Chemistry Reference
In-Depth Information
Widom's method presents problems when dealing with very dense and strongly
interacting fluids, because inserted test molecules almost always overlap with
“real” molecules, which leads to extremely large values for the potential energy
c
i
. These insertions contribute little information, resulting in poor statistics [
56
].
Therefore, advanced methods have been proposed in the literature. An example is
the gradual insertion method [
208
-
210
], where a fluctuating molecule is introduced
into the simulation. The fluctuating molecule undergoes a stepwise transition
between non-existence and existence, which allows the determination of the chem-
ical potential. This method has been applied successfully to vapor-liquid equilib-
rium calculations of numerous binary and ternary mixtures [
40
,
41
,
174
]. Many
other methods, such as configurational biased insertion [
211
] or minimum mapping
[
212
], have been proposed in the literature. A detailed description and comparison
thereof can be found, e.g., in [
213
].
The Henry's law constant can be obtained from molecular simulation using
several approaches [
214
,
215
]. It is related to the residual chemical potential of the
solute
i
at infinite dilution
m
i
by [
216
]:
ðm
i
H
i
¼ r
k
B
T
exp
=ð
k
B
T
ÞÞ;
(31)
where
r
is the density of the solvent.
5.4 Methods for Determining Transport Properties
Transport properties, such as diffusion coefficients, shear viscosity, thermal or
electrical conductivity, can be determined from the time evolution of the autocor-
relation function of a particular microscopic flux in a system in equilibrium based
on the Green-Kubo formalism [
217
,
218
] or the Einstein equations [
219
]. Autocor-
relation functions give an insight into the dynamics of a fluid and their Fourier
transforms can be related to experimental spectra. The general Green-Kubo expres-
sion for an arbitrary transport coefficient
g
is given by:
ð
1
1
G
h
A
ð
Þ
A
ð
g ¼
d
t
t
0
Þi;
(32)
0
and the general Einstein or square displacement formula can be written as
1
2
Gt
h½
A
Þ
A
2
g ¼
ð
t
ð
Þ
i:
0
(33)
A
Therein,
G
is a transport property specific factor, A the related perturbation, and
its time derivative. The brackets
denote the ensemble average. It was shown
that (33) can be derived from (32); thus both methods are equivalent [
220
].
In case of the self-diffusion coefficient, A(
t
) is the position vector of a given
molecule at some time
t
and
<...>
A(
t
) is its center of mass velocity vector. In this way,
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