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Last Stage Solidification of Alloys: Theoretical Model of Dendrite-Arm and Grain Coalescence



M Rappaz, A Jacot, William J. Boettinger


Hot tearing in castings is closely related to the difficulty of bridging or coalescence of dendrite arms during the last stage of solidfication. The details of the process determine the temperature at which a coherent solid forms; i.e., a solid that can sustain tensile stresses without rupture. Based on the disjoining pressure concept used in fluid dynamics, a theoretical framework is established for the coalescence of primary phase dendritic arms within a single grain or at grain boundaries. For pure subtances, approaching planar liquid solid interfaces coalesce to a grain boundary at an undercooling,Δ}Tb, given by:Δ}Tb= (Δ}Γb)/γ = ((γgb-2γsl)/Δ}Sf)x(1/δ where δ is the thickness of the solid-liquid interfaces, Δ}Γb is the difference between the grain boundary energy and twice the solid-liquid interfacial energy, divided by the entropy of fusion. If γgb<2γsl, then Δ}Tb<0, and the liquid film is unstable and coalescence occurs as soon as the two interfaces get close enough (at a distance on the order of δ): this situation, typical of dendrite arms belonging to the same grain (i.e. γgb=0), is referred to as attractive. The situation where γdgb^=2γsl is referred to as neutral; i.e., coalescence occurs at zero undercooling. If γgb>2γsl, the two liquid solid interfaces are repulsive. In this case, a stable liquid film betwen adjacent dendrite arms located across such grain boundaries can remain until the undercooling exceeds Δ}Tb. For alloys, coalescence is also influenced by the concentration of liquid film. The temperature and concentration of the liquid film must reach a 'coalescence line' parallel to, but Δ}Tb below, the liquidus line before coalescence can occur. Using one dimensional (1D) interface tracking calculations, diffusion in the solid phase perpendicular to the interface (back-diffusion) is shown to aid the coalescence process. To study the interaction of the interface curvature and diffusion in the liquid film parallel to the interface, a multi-phase field approach has been used. After validating the method with the 1D interface tracking results for pure substances and alloys, it is then applied to 2D situations for binary alloys. The coalescence process is shown to orginate in small necks and involve rapidly changing liquid-solid interface curvatures.
Metallurgical and Materials Transactions A-Physical Metallurgy and Materials Science
No. 3


coalescence, dendrite bridging, grain boundaries, solidifaction


Rappaz, M. , Jacot, A. and Boettinger, W. (2003), Last Stage Solidification of Alloys: Theoretical Model of Dendrite-Arm and Grain Coalescence, Metallurgical and Materials Transactions A-Physical Metallurgy and Materials Science (Accessed April 17, 2024)
Created February 28, 2003, Updated October 12, 2021