Chemistry Reference
In-Depth Information
united atom model [
56
,
61
] and use as a generic case methane (CH
4
) as solvent, it
would not make sense to treat the hydrogen atoms in the model of CH
4
explicitly
while the CH
2
and CH
3
-groups of polyethylene are treated as superatoms. Thus,
when CH
4
is treated as a point particle as well, the only interaction between the
solvent molecules that is left is also of the LJ type (
4
); remember that CH
4
is
a neutral molecule that possesses neither a dipole moment nor a quadrupole
moment. If one adjusts the LJ parameters
e
SS
; s
SS
for CH
4
such that the experimen-
tal critical temperature
T
c
and experimental density
r
c
are reproduced by the model,
T
c
¼
190
:
6 K and
r
c
¼
10
:
1 mol L
1
[
130
], then the vapor liquid coexistence data
both in the temperature density plane (Fig.
1a
) and in the pressure temperature
plane (Fig.
1b
) are indeed reproduced over a wide temperature regime [
56
], as is the
temperature dependence of the interfacial tension (Fig.
1c
). For the sake of com-
putational efficiency of the MC simulations needed to establish the phase diagram
of the considered LJ model with sufficient accuracy, it was decided [
131
]to
simulate a LJ model with truncated and shifted interactions rather than using the
full potential (
4
):
2
7
=
6
U
ij
ð
r
Þ¼
U
LJ
ð
r
Þþ
127
e
SS
=
4096
;
R
r
c
¼
s
ss
;
(5)
whereas
U
ij
ð
r
>
r
c
Þ
0. Note that the additive constant in (
5
) is chosen such that
the potential
U
ij
ð
r
c
. Figure
1
shows
that the united atom approximation for methane does reproduce the liquid vapor
coexistence of this fluid with very good accuracy. When one uses the LJ parameters
e
SS
; s
SS
for this solvent (determined by the fit of the critical point as obtained from
MC simulation to the experimental critical point data) as an input for approximate
equation of state theories such as first-order TPT combined with the mean spherical
approximation (TPT1-MSA) [
50
,
52
], one obtains [
56
] a reasonable agreement with
experimental data for
T
r
Þ
between particles
i
and
j
is continuous at
r
¼
170 K. At higher temperatures deviations appear; in
particular, TPT1-MSA overestimates the critical temperature significantly, and
predicts a parabolic shape of the coexistence curve in the critical region (the
difference between the coexisting liquid and vapor densities scales as
r
l
r
v
/
1
p
b
T
=
T
c
, while the actual coexistence curve is flatter,
r
l
r
v
/ð
1
T
=
T
c
Þ
with a non-mean-field exponent
b
326 [
132
,
133
]). This discrepancy between
TPT1-MSA and experiment (and simulation results) illustrates a general shortcom-
ing of all mean-field-type equation of state descriptions in the critical region, as
emphasized already in the Introduction.
The accuracy of the united atom description for polyethylene has also been
carefully tested in the literature [
60
,
71
], both by comparison of simulation results
with experiments and with simulations dealing with an all-atom model where
hydrogen atoms are explicitly considered. Of course, for polyethylene the vapor
liquid critical point would occur at very high temperatures, where the macromole-
cule would no longer be chemically stable, and is of no physical interest; thus one
uses data for single chain and collective structure factors to gauge the accuracy of
the simulation models in this case.
0
:
