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IMA Journal of Applied Mathematics Advance Access published online on December 26, 2008
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxn038
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© The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Micro/nanoparticle melting with spherical symmetry and surface tension

Scott W. McCue{dagger}

School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane QLD 4001, Australia

Bisheng Wu and James M. Hill

Nanomechanics Group, School of Mathematics and Applied Statistics, University of Wollongong, Wollongong NSW 2522, Australia

{dagger} Email: scott.mccue{at}qut.edu.au

Received on February 20, 2008; Revision received September 16, 2008. Accepted on October 23, 2008

The process of melting a small spherical particle is treated by setting up a two-phase Stefan problem. Surface tension is included through the Gibbs–Thomson condition, the effect of which is to decrease the melting temperature as the particle radius decreases. Analytical results are derived via a small-time expansion and also through large Stefan number asymptotics. Numerical solutions are computed with a front-fixing scheme, and these results suggest that the model exhibits finite-time blow-up, in the sense that both the interface speed and the temperature gradient in the solid phase (at the interface) will become unbounded at some time before complete melting. The near-blow-up behaviour appears to be similar to that encountered in the ill-posed problem of melting a superheated solid (without surface tension), and may help explain the onset of abrupt melting observed in some experiments with nanoscaled particles.

Keywords: two-phase Stefan problem; size-dependent melting; surface tension; finite-time blow-up; superheating.


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