CONVECTION INDUCED OSCILLATORY STRUCTURES

        Expoerimental results presented earlier in large diameter samples clearly showed that convection significantly influences the formation of microstructures. To explain this important effect of convection on the shape of the a -b interface, we shall first describe the physical mechanism developed by Mazumder et. al. [38] for the evolution of an oscillatory microstructure. Subsequently, a detailed numerical modelling that support this mechanism [15] will be discussed.

        In several experimental studies, different diameter samples were directionally solidified under identical processing conditions so that the tube diameter was the only variable. As the Rayleigh number changed six orders of magnitude and hence the degree and modes of convection [10,15], it strongly indicates that convection was the major cause of layered structure formation. The fundamental concept behind the mechanism is that the complex peritectic microstructure development in the convective regime is a result of the steady or time dependent lateral segregation of solute at the interface and the possibility of formation and growth of two phases at the same interface In this section we shall elaborate this concept.

        For simplicity we consider solidification inside a two dimensional rectangular vertical slot in which x and y are vertical and horizontal coordinate directions. The bottom horizontal wall is the instantaneous solid-liquid interface growing in the positive x direction, while gravity vector acts in the negative x direction. The vertical walls are rigid solid walls maintained at prescribed temperatures and impervious to mass flux. The temperatures of both the vertical walls increase linearly upwards with a gradient G_V. However a small but finite temperature gradient G_H is assumed to exist between the two vertical walls to model the real experimental imperfection. The horizontal temperature gradient is defined as

The horizontal temperature gradient is responsible for initiating the convective flow which is interfered cooperatively or uncooperatively with the solute composition gradient after certain lapse of time Although the exact dynamics of thermo-solutal convection can be very complex, the general effect of the flow is to establish a horizontal composition gradient at the interface. For example if we consider flow driven by horizontal thermal gradient alone, the direction of flow is usually up along the hot wall and down along the cold wall. This establishes an almost parallel flow along the interface (except near the corners) from the cold side towards the hot side. Such a flow causes lateral segregation or nonhomogeneity of solute composition at the interface (and also in the melt). In general the region near the warmer wall gets richer in solute [15,19]. The liquid composition profiles along the interface is shown schematically in Fig. 10a for three different times, with t₁ < t₂ < t₃. The solute profile at t₁ intersects the peritectic composition Cp at a point P(t₁) at a distance R(t₁) from the colder wall. At the point P(t₁) the three phases are in contact and in local equilibrium. Since the composition of the liquid falls below and above the peritectic composition C_p on the two sides about y = R(t₁), b phase will be present in the region y > R(t_i) and the a phase will exist over the region y < R(t_i). As solidification proceeds, more and more solute is rejected from the interface and the interface composition increases as shown by the schematic curves for times t₂ and t₃ in Figure 10a. Thus, the point P(t_i) gradually moves inwards or the distance R(t_i) monotonically decreases with time. Since, the point P(t_i) also has a velocity component in the vertical x direction due to the constant growth rate Vp, the locus of P(t) will delineate a smooth curve in the solidified sample and appear as a curved interface between the two solid phases a and b , as shown schematically in Fig. 10b For cylindrical geometry this curve will appear as a surface of revolution separating the two solid phases. We expect to see similar microstructures in systems where convection is steady and moderate, as in ampoules of diameter of the order of 800 µms - 1mm.

Fig. 10. A mechanism of nonplanar interface formation between primaty and paritectic solid phases due to solute-segregation along the interface, while the solid-liquid interface remains planar (a) Liquid composition profiles at three time levels. (b) Formation and growth of b- hase from the warmer wall due to the increase of composition beyond the peritectic composition.

        During the freezing of a double-diffusive system of binary melts, the steady convection may become unstable and oscillatory or quasi-periodic convection may set in [24]. The velocity, temperature and composition field oscillates in a periodic or quasi-periodic manner [35,36] and hence the composition profile at the interface may have a complex time-dependent behavior. This in turn will cause a periodic or quasi-periodic fluctuating motion of P(t) in the y direction while being displaced at a constant rate of Vp along the vertical direction. Hence the locus of P(t) in the solidified sample will now delineate a curve that may look like an Oscillating Structure with arms stretching out in the horizontal direction. This is schematically demonstrated in Fig. 11. The length and thickness of the arms or layers will depend on the detail characteristic of convective fluctuation and the growth velocity. Again in a cylindrical geometry this locus will be a surface of revolution and will appear as a tree like structure. We expect to see similar structures in situations where convection is intense and time periodic, as in ampoules of diameter ~ 6 mm.

Fig. 11. A mechanism of oscillating layered structure formation due to oscillating segregation profile at the interface. The four figures represent four different stages of the layered structure formation. The solid-liquid interface is still planar, but the solid-solid interface between primary and peritectic phase fluctuates due to the oscillation of interface composition about the peritectic composition. The figure at the right is an experimental micrograph showing a quenched interface which is planar.

        The model presented above is for high G/V_p ratio for which planar interface growth is predicted for both the a and b phases. We thus assume that the interface is planar, and remains planar when the two phases form and grow. In order to examine the validity of this assumption, experiments were conducted in which the sample was quenched to observe the shape of the interface. Figure 11 shows that the planar interface assumption is justified for the experimental conditions used.

        The mechanisms described above were first checked with a semi-analytical model [38] which was found to be successful in predicting complex microstructures observed in peritectic systems. In this model, the solute transport equation was solved with analytical but approximate models for prototype flows pertinent to steady and unsteady convection. The steady flow was modeled by the analytical solution due to Batchelor [39]. A normal mode type perturbation [40] was superimposed on the steady flow to investigate whether oscillation in fluid flow could lead to oscillating patterns in microstructure. This simple model proved that oscillating convection would give rise to fluctuating segregation profile at the interface and cause the formation of oscillating layered structures [38]. Thus, a more rigorous model was developed to predict the occurrence and nature of convection in the melt in the context of solidification [38]. The exact detail of the flow and its effect on the shape of a -b interface were accurately modeled through the full scale numerical solution of the constitutive transport equations. This involved simultaneous solving of Heat Transport Equation, Solute Transport Equation, Navier Stokes Equations for velocity components in x and y directions including both thermal and solutal buoyancy effects and the Continuity Equation along with the proper initial and boundary conditions.In this model the actual characteristics of convection in a differentially heated two dimensional cavity (with imposed lateral temperature gradient between vertical walls) were calculated, and its effect on solute transport was investigated.

        The model considers the directional solidification of a binary alloy of initial composition C₀ inside a two-dimensional vertical ampoule.A constant lateral temperature gradient is prescribed between the vertical walls to model the actual radial temperature gradient that exist in all vertical Bridgman systems that employ three zone heating assembly. The actual lateral temperature gradient for a particular design can only be determined through full scale thermal simulation of the Bridgman Growth process, and it turns out to be a complex function of the design variables and also the axial temperature gradient [15,19,20]. However, direct application of a prescribed lateral temperature gradient, consistent with the results from full scale simulation and experimental measurements, considerably simplifies the analysis and produce physically relevant results. The field variables, velocity, pressure, temperature and solute composition, are described in a dimensionless coordinate reference frame (x,y) that is fixed with the lower end of the sample. The field variables are put in dimensionless form by scaling the coordinates (x,y) and lengths with the inner width of the ampoule ‘d’, time t with the scale for heat diffusion d²/a_l, velocity u(x,y,t) with characteristic velocity of heat diffusion a_l/d. The dimensionless composition and temperature fields are defined as

where T_H and T_C are the temperatures of the isothermal hot and cold walls. The temperatures on the vertical walls are related through the application of constant lateral gradient as

The temperatures on each wall however increase linearly with a constant gradient G_V, or

Table 1: Dimensional Groups used in simulation

        The two-dimensional time-dependent equations describing convection, and heat and solute transport are

where q₀ is a reference temperature. The dimensionless groups that appear in the equations are defined in Table 1. The boundary conditions on the velocity field on the ampoule and at the melt/crystal interface specify that there is no slip relative to the solid surfaces. The vertical ampoule walls are impervious to mass flux. The constant linear temperature gradient is prescribed at the bottom and top surfaces. The bottom wall (solid-liquid interface) is continuously rejecting solute and hence has a mixed boundary condition. The concentration boundary condition on this wall is:

For a single phase solid, the nucleation condition is required to form the other phase. When the liquid composition at any point on the interface falls between and the phase is selected from the direction of composition variation at that point. If the composition is increasing from belowthe point remains as a phase until it crosses , whereas the point remains as b phase as the composition decreases from above until it reaches . When two solid phases are present, the condition of equation (2) is modified as: v = a if c < c_p, and v = b if c > c_p. The set of equations are solved by Alternate Direction Implicit (ADI) Finite Difference Method [41,42] on a uniform Cartesian grid. The details could be found elsewhere [15]. The results of the numerical model are presented mainly in terms of plots of composition profiles, shape of the a -b interface and 2D color-coded composition field-maps in the solid phase.

        The dynamical characteristics of the convection in different ampoules of varying diameters, are summarized in Table 2. This set of computation was done for a Sn-1.4 wt % Cdalloy with growth rate of 3.0 µm/sec. A vertically stabilizing temperature gradient of 10.0 K/mm and a horizontal temperature gradient ~ 0.4 K/mm were used [15]. The maximum convective velocity in the melt was taken as a measure of convection scale U_c, i.e. U_c=max{U,V}, where U, V are the dimensional axial and horizontal components of the fluid velocity. Hence, the ratio of convective transport to diffusive transport of solute is scaled as

where D is the diffusivity in the liquid phase, d is the lateral thickness of the ampoule and Vp is the imposed growth rate. It is found that for tube diameters of the order of 500 microns or smaller, the convective effects are insignificant and diffusion is the dominant process of transport. For tube diameters of the order of 800 µm -1.0 mm the convective transport is of the same order of magnitude as the diffusive transport and the flow is steady. For tube diameters greater than 3.0 mm the convective transport is significantly higher than diffusive transport and the dynamics of the flow is unsteady. The flow in the melt shows periodic or quasiperiodic oscillatory behaviors. A similar full scale analysis for the Pb-Bi system indicated that convection is significant even in a 600 µm diameter tube. Since most experimental work that observed banded structure in the Sn-Cd and Pb-Bi systems were done in ampoules of diameter 3.0 mm or greater [1-5,7-8], it is expected that the dynamics fell in the regime of complex time dependent convective transport. We expect that both the intensity and mode of convection (steady or unsteady) must have had profound effect on microstructures observed in these experiments. We shall now demonstrate how the oscillatory convection in the melt give rise to formation of oscillating layered structure in the solid. We will refer to two sets of calculations done on the diffusive and convective regimes to prove the point.

        Numerical calculations have been carried out for conditions characteristic of solidification of Sn-1.3%Cd at a growth rate of 3.0 m m/s in ampoules of inner diameter 0.6 mm and 6 mm. It must be noted that the composition 1.3 %Cd is exceedingly outside the banding window and should show only single transition from a to b if the transport is only by one dimensional molecular diffusion. The average interface compositions for Ra_T = 0.07, Ra_S = 1.5 (0.6 mm tube), and Ra_T = 200, Ra_S = 1440 (6.0 mm tube) are plotted in Figure 12a, which shows steady and oscillatory development respectively. The dashed line C_p is the peritectic composition. In Figure 12b, the concentration at three different locations on the interface for 6 mm tube are plotted. The composition near the hotter and colder walls (y=1 and 0 fluctuates either over or below C_p while all the points in-between fluctuates about C_p. The color-coded concentration maps in solid are plotted in Figure 13 for corresponding to the earlier calculations. The microstructure in 0.6 mm dia tube is diffusion controlled and shows a sharp transition from a to b phase corresponding to the point where C_av intersects C_p in Figure 12a. However, for the second case, the oscillating segregation profile at interface gives rise to oscillatory coupled growth of both the phases. This leads to the appearance of a tree-like microstructure as shown in figure 12b. The arms of the a phase may appear as isolated bands on a partially polished sample. Thus, the oscillating microstructure observed in experiments are explained in terms of oscillating convection in the melt.

     (a)                                                     (b)

Fig. 12. Numerical results of (a) average concentration variation along the interface as a function of time. (b) Interface composition variation with time at three different locations along the interface, y= 0 (center) , 0.5 and 1.0 (wall).

        The effect of simple oscillatory convection on the a à b interface is illustrated in Figure 13 along with liquid composition at two end points (y=0 and y = d) on the interface. The computation is done with C₀=1.4%, V_P = 1.4m /sec, a and b undercoolings of 0.1°C, U_c/V_P = 100, frequency of oscillation f = 0.004 Hz, and fractional amplitude e = 0.5. The composition at y = d reaches C_l^M at x ~ 0.05 cm and keeps increasing (Figure 13 a). The same behavior is observed for other points near y = d, though each point had different transition distances. Hence, near the edge y = d, the b phase is continuous along the length of the sample (Figure 13 b). On the other hand the composition at y = 0 shows multiple transitions of a:b and b:a as the composition fluctuates above and below C_l^M and C_l^m respectively until the composition increases to a value that it does not get below C_l^m thereafter (x ~ 0.65). This appears as oscillatory layers of a and b along the length of the sample at y = 0 (Figure 13 b) until it is completely taken over by the continuous b phase. Other points between y = 0 and y = d show similar behavior but with different phases and the locus of the transition points for all the points delineate an oscillatory layered structure where a phase appears to grow as a tree in the continuous b matrix. The layers have the shape which is thick at the center and tapers down at the edges. The layers do not extend over the entire cross section and the lateral width and axial thickness gradually decrease along the length of the sample.