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The structure and thermodynamics for model 2–2 electrolytes at a charged interface have been determined by the so-called “pair” approximation of integral equation theory. In addition to Coulombic interactions, the potential models for the ion–ion and ion–wall interactions employ “soft” continuous potentials rather than “hard”-sphere or “hard”-wall...
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... J 1 is the first order Bessel function. The points in the radial direction ( k 1 ̄ k N and s 1 ̄ s N ) are determined by the conditions that J 0 ( k m s N ) ϭ J 0 ( k N s j ) ϭ 0 and s N is suit- ably chosen so that all functions f ( s i ) approach zero as i approaches N . For the majority of calculations, N ϭ 90, with s N ϭ 30 Å. The numerical details of the pair approximation calculations are further discussed in Ref. 34. To our knowledge, no simulation data exist for the realistic ‘‘soft’’ potentials we have studied with pair and singlet theory. As a guide, we have chosen to make comparisons with the Monte Carlo ͑ MC ͒ data of Torrie and Valleau. 46,47 In their simulations, Torrie and Valleau use a different set of potentials than us, namely charged hard-spheres ͓ e.g., the potential in Fig. 3 ͑ a ͒ , dashed lines ͔ for the model electrolyte, and a short-range hard-wall potential, as shown in Fig. 3 ͑ b ͒ ͑ dashed line ͒ . Figure 4 shows the cationic and anionic normalized concentration profiles for 0.5 M 2–2 electrolyte at a surface charge density of Ϫ 21.29 C cm Ϫ 2 . The normalized concentration profiles for the singlet approximation have been included for comparison. The corresponding surface potential for each of the integral equation approximations is listed below in brackets. The circles are the digitized MC computer simulation results. The solid lines are the pair results using the AHNC ( Ϫ 99 mV) approximation. The long-dashed lines are the singlet approximation ( Ϫ 100 mV). Taking into consideration the differences in the potential functions, the approximate integral equation theories predict qualitatively the same behavior as the MC data. The pair approximations show quantitative agreement with MC data, particularly for the anionic profile, and a measurable im- provement over the singlet approximation. For divalent electrolytes at lower concentrations it is no longer possible to obtain singlet solutions at the surface charge densities used in the MC simulations. In Fig. 1 above we compare the AHNC approximation ͑ solid lines, Ϫ 69.6 mV) with the MC results ͑ circles ͒ for 2–2 electrolytes at 0.05 M and a surface charge density of Ϫ 8.65 C cm Ϫ 2 . We also show the pair using the LMBW approximation ͑ dashed lines, Ϫ 70.3 mV) for comparison. Allowing for the differences in potential models, the pair approximations predict qualitatively the same behavior as the MC data. We now consider comparisons of normalized concentration profiles for the singlet approximation, using our ‘‘soft’’ potential model, and the pair approximation using the AHNC density equation. In Fig. 5, comparisons for cationic and anionic normalized concentration profiles are made between pair results using the AHNC approximation ͑ solid lines ͒ and the singlet approximation ͑ dashed lines ͒ , for 2–2 electrolytes at 0.5 M with surface charge densities of 0, Ϫ 10 and Ϫ 30 C cm Ϫ 2 . A surface charge density of Ϫ 30 C cm Ϫ 2 represents the upper limit for obtaining solutions for the singlet approximation. With no surface charge on the wall, the profiles of the singlet and pair approximations are similar, with the pair approximation result showing slightly less structure. At a surface charge density of Ϫ 10 C cm Ϫ 2 , there is very little difference in the two profiles, though, as the magnitude of the surface charge increases, the singlet approximation predicts significantly more structure than the pair approximation, with secondary peaks developing in both the cationic and anionic profiles at a surface charge density of Ϫ 30 C cm Ϫ 2 . While both integral equations carry approximations, and hence conclusions drawn from differences in their results must be made with some caution, the differences in density profiles would seem to be a consequence of the combination of approximations in the singlet plus HNC approximation, 28 namely that the inhomogeneous correlation functions are equivalent to their bulk counterparts. The singlet approximation exaggerates the density profiles since the charged liquid is not allowed to relax from its bulk structure even very close to the charged surface. Indeed, for all concentrations studied, the general trend is that, for ...
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Classical density functional theory is combined with the extended primitive model and solvent primitive model to investigate how differential capacitance Cd of electrical double-layer formed inside cylindrical pore is influenced by solvent granularity, bulk concentration, counter-ion diameter, and neutral and non-hard sphere (HS) potential utail_αβ(r) between ion species, between solvent and counter-ion (and co-ion). Several main conclusions are drawn. (i) The Cd−surface charge strength (|σ|) curve generally rises as a result of consideration of the solvent granularity. (ii) An interionic attractive utail_αβ(r) helps in raising the Cd curve whereas a repulsive utail_αβ(r) tends to lower the Cd curve. Moreover, a secondary Cd peak appears when |σ| increases to an appropriately large value, and becomes increasingly evident with the strength of the attractive utail_αβ(r) and eventually disappears as the attractive utail_αβ(r)reduces in attraction or changes into repulsion at all. (iii) The Cd − |σ| curve is raised by increasing the repulsion between the solvent and counter-ion; however, both counter-ion diameter and bulk concentration cause obvious changes of the Cd − |σ∗| curve morphology. Concretely speaking, the Cd peak height and the peak position rise with the counter-ion size decreasing. Moreover, changing the utail_αβ(r) between solvent and counter-ion from attraction to repulsion facilitates a transition from bell-shaped curve to camel-shaped curve. (iv) Effects of all of the factors considered in influencing the Cd curve diminish increasingly with |σ|. All of the above observations can be explained by considering the interionic depletion potential induced by the solvent and its changes with the system parameters.
This chapter describes theoretical approaches of the classical statistical mechanics and modern computer simulation methods to study the microscopic structure and thermodynamic and electrical properties of liquid electrolyte solutions at solid surfaces. Theoretical approaches include integral equations and density functional methods. We analyze their development to reach an accurate description of the interfacial structure and thermodynamics of nonuniform electrolyte solutions. The computer simulation techniques are reviewed. Methodological aspects concerned with the key issue of simulations, namely, accounting for the long-range electrostatic interactions, are given. Finally, we present some selected results for simple models and next for some real and experimentally important systems. Some prospects for future studies are discussed in the summary.
Inhomogeneous correlation functions for model ‘ soft’ 2–2 and 1–1 electrolytes at a charged interface have been determined by the so–called ‘pair’ approximation of integral equation theory. The solvent is modeled as a structureless dielectric continuum at 25°C. The wall–ion–ion structure is calculated using the inhomogeneous Ornstein–Zernike relation, together with the hypernetted chain closure, and one of two choices for the functional relationship between the singlet and pair correlation functions. Both the interfacial density profiles and the inhomogeneous pair correlation functions are calculated. For most cases, the inhomogeneous pair correlation functions near the interface vary significantly from the homogeneous pair correlation functions. This deviation generally becomes stronger as the charge on the surface increases, and the deviation generally extends out further from the interface as the
surface charge increases. The density profiles predicted by the pair approximations generally show less structure than the singlet approximation density profiles, whereas the inhomogeneous pair correlation functions generally predict more structure than would be expected by simply assuming bulk pair correlation functions. The density profiles and inhomogeneous correlation functions are also found to agree qualitatively with previous simulations which used ‘charged hard–sphere/charged hard—wall’ potentials.
A study has been made of the electrical 'double layer' structure of molten salts, in particular a model of molten potassium chloride, using an integral equation approximation. This is in contrast to most statistical mechanical treatments of the double layer, which have concentrated on aqueous elecrolyte solutions. The results are compared with the output of computer simulations. In addition to the structural information contained in the density profiles, the calculations yielded charge profiles, the mean electrostatic potential and the double layer capacitance.
The hypernetted chain/mean spherical approximation (HNC/MSA) integral equation for a totally asymmetric primitive model electrolyte around a spherical macroparticle is obtained and solved numerically in the case of size-asymmetric systems. The ensuing radial distribution functions show a very good agreement when compared to our Monte Carlo and molecular-dynamics simulations for spherical geometry and with respect to previous anisotropic reference HNC calculations in the planar limit. We report an analysis of the potential versus charge relationship, radial distribution functions, mean electrostatic potential, and cumulative reduced charge for representative examples of 1:1 and 2:2 salts with a size-asymmetry ratio of 2. Our results are collated with those of the modified Gouy-Chapman (MGC) and unequal radius modified Gouy-Chapman (URMGC) theories and with those of HNC/MSA in the restricted primitive model (RPM) to assess the importance of size-asymmetry effects. One of the most striking characteristics found is that, contrary to the general belief, away from the point of zero charge the properties of an asymmetric electrical double layer (EDL) are not those corresponding to a symmetric electrolyte with the size and charge of the counterion, i.e., counterions do not always dominate. This behavior suggests the existence of a new phenomenology in the EDL that genuinely belongs to a more realistic size-asymmetric model where steric correlations are taken into account consistently. Such novel features cannot be described by traditional mean-field theories such as MGC, URMGC, or even by enhanced formalisms, such as HNC/MSA, if they are based on the RPM.
