Chemistry Reference
In-Depth Information
MC method can also be implemented for off-lattice models, e.g., united atom
models for alkanes [
59
,
252
] have been studied up to C
70
H
142
. Many such studies
of liquid vapor-type phase equilibria, however, do not use the grand canonical
ensemble but rather apply the Gibbs ensemble [
253
]. This method considers two
simulation boxes with volumes
V
1
,
V
2
and particle numbers
N
1
; N
2
such that
V
1
þ
const, while both particles and volume can be
exchanged between the boxes. In this way, it is ensured that both boxes are not only
at the same temperature, but also at the same pressure and the same chemical
potential. Thus, this method has been very popular for the estimation of vapor
liquid coexistence curves, both for small molecules [
253 257
] and for the alkanes
[
59
,
252
]. However, most of this work has yielded rather inaccurate data near the
critical point, due to finite size effects. If one combines grand canonical simulations
with histogram reweighting [
258
,
259
] and umbrella sampling [
260
,
261
] or multi-
canonical MC [
262
], one can obtain precise results including in the region of the
critical point (see, e.g., [
263 265
]). If one uses such simulation data in a finite size
scaling analysis [
18 20
], one can obtain both the coexistence curve and interfacial
tension near the critical point very precisely, as has been demonstrated for many
systems (e.g., [
52
,
53
,
55
,
56
,
131
,
170 173
,
204
,
266
,
267
]). Since a detailed
review of these techniques can be found in the literature [
10
], we refer the reader to
this source for technical details on these methods. We mention, however, that in
some cases special algorithms are needed to allow the use of grand canonical
simulations. For example, for the Asakura Oosawa model and related models of
colloid polymer mixtures [
204
], in the regime of interest the density of the polymer
coils (that may overlap each other strongly with no or little energy cost) can be so
high that it is almost impossible ever to successfully insert a colloidal particle,
which must not overlap any polymer or any other colloidal particle. To allow
nevertheless successful colloid insertions, an attempted “cluster move” [
268
]
needs to be implemented. In this move,in a spherical region a number
n
of polymers
is removed and a colloidal particle inserted, or the reverse move is attempted, and
transition probabilities are defined such that detailed balance (
44
) is obeyed.
A completely different nonstandard technique to obtain a first overview of the
equation of state was recently proposed by Addison et al. [
269
], whereby a
gravitation-like potential is applied to the system, and the equilibrium density
profile and the concentration profile of the center of mass of the polymers is
computed to obtain the osmotic equation of state. In this “sedimentation equilib-
rium” method one hence considers a system in the canonical
V
2
¼
const and
N
1
þN
2
¼
N
VT
ensemble using a
box of linear dimensions
L
H
, with periodic boundary conditions in
x
and
y
directions only, while hard walls are used at
z
L
¼
0 and at
z
¼
H
. An external
potenti
al is applied everywhere in the system:
U
external
ð
z
Þ¼
mgz
¼ð
k
B
T
=
a
Þl
g
z
:
(47)
Here,
m
is the mass of a monomer,
g
is the acceleration due to the gravity-like
potential, and
a
is the length unit (equal to lattice spacing if a lattice model is used).
