Precipitation nucleus energy

Up to this point we have dealt with random nucleation processes in a homogeneous phase. However, in solids with many structural imperfections, it is very likely that nonrandom, heterogeneous nucleation takes place. The basic idea of this mode of nucleation is that the energy of the imperfection is brought into the energy balance of the critical nucleus. Let us demonstrate the basic idea with a dislocation line as the preferred nucleation site. We assume that a cylindrical precipitate (p) forms along the dislocation line and, in the spirit of Eqn. (6.2), we obtain per unit length of the nucleus... 

Two-Component System with Isotropic Interfaces and Strain Energy Present. An example of this case is the solid-state precipitation of a 5-rich (i phase in an A-rich a-phase matrix. For steady-state nucleation, Eq. 19.16 again applies. However, for a generalized ellipsoidal nucleus, the expression for AQ will have the form of Eq. 19.28. Also, /3 must be replaced by an effective frequency, as discussed in Section 19.1.2. 

The simplest case occurs when the a-phase and /3-phase crystals have different compositions but still match almost exactly in all three dimensions. The critical nucleus can then form with a coherent interface and is therefore of relatively low energy.1 Also, any strain energy will be small. This condition is met during the precipitation of Ag-rich precipitates in a A1 + 4 at. % Ag matrix [8] and Co-rich precipitates in a Cu + 1 at. % Co matrix [9] where the precipitates are coherent and essentially spherical in shape. 

In order for a solid to precipitate from homogenous solution, first a nucleus has to form. The formation of a particle is governed by the free energy of agglomerates of the constituents of the solution. The total free energy change due to agglomeration, AG, is determined by... 

If a substance becomes less soluble by a change of some parameter, such as a temperature decrease or the addition of a nonsolvent, the solution may enter a metastable state with the formation of some precipitate or nuclei. The classical theory considers the nucleus to consist of a bulk phase containing Nf molecules and a shell with Nf molecules which have a higher free energy per molecule than the bulk. This is shown schematically in Figure 13.1. The Gibbs free energy of the nucleus G is made of a bulk part and a surface part [4] ... 

In the section below, we discuss the correlation between the critical nucleus and surface energy (23). A more complete analysis of precipitation must take into account the chemical potential of the nucleus formed in equilibrium with the reactional media. In such a condition, the nucleus possesses free Gibbs energy, as follows [23,98,108,124] ... 

The classic example of precipitate nucleation in metals is the formation of GP zones in Al-Cu alloys. In ceramics, analogous examples include spinel in NiO, rutile in sapphire, or platelets of nitrogen in diamond. When particles are very small, the surface energy dominates. The calculation in Eqs. Box 15.1-Box 15.4 is instructive. Remember that the calculation is for a spherical nucleus and it ignores kinetics kinetics are actually important as we saw in Figure 15.5. 

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