Chemistry Reference
In-Depth Information
Eqs. (2.3) and (2.5), we can easily fi nd the ratio of the energy differences
caused by the shape and the molecular anisotropies:
∆
∆
W
W
S
DD
χ
mol
shape
=−
a
(2.6)
(
)
(
)
χχ
−
−
χ
a
c
a
The calculation of this ratio is impeded by the lack of experimental data of
the diamagnetic susceptibilities and anisotropy. However, one can estimate
th
e latter by using the additive scheme of atomic susceptibilities. Thus, for
χχ χ
(
)
2/
the following values were obtained [in centimeter-gram-sec-
ond (CGS) units/mole]: for NaDS, 165
=+
⊥
1 0
− 6
; for Kl, 155
1 0
− 6
; and for DeOH,
×
×
1 0
− 6
. By
u
sing the known compositions (see above), one can adopt the
typical values
124
×
610
7
CGS units/cm
3
, and
1 0
− 7
CGS unit/cm
3
.
χ=− ×
−
χ
0
=
−
7
×
Estimates of
0
have been obtained under the assumption that the diamag-
netic anisotropy arises only from the anisotropy of the methylene groups.
Typical values for all the systems are equal to about 0.1. Upon using these
values of the diamagnetic constant as well as the typical values of
S
equal to
0.5- 0.6 we fi nd that
χχ
∆
∆
W
W
mol
shape
≈
10
4
(2.7)
Hence it is clear that the orientation of a lyotropic nematic is caused by the
molecular diamagnetic susceptibility (Boden et al., 1985 ; Sonin, 1987 ).
2.2
FLOW MODELING OF NEMATIC LIQUID CRYSTALS
2.2.1
Quadrupolar Order Parameter
The Landau-de Gennes theory of liquid crystals describes the viscoelastic
behavior of nematic liquid crystals by means of the second moment of the
orientation distribution function, known as the tensor order parameter
Q
(de
Gennes and Prost, 1993; Rey, 2007, 2009, 2010):
Q u
I
∫
()
2
=
−
f
uu
d
(2.8)
3
where
u
is the molecular unit vector, and
I
is unit tensor. The tensor order
parameter
Q
is expressed in terms of the orthonormal director triad (
n, m, l
)
and the scalar order parameters (
S
,
P
):
(
)
+
(
)
Q nI
=
S
−
1
3
1
3
P
mll
−
1
3
(2.9)
Q
T
;
Q
:
I
where the following restrictions apply:
Q
=
=
0 ;
−
≤
2
S
≤
2 ;
−
3
≤
2
P
≤
3. The uniaxial director
n
corresponds to the maximum eigenvalue
μ
n
=
2
S
/3,
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