Etudes analytiques et numériques du procédé de Bridgman démouillage: capillarité, transfert de chaleur et stabilité

Abstract : The phenomenon of dewetting is characterized by the Vertical Bridgman growth of a crystal without contact with the walls of the crucible due to the existence of a liquid meniscus at the level of the solid-liquid interface which creates a gap between the crystal and the inner crucible walls. One of the immediate consequences of this phenomenon is the drastic improvement of the crystal quality. This improvement is essentially related to the absence of wall-crystal interaction, so that no grain or twin spurious nucleation can occur and no differential dilatation stresses exist, which could generate dislocations. In order to bring crucial information concerning dewetted phenomenon, detailed theoretical results and numerical simulations are necessary, on the basis of the mathematical models able to reflect better the real phenomenon which should include all essential processes appearing during the growth. The main problem of the dewetting growth and the related improvements of the material quality is the stability of the growth process. In this context, the main purpose of the present work is to perform analytical and numerical studies for capillarity, heat transfer and stability problems of the dewetted Bridgman process. Firstly, the mathematical formulation of the capillary problem governed by the Young-Laplace equation has been presented, followed by analytical and numerical studies for the meniscus equation for the cases of zero and normal terrestrial gravity. Secondly, the heat transfer problems have been treated. Thus, in order to find analytical expressions of the temperature distribution and the temperature gradients in the melt and in the solid, analytical and numerical studies for the non-stationary one-dimensional heat transfer equation have been performed. The melt-solid interface displacement equation was also derived from the thermal energy balance at the level of the interface. Further, for studying the effect of the crystal-crucible gap on the curvature of the solid-liquid interface for a set of non-dimensional parameters representative of classical semiconductor crystal growth, an analytical expression for the interface deflection, based on simple heat fluxes arguments was found. In order to check the accuracy of the obtained analytical formula and to identify its limits of validity, the heat transfer equation was solved numerically in 2D axial symmetry, stationary case, using the finite elements code COMSOL Multiphysics 3.3. Further, the stability analysis has been developed. Different concepts of Lyapunov stability which can occur in shaped crystal growth: classical, uniform, asymptotic, and exponential Lyapunov stabilities of a steady-state; partial Lyapunov stability of a steady-state; and the same types of Lyapunov stabilities for time-dependent regimes, have been presented. In what follows, after the concept of practical stability over a bounded time period has been introduced, analytical and numerical investigations of the practical stability over a bounded time period of the nonlinear system of differential equations describing the melt-solid interface displacement and the gap thickness evolution for dewetted Bridgman crystals grown in terrestrial conditions have been performed.
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Mechanics [physics.med-ph]. Université de Grenoble, 2011. French


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Submitted on : Friday, May 27, 2011 - 6:09:17 PM
Last modification on : Monday, May 30, 2011 - 11:12:28 AM

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Simona-Mihaela Epure. Etudes analytiques et numériques du procédé de Bridgman démouillage: capillarité, transfert de chaleur et stabilité. Mechanics [physics.med-ph]. Université de Grenoble, 2011. French. <tel-00596581>

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