Spherical and Non-spherical Bubble Dynamics in Soft Matter

Abstract

Cavitation and bubble dynamics play an important role in a wide range of applications including material characterization and therapeutic ultrasound. A specific problem in bubble dynamics is inertially dominated collapse, which occurs when a bubble reaches a critical size compared to its equilibrium size. This inertial collapse of cavitation bubbles enables us to characterize soft materials at high strain-rates or to ablate malignant tissue in therapeutic ultrasound. This thesis contributes to the understanding of spherical and non-spherical bubble dynamics in soft matter relevant to these applications.

We first study ultrasound-induced oscillations of a gas bubble in soft matter. Oscillations of a spherical bubble in water are described by a classical Rayleigh-Plesset-type model. Although this model has been extended to include viscoelasticity necessary to represent soft matter, experiments of nonlinear bubble oscillations in soft matter are scarce, thus limiting opportunities for validating the models and understanding the role of viscoelasticity on bubble oscillations. We experimentally and numerically study ultrasound-induced bubble oscillations in a gelatin gel. Comparison of finite-amplitude oscillations between experiments and numerical solutions show good agreement, implying the validity of the viscoelastic modeling. A resonance curve of the ultrasound-induced bubble oscillations is obtained and shows the nonlinear feature of spring softening, where viscoelasticity has an impact on the resonant radius and peak amplitude.

Since cavitation bubbles in soft materials consist of a finite amount of non-condensible gas as well as vapor, we investigate the role of gas-vapor mixture transport on the bubble dynamics. From a modeling standpoint, past studies relied on the assumption that the ratios of specific heats are the same for the gas and vapor. We develop a new model that accounts for the mixture with different ratios of specific heats. Numerical solutions show that vapor is trapped by an air shell during inertial collapse. This trapped vapor reduces the bubble collapse velocity and thus energy losses via acoustic radiation, leading to a larger bubble rebound. This analysis is further validated against experiments of laser-induced bubble collapse in glycerol. Comparison between the new and conventional models shows experimentally measurable discrepancies of several percent.

Finally, we study the shape stability of a bubble in soft matter. Bubbles deviate from a spherical shape in practice, typically due to one of two mechanisms: parametric instability or Rayleigh-Taylor-type instability. We develop a model for non-spherical bubble dynamics in soft matter by extending classical perturbation analysis on a plane interface to a spherical interface between a gas and a soft solid. Parametric instability occurs during ultrasound-induced oscillations. The natural frequency and a Mathieu equation obtained from the non-spherical model predict unstable modes in parametric instability, which agrees with experimental observations in gelatin gels. Rayleigh-Taylor-type instability is induced by the large acceleration during inertial collapse. Numerical solutions of the non-spherical model show that viscoelasticity influences the bubble wall acceleration and thus is a key factor to determine this instability.