Signal propagation in diffusive molecular communications over a spherical surface

Diffusive molecular communications (DMC) relies on the Brownian motion of dedicated molecules for transmitting information. The signal propagation characteristics in a DMC channel is determined by the geometries of the environment, receiver and the transmitter and their physical and chemical properties, which impose the boundary- and initial conditions on the partial differential equation (PDE) describing the diffusion process. In this work, we investigate the directional signal propagation characteristics for a point source over (or, as a special case, on) a spherical surface, that may be reflective, perfectly absorptive or partially absorptive. We derive the general solution of the PDE for this case, and validate it via particle based simulations. Our results can be employed to determine the signal propagation characteristics in a wide range of practically relevant scenarios, for which we investigate the effect of the system parameters on the directivity of the signal propagation.