On the rate of dendrite arm coarsening
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I.
INTRODUCTION
IT is well established that surface-tension driven coarsening determines the final scale of cast dendritic microstructures, which is usually characterized by the secondary dendrite ann spacing, A. t~'2] The rate of coarsening is limited by the rate of heat conduction in pure materials and the rate of mass diffusion in alloys. As a consequence, h is generally related to time, t, by the usual law h 3 = Kt. Predicting the rate of coarsening of secondary dendrite arms hinges, therefore, upon the ability to calculate K from material constants and casting parameters. Despite significant engineering relevance of the phenomenon, the geometrical complexity of arrays of coarsening dendrite arms has precluded the derivation of exact expressions for K. Several approximate expressions have been proposed in the literature, t3"4"51the most satisfying of which is given in a recent article by Kirkwood. t21 Most of these calculate the time for remelting of a dendrite arm from its tip, with the assumption that diffusion to the remelting arm tip is according to the Zener approximation for diffusion from a sphere into an infinite bath of liquid. This assumption limits validity of these derivations to low volume fractions of the solid phase, because neighboring dendrite arms will influence the temperature or concentration profile around remelting dendrite arm tips, rendering the Zener approximation untenable. This paper proposes a different approximate derivation of K. Focus is placed on growing arms, because they grow continuously. This does away with the need to view dendrite ann coarsening as a sequential process of suddenly disappearing arms, as has been done so far. A finite volume fraction soli d is assumed in calculating the rate of diffusion, which renders the present expressions valid over most of the solidification process. Isothermal coarsening is considered first, and resulting expressions are then integrated to treat coarsening during solidification. II.
ISOTHERMAL COARSENING
Consider an array of tertiary and secondary dendrite arms held for a certain time t at a constant average ternA. MORTENSEN, ALCOA Associate Professor, is with the Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139. Manuscript submitted April 18, 1990. METALLURGICAL TRANSACTIONS A
perature, T. It is well known that several mechanisms contribute to coarsening of dendrite ann arrays: remelting of the smallest arms from their tip, radial thinning of the slimmest arm, or coalescence of dendrite arms by solidification of the liquid trough that separates two arms. We ignore here coalescence mechanisms and simplify the geometry of ripening secondary and tertiary dendrite arms to an array of parallel circular cylinders. On average, the dendrite arm spacing increases, and hence, the dendrite arms get thicker. For cyclindrical dendrite arms surrounded by a pool of liquid of uniform width, the average dendrite arm diameter, D, is related to the average dendrite arm spacing A by D = ;t ~
