Depletion force between two large spheres suspended in a bath of small spheres: Onset of the Derjaguin limit

TL;DR

This paper investigates the depletion force between large colloids in a small sphere solvent, revealing that the Derjaguin limit is approached more slowly than previously thought, and highlights limitations of current density functional theory methods at large size ratios.

Contribution

The study provides a detailed analysis of the depletion interaction near the Derjaguin limit and improves DFT techniques by enforcing test particle consistency, but finds these methods unreliable at high size ratios.

Findings

Derjaguin limit approached more slowly than previously believed

Improved DFT with test particle consistency does not converge to the Derjaguin limit

Standard DFT methods become unreliable for size ratios larger than 10

Abstract

We analyze the depletion interaction between two hard colloids in a hard--sphere solvent and pay special attention to the limit of large size ratio between colloids and solvent particles which is governed by the well--known Derjaguin approximation. For separations between the colloids of less than the diameter of the solvent particles (defining the depletion region), the solvent structure between the colloids can be analyzed in terms of an effective two--dimensional gas. Thereby we find that the Derjaguin limit is approached more slowly than previously thought. This analysis is in good agreement with simulation data which are available for a moderate size ratio of 10. Small discrepancies to results from density functional theory (DFT) at this size ratio become amplified for larger size ratios. Therefore we have improved upon previous DFT techniques by imposing test particle consistency which connects DFT to integral equations. However, the improved results show no convergence towards the Derjaguin limit and thus we conclude that this implementation of DFT together with previous ones which rely on test particle insertion become unreliable in predicting the force in the depletion region for size ratios larger than 10.