Statistical mechanics of particulate materials: Surface instabilities, condensation, and segregation

Four problems in the statistical mechanics of particulate materials are treated in this dissertation. First, we investigate the formation of ripples on the surface of windblown sand via the one-dimensional model of Nishimori and Ouchi. We carry out a nonlinear analysis to determine the propagation speed of the restabilized ripple patterns, and the amplitudes and phases of their first, second, and third harmonics. We find that the agreement between the theory and numerical solution is excellent near the onset of the instability.; Second, we present a one-dimensional model for the development of corrugations in roads subjected to compressive forces from a flux of cars. The cars are modeled as damped harmonic oscillators translating with constant horizontal velocity across the surface, and the road surface is subject to diffusive relaxation. We derive dimensionless coupled equations of motion for the positions of the cars and the road surface. Linear stability analysis of equations shows corrugations grow if the speed of the cars exceeds a critical value, which decreases if the flux of cars is increased Modifying the model to enforce the fact that the normal force exerted by the road can never be negative leads to restabilized, quasi-steady road shapes with fixed corrugation amplitude and phase velocity.; Third, the onset of condensation of hard spheres in a gravitational field is studied using density functional theory (DFT). We find that the local density approximation yields results identical to those obtained previously using kinetic theory and a weighted density functional theory gives qualitatively similar results, namely, that the temperature at which condensation begins at the bottom scales linearly with weight, diameter, and number of layers of particles.; Finally, density functional theory (DFT) for non-dissipative hard spheres and disks is used to show that dynamically excited granular materials under gravity may segregate not only in the widely known “Brazil nut” fashion, i.e. with the larger particles rising to the top, but also in reverse “Brazil nut” fashion. Specifically, the local density approximation of DFT is used to investigate the crossover between the two types of segregation occurring in the liquid state.