# The Electrostatic Component of Disjoining Pressure

Review by Bryan Hassell: AP 255 Fall 11

Keywords: disjoining pressure

## Introduction

Investigating the mechanisms of spontaneous charging of interfaces between two phases, one being an electrolyte solution, leads to an equilibrium diffuse atmosphere. If two interfaces are brought together then their ionic atmospheres will overlap and this gives rise to the electrostatic component of the disjoining pressure.

## Methods of Calculation of the Disjoining Pressure $\Pi_e$

There are several approaches for calculating $\Pi_e$. One by Derjaguin [1] directly determined the electrostatic disjoining pressure acting on the surfaces of parallel plates in an electrolyte solution. He assumed that at equal potentials of the external and internal surfaces of the plate the hydrodynamic pressure is the same and so the disjoining pressure is just the difference between the Maxwell stresses at the external and internal surfaces of the plates:

$\Pi_e = \epsilon E_{ex}^2/8\pi - \epsilon E_{in}^2/8\pi$

where Ein and Eout are the strength of the electric field at the internal and external surfaces found from the solution of the Poisson-Boltzmann equation with appropriate boundary conditions. Langmuir also proposed [2] a formula based on ionic concentrations in the plane where the potential was a maximum and the electric field is zero. But for the formulation without any assumptions, the calculation of $\Pi_e$, begins with general thermodynamic arguments. First, looking at the differential of Gibbs free energy, taking into account work of external forces that maintain the equilibrium thickness $h$ of the electrolyte layer, and of charge sources which maintain the equilibrium charge densities $\sigma_1$ and $\sigma_2$ on the two surfaces:
Figure 1. Model for the derivation of the expression of $\Pi_e$. Potential distribution in the gap between plates

$dG_{T,P,\mu_i} = S\Pi_e dh + S(\phi_1 d\sigma_1 + \phi_2 d\sigma_2)$

where $\mu_i$ are the chemical potentials, T is the temperature, P pressure, S is the surface area of the plates and $\phi_1$ and $\phi_2$ are the surface potentials as seen in the figure.