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effective LJ model (see Sect.
2.3
) overcomes both of these limitations if the
effective dipole moment
m
is adequately adjusted.
2.2 Coarse-Grained Models in the Continuum and on the Lattice
For long flexible polymer chains it has been customary for a long time [
1
,
2
]to
reduce the theoretical description to the basic aspects such as chain connectivity
and to excluded volume interactions between monomers, features that are already
present when a macromolecule is described by a self-avoiding walk (SAW) on a
lattice [
3
]. The first MC algorithms for SAW on cubic lattices were proposed in
1955 [
164
], and the further development of algorithms for the simulation of this
simple model has continued to be an active area of research [
77
,
96
,
165 169
].
Dynamic MC algorithms for multichain systems on the lattice have also been
extended to the simulation of symmetric binary blends [
15
,
16
]; comprehensive
reviews of this work can be found in the literature [
6
,
81
,
82
]. It turns out, however,
that for the simulation both of polymer blends [
6
,
9
,
21
,
82
,
170
,
171
] and of
solutions of semiflexible polymers [
121 123
], the bond fluctuation model [
76
,
79
,
80
] has a number of advantages, and hence we shall focus attention only on this
lattice model.
Using the lattice spacing of the simple cubic lattice as the unit of length,
a
1,
each coarse-grained macromolecule is represented as a chain of effective mono-
mers connected by bond vectors, which can be taken from the set
¼
fð
2
;
0
;
0
Þ;
ð
, including also
all permutations between these coordinates. Altogether 108 different bond vectors
occur, which lead to 87 different angles between successive bonds. Each effective
monomeric unit is represented by an elementary cube of the lattice, blocking all
eight sites at the corners of this cube from further occupation, thus realizing the
excluded volume interaction between the monomers. Allowing for two types
of polymers (A and B) in the system, it then is natural to also allow for (attractive)
interactions of somewhat longer range between any pair of monomers
2
;
1
;
0
Þ; ð
2
;
1
;
1
Þ; ð
2
;
2
;
1
Þ; ð
3
;
0
;
0
Þð
3
;
1
;
0
Þg
. These
interactions in most cases were assumed to have the simple square well (SW) form:
ða; bÞ
Þ¼
e
ab
2
r
r
c
;
U
a
SW
ð
r
(7)
0
r
>
r
c
:
p
was used (so all neighbors in the first-neighbor shell in a
dense melt, defined from the first peak position in the radial pair distribution
function
g
In most cases,
r
c
¼
between monomers, are included [
170 172
]). The extremely short-
range case
r
c
¼
ð
r
Þ
2 was also used [
21
]; then monomers attract each other only when
they are nearest neighbors on the lattice. Of course, (
7
) also includes, as a special
case, the case of a polymer solution
ða ¼ bÞ
where only a simple species of polymer
is present [
173
].
