Dendrites in the Small Undercooling Limit

Dendrites of a solid growing steadily in an undercooled melt have a shape given by the solution of certain nonlinear integrodifferential equations. If one does not take account of the Gibbs‐Thomson condition, these equations have an exact solution due to Ivantsov and to Cahn and Horway. For elementary dimensional reasons, the tip velocity in these solutions is left undetermined. This degeneracy is removed by the effect of interface curvature on the melting temperature (Gibbs‐Thomson effect). We show that, in the physically relevant limit of small undercooling, the shape of the dendrite can be deduced from parameterless similarity equations, where the Gibbs‐Thomson effect is included. Every physical quantity is known in this limit, up to a nonlinear eigenvalue that has to be found numerically.