The simulation used in this module performs what is known as a molecular dynamics (MD) simulation. (If you haven't already taken a look at the simulation applet, you may want to try it out before reading this section.) This is easily understood from freshman physics. The molecules start off at time zero with given positions and velocities. The velocities are assigned randomly to the molecules so that the overall temperature is the one requested by you. From this initial state, each molecule moves according to the Newtonian equations of motion: F = ma

where F is the force that the molecule feels, m is its mass, and a is the acceleration that it experiences due to the forces. Both F and a are, of course, vector quantities. The forces that the molecules feel come from the other molecules, since we don't consider here any external forces.

• Question 1: How large is the force due to gravity acting on an argon atom compared to the attraction from a second argon atom at a distance of, say, 5 Angstroms?

Most molecules will experience an attraction toward other molecules due to van der Waals dispersion forces. However, if the molecules come too close together, they will, of course, repel one another because of the overlap of their electron clouds. The figure at right demonstrates this phenomenon, where the minimum of the curve is the equilibrium separation between the two molecules. If this separation is increased, the attraction between the two decreases until the two molecules have reached a distance where they no longer "feel" each other, and the potential approaches zero. On the other hand, reducing the separation results in a sharp increase in the potential between the two molecules, and they quickly repel each other. In order to make the simulation run faster, the molecules on your screen do not experience any attraction toward one another. In addition, they are infinitely repulsive if they should try to overlap with one another. Thus the only forces that the molecules feel are when they collide with one another, when they bounce off of each other like perfect billiard balls such that the collisions conserve t senergy and linear momentum. Otherwise, the molecules travel in straight lines at constant velocities determined by their last collision.

Such `hard sphere' simulations (or hard disk here) have a long history, and actually are quite informative. The reason that this simple model is useful is that it is mostly the repulsions that govern the structure of liquids, and the attractions can be considered a small perturbation to this. Today, with fast computers, one can simulate much more realistic models that include attraction between molecules, that account for the individual atoms of the molecules (instead of considering only spheres), and that include many more molecules – and three dimensions instead of just two. The simple two-dimensional, hard-disk model considered here is only to illustrate some points for educational purposes and would not be used to actually calculate properties of real materials.

• Question 2: How large is the kinetic energy of a hard disk compared with the potential energy of a hard disk at room temperature? (there is no intermolecular interaction in a hard disk system so the only potential energy is gravitational potential energy)

From the discussion above, it should be clear that the forces the molecules feel depend on the positions (x, y) of the molecules. You will also recall that the acceleration is the second derivative of the position with respect to time and velocity is the first derivative. The Newtonian equations are thus a set of second-order ordinary differential equations that can be integrated forward in time given a set of initial positions and velocities of the molecules. This is a very powerful and general technique. You can read more about it in references such as Allen & Tildesley (1987) if you are interested, but for now, this is enough to move forward.

An MD simulation can provide many thermodynamic and transport properties for the system under investigation. The link between the microscopic level of the simulation and the macroscopic thermodynamics and transport properties is provided by the field of statistical mechanics. This is a very large subject, which you may want to study later, but we will only need a few results from statistical mechanics for our purposes here. Thermodynamic quantities accessible include PVT properties (e.g. the pressure at a given temperature and specific volume), the heat capacity, and the average internal energy. Other types of molecular simulation can also provide phase equilibrium properties, as discussed in other modules of this series. Transport properties that can be calculated from MD include the viscosity, self-diffusion coefficients, Fickian diffusion coefficients, and, of special interest here, the thermal conductivity.

Before discussing the calculation of thermal conductivity, it's worthwhile to look at the self-diffusivity, which is easier to compute. The self-diffusivity D_s is a measure of the average mobility of individual molecules and is usually defined for a system at equilibrium from the Einstein equation

where d is the dimensionality of the system, N is the number of molecules, t is the time, r_i(t) is the position of molecule i at time t and r_i(0) is the position of the molecule at time zero. The equation is valid at long times. In statistical mechanics, pointed brackets indicate an ensemble average, which means to average over all particles and ideally over many different systems to get a good average. The quantity inside the pointed brackets here is referred to as the mean-squared displacement (MSD) of the molecules during time t. At long times, a graph of the MSD versus time gives a straight line, and the slope is 2dD_s.

• Question 3 : Does this mean the self-diffusivity depends on time or not?

The module prints out values of MSD/t, which we can interpret as basically the self-diffusivity, aside from the constant factor of 2d, where d for this system is, of course, two. Some of the exercises below will allow you to explore how the self-diffusivity depends on factors such as the density and temperature. A future module may explore aspects of diffusion in more detail.

The next question is how to calculate the thermal conductivity from an MD simulation.