Determination of the angle $\alpha_o(x_1,h_1;p)$ appearing in the ribbon growth, when the profile curve is convex

  • Agneta Maria Balint
  • Robert Szabo


In this paper a procedure for the determination of the angle $\alpha _{o}(x_1,h_1;p)$, between the horizontal axis 0X and the tangent line to the free surface of a meniscus (having convex profile curve), at the three-phase point, of coordinates $(x_1,h_1)$, is presented. This angle appears in the system of differential equations, which describes the evolution of the half thickness $x_1$ of the ribbon, grown from the melt by edge-defined film-fed growth (EFG) method. The angle can fluctuate during the growth, due to the fluctuations of the ribbon half-thickness $x_1$, the fluctuations of the crystallization front height $h_1$ and the pressure fluctuations $p$ (of the gas flow introduced in the furnace for release the heat). The knowledge of the function $\alpha _{o}$ is necessary for the simulation of the half thickness evolution of the ribbon, to be grown with a given pulling rate $v$ in specified thermal and pressure conditions. This kind of results can be useful in experiment and manufacturing technology design, since simulation is less expensive as experiment. Numerical results are given for a silicon ribbon.