# Like-Charge Attraction between Metal Nanoparticles in a 1:1 Electrolyte Solution

## TL;DR

This paper demonstrates that metal nanoparticles with like charges can attract each other in a 1:1 electrolyte, a phenomenon analyzed through numerical solutions of the Poisson-Boltzmann equation and a new approximation method.

## Contribution

The authors introduce a modified Derjaguin approximation for efficient calculation of interactions between metal nanoparticles in electrolytes, revealing like-charge attraction due to curvature effects.

## Key findings

- Like-charged metal nanoparticles can attract each other in electrolyte solutions.
- Curvature plays a crucial role in enabling like-charge attraction, unlike flat metal surfaces.
- The new approximation method improves computational efficiency for nanoparticle interactions.

## Abstract

We calculate the force between two spherical metal nanoparticles of charge Q 1 and Q 2 in a dilute 1:1 electrolyte solution. Numerically solving the non-linear Poisson-Boltzmann equation, we find that metal nanoparticles with the same sign of charge can attract one another. This is fundamentally different from what is found for like-charged, non-polarizable, colloidal particles, the two body interaction potential for which is always repulsive inside a dilute 1:1 electrolyte. Furthermore, existence of like-charge attraction between spherical metal nanoparticles is even more surprising in view of the result that such attraction is impossible between parallel metal slabs, showing the fundamental importance of curvature. To overcome a slow convergence of the numerical solution of the full non-linear Poisson-Boltzmann equation, we developed a modified Derjaguin approximation which allows us to accurate and rapidly calculate the interaction potential between two metal nanoparticles, or between a metal nanoparticle and a phospholipid membrane.