# Non-equilibrium growth of a binary alloy

We present an interpretation of the phase diagram for a binary alloy from the point of view of reaction dynamics. We consider a model system in which the two-component liquid phase is an ideal solution, but the solid phase can be non-ideal, with its non-ideality increasing with reducing temperature. We show how a ‘batch’ model for the evolution of the two-component solid–liquid system, in which the interfacial growth rates are proportional to free energy differences, leads to a set of four differential equations, whose equilibria correspond to the familiar solidus and liquidus curves. In addition, we explain how the transitions between them depend on the bifurcation structure of the solutions. We show that locally stable mixed-phase solutions can exist below the eutectic temperature, thus providing an alternative explanation for the observation of super-cooled liquids, and we also explain why complete solidification ‘normally’ occurs below the eutectic temperature, despite the fact that this temperature has no intrinsic dynamic significance, being simply a value at which two completely different equilibrium states happen to share a common liquid concentration

## Funding

### Applied mathematical modelling applied to enterprise, science and technology (MACSI)

Science Foundation Ireland

Find out more...## History

## Publication

Journal of Applied Mathematics 87(3), pp. 354–379## Publisher

Oxford University Press## Also affiliated with

- MACSI - Mathematics Application Consortium for Science & Industry

## External identifier

## Department or School

- Mathematics & Statistics