Solid-liquid equilibria in nanoparticles of Pb-Bi alloys

The melting behavior of isolated nanoparticles of Pb-Bi alloys was observed by hot stage transmission electron microscopy. Dark-field images acquired as a function of temperature permitted the melting behavior of individual nanoparticles to be characterized as a function of their composition and size. In this way, the experimental measurements could be used to find the temperature at which the onset of melting occurred and the temperature at which the transition from a liquid-solid two-phase particle to a complete liquid particle occurred. From these data, phase diagrams of individual, isolated nanoparticles were constructed as a function of the size of the nanoparticle. Several deviations from the melting behavior of bulk materials were observed. For particles of a given composition the melting temperature is lowered relative to the melting temperature of bulk material and the depression of temperature increases with the inverse of the particle radius. The liquid and solid phases follow melting paths that form liquidus and solidus bands on the temperature-composition phase diagram. The range of two-phase coexistence shrinks as the solute concentration decreases, and the liquidus and solidus bands finally coalesce into a single line at low solute concentrations in apparent violation of the Gibbs phase rule. Of particular interest is the observation that melting begins and ends abruptly as temperature is increased. Considering the isolated particle as a thermodynamic system in contact with a heat reservoir and a pressure reservoir, the conditions for two-phase equilibrium in the particles were calculated from the total internal energy of the particle and the reservoirs. The loci of parameters of the calculated minimum points of the total energy were compiled to calculate a phase diagram for alloy nanoparticles as a function of their size. This calculated diagram contains all of the special features observed experimentally and agrees quantitatively with the experimental measurements. These features should be sufficiently general to apply to other alloys. The quantitative conformance of the experimental with the calculated results is sensitive to the material properties, in particular to the composition dependence of the surface energy of the liquid.