A vertical Bridgman system used for directional solidification consists of crystal and melt contained in a cylindrical ampoule surrounded by a thermal assembly that is pulled vertically upwards with constant velocity V_P. In the simplest analysis, one can assume that the hot and cold zones are isothermal at T_H and T_C and are separated by an adiabatic zone designed to establish high axial temperature gradient in the liquid at the interface. In a perfect vertical Bridgman system the thermal gradient should be perfectly vertical and positive upwards. Since, for most fluids the density decreases with increasing temperature, lighter liquid overlays heavier liquid and the density gradient is exactly parallel to the gravity vector. As the binary liquid freezes, solute is preferentially rejected at the interface (for partition coefficient, k < 1), so that a solute rich layer forms adjacent to the interface. Since the density of the liquid also depends on the solute concentration, the rejection of solute modifies the density field within this solute layer. If the solute is heavier than the solvent (as in Sn-Cd) then both the solutal and thermal buoyancy forces are parallel to the gravity vector. Under the ideal case of no horizontal variation of temperature, this arrangement is stable for stationary (no-flow) condition and the transport of solute must be solely due to molecular diffusion along the growth direction. The thermal field can be idealized as a nearly constant axial gradient and heat is transported from the hot zone to the cold zone only by conduction through the melt and the crystal. On the other hand if the solute rejected is lighter (as in Pb-Bi), the liquid becomes hydrostatically unstable when certain parameters exceeds critical values and natural convection sets in [15-17]. Under the condition of vigorous convection, the heat and solute are transported mainly by convection.

        In the vertical Bridgman system technique, in which the hot and cold zones are isothermal at T_H and T_C and are separated by the adiabatic zone designed to establish high axial temperature gradient at the interface, a radial temperature gradient always exists in the liquid [15,18-20]. It is well known that any horizontal temperature difference, however small, will initiate some convective motion [21,22]. Even in the presence of a stabilizing axial gradient, i.e. the melt over the crystal and temperature increasing upwards, the flows induced by radial temperature gradients can be extremely intense [15,18-20]. The convective flow in turn makes the composition field laterally nonuniform and this gives rise to solutal buoyancy. The complex interactions among the vertical temperature gradient (usually stabilizing), vertical composition gradient (either stabilizing or destabilizing), horizontal temperature gradient and horizontal composition gradient can lead to very complex dynamical behavior; the general trend being transition from minimal convection to steady cellular convection to steady multicellular convection to time-periodic convection to quasi-periodic convection to chaotic convection characteristic of turbulence [16,17, 23-36]. The actual dynamics is completely governed by the Thermal Rayleigh Number (Ra_T,) Solutal Rayleigh Number (Ra_S), Prandtl number (Pr), the aspect ratio (l/d), the Lewis number (Le), the growth rate (Vp) and the ratio of the vertical temperature gradient to the horizontal temperature gradient (z = G_V/G_H), where, d and l are the diameter and vertical height of the melt column. All the terms are defined in Table 1.

        For Sn-Cd system, since both the vertical gradients of temperature and composition are stabilizing, radial temperature gradient is the only force that drives convection. For a perticular design of the directional solidification unit, the radial temperature gradient depends on the thermal conductivities of the melt, solid crystal and ampoule, the heat transfer coefficients between the furnace and the outer surface of the ampoule. For a particular radial temperature gradient, the intensity and mode of convection depends on the value of thermal Rayleigh number (Table 1) which is the ratio between the driving buoyancy force to the resistive viscous force. Since Rayleigh number is directly proportional to the fourthe power of inner diameter of the ampoule for a given lateral temperature gradient, the melt convection increases with increasing diameter. Through numerical simulations it has been established that the convection becomes intense and oscillatory in ampoules of diameters greater than 3 mm [15]. The convection becomes insignificant (though present) in tubes of diameters less than 1 mm. It was concluded that the transport processes in directional solidification of Sn-Cd alloy in tubes of diameter less than 1 mm was diffusion dominated, whereas that in tubes of diameter greater than 3 mm was convection dominated.

        We shall now discuss the diffusive model which is valid for sample sizes less than 1 mm in diameter under 1g conditions. Next, we shall develop a model for microstructure formation in larger diameter samples where convection effects will be dominant.

Fig. 7. Schematic drawing of the oscillating composition and temperature cycle during the formation of a banded stucture under diffusive growth conditions. Stable a-phase and metastable b-phase equilibrium lines are shown. DT_N^a and DT_N^b are the mminimum undercoolings required for the nucleation of the a and the b phases, respectively. The arrow shows the banding cycle. DC₀ uis the composition range for which diffusive banding is predicted.