Cell/dendrite distribution in directionally solidified hypoeutectic Pb-Sb alloys
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I. INTRODUCTION
PRIMARY dendrites and their distribution influence the mechanical properties of cast components. Their processingparameter dependence has therefore been extensively investigated. Theoretical analyses have generally assumed a twodimensional (2-D) axisymmetric distribution of dendrites.[1,2] Only recently has three-dimensional (3-D) analysis of dendrites been attempted by phase-field methods.[3,4] An extensive body of literature exists on the dependence of the average primary spacing on the composition, growth speed, and thermal gradients for both the cellular and dendritic morphologies. Primary dendrite spacing has generally been measured as (A)0.5/N, where N is the number of dendrites in a given area A on a cross section that is transverse to the growth direction. This technique inherently assumes a square distribution of dendrites on the transverse cross section. A square distribution of dendrites has been assumed for modeling mushy-zone permeability.[5] A rectangular distribution has been assumed for examining the interaction among neighboring dendrites.[6] Each primary dendrite does have a fourfold symmetry in terms of its side-branch formation, but the distribution of many dendrites with respect to each other on the transverse section may not be square or rectangular. No detailed statistical examination of the primary dendrite distribution has been reported. There is only one such study reported in the literature, and that is for the cellular array in directionally solidifying Pb-Tl alloy.[7] Here, the cellular distribution was examined using a Wigner–Seitz construction of cell boundaries followed by minimum spanning tree analysis. This analysis demonstrated that the cells have a hexagonal distribution with random superimposed noise. Theoretical analyses of the mechanisms responsible for determining the range of primary spacings observed during directional solidification at a given speed, taking into account the recently discovered history dependence of the average
primary spacing,[8,9] would require a detailed statistical analysis of the distribution of the nearest and higher-order primary dendrite spacings. The purpose of this research was to compare the cells and dendrites in directionally solidified metallic alloys for statistical distribution of nearest and higher-order neighbor spacings, and also to compare them for the frequency distribution of their coordination numbers (i.e., their number of nearest neighbors) During directional solidification of metallic alloys, with melt on top and solid below, the thermal profile is expected to cause stability against convection. However, the solutal profile in the mushy zone and in the overlying melt immediately ahead of the dendritic array would be expected to cause stability only if the solute enrichment increases the melt density (for example, hypoeutectic Al-Cu alloys). In binary alloys where the solute enrichment decreases the melt density (e.g., hypoeutectic Pb-Sn alloys), significant convection is expected. The logical choice for our study wo
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