Pb Nanoprecipitates in Al: Magic-Shape Effects due to Elastic Strain

We present a theory for size-dependent shapes of Pb nanoprecipitates in Al, introducing the concept of “magic shapes,” i.e., shapes having near-zero homogeneous elastic strains. Our quantitative atomistic calculations of edge energies show their effect on precipitate shape to be negligible, thus it appears that shapes must be due to the combined effect of strain and interface energies. By employing an algorithm for generating magic shapes, we replicate the experimental observations by selecting magic-shape precipitates with interfacial energies less than a cutoff value.

Figure 1

The upper right-hand inset shows a typical Pb precipitate in Al with the precipitate size defined as $\overline{A}=(A_1+A_2)/2$. The red points show experimentally measured aspect ratios for a large experimental sample of such nanoprecipitates. The blue line shows the aspect ratio predicted by edge energies from the quantitative atomistic calculations described in the text. The square symbols plot the predictions of the magic-shape theory described in the text. The upper left inset shows two precipitates with a notch along one edge, also consistent with the magic-shape theory.

Figure 2

Summary of EAM calculations for Pb in Al. The upper plot shows $E_{\mathrm{precip}}$ as a function of the cube root of the number of Pb atoms in the precipitate. The insets show the three shapes considered: octahedra, tetrakaidecahedra, and cuboctahedra. The relative precipitate energies for these three shapes can be compared at a particular value of $N_{\mathrm{Pb}}^{1/3}$ (corresponding to a given volume). The lower plot is constructed from the octahedra calculations by subtracting $E_{\mathrm{inter}}=A_{111}{\gamma }_{111}$ from the precipitate energy, leaving $E_{\mathrm{strain}}+E_{\mathrm{edge}}+E_{\mathrm{vertex}}$ (plotted as red points) and fit by a parabola for each value of $\Delta$ (see Table ). The minima of the parabolas represent the energies of zero-strain octahedral precipitates. The slope of the black line tangent to the parabolas gives the edge energy ${\epsilon }_{111\mathrm{\text{−}}111}=-19.5\mathrm{meV}/\AA$.