Theoretical and Numerical Study of Microstructure Formation in Multi-component Alloys
The length scale, composition, orientation of the phases in a material constitutes its microstructure. In order to appreciate the whole gamut of microstructures observed in multi-component alloys, an identification of the functional dependencies of different micro structural parameters like the length scales, phase compositions, etc., on thermodynamic (like driving forces) and kinetic (like solute diffusivities) factors is critical. In this thesis, we seek to understand such dependencies theoretically (analytically and numerically) for solidification (diffusion-controlled transformations) involving two and more phases in multi-component alloys. We begin with a problem where a single solid phase forms from the multi-component liquid melt with a stable solidification front. We develop an analytical theory as an extension to Zener's theory for binary alloys to predict the interfacial compositions in either phase, alongside the velocity of the interface and the composition pro le in the liquid. The solidification interface is usually susceptible to perturbations leading to the most commonly observed structures of dendrites. We attempt to understand this phenomenon by first developing a multi-component extension to the Mullins-Sekerka type linear stability analysis of the solidification front. Here, we present analytical expressions which allow us to calculate the length scales selected due to the instability as functions of solute diffusivities and driving forces. A theoretical study of the product of such an instability is presented next, where we extend the LGK theory to multi-component systems to study the steady-state growth of dendrites into a uniformly undercooled melt. Our theory is able to predict the selection of the dominant length scale in the problem, which is the dendrite tip radius, in addition to the tip-growth velocity and the compositions of the solid and the liquid phases. For multi-phase solidification we follow a similar route and begin by developing a generalized Jackson-Hunt type analysis to predict the undercoolings, solid-phase fractions and phase compositions as functions of lamellar widths, during eutectic solidification of any number of solid phases from a multi-component melt. The instability of the eutectic solidification front leads to structures known as eutectic colonies. In this thesis, we present a study of the colony dynamics and the constituent lamellar morphologies in the presence of anisotropic solid-liquid and solid-solid interfacial energies through phase- field simulations.