The morphology and growth process of oxide precipitates in Czochralski silicon have been studied with prolonged thermal treatments up to 700 h at intermediate temperatures (700–900 °C). It was found with transmission electron microscopy observation that (i) the morphology of precipitates changes from platelet to aggregation of polyhedra at both 800 and 900 °C during isothermal heat treatment, and (ii) the growth of platelet precipitates follows a t1/2 law.

SynopsisTo quantify the solute redistribution during solidification, an approximate solution of the Brody-Flemings model of dendritic solidification has been derived on the assumption of parabolic solidification and a constant partition ratio, and compared with an exact solution of the model. It has been shown that the approximate solution has good accuracy.Further, on the basis of the derived equation, an extended mathematical model is proposed, incorporating a thermal model of solidification into the analysis without assumption on the solidification rate. The extended model includes the treatment of multi-component alloy and of 1 transformation, the temperature dependence of diffusion coefficients and the estimation of the diffusion path. The extended model provides the results consistent with the available data.Key words: mathematical modeling; solidification; segregation; sion; carbon steel. dif f u- I. IntroductionIn the earliest description with the Scheil equation for solute redistribution during solidification, the diffusion of solute into solid phase (back diffusion) was neglected.1'2~ Brody and Flemings3~ have extended the theory to include the back diffusion effects based on their solidification model; the model assumed 1) complete diffusion in liquid phase and incomplete back diffusion, 2) a plate-like dendrite geometry, 3) a constant diffusion coefficient, and 4) parabolic or linear solidification rate. Both approximate equations derived by Brody and Flemings for linear and parabolic solidification rate are, however, invalid for rapid diffusion in the solid phase.Clyne and Kruz,4~ and Ohnaka51 have modified the Brody and Flemings equation for parabolic solidification rate; both the modifications are valid for infinite and infinitestimal diffusion coefficient. Further, the present author has derived an exact solution of the solidification model of Brody and Flemings for parabolic solidification rate, and has shown that the accuracy of all the approximate solutions mentioned above are often bad.6~ Although the solidification model of Brody and Flemings with parabolic growth has been rigorously solved, several problems still remain in applying the model to solidification processes : 1) calculation of the exact solution is uneasy because the solution has an infinite series of the confluent hypergeometric functions, 2) no realistic method has been found to estimate the solidification rate or the local solidification time, of which value is necessary for the calculation,

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An exact solution and an approximate solution have been derived for the columnar dendrite model proposed by Ohnaka,8~ in which a solidification rate, equilibrium distribution coefficient and diffusion coefficient in solid are assumed to be constant throughout the solidification. By comparing with the exact solution, it has been shown that the approximate solution has good accuracy. Further the model is extended on the basis of the derived approximate solution by incorporating a thermal model of solidification. The extended model includes multi-component alloy accompanying phase transformation in solid and temperature dependence of diffusion coefficients, and excludes the assumption of a given solidification rate. The calculated results of the extended columnar dendrite model agreed well both with the available experimental data and the calculated results of the extended planar dendrite model proposed previously. The assumption of the solidification geometry is unimportant for quantifying the microsegregation effects.