Environmental Engineering Reference
In-Depth Information
different molecules are distinguishable from each other [15]. Take
the diamond-structured s
il
icon for instance, its lattice constant is
L
0
=
5.43 A, and
a
=
√
3
L
0
/
4
=
2.35
˚
A.ThedeBrogliethermal
wavelength at room temperature is
=
0.19 A, which is one
order of magnitude smaller than the nearest neighbor separation.
Therefore,forthegroupIVsemiconductorswithdiamondstructure,
such as silicon and germanium, the classical approximation is
appropriate above room temperature from the dilute gas point
of view. Furthermore, quantum effects become important for any
system when the temperature is su
ciently low. The drop of the
heat capacity for crystals below the Debye temperature is a well
know example of measurable quantum effects in solids. This is of
particular importance in the study of thermal transport [16].
Inthecontextofthermaltransport,MDsimulationisastatistical
mechanics approach, which relates the microscopic behavior in a
systemwithitsthermodynamics.Accordingtostatisticalmechanics,
physical quantities can be evaluated by averaging over configura-
tions distributed according to a certain statistical ensemble. For the
N
-particle system, MD simulation calculates the trajectory in a 6
N
-
dimensional phase space (3
N
positions and 3
N
momenta) at each
instantaneous time. This trajectory obtained from MD simulation
provides such a set of configurations. To get the thermodynamic
variables such as temperature, it relies on the ergodicity hypothesis
of statistical mechanics, which asserts that the phase space can
be fully recovered in the long-time limit and the time average
is equivalent to the ensemble average. Therefore, thermodynamic
variables of interest can be obtained by the time average along the
trajectory in MD simulation.
A typical procedure of MD simulation can be described by the
following key steps. To begin with, an appropriate force field that
can accurately describe the interatomic interaction in the system of
interest is required. The next step is to construct the initial position
and velocity of each atom in the system, and generate a list for
the nearest neighbor interaction. Usually atoms are initially located
at their equilibrium positions and assigned with random velocity
according to Gaussian distribution. After that, one calculates the
force applied to each atom according to the force field, numerically
solves Newton's equation of motion, and updates the position and
