Dendrite growth directions in aluminum-zinc alloys
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DENDRITES are very common in most metallic alloys produced by solidification. Resulting from an instability of the solid-liquid interface, they have a tendency to develop along directions corresponding to convexities (i.e., maximum curvature regions) of the equilibrium shape of a crystal, at least at low speed (i.e., if the attachment kinetics contribution can be neglected). Using Herring’s relations,[5] these convexities correspond to minima of the so-called ‘‘stiffness’’ of the solid-liquid interfacial energy. The stiffness of the interface is defined as the interfacial energy, g sl, plus the second derivative of gsl.* In many cases, but not @ 2 g s‘ *The stiffness of the interface is a tensor given by g s‘ 1 , where the @ti @tj variation of g s‘ with respect to t1 and t2 are measured along two orthogonal directions in the tangent plane. When expressed in the directions of principal curvature of the g plot, the stiffness has two principal values given by @2g g s‘ 1 2s‘ : @ti
always, as shown recently by Karma,[1] the minima of the stiffness plot correspond to the maxima of the gsl plot. Based upon such considerations, it is commonly admitted that dendrite growth directions in cubic metals (fcc or bcc) correspond to ,100., while they are ,110. in bcc tetragonal tin and ,1010. in zinc or ice.[4,6] (Note that hcp dendrites can also grow along the c-axis in CoSm alloys[6] and in zinc, but in this last case at a slower velocity than ,1010. dendrites[4]). Metallic alloys such as fcc nickel exhibit dendrites that are very well constrained to grow along ,100. directions, thus indicating that gsl in this case has a fairly large anisotropy. For example, an anisotropy of 4 pct has been used in the first phase field computations of a solutal dendrite of Ni-Cu.[7] Beyond 6.7 pct anisotropy, ,100. dendrites F. GONZALES, Postdoctoral Student, and M. RAPPAZ, Professor, are with the Computational Materials Laboratory, Institute of Materials School of Engineering, Ecole Polytechnique Fe´de´rale de Lausanne, CH-1015 Lausanne, Switzerland. Contact e-mail: frederic.gonzales@epfl.ch, michel. rappaz@epfl.ch Manuscript submitted February 24, 2006. METALLURGICAL AND MATERIALS TRANSACTIONS A
become facetted in cubic systems. At low anisotropy, however, seaweed microstructures seem to form, at least in two dimensions, very often with the presence of doublons, i.e., deep and narrow liquid channels remaining in between two solid parts. Predicted by theory[8] and phase field simulation,[9] such microstructures have also been observed in situ in organic alloys.[10] Aluminum alloys, which are of great technological interest, seem to have a very low solid-liquid interfacial energy anisotropy. Indeed, in the well-studied Al-Si or Al-Cu alloys, primary dendrite trunks grow along ,100. directions, but the secondary arm orientation is strongly biased toward the temperature gradient. By analyzing the shape of remaining liquid pockets in aluminum alloys, Napolitano et al.[2,3] estimated an anisotropy of gsl of the order of 1 pct. This low anisotrop
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