Macroscopic models of superconductivity - Chapman, SJ (University of Oxford, 1991)
After giving a description of the basic physical phenomena to be modelled, we begin by formulating a sharp-interface free-boundary model for the destruction of superconductivity by an applied magnetic field, under isothermal and anisothermal conditions, which takes the form of a vectorial Stefan model similar to the classical scalar Stefan model of solid/liquid phase transitions and identical in certain two-dimensional situations. This model is found sometimes to have instabilities similar to those of the classical Stefan model. We then describe the Ginzburg-Landau theory of superconductivity, in which the sharp interface is `smoothed out' by the introduction of an order parameter, representing the number density of superconducting electrons. By performing a formal asymptotic analysis of this model as various parameters in it tend to zero we find that the leading order solution does indeed satisfy the vectorial Stefan model. However, at the next order we find the emergence of terms analogous to those of 'surface tension' and 'kinetic undercooling' in the scalar Stefan model. Moreover, the 'surface energy' of a normal/superconducting interface is found to take both positive and negative values, defining Type I and Type II superconductors respectively. We discuss the response of superconductors to external influences by considering the nucleation of superconductivity with decreasing magnetic field and with decreasing temperature respectively, and find there to be a pitchfork bifurcation to a superconducting state which is subcritical for Type I superconductors and supercritical for Type II superconductors. We also examine the effects of boundaries on the nucleation field, and describe in more detail the nature of the superconducting solution in Type II superconductors - the so-called 'mixed state'. Finally, we present some open questions concerning both the modelling and analysis of superconductors.