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 complex eigenfrequencies and the displacement for the corresponding eigenmodes of a viscoelastic (nano)sphere in the presence of linear damping. Where possible, we provide closed expressions for damping rates, including the contributions from viscosity, as well as thermal conductivity and solutions of the heat equation. We assume an isolated system, such that no energy/heat transfer to the environment is allowed. We find monotonic behavior of the damping as a function of frequency for breathing and torsional modes, however, for spheroidal modes we find non-monotonicity. Furthermore, we analytically analyze the thermodynamic limit for all mode types. We also investigate the frequency shift and find expected behavior, i.e., a reduced eigenfrequency with damping than without damping for breathing and torsional modes. For spheroidal modes, however, we find non-monotonic shifts, corresponding to the damping. For some eigenfrequencies, we find anomalous frequency shifts.