A procedure for the determination of the angles α? _e(r_e,h;p_e) and α?_i(r _i,h;p_i) which appear in the nonlinear system of differential equations describing the dynamics of the outer and inner radii of a tube grown by the edge-defined film-fed growth (EFG) technique

摘要

In this paper, a procedure for the determination of the angles α_e(r_e,h;p_e)(α_i(r _i,h;p_i)) between the tangent line to the outer free surface (inner free surface) of the meniscus at the three-phase point, of coordinates (r_e,h)((r_i,h)) and the horizontal axis 'Or' is presented. These functions appear in the system of differential equations which describe the evolution of the outer radius re (inner radius r_i) of a tube grown from the melt by the edge-defined film-fed growth (EFG) technique. These angles can fluctuate during the growth. The deviation of the tangent to the crystal outer (inner) free surface at the triple point from the vertical is the difference α_e(r_e,h;p_e)-(π/2- αg), (α_i(r_i,h;p_i)-(π/2-αg)), where αg is the growth angle. The deviation can also fluctuate, and the outer (inner) radius r_e(r_i) is constant when the deviation is constant equal to zero. The knowledge of α_e(r _e,h;p_e), α_i(r_i,h;p _i) is necessary for the analysis of the nonlinear system of differential equations describing the dynamics of the tube growth and for the determination of the interface height h and of the outer and the inner radii r_e, r_i, respectively, of the tube to be grown with a given pulling rate v in specified thermal and pressure conditions pe, pi. These kinds of result are useful in experiment planning and manufacturing technology design. Numerical results are given for a silicon tube.