On the Karma model for spontaneous grain refinement at high solid fractions

In a paper published recently in this journal, Karma set forth the details of a model for spontaneous grain refinement in undercooled melts. In this m odel an isolated cylindrical dendrite undergoes spheroidisation due to the growth of a Rayleigh-like morphological instability with wavelength lambda. In this paper we present an extension to this model in which we assume the dendrite to be part of an array rather than isolated, thus simulating a no n-zero solid fraction following recalescence. We find that the wavelength o f the fast growing perturbation increases with increasing solid fraction, e ventually becoming infinite above some critical value. This defines an unde rcooling, Delta T-c, above which instabilities of the type postulated would no longer grow. In the widely studied system Ni-Cu we find that Delta T-c is close to the upper grain refinement transition temperature, Delta T-0*. Consequently we question whether the Karma model can in fact explain sponta neous grain refinement at high undercoolings.