Biomedical Engineering Reference
In-Depth Information
Then Newton's equations of motion can be solved for every atom of the investigated
system,
2
r
i
(
)
m
i
∂
t
F
i
=
.
(11.3)
∂
t
2
The starting velocities
v
i
(
of all atoms are assigned through the well known rela-
tion between the average kinetic
E
kin
energy of an atomistic system and its temper-
ature
T
,
0
)
N
i
=
1
1
2
m
i
v
i
3
N
−
6
E
kin
=
=
k
B
T
2
where
k
B
is the Boltzmann constant,
is the number of degrees of freedom of
a
N
-atom model considering the fact that in the given case the centre of mass of the
whole model with its 6 translation and rotation degrees of freedom does not move
during the molecular dynamics simulation. Using this approach it is then possible to
follow the motions of the atoms of a polymer matrix and the diffusive movement of
penetrant molecules at a given temperature over a certain interval of time.
Since Eq. (11.3) represents a system of several thousand coupled differential
equations of second order, it can be solved only numerically in small time steps
Δ
(
3
N
−
6
)
t
. This time step must not be greater than about a tenth of the oscillation time of
the fastest vibrations in the investigated system, otherwise serious numerical errors
occur. Since the fastest oscillation time (covalent stretching of a C-H bond) in a poly-
mer is approximately 10 fs, the integration time step
t
is usually assumed as 1 fs.
This imposes a further restriction in the processes that can be treated by molecular
dynamics simulations. With currently available coputational power, it is possible to
solve Eq. (11.3) for a system composed of several thousands of atoms for a few mil-
lion time steps
Δ
t
, i. e. for a few nanoseconds. That means, any simulated process
in a polymer bulk has to be fast enough that a few nanoseconds are sufficient to get
the relevant information.
The restriction of the possible number of atoms
N
leads to a characteristic side
length of the atomistic model in the order of a few nanometers. Thus, to avoid severe
surface effects it is necessary to introduce periodic boundary conditions, i. e. the
basic volume element is surrounded by virtual volume elements of identical contents.
The atoms of the surrounding cells only contribute to the nonbond interactions of the
atoms in the basic cell, which basically establishes conditions of an infinite solid.
To avoid the consideration of an infinite number of pair interactions and artificial
translation symmetry, for nonbond interactions it is indrocuced a cut-off distance,
r
cut
, smaller than half the shortest side length of the characteristic volume element.
In this way, van der Waals and electrostatic interactions are considered to be zero
for interatomic distances greater than
r
cut
.
A promising strategy to overcome the computational limitations of molecular dy-
namics simulations is the use of Graphical Processing Units (GPUs). In recent years,
commercial GPUs have acquired non-graphical, general-purpose programmability.
The most mature programming environment is the so called Compute Unified Device
Architecture (CUDA) and have been the focus of the majority of investigation in the
computational science field. Several groups have lately shown results for molecular
Δ
