The mathematical threshold at given by Eq. (17) plotted as a function of the term for with fixed . We observe that if is large enough, the threshold for a coated bubble can decrease below the threshold of a free gas bubble despite its additional shell damping. The damping for the free gas bubble is determined by the reradiation damping and the liquid viscosity, for this bubble . For the coated bubble model the shell damping introduces and extra damping described by the shell viscosity which is taken resulting in a total damping of .
The mathematical threshold and the instability threshold as a function of for . The damping for the coated and the free bubble are the same as in Fig. 1, i.e., the damping coefficient for the coated bubble is five times as large as for the uncoated bubble. Even so, the threshold for a coated bubble is only 6 kPa, much lower than for an uncoated bubble which has a threshold of 90 kPa. This decrease of the threshold for the coated bubble results from the rapid change of in the effective surface tension as a function of described by and .
Top figures: An example of the driving pressure waveform (a), and (b) its corresponding power spectrum. Bottom figures: The radius time curve (c) and the corresponding Fourier transform amplitude (d) for two bubbles with a different initial surface tension driven with a driving pressure pulse of 40 kPa with a frequency of 2.4 MHz. The dotted line represents the numerical simulation for a bubble with and the solid line corresponds to a bubble with . The initial bubble radius and the other shell parameters are the same for both bubbles, , and .
The absolute value of the Fourier transforms of a parametric study on the simulated radius-time curve presented in Fig. 3. The fundamental response to the driving pressure of 2.4 MHz is clearly visible in all three figures while the subharmonic response is observed to strongly vary for each shell parameter varied independently. (a) For varied between 0 and the subharmonic response is only visible for the initial condition of the bubble satisfying or . (b) As expected the subharmonic response is observed to decrease for increasing from 0 to and (c) for increasing from 342 to 10 000 N/m the subharmonic is observed to increase but for the amplitude of the subharmonic response saturates.
A schematic overview of the experimental setup that was used to optically record the radial dynamics of coated microbubbles located inside an optically and acoustically transparent OptiCell chamber. The driving pressure waveform produced by an arbitrary waveform generator (AWG) was amplified and transmitted by a focused transducer. The radial dynamics were recorded through a objective coupled through an inverted microscope into the Brandaris ultra high-speed camera.
The radius-time curves (left column) of a microbubble excited with twelve different driving pulses all with an amplitude of 40 kPa and different frequencies. In the corresponding absolute value of the Fourier transform (sampling rate 50 MHz, length pulse 501 points) of the radius-time curves (right column) we observe clear subharmonic behavior. We can identify a subharmonic resonance curve that peaks at a driving frequency of 2.4 MHz, about twice the resonance frequency of the bubble.
The best fit of the fifth radius-time curve from Fig. 6(e) with the model proposed by Marmottant et al. with the shell parameters , and N/m both in (a) the time domain and (b) in the frequency domain (sampling rate both curves 50 MHz, 501 points).
The amplitude of the Fourier transform of the radial response of three differently sized bubbles as measured with the Brandaris ultra high-speed camera represented by a color. The horizontal axis represents twelve different driving pressure frequencies with a fixed driving pressure amplitude of 40 kPa. The response frequency is represented by the vertical axis.
Simulated subharmonic resonance behavior of coated microbubbles with the same initial bubble radii as in Fig. 8 using the best fit shell parameters found in Fig. 7
The maximum amplitude of the subharmonic oscillations of a (a) , (b) and (c) bubble as a response to different driving pressure amplitudes. The measured responses are compared with the subharmonic responses for the same initial bubble radii predicted by three different models. The model proposed by Marmottant et al. 38 (solid line), and a purely linear viscoelastic shell model (dashed line) and a free gas bubble model (dotted line).
In the model of Marmottant et al. 38 the second derivative of with respect to is undefined in the transitions from the buckled regime to the elastic regime, and from the elastic regime to the free gas bubble regime. To correct this, we propose to expand the original model with two quadratic functions and that describe the two transition points.
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