The Boundary Integral Equation for Kinetically Limited Dendrite Growth

The boundary integral equation defining the interface function for a curved solid/liquid phase transition boundary is analytically solved in steady-state growth conditions. This solution describes dendrite tips evolving in undercooled melts with a constant crystallization velocity, which is the sum of the steady-state and translational velocities. The dendrite tips in the form of a parabola, paraboloid, and elliptic paraboloid are considered. Taking this solution into account, we obtain the modified boundary integral equation describing the evolution of the patterns and dendrites in undercooled binary melts. Our analysis shows that dendritic tips always evolve in a steady-state manner when considering a kinetically controlled crystallization scenario. The steady-state growth velocity as a factor that is dependent on the melt undercooling, solute concentration, atomic kinetics, and other system parameters is derived. This expression can be used for determining the selection constant of the stable dendrite growth mode in the case of kinetically controlled crystallization.