Granular Materials

Granular materials are ubiquitous in our everyday lives and have been for as long as humans have walked the earth. From the sugar we put in our coffee to the aggregate used to pave our roads. From the twigs and branches that humans have used to start fires to the sand and dirt upon which we walk. From the powders that make up the myriad medicines we use to the clumps of hair on a barber's floor.

Many of the studies of granular materials using photoelastic particles focus on looking at stable packings of particles, that is, configurations of grains that minimize the potential energy between grains. A prototypical example of such a study consists of a collection of grains or disks contained in some confining geometry, often a square or rectangular box. The walls of the box are then sheared or compressed into the particles, exerting force on them. This shear or compressive motion of the walls is generally quite small and does not cause radical changes in the configurations of the particles. However, even with these small changes, there are large variations in the forces between particles.

The PESs of these systems can have an arbitrary number of dimensions, which makes them difficult to explore and study. However, if instead of looking at the PES as an n-dimensional surface and we instead look at the network of connected minima, we can greatly reduce the dimensionality of the PES and make it more tractable to analyze. In such a network, the nodes represent minima and the edges represent the energy barriers or saddle points traversed when transitioning between stable configurations. The heights of the energy barriers can be encoded as weights of this network in a directed network.

In my study of granular materials, I modeled them as a system of two-dimensional soft-spheres. These are effectively disks which are very similar to those studied in experimental set-ups such as the ones being studied at NCSU. The force between disks in contact is a linear spring force proportional to the overlap between the disks. In experimental systems, this would be the same as the amount the disks are "squished". This gives rise to an immensely complex PES with a dimensionality equal to 2N where N is the number of disks since each disk can move in the x- or y-directions. Three dimensional systems or those with additional forces between the particles, such as friction or electric forces, can have even more complex PES.