Chemistry Reference
In-Depth Information
where
t
s
is the viscoelastic stress tensor,
H
is the dimensionless molecular
fi eld, and
ˆ
Q
is the Jaumann derivative of the tensor order parameter. The
molecular fi eld
H
is the negative of the variational derivative of the free
energy density
f
:
f
ckT
1
2
1
3
1
3
1
4
(
)
+
(
)
2
=−
1
U
Q:Q
−
U
Q: Q Q
U
Q:Q
i
(2.34)
L
ckT
L
ckT
1
(
)
T
2
(
)
(
)
+
∇∇
Q:
Q
+
∇
QQ
∇
i
i
i
2
2
where the fi rst line is the homogeneous (
f
h
) and the second is the gradient (
f
g
)
contribution;
L
1
and
L
2
are the Landau coeffi cients. In this format, comparing
Eqs. (2.12) and (2.29) gives
L
1
K
33
.
The presence of the homogeneous energy allows the resolution of defect cores
and the prediction of defect nucleation and coarsening. Expanding the forces
(
t
s
,
ˆ
Q
) in terms of fl uxes (
A
,
ckT
H
) in Eq. (2.33), and taking into account
thermodynamic restrictions and the symmetry and tracelessness of the forces
and fl uxes, the equations for
t
s
and
ˆ
Q
=
K
22
/2S
2
,
L
2
=
K
−
K
22
/
S
2
, and
K
=
K
11
=
can be obtained. The dynamics of the
tensor order parameter is given by
2
3
2
3
!
!
"
ˆ
Q
(
)
Er
=
Er
ββ
A
∗
+
A
∗
i
Q
+
Q
i
A
∗
−
A
∗
: Q I
1
2
[
(
)
−
β
A:QQ A
∗
+
∗
i
QQQ QA Q
i
+
i i
∗
}
]
{
(
)
+
QQA
i i
∗
−
QQ:A I
i
∗
"
{
}
3
1
3
1
3
!
U
"
(
)
(
)
−⋅
1
−
QQQ
−
U
i
+
U
Q: QQ QQI
+
:
(
)
2
U
1
−
3
2
Q:Q
3
!
1
2
!
(
)
∗
2
*
∗
∗
+
∇+
Q
L
∇
∇
i
Q
2
(
)
2
1
−
3
2
Q:Q
2
3
tr
"
"
{
(
)
}
{
(
)
}
T
+∇ ∇
∗
∗
i
Q
−
∇∇
∗
∗
i
Q
I
(2.35)
/
γ
,
γ
where
t
*
= γ
t
,
AA
∗
=
,
WW
∗
=
/γ
,
∇
*
=
H
∇
,
*
is a characteristic
LLL
2
=
/
2
1
is a shape parameter. The fi rst brackets denote fl ow - induced
orientation, the second phase ordering, and the third gradient elasticity. The
total extra stress tensor
t
t
for liquid crystalline materials is given by the sum
of symmetric viscoelastic stress tensor
t
s
, antisymmetric stress tensor, and
Ericksen stress tensor
t
Er
(Grecov and Rey, 2004; Rey, 2007, 2009, 2010):
shear rate, and
β
t
s a
=++
Er
(2.36)
t
t
t
t
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