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1, . . . , 6 are the six Leslie
viscosity coeffi cients;
A
is the symmetric rate of deformation tensor;
N
is the
Zaremba-Jaumann time derivative of the director; and
W
is the vorticity
tensor. The Frank elastic energy density
f
g
is given by
where
p
is the pressure,
I
is the unit tensor;
α
i
,
i
=
2
2
2
2
f
=∇ +
K
(
i
n
)
K
(
n
i
×+
n
)
K
(
n
¥¥
∇
n
)
(2.17)
g
11
22
33
where {
K
ii
;
ii
11, 22, 33} are the temperature-dependent three elastic con-
stants for splay, twist, and bend, respectively. Anisotropies and thermal depen-
dence of the elastic constants are discussed in the literature (de Gennes and
Prost, 1993; Larson and Doi, 1991; Rey 2007, 2009, 2010; Rey and Denn, 1987,
1998a-c, 2002). The director torque balance equation is given by the sum of
the viscous
=
Γ
v
and the elastic
Γ
e
torque:
GG G
v
+
e
=
0
v
=
n
×
h
v
≡ −
n
×
(
γ
N +
γ
A n
⋅
)
G
e
1
2
(2.18)
∂
∂
f
−∇⋅
∂
∂∇
f
g
g
=
nh
¥
e
≡ −
n
×
T
n
()
n
where
h
v
is the viscous molecular fi eld,
h
e
is the elastic molecular fi eld,
γ
2
is the irrota-
tional torque coeffi cient. For thermotropic LCs, the rheological behavior is
controlled by the temperature-dependent reactive parameter
1
=
α
3
−
α
2
is the rotational viscosity, and
γ
2
=
α
6
−
α
3
=
α
3
+
α
λ
, given by
γ
γ
αα
αα
+
−
2
2
3
λ
=−
=−
(2.19)
1
3
2
2.2.3.2 Rheological Functions
The LE nematodynamics predicts the fol-
lowing rheological functions (de Gennes and Prost, 1993; Larson and Doi,
1991; Rey, 2007, 2009, 2010; Rey and Denn, 2002):
Shear Flow Alignment
At suffi ciently large Er, when
λ
>
1 (rods) or
λ
<
−
1
(disks), the stable shear fl ow alignment angle
θ
al
is given by (de Gennes and
Prost, 1993 )
1
2
−
1
(2.20)
θ
=
cos
al
λ
where
θ
al
is the angle between
n
and the velocity
v
in the shear plane (
v
−
∇
v
plane).
Shear Viscosities
η
3
) that
characterize viscous anisotropy are measured in a steady simple shear fl ow
between parallel plates with fi xed director orientations along three character-
istic orthogonal directions:
The three Miesowicz shear viscosities (
η
1
,
η
2
, and
η
1
when the director is parallel to the velocity
direction,
η
2
when it is parallel to the velocity gradient, and
η
3
when it is paral-
lel to the vorticity axis, given by
ααα
++
−+ +
ααα
α
3
4
6
2
4
5
4
(2.21)
η
=
η
=
η
=
1
2
3
2
2
2
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