Aaron M. Johnson and Dr. Randall B. Shirts, Chemistry and Biochemistry
Computer modeling has become an important aspect of modern chemistry. As chemists try to better understand the how and why of chemical reactions, computer modeling and simulation is unparalleled in value. We are able to learn the conditions that best facilitate effective collisions, and therefore, the best way to design chemical synthesis. However, in order for these simulated results to be most accurate, chemists need to understand how the simulations relate to real systems. Limited by computation speed and programming complexity, simulated systems are always much smaller than real systems in both volume and number of particles. The purpose of my research has been to understand the corrections that are necessary to translate the data received from simulations of a few particles into meaningful predictions of systems with macroscopic parameters.
Traditionally, this discrepancy has been corrected by using periodic boundary conditions. This involves allowing particles to enter and leave the confines of the region being simulated, simulating a small segment of a large system as opposed to trying to simulate the entire system. In essence, using periodic boundary conditions is like simulating a drop of water in the middle of the ocean. According to statistical mechanics, by observing a small portion of a system for a long enough period of time, one can obtain derive the properties of the entire system (watch that drop long enough, and you’ll learn everything you need to know about the entire ocean). However, we have proven that when classical, finite systems are simulated, simply using periodic boundary conditions is not enough to relate these microscale systems to macroscale systems. It was observed that these deviations could be expressed as dependant on two factors, the number of particles and the ratio of the size of the particles to the size of the container.
Our research focused on developing an accurate equation of state for hard spheres (particles with volume, no attractive potential and perfectly elastic collisions). We found it easiest to relate simulations to macroscopic properties in terms of “corrections” to the coefficients of the virial expansion of pressure, P, in terms of density, .
where equals the number of particles, N, divided by the volume, V; Bn(T) are the temperature, T, dependent virial coefficients; and k is Boltzmann’s constant. This was executed by performing thousands of simulations and deriving the corrections need to reproduce the data from theory. The most important correction was in expressing the number of particles in the density. Because pressure is related to the rate of collisions, it became apparent early that N should not simply be the total number of particles, but N-1 (the number of particles available for collision), did not work either. We found that the proper expression was the geometric mean of these two, [N*(N-1)]1/2. Empirical fitting was successful in predicting first order corrections, but its usefulness ended there.
During the winter of 2004, my research on the virial expansion and equation of state of hard-sphere fluids lead to a change of approach. This past semester, we concluded that empirical fitting of the data would not produce higher order virial coefficients to the desired accuracy. Our current method is to focus on systems of very few particles (3 or 4) and analytically derive the virial coefficients from the corresponding configuration integrals. The success of this approach has been fruitful. I performed many hundred simulations of systems with 3, 4, or 5 particles and used this simulation data to verify Dr. Shirts’ determination of the exact dependence of the third and higher virial coefficients on the number of particles in a system. Furthermore, we are able to explain the particle number dependence of the second virial coefficient in periodic boundaries by properly understanding the relationship between the average and relative velocities in a system with a constrained center of mass. I have shown that the relationships developed by my group are able to better predict the behavior of macroscale systems from microscale computer simulations.
This project helped me learn how to recognize the limitations of a current procedure, and tested my imagination in developing new techniques. Developing an analytical expression for configuration integrals required creative uses of calculus and mathematical tools. As I sought an expression for the overlap of three randomly placed circles in terms of their relative positions, I began seeking out new methods of numerical analysis. I tried using Matlab and Excel, but in both cases realized the complexity of the problem was greater than I had realized.