percent cobalt), and in other face-centered cubic alloy systems (e.g., copper-1.9 atomic percent titanium and nickel-14 atomic percent aluminum), the nucleation rates may be orders of magnitude larger, and the corresponding number densities of precipitated particles increase into the range where the atom probe/field-ion microscope becomes an exceptionally powerful instrument for quantifying the early stages of precipitation.87 In addition to number densities, this research technique is also capable of measuring small-particle compositions and particle-size distributions below 10 angstroms. However, in this high-particle-density regime, the nucleation stage is overlapped by solute depletion of the parent phase and by Ostwald ripening, both of which act to decrease the observed particle densities by re-solution of some of the previously precipitated particles. Nevertheless, the earliest detectable precipitates are found to have their equilibrium composition.

In a salient advance, these complications have been sorted out analytically, and calculated and experimental results for the critical nucleus size, mean particle size, and number density are compared in Figure 47 for a nickel-47 atomic percent aluminum alloy.87,88 Calculated nucleation rates, based on homogeneous nucleation theory, are also shown in Figure 47. The interfacial energy (σ) and diffusivity (D) adopted for these computations can be derived directly from the longer aging times where only Ostwald ripening is operative. There is some uncertainty, however, as to whether the state of coherency characteristic of the nucleation stage is fully maintained throughout the coarsening period, but this does not interfere with the overall interpretation that follows.

It is evident from Figure 47 that the earliest measurements are already at a stage of decreasing particle-number density (Nv) despite the initially increasing transient nucleation rate (J). This means that Ostwald ripening sets in almost at the beginning of precipitation in this alloy and limits the fineness of the second-phase dispersion that might otherwise be available for maximum precipitation strengthening. This circumstance may also account for the rather constant mean particle size during the initial precipitation process, the growth of some nucleated particles being offset by the re-solution of others. In any event, the subsequent decrease in the calculated nucleation rate reflects the decreasing supersaturation of the parent solid solution as the transformation proceeds, and the critical nucleus size (R*) increases accordingly.

Because of the complexities of the foregoing kinetic analysis, it has been applied so far only to homogeneous nucleation in face-centered cubic systems with coherent interphase interfaces. The next big step must cope to an increasing extent with dissimilar parent/product phases, semicoherent interfaces, and heterogeneous nucleation to establish a more general understanding and a more useful control of precipitation processes in a wider variety of alloy systems. It is a formidable long-range task, to be sure, but recent

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