Scaling analysis of convective solute transport and segregation in Bridgman crystal growth from the doped melt

Abstract

To cite this version:D. Camel, J.J. Favier. Scaling analysis of convective solute transport and segregation in Bridgman crystal growth from the doped melt. Journal de Physique, 1986, 47 (6)

Thermodynamics and kinetics of dissolutive wetting of Si by liquid Cu

“…It therefore has to be assumed that the flow cannot be laminar, in which case the proportionality constant between u N (d) and d 2 is given as the ratio of the bulk velocity u B to the square of the hydrodynamic boundary layer d H . Invoking again the scaling arguments developed in Camel and Favier [28] the ratio d H /h should scale as Re À1/2 . Setting h = 600 lm and m = 4 Â 10 À7 m 2 s À1 , the resulting algebraic equation yields a value for u B of about 0.35 m s À1 , which results in a Reynolds number slightly above 500, confirming that the flow in the drop is indeed inertial.…”

“…Therefore, consistency arguments require the bulk velocity to be much larger than the above mentioned 2-5 Â 10 À3 m s À1 . More precisely, as the tangential component of the convective velocity is expected to increase linearly with the distance to the solid front d, continuity arguments developed for a related solidification problem by Camel and Favier [28] show that the normal velocity should scale with d 2 . In laminar flows, the proportionality coefficient between u N (d) and d 2 is given as the ratio of the bulk velocity u B to the square of the smallest macroscopic dimension of the drop, here its height h. Even with conservative estimates h = 600 lm, d = 2 lm and u N (d) = 2 Â 10 À3 m s À1 would result in a u B value of 180 m 2 s À1 .…”

Directional solidification of Al–1.5wt% Ni alloys under diffusion transport in space and fluid-flow localisation on earth

“…Indeed, it is mainly the flow in the solute boundary layer that, by penetrating into a thin region of the mush beneath the envelope of the cell/dendrite tips, also washes nickel out of the mush. Knowing Dh, estimates can be obtained for the radial temperature gradient, G R ¼ G Dh=R, and the Grashoff number which gives the strength of thermal convection [23], Gr ¼ b T gG R R 4 =n 2 with b T ( ¼ 1 Â 10 À4 /K) the coefficient of thermal expansion of the melt, g the acceleration of gravity and n (¼ 4 Â 10 À7 m 2 =s) the kinematic viscosity. For G ¼ 25 K=cm, G R ¼ 1:7 K=cm and Gr ¼ 650, so that the order of magnitude of the flow velocity in the bulk nGr 1=2 =R ¼ 0:2 cm=s.…”

Enhanced Morphological Stability in Sb-Doped Ge

“…[46] Experimental minimum values of the melt height (5 mm) and liquid temperature gradient (10 K/ cm) give an experimental lower limit for Gr of 4.0 · 10 4 . This indicates that convection was present during the growth, supporting the smaller boundary layers calculated with the FR analysis.…”