Parameter regions have been found experimentally [7, 8], where the clocklike collapse of an air seeded bubble in water looses its regularity. The variability of the collapse time, defined as the interval between some fixed phase of the driving and the moment of light emission has been determined to be mostly of nano second magnitude [7, 2]. But sometimes the distribution spans a couple of micro seconds (Fig. 7 in [8]). The collapse times may not be distributed randomly as there is a frequency associated with this process much lower and quasiperiodic with respect to the driving. The reason of this is unknown.
To solve this phenomenon, a numerical model is developed that includes as much details as is needed to include most relevant physical aspects while maintaining the possibility to calculate several tenthousand cycles (seconds) of oscillations. As the measured frequency of the modulations is 5 orders of magnitude lower than the main bubble oscillation frequency, computational time restrictions imply simulation of the most important features only. The radial dynamics of a bubble in a compressible liquid is calculated with the Gilmore model [6].
 (1) 
 (2) 
 (3) 
The temperature T_{B} is taken to be uniform within the bubble. It is calculated via compression of a van der Waals gas by T_{B} = T_{0}^{γ1} with the ambient liquid temperature T_{0} and the effective polytropic exponent γ.
The number of moles of gases in the bubble n_{i}, i = N_{2},O_{2},Ar is changed by diffusion of air through the bubble wall. Because of the slow diffusional time scale an adiabatic approximation [13, 12, 14] can be employed and the change per period T is
 (6) 
n_{0} is the sum of moles of all molecules in the bubble, M_{i} is the molar mass, D_{i} and C_{i} are the diffusion constants and concentration fields of the gas species. 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}. The same law holds true for r = ∞. _{i} = ∫ _{0}^{T}f(t)R^{i}(t)dt^{1} are weighted time averages.
Evaporation and condensation of water molecules at the bubble wall [15, 3, 16] are included in the model for the bubble dynamics, as experimental results [3] stress the importance of a decrease of the polytropic exponent induced by water vapor at bubble collapse. A Hertz/Knudsen model for the change of moles n_{H2O} of water vapor in the bubble is ṅ_{H2O} = ṅ_{H2O}^{evap} ṅ_{H2O}^{cond} and
with the constant evaporation coefficient α (also called accommodation coefficient or sticking probability), and the average velocity of molecules of a MaxwellBoltzmann distribution (T_{s}) = ^{1∕2}. ρ_{g,H2O} is the density of water vapor of molar weight M_{H2O} in the bubble, ρ_{g,H2O}^{sat} the saturated vapor density [10] and R_{gas} = 8.3143 J mol^{1}K^{1} is the gas constant. The bubble surface temperature is taken as T_{s} = T_{0}. The simple model (7) takes the temperature distributions in the bubble and liquid as fixed. Γ is a correction factor for nonequilibrium conditions induced by mass motion of vapor and bubble wall movement [17, 16, 18]. Calculations show, that Γ varies by as much as 20% around unity during the collapses. However, in the observed parameter range almost no notable difference exists in the amount of water vapor at collapse time when compared to a fixed value. Therefore constant values of Γ = 1, α = 0.4 [6] are taken in the following calculations.

Chemical dissociation occurs for noninert gases [4, 16, 15, 3, 19, 6], reaction products are immediately diffused into the liquid. The dissociation per period is calculated as a second order reaction by a modified Arrhenius law:
 (8) 
E_{A}^{i} are activation energies, = n_{0}∕πR^{3} is the molar concentration of all molecules, i = N_{2},O_{2}. A_{i} and β_{i} are Arrhenius constants [20] . The values of the Arrhenius constant A_{i} has been shown to depend on the gas mixture (3^{rd} body), especially on the argon content and is changed accordingly.
A drop of the bubble temperature due to water vapor condensation and endothermic chemical reactions is neglected. This contribution is small compared to the heating over a temperature range of several thousand degrees Kelvin. The same holds true for reaction enthalpies, as only small amounts are dissociated during a single collapse. Spatial translations of the bubble in the sound field are implemented [6].
Experiments [8] have determined parameter regions, where a quasiperiodic modulation of bubble characteristics occurs: Low frequency oscillations are seen below the pressure range for stable argon bubbles. Calculations show that in this region incomplete dissociation of noninert gases exists. The rates and the change of rates of diffusion and dissociation may oscillate antisymmetrically and never reach a static equilibrium. Fig. 1 shows a normalized distribution of calculated collapse times, defined as the interval between the moment the driving pressure crosses zero going to rarefaction and the moment of minimum radius. The broad distribution spanning a couple of micro seconds is substantially the same as in experiments [7], Fig. 7 in [8].
Fig. 2 shows changes of some characteristics of a bubble on a long time scale. The numerical calculations use parameter settings as in published experimental results [8]. The driving amplitude is 1.194 bars, the ambient pressure 1.033515 bars augmented by a hydrostatic pressure of a water column of 4.36 cm. The water is containing air degassed to 20% ambient concentration. The wave number of the standing wave mode in the resonator equals the published value k = , λ = c_{0}∕f.

In consistency with the experiments, a slow modulation with a period of 3 s is visible in the collapse time and in the maximum radius taken during a single driving period. The collapse time varies by ≈ 3μs, the maximum radius has a variability of about 20 μm. The levitation position [6] in the resonator varies by approximately 15 μm in Fig. 2c. This accounts for a very small change in driving pressure of 0.2 Pa during the modulation period. This change cannot be made responsible for the modulation, in agreement with arguments in [8]. The radiustime graphs in Fig. 3 show that the bubble oscillates inertially well above the blake threshold.

To get further insight into the reasons for these modulations, the gas content of the bubble is analyzed (Fig. 4). Here, modulations of different species and long term changes of the maximum temperature are observed: The variation of gas content species in the bubble is shown in Fig. 4a and b. Nonnoble gas contents vary synchronous with the maximum radius. However, the number of molecules of argon changes in an anti correlated manner during parts of the modulation oscillation (Fig. 4b, dashed line). The same is true for the diffusion and dissociation rates of nitrogen in Fig. 4c. Their values cross several times and the changes of rates have different signs: while the diffusion rate decreases, the dissociation rate increases by a large factor. The oxygen diffusion and dissociation rates are equal (dotted line in Fig. 4c) and do not dynamically influence the oscillations. Fig. 4d shows the diffusion rate of argon as it oscillates around zero. The temperature and the density in the bubble (Fig. 4e) show vast changes during the modulation oscillations.
When analyzing Fig. 4 in detail the following explanation of the long term modulations is plausible: When the bubble is small (at t ≈ 0 s) (see also Fig . 3) nitrogen is diffusing into and argon out of the bubble because only the nitrogen concentration is large enough to establish a net influx. As the temperature decreases due to the smaller average polytropic exponent within the bubble, the dissociation rate of nitrogen does not keep up with the diffusion rate. Due to the increasingly larger ambient and maximal bubble radius (t ≈ 1 s) the argon diffusion rate increases to a net influx. This results in a temperature and nitrogen dissociation rate increase. Shortly before t ≈ 3 s the nitrogen dissociation rate overtakes its diffusion rate while the argon rate still increases as the bubble oscillations are fairly large now. At t ≈ 3.2 s the temperature increases to 9000 K which, together with a high density of 0.7 g/cm^{3} results in a sudden nitrogen loss of 2 orders of magnitude. The bubble now contains mostly argon. Its oscillations get small, the temperature, the density and the nitrogen dissociation rate drop sharply. The cycle repeats itself with an increasing influx of nitrogen. The modulation oscillation occurs as a limit cycle in a bistable system, where both equilibrium states are unstable.

The dependence of the slow oscillations on the driving pressure is seen in the time dependence of the modulation of the maximum radius of the bubble taken during a single driving period (Fig. 5). With decreasing driving pressure the frequency of the modulation decreases and its amplitude increases. The same behavior has been reported in the experiments in [8]. In Fig. 5a the bubble settles to a non modulated oscillating volume containing an N_{2}∕Ar mixture (ratio ≈ 60∕40) with some O_{2} and H_{2}O. With decreasing driving amplitude modulation oscillations set in. The dotted line in Fig. 5d shows the lower pressure limit of the modulations. The long term transient of the maximum radius shows that the bubble is getting so large that it enters the region of parametric instability. Here, micro bubble pinchoff can occur [6]. Calculations verify, that the reported modulations are robust with respect to changes in the amount of allowed water vapor and to a more complete model of thermal damping [3]. The observed effects may well play a role during path instabilities of bubbles in sulfuric acid [21].
Oscillations occur in a large number of chemical systems in different configurations [22]. The system described here is a forced system which is open by continuous inflow of reactants, nonisothermal and nonisobaric, whereby the Arrhenius rates are changed nonlinearly. It has analogies to the nonlinear dynamics in a continuously stirred tank reactor (CSTR), where Hopfbifurcations, saddlenode bifurcations and multistabilities have been found [23]. This system is unique in that it shows a slow physicochemical oscillation initiated by nonlinear dynamics on a fast time scale of a bubble.
The author thanks C. R. Thomas and R. G. Holt for making available their data prior to publication.
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