Singular electrostatic energy of nanoparticle clusters

TL;DR

This paper derives a general formula for the electrostatic energy of nanoparticle clusters at small separations, revealing a singular logarithmic dependence and showing that more compact clusters are energetically favored.

Contribution

It introduces a new approach to calculate electrostatic energy in nanoparticle clusters considering quantum charge constraints and singular capacitance effects.

Findings

Energy exhibits a logarithmic divergence as particles approach contact.

More compact clusters are energetically more stable.

The results explain the relative abundance of certain nanoparticle clusters.

Abstract

The binding of clusters of metal nanoparticles is partly electrostatic. We address difficulties in calculating the electrostatic energy when high charging energies limit the total charge to a single quantum, entailing unequal potentials on the particles. We show that the energy at small separation $h$ has a singular logarithmic dependence on $h$. We derive a general form for this energy in terms of the singular capacitance of two spheres in near contact $c(h)$, together with nonsingular geometric features of the cluster. Using this form, we determine the energies of various clusters, finding that more compact clusters are more stable. These energies are proposed to be significant for metal-semiconductor binary nanoparticle lattices found experimentally. We sketch how these effects should dictate the relative abundances of metal nanoparticle clusters in nonpolar solvents.