Modeling of micro- and macrosegregation and freckle formation in single-crystal nickel-base superalloy directional solidification

The formation of macrosegregation and freckles by multicomponent thermosolutal convection during the directional solidification of single-crystal Ni-base superalloys is numerically simulated. The model links a previously developed thermodynamic phase equilibrium subroutine with an existing code for simultaneously solving the macroscopic mass, momentum, energy, and species conservation equations for solidification of a multicomponent alloy. Simulation results are presented for a variety of casting speeds and imposed thermal gradients and for two alloy compositions. It is found that for a given alloy composition, the onset of convection and freckle formation occurs at a critical primary dendrite arm spacing, which agrees well with previous experimental findings. The predicted number and shape of the freckle chains in the unstable cases also agree qualitatively with experimental observations. Finally, it is demonstrated how the onset and nature of convection and macrosegregation vary with alloy composition. It is concluded that the present model can provide a valuable tool in predicting freckle defects in directional solidification of Ni-base superalloys.     

Effect of fluid flow and hafnium content on macrosegregation in the directional solidification of nickel base superalloys

Nickel base superalloys of the MAR-M200 type with additions of hafnium from 0 to 2.6 pct were directionally solidified in investment molds with an abrupt change in cross-section. Segregation patterns of several alloying elements (Hf, W, Ti, and Co) were determined and related to partition ratios, hafnium content, and casting geometry. The segregation is more pronounced near the section change, and hafnium substantially increases the segregation tendency of these alloying elements. Variation in eutectic volume percent was related to the observed macrosegregation. A model based on the flow of interdendritic liquid to feed solidification shrinkage was Actapted to describe the macrosegregation behavior of these complex superalloys. The model accurately predicts the effect of casting geometry on macrosegregation.     

Mushy-zone rayleigh number to describe macrosegregation and channel segregate formation during directional solidification of metallic alloys

A recently defined mushy-zone Rayleigh number (R_aM) that includes side-branching contributions to the mushy-zone permeability has been examined for its correlation with the longitudinal macrosegregation and channel segregate formation. The Rayleigh number shows (1) a strong correlation between the extent of longitudinal macrosegregation and increase in the mushy-zone convection and (2) a good ability to predict the formation of channel segregates during directional solidification.     

Primary spacing in directional solidification

A new analytical model is developed to explain the variation in primary spacing λ with growth velocity V. In this model, dendrite growth is resolved into two parts: the growth of the center core and that of the side arms, which are separately treated. In contrast to the assumption in the current models, it is only the dendrite core, not the entire dendrite, whose curvature radius at the tip is directly related to dendrite tip radius R. The primary spacing is considered to be the sum of core diameter and twice the sidearm length. As long as the growth of side arms is suppressed, it becomes cellular growth. As a result, this model gives a reasonable dependence of cell and dendrite spacing on the process parameters. The proposed model has been applied to several alloys to compare its predictions both with experimental data and with the analytical expression of the Hunt-Lu model.     

Primary spacing in directional solidification

A new analytical model is developed to explain the variation in primary spacing λ with growth velocity V. In this model, dendrite growth is resolved into two parts: the growth of the center core and that of the side arms, which are separately treated. In contrast to the assumption in the current models, it is only the dendrite core, not the entire dendrite, whose curvature radius at the tip is directly related to dendrite tip radius R. The primary spacing is considered to be the sum of core diameter and twice the sidearm length. As long as the growth of side arms is suppressed, it becomes cellular growth. As a result, this model gives a reasonable dependence of cell and dendrite spacing on the process parameters. The proposed model has been applied to several alloys to compare its predictions both with experimental data and with the analytical expression of the Hunt-Lu model.     

Morphological instability in rapid directional solidification

Mullins and Sekerka showed for fixed temperature gradient that the planar interface is linearly stable for all pulling speeds V above some critical value, the absolute stability limit. Near this limit, where solidification rates are rapid, the assumption of local equilibrium at the interface may be violated. We incorporate non-equilibrium effects into a linear stability analysis of the planar front by allowing the segregation coefficient and interface temperature to depend on V in a thermodynamically-consistent way. The absolute stability limit of the cellular mode is modified. A new oscillatory state is formed which, in the absence of latent heat, has a critical wavenumber of zero; by itself this instability would lead to the formation of solute bands in the solid. This mode has its own absolute-stability limit determined by solute trapping and kinetics. Under certain conditions there exists a window of stability above the steady absolute-stability boundary and below the oscillatory-stability; here the planar, segregation-free state is restabilized.     

Analysis of the effect of shrinkage on macrosegregation in alloy solidification

Numerical calculations based on a continuum model are used to examine the effects of solidification shrinkage on the redistribution of solute in a Pb-19.2 pct Sn mixture which is convectively cooled at a sidewall. For each of three different cooling rates, separate calculations are performed for shrinkage and buoyancy-induced flows, as well as for the combined influence of shrinkage and buoyancy effects. The calculations reveal that flow and macrosegregation patterns are more strongly influenced by buoyancy effects over a wide range of solidification rates. Although extremely large solidification rates yield small regions near the chilled wall in which shrinkage-induced flows control the redis-tribution of solute, the overall effect on macrosegregation is small relative to that associated with buoyancy. Scaling analysis of the governing equations produces reference shrinkage and buoyancy velocities which can be used to extend the current numerical results to other binary systems.     

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.     

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.     

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.