Dendritic growth-A test of theory
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HE study of dendritic crystal growth has remained an active subject for experimental and theoretical inquiry for about a quarter century. The modern impetus to this area of crystal growth science may be traced to the experimental studies of Chalmers and his collaborators/-6 who demonstrated that dendrites represent the most advanced stage of interfacial instability in a wide class of materials, and to the theoretical work of Ivantsov,7 who showed that some aspects of dendritic growth could be described quantitatively with heat flow theory applied at a suitable microscopic level. Quite naturally, as the concepts of solidification and crystal growth were refined and the roles of thermal and solute diffusion (along with interfacial processes such as capillarity and molecular attachment) were delineated, a multiplicity of theories arose attempting to describe dendritic growth. Table I lists the most significant theories offered to date, with annotations to distinguish their essential differences and expose their similarities. It is notable, however, that these theories all involve steady-state descriptions of dendritic growth, and none are concerned with time-dependent features of the process, such as side-branching and remelting. The inherent limitations imposed on theory by the steady-state restriction have never been discussed adequately nor have these theories been critically tested by experiment. It is nonetheless well known that all steady-state theories of dendritic growth lack M. E. GLICKSMAN is Chairman, Materials Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12181. R. J. SCHAEFER and J. D. AYERS are Physicist and Metallurgist, respectively, Materials Science Division, Naval Research Laboratory, Washington, D.C. 20375, where M. E. Glicksman was formerly affiliated. Manuscript submitted May 12, 1976. MET ALLURGICAL TRANSACTIONS A
a truly predictive capability, insofar as they yield at most a range of permissible steady states at a given supercooling, each of which is characterized by some smooth, time invariant shape and a compatible tip speed. Prediction from theory of the specific growth rate-supercooling relationship always requires an additional criterion-typically it has been the ad hoc condition that dendritic crystals grow at the maximum velocity, although alternative criteria involving entropy or free energy production have also been suggested from time to time. Although it is beyond the scope of this paper to show precisely why all steadystate theories of dendritic growth suffer from this serious limitation, it shall suffice to say that the number of unknowns exceeds the number of inherently independent equations by one, and that (right or wrong) the ad hoc condition provides the missing mathematical relationship. Among the theories appearing in Table I, only three are in a form well suited to experimental testing: 1) Trivedi's theory-an advanced form of Temkin's original model; 2) the "modified" version of Ivantsov's original model; 3) Nash and Glicksman's self- consistent mod
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