Thermodynamics of Electrical Double Layers with Electrostatic Correlations

We present a systematic analysis to describe the free energy of electrical double layers (EDLs) based on variational calculus and demonstrate that the EDL free energy needs to be appropriately modified for different boundary conditions. We extend our formalism to study the electrostatic interaction potential between two plates and reconcile the two complementary methods utilized in the literature, i.e., the Gibbs-Duhem equation and the EDL grand potential. Next, we perform the same analysis while also including electrostatic correlations between the ions through the recently proposed modified Gauss's law involving a fourth-order differential equation for the electrical potential. The variational calculus formalism enables us to self-consistently derive the additional boundary conditions required for higher-order derivatives of the electrical potential, an approach that was previously overlooked. Next, we expand our analysis to predict the electrostatic interaction potential between two plates for any electrical potential, ion size, and correlation length scale. We calculate the potential distribution and estimate the interaction potential for a range of physical parameters. Our results reveal that the self-consistently derived boundary condition may significantly affect the properties of EDLs. Finally, we outline an alternative model to include electrostatic correlations that does not require additional parameters or boundary conditions. In summary, our analysis provides a robust framework to describe the thermodynamics of EDLs and will be useful for future fundamental and applied studies of EDLs, which occur across the sciences and engineering.