Resumen de Diffusion processes and nanoparticle growth
Claudia Fanelli
It is well-known that many properties of nanoparticles, such as luminescence, photostability, optical radiation efficiencies and electric properties, are size dependent. Hence, the ability to create nanoparticles of a specific size is crucial. In this thesis, we begin by developing mathematical models for the nanoparticle growth process and so obtain guidelines for efficient growth strategies. Once the growth process is understood we move on to a specific practical application of nanoparticles, namely targeted drug delivery. In the first part, the mathematical model analysed is a non-standard Stefan problem where the moving boundary is the surface of the particles. In the second part, the model involves the motion of a non-Newtonian nanofluid subject to an external magnetic field and an advection-diffusion equation for the concentration of the nanoparticles in the fluid. In both cases we employ several mathematical tools, such as similarity solutions, asymptotic analysis and numerical techniques.
In Chapter 2 we work on a simple but representative Stefan problem with constant boundary values by means of analytical and numerical methods in order to identify the key mathematical aspects of this type of problem. In Chapter 3 the standard model for the growth of a single nanoparticle in solution is presented and analysed using the techniques developed in the previous chapter. Particular attention is paid to the validity of the assumptions regularly made in literature. Specifically, the analysis of the diffusion boundary layer shows how the standard model does not hold at early times, while the pseudo-steady assumption is found to be valid. Moreover, within experimental error a new analytical solution for the particle radius depending only on two independent parameters is determined. This demonstrates that the model is unable to distinguish between diffusion and reaction driven growth. In Chapter 4 the model of Chapter 3 is extended for a system of N particles, where N is arbitrarily large. By non-dimensionalising the system and identifying dominant terms, the problem is reduced and solved by analytical and numerical techniques. The Gibbs-Thompson equation for the solubility of the particles shows the importance of this effect in order to control Ostwald ripening, which is driven by the delicate balance between the bulk concentration and the particles solubility. The comparison with experimental data and the analytical solution found in the previous chapter shows excellent agreement, giving an important tool to control the particle size distribution and optimise strategies for the growth.
The second part of the thesis deals with a practical use of nanoparticles, the promising medical technique of magnetic drug targeting. In Chapter 5 a mathematical model for the transport of drug nanocarriers in the bloodstream under the influence of an external magnetic field is presented. Simplifications of the geometry allows the reduction of the Navier-Stokes equations for the blood flow. Within the restrictions of these simplifications analytical solutions are obtained. The comparison between the Newtonian and non-Newtonian approximations shows the importance of taking into account the shear-thinning behaviour of the blood when modelling drug delivery. In this scenario, the viscosity of the blood, which changes depending on the shear rate, is crucial in the calculation of the velocity of the magnetic particles in the vessel and non-Newtonian models need to be used. The ultimate goal is to determine strategies to maximise drug delivery to a specific site.
