The effects of an elastic solid surface layer on the radial pulsations of gas bubbles

Most previous theoretical investigations of gas bubble dynamics have assumed an uncontaminated gas–liquid interface. Recently, however, the potential importance of layers of surface active agents on bubble dynamics has been increasingly recognized. In this work it is assumed that a continuous layer of incompressible, solid elastic material separates the gas from the bulk Newtonian liquid. Elasticity is modeled to include viscous damping. A Rayleigh–Plesset-like equation describing the dynamics of such surface-contaminated gas bubbles is derived. The equation predicts that the surface layer supports a strain that counters the Laplace pressure and thereby stabilizes the bubble against dissolution. An analytical solution to this equation which includes both the fundamental and second-harmonic response is presented. The dispersion relation describing the propagation of linear pressure waves in liquids containing suspensions of these bubbles also is presented. It is found that (1) the resonance frequencies of individual bubbles tend to increase as the modulus of rigidity increases; (2) the damping provided by the viscosity of the shell dominates thermal effects for bubble radii less than ∼10 μm; (3) the attenuation coefficient in a bubbly liquid decreases as either the rigidity or the viscosity of the surface layer increases; (4) encapsulated bubbles with shell rigidity greater than ∼85 MPa provide a greater total scattering cross section per unit attenuation in the lower biomedical frequency range than do free bubbles of the equivalent size.