8 Fréedericksz-Like Positional Transition Triggered by An External Electric Field
325
istence system, it is observed that external field is able to drive particles apart [12],
cause rotation [18] and alignment [43] of LC molecules, and even manipulate the
equilibrium position of microdroplet [44]. Although interactions of two particles in
a NLC are very well understood and the particle-wall interaction has been widely
observed experimentally for a single particle immersed in a nematic cell [41, 45,
46], the properties of a single particle in a uniform NLC cell in the presence of an
external electric field theoretically have not been fully addressed. Thus, it is of crucial
importance to investigate the nature of a single particle in a uniform NLC cell in the
presence of an external electric field.
8.2 Fréedericksz Transition in NLC
In a uniaxial nematic liquid crystal, the anisotropy of nematic phase is characterized
by a symmetric and traceless tensor order parameter Q αβ which can be written as
Q αβ = S(n α n β −
1
3
δ αβ ).
(8.1)
Here n α and n β are components of the director n, which is a unit vector with the
property n = −n, describing the direction along which the molecules are aligned.
And S is the scalar order parameter that describes the degree of nematic order. It also
represents how well the molecules are aligned along n. If S equal to 0, there is no
alignment, which means that the system is in an isotropic phase; If S equal to 1, it
corresponds to a perfect alignment. When the director field n(r) changes drastically
due to the distortion from undeformed ground state in nematic liquid crystal, it costs
elastic energy for the deviation of the director, which can be classified into three
types, namely splay, twist and bend, making the Frank-Oseen free energy density for
elastic distortions reads as [47]
f el =
1
2
K 11 (∇ · n)
2
+
1
2
K 22 [n · (∇ × n)]
2
+
1
2
K 33 [n × (∇ × n)]
2
,
(8.2)
where K 11 , K 22 and K 33 are Frank elastic constants corresponding to splay, twist
and bend constants, respectively. The bulk free energy of the nematic liquid crystal
sample can be obtained by integrating f el over the sample volume
F el =
f el dV.
(8.3)
LC molecules are sensitive to weak external stimuli, such as electric field, magnetic
field and light, due to the anisotropic property of NLC. The facile response to weak
external stimuli results in the easy distortion of director field when a magnetic or
an electric field is applied. If an external magnetic filed is applied to the NLC, the
following extra term should be added to the free energy
325
istence system, it is observed that external field is able to drive particles apart [12],
cause rotation [18] and alignment [43] of LC molecules, and even manipulate the
equilibrium position of microdroplet [44]. Although interactions of two particles in
a NLC are very well understood and the particle-wall interaction has been widely
observed experimentally for a single particle immersed in a nematic cell [41, 45,
46], the properties of a single particle in a uniform NLC cell in the presence of an
external electric field theoretically have not been fully addressed. Thus, it is of crucial
importance to investigate the nature of a single particle in a uniform NLC cell in the
presence of an external electric field.
8.2 Fréedericksz Transition in NLC
In a uniaxial nematic liquid crystal, the anisotropy of nematic phase is characterized
by a symmetric and traceless tensor order parameter Q αβ which can be written as
Q αβ = S(n α n β −
1
3
δ αβ ).
(8.1)
Here n α and n β are components of the director n, which is a unit vector with the
property n = −n, describing the direction along which the molecules are aligned.
And S is the scalar order parameter that describes the degree of nematic order. It also
represents how well the molecules are aligned along n. If S equal to 0, there is no
alignment, which means that the system is in an isotropic phase; If S equal to 1, it
corresponds to a perfect alignment. When the director field n(r) changes drastically
due to the distortion from undeformed ground state in nematic liquid crystal, it costs
elastic energy for the deviation of the director, which can be classified into three
types, namely splay, twist and bend, making the Frank-Oseen free energy density for
elastic distortions reads as [47]
f el =
1
2
K 11 (∇ · n)
2
+
1
2
K 22 [n · (∇ × n)]
2
+
1
2
K 33 [n × (∇ × n)]
2
,
(8.2)
where K 11 , K 22 and K 33 are Frank elastic constants corresponding to splay, twist
and bend constants, respectively. The bulk free energy of the nematic liquid crystal
sample can be obtained by integrating f el over the sample volume
F el =
f el dV.
(8.3)
LC molecules are sensitive to weak external stimuli, such as electric field, magnetic
field and light, due to the anisotropic property of NLC. The facile response to weak
external stimuli results in the easy distortion of director field when a magnetic or
an electric field is applied. If an external magnetic filed is applied to the NLC, the
following extra term should be added to the free energy