Sintering behavior of two roughened crystals just after contact

We consider two spherical, roughened crystals with approximately isotropic surface free energy which are brought into contact and begin to sinter. We argue that the geometry immediately post-contact is two dimensional and Cartesian and can be approximated by the evolution of a slot-shaped cavity. On this basis, we construct travelling wave solutions for the crystal shape in the limits of bulk diffusion limited and surface diffusion limited kinetics. These solutions are then used to calculate novel scalings for the neck size as a function of time $t$ after contact: We predict that neck size is proportional to $t^{1/4}$ for the bulk diffusion limited case and (following a single pinch-off event) approximately proportional to $t^{1/3}$ for the surface diffusion limited case.