where
θ H
n cos
H ˆ
,
T k
H h
B 2
χ Δ
is the dimensionless parameter of the magnetic field,
and
S cos
2 1
θ 3
1 2
. If this is applied to the swollen LCE, which has two kinds of mesogen with
different volume fraction, then the total energy is the sum of each component as follows :
14 2
1 φ
2 1
φ 13
2 1
φ 2
1 φ
2 2
3 1
1 1
3 1
2 1
2 2
3 1
1 1
3 1
2 1
2 1
S S
h f
S S
h T
k N
F F
F F
mag B
t mag
m m
mag
2.2. Order Parameter
Orientational order is the most important feature of liquid crystals. The average directions of the long axes of the
rod-like molecules are parallel to each other. Because of the orientational order, liquid crystals possess anisotropic
physical properties; that is, in different directions, they have different responses to external fields such as an
electric field, a magnetic field, and shear [23, 24]. In mixtures of liquid crystals, the molecules of different
components may possess different degrees of nematic
ordering. In the mixture, the order parameter expressed by [25, 26]:
where
1 3
2 1
2
x x
P
is the second Legendre polynomial,
i
θ
is the angle between a reference axis and the director of a mesogen belonging to component i i=1
for solute
and i=2
for solution,
and
θ θ
π 2
d sin
d
i
Ω
. The function
i
f θ
is the normalized orientation distribution function which may
be expressed by:
17 θ
1 θ
T k
V exp
Z f
B i
i i
where
i
Z
is partition function defined as:
18 θ
π
d T
k V
exp Z
i B
i i
Ω
and
i
V θ
is the potential field describes intermolecular interaction. In the Maier-Saupe model, the potential
i
V θ
expressed by:
21 α
20 1
θ θ
19 1
θ α
θ
2 2
3 2
2 3
T k
S m
cos T
k m
V cos
S V
B i
i i
B i
i i
i i
where
α
is orientational interaction constant and
i
m
is dimensionless mean field parameter. Substitution
Eqs.17, 18, 19, and 21 into Eq.16, we obtain :
2 2
2 2
2 2
cos cos
exp sin d
cos exp
sin d cos
exp cos
sin d 3
exp cos
sin d
i i
i i
i i
i i
i i
i i
S P P
kT S
S P kT
P m P
S m P
22
2
The order parameter
i
S can be obtained by numerically solving of Eq.23. The average
value of order parameter
S
in binary mixture is given by [27]:
4 2
φ φ
2 2
1 1
S S
S
where
1
S
is order parameter of mesogen in solute and
2
S
is order parameter of mesogen in solvent.
2.3. Phase Transition under Magnetic Field
