Oscillations, Waves, and Interactions - GWDG

The single bubble – a hot microlaboratory 165
At present we are not able to look into the interior of bubbles at collapse to measure
pressures and temperatures there, notably at the center. However, numerical investigations
can be done to get an impression of the phenomena to expect and to couple
them to the observed outside effects. One speculation that has been around for quite
a long time [28] is that converging microshocks are launched in a bubble collapsing
at supersonic speed, which are held responsible for the extreme heating. After the
discovery of SBSL these ideas have been reconsidered and elaborated upon. Initial
fluid-dynamical calculations revealed astonishing peak temperatures at the bubble
center (greater than 1 million Kelvin [29]). Such results are now considered obsolete,
and more sophisticated models [30–32] have yielded an interior dynamics that
features inward-travelling compression waves, mixture segregation, vapour trapping,
chemical reactions and entropy effects, a part of which may actually prevent the
formation of focussing shock waves and limit the peak temperatures.
Notwithstanding theses successes the numerical modelling of the bubble medium
remains a formidable problem. A host of physical and chemical mechnisms have to
be taken into account, while the bubble medium undergoes extreme changes of its
thermodynamic state. The equation of state, chemical reactions and transport phenomena
within the medium and across the phase boundary (heat and mass diffusion,
phase change) have to be captured. It is clear that detailed models become complicated
and difficult to solve, much more difficult to validate. Concerning the question
of shock formation, continuous hydrodynamic solvers have to be used that are able to
handle the shock discontinuity. This may become problematic for converging shocks
whose real thickness near the origin approaches the size of the region to be modelled.
To circumvent this problem, molecular dynamics (MD) simulations of the bubble
medium have been proposed [33,34] as an alternative numerical approach. In fact,
micrometer-sized bubbles contain relatively few (i. e., on the order of 10 10 ) particles.
With suitable coarse-graining, the number of particles (pseudo-molecules) that can
be expected to give a faithful representation of reality can be reduced to about one
million, a number that is tractable by present-day computers.
In our MD model the gas and vapour molecules are treated as hard spheres that
bounce around in the bubble. The hard-sphere system can be advanced in time
very efficiently by an event-driven algorithm [35]. The bubble wall is prescribed as
a spherical container whose radial dynamics is governed by a Rayleigh–Plesset-type
differential equation. It is coupled to the MD model by the pressure exerted on the
wall by particle impacts. The physics at the phase boundary is implemented by an
appropriate set of wall collision rules [34].
The MD approach naturally includes linear and nonlinear diffusive processes as
heat conduction or molecular species diffusion. Chemical reactions can be modelled
relatively easily by introducing different particle species and modifying collision rules
to reflect possible chemical reactions. Being inherently three-dimensional the general
case of aspheric collapse can be simulated without redesign of the algorithm when
the bubble wall dynamics is known or can be calculated.
Figure 29(top) depicts the temperature in the interior of a typical sonoluminescing
bubble of radius Rn = 4.5µm driven at 26.5 kHz and 130 kPa as obtained with a
MD simulation with 10 7 particles. The bubble contains argon and water vapour.
Water dissociation and corresponding chemical reactions have been included in the