Bubbles encapsulated with a shell composed of polymers, lipids, proteins, or a combination of these are called encapsulated bubbles (EBs). EBs are widely being used as contrast agents for ultrasound medical imaging, for targeted drug delivery, and so on. For a gas-filled EB suspended in a fluid, the interface between the gas-encapsulation and encapsulation-fluid carries sufficient interface energy that significantly affects the mechanics and cannot be neglected. However, the influence of this interface energy on EB dynamics is not well studied in the existing EB models.
A mathematical model based on interface energy is developed within the framework of surface continuum mechanics to study the dynamics of an EB. The proposed model naturally induces residual stress field into the bulk of the bubble, with possible expansion or shrinkage from a stress- free configuration to a natural equilibrium configuration. The significant influence of interface area strain and coupled effect of stretch and curvature is observed in the numerical simulations based on constrained optimization. This model provides a better fit of the experimental data and resolves the spurious dependency of shell viscoelastic parameters on the initial size of the bubble which remained hitherto unexplained. A path towards a thorough modelling of these lipidic shells through both elastic and surface tension contributions may also have been opened by this model, through a curvature- dependent surface tension to describe the spherical oscillations of EBs, allowing for an interesting analysis of existing experimental data for radial oscillations of EBs.
This study is extended to describe the dynamics of an EB with a different nonlinear viscoelastic shell, analysing the dynamic behavior of the bubble through the time series curves, phase space analysis, and the nonlinear frequency response of the bubble. The interface energy model is further used to study the nonspherical oscillations of an encapsulated microbubble under the acoustic field. This has enabled a deeper understanding of the dynamics and stability of the EB to exhibit finite amplitude shape mode oscillations.