Resumen de Mathematical modelling of diffusion processes at the nanoscale
Helena Ribera Ponsa
In this thesis we study diffusion processes of nanoparticle evolution and develop appropriate models with the aim of being able to optimise their functions according to the needs of industry. Two distinct diffusion processes are studied in detail throughout this thesis: phase change and atomic interdiffusion. To do this we employ various mathematical techniques. The list includes asymptotic analysis, the Heat Balance Integral Method (HBIM), the opTimal HBIM (TIM), similarity variables, separation of variables and numerical methods.
In Chapters 3, 4 and 5 we focus on the phase change problem, also termed the Stefan problem. In Chapter 3 we explore the application of the HBIM to Stefan problems in spherical and cylindrical coordinates. Working with a reduced one-phase model, we use the standard version of this method and one designed to minimise the error. Furthermore, we define coordinate transformations with the aim of improving their accuracy. We compare the results obtained against numerical and perturbation solutions. It is shown that, whilst the results for the cylindrical problem are not excellent, for the spherical case it is possible to obtain highly accurate approximate solutions. In Chapter 4 we present a model for the melting of a spherical nanoparticle that differs from previous ones. This model includes the size dependence of the latent heat and a cooling condition at the boundary. The latent heat variation is modelled by a new relation, which matches experimental data better than previous models. A novel form of Stefan condition is used to determine the position of the melt front. Other features that the model includes are melting point depression and density change in the different phases. For large Stefan numbers we compare the perturbation solution with a numerical one and show that the agreement between them is excellent. Results show faster melting times than previous theoretical models, primarily due to latent heat variation. Chapter 5 links the previous two chapters; we use the optimal exponents found in Chapter 3 in the approximate solution for a simplified one-phase reduction of the model presented in Chapter 4. We study different outer boundary conditions, and then compare the solution given by the TIM with numerical and perturbation solutions for the same problem. Results indicate that the TIM is more accurate than the first order perturbation for all cases studied.
In Chapters 6 and 7 we shift our focus to binary diffusion in solids. In Chapter 6 we detail the mechanisms that drive substitutional binary diffusion via vacancy exchange, and derive appropriate governing equations. Our focus is on the one-dimensional case with insulated boundary conditions. We are able to make analytical progress by reducing the expressions for the concentration-dependent diffusion coefficients for different limiting cases related to the ratio of diffusion rates between species. After carrying out an asymptotic analysis of the problem, and obtaining analytical solutions, we compare them against a numerical solution. We find that these reductions are in excellent agreement in the limiting cases. Moreover, they are valid, within 10%, to the general solution. In Chapter 7 we develop a cellular automata (CA) model to study the problem presented in the previous chapter. Using a very simple state of change rule we are able to define an asynchronous CA model that shows excellent agreement when compared to the solution of the continuum model derived in Chapter 6. This is proven further by taking the continuum limit of the CA model presented and showing that the governing equations are the same as the ones rigorously derived before, for one of the limiting cases. This provides us with a new, simple method to study and model binary diffusion in solids. Further, since the computational expense of the CA model increases with the number of cells, this approach is best suited to small materials samples such as nanoparticles.
