A Kinetic Model of Precipitate Evolution
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assumed to be isotropic and particles were assumed to be spherical. 3) Only binary alloys decomposing into stoichiometric product phases were considered. 4) Atomic rearrangements associated with forming the product phase were assumed to occur quickly relative to the solute arrival rate. 5) Consistent with classical nucleation theory [31, it was assumed that the fastest forming embryos are those following the lowest free energy path and that this path is reasonably defined by the bulk thermodynamic free energy of the product phase together with interfacial and strain energy contributions. Embryos which form along alternative energy paths were considered to have a negligible effect on the overall transformation rate. Consistent with these simplifications, a general set of master equations describing the precipitate size evolution can be written as >N(i = _ (g(i,j) N(1) + e(i,j) ) N(i,j)
(I
+ g(i- l,j) N(1) N(i- l,j)+ e(i + l,j) N(i + 1,j) d(N(t))
-
= -2 g(l,j) N(I) N(l) + 2 # phases Y-e(2,j) N(2,j) (Itl # phases o0 YE [g(k - 1,j) N(l) N(k - 1,j) - e(k,j) N(k,j)] j=l k-3
477 Mat. Res. Soc. Symp. Proc. Vol. 398 01996 Materials Research Society
(1 b)
where N(i,j) is the number density (m- 3 ) of phase j particles containing i solute atoms plus the appropriate number of solvent atoms to maintain the stoichiometry of the specified phase. The rate coefficients, g(i,j) and e(ij), indicate the rates of attachment and emission, respectively, of one monomer to or from a phase j particle of size i. The evolution of the monomer number density is written separately in Eq. 1b because no emission is possible from a monomer and emission from every greater size class results in the formation of one monomer along with a particle of the original size minus one. The factor of two in the first terms of Eq. lb arises because dimer emission forms two monomers and, likewise, dimer formation takes two monomers. The rate coefficients, g(i) and e(i) (with the phase designation omitted for clarity), were defined separately for precritical "embryos" and for stable product phase precipitates in the spirit of the classical theories of nucleation and growth. Precipitates were considered stable once they had exceeded the critical radius, r*, plus a thermal factor, 6, defined as in [4, 51. Particles with radii less than r* + 8 were considered to be thermodynamically unstable with respect to the matrix phase and to evolve by local atomic jumps with an energy barrier equal to the free energy change upon transforming a size i embryo to an embryo of size i+l. The growth rates for embryos of size i>l were defined similarly to the derivation of Kelton and coworkers [6] gi)=0(6(y~ 6~) ,,)) De
a/kTv
)(l)~A,+,je~ (AGv + AGs)(Vi+l ~Vi) +y(Ai±1 A exp (kT)(2
-
Ai)J 2
where a is the composition dependent matrix lattice parameter and a / 2 is the atomic jump length for an fcc lattice. The solute diffusivity is represented by the diffusivity prefactor, Do, the barrier to solute migration via the monovacancy mechanism, Ea, and the monovacancy concentrat
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