11.ppt.growth

11.ppt.growth - T Y Tan 11 KINETICS OF PHASE CHANGES II...

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T. Y. Tan 1 11. KINETICS OF PHASE CHANGES II: PRECIPITATE GROWTH In this chapter we discuss the precipitate growth process, which falls into two broad catego- ries. For cases for which the α solution and the β precipitate composition differs little, the pre- cipitate growth process is limited or controlled by the interface reaction. For cases for which the α solution and the β precipitate compositions differ considerably, precipitate growth is controlled by solute diffusion. In the absence of a volume misfit, the interface reaction controlled precipi- tate growth process is always faster than that controlled by the solute diffusion process. 11.1 Interface Reaction Controlled Growth In this case, precipitate growth is controlled by processes occur in the immediate vicinity of the interface. This requires the assumption that the compositions of the α solution and the β pre- cipitates are the same. This condition is nearly satisfied in many cases, owing to either the com- position difference of the two phases being small, such as the case for which the α solution com- position is chosen to be very close to that of the β phase, or the solute diffusion process in the α solution being very fast so that the effect of the solute concentration gradient in the solution may be ignored. We assume that there is no (volume) misfit between the two phases, or, if there is a misfit, it is relaxed to an extent that the strain energy becomes ignorable. We further assume that the inter- face is isotropic in that the interfacial energy density σ is independent of crystallographic orien- tations. Under these conditions, the β precipitate will be spherical in shape. The system Gibbs free energy is given by G = n g β + n' g α + 4 π r 2 σ , (11.1) where n is the number of atoms in the β precipitate, n' is the number of atoms in the α phase, g β is the formation Gibbs free energy of an atom in the β phase, g α is that in the α phase, and r is the radius of the β precipitate. The number n is given by n = 4 π r 3 3 Ω β , where Ω β is the volume of an A x B molecule in the β phase. The process of precipitate growth may be described by the reaction
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T. Y. Tan 2 B B ' , (11.2) where B is an atom in the α phase and B ' an atom in the β phase. The chemical potential or driv- ing force per atom for β precipitate to grow is μ = dG dn = δ g βα + 2 Ω β σ r , (11.3) where δ g βα =g β -g α <0. Furthermore, in the precipitate growth regime μ <0 should also hold. With Eq. (11.3), the Gibbs free energy per atom associated the two phases is shown in Fig. 11.1. If each α phase atom at the interface will make ν attempts per sec to jump across the interface to become a β phase atom at the interface, the interface will advance at the rate of ∨λ exp - g m k B T , while the reverse process will make the interface retreat at the rate of ∨λ exp - g m + μ k B T , where g m is a migration energy barrier preventing the atom jump from occurring. Thus, the net
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11.ppt.growth - T Y Tan 11 KINETICS OF PHASE CHANGES II...

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