Single bubble sonoluminescence (SBSL) is the phenomenon of a single bubble in water driven in an ultra sound field emitting light flashes upon a violent collapse [1, 2]. One of the astonishing facts is that the pulsations and light emissions can be made very stable and lasting for hours. This stability is astonishing since it is known for a long time that during bubble oscillations diffusion of gases from the surrounding fluid through the bubble wall is present. Because the pressure in the bubble is oscillating by more than 10 orders of magnitude the partial pressure of gases in the bubble should also be changing. Early theoretical papers [3] realized that in the SBSL parameter space a bubble should be constantly growing due to rectified diffusion since the high pressures only occur during a very short time interval of the whole oscillation period. Experiments showed that an air bubble is stable if the gas concentration in the liquid would be 1% of the value set in the experiment [4]. In [5] it was proposed that due to the high temperatures at bubble collapse chemical dissociation of noninert gases gives rise to the formation of highly soluble species that immediately dissolve in the surrounding water leaving only argon (1% of air) in the bubble. Further experiments [6] underlined the finding that mostly argon is present. Parameter regions are identified where a bubble is stable against diffusion and dissociation. In these experiments the gas concentration in the liquid and driving sound pressure are fixed and it is seen that the bubble chooses an adequate ambient radius that can be measured. Stability curves are drawn in the parameter space representing a stable oscillating bubble.
In the upper driving pressure range at higher dissolved gas concentrations the bubble is no longer stable [1, 7]. The ambient radius of the fast oscillating bubble is growing due to diffusion on a slow time scale until the bubble breaks up by micro bubble splitoff. Normally a bubble with small ambient radius survives which is then increasing until the whole process repeats. The breakup is explained by the fact that an argon bubble reaches a parametric instability threshold where surface waves are amplified leading to a sudden removal of bubble volume.
A close inspection of the time evolution near this instability however shows that the process is more involved (FIG. 1): The relative collapse time, the time difference between collapse and a constant phase of the driving, is calculated from shock wave recordings [8]. It is a measure for the ambient bubble radius. FIG. 1 shows the slow increase in relative collapse time followed by bubble splitoff. Immediately after the breakup a fast increase of the collapse time followed by a rapid decrease is seen, after which it increases again monotonically .
This peak during diffusionally unstable SBSL is shown to be the result of a combined instability effect: A sonoluminescing bubble consisting solely of argon looses some of its volume at micro bubble splitoff thereby undergoing a sudden spatial dislocation (recoil). Translating back to the antinode of the sound pressure field it is moving through a complex phase space during which the chemical composition of its gas contents change drastically. This effect demonstrates a dynamical variant of the argon hypothesis [5].


For a further explanation we have to look closer at bubbles during bubble splitoff. FIG. 2 shows a series of images of a big bubble emitting a small bubble near the lower SBSL threshold. The small bubble is leaving the splitoff site at a decreasing speed and dissolves. The bigger part of the bubble experiences a recoil and subsequent return, noted by some authors as “dancing”. FIG.3 shows a double exposed image of 2 shock waves emitted during bubble collapse, one before and one after the micro bubble splitoff at the upper SBSL threshold. As the centers of the circular structures are shifted a recoil or spatial translation has happened. The radii of the shock waves differ. A larger radius means a larger time between shedding of the shock at collapse and the following phase locked illumination flash. Smaller bubbles have an earlier collapse and are thus identified by a larger shock wave radius. From the position of the centers of the two shock waves of FIG.3 a lower bound of the translational velocity during split off of 0.5 m s^{1} can be calculated [8], the change of shock wave radius resulting in a change of collapse time of 1μs shows the volume loss.
A numerical analysis of the dynamics of a diffusionally unstable SBSLbubble uses a model consisting of equations for radial motion of the bubble wall, gas diffusion through the bubble wall, chemical dissociation of noninert gases and translational movement of the bubble in a sound pressure gradient. The Gilmore model [9] describing the radial motion of a bubble in a compressible liquid is integrated numerically.

The number of moles of the different species in the bubble n_{i} is changed by diffusion of air dissolved in the water. The solution of the diffusion equation is
 (4) 
M_{i}, D_{i}, and C_{i} are the individual molar masses, diffusion constants and concentration fields, i = N_{2}, O_{2}, Ar. The concentration of species at the bubble wall is assumed to connect to the partial pressures p_{i} inside according to Henry’s law: C_{i}_{r=R} = C_{i}^{0}p_{i}(R)∕p_{0}. Because of the slow diffusional time scale an adiabatic approximation [13] can be employed and the change per period T is
 (5) 
n_{0} is the sum of moles of all molecules in the bubble,
⟨f(t)⟩_{i} = ∫
_{0}^{T}f(t)R^{i}(t)dt∕∫
_{0}^{T}R^{i}(t)dt are weighted time
averages.

Chemical dissociation occurs for noninert gases [5, 14, 15, 16], reaction products are immediately diffused into the liquid. The dissociation per period is calculated as a second order reaction by a modified Arrhenius law:
 (6) 
A_{i} and β_{i} are Arrhenius constants [17], E_{A}^{i} activation energies. R_{gas} = 8.3143 J Mole^{1}K^{1} is the gas constant, T_{B} = T_{0}^{γ1} the bubble temperature, i = N_{2}, O_{2}. w = (1  x)e^{x∕(1x)}, x = λR^{3}∕b^{3} limits the dissociation rate to reflect an excluded volume [16]. λ [0,1] is introduced to gradually control this high pressure limit. A value of λ = 0.85 is taken, as justified by the results.
Evaporation and condensation of water molecules at the bubble wall [15, 14] is included in the model, as experimental results [18] stress the importance of a decrease of the polytropic exponent at bubble collapse. A simple Hertz/Knudsen model for the change of moles of water vapor is
α = 0.4 is the evaporation coefficient [19],To model bubble translations (FIGs. 2,3) an equation for radius and time dependant buoyancy and Bjerknes force, change of effective bubble mass in the liquid and drag force is added
The numerical simulation starts after bubble splitoff at the upper threshold for SBSL, as a small remaining argon bubble has been shifted by recoil and is translating back into the pressure antinode (FIGs. 4,5): In the beginning the sound pressure is low and the bubble temperature is small such that N_{2} and O_{2} molecules diffusing into the bubble are not dissociated. As a consequence the bubble volume (ambient radius) increases (FIG.5). On its way to the center, driving pressure and temperature increase and dissociation sets in (FIG.4). The ambient bubble radius is decreasing as the reaction products are diffusing into the liquid. Only the inert argon remains, while the ambient bubble radius is still growing due to diffusion. When the upper SBSL threshold for parametric instability [22] is reached (FIG.5), the process can repeat. FIG.5 shows bubble paths for different start sizes and recoil distances. Larger recoil distances display a larger variation in equilibrium size/collapse time. While the calculated time scales of the width of the peak agree with the experiment, the dissociation dominated side is less steep in the experiment (FIG.1) as in the simulation (FIG.4). This suggests that a substantial amount of reaction products with small diffusion constants is produced. The dissociation rates have to be limited at the very high pressures involved around the peak (from 0.3 to 138 kbars dropping to 90 kbars at splitoff), else the peak duration is an order of magnitude too small. The λ factor in the dissociation limiting function has to be smaller than 1, else the high pressures inhibit dissociation of air in this model. Bubbles with an upwards pointing recoil display higher temperatures and shorter peaks at the decreased hydrostatic pressure. The results show, that diffusion together with chemical kinetics and translatory dynamics as a dynamical application of the dissociation theory [5] explains the details of a diffusionally unstable SBSL bubble. The results suggest that spectra of the light emission of diffusionally unstable sonoluminescing bubbles change during the cycle from a line dominated spectrum to bremsstrahlung. It is supposed that this also holds true for the spectra difference of stable SBSL argon bubblesand multi bubble sonoluminescence. Small bubbles approaching pressure anti nodes along streamers grow by diffusion of air and stay too cold to dissociate much of their contents before they get surface unstable, split and recycle.
The author acknowledges the collaboration with M. Rüggeberg and the scientific exchange with R. G. Holt, S. Putterman, K. Suslick and A. Szeri. Part of this work has been funded by the SFB 185 “Nichtlineare Dynamik” of the DFG.
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