Chemistry Reference
In-Depth Information
Hence, under shear fl ow in the
x
direction and three-dimensional (3D) orienta-
tion
n
=
(
n
x
,
n
y
,
n
z
), a velocity of the form
v
=
(
v
x
, 0,
v
z
) must be considered at
a minimum.
First Normal Stress Difference N
l
For nematic liquid crystals
N
l
is a strong
function of orientation and can have positive or negative values. Expressions
for
N
l
in terms of the director components
n
x
and
n
y
are (Grecov and Rey,
2003a,b; Rey, 2007, 2009, 2010)
(
(
)
)
γ
Nt
=−=
t
nn
γ
+
α
nn
2
−
2
(2.28)
1
xx
yy
x
y
2
1
y
x
As the director circles the shear plane, the total number
N
T
of sign changes in
N
l
is (de Andrade Lima and Rey, 2003b,c, 2004d)
(
(
(
)
)
)
=
(
(
(
)
)
)
()
()
NNnNnN
=
γα
+
nn
2
−
2
4
N
γα
+
nn
2
−
2
T
SCx
SC
y
SC
2
1
y
x
SC
2
1
y
x
(2.29)
where
N
SC
denotes the number of sign changes. The number of sign changes
in
n
x
is 2, and similarly for
n
y
. The two following material property-dependent
outcomes are found:
γ
<
α
:
N
=
4
γ
>
α
:
N
=×=
4
2
8
(2.30)
2
1
T
2
1
T
α
1
can increase the frequency of
sign changes from four to eight. Equation (2.30) embodies the orientation-
driven fi rst normal stress sign change mechanism (ONSC). In the fl ow align-
ment regime, Eq. (2.28) becomes
The nonlinearity orientation introduced by
α
λ
λ
2
−
1
lim
E
(2.31)
1
NN
=
=
γγ
−
>
Er
1
1
al
2
ST-FA
2
λ
and is proportional to the shear rate. Shearing an LC, with a heterogeneous
director fi eld and suffi ciently high material nonlinearity (i.e., large |
1
|), at
increasing rates, it will narrow and shift the orientation distribution function
toward the Leslie angle, an orientation process that causes
N
l
to change sign
(Grecov and Rey, 2003a,b):
α
(
)
<→
(
)
>
Nx
n
(),
E
0
Nx
n
(),
EE
>
0
(2.32)
11
1
12
2
2
2.2.4
Landau de Gennes Nematodynamics
2.2.4.1
Bulk and Interfacial Equations
The governing equations for LC
fl ows follow from the dissipation function
(de Andrade Lima et al., 2006;
Farhoudi and Rey, 1993a-c; Murugesan and Rey, 2010; Rey 2007, 2009, 2010;
Soule et al., 2009):
Δ
ˆ
(2.33)
∆=
t:A
s
+
ckT
H:Q
Search WWH ::
Custom Search