Effects of encapsulation damping on the excitation threshold for subharmonic generation from contrast microbubbles

A recent study [Katiyar and Sarkar (2011). J. Acoust. Soc. Am. 130, 3137-3147] showed that in contrast to the analytical result for free bubbles, the minimum threshold for subharmonic generation for contrast microbubbles does not necessarily occur at twice the resonance frequency. Here increased damping-either due to the small radius or the encapsulation-is shown to shift the minimum threshold away from twice the resonance frequency. Free bubbles as well as four models of the contrast agent encapsulation are investigated varying the surface dilatational viscosity. Encapsulation properties are determined using measured attenuation data for a commercial contrast agent. For sufficiently small damping, models predict two minima for the threshold curve-one at twice the resonance frequency being lower than the other at resonance frequency-in accord with the classical analytical result. However, increased damping damps the bubble response more at twice the resonance than at resonance, leading to a flattening of the threshold curve and a gradual shift of the absolute minimum from twice the resonance frequency toward the resonance frequency. The deviation from the classical result stems from the fact that the perturbation analysis employed to obtain it assumes small damping, not always applicable for contrast microbubbles.