Microbubble shape oscillations excited through ultrasonic parametric driving

An air bubble driven by ultrasound can become shape-unstable through a parametric instability. We report time-resolved optical observations of shape oscillations (mode $n=2$ to 6) of micron-sized single air bubbles. The observed mode number $n$ was found to be linearly related to the ambient radius of the bubble. Above the critical driving pressure threshold for shape oscillations, which is minimal at the resonance of the volumetric radial mode, the observed mode number $n$ is independent of the forcing pressure amplitude. The microbubble shape oscillations were also analyzed numerically by introducing a small nonspherical linear perturbation to a Rayleigh-Plesset-type equation, capturing the experimental observations in detail.

Figure 1

Growth of a surface mode vibration for a bubble with a radius of $36\mu \text{m}$ as captured with the Brandaris high-speed camera. First, the bubble oscillates in a purely volumetric radial mode (a), then after 5 cycles of ultrasound (b) the bubble becomes shape-unstable and a surface mode $n=4$ is formed [28]. (c) A selection of surface modes observed for various bubble radii.

Figure 2

Experimental analysis. In (a) the driving pressure pulse $P_a\left(t\right)$ is shown, which drives a bubble with an ambient radius of $33\mu \text{m}$. (b) shows an image of the shape-unstable bubble at $t=80\mu \text{s}$. In (c) the radial excursion is plotted against the polar angle $\theta$, at $t=80\mu \text{s}$. (d) shows the volumetric response of the bubble. (e) displays the amplitude of mode number $n$ which is derived from the Fourier transform of (c), and the time evolution of the growth of the shape instability of order 4 is shown in (f) and (g), in a linear-linear and a log-linear scale, respectively. The dotted lines in (a), (d), (f), and (g) indicate $t=80\mu \text{s}$.

Figure 3

(Color) Bottom part. Colored dots: the observed mode number $n$ as a function of the ambient bubble radius $R_0$ for the driving frequency of $f=130\text{kHz}$. Grey dots: the resonant mode number $n$ following Lamb’s expression [Eq. (2)]. Upper part: Solid lines represent the pulsation amplitude threshold $C_n$ following Ref. [33]. The lowest threshold preferentially selects the mode number $n$ indicated by the colored bars.

Figure 4

(Color) Phase diagram in $R_0$ vs $P_a$ space for the driving frequency of $f=130\text{kHz}$. Every experimental data point is included as a circle: white—no shape mode detected, red—mode 2, green—mode 3, blue—mode 4, turquoise—mode 5, magenta—mode 6, and yellow—mode 7. For comparison, also the theoretical result is shown. The color coding indicates the specific mode preference and corresponds to the coding used for the experimental data.

Figure 5

(Color online) Numerical simulation of the onset of the shape oscillation. (top) An ultrasound pulse of 10 cycles of 130 kHz at a driving pressure of 80 kPa. (a): mode amplitude $a_n\left(t\right)$ for $n=2$ to 7 for a $30\mu \text{m}$ radius bubble. The mode $n=3$ appears to be the most unstable mode. (b): the very same result plotted on a log-linear scale showing the exponential growth of the various modes. (c) and (d): similar calculations for a $45\mu \text{m}$ bubble, where $n=5$ is the most unstable mode.

Figure 6

Splitting of a shape-unstable bubble of surface mode $n=3$: Three fragments are split off.