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another approach, introduced by Leikin et al. (1996), that insists that an effec-
tive lipid molecule volume should be used:
VV xV
l
=+
(3.6)
A
B
Here
V
A
is the molar volume of lipid that builds the rods and
V
B
is the
molecular volume of the second lipid. Equation (3.6) is used to standardize
the analysis of lipid mixture and to evaluate the pivotal parameters (see
Section 3.2.2 ).
3.2.2
Pivotal Plane
The pivotal plane is the plane that has a molecular cross-sectional area
invariant upon isothermal bending. The position of the pivotal plane (Fig. 3.1)
depends on the relationship between area compressibility and bending of the
monolayer (Leikin et al., 1996). In other words, the pivotal plane position and
elastic constants are specifi c to a particular deformation. Especially in the H
II
phase, the pivotal plane is the surface at which the area remains constant as
the curvature in the phase is changed by varying the water content. The last
defi nition connects the area at the pivotal plane to the process of hydration
of lipid head groups, but it is not very usable because it does not incorporate
the natural plane position (Marsh, 2011).
For the pivotal plane, Eq. (3.1) could be transformed as follows (Helfrich,
1973; Kirk et al., 1984):
2
1
2
11
FAk
RR
(3.7)
≈
−
pcp
p
0
p
where
A
p
is the area per molecule at the pivotal surface,
k
cp
is the monolayer
bending modulus,
R
p
is the radius of the plane, and 1/
R
0p
is the spontaneous
curvature.
3.2.2.1 Bicontinuous Inverse Cubic Phases
Note: This section is
included only for comparison purposes. It is not a part of the study of H
II
phase
and its goal is to help the reader to better understand the process of data
management.
The models of bicontinuous inverse cubic phases are based on the infi nite
periodic minimal surface model (IPMS). The minimal surface lies at the bilayer
midplane, and it has the property that the mean curvature, defi ned as half the
sum of the principal curvatures, is zero everywhere on the surface. The pivotal
surfaces of each monolayer are displaced from the midplane at a distance
l
p
(midplane is defi ned by IPMS), and the area per molecule at the pivotal plane
is given by the expression (Templer et al., 1998b):
2
l
a
2
V
p
l
A
=
σ
+
2
χ
(3.8)
p
0
a
φ
l
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