A procedure for the determination of the angles (alpha)over-tilde(e)(r(e), h; p(e)) and (alpha)over-tilde(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

作者：Balint Stefan; Tanasie Loredana^*

来源：Nonlinear Analysis: Real World Applications , 2010, 11(5): 4043-4053.

DOI：10.1016/j.nonrwa.2010.03.010

• Nonlinear system of differential equations
• Nonlinear boundary value problems
• Tube growth dynamics

In this paper, a procedure for the determination of the angles (alpha) over tilde (e)(r(e), h; p(e)) ((alpha) over tilde (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 r(e) (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 (alpha) over tilde (e)(r(e). h; p(e))-(pi/2 - alpha(g)), ((alpha) over tilde (i)(r(i), h; p(i))-(pi/2 - alpha(g))). where alpha(g) 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 (alpha) over tilde (e)(r(e). h; p(e)), (alpha) over tilde (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 is 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 p(e), p(i). These kinds of result are useful in experiment planning and manufacturing technology design. Numerical results are given for a silicon tube.

• 出版日期2010-10