Vacancy supersaturation in rapidly solidified metal droplets

A self-consistent theory for vacancy entrapment in rapidly solidified metal droplets is presented. Supersaturation occurs when excess (nonequilibrium) vacancies created at the solidification front by the liquid-solid density difference are unable to diffuse back to the interface before the droplet solidifies. The model consists of heat-conduction equations for the liquid and solid phases, a vacancy-diffusion equation, and boundary conditions at the internal and external surfaces that provide coupling between regions and generate the interface dynamics and vacancy entrapment. Solutions to this system of equations are derived in the form of a set of integral equations that incorporate the boundary conditions as integral kernels. These are evaluated numerically to produce temperature and vacancy-concentration profiles for rapidly solidified droplets of various sizes, initial undercoolings, and convective cooling rates. For undercooled, micrometer-sized metal droplets, the model gives vacancy concentrations at solidification far in excess of equilibrium values.