Scripta METALLURGICA Vol. 18, pp. 463-466, 1984 Pergamon Press Ltd. Printed in the U.S.A. All rights reserved STEADY-STATE DENDRITIC GROWTH AT NON-ZERO CAPILLARITY Richard C. Brower Natural Sciences II, University of California, Santa Cruz, CA 95064 David A. Kessler Department of Physics, Rutgers University, New Brunswick, NJ 08903 Joel Koplikand Herbert Levine Schlumberger-DollResearch, P.O. Box 307, Ridgefieid, CT 06877 (Received October 24, 1983) (Revised February 27, 1984) Recently there has been renewed interest in the dendritic growth exhibited by substances solidifying from a supercooled melt(I,2). The growth process gives rise to a complex morphology of the resulting solid, making solidification a prime example of non-equilibrium pattern formation. Under the assumption that the interface motion is completely controlled by heat diffusion, one can derive a set of equations which should be able to predict the long-time evolution of the growing crystal. To date, most attention has focused on the rate at which the dendritic tip grows and the spatial distribution of the branches generated along the initial needle crystal(3). The purpose of this note is to point out that the long standing assumption(3) that one can find non- isothermal steady-state solutions to the heat diffusion equations may not be true. To do this, we first explain why the numerical procedure of Nesh and Glicksman(4), which purported to demonstrate such solutions, is in fact incapable of testing whether or not such a solution exists. We then give a simple example of an analogous system where one can demonstrate the lack of any steady-state needle crystal and explain why the dendritic growth equations might behave in a similar fashion. Finally, we will comment briefly on the implications this would have for the current theory of the rate of the dendritic growth (5). Many researchers have attempted to generalize the isothermal needle crystal solution of Ivantsov(6) and Horvay and Cahn(7) to include the effect of capillarity. As discussed in Ref. (4), this amounts to solving the non-linear integro-differential equation AO + kK(r) = l~xe-l'¢O-*(x)] O(r,~(r);x,co(x)) dx (1) where AO is the (dimensionless) supercoolng, ~, is the (dimensionless) capillarity, and K(r) - oJ"(r) + co'(r) [1 +co'2(r)]3/2 [1 +o'2(r)]l/2r is twice the local mean curvature of the interfacial position (obtained by rotating the curve z-co(r) around the z axis with r the radius in a cylindrical coordinate system). G is the heat diffusion kernel 1 G(r,co (r);x,~ (x)) -- [ exp- ([r2+x2+ (co(r)-°J (x))2+2rxu]'h) du (2) - -'1 [r2+x2 + (co (r)-co (x))2+2rxu] '~ A dendrite satisfying (1,2) would uniformly translate in the ~ direction with unit velocity. Equation (1) is the demand that the temperature at the liquid-solid interface be correctly given by the local equilibrium Gibbs-Thomson condition. In the k-0 limit, the interface is isothermal and the parabolic profile, denoted by ~motherm*d,solves the resulting equation. 463 0036-9748/84 $3.00 + .00 Copyright (c) 1984 Pergamon Press Ltd.