The dynamics of hard spheres in a collapsing spherical container

Abstract

The properties of a collapsing spherical bubble have been studied in the past using the methods of continuum gas dynamics. In this report the problem is approached in an entirely new way by using the molecular dynamics of hard spheres to represent a dense gas during the final stages of cavity collapse. The advantage of this method is that it is only necessary to assume that the laws of classical mechanics are valid in order to ensure that all the transport properties and equation of state for the gas are properly taken into account. The disadvantage is that the trajectories of only a relatively small number of particles can be followed on even today's fastest computers, so that the calculations are limited to small systems. The methods used are essentially the same as those already employed to study hard spheres in a channel or a spherical cavity. Four cases are examined, with wall Mach numbers of 0.5, 1.0, 1.5 and 15.0. The temperature, pressure and density distributions are calculated inside the cavity for several average densities during the course of bubble collapse. Our simulation also indicates that a heavier gas will gain more energy from the wall than a light gas, possibly explaining the observation that heavy, dissolved gases yield more luminosity than light gases during cavity collapse.

Molecular dynamics simulations of cavitation bubble collapse and sonoluminescence

“…As only one MD simulation [31] existed for spherical symmetry at the beginning of this project, the validation of the code developed had to be ensured. This was done first by extending the MD calculations with a constantly contracting sphere to particle numbers of 10 6 ([32], figure 21).…”

“…These results are well within expectations, considering that the speed of sound decreases from helium via argon to xenon. However, another effect is to be considered: the heavier the particles, the more energy can be transferred from the bubble wall motion to the inside of the bubble (see table 4 and compare [13,31]). The wall velocity v W shrinks as molecular weight is increased, because the heavier particles induce a higher pressure at the bubble wall (example given at t = −200 ns).…”

“…Reflecting and heat bath boundaries are applied, and besides hard spheres, variable soft spheres [40,41] are also used. Although no realistic temperature values can be given, as, for instance, water vapour is not included, calculations confirm the trend in temperature rise from light to heavy particles (He to Xe) [13,31], the build-up of strong inhomogeneities in the bubble upon strong collapse [32,35,36] and mixture segregation [25,28,36]. From the duration of high temperature inside the bubble in dependence on the number of particles (three cases: 10 4 , 10 5 and 10 6 ) a proportionality of bubble luminescence pulse width to the bubble radius is derived as found experimentally ( [32], figure 38; see also [42], figure 10).…”

“…No realistic conclusions can be drawn from this start-up model, for instance on temperatures, but some trends can be predicted. The temperature inside the bubble increases 'dramatically' [31] with increasing Mach number. Heavy, non-reacting gases (particles) gain more energy from the wall collisions than lighter ones.…”

“…The first significant approach to bubble collapse and bubble luminescence with an (MD) particle model known to the authors starts with 725 hard sphere particles in a spherical volume (the bubble) made shrinking at a prescribed constant velocity with Mach numbers of 0.5, 1.0, 1.5 and 15 with respect to the sound velocity in the undisturbed gas [31]. The particles move according to Newton's laws.…”