8 Fréedericksz-Like Positional Transition Triggered by An External Electric Field
329
with Euler-Lagrange equations of Laplace type
∇
2 n μ = 0.
(8.15)
Here n μ (μ = x, y) represents the components of the director field n perpendicular
to n 0 . Expanding the solutions into multipoles and we have the form of the solutions
as follows [53, 55]
n x = p
x
r 3 + 3c
xz
r 5 ,
(8.16)
n y = p
y
r 3 + 3c
yz
r 5 ,
(8.17)
where p and c are the magnitude of the dipole and quadrupole moments respectively.
If a particle is immersed in a NLC with confinement (i.e., NLC cell) and in the
presence of external electric field, another surface anchoring free energy at the two
plates of the cell and free energy arising from the applied electric field are need
to be added. Thus our task is to minimize the more complicated total free energy
functional. Unfortunately, it becomes more difficult to find the analytical solutions
for the system of a particle suspended in a NLC cell in the presence of external
field. Therefore, it is necessary to develop a phenomenological method to address
this problem [40, 55], and this approach is introduced in the next section.
8.3 Theoretical Modeling
In order to introduce the phenomenological method mentioned above in details, we
take the system that a spherical microparticle of radius r suspended in a NLC cell
sandwiched between two parallel plates a distance L apart in the presence of an electric field as an example. The polarization of the particle is neglected compared with
the influence of external field on the alignment of liquid crystal molecules. Figure 8.2
illustrates two systems schematically under external field with a homeotropic anchoring (Fig. 8.2a) and a homogeneous planar anchoring (Fig. 8.2b) respectively at the
two cell walls. The suspended microparticle induces the director distortion and the
director deviations n μ (μ = x, y) from the undeformed director field n 0 = (0,0,1) are
small at the region far from the microparticle. In order to use the same set of symbol
subscripts (n μ (μ = x, y)) in our theoretical modelling for the two surface anchoring
conditions, two different coordinate frames are deliberately used here, as illustrated
in Fig. 8.2a and b. Assuming n ≈ (n x ,n y ,1) with one Frank constant approximation,
the effective elastic energy for the system reads [42]
U e = K
d
3 x
(∇n μ )
2
2
−
k
2
2
(e · n)
2
− 4π P(x)∂ μ n μ − 4πC(x)∂ z ∂ μ n μ
, (8.18)
329
with Euler-Lagrange equations of Laplace type
∇
2 n μ = 0.
(8.15)
Here n μ (μ = x, y) represents the components of the director field n perpendicular
to n 0 . Expanding the solutions into multipoles and we have the form of the solutions
as follows [53, 55]
n x = p
x
r 3 + 3c
xz
r 5 ,
(8.16)
n y = p
y
r 3 + 3c
yz
r 5 ,
(8.17)
where p and c are the magnitude of the dipole and quadrupole moments respectively.
If a particle is immersed in a NLC with confinement (i.e., NLC cell) and in the
presence of external electric field, another surface anchoring free energy at the two
plates of the cell and free energy arising from the applied electric field are need
to be added. Thus our task is to minimize the more complicated total free energy
functional. Unfortunately, it becomes more difficult to find the analytical solutions
for the system of a particle suspended in a NLC cell in the presence of external
field. Therefore, it is necessary to develop a phenomenological method to address
this problem [40, 55], and this approach is introduced in the next section.
8.3 Theoretical Modeling
In order to introduce the phenomenological method mentioned above in details, we
take the system that a spherical microparticle of radius r suspended in a NLC cell
sandwiched between two parallel plates a distance L apart in the presence of an electric field as an example. The polarization of the particle is neglected compared with
the influence of external field on the alignment of liquid crystal molecules. Figure 8.2
illustrates two systems schematically under external field with a homeotropic anchoring (Fig. 8.2a) and a homogeneous planar anchoring (Fig. 8.2b) respectively at the
two cell walls. The suspended microparticle induces the director distortion and the
director deviations n μ (μ = x, y) from the undeformed director field n 0 = (0,0,1) are
small at the region far from the microparticle. In order to use the same set of symbol
subscripts (n μ (μ = x, y)) in our theoretical modelling for the two surface anchoring
conditions, two different coordinate frames are deliberately used here, as illustrated
in Fig. 8.2a and b. Assuming n ≈ (n x ,n y ,1) with one Frank constant approximation,
the effective elastic energy for the system reads [42]
U e = K
d
3 x
(∇n μ )
2
2
−
k
2
2
(e · n)
2
− 4π P(x)∂ μ n μ − 4πC(x)∂ z ∂ μ n μ
, (8.18)