The Maxwell stress tensor and the thermodynamics of the diffuse double layers at surfactant-loaded interfaces

The stress tensor or its negative counterpart, the pressure tensor, is not uniquely defined for a small system of molecular dimensions. Hence there are several possible definitions of the local stress tensor, among them the well known Kirkwood–Buff and Irving–Kirkwood definitions. In the case of electrostatic intermolecular forces we have also the so-called Maxwell tensor. The various definitions may differ from one another mathematically by an arbitrary, divergenceless tensor and this raises questions, in particular as to the validity of the surface thermodynamic approach to the theory of surfactant micelles. All the stress tensors give identical results for the total force acting on a given volume element. However, when calculating the deformation work, e.g. of a spherical cone that is fundamental in surface thermodynamics, the different pressure tensors may yield differing results when the intermolecular forces are long-range. Thus, the electrostatic contribution to the surface tension of a surfactant micelle, γel, as calculated from the pressure tensor, is not a priori a well defined property. In the present paper we show that there exist cases when γel(and some closely related quantities) can really be calculated from the pressure tensor in a unique way. For reasons of simplicity, the Maxwell tensor is preferred in such cases for the localization of the electrostatic part of the pressure tensor. Notable among the convention-invariant (Cl) cases are the planar lamella and the infinite cylinder where the integral expressions always yield the same values for γel, irrespective of which definition is used for the pressure tensor. We also demonstrate that within the realm of the Poisson–Boltzmann approximation, the Maxwell tensor gives the correct γel, and likewise the correct electrostatic contribution to the reversible work of formation of a spherical micelle.