Static response and stability of coated microbubbles—multiplicity of solutions and parameter estimation

Abstract

The static response of a coated microbubble subject to an external pressure distribution is investigated, in order to identify different response patterns with varying viscoelastic properties of the shell. Theoretical and numerical analysis of the axisymmetric response of a microbubble is performed via the static force balance, in order to obtain the radial and tangential (polar) displacements of a shell subject to a uniform or point load. The stretching and bending stiffnesses of the shell, along with the compressibility of the internal gas, comprise the resistance to deformation of the microbubble. The finite element methodology, with B-splines as basis functions, is employed for the solution of the nonlinear static problem while Newton’s iterations provide the converged solution. The Jacobian matrix provides necessary information regarding stability of the emerging static configurations. The buckling instability of a uniformly loaded shell results in a subcritical bifurcation that is characterized by symmetric/asymmetric shapes for the parameter range pertaining to polymeric/phospholipid shells. As the relative importance of bending stiffness with respect to stretching decreases symmetric shapes determine the primary buckling instability. Strain softening shell behavior conforms to this pattern due to the increase of the effective area dilatation modulus during compression. Increasing the resistance to compression forces the asymmetric and symmetric solution families to terminate at larger bubble volumes. When a point load is considered the force deformation curve is characterized by a transition from a linear Reissner-type to a nonlinear Pogorelov-type response, followed by a regime where resistance to compression dominates. Identifying these regimes in atomic force microscopy measurements can be used for estimating the area dilatation and bending modulus of the shell.