A new criterion for the appearance of hot tears in metallic alloys is proposed. Based upon a mass balance performed over the liquid and solid phases, it accounts for the tensile deformation of the solid skeleton perpendicular to the growing dendrites and for the induced interdendritic liquid feeding. This model introduces a critical deformation rate ( p,max ) beyond which cavitation, i.e., nucleation ⅐ ε of a first void, occurs. As should be expected, this critical value is an increasing function of the thermal gradient and permeability and a decreasing function of the viscosity. The shrinkage contribution, which is also included in the model, is shown to be of the same order of magnitude as that associated with the tensile deformation of the solid skeleton. A hot-cracking sensitivity (HCS) index is then defined as . When applied to a variable-concentration aluminum-copper alloy, this HCS ⅐Ϫ1 ε p,max criterion can reproduce the typical ''⌳ curves'' previously deduced by Clyne and Davies on a phenomenological basis. The calculated values are in fairly good agreement with those obtained experimentally by Spittle and Cushway for a non-grain-refined alloy. A comparison of this criterion to hot cracks observed in ring-mold solidification tests indicates cavitation depression of a few kilo Pascal and tensile stresses in the coherent mushy zone of a few mega Pascal. These values are discussed in terms of those obtained by other means (coherency measurement, microporosity observation, and simulation). Even though this HCS criterion is based only upon the appearance of a first void and not on its propagation, it sets up for the first time a physically sound basis for the study of hot-crack formation.

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Rapid advances in atomistic and phase-field modeling techniques as well as new experiments have led to major progress in solidification science during the first years of this century. Here we review the most important findings in this technologically important area that impact our quantitative understanding of: (i) key anisotropic properties of the solid-liquid interface that govern solidification pattern evolution, including the solid-liquid interface free energy and the kinetic coefficient; (ii) dendritic solidification at small and large growth rates, with particular emphasis on orientation selection; (iii) regular and irregular eutectic and peritectic microstructures; (iv) effects of convection on microstructure formation; (v) solidification at a high volume fraction of solid and the related formation of pores and hot cracks; and (vi) solid-state transformations as far as they relate to solidification models and techniques. In light of this progress, critical issues that point to directions for future research in both solidification and solid-state transformations are identified.

D endritic alloy microstructures are formed during a wide range of solidification processes from casting to welding. These microstructures result from a morphological instability of the solid-liquid interface that produces dendrites, which are highly hierarchical branched patterns with primary-, secondary-and higher-order branches. As alloy impurities segregate in the interdendritic liquid during solidification, the spatially inhomogeneous distribution of impurities in the completely solidified alloy is a direct footprint of the dendritic network that formed and coarsened during the solidification process. It also determines the formation and distribution of secondary phases, and thus has a profound influence on the properties of a wide range of technologically important structural materials, from light-weight aluminium alloys used in the automotive industry to nickel-based superalloys used for turbine blades. The study of dendritic growth 1,2 has also been of long-standing fundamental interest because of the ubiquity of branched structures exhibited by diverse interfacial pattern formation systems [3][4][5] .Major theoretical and computational advances over the past two decades have improved our fundamental understanding of dendrite growth, as well as new capabilities to simulate and predict dendritic microstructures on experimentally relevant length and timescales 6 and to elucidate new pattern formation mechanisms 7,8 that enlarge the scope of our understanding of these structures. The commonly accepted microscopic solvability theory of steadystate dendrite growth 9-12 , which builds on the earlier diffusive transport theory of Ivantsov 13 , has led to the understanding that crystalline anisotropy is a crucial parameter that uniquely determines the growth rate and tip radius of dendrites, which is the basic scaling length for the entire dendritic network. Predictions of this theory have been largely validated by phase-field simulations of dendritic evolution over the past few years for both small [14][15][16][17][18][19] and large 20 growth rate. Moreover, molecular dynamics (MD) simulation methods [21][22][23][24][25][26][27][28][29][30] as well as experimental techniques 31,32 have recently been developed to accurately compute anisotropic interfacial properties that control dendritic evolution.Despite this progress, dendrite growth theory remains limited to predicting the steady-state characteristics of dendrites growing along simple crystallographic directions, such as the 100 directions that correspond to the main crystal axes for materials with cubic symmetry, or the six directions in the basal plane of

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Hot tearing in castings is closely related to the difficulty of bridging or coalescence of dendrite arms during the last stage of solidification. The details of the process determine the temperature at which a coherent solid forms; i.e., a solid that can sustain tensile stresses. Based on the disjoining-pressure concept used in fluid dynamics, a theoretical framework is established for the coalescence of primaryphase dendritic arms within a single grain or at grain boundaries. For pure substances, approaching planar liquid/solid interfaces coalesce to a grain boundary at an undercooling (⌬T b ), given bywhere ␦ is the thickness of an isolated solid-liquid interface, and ⌬⌫ b is the difference between the grain-boundary energy, ␥ gb , and twice the solid/liquid interfacial energy, 2␥ sl , divided by the entropy of fusion. If ␥ gb Ͻ 2␥ sl , then ⌬T b Ͻ 0 and the liquid film is unstable. Coalescence occurs as soon as the two interfaces get close enough (at a distance on the order of ␦ ). This situation, typical of dendrite arms belonging to the same grain (i.e., ␥ gb ϭ 0), is referred to as "attractive". The situation where ␥ gb ϭ 2␥ sl is referred to as "neutral"; i.e., coalescence occurs at zero undercooling. If ␥ gb Ͼ 2␥ sl , the two liquid/solid interfaces are "repulsive" and ⌬T b Ͼ 0. In this case, a stable liquid film between adjacent dendrite arms located across such grain boundaries can remain until the undercooling exceeds ⌬T b . For alloys, coalescence is also influenced by the concentration of the liquid film. The temperature and concentration of the liquid film must reach a coalescence line parallel to, but ⌬T b below, the liquidus line before coalescence can occur. Using one-dimensional (1-D) interface tracking calculations, diffusion in the solid phase perpendicular to the interface (backdiffusion) is shown to aid the coalescence process. To study the interaction of interface curvature and diffusion in the liquid film parallel to the interface, a multiphase-field approach has been used. After validating the method with the 1-D interface tracking results for pure substances and alloys, it is then applied to twodimensional (2-D) situations for binary alloys. The coalescence process is shown to originate in small necks and involve rapidly changing liquid/solid interface curvatures.