Convection effects on radial segregation and crystal–melt interface in vertical Bridgman growth

Abstract

The influence of convection caused by horizontal heat transfer through the sides of a vertical Bridgman apparatus is studied analytically. The case when the heat transfer across the side walls is small is considered, so that the resulting interfacial deformation and fluid velocities are also small. This allows one to linearize the Navier–Stokes equations and express the interfacial conditions about a planar interface through a Taylor expansion. Using a no-tangential stress condition on the side walls, asymptotic expressions for both the interfacial slope and radial segregation at the crystal–melt interface are obtained in closed form in the limit of large thermal Rayleigh number. It is suggested that these can be reduced by appropriately controlling a specific heat transfer property at the edge of the insulation zone on the solid side.

An asymptotic analysis for directional solidification of a diffusion-dominated binary system

“…Consistent with the previously stated comments concerning the expression b zz þ Pe b z , we conclude that the flow field in the melt region away from the interface is driven by the heat exchange between the ampoule and melt. Furthermore, this heat exchange can change sign along the ampoule wall which leads to a stacked eddy profile in the ampoule [11][12][13][14][15][16]21,23].…”

“…Since there is a close relationship between growth conditions and the microstructure and properties of solidified materials, there has been an extensive amount of investigation [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] of sharp-interface, Stefan-type problems and directional solidification configurations. Exact solutions to these classes of problems are generally restricted to unbounded domains and are subject to limitations on the boundary conditions.…”

“…Tanveer [14] and Foster [11][12][13] used a version of the small Biot number/small thermal Peclet number approach to obtain results for an Oð1Þ-aspect ratio of the ampoule. The heater profile was assumed to be dependent either on the axial variable only [11,14] or on both axial and azimuthal variables [12,13]. In the limit of a large Rayleigh number, Ra, these workers developed the first asymptotic descriptions of the flow field both in slip and no-slip regimes.…”

Asymptotic solutions for an axisymmetric, stagnant film model of directional solidification

“…Brattkus and Davis [7] thoroughly discuss this point and ultimately investigate a parameter regime that uncouples the solute field from the flow field. Tanveer [8] and Foster [9][10][11] used a version of the small Biot number/ small thermal Peclet number approach to obtain results for an Oð1Þ-aspect ratio of the ampoule. The heater profile was assumed to be dependent either on the axial variable only [8,9] or on both axial and azimuthal variables [10,11].…”

“…Tanveer [8] and Foster [9][10][11] used a version of the small Biot number/ small thermal Peclet number approach to obtain results for an Oð1Þ-aspect ratio of the ampoule. The heater profile was assumed to be dependent either on the axial variable only [8,9] or on both axial and azimuthal variables [10,11]. In the limit of a large Rayleigh number, Ra, these workers developed the first asymptotic descriptions of the flow field both in slip and no-slip regimes.…”