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A simple generalized model (GM) for coated bubbles accounting for the effectof compressibility of the liquid is presented. The GM was then coupled withnonlinear ODEs that account for the thermal effects. Starting with mass andmomentum conservation equations for a bubbly liquid and using the GM, nonlinearpressure dependent terms were derived for energy dissipation due to thermaldamping (Td), radiation damping (Rd) and dissipation due to the viscosity ofliquid (Ld) and coating (Cd). The dissipated energies were solved for uncoatedand coated 2- 20 $\mu m$ bubbles over a frequency range of $0.25f_r-2.5f_r$($f_r$ is the bubble resonance) and for various acoustic pressures(1kPa-300kPa). Thermal effects were examined for air and C3F8 gas cores in eachcase. For uncoated bubbles with an air gas core and a diameter larger than 4$\mu m$, thermal damping is the strongest damping factor. When pressureincreases, the contributions of Rd grow faster and become the dominant dampingmechanism for pressure dependent resonance frequencies (e.g. fundamental andsuper harmonic resonances). For coated bubbles, Cd is the strongest dampingmechanism. As pressure increases Rd contributes more to damping compared to Ldand Td. In case of air bubbles, as pressure increases, the linear thermal modellargely deviates from the nonlinear model and accurate modeling requiresinclusion of the full thermal model. However, for coated C3F8 bubbles ofdiameter 1-8 $\mu m$, typically used in medical ultrasound, thermal effectsmaybe neglected even at higher pressures. We show that the scattering todamping ratio (STDR), a measure of the effectiveness of the bubble as contrastagent, is pressure dependent and can be maximized for specific frequency rangesand pressures.

Nonlinear energy loss in the oscillations of coated and uncoated  bubbles: Role of thermal, radiation damping and encapsulating shell at  various excitation pressures