# How Do Molecules Move in a Liquid?

September, 2005.

My research uses physics to understand how molecules move in a liquid. I use pencil and paper mathematical techniques to make predictions about the movement of a molecule and then test these predictions with computer simulations. The research aims to understand the basic dynamics of molecules in liquids as well as suggests engineering applications such as how to better mold a plastic.

The field of statistical mechanics is a branch of physics that allows one to determine how a molecule moves in a liquid from the interactions between molecules. The most simple interaction between two molecules is that they can't occupy the same position in space. In order to capture this aspect of liquids, we consider a model where a molecule is represented by a particle that occupies a single site on a grid. The only interaction between a pair of particles is that they can't occupy the same site on the grid. To mimic the dynamics of a liquid, the particles can move in a random direction to any neighboring site but only if the site is unoccupied.

Understanding the movement of a particle in our grid model requires answering the following question of dynamics: if a particle occupies position A at an initial time, then what is the probability that it occupies position B at a later time? In order to answer this question for the model, I use theoretical techniques that require only pencil and paper; this is the process of developing a theory. One theory called the binary collision theory uses the idea that a particle can block the motion of another particle by occupying a neighboring site. This is the manner in which two particles 'collide'. We want to determine how well this idea can describe how a particle moves in our model. A second method of answering the question of dynamics is the simulation of the model in a computer. To move a particle, the computer generates a random number to select an empty neighboring site. Since this is exactly how a particle moves in the model, the computer simulation answers the question of dynamics correctly. By comparing the theory to the results of computer simulations, we can determine how well the theory answers our fundamental question of dynamics.

The theory describes the movement of particles well for low concentrations of particles but not as well for high concentrations. This tell us that the blocking effect of two particles describes the movement of the particles well at low concentration, but this is not the only important effect at higher concentrations. This analysis suggests that we need improvements to the theory to describe the dynamics at higher concentrations; one idea is to include the effect of more than two particles in the theory. This research allows us to understand how well we can describe the movement of a particle by including the effect of 'collisions' between pairs of particles in the theory. Another benefit of research on this simple model is that it suggests how we could develop a theory for a more realistic model of a liquid. One such model would allow particles to move continuously in space and would include attractive interactions between pairs of particles.

My current work involves extending these ideas to more complicated grid models of liquids. Imagine connecting a group of particles in a chain by inserting springs between the particles. Each spring could only compress or stretch a limited amount. This is a model for a polymer molecule, which constitute important materials such as plastics and teflon. Developing theories for polymer molecules would not only answer scientific questions about dynamics but would also suggest industrial applications. One such application occurs when a liquid consists of only polymer molecules; the dynamics are complicated since the polymer molecules get entangled. Since plastics go through this liquid state before they are molded, understanding the dynamics of these liquids would aid in the industrial process of molding plastics.