Aspherical growth of the inner core has been suggested as a mechanism to produce seismic anisotropy through alignment of crystal lattices. This mechanism is viable if the response to aspherical growth occurs by slow viscous deformation. The inner core can also respond by melting and solidification at the boundary if flow in the liquid core can redistribute latent heat over the surface. We use a numerical geodynamo model to quantitatively assess the process of melting and solidification, and find that the response to aspherical growth occurs primarily through phase transitions when the viscosity of the inner core is 1021Pas or higher. A lower inner-core viscosity favors viscous adjustment, but the associated stresses may be too low to produc...