Parametric stability and dynamic buckling of an encapsulated microbubble subject to acoustic disturbances

Abstract

Stability analysis of the radial pulsations of a gas microbubble that is encapsulated by a thin viscoelastic shell and surrounded by an ideal incompressible liquid is carried out. Small axisymmetric disturbances in the microbubble shape are imposed and their long and short term stability is examined depending on the initial bubble radius, the shell properties, and the parameters, i.e., frequency and amplitude, of the external acoustic excitation. Owing to the anisotropy of the membrane that is forming the encapsulating shell, two different types of elastic energy are accounted for, namely, the membrane and bending energy per unit of initial area. They are used to describe the tensions that develop on the shell due to shell stretching and bending, respectively. In addition, two different constitutive laws are used in order to relate the tensions that develop on the membrane as a result of stretching, i.e., the Mooney-Rivlin law describing materials that soften as deformation increases and the Skalak law describing materials that harden as deformation increases. The limit for static buckling is obtained when the external overpressure exerted upon the membrane surpasses a critical value that depends on the membrane bending resistance. The stability equations describing the evolution of axisymmetric disturbances, in the presence of an external acoustic field, reveal that static buckling becomes relevant when the forcing frequency is much smaller than the resonance frequency of the microbubble, corresponding to the case of slow compression. The resonance frequencies for shape oscillations of the microbubble are also obtained as a function of the shell parameters. Floquet analysis shows that parametric instability, similar to the case of an oscillating free bubble, is possible for the case of a pulsating encapsulated microbubble leading to shape oscillations as a result of subharmonic or harmonic resonance. These effects take place for acoustic amplitude values that lie above a certain threshold but below those required for static buckling to occur. They are quite useful in providing estimates for the shell elasticity and bending resistance based on a frequency/amplitude sweep that monitors the onset of shape oscillations when the forcing frequency resonates with the radial pulsation, cor coo, or with a certain shape mode, omega(f)=2 omega(n). An acceleration based instability, identified herein as dynamic buckling, is observed during the compression phase of the pulsation, evolving over a small number of periods of the forcing, when the amplitude of the acoustic excitation is further increased. It corresponds to the Rayleigh Taylor instability observed for free bubbles, and has been observed with contrast agents as well, e.g., BR-14. Finally, phase diagrams for contrast agent BR-14 are constructed and juxtaposed with available experimental data, illustrating the relevance and range of the above instabilities. (C) 2011 American Institute of Physics. [doi:10.1063/1.3536646]