1. Coherency loss When a spherical precipitate nucleates and begins to grow in a a matrix, they have identical atomic structure but with a misfit given by S. (a) Write down two expressions for the Gibbs free energy (one for a coherent interface and one for a noncoherent interface) in terms of r, & and any other materials constants that you think are necessary. Please be specific about the types of interfacial energy. (b) Following is a graph of the Gibbs free energy versus radius of the b precipitate, showing two different types of interfaces. Label each curve with the type of interface. AG 0 "crit (c) Provide both a mathematical and physical argument for why the interface evolves from one type to another type with increasing r. (d) Find r critical, the value of the radius when the crossover from one type of interface to the other type takes place.