Transition from a Planar Interface to Cellular and Dendritic Structures during Rapid Solidification Processing
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where G=2DLGL/kRAT° and a = DLGL/mLRCo are dimensionless groups introduced by Trivedi and Flemings respectively [13,14]. Other symbols used here are explained in Appendix A and are the same as employed in Ref. 12. All three parameters, s, a and G reflect the importance of the thermal gradient in determining the stability of the interface. Of the three, however, s is most conveniently used to describe the type of solidification microstructure obtained. Thus, when GL/R is very large, s +l. At the limit of stability of the planar interface s = 1 and R = Rn = DLGL/ATo0 For R > R , the planar interface breaks down to form a cel ular structure, at first,and eventually a dendritic structure when R >> Rn or GL/R - 0. It follows that 0 < s < 1 during dendritic and cellular growth. It is proposed that s be called the Chalmers number, since it was the work of Chalmers and co-workers (duly commemorated in a special issue of Materials Science and Engineering [15]) which lead to the modern understanding of the conditions leading to planar, cellular and dendritic growth in alloy systems.
Mat. Res. Soc. Symp. Proc. Vol. 58. ý 1986 Materials Research Society
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CELL-DENDRITE TRANSITION (CDT) The transition from a cellular to a dendritic microstructure or viceversa is as yet not very well understood. Kurz and Fisher [16] were the first to propose that CDT occurs when s = k, where k is the equilibrium partition ratio. Although recent experimental observations in succinonitrile-acetone (SCN-Ace) alloys seem to confirm this prediction [10, 17], it may be shown quite easily that relaxing certain assumptions and approximations made by Kurz and Fisher would lead to a very different result: s = k/2 at CDT, as shown in Appendix B. In other words, their theoretically predicted transition velocity would differ by almost a factor of 2 from the experimentally observed value, see later in Fig. 2. Also, experimental results in other alloy systems do not seem to agree with this prediction. For example, cellular structures have been reported in SCN-salol for s/k l/e (where e = 2.718 ... is the natural logarithmic base) only "unstable" cellular states are possible. "Stable" cellular states exist only when s < l/e, or s < 0.368. Cellular and dendritic growth can, therefore, only occur for 0.368 < s < 0. Trivedi and co-workers [17] have suggested yet another criterion for CDT. According to these authors, the transition in microstructure occurs when the solute Peclet number p = Rrt/ 2 DL goes through a minimum, as the growth velocity is decreased, at constant GL (i.e. when ap/AR = 0 at constant GL, k, C0 ). Correspondingly, the primary arm spacings, x , have been noted to go through a maximum. (Note that there is no firm theoretical justification for this criterion). The mathematical complexity of the Trivedi analysis makes it difficult to examine, theoretically, the conditions leading to a minimum in p, or a maximum in x . The simpler model for cellular and dendritic growth presented by the author [12] may, however, be used quite conveniently
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