Biomedical Engineering Reference
In-Depth Information
11.3 Molecular modelling of polymers
In molecular modelling of amorphous polymers a volume element is filled with poly-
mer chain segments and small permeant molecules (e.g, water, gases). In the molec-
ular modelling paradigm all atoms are considered to be spheres of given diameter
di and mass
m
i
. The bonded interactions between atoms resulting in bonds, bond
angles, and conformation angles are then described by springs with spring constants
related to the bond strength known experimentally or by quantum mechanics calcu-
lations. Non-bond interactions between atoms are considered via e. g. Lennard-Jones
and Coulomb potentials. The sum of all interatomic interactions written as the po-
tential energy of the atomistic model is then called a forcefield. Below is shown the
typical forcefield structure for a system of
N
atoms with the cartesian atomic position
vectors
r
i
,
k
l
l
l
0
2
k
θ
θ
−
θ
0
2
1
2
1
2
∑
bounds
∑
angles
V
(
r
1
,...,
r
N
)=
−
+
q
i
q
j
4
N
i
=
1
N
∑
k
φ
1
φ
−
δ
)
+
A
ij
r
12
C
ij
r
ij
∑
dihedrals
+
+
cos
(
n
r
ij
+
ij
+
πε
j
=
i
+
1
where
l
is the bond length between two atoms,
k
l
is the bond spring constant and
l
0
the reference bond length;
θ
is the angle between three atoms,
k
θ
is the angle spring
constant and
represent the cis and trans dihedral
angles,
n
stands for the periodicity of the dihedral term and
k
φ
is the dihedral spring
constant. The parameters in the last term regard non-bonded interactions where
A
ij
and
C
ij
are the van der Waals parameters for the interacting pair of atoms,
r
ij
is
the non-bonded distance between the atoms,
q
i
and
q
j
are their charges and
θ
the reference angle;
φ
and
δ
ε
is the
dielectric constant.
The evaluation of the nonbonded energy terms is the numerically most expensive
part in molecular modelling calculations, because these terms include contributions
from each pair of atoms in a model. This leads to restrictions on the maximum pos-
sible size of a simulated molecular system. With current computational resources,
the number of atoms
N
can not be much higher than 10'000-20'000 for simulations
on modern workstations, whereas
N
may be up to about a factor of ten higher for se-
lected simulations on computer clusters. Therefore, only polymeric materials being
homogeneous enough that a volume element of several thousand atoms is represen-
tative for the whole polymer structure can be modelled with atomistic simulations.
Forcefields may be used in two ways. On one hand, model systems can be sub-
jected to a static structure optimization, i.e. the geometry of the simulated system is
changed until the potential energy reaches the closest minimum value. This is per-
formed for the reduction of unrealistic local tensions in a model structure. On the
other hand, from the potential energy of a model system it is possible to calculate
the forces
F
i
acting on each atom of the model via the gradient operator,
F
i
=
∂
V
(
r
1
,...,
r
N
)
.
∂
r
i
