Saturation of shape instabilities in single-bubble sonoluminescence

Excitation of shape instabilities represents one route to bubble death in single-bubble sonoluminescence. This feature is satisfactorily explained by an expansion to first order in the amplitude of a shape distortion in the form of a spherical harmonic. By taking the expansion to second order, it is found that regions of parameter space exist where the exponential growth into bubble disruption is checked and a saturated stable state of shape distortion is possible. Experimental evidence provided by Mie scattering is presented, and a possible connection to simultaneous spatially anisotropic light emission is discussed.

Figure 1

Result of simulation showing locations of stable period 1 and period 2 shape distortion in the rest radius $R_0$, drive pressure $P_a$ parameter space. Also shown are data for the highest possible stable sonoluminescing state taken from Simon et al. [28].

Figure 2

Locations of stable period shape distortion in the argon concentration, drive pressure $P_a$ parameter space from Fig. 1 but also with regions of higher order and seemingly aperiodic modes filled in.

Figure 3

Simulated rest radius $R_0$ versus drive pressure $P_a$ for an air bubble with relative argon concentration 0.00255. In the green section (+) stable period 1 shape distortion is present. Stable period 2 shape distortion is present in the blue (x) sections. No stable solutions are found in between. The arrows show the operating points for the next figures.

Figure 4

Upper panel: The undisturbed bubble oscillation. Middle panel (scale $\mu$m): Growth and saturation of the excited period 2 shape distortion $a_2$ computed at the arrow pointing to the blue section in Fig. 3. The vertical lines show the timing of the main collapse. Lower panel (scale nm): Magnified view of $a_2$.

Figure 5

Expanded view of the saturation part of the excited period 2 shape distortion shown in Fig. 4. Both Rayleigh-Taylor (the first sharp peaks) and later after-bounce shape oscillations are present.

Figure 6

Saturation of the excited period 1 shape distortion computed at the arrow pointing to the green section in Fig. 3.

Figure 7

Position along the diffusional stability curve of Fig. 3 of the period 1 (denoted P1) and period 2 (denoted P2) shape distortions as function of the parameter ${\delta }_0$. Notice that as ${\delta }_0$ goes to zero the position proceed to higher values of $R_0$ (and $P_a$) and the order of P1 and P2 is exchanged.

Figure 8

Experimental data showing the position of period doubled emission in $P_a,R_0$ parameter space from a sonoluminescing bubble (data from Ref. [7]), compared to simulation with input values ${\delta }_0=0.25$ and ${\delta }_0=0.1$.

Figure 9

Mie scattering [12] from a sonoluminescing bubble ($c\sim 0.25$, $T=9^{\circ }$) that simultaneously exhibits period double emission compared to simulation with input values ${\mu }_{\mathrm{eff}}=4\mu$, $\alpha =0.25$ used for the simulation of $R\left(t\right)$ while $\mu$ is used in the simulation of $a_2$ together with $\delta =0.3$. Experimental data are time averaged over 20 consecutive sets, each lasting four periods to accentuate the period doubling of the structures in the after-bounce regions.