Biomedical Engineering Reference
In-Depth Information
10.3 Molecular Dynamics Methodology
In an ideal gas, the particles do not interact with each other and so the potential Φ is zero.
Deviations from ideality are due to the interparticle potential, and most of the early studies
were made on just three types of particle: the hard sphere model discussed in Chapter 9,
the finite square well and the Lennard-Jones model.
10.3.1 Hard Sphere Potential
Just to remind you, the hard sphere potential is the simplest one imaginable; the system
consists of spheres of radii σ and
U
AB
(
R
AB
) is zero everywhere except when the two spheres
A and B touch, when it becomes infinite.
The hard sphere potential is of great theoretical interest not because it represents the
intermolecular potential of any known substance, rather because any calculations based on
the potential are simple.
Alder andWainwright (1957) introduced the modelling technique now known as molecu-
lar dynamics to the world in a short article in 1957. They reported a study of hard discs,
the two-dimensional equivalent of hard spheres.
10.3.2 Finite Square Well
A subsequent paper by Alder and Wainwright (1959) is usually regarded as the keynote
paper in the field, and I quote the abstract:
A method is outlined by which it is possible to calculate exactly the behaviour of several
hundred interacting classical particles. The study of this many-body problem is carried out
by an electronic computer that solves numerically the simultaneous equations of motion.
The limitations of this numerical scheme are enumerated and the important steps in making
the program efficient on computers are indicated. The applicability of this method to the
solution of many problems in both equilibrium and nonequilibrium statistical thermodynamics
is discussed.
In this second paper they chose a three-dimensional system of particles and the finite
square well potential shown in Figure 10.5. This potential is especially simple because a
given particle does not experience any change in velocity except when it is separated from
another particle by σ
2
(when it undergoes an attractive collision) or σ
1
(when it undergoes
a repulsive collision).
On collision, the velocities are adjusted and the calculation restarts. Statistical data is
collected every collision.
In their dynamical calculation, all the particles were given initial velocities and positions.
Typically, equal kinetic energies with the three direction cosines of the velocity vector were
chosen at random and the initial positions corresponding to a face-centred cubic lattice.
Once the initial configuration was set up, they calculated exactly the time at which the first
collision occurs. The collision time can be found by evaluating for every pair in the system,
the time taken for the projected paths to reach a separation of σ
1
or σ
2
.
