Daniel C. Hong

Reverse Brazil Nut Problem: Competition between Percolation and Condensation

In the Brazil nut problem (BNP), hard spheres with larger diameters rise to the top. There are various explanations (percolation, reorganization, convection), but a broad understanding or control of this effect is by no means achieved. A theory is presented for the crossover from BNP to the reverse Brazil nut problem based on a competition between the percolation effect and the condensation of hard spheres. The crossover condition is determined, and theoretical predictions are compared to molecular dynamics simulations in two and three dimensions.

Hydrodynamic description of granular convection.

We present a hydrodynamic model that captures the essence of granular dynamics in a vibrating bed. We carry out the linear stability analysis and uncover the instability mechanism that leads to the appearance of the convective rolls via a supercritical bifurcation of a bouncing solution. We also explicitly determine the onset of convection as a function of control parameters and confirm our picture by numerical simulations of the continuum equations.

Thermodynamic Theory of Weakly Excited Granular Systems

We present a thermodynamic theory of weakly excited two-dimensional granular systems from the view point of elementary excitations of spinless Fermion systems. We introduce a global temperature T that is associated with the acceleration amplitude \Gamma in a vibrating bed. We show that the configurational statistics of weakly excited granular materials in a vibrating bed obey the Fermi statistics.

Condensation of hard spheres under gravity

Starting from the Enskog equation of hard spheres of mass m and diameter D under the gravity g, we first derive the exact equation of motion for the equilibrium density profile at a temperature T and examine its solutions via the gradient expansion. The solutions exist only when βμ⩽μ0, where μ is the dimensionless initial layer thickness and β=mgD/T, and the precise value of the upper bound μ0 depends on the underlying packing. When this inequality breaks down, a fraction of particles condenses from the bottom toward the surface.

Increasing the size of a piece of popcorn

Popcorn is an extremely popular snack food in the world today. Thermodynamics can be used to analyze how popcorn is produced. By treating the popping mechanism of the corn as a thermodynamic expansion, a method of increasing the volume or size of a kernel of popcorn can be studied. By lowering the pressure surrounding the unpopped kernel, one can use a thermodynamic argument to show that the expanded volume of the kernel when it pops must increase. In this project, a variety of experiments are run to test the qualitative validity of this theory. The results show that there is a significant increase in the average kernel size when the pressure of the surroundings is reduced.

Corrugation of roads

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 H(x,t), which contain two phenomenological variables: an effective diffusion constant Δ(H) that characterizes the relaxation of the road surface, and a function a(H) that characterizes the plasticity or erodibility of the road bed. Linear stability analysis shows that 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 simple fact that the normal force exerted by the road can never be negative seems to lead to restabilized, quasi-steady road shapes, in which the corrugation amplitude and phase velocity remain fixed.

Surface instability in windblown sand.

We investigate the formation of ripples on the surface of windblown sand based on the one-dimensional model of Nishimori and Ouchi [Phys. Rev. Lett. 71, 197 (1993)], which contains the processes of saltation and grain relaxation. 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. The agreement between the theory and our numerical simulations is excellent near the onset of the instability. We also determine the Eckhaus boundary, outside which the steady ripple patterns are unstable.

Liquid-solid transition of hard spheres under gravity

We investigate the liquid-solid transition of two-dimensional hard spheres in the presence of gravity. We determine the transition temperature and the fraction of particles in the solid regime as a function of temperature via event-driven molecular-dynamics simulations and compare them with the theoretical predictions. We then examine the configurational statistics of a vibrating bed from the viewpoint of the liquid-solid transition by explicitly determining the transition temperature and the effective temperature T of the bed, and present a relation between T and the vibration strength.

Density-functional theory of hard-sphere condensation under gravity

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 the kinetic theory [Physica A 271, 192, (1999)], and a weighted density-functional theory gives qualitatively similar results, namely, that the temperature at which condensation begins at the bottom scales linearly with the weight, diameter, and number of layers of particles. We find also that the different DFT approaches give quantitatively different results for the density profiles at low temperatures. In particular, the weighted density-functional approach reveals the layering of hard spheres in the solid regime.

Controlling the size of popcorn

We present a thermo-statistical model of popcorn production and propose a way to control the final size of the popcorn by monitoring only the chamber pressure.