Inequalities for Single Crystal Tube Growth by Edge-Defined Film-Fed Growth Technique

The axi-symmetric Young-Laplace differential equation is analyzed. Solutions of this equation can describe the outer or inner free surface of a static meniscus the static liquid bridge free surface between the shaper and the crystal surface occurring in single crystal tube growth. The analysis concerns the dependence of solutions of the equation on a parameter p which represents the controllable part of the pressure difference across the free surface. Inequalities are established for p which are necessary or sufficient conditions for the existence of solutions which represent a stable and convex outer or inner free surfaces of a static meniscus. The analysis is numerically illustrated for the static menisci occurring in silicon tube growth by edge-defined film-fed growth EFGs technique. This kind of inequalities permits the adequate choice of the process parameter p. With this aim this study was undertaken.