Thermodynamics of Electrical Double Layers with Electrostatic Correlations

Abstract

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.

Charging dynamics of electrical double layers inside a cylindrical pore: predicting the effects of arbitrary pore size

“…We emphasize that this crucial step enables us to derive a solution for arbitrary a p l , thereby bridging the previously reported trends in thin and overlapping double layer limits. 2,27,28,36 Now, substituting eqn (18) and (19) into eqn (21), we obtain…”

“…For higher electrolyte concentrations, other effects such as multi-component diffusion given by Stefan-Maxwell fluxes, 15 and ion correlations [16][17][18] have been studied, for instance in some continuum models. [19][20][21] The porous geometry is incorporated in the form of equivalent circuit representations, widely used in the modeling of pore charging in supercapacitor electrodes, [22][23][24][25][26] stemming from the pioneering work of de Levie. 27,28 Later, Biesheuvel and Bazant 2 extended the circuit to high potentials for capacitive deionization applications.…”

On Capacitance and Energy Storage of Supercapacitor with Dielectric Constant Discontinuity

“…It is well known that charged colloids (i.e., macroions) have typically a low relative dielectric constant ( ≈ 2−5) which is much smaller than that of the surrounding solvent (e.g., for water ≈ 80). In most of the simulation and theoretical works [ 12 , 13 , 14 , 15 , 16 , 17 , 18 ], it is generally assumed that the electrolyte solution and electrode have the same relative dielectric constant.…”

Charging Dynamics of Electrical Double Layers Inside a Cylindrical Pore: Predicting the Effects of Arbitrary Pore Size

“…Kilic and Bazant [13,14] derived modified PNP equations including ion-radius effects to study the influence of ion crowding in charge storage. For higher electrolyte concentrations, other effects such as multi-component diffusion given by Stefan-Maxwell fluxes [15], and ion correlations [16][17][18] have been studied, for instance in some continuum models [19][20][21].…”