Simulation of the pressure distribution in the melt for sapphire ribbon growth by the Stepanov (EFG) technique
Introduction
In order to accomplish an automated crystal growth process by the Stepanov (EFG) technique we studied the influence of the growth process parameters on force sensor readings [1]. It was established that an alteration of the heating power and of the pulling rate which did not lead to freezing of the growing crystal to the die top face caused an abrupt change of force sensor readings that vastly exceeded the increasing weight of a crystal.
From the obtained data it is concluded that along with the forces such as the crystal and meniscus weights, the surface tension force and the force stipulated by hydrostatic pressure, the growing crystal undergoes extra forces caused by hydrodynamic flow of melt in meniscus and capillary regions.
Indeed, to cause melt flow, accompanying growth, it is necessary to apply some force on the side of a crystal. An alteration of pulling rate brings about a redistribution of melt velocity field in the die capillary and in meniscus. Heating power variation influences the meniscus temperature, and, consequently, its height which, in turn, also defines melt velocity field and the corresponding hydrodynamic pressure. Having integrated the hydrodynamic pressure along the melt–solid interface one can find the force stipulated by hydrodynamics.
Stokes equations and appropriate boundary conditions describe the velocity field and the corresponding pressure distribution in the capillary and in the meniscus. The resulting differential equations have been solved by the finite-element method with an appropriate class of functions. When deciding the problem we have made the following assumptions: the crystallization front is flat, hydrodynamic flow has no influence upon the form of fluid meniscus, and melt viscosity is constant.
Section snippets
Mathematical model
Stress acting on a melt surface area on the side of the crystal is written as followswhere is the unit vector, p is the pressure in the melt, μ is the dynamic viscosity of the melt, the flow velocity of melt and Def a deformation rate tensor. Our calculations demonstrate that in Eq. (1)the second item is small so the force stipulated by hydrodynamic pressure p has the main effect on growing crystal.
Let us consider steady-state growth of a ribbon crystal of 2b
Numerical analysis
The convenient numerical approach is the finite element method. This method requires to divide the overall region into subregions of more simple structure and to construct the related basis functions system. While solving the problem of melt transfer it is convenient to use the rectangular elements for approximating the solution using the appropriate set of basis functions [2]. However, in typical problems the overall region is not of a rectangular shape, and mapping it to the rectangular leads
Conclusions
- 1.
The deviation from steady state growth introduces a redistribution of the hydrodynamic pressure in the meniscus and, accordingly, causes the change of the force acting on the crystal. There is a range of meniscus heights where a change of the force stipulated by hydrodynamic pressure redistribution is considerably more than a deviation which the static model describes.
- 2.
In this range the force stipulated by hydrodynamics has the main effect on the crystal, and, accordingly on the force sensor
References (5)
- et al.
J. Crystal Growth
(1999) - et al.
J. Crystal Growth
(1983)
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