Chemistry Reference
In-Depth Information
σ
0
is the dimensionless area per cubic unit
cell of the minimal surface,
V
l
is the volume of lipid molecule, and
Here,
a
is the cubic lattice constant,
is the
Euler characteristic of the surface, per cubic unit cell. It should be noted that
χ
χ
has a role of a
topology index
of the surface. The volume fraction between
pivotal and minimal surfaces can be expressed as follows (Marsh, 2011; Turner
et al., 1992):
3
V
V
l
a
4
3
π
χ
l
a
p
p
p
(3.9)
φσ
=
2
+
l
0
l
Here
V
p
is the pivotal volume of the molecule and
V
l
is the volume of the
molecule. Equation (3.8) is used primarily to calculate
l
p
and its only physically
meaningful solution is (Marsh, 2011)
V
V
2
σ
πχ
πϑ
+
9
8
πχ
σ
0
p
l
=
a
cos
ϑ
=
arccos
φ
(3.10)
p
l
3
3
l
0
Twice the value of
l
p
determines the separation of the pivotal surfaces in
bicontinuous cubic phases—2
l
p
. By combining Eqs. (3.8) and (3.10) it is easy
to derive the expression for the lattice parameter of bicontinuous inverse cubic
phases:
2
σ
V
πϑ
+
0
l
a
=
14
−
cos
(3.11)
A
φ
3
pl
The most common use of Eq. (3.11) in data management is to fi t a set of
experimental data (
a
,
ϕ
l
) and to fi nd the corresponding optimal values of
σ
0
,
A
p
,
V
p
, and
(using the nonlinear least-squares method). Due to the large
number of fi t parameters (four), a large set of experimental data must be used.
The nonlinear optimization, based on the nonlinear least-squares method,
could be transferred into an economizing one by reducing the number of the
variable parameters by one. Such a reduction is possible because
a
has a linear
dependence on 1/
A
p
. Therefore, 1/
A
p
can be calculated (as an internal/local
solution) at each step of the nonlinear routine for fi nding
χ
. This
type of hybrid optimization reduces the computational error and speeds up
the entire process of optimization.
Due to the importance of the optimization design for the processing of
experimental data, a brief technical description is presented. If the given set
of data (
a
,
σ
0
,
V
p
, and
χ
ϕ
l
) consists of
N
data pairs, and
σ
2
(
a
i
) is the standard error of the
i
th measurement of
a
(
i
=
1 , . . . ,
N
), then the merit function can be defi ned as
(
)
2
N
πϑ
+
V
,
σχ
,
1
=
1
2
σ
V
∑
0
l
i
p
0
2
Φ
,
V
,
σχ
,
a
−
14
−
s
p
0
i
()
A
σ
2
a
A
φ
3
p
i
pl
,
i
i
=
1
(3.12)
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