Monte Carlo Study of a Planar Electric Double Layer Formed by Ions

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Monte Carlo Study of a Planar Electric Double Layer Formed by Ions with Off-Center Charge Stanisław Lamperski,*,† Lutful Bari Bhuiyan,‡ Douglas Henderson,§ and Monika Kaja† †

Faculty of Chemistry, Adam Mickiewicz University in Poznań, Umultowska 89b, 61-614 Poznań, Poland Laboratory of Theoretical Physics, Department of Physics, University of Puerto Rico, San Juan, Puerto Rico 00931-3343 § Department of Chemistry and Biochemistry, Brigham Young University, Provo, Utah 84602-5700, United States ‡

ABSTRACT: Grand canonical Monte Carlo simulation results are reported for an electric double layer (EDL) modeled by a planar charged hard wall, hard sphere cations with an off-center charge, and spherical anions with a charge at the center of the sphere. The ion charge numbers are Z+ = +1 and Z− = −1, and the diameter, d, of a hard sphere is the same for anions and cations. The ions are immersed in a solvent mimicked by a continuum dielectric medium at standard temperature. The results are obtained for three values of charge displacement, s+0 = d/16, d/4, 7d/16 from the center of the sphere and the following electrolyte concentrations: 0.5, 1.0, 2.0, and 3.0 M. The profiles of electrode−ion singlet distributions, cation reduced charge density, angular function, and mean electrostatic potential are reported for an electrode surface charge density σ = −0.30 C m−2, whereas the electrode potential and the differential capacitance of EDL are shown as functions of the electrode charge density varying from −1.00 to +1.00 C m−2. At negative electrode charges and with increasing values of the charge separation, the differential capacitance curve rises. As the electrolyte concentration increases, the shape of the differential capacitance curve changes from that of a minimum surrounded by two maxima into that of a distorted single maximum.

1. INTRODUCTION A theoretical description of the structural and thermodynamic properties of the electric double layer (EDL) is important for many biological and technological systems.1 The first statistical theory of EDL was proposed by Gouy,2 Chapman,3 and Stern4 (GCS). Although ions were modeled by point electric charges, the theory described relatively well the experimental results at small electrode charges and low electrolyte concentrations. The ion volume was taken into account in the restricted primitive model (RPM) (equally sized charged hard spheres in a continuum dielectric medium). The model was intensively used in different EDL theories. (See ref 5.) The correctness of the theory was tested by comparing the theoretical and molecular simulation results. The computer molecular simulation technique for the grand canonical ensemble was introduced by Torrie and Valleau.6,7 Recent experimental studies of EDL involve ionic liquids. The dependence of the differential capacitance on the electrode potential or surface charge density shows a single maximum (called the dromedary camel shape), double maximum (bactrian camel shape), or distorted double maximum (e.g., refs 8−11). Theoretical studies of ionic liquid EDLs have been realized mainly by molecular computer simulations. Fedorov et al.12,13 and Trulsson et al.14 have used dimer, trimer, and linear pentamer models with one sphere charged to investigate the © XXXX American Chemical Society

properties of the electrical double layer formed by ionic liquids. Vatamanu and co-workers15 have noticed a substantial influence of the electrode surface topography on the shape of differential capacitance. Classical density functional theory (DFT) studies are also very promising. Early works on the application of DFT to the RPM electrical double layer have been reported by Rosenfeld,16,17 Tang et al.,18 Mier-y-Teran et al.,19 and others. The DFT has evolved over the past several decades,20,21 and its present versions are able to describe precisely the properties of EDLs22−24 and complex molecular systems.25 Recently, we have studied the structural and thermodynamic properties of EDL formed by shape-anisotropic organic cations such as N-methyl-pyridinium and 1-butyl-3-methylimidazolium. In much of our former work, we modeled these ions by a dimer consisting of two tangentially tethered hard spheres, one of which has a point electric charge immersed at its center and the other being neutral.26−31 A natural extension of this dimer Special Issue: Tribute to Keith Gubbins, Pioneer in the Theory of Liquids Received: May 18, 2017 Revised: July 10, 2017

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center charge. The ion charge is Ze, where Z is the ion charge number or the valency and e is the magnitude of the elementary charge. The ions are immersed in a homogeneous and a continuous dielectric medium of relative permittivity ϵr. In this study, we consider binary symmetric ionic valencies, where Z+ = |Z−| = Z. The electrode is modeled by a hard planar wall with a uniformly distributed charge of surface density σ. The image effect is not considered, which means that the electrode material and solvent have the same ϵr value. It is of interest to note that in our previous study34 the off-center charge model was conceived by letting two unequally sized spheres of the dimer fuse into one another. An off-center charge cation resulted when the smaller positively charged sphere of the dimer disappeared completely into the larger neutral sphere such that the two sphere centers did not coincide. In the present case, the nonzero distance between the charge center and the geographical center of the cation sphere is built in. Consistent with the model of the off-center charge cation as illustrated in Figure 1, it is clear that the minimum separation between the charge centers of a cation and an anion, r+− mim, lies in the range of d − s+0 < r+− min < d + s+0, whereas the minimum separation between the ++ charge centers of two cations, r++ min, lies in the range of d − 2s+0 < rmin < d + 2s+0. The potential energy of various ion−ion interactions can thus be written as

model is the situation when the two spheres can fuse into one another, leading to dumbbell-shaped ions.32 In an another group of asymmetric ion models, the point electric charge is displaced from the center of a sphere. This offcenter charge model may be a useful model for theory because, generally, it is more convenient to deal with spheres in a theory. The spherical hard core could be treated by means of the wellknown Percus−Yevick theory, for which convenient analytic expressions are available. The electrostatic component could then be treated by using the charge centered at the spherical center as a starting point and then expanding in a series expansion in powers of charge displacement. The simulation results reported here would be useful for testing such a theory. The properties of EDL formed by such systems have been studied earlier.33 The off-center charge geometry was achieved by letting the two dimer spheres fuse into each other. In that preliminary study, we concentrated primarily on the structural aspects of the EDL, comparing and contrasting results from the off-center charge model, the dimer/dumbbell models, and the restricted primitive model. The manifest departure in the results from the habitual spherically symmetric ion models seemed promising. In the present work, we will take up the case of the off-center charge model again and focus on a more detailed analysis of both structure and thermodynamics. A physical relevance for such models can be seen from the fact that some organic cations such as pyrrolidinium, piperidinium, and pyridinium, which are the ring compounds composed of four to five carbon atoms and one nitrogen atom, have a positive charge localized on the nitrogen atom. Such cations can be modeled by a hard sphere with the off-center point charge. We assume that the anion charge is located at the center of the hard sphere. In the present off-center charge model, we will consider three different displacements of the point charge in the sphere. These three sets of results can be combined with our previous results for centrally placed charges, s+0 = 0,33 to yield results for four values of the displacement. The structural results in terms of the electrode−ion singlet density profiles will be supplemented by the orientational and charge distribution profiles. We will also study the differential capacitance of the EDL, which is an experimentally measurable property.

+− ⎧ ∞ r′ < rmin ⎪ u+−(r′) = ⎨ e 2Z Z + − +− ⎪ ⎪ 4π ϵ ϵ r′ r′ > rmin ⎩ 0 r

(1)

++ ⎧ ∞ r′ < rmin ⎪ u++(r′) = ⎨ e 2Z Z + + ++ ⎪ ⎪ 4π ϵ ϵ r′ r′ > rmin ⎩ 0 r

(2)

⎧ ∞ rd ⎪ ⎩ 4π ϵ0ϵrr

(3)

Here, ϵ0 is the vacuum permittivity and r and r′ are the distances between the centers of two hard spheres and between ion charges, respectively, With regard to the electrode−ion interactions, we note that although the distance of minimum approach to the planar electrode for the anion and cation spheres is d/2, the charge center of the cation can be located within [0, d]. Thus, whereas the minimum separation between the electrode and the anion charge center is d/2, that between the electrode and the cation charge center, xw+ min, lies in the range of d/2 − s+0 < xw+ min < d/2 + s+0. The electrode−ion interactions can therefore be written as

2. MODEL AND METHODS We will treat the model of an electrolyte including one species of ions, viz., the cations, with the off-center charge. The degree to which the cation charge is off-center is controlled by the displacement or distance s+0 between the charge location and the center of the hard sphere (Figure 1). The ion diameter d is the same for anions and cations. Essentially, this is the RPM with the caveat that the cation has an off-

⎧ ∞ x < d /2 ⎪ uw −(r ) = ⎨ σZ −ex , x > d /2 ⎪− ⎩ ϵ 0 ϵr

(4)

w+ ⎧ ∞ x < xmin ⎪ uw +(r ) = ⎨ σZ+ex w+ , x > xmin ⎪− ⎩ ϵ 0 ϵr

(5)

with x being the perpendicular distance from the electrode surface to an ion charge. The long-range electrostatic interactions were determined with the charged sheets method proposed by Torrie and Valleau.6 Simulations were carried out using the grand canonical Monte Carlo technique. This technique is recommended for heterogeneous systems.34 The ionic activity coefficients required by the GCMC technique as input were obtained from the inverse GCMC method.35

Figure 1. Graphical illustration of the off-center charge model. B

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Langmuir The results obtained directly from the simulation are (i) the number density, ρs(x), of the s species of ions, (ii) the ion charge density, ρQs(x), and (iii) the angular function, ⟨cos θ⟩, or the orientational profile of ions with the off-center charge. In the discussion of the results, it is convenient to use the reduced density, gs(x) = ρs(x)/ρ0s , also called the singlet distribution function, where ρ0s is the bulk number density of species s. The ion charge densities are used to calculate the mean electrostatic potential profile

ψ (x) =

1 ϵ 0 ϵr

∑∫ s

x

ion curves show the repulsion of these ions in the vicinity of the electrode. When the ion charge is located at the center of a hard sphere, the reduced charge density distribution is the same as the singlet distribution. This is not the case for the off-center charge ions. The ρ*Q+ curve of the off-center charge counterions has a rounded maximum near the electrode surface. With increasing values of s+0, the maximum decreases and is shifted toward the electrode surface. At the same time, the contact distance decreases. The shape of the ρ*Q+ curve is determined by the following four effects: (i) ion off-center charge (the larger the value of s+0, the closest approach of an ionic charge cloud to the electrode surface), (ii) electrostatic interactions (the ion charge distribution on a sphere of radius s+0 is the largest for θ = 0 and the smallest for θ = π), (iii) off-center charge orientation (when the electrostatic interactions are neglected, the largest contribution to ρ*Q+ gives the orientation for θ = π/ 2), and (iv) ion number density (with increasing ρ*+ , the value of ρQ+ * increases). Profiles of the angular function ⟨cos θ⟩ with the origin of the θ angle at the center of the cation sphere for c = 1.00 M and σ = −0.30 C m−2 are shown in Figure 3. The negative values of

∞

dx′(x − x′)ρQs (x′)

(6)

When the electrode potential, ψ(0), is known, the differential capacity can be evaluated form the formula

Cd = dσ /dψ (0)

(7)

The angular function, ⟨cos θ⟩, is the average value of cos θ, where θ is the angle between the normal to the electrode surface and the straight line joining the center of the sphere and the charge.

3. RESULTS AND DISCUSSION In the present studies, the diameters of spherical anions and cations are the same and are equal to 425 pm. The ion charge numbers are Z+ = +1 and Z− = −1. The following values of charge displacement, s+0, with respect to the sphere center are considered: d/16, d/4, and 7d/16. For s+0 = d/16, the positive charge is located close to the center of the hard sphere, for d/4, it is halfway between the center and the sphere surface, and for 7d/16, it is close to the sphere surface. The simulations were carried out for electrolyte concentrations of c = 0.50, 1.00, 2.00, and 3.00 M at temperature T = 298.15 K, relative permittivity ϵr = 78.5, which is characteristic of water, and the surface charge density σ varying in the range from −1.00 to +1.00 C m−2. Figure 2 shows the profiles of the singlet distribution functions and the cation reduced charge density

Figure 3. Dependence of the angular function ⟨cos θ⟩ on the distance, x, from the electrode surface for three values of charge displacement, s+0. Parameters and symbols are the same as in Figure 2.

⟨cos θ⟩ mean that the most probable position of a charge at the off-center charge ions is that close to the electrode surface. This effect increases with increasing value of s+0. These results confirm the earlier findings,33 according to which with increasing s+0 the charge density is shifted toward the electrode surface and hence the consistency of the present distribution functions is in agreement with that in ref 33. Figure 4 shows the influence of the electrolyte concentration on the cation charge density distribution ρQ+ * (panel A) and the angular function ⟨cos θ⟩ (panel B) at s+0 = d/4 and σ = −0.30 C m−2. The ρ*Q+ curves have a rounded maximum. However, the position of the maximum does not depend on the electrolyte concentration. Although an increase in the electrolyte concentration lowers the value of the maximum, the shape of the curves remains similar. The curves of ⟨cos θ⟩ for c = 0.50, 1.00, and 2.00 M are negative and tend to zero. They do not show any extrema and remain monotonic throughout. The behavior of the curve for 3.00 M is different. It has a planar positive maximum at about x/d = 1.8. In the range of positive values of ⟨cos θ⟩, the off-center charge takes an outward orientation with respect to the electrode surface. This is related to the charge reversal (overscreening) effect36 observed earlier at high electrolyte concentrations.37

Figure 2. Dependence of the singlet distribution function, g(x), and * (x), on the distance, x, from the the cation reduced charge density, ρQ+ electrode surface for three values of charge displacement, s+0 = d/16 (circles), d/4 (triangles), and 7d/16 (squares) at an electrode charge density of σ = −0.3 C m−2 and an electrolyte concentration of c = 1.00 M. Filled symbols show g(x) of cations; filled with crosses, g(x) of anions; and empty symbols, the cation reduced charge distribution *. ρQ+

* = ρQ+/(z+eρ0+) for the three considered values of s+0 at c ρQ+ = 1.00 M and σ = −0.30 C m−2. We selected the negative electrode charge to concentrate the analysis on the properties of cations with the off-center charge. The singlet distribution functions of counterions, viz., centers of positively charged hard spheres, are typical, showing the sharp maximum at a contact distance. The curves for different values of s+0 overlap. The coC

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Figure 6. Mean electrostatic potential, ψ(x), as a function of the distance from the electrode surface, x/d, for three values of charge displacement, s+0. The parameters and symbols are the same as in Figure 2.

Figure 4. Dependence of the cation reduced charge density, ρQ+ * (panel A, empty symbols) and the angular function ⟨cos θ⟩ (panel B, filled symbols) on the electrolyte concentration, c/M = 0.5 (circles), 1.0 (diamonds), 2.0 (squares), and 3.0 (triangles) at s+0 = d/4 and σ = −0.3 C m−2.

Figure 7 shows the dependence of the electrode potential on the surface charge density of the electrode. At negative values of

Figure 5 illustrates the influence of the electrode surface charge σ on the cation charge density distribution ρQ+ * (panel

Figure 7. Electrode potential, ψ(0), as a function of the electrode charge density, σ, for three values of charge displacement, s+0. The symbols have the same meaning as in Figure 2.

Figure 5. Dependence of the cation reduced charge density, ρ*Q+ (panel A, empty symbols) and the angular function, ⟨cos θ⟩ (panel B, filled symbols) on the electrode surface charge density, σ/C m−2 = −0.1 (circles), −0.3 (diamonds), −0.5 (squares), −0.7 (triangles), and −0.9 (asterisks) at s+0 = d/4 and c = 1.00 M.

σ with increasing values of s+0, the negative values of the electrode potential decrease. The discrepancy extends with increasing negative charge at the electrode. At the positive values of σ, when cations are desorbed, the curves have very similar patterns. The above behavior of the electrode potential determines the nature of the EDL capacitance curves. Figure 8 shows the dependence of the differential capacitance of EDL, Cd, on the

A) and the angular function ⟨cos θ⟩ (panel B) at s+0 = d/4 and c = 1.00 M. With an increasingly negative value of σ, the maximum in the ρQ+ * curve increases and is shifted toward the electrode surface. This effect is a result of increasing the electrode-off-center charge electrostatic interactions. At σ = −0.70 and −0.90 C m−2, the second maximum on the ρQ+ * curve is seen. This effect indicates the second ion layer formation. The ⟨cos θ⟩ results show that with decreasing σ the position of the charge at the off-center charge ions moves toward the electrode surface. The dependence of the mean electrostatic potential on the distance from the electrode surface at c = 1.00 M and σ = −0.30 C m−2 is presented in Figure 6. At short distances from the electrode surface, with increasing values of s+0, the negative values of the electrostatic potential decrease, but the effect is small. At large distances, the curves overlap.

Figure 8. Differential capacitance, Cd, of the electrical double layer as a function of the electrode charge density, σ, for three values of charge displacement, s+0. The symbols have the same meaning as in Figure 2. D

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Langmuir electrode charge density for an electrolyte concentration of 1.00 M and three considered values of s+0. In the range of cations adsorption, the differential capacitance increases with increasing value of s+0 (when the point electric charge is shifted away from the center of a sphere). The Cd curve has the so-called disordered camel shape with the minimum close to the potential of zero charge (PZC). The asymmetry of Cd for the s+0 = d/16 curve is small. The next three figures show the influence of electrolyte concentration on the Cd curves for three values of s+0. With increasing concentration, the minimum close to PZC transforms into its maximum and the curve has a bell-like shape. For the smallest value of s+0 (Figure 9), the asymmetry is barely

Figure 11. Differential capacitance, Cd, of the electrical double layer as a function of electrode charge density, σ, for s+0 = 7d/16 and different electrolyte concentrations, c. The parameters and symbols are the same as in Figure 9.

one neutral. In one instance, these spheres were also assumed to be able to fuse into each other.33 The charged dimer is the simple model of charged surfactants and ionic liquids. In the present study, we have applied the grand canonical Monte Carlo simulation to investigate the structural and thermodynamic properties of EDL comprising monomer cations with an off-center charge and monomer anions with a centrally located charge. The off-center charge cation has relevance for ring compounds composed of four to five carbon atoms and one nitrogen atom. We have analyzed the EDL properties for different values of charge separation, electrode surface charge density, and electrolyte concentration. It was found that the curves of the cation singlet distribution function and reduced charge distribution do not overlap. The ρQ+ * curve has a rounded maximum near the electrode, shifted toward the electrode surface, and the g+ curve has a sharp maximum at the contact distance. With increasing negative electrode charge, the maximum in the ρQ+ * curve becomes pronounced and is shifted toward the electrode surface. The position of the maximum is, however, independent of the electrolyte concentration. The orientational profile has negative values, which suggests that the most probable position of an off-center charge ion is close to the electrode surface. This probability increases with increasing charge displacement, electrolyte concentration, and magnitude of the negative electrode charge. The differential capacitance behavior is consistent with what is well-known in the literature, viz., a transition from a double-humped shape to a singlehumped shape as the concentration increases.26,29 However, the striking features seen in this work are the increases in the magnitude of the capacitance and in the asymmetry of the curves with increasing charge displacement. The off-center charge model introduces asymmetry into the spherically symmetric primitive models, although the ions retain their spherical shape. This asymmetry has significance for many organic electrolytes and ionic liquids and as such should be useful in understanding the differential capacitance behavior of many such systems.

Figure 9. Differential capacitance, Cd, of the electrical double layer as a function of the electrode charge density, σ, for s+0 = d/16 and different electrolyte concentrations, c/M = 0.5 (circles), 1.0 (diamonds), 2.0 (squares), and 3.0 (triangles).

Figure 10. Differential capacitance, Cd, of the electrical double layer as a function of the electrode charge density, σ, for s+0 = d/4 and different electrolyte concentrations, c. The parameters and symbols are the same as in Figure 9.

visible as the model begins to resemble the RPM rather closely. For the largest s+0 (Figure 11) and c = 3.00 M, a single maximum is observed at small negative electrode charges.

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4. CONCLUSIONS In recent years, the use of more complex models of the electrolyte beyond the primitive models has aroused further interest in the theory of EDL. 38 Cations have been approximated by beads,12,13 coarse-grained models,39,40 and pentameric ions.14 In our recent studies of EDL, we have also used an advanced model of an electrolyte.26−32 Its molecules were assumed to be charged dimers composed of two tangentially touching hard spheres, one charged and the other

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Stanisław Lamperski: 0000-0003-4240-3724 Notes

The authors declare no competing financial interest. E

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electrical double layerPart I: Symmetrical electrolytes. Mol. Phys. 1990, 71, 369−392. (19) Mier-y-Teran, L.; Tang, Z.; Davis, H. T.; Scriven, L. E.; White, H. S. Non-local free-energy density-functional theory applied to the electrical double layerPart II: 2:1 electrolytes. Mol. Phys. 1991, 72, 817−830. (20) Walther, C. F. J.; Patchkovskii, S.; Heine, T. Grand-canonical quantized liquid density-functional theory in a Car-Parrinello implementation. J. Chem. Phys. 2013, 139, 034110. (21) Jeanmairet, G.; Levesque, M.; Borgis, D. Molecular density functional theory of water describing hydrophobicity at short and long length scales. J. Chem. Phys. 2013, 139, 154101. (22) Zhou, S. Density Functional Analysis of Like-Charged Attraction between Two Similarly Charged Cylinder Polyelectrolytes. Langmuir 2013, 29, 12490. (23) Naji, A.; Kanduc, M.; Forsman, J.; Podgornik, R. Perspective: Coulomb fluids-Weak coupling, strong coupling, in between and beyond. J. Chem. Phys. 2013, 139, 150901. (24) Kim, E. Y.; Kim, S. C.; Han, Y. S.; Seong, B. S. Structure of a planar electric double layer containing size-asymmetric ions: density functional approach. Mol. Phys. 2015, 113, 871−879. (25) Wu, J. In Molecular Thermodynamics of Complex Systems; Lu, X., Hu, Y., Eds.; Structure and Bonding; Springer: Berlin, 2009; Vol 131. (26) Lamperski, S.; Kaja, M.; Bhuiyan, L. B.; Wu, J.; Henderson, D. Influence of anisotropic ion shape on structure and capacitance of an electric double layer: A Monte Carlo and density functional study. J. Chem. Phys. 2013, 139, 054703. (27) Kaja, M.; Silvestre-Alcantara, W.; Lamperski, S.; Henderson, D.; Bhuiyan, L. B. Monte Carlo investigation of structure of an electric double layer formed by a valency asymmetric mixture of charged dimers and charged hard spheres. Mol. Phys. 2015, 113, 1043−1052. (28) Silvestre-Alcantara, W.; Henderson, D.; Wu, J.; Kaja, M.; Lamperski, S.; Bhuiyan, L. B. Structure of an electric double layer containing a 2:2 valency dimer electrolyte. J. Colloid Interface Sci. 2015, 449, 175−179. (29) Kaja, M.; Lamperski, S.; Silvestre-Alcantara, W.; Bhuiyan, L. B.; Henderson, D. Influence of anisotropic ion shape, asymmetric valency, and electrolyte concentration on structural and thermodynamic properties of an electric double layer. Condens. Matter Phys. 2016, 19, 13804. (30) Henderson, D.; Silvestre-Alcantara, W.; Kaja, M.; Lamperski, S.; Wu, J.; Bhuiyan, L. B. Structure and capacitance of an electric double layer of an asymmetric valency dimer electrolyte: A comparison of the density functional theory with Monte Carlo simulations. J. Mol. Liq. 2017, 228, 236. (31) Lamperski, S.; Bhuiyan, L. B.; Henderson, D.; Kaja, M.; Silvestre-Alcantara, W. Influence of a size asymmetric dimer on the structure and differential capacitance of an electric double layer. A Monte Caro study. Electrochim. Acta 2017, 226, 98−103. (32) Silvestre-Alcantara, W.; Kaja, M.; Henderson, D.; Lamperski, S.; Bhuiyan, L. B. Structure and capacitance of an electric double layer formed by fused dimer cations and monomer anions: A Monte Carlo Simulation study. Mol. Phys. 2016, 114, 53−60. (33) Silvestre-Alcantara, W.; Bhuiyan, L. B.; Lamperski, S.; Kaja, M.; Henderson, D. Double layer for hard spheres with an off-center charge. Condens. Matter Phys. 2016, 19, 13603. (34) Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications; Academic Press: San Diego, 1996; p 126. (35) Lamperski, S. The individual and mean activity coefficients of an electrolyte from the inverse GCMC simulation. Mol. Simul. 2007, 33, 1193−1198. (36) Jiménez-Á ngeles, F.; Lozada-Cassou, M. A Model Macroion Solution Next to a Charged Wall: Overcharging, Charge Reversal, and Charge Inversion by Macroions. J. Phys. Chem. B 2004, 108, 7286− 7296. (37) Bhuiyan, L. B.; Outhwaite, C. W. Comparison of the modified Poisson-Boltzmann theory with recent density functional theory and simulation results in the planar electric double layer. Phys. Chem. Chem. Phys. 2004, 6, 3467−3473.

ACKNOWLEDGMENTS S.L. and M.K. gratefully acknowledge financial support from the Faculty of Chemistry, Adam Mickiewicz University of Poznań, Poznań, Poland.

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DEDICATION The authors are pleased to dedicate this article to our good friend Keith Gubbins on the occasion of his 80th birthday. We have followed his work at the University of Florida, Cornell University, and now North Carolina State University. He has accomplished much for such a youngster. Now that he has reached maturity, we look for even more significant accomplishments.

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DOI: 10.1021/acs.langmuir.7b01677 Langmuir XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.langmuir.7b01677 Langmuir XXXX, XXX, XXX−XXX