Chemistry Reference
In-Depth Information
present review. Also, we will not consider quantum fluctuations due to the nuclei
(e.g., in orthorhombic crystalline polyethylene the quantum-mechanical zero-point
motion of the atoms does affect the thermal expansion and elastic constants of the
material [
126
]).
Thus, the starting point of both MD [
87 91
] and MC methods [
18 20
]isa
classical potential
U
ðf
~
(often it is referred to as force field [
71
,
81
,
82
,
127
]),
which contains only the positions of all the atoms
r
i
gÞ
f
r
i
g
as variables. Typically,
ðf
r
i
gÞ
U
is decomposed into contributions describing intramolecular forces along a
polymer chain, which are described by bond stretching potentials
U
'
ðj
r
ij
jÞ;
r
ij
¼
r
i
r
j
(where
is the bond length), bond angle potentials
U
bend
ðY
ijk
Þ
(describing local bond angles), torsional potentials
U
tors
ðf
ijk'
Þ
'
, and last but not
least nonbonded interactions
U
nb
ð
~
. The latter are typically assumed to be pair-
wise additive. For example, the bond length potential is often assumed to have a
simple harmonic form:
r
ij
Þ
2
1
2
k
'
j
r
ij
j'
0
U
'
ðj
r
ij
jÞ ¼
;
(1)
where
k
'
is a “spring constant” for the chemical bond between the two neighboring
atoms in a polymer chain, and
'
0
is their (classical) ground-state distance. Also, the
bending potential often is assumed to be harmonic in the angle
Y
ijk
formed between
two successive bonds r
ij
andr
jk
along a chain:
1
2
k
Y
ðY
ijk
Y
0
Þ
2
U
bend
ðY
ijk
Þ¼
;
(2)
where again
k
Y
is a spring constant but now for chain bending, and
Y
0
the classical
ground state value for the bond angle. Finally, the torsional potential (defined in
terms of the angle
~
r
k'
makes with its projection into the plane
formed by the bonds r
ij
andr
jk
) can be parameterized as:
Y
ijk'
that the bond
h
i
;
2
X
n
max
1
U
tors
ðf
ijk'
Þ¼
k
n
1
cos
ð
nf
ijk'
Þ
(3)
n
¼
1
where further constants
, and
n
max
enter. For neutral polymers, for which
Coulomb interactions can be disregarded, the nonbonded interactions are often
assumed to have the simple Lennard Jones (LJ) form:
f
k
n
g
12
6
r
r
U
LJ
ð
r
Þ¼
4
e
;
(4)
with
e
describing the strength and
s
the range of this potential. Note that
U
LJ
ð
acts
both between monomers of different chains and between monomers of the same
chain if they are neither nearest, nor next-nearest, nor third-nearest neighbors along
the chain (so that none of the interactions in (
1
) (
3
) would apply).
r
Þ
