8.6), which creates a force
directing toward the midplane much larger than the gravitational contribution and
thus denies any positional transition.
8.4.1.2 External Field Parallel to the Two Plates
For the case of an electric field parallel to the two plates, i.e., Ex in Fig. 8.2a, the
Euler-Lagrange equations for n x and n y are written as Eq. (8.20). With Dirichlet
boundary conditions n μ (z = 0) = n μ (z = L) = 0, the related Green’s functions G x
and G y are given by
G x (x, x
) =
4
L
∞
n=1
∞
m=−∞
e
im(ϕ−ϕ
) sin
nπ z
L
sin
nπ z
L
I m (ν n ρ < )K m (ν n ρ > ),
G y (x, x
) =
4
L
∞
n=1
∞
m=−∞
e
im(ϕ−ϕ
) sin
nπ z
L
sin
nπ z
L
I m (μ n ρ < )K m (μ n ρ > ), (8.27)
where ν n = [(nπ/L)
2
− εE
2
/4π K ]
1/2 and μ n = nπ/L. In analogue to the previous case, we can obtain the elastic energy U
I I
e and thereby the total energy U
I I
total is
written as
U I I
total = U I I
e + U g
= − 2π K p 2
−
2
L
∞
n=1
sin 2 (
nπ z
L
)(α n + β n ) +
1
ρ 3
ρ→0
−
4
3
πr 3 (ρ LC − ρ mp )gz,
(8.28)
where α n = ν
2
n K 0 (ν n ρ) + μ
2
n K 0 (μ n ρ), and β n = ν
2
n K 2 (ν n ρ) − μ
2
n K 2 (μ n ρ).
In order to seek for the effect of electric field on the equilibrium position of
microparticle, plots for the elastic energy and total energy against the microparticle
position for different electric fields are presented in Fig. 8.7. When ε > 0, the narrows down of the potential well of elastic energy and total energy with the increase
of electric field in Fig. 8.7a indicates that the microparticle tends to be trapped in
the midplane of the NLC cell. When ε < 0, the increases of electric field tends
to deepen and narrows down (which can hardly tell) the potential well of elastic
energy and total energy, as illustrated in Fig. 8.7b. Due to the deep potential well,
the microparticle is trapped in the midplane of the NLC cell regardless of the sign of
the molecular dielectric anisotropy, indicating that an application of external elec-