Weakly Nonlinear Acoustics of Gas–Liquid–Shell Three-Layer Microbubbles

Microbubbles encapsulated by shells have been utilized in ultrasound diagnosis to enhance imaging resolution. Additionally, liquid droplets covered with shells are attracting considerable attention for their potential applications in targeted drug delivery systems. While theoretical studies have been conducted on the mechanical properties of single-bubble dynamics, research on pressure ultrasound propagation in liquids containing multiple bubbles remains limited. In this study, we derive an evolution equation for the first-order perturbation of liquid pressure by combining a volume-averaged, two-fluid model, which accounts for the effects of multiple bubbles in a liquid, using the equation of motion for a single liquid droplet obtained from previous studies [Gubaidullin and Fedorov, 2020] and the perturbation method. We impose volume conservation on both the shell and the inner liquid, allowing the shell thickness to vary with radius rather than assuming it constant. The nonlinearity and attenuation of pressure in the liquid medium are qualitatively evaluated using the evolution equation. The derived evolution equation provides explicit coefficients for ultrasound nonlinearity, viscous and thermal attenuation, and dispersion. It shows that shell-coated liquid droplets reduce the effective nonlinearity relative to free bubbles, whereas viscous attenuation increases owing to the shell and the inner liquid layer. For representative material properties, the attenuation budget is dominated by shell viscosity, with smaller contributions from the internal and surrounding liquids.