Chemistry Reference
In-Depth Information
In this model, one can also introduce effective potentials that depend on the bond
length
b
such as [
121
]:
2
U
b
ð
b
Þ¼e
0
ð
b
b
0
Þ
;
(8)
and on the bond angle
Y
between successive bonds, e.g., [
121
,
123
]:
U
bending
ðYÞ¼
fcos
Yð
1
þ
ccos
YÞ:
(9)
In [
121 123
], the constants
e
0
;
b
0
;
f
, and
c
were chosen quite arbitrarily as
e
0
¼
4
;
¼
0
:
03 (
e
0
and
f
are quoted in units of absolute temperature
k
B
T
,
k
B
being Boltzmann's constant). On the other hand, if one chooses a bond
length potential [
174
,
175
] defined as
U
b
ð
b
0
¼
0
:
86 and
c
1
p
and
U
b
ð
1 other-
wise, a rather good model for the glass transition of polymers is obtained [
84
] due
to the resulting “geometric frustration” [
174
,
175
]. Finally, we mention that occa-
sionally one finds in the literature (e.g., [
176 179
]) another version of the bond
fluctuation model, in which monomers take a single lattice site only and the bond
vectors are allowed to be
b
Þ¼
0if
b
¼
b
Þ¼
fð
1
;
0
;
0
Þ; ð
1
;
1
;
0
Þ
and sometimes also [
178
]
ð
. All permutations between these coordinates are included, but this
model will not be discussed further here because it has mostly only been applied to
study mesophase ordering of block copolymers. We stress that the advantage of the
bond fluctuation model [
76
,
79
,
80
] as described above is that at a volume fraction
of 1
1
;
1
;
1
Þg
5 of occupied lattice sites, one reproduces both the single chain structure
factor (as described by the Debye function [
8
]) of polymer chains as well as the
collective structure factor of dense melts [
175
] qualitatively in a reasonable way.
If one uses a dynamic algorithm in which monomers are chosen at random, and
a lattice direction
F ¼
0
:
is chosen at random, and a move of the monomer
by one lattice unit is attempted as a trial move according to the Metropolis MC
algorithm [
18 20
], then a qualitatively reasonable description of the polymer
dynamics is also obtained [
80
,
82
,
175
]. For short chains, the dynamics correspond
to the Rouse model [
5
,
8
,
31
] whereas, for long chains, reptation [
5
,
8
,
31
]is
observ
ed since for the chosen bond lengths no bond crossing is possible [
79
,
80
]. On
the other hand, in order to allow for a fast equilibration, moves can be introduced
(such as the slithering snake algorithm [
82
,
83
] or monomeric jumps over larger
distances that allow for bond crossing [
95
]) that have no counterpart in the real
dynamics of polymers, but do not alter the static properties of the model. Therefore,
the bond fluctuation model has also been broadly used (e.g., [
9
,
180
]) to simulate
the dynamics of spinodal decomposition [
181
,
182
] of polymer blends.
We now turn to coarse-grained off-lattice models. One strategy is to stay as close
to the atomistic model as possible but to eliminate many degrees of freedom, e.g.,
for modeling alkane chains [
183 185
], both the bond length
ð
x
;
y
;
z
Þ
'
of C C bonds and
54 A
112
), but the torsional angle
f
ijk'
the bond angle
(
3
)
is kept as a variable. The advantage of such a model (with suitable choices of the
torsional and nonbonded potentials [
185
]) is that one can still make a direct
connection with polyethylene melts. Both local MC moves (where two subsequent
Y
is fixed (
' ¼
1
:
; Y ¼
