Chemistry Reference
In-Depth Information
By combining Eqs. (3.16) and (3.18), an expression for
α
can be derived
(Templer et al., 1998b):
V
A
2
2
π
V
V
p
l
α
=
1
−
φ
(3.19)
l
φ
3
pl
l
Equation (3.19) can be used to construct a target (merit) function and to fi t a
set of experimental data (
ϕ
l
) in order to obtain the values of
A
p
and
V
p
. The
idea behind the optimization routine here is similar to those applied for bicon-
tinuous cubic phases before [Eqs. (3.11)-(3.15)]. The corresponding merit
function takes the form
α
,
2
N
=
V
V
1
1
V
A
2
2
π
∑
1
p
l
Φ
,
V
α
−
1
−
φ
(3.20)
p
i
l
,
i
()
A
σα
2
φ
3
p
i
pl
l
i
=
The respective internal solution for
A
p
, for a given value of
V
p
, could be
expressed as
1
V
V
p
∑
N
1
−
φ
i
=
1
l
,
i
()
8
π
σαφ
2
2
()
=
i
l
,
i
l
AV
(3.21)
V
pp
l
3
V
V
α
σα
i
p
∑
i
N
1
−
φ
=
1
l
,
i
()
2
φ
i
l
,
i
l
and it can help to reduce by one the number of parameters of the merit
function:
2
N
1
V
AV
2
2
π
V
V
∑
()
=
l
p
(3.22)
Φ
V
α
−
1
−
φ
p
i
l
,
i
()
σα
2
()
φ
3
i
ppl
l
i
=
1
The reduced expression, Eq. (3.22), represents a merit function with only one
parameter and its minimum can be easily calculated by using the golden
section method (Press et al., 2007). Figure 3.3 gives an idea of the quality of a
similar optimization using the nonreduced model from Eq. (3.20). Table 3.2
contains calculated values of
R
p
and
A
p
for different lipids at different tem-
peratures and water mole ratios, as well as the
V
p
/
V
l
ratio. The routine explained
here could be extended by adding
V
l
to the list of the fi t parameters when
necessary.
There is another way to reduce the number of fi t parameters by setting
V
l
/
A
p
and
V
p
/
V
l
as independent parameters in Eq. (3.20) (March, 2011).
There is an alternative method for obtaining
R
p
,
A
p
, and
V
p
that is applicable
to the H
II
phase. It is simpler, but works well only if the dependence between
A
2
and
A
w
/
R
w
is linear (Fuller et al., 2003; Leikin et al., 1996):
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