Weak Stefan Formulation for Bulk Crystal Growth with Non-smooth Interfaces

TL;DR

This paper introduces a weak formulation of the Stefan problem for bulk crystal growth, enabling accurate simulation of non-smooth interfaces and revealing new insights into the limitations of pull speed and cooling methods.

Contribution

It proposes a novel weak Stefan formulation with finite volume discretization to handle non-smooth interfaces in crystal growth models.

Findings

Weak formulation avoids singularities at non-smooth interfaces.

Simulation shows a maximum pull speed limit in silicon ribbon growth.

Gas cooling enhances the maximum achievable pull speed.

Abstract

Most heat transfer models for bulk crystal growth rely on the classical Stefan formulation to evaluate interface motion during phase change. However, when the interface is non-smooth the use of the classical Stefan formulation may lead to singularities. To address this problem, we propose a simulation model based on the weak formulation of the Stefan problem. Numerical solutions of the weak Stefan formulation are obtained using the finite volume method. This approach provides an energy conserving discretization scheme that accurately evaluates heat transfer around non-smooth interfaces. We apply the weak formulation to numerically simulate the solidification of silicon in the horizontal ribbon growth process. Results exhibit a limitation on the ribbon's pull speed, which previous classical Stefan models have failed to demonstrate. A comparison of heat transfer between radiation and gas cooling shows that gas cooling increases the pull speed limit for the same amount of heat removed.