# Uses of Applied Mathematics: Computational Materials Science

The development of new technology is increasingly dependent on our ability to understand matter at the atomic level. Liquid crystal displays for computers, semiconductors for solar cells, and stronger, lighter plastics are all examples of materials that have come into being because we know how the smallest pieces of matter, atoms and molecules, assemble to produce substances with specific properties.

The mathematical equations that describe the properties and motions of an individual atom in a solid or liquid can be quite simple. The problem is that there are a huge number of atoms in any sample of material. If we want to predict what the properties of the material will be from the equations that describe individual atoms, we must solve a huge number of equations all at the same time.

Fortunately modern computers have the ability to carry out such large numbers of calculations. As a result, tremendous progress has been made using computers to calculate the properties of specific substances at the microscopic level. These computer simulations are changing the way matter is studied. They provide us with an "atom's-eye-view" of the behavior of matter at microscopic scales, changing the way new theories are developed and tested. They also provide a way to predict the properties of new materials even before they are created in the laboratory.

As an example, shown below are results from computers simulations of one of the most common substances in our everyday experience, the solid and liquid forms of water.

Ice

To the right is an image from a molecular-level computer simulation of an ice crystal. The red spheres represent oxygen atoms, while the white spheres are hydrogen atoms. The special structure of this crystal is responsible for the unique physical properties of ice. For example, in this view, open "channels" can be seen running through the crystal along our line of sight. The presence of these channels means that the crystal contains a lot of empty space, giving it a much lower density than is usual for a crystal. The result is that ice floats on water rather than sinking. For almost all substances other than water, the crystalline form sinks it the liquid form. Though we take this odd property of ice for granted, it is important. For example, if ice sank in water then lakes would freeze from the bottom up, making it much more difficult for fresh-water aquatic life to survive the winter.

Water

In water, the regular structure of the ice crystal has been destroyed because the molecules are no longer just vibrating in place, but are also colliding with and moving around one another. The result is that the arrangement of the molecules is disordered and their motions are much more complicated. Using computer simulations we can calculate the kind of path followed by a water molecule as it moves around in the liquid. The results provide insights into the microscopic properties of the liquid that cannot be obtained in any other way, even with the most powerful experimental tools.