Fluid Flow and Defect Formation in the Three-Dimensional Dendritic Structure of Nickel-Based Single Crystals
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DIRECTIONAL solidification provides control of the orientation and growth of columnar grain and single-crystal materials.[1,2] This advancement in processing has yielded a significant improvement in the high-temperature creep and fatigue properties of nickelbased superalloys.[1–4] Unfortunately, depending on alloy and process conditions, directional solidification may result in the formation of grain defects such as misoriented, high-angle grain boundaries and/or freckle chains.[4–7] The frequency of such defects has been shown to be related directly to slower cooling rates that characteristically produce coarser dendritic structures and larger interdendritic porosity.[7] Advanced alloys with higher levels of refractory alloying elements and large, more geometrically complex cast components are even more prone to defect formation. Thus, accurate solidification models that can predict the conditions under which these defects form as well as provide input
to alloy and component design can clearly provide substantial benefits.[5,6,8–16] During directional solidification, an inversion in the solute density gradient can lead to localized fluid flow, resulting in flow channels compositionally rich in interdendritic solute.[5–11] These channels or ‘‘chimneys’’ are the precursor event to the breakdown of singlecrystal solidification, as the local transport of molten alloy upward through the melt can fragment, erode, and transport solidified material to various locations in the melt. These transported fragments migrate primarily toward the walls of the casting and may form large misoriented grains or chains of fragments, termed ‘‘freckle chains’’ shown in Figure 1. These grain defects develop when a critical soluteinduced density gradient allows the buoyant forces in the melt to overcome the frictional drag forces of the environment; therefore, the Rayleigh number (Rah) is often used as a quantitative predictor for the conditions associated with defect formation.[8,11,13,14] Rah ¼
J. MADISON, Senior Member of Technical Staff, is with the Computational Materials Science & Engineering Department, Sandia National Laboratories, Albuquerque, NM 87185. Contact e-mail: [email protected] J.E. SPOWART, Senior Materials Research Engineer, is with the Air Force Research Laboratory/RXBC, Wright Patterson AFB, OH 45433. D.J. ROWENHORST, Metallurgist, is with the Naval Research Laboratory, Washington, DC 20375. L.K. AAGESEN, Post-Doctoral Associate, and K. THORNTON, Associate Professor, are with the Department of Materials Science & Engineering, University of Michigan, Ann Arbor, MI 48109. T.M. POLLOCK, Professor, is with the Department of Materials, University of California at Santa Barbara, Santa Barbara, CA 93106. Manuscript submitted January 31, 2011. Article published online July 22, 2011 METALLURGICAL AND MATERIALS TRANSACTIONS A
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The density gradient in the liquid is denoted by the term (Dq/qo), g is the acceleration because of gravity, a is thermal diffusivity, m is kinematic viscosity, and K is the average permeabil
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