J. Phys. Chem. B 2009, 113, 8925–8929
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The Electric Double-Layer Differential Capacitance at and near Zero Surface Charge for a Restricted Primitive Model Electrolyte Stanislaw Lamperski,† Christopher W. Outhwaite,*,‡ and Lutful B. Bhuiyan§ Department of Physical Chemistry, Faculty of Chemistry, Adam Mickiewicz UniVersity, Grunwaldzka 6, 60-780 Poznan´, Poland, Department of Applied Mathematics, UniVersity of Sheffield, Sheffield S3 7RH, U. K., and Department of Physics, Laboratory of Theoretical Physics, UniVersity of Puerto Rico, San Juan, Puerto Rico, 00931-3343 ReceiVed: January 3, 2009; ReVised Manuscript ReceiVed: May 6, 2009
The behavior of the differential capacitance of a planar electric double layer containing a restricted primitive model electrolyte in the neighborhood of zero surface charge is investigated by theory and simulation. Previous work has demonstrated that at zero surface charge the differential capacitance has a minimum for aqueous electrolytes at room temperature but can have a maximum for molten salts and ionic liquids. The transition envelope separating the two situations is found for a modified Poisson-Boltzmann theory and a Poisson-Boltzmann equation corrected for the exclusion volume term. Good agreement is found between simulation and the modified Poisson-Boltzmann theory in the neighborhood of the envelope at the reduced temperature of 0.8, while the exclusion volume corrected Poisson-Boltzmann theory shows correct qualitative trends. Introduction The differential capacitance CD of ionic liquids and molten salts at an electrified interface is coming under increased theoretical scrutiny.1-4 Problems arise in choosing a suitable model for a particular system, besides the difficulties in implementing any theoretical approach. A standard model of an aqueous electrolyte in the planar electric double layer is a restricted primitive model (RPM) electrolyte next to a uniformly charged, planar hard wall; the RPM being a system of equisized, charged hard spheres moving in a dielectric medium. Ionic liquids and molten salts are solvent free with the ionic liquid being regarded as a low-temperature molten salt, and so they can be modeled by a RPM with a low value of the permittivity representing the electronic polarization. Clearly this is an extreme simplification of the intermolecular potentials, but, along with the model of a planar, uniformly charged electrode, the RPM is a useful benchmark. With a RPM molten salt, Monte Carlo (MC) simulations and a modified Poisson-Boltzmann (MPB) theory5 have shown that at zero surface charge the CD has a maximum.4 Kornyshev1 and Kilic et al.2 have also shown analogous results for an ionic liquid using a Poisson-Boltzmann (PB) lattice-gas model. The complex nature of the electric double layer means that an overall interpretation of the experimental capacitance is unclear for both molten salts6-8 and ionic liquids.7,9-12 The classical GouyChapman-Stern (GCS) theory13-15 always predicts a minimum in CD at zero surface charge for the RPM and, hence, is unable to capture any such maximum in CD. However, Oldham3 with a GCS type approach has predicted a maximum. Our interest in this paper is an investigation of CD for a RPM planar double layer at and near zero surface charge for a wide set of physical states, encompassing the variation of CD from * Corresponding author. E-mail address:
[email protected]. † A. Mickiewicz University. ‡ University of Sheffield. § University of Puerto Rico.
having a minimum to a maximum. Using the MPB theory, and the PB theory, including an exclusion volume term, we can derive a transition envelope in the reduced temperature and density plane which separates the region into two parts. In one part where the reduced density is relatively small, the CD has a minimum at zero surface charge corresponding to the GCS behavior, while in the other, the CD has a maximum at zero surface charge. The existence of the RPM envelope is verified by MC simulation. Theoretical Formalism The ionic solution is modeled by the RPM, the ions being charged hard spheres of diameter d with a point charge ei at their center, moving in a dielectric medium of constant permittivity. The electrode is nonpolarizable and has a hard plane surface with a uniform surface charge density σ. A. MC Simulation. The MC simulations of the RPM were carried out in the grand canonical (GC) ensemble. In the GC ensemble, the volume, temperature, and chemical potential are held constant while the number of molecules fluctuates. The GC Monte Carlo (GCMC) technique is usually recommended for inhomogeneous systems16 as it eliminates the uncertainties associated with the determination of the bulk concentration, which appears in the canonical ensemble MC simulations. In the GCMC technique there are three equally probable moves: displacement, insertion, and removal. The conditions for acceptance of these moves are described in standard handbooks on the molecular simulations.16 To implement, the GCMC requires the specification of the chemical potential or, equivalently, the activity coefficient of the fluid being treated. Unfortunately, this means that difficulties arise when numerous calculations are needed for a particular concentration. To circumvent this problem, the activity coefficient for a given bulk concentration can be found from the inverse GCMC technique (IGCMC).17 This technique consists of a large number of short GCMC simulations with the resulting average number of ions obtained after each short simulation being used to modify the
10.1021/jp900037h CCC: $40.75 2009 American Chemical Society Published on Web 06/04/2009
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mean activity coefficient γ(. All of these modified activity coefficients are then averaged to give the final mean activity coefficient, ln γ(. Further details of the simulation procedure have been given elsewhere.17 B. MPB Theory. The theoretical results are given for both the MPB approach and the PB theory with an exclusion volume correction. In the electric double layer, the mean electrostatic potential ψ(x), a perpendicular distance x from the electrode into the solution, satisfies
d2ψ 1 )ε0εr dx2
∑ eini0gi(x),
x > d/2
(1)
i
ψ(x) ) ψ(0) - σx/ε0εr,
0 < x < d/2
(2)
where n0i is the bulk number density of ions type i, ε0 and εr are the vacuum and relative permittivities, respectively, and gi(x) is the ion-electrode distribution function. The charging process of Kirkwood18 for homogeneous solutions enables gi(x) to be cast in the form:
gi(x) ) ξi(x) exp{-βei[ψ + ηi]}
(3)
where ξi(x) ) gi(ei ) 0) is the exclusion volume term (EVT), which is zero for x < d/2, β ) 1/kBT, and ηi is the fluctuation term. The singlet distribution function in the GCS theory is given by ξi ) 1 for x > d/2 and ηi ) 0. Here, we consider the MPB (MPB5 version),5 which takes into account both the exclusion volume and fluctuation terms. The MPB5 version is presently the most accurate formulation of the MPB equation for the planar double layer. Based on Loeb’s closure19 to estimate the fluctuation term, previous analysis for no imaging gives5
[
gi(x) ) ξi(x) exp -eiβL(ψ) -
]
βei2 (F - F0) 8πε0εr
(4)
parable accuracy have been developed for the exclusion volume term. The approximation used in the MPB5 theory is derived from the Bogoliubov-Born-Green-Yvon (BBGY) hierarchy, namely: x
ξi(x) ) exp{-2π
∞
{∑
ξi(x) ) ξi(σ ) 0) exp
∫
ψ(y) dy
x-d
(4a) F ) 4/[4 + κ(d + 2x)],
d/2 < x < 3d/2
F ) 1/[1 + κd], κ2 )
β ε0εr
x g 3d/2
∑ ei2ni0gi(x)
(4b) (4c) (4d)
i
and
κ0 ) lim κ, xf∞
F0 ) lim F ) 1/[1 + κ0d] xf∞
(4e)
Here, κ0 is the classical Debye-Hu¨ckel constant, and κ is a local Debye-Hu¨ckel parameter. Two approximations of com-
Fs
s
(X - y)gs(X) ×
∫ cs0[gs - gs(σ ) 0)] dV}
(6)
where cs0 is the bulk uncharged hard sphere direct correlation function. The functions ξi(σ ) 0) and cs0 are approximated by the corresponding Percus-Yevick uncharged hard sphere values.21 A common approximation to improve on the PB distribution is to neglect the fluctuation term and solely consider exclusion volume corrections based on a lattice approach. These Langmuir-type theories (for example, see reference 1) are very useful in situations where the steric effect of the ions dominates the concentration and adsorption of the ions in the neighborhood of the electrode. To appreciate the influence of the EVT in the MPB approach, we also consider the situation in which the fluctuation term ηi is neglected so
gi(x) ) ξi(x) exp[-βeiψ(x)]
(7)
With ηi ) 0, eq 5 simplifies to
∑ ns0 s
x+d
∫
max(d/2,y-d)
where φ(1;2) is the fluctuation potential at r2 for the ion i at r1. An alternate exclusion volume term is20
{ [
where
s
y+d
exp[-βesφ(1;2/ei ) 0, r12 ) d)]dXdy} (5)
ξi(x) ) exp π
F (F - 1) L(ψ) ) [ψ(x + d) + ψ(x - d)] 2 2d
∫ ∑ ns0
x+d
∫
max(d/2,x-d)
[(X - x)2 - d2]
]}
gs(X) dX + 4d3 /3
(8)
We will denote by PB+EVT the combination of eq 7 with the exclusion volume term of eqs 6 or 8. A closed set of equations for ψ(x) is now obtained by substituting eqs 3 or 7, along with an appropriate EVT, in the Poisson eq 1. The associated boundary conditions are either ψ(0) or σ specified and ψ(x) f 0 as x f ∞, while both ψ(x) and dψ(x)/dx are continuous for x > 0. The resulting differential equations for ψ(x) are efficiently solved using a previously developed quasilinearization technique.22 We expect the PB+EVT theory to make the correct qualitative prediction of CD at high ionic concentrations when steric effects are important.23 However, the PB+EVT theory is unsuitable for situations in which the exclusion volume and fluctuation terms are of equal importance. A typical situation would be room temperature aqueous electrolytes at low concentration and surface charge. The MPB theory can predict damped charge density oscillations in gi(x), but the neglect of the fluctuation term means that the PB+EVT theory cannot predict these oscillations.
Electric Double-Layer Differential Capacitance
Figure 1. The transition envelope of the MPB and PB+EVT theories. The maximum and minimum refer to the behavior of CD at σ ) 0.
J. Phys. Chem. B, Vol. 113, No. 26, 2009 8927
Figure 2. The simulation and MPB differential capacitance CD at T* ) 0.8 for F* varying from 0.14 to 0.24 in intervals of 0.02. The MPB is given by the solid line, and the simulation points for a specific F* are joined by a dashed line.
Results and Discussion Previous simulation and theoretical work has indicated that the RPM CD has a minimum at zero surface charge for room temperature aqueous electrolytes and a maximum for molten salts and ionic liquids. To cover such a range of physical parameters, we present our results in terms of the reduced temperature T* ) 4πε0εr dkBT/e+e- and reduced density F* ) d3(n0+ + n0-). Calculations were carried out for 1:1 ionic solutions with the ion diameter d ) 0.4 nm. With this valence and ion size, we have considered various states covered by the approximate intervals 0.02 < F* < 0.7 and 0.1 < T* < 1.1. The GCMC CD was calculated using the algorithm applied successfully to the solvent primitive model electrolyte interface.24 The points (σ, ψ(x ) 0)) were arranged in order of increasing σ with -0.3 C/m2 e σ e 0.3 C/m2 at intervals of 0.05 C/m2, except at σ ) 0 when σ ) ( 0.02 C/m2 was used, and the second-order polynomial fitted to the first five points by the least-squares method. The derivative of the polynomial was calculated for the middle point, its inverse giving the differential capacitance. The algorithm was repeated for the five consecutive points, starting with the second one, and then these calculations continued until the last five points were reached. The MPB and PB+EVT CD was calculated using the 5-point difference formula with a step length of 0.01 C/m2. The reported PB+EVT results are those for the EVT in eq 6, as both exclusion volume expressions make similar predictions. In Figure 1 we plot the transition or envelope curve separating the (T*, F*) plane into the regions of maximum and minimum CD behavior at σ ) 0 for the MPB and PB+EVT theories. Convergence problems meant that the investigation of the envelope curve was restricted to 0.1 < T* < 1.1. The GCS minimum behavior is mainly confined to the lower values of F*, the region increasing with F* as T* decreases. The PB+EVT numerical solution breaks down at approximately F* ) 0.7 for all T*, with the MPB theory breaking down at lower values of F*. A detailed investigation of the behavior of GCMC CD across the envelope at T* ) 0.8 as F* increases from 0.14 to 0.24 is shown in Figure 2. The mean activity coefficients corresponding to these F* needed for the GCMC simulations were calculated from the IGCMC technique17 with the values being ln γ ( ) 0.2498, 0.3474, 0.4514, 0.5617, 0.6790, and 0.8025, respectively. The MPB predictions for CD are also given in the same figure, and they are seen to compare favorably with the simulation data. At F* ) 0.14, the GCMC and MPB results are in good agreement, then as F* increases the MPB CD at σ ) 0 increases at the greater rate. The GCMC transition envelope at
Figure 3. The PB+EVT differential capacitance CD at T* ) 0.8 for F* varying from 0.16 to 0.28 in intervals of 0.02.
T* ) 0.8 is approximately at F* ) 0.18 with CD ∼ 2.01 F/m2, while the MPB value of CD is similar with F* taking a slightly lower value. Figure 3 gives the corresponding PB+EVT CD behavior at T* ) 0.8 in the neighborhood of its envelope. From Figures 1 and 3 we have that the PB+EVT envelope occurs at a higher value of F* than either the GCMC or the MPB, with a smaller value of the differential capacitance at σ ) 0 in the transition region. An explanation of the behavior of the two envelopes in Figure 1 can be essentially given by considering the three types of interactions delineated in gi(x): an ion subjected to the mean electrostatic potential, a hard-core interaction, and the interionic electrostatic correlations. With the first interaction, the counterions are attracted to the electrode surface, the thickness of the electric double layer is reduced, and the capacitance increases. The differential capacitance curve is parabolic like, in accordance to the GCS theory. The hard sphere interactions are important in high ion density systems, like the ionic liquids or molten salts. Here, even at σ ) 0, the ions adjacent to the electrode surface form a densely packed layer, so as σ departs from zero, this layer impedes the access of counterions to the electrode surface. Thus, these ions are located at the second or higher layers, the thickness of the electric double layer increases and the capacitance decreases. Although simplified, this picture explains the capacitance maximum at σ ) 0. The interionic correlations stabilize the first layer structure and so support the effect of the hard sphere interactions. Hence, for a particular T*, the MPB envelope is at a lower density than the PB+EVT envelope. The behavior of both curves depends on the temper-
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ature. The hard-core effect is weakly temperature dependent, so the curvature of the envelopes must be mainly due to the electrostatic interactions typical for each theory. The shape of the transition envelope is essentially determined through the competing effects of the electric field and the electrolyte concentration, this being borne out of the qualitative similarity of the PB+EVT and MPB envelopes. The electric field gives the GCS capacitance minimum while the concentration supports a maximum. Increasing the temperature reduces the electric field effect in relation to that of the concentration. All the theories of the electric double layer predict a capacitance minimum at low electrolyte concentrations. This minimum is observed experimentally for dilute electrolyte solutions (F* f 0), while the location of the minimum is used to determine the potential of zero charge at solid electrodes. More concentrated electrolytes (c ∼ 1 M) do not manifest the capacity minimum due to the predominance of the compact (inner) layer capacitance. An aqueous 1:1 electrolyte of such a concentration with the ion diameter d ) 0.4 nm at T ) 298 K and εr ) 78.5 has T* ) 0.56 and F* ) 0.077. The point specified by these coordinates lies to the left of the transition envelope. At the opposite end of the density scale, we have the solventfree liquid electrolytes: molten salts and ionic liquids. According to Larsen,25 the density of molten salts can be taken as F* ) 0.669, and we expect that a similar value is applicable to ionic liquids. Boda et al.26 and Lamperski and Klos4 have found in MC simulations that the reduced melting temperature for a primitive model electrolyte is T*m ) 0.282, so the reduced melting temperature of solvent-free electrolytes is two times lower than the typical value obtained for aqueous electrolytes. The coordinate point F* ) 0.669 and T* ) 0.282 clearly lies to the right of the transition envelope. Assuming εr ) 10 for molten salts,4 we obtain Tm ) 1179 K. This value is in agreement with the experimental data as the melting temperature of a simple 1:1 salt varies from 742 K for LiI to 1269 K for NaF.27 The diameter of organic ions in ionic liquids is much larger than we considered above, while the dielectric spectroscopy measurements show28,29 that εr of ionic liquids is in the range of 9-15. Thus, assuming that the average ion diameter is 1.0 nm and εr ) 15, we obtain Tm ) 314 K. This rough result shows, evidently, that when the ion diameter increases, the melting temperature decreases. The density of a solvent-free electrolyte can only be varied in a small range, so it seems that the transition envelope parameters can only be reached by highly concentrated aqueous electrolytes. From the MPB envelope at T* ) 0.56, we have F* ) 0.198 which corresponds to c ) 2.57 M for d ) 0.4 nm or to c ) 6.09 M for d ) 0.3 nm. Such concentrations are available for many aqueous electrolytes. Using a PB lattice-gas model for ionic liquids, Kornyshev1 has found that, in terms of the lattice saturation factor γ ) F*/ F*max (where F*max is the maximum local reduced density of ions), the CD is bell shaped for γ > 1/3 and “camel” shaped for γ < 1/3. The expression “camel” shaped signifies that the curve has a maximum on either side of the minimum. This analysis of Kornyshev is of interest in that it gives a criterion for CD changing from a minimum to a maximum, as the density increases, for those lower values of T* which applies to ionic liquids. Furthermore, it suggests how the CD may change shape as the density varies. Contrary to our predictions, the critical value of γ controlling the transition envelope is independent of the temperature. This is not surprising as Kornyshev, in his review article, used the mean field lattice-gas model to illustrate how purely packing effects could alter the classical GCS prediction. Presumably a more rigorous application of the
Lamperski et al.
Figure 4. The MPB differential capacitance CD at T* ) 0.5 for F* varying from 0.02 to 0.26 in intervals of 0.04.
Figure 5. The MPB differential capacitance CD at F* ) 0.02 for T* ) 0.3, 0.5, and 0.7.
Langmuir-type theory, which takes into account the density temperature dependence, will lead to a transition envelope dependent on both F* and T*. The predicted critical value of γ is not supported by our calculations. The random close packing value for hard spheres of F*max ∼ 1.23 gives F* ) 0.41 for γ ) 1/3. This value of F* is not possible on the MPB envelope and is only achievable on the PB+EVT envelope for T* approximately less than 0.2. The close-packed face-centered cubic value of F*max ∼ 1.413 leads to a higher envelope value of F* ) 3 0.471. Oldham’s theory is based on a GCS model of the electric double layer and cannot predict any transition envelope. Only bell-shaped CD curves were found, while contact with the work of Kornyshev is only possible at the particular case of γ ) 1. A camel-type behavior is seen in the MPB CD at a lower density than the transition F*, while at a higher density a parabola-like dependence is observed. This is illustrated in Figure 4 at T* ) 0.5 for F* increasing from 0.02 to 0.26. The theoretical prediction, whereby the MPB CD transforms from a minimum to a maximum for varying F* at fixed T*, differs from that suggested by Kornyshev.1 In the PB lattice-gas model the CD at the position of zero charge is unaltered for varying γ, while the MPB CD is found to be nearly constant for some nonzero values of σ. For instance, in Figure 4 where T* ) 0.5 and F* varies from 0.02 to 0.26, the MPB CD curves intersect in the interval 2.30-2.38 F/m2 with |σ| lying between 0.27 C/m2 and 0.38 C/m2. The MPB camel behavior is also sensitive to variations in T*, as can be seen in Figure 5 for F* ) 0.02. Reducing T* at a fixed F* leads to an increase in CD for all σ and distinct maxima. At a large surface potential, the lattice theory approach1 predicts that CD is proportional to ψ(0)-R, where R ) 1/2. A check was made on the corresponding
Electric Double-Layer Differential Capacitance behavior of the MPB and PB+EVT theories at the F*and T* values of Figures 3-5. Unfortunately, the numerical limitations meant that no definite conclusions could be made regarding the limiting behavior of CD at large ψ(0). For example, in Figure 5 for T* ) 0.7 and F* ) 0.02, the MPB convergence was restricted to σ < 0.5 C/m2 and the PB+EVT convergence to σ < 1 C/m2. Both theories suggested that R < 1/2 with R varying with the physical parameters. Furthermore, in Figure 5, it is unclear whether or not maxima occur in the MPB CD at T* ) 0.7 and F* ) 0.02 for large surface charge. Careful simulations are required to analyze the shape of differential capacitance curves in the vicinity of, and well away from, the transition envelope.
J. Phys. Chem. B, Vol. 113, No. 26, 2009 8929 equation must be treated for an accurate prediction of the transition region. Both the theories predict that along the envelope, F* increases as T* decreases. The PB lattice transition occurs at too high a value of F*, and the simple transition value of γ appears unrealistic. The mechanism whereby the CD changes from a GCS-type minimum to a maximum differs between the PB lattice-gas model and the MPB or PB+EVT theories. More detailed simulations are required to identify the full transition envelope and clarify for a given T* how, in the RPM framework, the CD mutates in going from low to high densities. Acknowledgment. Financial support from the Adam Mickiewicz University, Faculty of Chemistry, is appreciated.
Conclusion The classical PB equation of the GCS theory only predicts a parabolic-type behavior with a minimum at σ ) 0 for the RPM CD. Kirkwood identified that the PB equation neglected both an exclusion volume and a fluctuation term. Addition of only the exclusion volume term, giving the PB+EVT theory, shows that a qualitative change can be brought about in the CD at σ ) 0. A similar qualitative change can also be predicted by latticetype exclusion volume corrections to the classical theory. The PB lattice theory predicts a change in CD at σ ) 0, at the critical value of the saturation factor γ ) 1/3, but only predicts the GCS result for γ f 0. The MPB includes approximations to both the exclusion and fluctuation terms. The addition of the fluctuation term results in the MPB transition envelope occurring, for fixed T* at a lower value of F* rather than in the corresponding PB+EVT envelope. Experimental capacitance can display a wide variety and range of shapes. This is not surprising since for solvent-free ionic liquids and molten salts, effects such as nonspherical ions, different valences and temperature, ion adsorption, electric polarization, and electrode structure, can radically change the double-layer properties. In general, electrolyte solutions have the further complicating feature of the solvent. Maxima in CD have been observed in various electrolyte solutions,30 but the mechanism is different to that seen here for the RPM. These maxima essentially arise from the behavior of the solvent in the compact layer rather than the high density of the ions. The analysis presented here has been for a symmetric valence RPM electrolyte so that the calculated CD is symmetric about σ ) 0 with the maximum or minimum also at σ ) 0. Simply relaxing the restriction to unequal ion sizes or asymmetrical valences leads to an asymmetrical CD.31,32 Fedorov and Kornyshev33 considered unequal ion sizes in the mean field lattice theory and with this extension have made a successful comparison with a molecular dynamics simulation calculation of CD. The PB+EVT theory is capable of treating unequal sizes and valences, but the present MPB formulation for unequal ion sizes is unsatisfactory.34 A detailed comparison of the MPB predictions with the simulation data, in the neighborhood of the envelope at T* ) 0.8, indicated that the MPB gives a good representation of the behavior of CD at σ ) 0. Thus, both corrections to the PB
References and Notes (1) Kornyshev, A. A. J. Phys. Chem. B 2007, 111, 5545. (2) Kilic, M. S.; Bazant, M. Z.; Ajdari, A. Phys. ReV. E: Stat., Nonlinear, Soft Matter Phys. 2007, 75, 021502. (3) Oldham, K. B. J. Electroanal. Chem. 2008, 613, 131. (4) Lamperski, S.; Kłos, J. J. Chem. Phys. 2008, 129, 164503. (5) Outhwaite, C. W.; Bhuiyan, L. B. J. Chem. Soc., Faraday Trans. 2 1983, 79, 707. (6) Graves, A. D.; Inman, D. J. Electroanal. Chem. 1970, 25, 357. (7) Parsons, R. Chem. ReV. 1990, 90, 813. (8) Kisza, A. Electrochim. Acta 2006, 51, 2315. (9) Nanjundiah, C.; McDevitt, S. F.; Koch, V. R. J. Electrochem. Soc. 1997, 144, 3392. (10) Lockett, V.; Sedev, R; Ralston, J.; Horne, M.; Rodopoulos, T. J. Phys. Chem. C 2008, 112, 7486. (11) Alam, M. T.; Islam, M. M.; Okajima, T.; Ohsaka, T. J. Phys. Chem. C 2008, 112, 16600. (12) Feng, G.; Zhang, J. S.; Qiao, R. J. Phys. Chem. C 2009, 113, 4549. (13) Gouy, G. J. Phys. (Paris) 1910, 9, 457. (14) Chapman D. L. Philos. Mag. 1913, 25, 475. (15) Stern, O. Z. Elektrochem. 1924, 30, 508. (16) Frenkel, D.; Smit, B. Understanding Molecular Simulation. From Algorithms to Applications; Academic Press: San Diego, 1996. (17) Lamperski, S. Mol. Simul. 2007, 33, 1193. (18) Kirkwood, J. G. J. Chem. Phys. 1934, 2, 767. (19) Loeb, A. L. J. Colloid Sci. 1951, 6, 75. (20) Outhwaite, C. W.; Lamperski, S. Condens. Matter Phys. 2001, 4, 739. (21) Hansen, J. P.; McDonald, I. R. The Theory of Simple Liquids, 2nd ed.; Academic Press: London, 1990. (22) Outhwaite, C. W.; Bhuiyan, L. B.; Levine, S. J. Chem. Soc., Faraday Trans. 2 1980, 76, 1388. (23) Lamperski, S.; Bhuiyan, L. B. J. Electroanal. Chem. 2003, 540, 79. (24) Lamperski, S.; Zydor, A. Electrochim. Acta 2007, 52, 2429. (25) Larsen, B. J. Chem. Phys. 1976, 65, 3431. (26) Boda, D.; Henderson, D.; Chan, K.-Y. J. Chem. Phys. 1999, 110, 5346. (27) CRC Handbook of Chemistry and Physics; 88th ed.; Lide, D. R., Ed.; CRC Press: Cleveland, OH, 2007. (28) Wakai, C.; Oleinikova, A.; Ott, M.; Weinga¨tener, H. J. Phys. Chem. B 2005, 109, 17028. (29) Daguenet, C.; Dyson, P. J.; Krossing, I.; Oleinikova, A.; Slattery, J.; Wakai, C.; Weinga¨tener, H. J. Phys. Chem. B 2006, 110, 12682. (30) Parsons, R. Electrochim. Acta 1976, 21, 681. (31) Valleau, J. P.; Torrie, G. M. J. Chem. Phys. 1982, 76, 4623. (32) Carnie, S. L.; Torrie, G. M. AdV. Chem. Phys. 1984, 56, 141. (33) Fedorov, M. V.; Kornyshev, A. A. J. Phys. Chem. B 2008, 112, 11868. (34) Outhwaite, C. W.; Bhuiyan, L. B. J. Chem. Phys. 1986, 84, 3461.
JP900037H
