The effect of surface tension and kinetic undercooling on a radially-symmetric melting problem

- We study a one-phase Stefan problem for melting a nanoscaled spherical ball.
- We include both surface tension and kinetic effects in the Gibbs-Thomson condition.
- Kinetic undercooling is required to keep the solution regular until complete melting.
- With kinetic effects taken into account, the melting temperature is no longer singular.
- The model with kinetic undercooling is consistent with observations of abrupt melting.

The addition of surface tension to the classical Stefan problem for melting a sphere causes the solution to blow up at a finite time before complete melting takes place. This singular behaviour is characterised by the speed of the solid-melt interface and the flux of heat at the interface both becoming unbounded in the blow-up limit. In this paper, we use numerical simulation for a particular energy-conserving one-phase version of the problem to show that kinetic undercooling regularises this blow-up, so that the model with both surface tension and kinetic undercooling has solutions that are regular right up to complete melting. By examining the regime in which the dimensionless kinetic undercooling parameter is small, our results demonstrate how physically realistic solutions to this Stefan problem are consistent with observations of abrupt melting of nanoscaled particles.