Last-stage solidification of alloys: Theoretical model of dendrite-arm and grain coalescence
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␥gb ⫺ 2␥sl 1 ⫽ ␦ ⌬sf ␦
where ␦ 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 ⬍ 2␥sl , then ⌬Tb ⬍ 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 ⫽ 0), is referred to as “attractive”. The situation where ␥gb ⫽ 2␥sl is referred to as “neutral”; i.e., coalescence occurs at zero undercooling. If ␥gb ⬎ 2␥sl , the two liquid/solid interfaces are “repulsive” and ⌬Tb ⬎ 0. In this case, a stable liquid film between adjacent dendrite arms located across such grain boundaries can remain until the undercooling exceeds ⌬Tb. 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 ⌬Tb 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.
I. INTRODUCTION
SOLIDIFICATION of metallic alloys has been extensively studied (dendrite-tip kinetics, microsegregation, coarsening of dendrite arms, etc.), but surprisingly, in the absence of a large fraction of eutectic (i.e., low-concentration alloys), little is known about the last stage of solidification when the primary-phase regions impinge. Yet, the details of the final stages of solidification of multigrain or dendritic materials have significant impact on casting defects such as hot tearing. Hot tearing is caused by a lack of tensile strength of the thin continuous liquid film that remains between dendrite arms up to a very high volume fraction of solid. It is, therefore, important to know when dendrite arms bridge M. RAPPAZ, Professor, is with the Faculty of Engineering, Physical Metallurgy Laboratory, Ecole Polytechnique Federale de Lausanne, CH1015 Lausanne, Switzerland. Contact e-mail: [email protected] A. JACOT, Research Associate, Faculty of Engineering, Physical Metallurgy Laboratory, Ecole Polytechnique Federale de Lausanne, is also with Calcom SA, CH-1015 Lausanne, Switzerland. W.J. BOETTINGER, NIST Fellow, is with the Metallurgy Division, Materials Science and Engineering Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899. This article is based on a presentation giv
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