Stability and natural frequency of nonspherical mode of an encapsulated microbubble in a viscous liquid

The dynamics of encapsulated microbubbles (EMBs) subject to an ultrasound wave have wide and important medical applications, including sonography, drug delivery, and sonoporation. The nonspherical shape oscillation of an EMB, termed as shape modes, is one of the core mechanisms of these applications and therefore its natural frequency is a fundamentally important parameter. Based on the linear stability theory, we show that shape modes of an EMB in a viscous Newtonian liquid are stable. We derive an explicit expression for the natural frequency of shape modes, in terms of the equilibrium radius of an EMB, and the parameters of the external liquid, coating, and internal gases. The expression is validated by comparing to the numerical results obtained from the dynamic equations of shape modes of an EMB. The natural frequency of shape modes shifts appreciably due to the viscosity of the liquid, and this trend increases with the mode number. The significant viscous effects are due to the no-slip condition for the liquid flow at the surface of an EMB. Our results show that when subject to an acoustic wave, the shape instability for an EMB is prone to appear if 2ωk/ωd = n, where ωk is the natural frequency of shape modes, ωd is the driving frequency of the acoustic wave, and n is a natural number. The effects of viscosity on the natural frequency is thus critical in setting the driving frequency of ultrasound to avoid or activate shape modes of EMBs, which should be considered in the applications of medical ultrasound.