Influence of Adhesion on the Chemical Potential of Supported Nanoparticles as Modeled with Spherical Caps

Abstract

The effect of adhesion on the chemical potential of supported nanoparticles is derived for the case of spherical caps. It is explicitly shown that for the minimum-energy particle shape (neglecting any anisotropy in surface energy), the chemical potential of a spherical cap attached to any flat support material reduces to μ = 2γVm/r, where r is the radius of curvature, γ is the surface energy, and Vm is the volume per mole of atoms. This is identical to the well-known Gibbs–Thomson relation derived instead for free-standing spherical particles. The chemical potential nevertheless depends on the adhesion energy Eadh because this radius r is a strong function of both adhesion (specifically, of Eadh/γ) and particle volume. The approximation of hemispherical particle shape, for which μ = (3γ – Eadh)Vm/r as proposed by Campbell and Hemmingson (ACS Nano 2017, 11, 1196) is exact for γ = Eadh, where it reduces to μ = 2γVm/r. Using hemispheres, or any fixed particle shape, is furthermore shown to be a linear approximation to the exact dependence of μ on Eadh for the minimum-energy particle shape, with error <10% for contact angles between 66 and 120° (i.e., for Eadh/γ = 0.5–1.4). Generally, these approaches only consider the limit of a large radius of curvature, where γ and Eadh are constant. It is known that both γ and Eadh increase with the decreasing r below 4 nm.