Chemistry Reference
In-Depth Information
X
00
X
0
!
If it is possible to generate a Markov chain of transitions X
!
! ...
,
one can show that, in the limit when the number
M
of configurations generated is
very large, i.e.,
M!1
, the canonical average of some observable
A
ð
X
Þ
:
1
Z
dX
A
h
A
ð
X
Þi
NVT
¼
Z
ð
X
Þ
exp
½
U
ð
X
Þ=
k
B
T
(45)
can be approximated by a simple arithmetic average over the
M
configurations
generated:
1
M
A
¼M
A
ð
X
i
Þ :
(46)
i¼
1
One of the big advantages of MC methods is that they can be readily generalized
to all statistical ensembles of interest. For example, in the grand canonical ensemble
of a single component system it is not the particle number
that is fixed, but rather
the chemical potential
m
. Thus, in order to realize the distribution
P
NVT
in the
grand canonical ensemble, the moves X
N
X
0
must include insertion and deletion
of particles (in practice this is easily realizable for small molecules, such as solvent
molecules, but becomes difficult for short polymers, and impossible for long
polymers because the acceptance rate of such “MC moves” becomes too small).
For polymer blends, a particularly useful ensemble is the semigrand canonical
ensemble. Suppose we have two types of polymers, A and B, having the same
chain length,
N
A
¼
!
N
B
(the extension to different chain lengths is discussed in [
170
,
171
]). Then, it is possible to consider a MC move where an A chain is replaced by a
B chain (with identical configuration) or vice versa, taking the chemical potential
difference
Dm ¼ m
A
m
B
properly into account in the transition probability [
6
,
15
,
16
,
21
,
82
,
170 172
]. An example for such an application, extending the method
to a mixture of homopolymers and block copolymers, will be presented as a case
study in Sect.
4.4
. We emphasize, that neither the grand canonical nor the semi-
grand canonical ensemble can be used in MD simulations.
The random numbers (actually no strictly random numbers are used, but rather
only pseudorandom numbers, generated on the computer by a suitable algorithm
[
20
]) are then used for two purposes: first a trial MC move X
X
0
is attempted. For
example, in a simulation of a polymer-plus-solvent system in the grand canonical
ensemble, coordinates of a point in space are chosen at random, and there one
attempts to insert an additional solvent particle; or one chooses a randomly selected
bead of a polymer chain and attempts to move it to a randomly chosen neighboring
position in a small volume region
dV
around its previous position; etc. Then, one
needs to expose this trial configuration to the Metropolis acceptance test. In the
canonical ensemble, one simply needs to compute the change in total potential
energy
!
D
U
caused by the trial move: if
D
U
<
0, the trial move is accepted; if
D
U
>
0, one compares
D
W
exp
½D
U
=
k
B
T
with a random number
x
uniformly
x
, the trial move is accepted, and X
0
distributed in the unit interval
f
0
;
1
g
.If
D
W
