STEFAN PROBLEMS FOR MELTING NANOSCALED PARTICLES

JULIAN M. BACK
2015 Bulletin of the Australian Mathematical Society  
Stefan problems are moving boundary problems that model how pure materials undergo phase transitions due to the conduction of heat and exchange of latent heat energy. Their solution involves satisfying the heat equation in each phase (liquid and/or solid, say), subject to specific boundary conditions on the moving phase interface. As the location of the moving boundary is unknown in advance, Stefan problems are nonlinear. Despite appearing relatively simple, the solutions can display highly
more » ... licated behaviour leading to many theoretical studies, but also give accurate results for a wide range of industrial applications in physics and engineering. McCue et al. [10] study a particular two-phase Stefan problem for a melting nano-sized particle. Their model includes the Gibbs-Thomson equation: a nanoscale condition where the melting temperature is dependent on the particle size, and includes surface tension effects (see also [5, 14] ). They find that the inner solid core becomes superheated in the sense that the temperature of the solid is everywhere greater than the size-dependent melting temperature (albeit less than the bulk melting temperature), and is quickly followed by finite-time blow-up of the solution. An infinite temperature gradient develops at the moving boundary and the melting speed becomes unbounded. As with all types of finite-time blow-up, these predictions are unphysical, which is strange as the original problem without the Gibbs-Thomson effect is well posed. This mathematical blow-up suggests the model is incomplete. In Chapter 3 (and [2]) we investigate two related moving boundary problems, both of which are particularly useful in demonstrating the key properties of a melting
doi:10.1017/s0004972715001008 fatcat:gp27b5fx3nfanlpxjrrfbtyj5u