Microfluidics and Nanofluidics

, Volume 18, Issue 2, pp 233–243 | Cite as

A mathematical model for nanoparticle melting with density change

Research Paper


The melting process of a spherical nanoparticle is analysed using a mathematical model derived from continuum theory. The standard model for macro-scale melting is modified to include melting point depression using the Gibbs–Thomson equation. The key difference between the current and previous work in the melting of nanoparticles is that the difference in densities between the solid and liquid phases is accounted for. This modifies the energy balance at the phase change interface to include a kinetic energy term, which then changes the form of the equation, and it also requires an advection term in the heat equation for the liquid phase. Approximate analytical and numerical solutions are presented for the melting of particles in the range 10–100 nm. It is shown that when the density difference is included in the model, melting is significantly slower than when density is assumed constant throughout the process. This is attributed to the flowing liquid providing a sink term, namely kinetic energy, in the energy balance. The difference in results is greatest for small particles; however, it is concluded that the varying density model will never reduce to the constant density model resulting in a difference of around 15 % even at the macro-scale.


Phase change Nanoparticle Expansion Melting 



The research of TGM was supported by a Marie Curie International Reintegration Grant Industrial applications of moving boundary problems Grant No. FP7-256417 and Ministerio de Ciencia e Innovación Grant MTM2011-23789. FF acknowledges the support of a Centre de Recerca Matemàtica PhD grant. SLM acknowledges the support of the Mathematics Applications Consortium for Science and Industry (MACSI, www.macsi.ul.ie) funded by the Science Foundation Ireland Mathematics Initiative Grant 12/IA/1683. The authors acknowledge Gloria Garcia for her contribution in the art work of the manuscript.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Centre de Recerca MatemàticaBellaterraSpain
  2. 2.Departament de Matemàtica Aplicada IUniversitat Politècnica de CatalunyaBarcelonaSpain
  3. 3.MACSI, Department of Mathematics and StatisticsUniversity of LimerickLimerickIreland

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