Dividing Planes of Hexagonal H Mesophase - Self-Assembled Supramolecular Architectures: Lyotropic Liquid Crystals - page 89

and the pivotal surface can be estimated from the calculated intercept of the
linear fi t. Equation (3.23) could be rewritten in a more general form:

2

A
V

2

A
V

V
V

A
VR

=−

p p

w w

2 (3.24)

2 2

l l l l w

This form can be used to standardize the linear fi t routine in the case of simul-
taneous fi t of multiple sets of data (Leikin et al., 1996).
Another model for determining both the pivotal radius and pivotal cross-
sectional area is based on geometrical considerations (Fuller et al., 2003):

V
V

1 φ

RR

l p

(3.25)

= 1 +

p w

− φ

l l

V
V 1 φ

l p

AA = 1 + (3.26)

p w

− φ

l l

In terms of A p and V p , it is possible to formulate another (recursive-like)
defi nition of the pivotal plane for H II phase: The pivotal plane is defi ned as a
surface inside the lipid phase such that both A p and V p are constants when the
distance between the rod axes of H II varies.
The radius of the pivotal plane is incorporated in the process of determining
the elastic parameters at the pivotal plane of the lipid mixture from osmotic
stress experiments (Gruner et al., 1986; Rand et al., 1990):

11

=−

1 1

2

Π Rk RR = 2 − 2 k + 2 k (3.27)

p cp cp cp

R R

p 0 p 0 p p

2
versus 1/ R p can be easily adapted to process the experimental data from H II
phase analysis. First, the initial data set (

where Π is the measured osmotic stress. The described linear model Π R

Π , d hex , ϕ l ) is taken from the experi-

R 2 , 1/ R p )
has to be determined by using a fi t routine and Eq. (3.27). Figure 3.4 illustrates
the quality of the performed linear fi t. The estimated value of the slope is twice
the value of k cp . To obtain the value of the spontaneous curvature at the pivotal
plane, 1/ R 0p , the calculated intercept is divided by the slope value.
The measured values of the spontaneous curvature are properties of the
lipid mixture. As the mixture consists of two components (the fi rst one builds
the hexagonal rods), 1/ R 0p should depend on the quantity of the two compo-
nents separately. For example, if the mixture consists of DOPE and DOG (an
intensively studied system), the spontaneous curvature could be expressed as
(Leikin et al., 1996):

ment and used to derive the set ( R p , ϕ l ). Next, the fi nal data set ( Π

1 1 1

( ) =− 1 m + m (3.28)

DOG DOG

R R

DOPE

R

DOG 0 p 0 0