Duality of Theoretical Approaches to Understand the Electrical Double Layer in Concentrated Electrolytes

Abstract

Understanding the electrical double layer (EDL), i.e, the distribution of electrolyte at an electrified interface, in concentrated electrolytes is important for various technologies, such as supercapacitors, batteries and electrocatalysis. Atomistic approaches offer unprecedented detail, but are too computationally expensive to exhaustively investigate the EDL of concentrated electrolytes, motivating the development of continuum theories. In these concentrated electrolytes, correlations between ions and solvents are strong, through electrostatic and specific interactions, as well as significant excluded volume effects of the complicated molecular species, making the development of theories challenging. Thus far, there are mainly two distinct \textit{simple} theoretical approaches to understand the EDL of concentrated electrolytes, with account of these correlations beyond mean-field. One is a local-density approximation (LDA) based on treating electrostatic and specific interactions beyond mean-field through the ionic aggregation and solvation; where a simple conceptual understanding can be gained and reasonable agreement with experiments in terms of integrated quantities, but poor agreement for ion profiles and Debye capacitance. The other approach is to treat electrostatic correlations and excluded volume effects more rigorously with beyond LDA approaches, but at the cost of simplifying the chemical interactions between species; where excellent agreement can be obtained for ion profiles, differential capacitance, etc., but mainly for the simplified hard-sphere systems that the theories are based on. Here, we describe the merits and downfalls of these two approaches, how they have contributed to understanding anomalous underscreening, and outline future directions for these theoretical approaches.

Figures (3)

• Figure 1: Summary of general trends from inoic aggregation theory in ILs. a) Schematic illustration of the EDL of ILs with consistent account of thermoreversible associations. Anions are shown in blue, cations in red. The dangling bonds represent the number of associations they can form, and aggregates are denoted by dotted lines surrounding them. Close to the interface, ionic associations are destroyed, and free species dominate. b) Volume fractions ($\phi$) of various species ($\phi_{10}$ is free cations, $\phi_{01}$ is free anions, $\phi_{11}$ is ion pairs, $\phi_{+/-}^gel$ are cations/anions in the gel, and $\phi_{lm > 11}$ are the remaining aggregates) as a function from the positively charged interface, where $\kappa$ is the inverse Debye length. In the titles are the association constants, $\lambda = \exp(-\beta \Delta f)$, where $\beta$ is inverse thermal energy and $\Delta f$ is the free energy of an association. c) Experimental differential capacitance (from Ref. Monchai2018) as a function of applied voltage, compared against the "new theory" from Ref. Goodwin2022EDL with the indicated association constant, compared against free ion theories (FIT) from Ref. Goodwin2017a. Figures reproduced from Ref. Goodwin2022EDL, under the CC-BY-4.0 license.
• Figure 2: Overview of ionic aggregation approach in describing the EDL of LiTFSI WiSE. a) Comparison between simulation and theory for 15 m LiTFSI WiSE at a surface charge of -0.2 Cm$^{-2}$. Top panels show how the volume fractions of various aggregates, as indicated, vary within the EDL. The bottom panels show association probabilities between the species change within the EDL. The gray region denotes where interfacial interactions become important, b) Comparison between experimental and theoretical differential capacitance of 15m and 12m LiTFSI WiSE. Figures reproduced from Ref. Markiewitz2025, under the CC-BY-4.0 license.
• Figure 3: Summary of recent direct ion-interaction approaches, and success at predicting ionic layering. a) MD simulation of charged Lennard-Jones spheres, with cations in red and anions in blue (top panel). Concentration of cations and anions ($c_i$, red and blue, respectively) over bulk concentration $c_0$, as a function of distance from the electrodes, relative to the ion diameter $d$ (bottom panel). Surface charge of 0.12 Cm$^{-2}$ at a temperature of 300 K. b) Ion layering profiles at a 0.01 Cm$-2$ charge surface from the theory and the simulations, where remarkable agreement between them is found. c) Differential capacitance as a function of applied voltage for the WDA of Ref. Pedro2020, compared against the simulations, which match well, and the LDA theory of Refs. Kornyshev2007kilic2007a, which predicted much larger values. a)-c) have been reproduced from Ref. Pedro2020 with permission. d) Ion profiles from the classical-DFT with neural network learnt correlation function, compared to classical simulations of the electrolytes. Examples of charge and size symmetric, charge symmetric but size asymmetric, and size symmetric but charge asymmetric electrolytes simulations all show remarkable agreement with the developed cDFT. Figure reproduced from Ref. Bui2025 with the CC-BY-4.0 license.