The effect of mass transfer on the temperature gradient of a crystal growing from melt

The change in the temperature gradient on the crystal side while the rate of crystal growth from melt is varied has long been debated. Abe and Takahashi have recently reported an unambiguous experimental demonstration that the temperature gradient is a decreasing function of the growth rate, which is different from previous theories, experimental results, and widely held notion of other researchers. The present paper provides a theoretical basis for this seemingly peculiar effect of the growth rate on the temperature gradient. The essential matter is the effect of mass transfer, the role of which had been commonly disregarded in old studies. Although the rate of mass transfer is not large compared to that of heat conduction, it is proven that the temperature gradient is subjected to the mass transfer in a definite manner. Our analysis shows that the effect becomes significant when the crystal diameter is large, which is consistent with the experimental observation. Another effect of the mass transfer is the change in the shape of melt/crystal interface. In old studies, the temperature gradient was determined by Stefan's equation; however, this treatment confuses the cause and effect. The temperature gradient should be determined by the fundamental equation of heat conduction. When the gradient is determined in this way, the shape of the melt/crystal interface spontaneously adjusts to satisfy Stefan's equation.