Calorimetric Energies and Chemical Potentials of Metal Atoms in Catalytic Nanoparticles on Oxide and Carbon Supports: Improved Size Dependencies and Adhesion Energies
The chemical potential of metal atoms, μM, in supported metal nanoparticles is an important descriptor related to both the catalytic activity and the stability of the nanoparticles. Here, we derive an expression relating μM to the radius of the particle’s contact area with the support and the adhesion energy at the metal/support interface (E adh ) that assumes the particles have the shape of spherical caps but of arbitrary contact angle with the support (θ c ) and includes an empirical correction for the increase in metal surface energy and adhesion energy with decreasing radius of curvature. We then show that, at any assumed contact angle, we can simultaneously fit previously reported measurements of both calorimetric μ M (from heats of metal vapor adsorption during nanoparticle growth by vapor deposition) versus metal coverage data and the He + low-energy ion scattering (LEIS) intensities for the metal and/or support versus metal coverage (using our recently developed spherical cap model for quantitative LEIS intensities), to determine the particle size versus coverage and E adh . Only one choice of contact angle gives a pair of values for contact angle and E adh , which is consistent with the Young–Dupré equation for the equilibrium shape of a spherical particle. At this equilibrium shape, we then applied this spherical cap model (SCM) to reanalyze microcalorimetric metal chemical potentials and LEIS signals versus coverage data for nine metal/support combinations that were previously analyzed by assuming that the particles had the shape of hemispherical caps, i.e., with a contact angle of 90°. We show that this revised approach gives close agreement with the calorimetric and LEIS data; the best-fit contact angles vary from 64 to 84°, correcting the earlier assumption of 90°. Furthermore, these results provide significant accuracy improvements in particle size versus coverage, metal chemical potential versus size and coverage, metal/support adhesion energies and contact angles for Cu, Ag and Au on CeO 2 (111), Ni on MgO(100), Ag on Fe 3 O 4 (111) and TiO 2 (100), and Ag, Ni and Pd on Ni-supported graphene. This revised approach is much more broadly applicable than the earlier hemispherical cap model (HCM).