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Scripta METALLURGICAVol. 251-254, 1968 Printed in the United StatesPergamon P r e s s, IncON THE AGEING OF DENDRITESManfred Kahlweit Max-Planck-Institut fuer Physikalische Chemie, Goettingen Germany(Received January 18, 1968) NH4C1 d e n d r i t e sin aqueous solution tipsage by dissolutiondown t o t h e i rroots,a processof thesidearms from t h e i rdu-ringwhich the radii of the arms remain practically unchanged.Approximating the arms by cylinders with spherical tips and assuming diffusion controlled dissolution the agreement between the measured and calculated dissolution rate is satisfactory. In some recent papers Flemings and coworkers (I) have studied the ageing of dendrites during the solidification of binary metallic melts. They observed a change in dendrite arm spacing with time, suggesting (2) this to come about either by the thinning of the thinner arms due to their lower melting point (model I), or by the melting off of the arms at their roots (model II).

The same process has previously been investigated by Papapetrou (3) and Kliya (4) during the precipitation of ionic crystals from aqueous solutions. For experimental investigations the latter systems seem to be the more convenient,since they are transparent,so that the change of mor-phology may be continuously observed under a microscope.

This avoids the problems of quenching and sectioning of the specimen. The system investigated was that of NHC1 precipitating from aqueous solution. The solution was saturated at 35°C, poured into a closed glass chamber of I mm height and then cooled to 25°C on a thermostated hot stage2519.52ON THE AGEING OF DENDRITESof a microscope.Vol. 5A seed crystal was then introduced leading to the forma-tion of NHC1 dendrites. Keeping the chamber at 25 0.05°C pictures of a chosen area were then taken at different times. It was found that model I of Flemings does not hold in this case, but that instead the arms dissolve from their tips down to their roots, a process during which the radii of the arms remain practically unchanged.

This observation is consistent with the fact that the driving force for the ageing process is the increased solubility at curved interfaces. Approximating an arm by a cylinder with a spherical tip, the increase i n solubility at the tip will then be twice as high as that at the walls of the cylinder. Furthermore,it is accepted that dendrites grow in the di-rections of the highest growth rates of the crystals, which a r in gener a also the directionsof the highest dissolutions rates.Since the mean concentration of the solute in the parent phase is determined by the mean radius of all curved interfaces in the system, the arms with the smallest radii will dissolve first. The mean radius thus increases, dissolve,overtaking the radii of the next thicker arms, causing them to and so forth. Accordingly the distribution curve over the armradii tends with time to zero at the lower end, while it remains practically unchanged at the upper end (during the time of observation).Thedistribution curve over arm lengths l, on the other hand, moves with time continuously towards smaller values of 1. A process following model II of Flemings was observed only when the solution was heated to bring the crystals back into solution.

Since the dissolution of ionic crystals i in general, trolled,diffusion con-one may write for the flux density at the tipsRD.in which D denotes the diffusion coefficient phase, and c s the solubility.of the solute in the parentThe number of moles dissolving from a halfsphere per unit time is then given byVol. 5ONTHEAGEINGOF253DENDRITES(2) The increase in solubility of an ionic crystal dissociating into two ions is given by 2 in(Cs(r) ) c®2aV = RTr;c= = Cs(r=®),(3)in which denotes the interfacial tension between the crystal and the parent phase, V the molar volume of the crystal, R the gas constant, the absolute temperature.

The logarithm,and TIn view of the size of the arms one may expandso thatCs(r ) - c s ( ).RT.(4)rThe mean radius can be determined from the distribution curve over r. But, since this curve is determined by the growth conditions drites and, furthermore,changes with ageing time,of the den-it seems to be suffi-cient to replace by the radius of the main arms of the dendrite (effectively setting Cs( ) = c®). Eq.(2) then reduces to 2-DaVc® -.(5)RT Setting for the number of moles per armn1 r2 -1(6)V and assuming,as above, that the radii of the arms remain unchanged duringtheir dissolution,one then finds for the dissolution rate dl -dt2 D V 2 c® I 'RTr2.(7)254ON THE AGEING OF DENDRITESVol. 5It may be noticed that the combination of factors determining the dissolution rate is the same as that found in Ostwald-ripening. Because of the assumptions made (neglecting the constrictions of the arms at their roots, and setting Cs( ) = c®) one would expect the dissolution rate to be a little smaller than the calculated value at the beginning, but tending towards this value with increasing ageing time.

The change of arm length can easily be measured, yielding as an average value in these experiments(r2)= 3. 7 10 - 1 3 c m 3sec -I. ExpFor comparison with eq.

(7) we have set D = 1.910 -5cm2sec -I; = 40(+10) ergcm -2; V = 35cm3mo1-1; c® = 5. 7 1 0 - 3 m o l c m - 3;and T = 2.98102 OK, giving2 D a V 2 c® = 4. 3 (+1.1) 10 -13 cm3 s e c -1 RT in satisfactory agreement. The number of arms dissolving, leading to an increase in arm spacing, depends on the distribution of arm radii and lengths, lIrther work, to predict this distribution, is in progress. I am indebted to Mr.

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Hanitzsch and Dr. Kirkwood for valuable assistance and discussions.I a) T.Z. Kattamis and M.C. Flemings, Trans. AIME 236, 1523 (1966).

Kattamis, J.C. Coughlin, and M.C. Flemings, ibid.

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29, 1504 (1967). Kattamis, U.T. Holmberg, and M.C. Metals 95, 343 (1967). Papapetrou, Z. 92, 89 (1935).4)M.O.

Kllya, Soviet Physics - Crystallography (Engl. Transl.) , 456 (1956).

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