Determining the critical temperature and number of frozen layers in a two-dimensional bed of vibrating hard spheres using a global equation of state

Granular materials at finite temperature exhibit rich phase behavior that can simultaneously encompass solid, liquid, and gaseous states. Despite simple particle–particle interactions, predicting the location between these phase boundaries remains a challenge. Here, we use a global equation of state to model the density profile of a two-dimensional hard sphere system, defined by particle diameter D and mass m under gravity for a given temperature T. We then compare our solutions to computational molecular dynamics data. Using the particle density profile, we solve for the critical temperature Tc, marking the onset of condensation in the system. If T is below Tc, there is some frozen solid portion of the system. We derive an analytical formula for the number of frozen layers μf, which accurately captures the number of frozen layers in the simulated data. These results provide a rigorous framework to determine the density profile and solid–liquid phase boundary in thermal granular materials under the influence of gravity.