Chemistry Reference
In-Depth Information
I
P
(-0.5, 1.5)
µ n = µ m
p=3s
m
(0, 1)
D
γ
B
µ m = µ 1
n
p=0
α
(1, 0) s
(-0.5, 0)
ζ
E
n
(0,-1)
µ n = µ 1
I
p = -3s
(-0.5, -1.5)
-
N -
N -
N -
N B
m i < m m = m n ,
m i < m m < m n ,
m i < m m = m n ,
m i < m m < m n ,
Line g
Region D
Line x
Region E
Figure 2.3
Ordering ( P , S ) triangle showing the different rod confi gurations.
pression ( y , z ) plane (middle and right panel). The three lines from each vertex
to the midpoint of the opposite side represent uniaxial states, and they meet
at the isotropic state, while the rest of the triangle represents the biaxial states.
By using symmetry considerations, equivalent states are found in the other
sectors. For example, at the corners of the triangle there is maximal positive
unixiality along each of the three directors. The relevant eigenvalue inequali-
ties and the superscript sign denotes the sign of the eigenvalue correspond-
ing to the unique eigenvector or for biaxial cases to prolate (
)
shapes. A thermodynamic or fl ow process is a trajectory in the ordering
triangle.
Another case is when temperature changes and provokes a modifi cation in
the micellar shape from uniaxial prolate to biaxial to uniaxial oblate. This is
represented as a trajectory from a point in the uniaxial
+
) or oblate (
α
line through the B
region, which ends on a point in the
3 S ) uniaxial line. In general, pro-
cesses involving ( n, m, l ) fl ow-induced orientation changes are represented as
trajectories in the orientation unit sphere (Boden et al., 1981; Rey, 2009). For
example, Figure 2.4 shows that imposing a strong uniaxial extension fl ow along
γ
( P
=
 
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