Chemistry Reference
In-Depth Information
The chemical potentials in both phases can, in principle, be “measured” in
a simulation by the Widom virtual particle insertion/deletion technique [
247
].
Determining, for both liquid and vapor at a chosen temperature, the chemical
potentials as function of pressure, one finds the coexistence pressure
p
coex
ð
Þ
from the intersection of both curves. This approach is readily generalized to
more-component systems. This technique was first demonstrated for simple models
of pure fluids [
248
,
249
] and then extended to more complicated models of
molecules [
153
,
250
] describing quadrupolar fluids, and to various mixtures
[
154
]. Again, this method is problematic near critical points. The angle under
which the two curves
m
I
ð
T
p
coex
becomes very
small when
T
is only slightly below
T
c
, and one has to deal with critical slowing
down [
229
], finite size effects, etc. It is also problematic for large molecules, for
which the acceptance of particle insertions becomes too low.
For fully atomistic all-atom models, it is often difficult to find efficient MC
moves to relax their configurations, and then MD is normally the method of choice.
We note, however, that for chemically realistic models of polymer blends equili-
bration by MD methods is extremely difficult, if at all possible. Dealing with such
systems is still an unsolved challenge.
p
;
T
Þ
and
m
II
ð
p
;
T
Þ
cross at
p
¼
3.2 Monte Carlo
MC simulations aim to realize the probability distributions considered in statistical
thermodynamics numerically using random numbers and to calculate the desired
averages of various observables in the system using these distributions [
18 20
].
There exist numerous extensive reviews describing the specific aspects of MC
methods for polymers [
77
,
82
,
84
,
90
,
96
], and thus we focus here only on some
salient features that are most relevant when one addresses the estimation of the
equation of state and phase equilibria of systems containing many polymers.
These MC methods then are based on the Metropolis algorithm [
251
], by which
one constructs a stochastic trajectory through the configuration space (X) of the
system, performing transitions
W
X
0
Þ
ð
!
. The transition probability must be
chosen such that it satisfies the detailed balance principle with the probability
distribution that one wishes to study. For example, for classical statistical mechan-
ics, the canonical ensemble distribution is given in terms of the total potential
energy
U
X
ð
X
Þ
, where X
ð~
r
1
;~
r
2
; ...;~
r
N
Þ
stands symbolically for a point in config-
uration space [
17
]:
1
exp
P
NVT
ð
Þ¼
Z
½
U
ð
Þ=
k
B
T
;
X
X
(43)
Z
being the partition function (remember that the free energy
F
then is [
17
]
F
ðN ;
V
;
T
Þ¼
k
B
TlnZ
Þ
. The detailed balance principle then requires that:
X
0
Þ¼
X
0
Þ
X
0
!
P
ð
X
Þ
W
ð
X
!
P
ð
W
ð
X
Þ :
(44)
