Chemistry Reference
In-Depth Information
is taken as the next configuration; otherwise, X
0
is rejected, the old configuration X
is counted once more for the average, and a new trial move is attempted.
Although the basic step of MC algorithms is very simple, much know-how is
needed to carry out successful MC simulations of dense polymeric systems in
practice. One must
realize that
the subsequently generated configurations
X
00
X
0
!
!
! ...
X
of such a stochastic trajectory in phase space are not statisti-
cally independent of each other, but in most cases strongly correlated. In fact, one
can give MC sampling a dynamic interpretation: one numerically realizes a master
equation for the probability
P
that a state X is found at the “time”
t
of the
sampling process [
18 20
]. So, if in a multichain system in the canonical ensemble
the attempted MC moves just consist of small random displacements of the
effective monomers, one generates a chain dynamics consistent with the simple
Rouse model of polymer dynamics (or reptation model, if the chains are entangled)
[
5
,
8
,
31
]. Of course, the time scale of the MC sampling process has no a priori
interpretation in terms of physical time units and one traditionally uses dimension-
less “time” units such as Monte Carlo step (MCS) per monomer. When one wishes
to connect this “time” to physical time, one needs to use extra information (e.g.,
from the energy barriers of torsional potentials, etc.) to map the MC time onto the
physical time via a “time rescaling factor” (which depends on temperature and
density [
101
,
104
]).
Thus, although MC applications to study the dynamics of polymers (e.g., near
their glass transition [
82
,
86
]) exist, an important advantage of MC is that one can
abandon the possibility of studying polymer dynamics in favor of a speedup of the
sampling by using MC moves that look artificial from the point of view of the real
dynamics of polymers in the laboratory, but which are perfectly permissible as a
means of creating a trajectory through the configuration space to sample probabilities
such as (
43
). For example, one may allow for moves of monomers over such large
distances that the covalent bonds connecting neighboring monomers along a chain
are crossed during the move [
95
]. Such moves do not occur in real polymer melts,
where chains can never cross each other, but in MC simulations such moves can be
implemented such that they satisfy (
44
) and hence are perfectly valid to study static
equilibrium behavior. This is also true for a large variety of other “artificial” moves,
such as the “slithering snake” algorithm [
82
] (one chooses a chain end of one of the
chains at random, and tries to remove the end monomer and attach it to the other
chain end in a randomly chosen direction), or algorithms involving chain fission and
fusion [
95
,
96
,
230
]. However, we shall not describe these algorithms here, but rather
direct the reader to the literature [
82
,
83
,
88
,
90
,
96
].
Similarly, “tricks of the trade” are also needed when one wishes to realize the
grand canonical ensemble: inserting a polymer chain of moderate length even in a
semidilute polymer solution has such a low acceptance probability that a straight-
forward simulation would never work. This problem can be overcome to some
extent by the configurational bias algorithm [
88
]; for the bond fluctuation model
[
76
,
79
,
80
], this algorithm has allowed a successful study of the phase diagram of
polymer solutions up to a chain length of
N
ð
X
;
t
Þ
¼
60 [
173
]. The configurational bias
