Assessing the accuracy of three classical density functional theories of the electrical double layer

Classical density functional theory (DFT) is a useful tool to compute the structure of the electrical double layer because it includes ion-ion correlations due to excluded-volume effects (i.e., steric correlations) and ion screening effects (i.e., electrostatic correlations beyond the electrostatic mean-field potential). This paper systematically analyzes the accuracies of three different electrostatic excess free-energy functionals, as compared to Monte Carlo (MC) simulations of the planar electrical double layer, over a large parameter space. Specifically, we tested the reference fluid density (RFD) [Gillespie et al., J. Phys.: Condens. Matter 14, 12129 (2002)10.1088/0953-8984/14/46/317], functionalized mean spherical approximation (fMSA) [Roth and Gillespie, J. Phys.: Condens. Matter 28, 244006 (2016)10.1088/0953-8984/28/24/244006], and bulk fluid (BF) [Kierlik and Rosinberg, Phys. Rev. A 44, 5025 (1991)10.1103/PhysRevA.44.5025; Y. Rosenfeld, J. Chem. Phys. 98, 8126 (1993)10.1063/1.464569] functionals. Previous work compared these DFT methods to MC simulations only for a small set of parameters. Here, a total of twelve different cations were studied, with valences of +1, +2, and +3 and ion diameters of 0.15, 0.30, 0.60, and 0.90 nm at bulk concentrations between 1 μM and 1 M. The anion always had valence -1 and diameter 0.30 nm. The structure of the double layer of these charged, hard-sphere ions was computed for surface charges ranging from 0 to -0.50C/m^{2}. All the DFTs were compared against each other for all the parameters, as well as to 378 MC simulations. Overall, RFD was the best of the three functionals, while BF was the least accurate. fMSA performed significantly better than BF, making it a reasonable choice that is less computationally expensive than RFD. For monovalent cations, all three functionals worked reasonably well, except BF was qualitatively different from MC at very low surface charges. For multivalent cations, BF underestimated charge inversion while fMSA overestimated it. All DFTs performed poorly for small multivalent ions.