Degree Name

Doctor of Philosophy


School of Mathematics and Applied Statistics, Faculty of Informatics


This thesis presents a mathematical modelling in nanotechnology. Many ex- periments and molecular dynamics simulations demonstrate that the melting point of nanoparticles shows a size-dependent characteristic in the nanoscale. Based on the assumption that the material is a pure one, the melting process of spherical and cylindrical particles, especially nanoparticles, is treated as a Stefan moving boundary problem. Analytical or semi-analytical approaches, such as small-time perturbation expansions with front-¯xing techniques, large Stefan number limit, integral iterative scheme, and numerical methods, such as enthalpy scheme and front-¯xing method, are applied to the one- and or two- phase Stefan problem in spherical and cylindrical domains by taking into account the e®ect of the interfacial or surface tension. The results from these methods are compared and show excellent agreement to some ex- tent. This thesis may provide a possibility of explaining some interesting phenomena occurring in the physical experiments, i.e. superheating and \abrupt melting", or work as a guide for the potential applications of nanoparticles, for example, drug delivery, nanoimprinting and targeted ablation of tumor cells In Chapter 1, a simply survey of the research background is given. Chapter 2 studies the full classical two-phase Stefan problems without surface or interfacial tension. By using the approach from large Stefan number limit and small-time perturbation methods, long- and short-time solutions are obtained, and the results from these methods are compared with the numerical enthalpy scheme. The limits of zero Stefan-number and slow di®usion in the inner core are also noted. Chapter 3 presents the melting of a spherical or cylindrical nanoparticle by including the e®ects of surface tension through the Gibbs-Thomson condition. A single-phase melting limit is derived from the general two-phase formulation, and the resulting equations are studied analytically in the limit of small time and large Stefan number. Further analytical approximations for the temperature distribution and the position of the solid-melt interface are found by applying an integral formulation together with an iterative scheme. All these analytical results are compared with numerical solutions obtained using a numerical front-¯xing method, and they are shown to provide good approximations in various regimes. In Chapters 4 and 5, the methods used in above sections are extended to the melting problem for spherical and cylindrical ii nanoparticles, respectively. The results from these approaches are compared with those from the numerical front-¯xing method. The original contributions of this thesis are: approximate analytical solutions are obtained for the classical two-phase Stefan problems in a spherical domain; a general single-phase limit for the melting of nanoparticles are derived and analyzed with the correct boundary conditions; a critical radius is found to exist for the blow-up of the one-phase melting; the melting process of spherical and cylindrical nanoparticles are studied analytically from the perspective of Stefan moving boundary problem by including the e®ect of surface tension; some interesting phenomena observed in physical experiments, i.e. superheating and \abrupt melting", are explained in terms of Stefan problems.

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