Solvability condition for needle crystals at large undercooling in a nonlocal model of solidification

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Abstract
We show explicitly that, in a realistic model of diffusion-controlled dendritic solidification, Ivantsov’s continuous family of steady-state needle crystals is destroyed by the addition of surface tension. Our starting point is the exact integro-differential equation for the one-sided model, in two dimensions, in a moving frame of reference. In the limit of large undercooling, where the range of the diffusion field is much smaller than the radius of curvature of the tip of the needle, we are able to reduce this problem to a linear, inhomogeneous differential equation of infinite order. We derive a solvability condition for this equation and show that solutions cease to exist for arbitrarily small but finite, isotropic surface tension.