Solid-Liquid Interface Energy between Silicon Crystal and Silicon-Aluminum Melt
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I.
INTRODUCTION
THE solid-liquid interface energy between a crystal and its melt is an important physical parameter for research of the nucleation and growth of crystal. This energy influences the nucleation undercooling,[1,2] the growth undercooling,[1,2] the nucleation rate,[1,2] the growth rate,[2] the solid-liquid interface structure,[3,4] the growth mode,[3,4] and the solidification structure.[5,6] Therefore, in order to comprehend the essences of the nucleation and growth of crystal, it is necessary to achieve a clear knowledge of the solid-liquid interface energy. Over the past several decades, more and more attention has been paid to the research of the solidliquid interface energy.[1,7–26] The nucleation-undercooling method[1,7,8] and grain-boundary method[9–15] are the common methods to measure the solid-liquid interface energy of materials. By measuring the nucleation undercooling, the solid-liquid interface energy can be estimated from the homogenous nucleation equation.[1,2,7,8] The accuracy of this method depends on whether the measured undercooling is for a homogenous nucleation. Unfortunately, for a long time, the process of predicting whether a nucleation is a homogenous one was not understood. Obviously, the result of the solidliquid interface energy from the nucleation-undercooling method has some uncertainty. By measuring the shape of the solid-liquid interface and the temperature, ZENGYUN JIAN and FANGE CHANG, Professors, and XIAOQIN YANG, Research Fellow, are with the School of Materials and Chemical Engineering, Xi’an Technological University, Xi’an 710032, People’s Republic of China. Contact e-mail: jianzengyun@ yahoo.com WANQI JIE, Professor, is with the State Key Laboratory of Solidification Process, Northwestern Polytechnical University, Xi’an, 710072, People’s Republic of China. Manuscript submitted September 15, 2009. Article published online April 22, 2010 1826—VOLUME 41A, JULY 2010
the solid-liquid interface energy can be calculated from the Gibbs–Thomson equation. This grain-boundary method has been wildly used to directly measure the solid-liquid interface energy of eutectic alloy at the eutectic temperature[9–15] and peritectic alloy at the peritectic temperature.[16,17] The limitation of the grainboundary method is that it can only be used to measure the solid-liquid interface energy of alloy at some specific composition at some restricted temperature. Recently, by introducing a parameter r0, which is the solid-liquid interface energy at the melting point under the condition that the interface is supposed as a perfectly smooth one, Jian et al.[3,26] developed a model of the solid-liquid interface energy for lateral growth materials such as semiconductors: 2rAs 1:5 ab2 2xð1 xÞab ¼ 1:5 þ ln 0:5 RT
½1
2r0 As 2DS 1:5 ¼ ab 2xð1 xÞa þ R 1:5 þ ln 0:5 RTm
½2
a¼
lnð1 xÞ ln x 1 2x
" b ¼ 1 þ 2xð1 xÞ þ 2
# x ln x þ ð1 xÞ lnð1 xÞ lnð1xÞln x 12x
DS ¼ Sl Ss DSf
½3
½4
½5
where r is the solid-liquid interface energy, x is the thermodynamic equilibrium fract
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