Determination of fluctuations of temperature supercooling and the solidification front in the directional crystallization of a melt under controlled conditions

Abstract

Relations are obtained to calculate transient temperature supercooling at the solidification front in the directional crystallization of a melt.Relations are also obtained to calculate fluctuations in the front with arbitrary perturbations of a prescribed thermal regime.Various modifications of directional crystallization, which includes zone refining, are widely used to obtain an oriented structure (columnar and monocrystalline) in ingots.Other conditions being equal, the degree of perfection of the crystal structure of the resulting material is determined by the stability of heat transfer in the solidification zone.Convenient parameters for analyzing and optimizing the prescribed thermal regime are temperature supercooling at the crystallization front and the position of the front due to this supercooling.The goal of the present study is to obtain relations to calculate these parameters with arbitrary known perturbations of the prescribed thermal regime. We also want to develop a method to evaluate the effect of the perturbing factors on the value of these parameters.We will analyze the conditions under which single crystals of magnetic alloys are grown.The USSR has developed a technology and industrial equipment for producing permanent magnets with a monocrystalline structure on the basis of Fe-Co--Ni--AI [i]. Here, monocrystalline ingots are obtained in special units using inoculation and controlled directional crystallization (CDC) [2] in vertically positioned tubular containers. Figure i shows a diagram illustrating heat exchange in the formation of a monocrystalline structure in the dynamic regime of CDC, i.e.~ with the motion of the thermal component (the heater--cooler block) at a constant speed relative Vo relative to the cylindrical container.The temperature field is assumed to be axisymmetric.The temperature distribution in the gap between the container and the tubular container located coaxially relative to it (for example, along the straight line Zm) can be described by a certain curve Q and a function approximating this curve. For the sake of determinateness, we assume that in the case of equilibrium solidification the crystallization front coincides with the isothermal surface corresponding to the solidus tso I. The remaining assumptions used in constructing the mathematical model of CDC are conrained in [3].The main factors affecting steady-state heat exchange are fluctuations in the power supplied to the heater and mechanical vibrations of the drive of the thermal component. The power oscillations correspond to sinusoidal oscillations of the curve Q in a direction perpendicular to the ingot axis, while the mechanical vibrations of the drive are along the axis [3].Temperature supercooling at the crystallization front will be represented in the form of the sum of two components: 6t = 6t m + 6t t.We will examine the case of crystallization of a unidimensional rod. The derivation of the equation of interest to us is based on the following assumptions: a) there is the nor-~ mal mechanism of crystal g...