Analytical Modeling of the Growth of Arrayed Needles during Unidirectional Solidification Process under High Temperature Gradient: Part I. Modeling of Array and Growth Theory

Abstract

An approximate analytical description of the unidirectional growth of arrays of needles is proposed for the solidification under high temperature gradients. Temperature and solute distribution for a needle are described by use of functions with exponential increase/decrease and integral exponential functions. Those for arrayed needles are described by the addition of distributions. Then, a modeling of the growth of arrays of needles is developed in order to predict needle dimensions. Local equilibrium conditions are applied to determine the tip radius and unknown coefficients included in descriptions of temperature and solute distribution. Minimum undercooling of needle tip is also applied to select the primary spacing of growing needles. Predicted dimensions are studied and compared with those given by other theoretical predictions. The model predicts the growth rate of transition from planar interface to cellular, which is very close to the growth rate given by the Mullins-Sekerka theory. The predicted dependency of tip radius of curvature on growth rate is similar to the dependency given by the Kurz-Fisher model for needle growth.