Scott Burt and Dr. Randy Shirts, Chemistry and Biochemistry
Chemical modeling is an increasingly important tool in modern research. Despite vast improvements in computational speed, most systems of interest are so complex that simulations must be carried out on very small model systems. However, such simplified systems often contain constraints that do not apply to the larger system. This causes the simulation results to deviate significantly from the true behavior of the corresponding macroscopic system. Such deviations are known as finite size effects. Understanding these effects are important for correctly extrapolating simulation results to the macroscopic system of interest. Chemical simulations are often performed under conditions of constant total energy (i.e. an isolated system), which makes understanding finite size effects in isolated systems important.
Several years ago, Dr. Shirts created the Boltzmann1 program to help undergraduates better visualize moleculat motion. This program simulated the behavior of gas molecules in a box. Its purpose was to demonstrate how, over time, the values produced by this simulation approached the values predicted by the Boltzmann energy and velocity distribution. However, it soon became apparent that for small systems, there are significant deviations from the predicted results. Dr. Shirts was able to explain these deviations as finite size effects of an isolated system as described above and subsequently calculated the expected energy and velocity distributions for such systems.
In 2002, I designed and wrote a new software package to simulate the motion of hard spheres in one, two and three dimensions with both reflecting and periodic boundary conditions while gathering a variety of statistical information about the system. This software allowed us to confirm Dr. Shirts’ previous results as well as exploring other areas where finite size effects might be important. By the end of 2002, our results led to several areas of interest that we desired to research during 2003. These areas are deviations in the mean free path and collision lifetime distributions, corrections to the virial coefficients and a new formulation of the pressure equation.
The mean free path and collision lifetime distributions are expected to have decreasing exponential relationships, thus indicating that the particles are randomly distributed. Surprisingly, our results show significant deviations from this expected result. One possible explanation that we found in the literature was an alternate definition of the mean free path described by Jeans.2 To test this hypothesis, I had to make significant changes to the simulation program in order to track the Jeans-Tait mean free path and time distributions. The simulation results, however, deviated significantly from Jeans’ distributions as well. Dr. Shirts then formulated the mean free path in terms of some integrals that cannot be solved for analytically. Our numerical solutions to these integrals show the same behavior as the distributions observed in our simulations. We are still studying these distributions to better characterize their exact dependance on other simulation conditions.
The virial equation of state is used to obtain a more accurate description of the behavior of a gas than is possible with the ideal gas equation: PV/nRT = 1 + B2ñ + B3ñ2 + … For a real gas, the virial equation makes corrections due to intermolecular forces as well as the volume of the molecules themselves. Since our simulations model hard spheres without attractive forces, the coefficients are expected to reflect corrections due to molecular volume. The virial equation of state for hard spheres was studied extensively in the 1960s,3 but these studies focused on large systems where finite size effects would become negligible. Our research quickly identified the need for finite size corrections to the virial coefficients. However, to characterize corrections for higher order virial coefficients, it was necessary to run simulations at high particle densities.
This presented a problem because the original simulation program used a simple algorithm to randomly place the particles in the box. This worked well for low densities, but quickly became problematic for intermediate and high densities. I had to create an algorithm that finds the most effective way to distribute an arbitrary number of particles in a box with arbitrary dimensions in such a way that it limits to the close packing density. I was successful in creating such an algorithm and it has allowed to us to charactarize the corrections to the first few coefficients.
The virial equation of state can also be expressed as a pressure equation.4 Dr. Shirts derived a novel formulation of the pressure equation that depends solely on the mean free path. We spent considerable time verifying the accuracy of this equation. Initially, we tested this equation in one dimension because the exact hard sphere equation of state can be solved for analytically.
This presented a problem, however, due to the fact that no energy mixing can occur between particles of the same mass in one dimension. To fix this, I had to implement a number of significant code changes to allow the program to simulate a mixture of particles with different masses. By using a mixture of particles with two different masses, the energy will mix and provide a good mean free path distribution much faster. This, however, prevents using the simple exact equation of state for a one dimensional system of hard spheres because it only applies to systems of identical particles. The equation of state for a mixture can be obtained from the pair distribution function, but the software was not set up to track that type of statistic.
I altered the program to allow it to track the distribution of collisions and relative velocities of each pair in the system. With the pair distribution function, we were able to show that Dr Shirts’ formulation of the pressure equation was exact for all three dimensions and worked in both reflecting and periodic boundary conditions. This is significant because determining the pressure of mixtures or systems with periodic boundary conditions is difficult, generally requiring the pair distribution function. The ability to obtain the pressure of a system directly from the mean free path is much simpler. We are currently investigating the applicability of this method to simulations of colloidal suspensions in conjuction with Dr. Doug henderson. This has also necessitated rewriting my ordered packing code in order to evenly distribute a mixture of spheres with very different radii in the box.
References
- R. B. Shirts, Boltzmann, A Kinetic Molecular Theory Demonstrator, Trinity Software, 1995.
- J. H. Jeans The Dynamical Theory of Gases. 4th ed. Cambridge University Press, 1925. p. 342.
- F. H. ree and W. G. Hoover, J. Chem Phys. 40, 939 (1964).
- D. Chandler, Introduction to Modern Statistical Mechanics. New York: Oxford University Press, 1987. p. 205.