Two-phase equilibrium in individual nanoparticles of Bi-Sn

Abstract

When evaluating the path of phase transformations in systems with nanoscopic dimensions, one often relies on bulk phase diagrams for guidance because of the lack of phase diagrams that show the effect of size. In order to provide insight into how phase diagrams can very when a high surface curvature exists, binary alloys of Bi and Sn were investigated as a collection of individual crystalline particles vapor deposited onto amorphous carbon substrates in ultra-high vacuum. These crystallites were annealed after deposition to equilibrate the phases and structures. After annealing, they were transferred to the transmission electron microscope for analysis of the phase state as a function of composition and surface curvature, i.e., particle radius. Individual crystallites were analyzed with respect to crystallinity, two-phase or one-phase coexistence, and composition. The data show that there is a critical size below which there is no limit to the solubility, in strong contrast to that found in the bulk system, which is a simple eutectic alloy with less than 0.3 pct solubility on the bismuth-rich solid solution side of the phase diagram and about 15 pct on the tin-rich side. The change in solubility limit with size was found to be equally strong in both the tin-rich terminal solid solution and the bismuth-rich terminal solid solution. A thermodynamic approach to using free-energy expressions modified to account for surface curvature can be successful in showing the shift in solubility with size. It is shown that the appropriate thermodynamic potential to minimize is a modified Helmholtz free energy.