Vibrations and damping of the eigenmodes of viscoelastic nanospheres with thermal conductivity

In this paper, we derive a series of exact analytical closed expressions to calculate the complexeigenfrequencies and the displacement for the corresponding eigenmodes of a viscoelastic (nano)sphere in thepresence of linear damping. Where possible, we provide closed expressions for damping rates, including thecontributions from viscosity, as well as thermal conductivity and solutions of the heat equation. We assume anisolated system, such that no energy/heat transfer to the environment is allowed. We find monotonic behaviorof the damping as a function of frequency for breathing and torsional modes, however, for spheroidal modes wefind nonmonotonicity. Furthermore, we analytically analyze the thermodynamic limit for all mode types. Wealso investigate the frequency shift and find expected behavior, i.e., a reduced eigenfrequency with damping thanwithout damping for breathing and torsional modes. For spheroidal modes, however, we find nonmonotonicshifts, corresponding to the damping. For some eigenfrequencies, we find anomalous frequency shifts.