On the Kinetics of Precipitate Dissolution

Abstract

A theory of precipitate dissolution has recently been proposed by Aaron (Metal Sci. J., 1968,2, 192)in which it is implied that a previous treatment by Thomas and Whelan (Phil. Mag., 1961, 6, 1103), where dissolution was considered to be approximately the reverse of growth, is in error in this assumption. Moreover, the time-dependence of the radius of a dissolving precipitate according to Aaron (R = Ro -Kv Dt) disagrees with that of Thomas and Whelan (dR2/dt = -kD). It is pointed out that the "disagreement" arises because the situations treated are themselves dissimilar. Aaron's result is essentially one-dimensional and is derived from the transient part of the diffusion field in one dimension. The result of Thomas and Whelan is for threedimensional diffusion and is obtained essentially from the steady-state part of the diffusion field around a spherical precipitate. The dissolution of a spherical precipitate, taking account of transient effects,is solved using an approximation for the diffusion field. The validity of the approximation is discussed and it is concluded that the time-dependence dR2/dl = -kD is reasonable for the case of a-phase precipitates in Al + 4 wt.-% eu alloy. (>(r) I I I I <-R(t)~I ,,~~S 1'" /2 Fig. 1 Schematic diagram of the solute concentration in the vicinity of a dissolving precipitate. pc is the solute concentration inside the precipitate, which is initially in equilibrium with a uniform concentration Pe of solute in the matrix. At time t = 0 the temperature is raised by AT and the equilibrium surface concentration becomes Ps. The full line gives the solute concentrationRo is the initial half-thickness or radius.For the one-dimensional case, the concentration profile Ap in Fig. 1 for r~R is then 3 Vol. 3Equating the flux of solute at the interface to the rate of loss of solute from the precipitate we obtain where D is the coefficient of diffusion of solute atoms in the matrix. Substitution of (1) into (2) gives •.. (4) · .. (5) · .. (3) · .. (2) · .. (1) k -R=Ro--yDt y1t dR (OAP) -(Pc -Ps) -= -Ddt or r=R ( r -R) Ap (r,t) = (Ps -Pe) erfc ---== 2yDtwhere k = 2(ps -Pe)/ (Pc -Ps) which integrates to Ro is the initial half-thickness of the precipitate. Equation (4) is substantially the same as Aaron's result.The constant multiplying y Dt in equation (4) is not exactly the same as that occurring in Aaron's equation. For PcP s, Pe (which is the case in practice), the discrepancy is a factor of ly1t, which is considered to be insignificant in view of the rather different details of the two treatments.Aaron 1 has proposed a theory of diffusion-controlled dissolution of a precipitate, the results of which are stated to be at variance with those of a previous treatment by Thomas and Whelan 2 and also, incidentally, with the experimental evidence advanced by them. The object of this paper is to point out that the supposed discrepancy is not a real one because the situations are basically different, Aaron's treatment being essentially one-dimensional while that of Thomas and Whelan is three-dim...

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