Macrosegregation during directional solidification of Pb-Sb alloys

Macrosegregation of Sb was investigated during directional solidification of binary Pb-Sb alloys containing 2.2 and 5.8 wt% Sb over growth rates varying from 0.8 to 30 μm s^?1. The cellular to dendritic transition was observed at a growth rate of 3.0 μm s^?1 in Pb-2.2 Sb alloy in contrast to a growth rate of 1.5 μm s^?1 in Pb-5.8 Sb alloy. The chemical analysis data revealed considerable macrosegregation of Sb along the longitudinal section of alloys. The degree of macrosegregation increased with a decrease in the growth rate. This behavior is discussed in light of thermo-solutal convection in the mushy zone as well as that in the melt ahead of the solid-liquid interface.     

Primary dendrite distribution and disorder during directional solidification of Pb-Sb alloys

Pb-2.2 wt pct Sb and Pb-5.8 wt pct Sb alloys have been directionally solidified from a single-crystal seed with its [100] orientation parallel to the growth direction, to examine the primary dendrite distribution and disorder of the dendrite arrays. The dendrite distribution and ordering have been investigated using analysis techniques such as the Gauss-amplitude fit to the frequency distribution of nearest and higher-order spacings, minimum spanning tree (MST), Voronoi polygon, and Fourier transform (FT) of the dendrite centers. Since the arrangement of dendrites is driven by the requirement to accommodate side-branch growth along the 〈100〉 directions, the FT images of the fully developed dendrite centers contain spots which indicate this preferred alignment. A directional solidification distance of about three mushy-zone lengths is sufficient to ensure a steady-state dendritic array, in terms of reaching a constant mean primary spacing. However, local dendrite ordering continues throughout the directional solidification process. The interdendritic convection not only decreases the mean primary spacing, it also makes the dendrite array more disordered and reduces the ratio of the upper and lower spacing limits, as defined by the largest 5 pct and the smallest 5 pct of the population.     

A mushy-zone rayleigh number to describe interdendritic convection during directional solidification of hypoeutectic Pb-Sb and Pb-Sn alloys

Based on measurements of the specific dendrite surface area (S _v), fraction of interdendritic liquid (φ), and primary dendrite spacing (λ ₁) on transverse sections in a range of directionally solidified hypoeutectic Pb-Sb and Pb-Sn alloys that were grown at thermal gradients varying from 10 to 197 K cm⁻¹ and growth speeds ranging from 2 to 157 μm s⁻¹, it is observed that S _v=λ ₁ ⁻¹ S*^−0.33 (3.38−3.29 φ+8.85 φ ²), where S*=D _l G_eff/V m ₁ C _o (k−1)/k, with D _l being the solutal diffusivity in the melt, G _eff being the effective thermal gradient, V being the growth speed, m _l being the liquidus slope, C _o being the solute content of the melt, and k being the solute partition coefficient. Use of this relationship in defining the mushy-zone permeability yields an analytical Rayleigh number that can be used to describe the extent of interdendritic convection during directional solidification. An increasing Rayleigh number shows a strong correlation with the experimentally observed reduction in the primary dendrite spacing as compared with those predicted theoretically in the absence of convection.     

Thermosolutal convection during directional solidification

During solidification of a binary alloy at constant velocity vertically upward, thermosolutal convection can occur if the solute rejected at the crystal-melt interface decreases the density of the melt. We assume that the crystal-melt interface remains planar and that the flow field is periodic in the horizontal direction. The time-dependent nonlinear differential equations for fluid flow, concentration, and temperature are solved numerically in two spatial dimensions for small Prandtl numbers and moderately large Schmidt numbers. For slow solidification velocities, the thermal field has an important stabilizing influence: near the onset of instability the flow is confined to the vicinity of the crystal-melt interface. Further, for slow velocities, as the concentration increases, the horizontal wavelength of the flow decreases rapidly — a phenomenon also indicated by linear stability analysis. The lateral in-homogeneity in solute concentration due to convection is obtained from the calculations. For a narrow range of solutal Rayleigh numbers and wavelengths, the flow is periodic in time. Formerly with the Mathematical Analysis Division, Center for Applied Mathematics, National Bureau of Standards, Washington, DC 20234. This paper is based on a presentation made at the symposium “Fluid Flow at Solid-Liquid Interfaces” held at the fall meeting of the TMS-AIME in Philadelphia, PA on October 5, 1983 under the TMS-AIME Solidification Committee.     

Formation of macrosegregation by multicomponent thermosolutal convection during the solidification of steel

The formation of macrosegregation by multicomponent thermosolutal convection during the solidification of steel is simulated by simultaneously solving macroscopic mass, momentum, energy, and species conservation equations with full coupling of the temperature and concentrations through thermodynamic equilibrium at the solid/liquid interface. The flow field, solid fraction evolution, and macrosegregation patterns for four cases are presented. The results show both the formation of channel segregates and the formation of islands of mush surrounded by bulk melt. In examining the solidification of a ten-element steel, the global extent of macrosegregation of an element is found to be linearly dependent on its partition coefficient (more severe segregation for small partition coefficient), although such scaling is not possible locally. Results for the solidification of a binary Fe-C alloy (with the same carbon content as the ten-element alloy) are similar to those for the ten-element alloy due solely to the large contribution of carbon to buoyancy driven flow in the ten-element steel chosen for study. While including only those elements that make significant contributions to buoyancy driven flow reproduces the global extent of macrosegregation seen in the ten-element alloy, local differences in the predictions are visible. Finally, comparison of results for the solidification of the same ten-element steel using two different sets of data to describe the partition coefficients and change in liquidus temperature with concentration of the elements shows completely opposite behavior,i.e., upward flow through the mushy zone for one case and downward flow for the other. Thus, the need to have accurate phase-equilibrium data when modeling multicomponent macrosegregation is illustrated. Together, the results give an indication of what areas require more careful examination if accurate modeling of multicomponent solidification is to be accomplished.     

Phase selection during directional solidification of peritectic alloys

Directional solidification studies have been conducted using Pb-Bi peritectic alloys over a wide range of compositions, temperature gradients, and growth velocities to characterize the primary α- to primary β-phase transitions, which have been observed at both very low and very high velocities. The critical conditions for these transitions correspond to the simultaneous growth of the α and β phases at or close to a single isotherm. The low velocity transition occurs under very specific conditions of composition, temperature gradient, and growth velocity. Since the transition conditions are composition dependent, they change continuously under terrestrial conditions where rejected solute is convectively mixed into the liquid. Detailed experimental studies have been carried out to examine the phase selection in the immediate vicinity of the critical velocity for the α to β transition, and the effect of convection on this transition is examined experimentally in the Pb-Bi system. The dynamic condition, at which both phases are present at the same isotherm, was shown to depend not only on velocities, temperature gradients, and bulk (nominal) alloy compositions, but also on the volume fractions of solid. A quantitative expression for the α- to β-phase transition condition was obtained by using the boundary layer model of fluid flow, which showed good agreement with the experimental results. It is shown that the transition occurs at the volume fraction where the bulk composition reaches the critical composition value predicted by the diffusive model. The modification in the microstructure map for the trailing planar or nonplanar β phase is discussed.     

Dendritic growth during directional solidification of hypoeutectic Fe-C-Si alloys

The liquid decanting technique has been used to study the morphology of dendrites in directionally solidified Fe-3.08 pct C-2.01 pct Si alloy. The experimental results indicated that the morphology of primary dendrites in the Fe-C-Si system is very similar to those obtained in some transparent metal model systems and in some other metal systems. In order to study the morphological transition between cellular and dendritic growth, directionally solidified samples were quenched in cold water at various stages of solidification and the morphology was examined on the polished and etched surface. It has been found that when the growth velocity decreased from 326.6 to 0.8 μn/s, the average dendrite tip radius increased from 1.12 to 33.1 μm. At a growth velocity of about 0.65 μm/s, a transition from dendritic to cellular growth occurred. Models for dendritic growth proposed by various investigators have been briefly reviewed and compared with the present experimental results. Significant disagreements were found for some of the available theoretical models. Possible explanations have been given for these disagreements.     

Cellular-to-dendritic transition during the directional solidification of binary alloys

The transition from a cellular to dendritic microstructure during the directional solidification of alloys is examined through experiments in a transparent system of succinonitrile (SCN)-salol. In a cellular array, a strong coupling of solute fields exists between the neighboring cells, which leads not only to multiple solutions of primary spacing, but also includes multiple solutions of amplitude, tip radius, and shape of the cell. It is found that these multiple solutions of different microstructural features in a cellular array, obtained under fixed growth conditions and compositions, play a key role in the cell-dendrite transition (CDT). The CDT is controlled not only by the input parameters of alloy composition (C ₀), growth rate (V), and thermal gradient (G), but also by microstructure parameters such as the local primary spacing. It is shown that the CDT is not sharp, but occurs over a range of growth conditions characterized by the minimum and maximum values of V/G. Within this transition range, a critical spacing is observed above which a cell transforms to a dendrite. This critical spacing is given by the geometric mean of the thermal, diffusion, and capillary lengths and is inversely proportional to composition in weight percent.     

The directional solidification of Pb-Sn-Cd alloys

The growing interest in composite structures for new material applications makes it necessary to determine just how generally we can apply existing solidification theory to controlled three-phase ternary solidification. The Pb-Sn-Cd ternary eutectic system was used as a suitable model system to completely map the phase morphology as a function of G/R and compositions. By carefully controlling the freezing rate and the thermal gradient in the liquid ahead of the solid-liquid interface (in the range 400 to 500 C/cm) the following areas of interest were investigated: 1) the effect of growth velocity and composition on coupled structures, 2) ternary impurities and their effect on the minimum G/R for coupled growth in a binary system, 3) the effect of growth velocity and composition on the nonplanar interface structures, and 4) the adaptability of present theories (the constitutional supercooling criterion and Cline’s binary analysis) in predicting the region of coupled growth in a three-component eutectic system growing at steady-state. It was found that much of the one and two-phase directional solidification theory and terminology can be directly extended to a ternary eutectic system. This suggests a further extension to n-phase, m-component systems (m ≥ n) with at least a qualitative understanding of the solidification process. The Authors wish to acknowledge the support of the National Science Foundation which made this study possible.     

Thermosolutal convection during dendritic solidification of alloys: Part II. Nonlinear convection

A mathematical model of thermosolutal convection in directionally solidified dendritic alloys has been developed that includes a mushy zone underlying an all-liquid region. The model assumes a nonconvective initial state with planar and horizontal isotherms and isoconcentrates that move upward at a constant solidification velocity. The initial state is perturbed, nonlinear calculations are performed to model convection of the liquid when the system is unstable, and the results are compared with the predictions of a linear stability analysis. The mushy zone is modeled as a porous medium of variable porosity consistent with the volume fraction of, interdendritic liquid that satisfies the conservation equations for energy and solute concentrations. Results are presented for systems involving lead-tin alloys (Pb-10 wt pct Sn and Pb-20 wt pct Sn) and show significant differences with results of plane-front solidification. The calculations show that convection in the mushy zone is mainly driven by convection in the all-liquid region, and convection of the interdendritic liquid is only significant in the upper 20 pct of the mushy zone if it is significant at all. The calculated results also show that the systems are stable at reduced gravity levels of the order of 10⁻⁴ g ₀ (g ₀=980 cm·s⁻¹) or when the lateral dimensions of the container are small enough, for stable temperature gradients between 2.5≤G _l≤100 K·cm⁻¹ at solidification velocities of 2 to 8 cm·h⁻¹.     

The planar to cellular transition during the directional solidification of alloys

The planar to cellular interface transition during the directional solidification of a binary alloy has been studied in the succinonitrile-acetone system. The interface velocity at which the planar interface becomes unstable and the wave numbers of the initially unstable interface have been precisely determined and compared with the linear stability analysis. Critical experiments have been carried out to show that the planar to cellular bifurcation is subcritical so that a finite amplitude perturbation below the critical velocity can also give rise to planar interface instability.     

Microstructural variations induced by gravity level during directional solidification of near-eutectic iron-carbon type alloys

The effects of gravity on the microstructure of directionally solidified near-eutectic cast irons are studied, using a Bridgman-type automatic directional solidification furnace aboard a NASA KC-135 aircraft which flies parabolic arcs and generates alternating periods of low-g (0.01 to 0.001 g, 30 seconds long) and high-g (1.8 g, 1.5 minutes long). Results show a refinement of the interlamellar spacing of the eutectic during low-g processing of metastable Fe-C eutectic alloys. Low-g processing of stable Fe-C-Si eutectic alloys (lamellar or spheroidal graphite) results in a coarsening of the eutectic grain structure. Secondary dendrite arm spacing of austenite increases in low-g and decreases in high-g. The effectiveness of low-gravity in the removal of buoyancy-driven graphite phase segregation is demonstrated.     

Macrosegregation in castings rotated and oscillated during solidification

The macrosegregation present in stationary, rotated, and oscillated castings of Al-3 wt pct Ag was determined by measuring the distribution of radioactive silver added to the melt. Considerable scatter was observed in the measurements, the scatter being dependent on the sampling technique used. It was found that no significant macrosegregation was present in the stationary and rotated castings. Extensive macrosegregation was detected in the oscillated casting. For the oscillating case the macrosegregation can be accounted for on the basis of the long range movement of dendrite fragments which break and/or melt off in the solid-liquid interface region. This movement is a direct result of turbulent waves associated with the oscillation. The maximum silver concentration is shown to be related to the columnar-to-equiaxed transition.     

Modeling of marangoni-induced droplet motion and melt convection during solidification of hypermonotectic alloys

A two-phase volume averaging approach to model Marangoni-induced droplet motion of the minority liquid phase and the convection in the parent melt during solidification of the hypermonotectic alloys is presented. The minority liquid phase decomposed from the parent melt as droplets in the miscibility gap was treated as the second-phase L ₂. The parent melt including the solidified monotectic matrix was treated as the first phase L ₁. Both phases were considered as different and spatially interpenetrating continua. The conservation equations of mass, momentum, solute, and enthalpy for both phases, and an additional transport equation for the droplet density, were solved. Nucleation of the L ₂ droplets, diffusion-controlled growth, interphase interactions such as Marangoni force at the L ₁-L ₂ interface, Stokes force, solute partitioning, and heat release of decomposition were taken into account by corresponding source and exchange terms in the conservation equations. The monotectic reaction was modeled by adding the latent heat on the L ₁ phase during monotectic reaction, and applying an enlarged viscosity to the solidified monotectic matrix. A two-dimensional (2-D) square casting with hypermonotectic composition (Al-10 wt pct Bi) was simulated. This paper focused on Marangoni motion, hence gravity was not included. Results with nucleation, droplet evolution, Marangoni-induced droplet motion, solute transport, and macrosegregation formation were obtained and discussed.     

Macrosegregation during solidification resulting from density differences in the liquid

Macrosegregation has been observed in directionally solidified Pb-20 pct Sn alloys, over a range of freezing rates and temperature gradients. The macrosegregation was shown to result from the upward flow of less dense, tin rich, interdendritic liquid during solidification, using radioactive tracer techniques. For comparison, it was shown that macrosegregation occurred in the opposite direction in a Sn-4 pct Pb alloy, where the interdendritic liquid was lead rich, and consequently more dense. Shrinkage trails and pipes were observed in some of the Pb-20 pct Sn ingots, similar to “freckles” observed in directionally cast superalloys. A mathematical model for macrosegregation in vertically solidified ingots is presented, the driving force being the density differences in the interdendritic liquid during solidification. Liquid flow through the dendritic array is estimated by considering the partially solidified alloy as a porous medium of variable porosity. For simplicity, the model neglects backflow due to volume shrinkage (inverse segregation). The experimental results are compared to the model predictions. Formerly Research Associate, Department of Metallurgy, University of British Columbia     

Cellular array morphology during directional solidification

Cellular array morphology has been examined in the shallow cell, deep cell, and cell-to-dendrite transition regime in Pb-2.2 wt pct Sb and Al-4.1 wt pct Cu alloy single-crystal samples that were directionally solidified along [100]. Statistical analysis of the cellular spacing distribution on transverse sections has been carried out using minimum spanning tree (MST), Voronoi polygons, radial distribution factor, and fast Fourier transform (FFT) techniques. The frequency distribution of the number of nearest neighbors and the MST parameters suggest that the arrangement of cells may be visualized as a hexagonal tessellation with superimposed 50 pct random noise. However, the power spectrum of the Fourier transform of the cell centers shows a diffused single-ring pattern that does not agree with the power spectrum from the hexagonal tessellation having a 50 pct superimposed random (uniformly distributed or Gaussian) noise. The radial distribution factor obtained from the cells is similar to that of liquids. An overall steady-state distribution in terms of the mean primary spacing is achieved after directional solidification of about three mushy-zone lengths. However, the process of nearest-neighbor interaction continues throughout directional solidification, as indicated by about 14 pct of the cells undergoing submerging in the shallow cell regime or by an increasing first and second nearest-neighbor ordering along the growth direction for the cells at the cell-to-dendrite transition. The nature of cell distribution in the Al-Cu alloy appears to be the same as that in the Pb-Sb. The ratio between the upper and lower limits of the primary spacing, as defined by the largest and the smallest 10 pct of the population, respectively, is constant: 1.43±0.11. It does not depend upon the solidification processing conditions.