Dividing Planes of Hexagonal H Mesophase - Self-Assembled Supramolecular Architectures: Lyotropic Liquid Crystals - page 80

plane—the radius of the water cylinder. See the respective formulas in
Kulkarni et al. (2010) about the minimal and maximal lengths of the lipid
molecule that are also involved in the determination of the cone height.
But when it comes to a mechanical description of the hexagonal inverse H II
phase, the abstraction of the Luzzati plane is insuffi cient because it cannot be
directly connected to the mechanical characteristics of the rod structure—the
monolayer bending modulus and the spontaneous curvature radius. When
the rod structure deformation depends on the experimental technique used,
the monolayer bending modulus and the spontaneous curvature radius defi ni-
tions have to be taken into account. Their defi nitions are related to a specifi c
abstracted plane—the pivotal plane (Fig. 3.1). If the deformation does not
depend on experimental technique, both the monolayer bending modulus and
the spontaneous curvature radius must be defi ned relative to another abstract
plane—the natural plane (Fig. 3.1). Both the pivotal and natural planes are
connected to each other—the position of the natural plane could be calculated
only if that of the pivotal plane is known. In turn, the position of the pivotal
plane (the radius of the pivotal plane) depends on the lattice constant of the
H II mesophase. The invariant values of the monolayer bending modulus and
the spontaneous curvature radius are those at the natural plane.
The aim of the present chapter is to help the reader make the required
sequence of assumptions and calculations that leads to deriving the radii of
the three planes, their related areas of the lipid molecule, and the correspond-
ing mechanical parameters. It also draws attention to the correct statistical
treatment of the data and plane models.

3.2 DETERMINATION AND ROLE OF THE PLANES

Inside the lipid structure of the H II phase, it is possible to defi ne three cylindri-
cal Gibbs dividing planes/surfaces (ordered by their distance from the water
channel): Luzzati plane, neutral plane, and pivotal plane (Fig. 3.1). According
to Gibbs terminology, the defi ned planes should have zero surface excess of
water and lipid . Kozlov and Winterhalter (1991a,b) proposed the following
expression of the elastic energy per lipid molecule, at any of the above listed
dividing surfaces of H II phase:

2

( )

2

1
2

11 11 1

2

AA
A

−

+

( )

0

(3.1)

FAE RR ≈ − + EAA RR − − E

0 RR AR 0 AA

0 0 0

where E RR , E AR , and E AA are the corresponding elastic moduli. The values and
defi nitions of A and R depend on the selection of the dividing surface. It is
very important to note that the neutral and pivotal planes are well defi ned
only for small deformations within the limits of validity of the quadratic free
energy expansion (Leikin et al., 1996). In other words, Eq. (3.1) defi nes the
curvature-independent energy of the interstices only for small deviations from