径向收敛流中Burton-Prim-Slichter方法的失效和横向溶质偏析
Breakdown of Burton-Prim-Slichter approach and lateral solute segregation in radially converging flows
摘要：
本文从理论上研究了在简单的凝固模型中，径向收敛的熔体流（即远离凝固前沿的熔体流）对径向溶质偏析的影响。我们表明，经典的伯顿-普里姆-Slichter（BPS）的解决方案描述的溶质纳入到凝固材料的发散流的效果打破了沿凝固前沿收敛的流量。该故障是由定义有效边界层厚度的积分发散引起的，有效边界层厚度是BPS理论的基本概念。虽然这样的发散可以正式避免限制熔体的轴向延伸到一层有限的高度，径向均匀的溶质分布是可能的，只有弱熔体流的轴向速度远离凝固前沿可比的增长率。对于每种生长速率都有一个临界熔体速度，在该速度下，溶液通过奇点，并在更强的熔体流动中变得物理上不一致。为了解决这些矛盾，我们考虑了一个有限半径$R_0$圆盘在强收敛熔体流作用下的凝固前沿，得到了一个解析解，表明径向溶质浓度随$R_0$半径的变化关系为$\sim\ln^{1/3}(R_0/r)$和$\sim\ln(R_0/r)$，浓度的对数增长在对称轴附近受到扩散的限制，在对称轴附近的扩散系数变为在与溶质边界层的特征厚度相当的距离处有效。会聚流导致溶质堆积，在对称轴处形成对数浓度峰，这对于晶体生长过程可能是不期望的特征。
英文摘要：
A theoretical study is presented of the effect of a radially converging melt flow, which is directed away from the solidification front, on the radial solute segregation in simple solidification models. We show that the classical Burton-Prim-Slichter (BPS) solution describing the effect of a diverging flow on the solute incorporation into the solidifying material breaks down for the flows converging along the solidification front. The breakdown is caused by a divergence of the integral defining the effective boundary layer thickness which is the basic concept of the BPS theory. Although such a divergence can formally be avoided by restricting the axial extension of the melt to a layer of finite height, radially uniform solute distributions are possible only for weak melt flows with an axial velocity away from the solidification front comparable to the growth rate. There is a critical melt velocity for each growth rate at which the solution passes through a singularity and becomes physically inconsistent for stronger melt flows. To resolve these inconsistencies we consider a solidification front presented by a disk of finite radius $R_0$ subject to a strong converging melt flow and obtain an analytic solution showing that the radial solute concentration depends on the radius $r$ as $\sim\ln^{1/3}(R_0/r)$ and $\sim\ln(R_0/r)$ close to the rim and at large distances from it. The logarithmic increase of concentration is limited in the vicinity of the symmetry axis by the diffusion becoming effective at a distance comparable to the characteristic thickness of the solute boundary layer. The converging flow causes a solute pile-up forming a logarithmic concentration peak at the symmetry axis which might be an undesirable feature for crystal growth processes.
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