J. Phys. Chem. B 2007, 111, 13075-13081
13075
Series Approach to Modeling Ion Size Effects for Asymmetric Electrolytes in the Diffuse Double Layer Thomas G. Smagala and W. Ronald Fawcett* Department of Chemistry, UniVersity of California, DaVis, California 95616 ReceiVed: May 18, 2007; In Final Form: September 2, 2007
The theory of the diffuse layer for asymmetric electrolytes is reconsidered with emphasis on the effects of ion size on the diffuse layer potential drop and differential capacity. For asymmetric 2:1 and 1:2 electrolytes, this potential drop is expressed in terms of a polynomial with a linear, quadratic, and cubic term in the corresponding estimate in the Gouy-Chapman theory. Optimal polynomial coefficients and model validation for 2:1 electrolytes are provided by least-squares regression of Monte Carlo data obtained for a restricted electrolyte in a primitive solvent. These coefficients are then expressed as simple functions of the parameters commonly associated with the mean spherical approximation. The series approach accurately describes potential drops and differential capacities of the diffuse layer for 2:1 and 1:2 electrolytes for the chosen assumptions.
Introduction In our previous work on the series approach,1-4 the focus was on symmetric electrolytes. The goal was to develop analytical improvements to the classical Gouy-Chapman (GC) theory5-7 that take into account ion size effects and yet are still easily applied by experimentalists. An expression for the potential drop across the diffuse layer was developed as a series in the corresponding GC estimate.1-4 In the most recent version,4 the expressions obtained for the series coefficients were based upon the estimates of quantities from the hypernetted chain/ mean spherical approximation (HNC/MSA) integral equation approach8-10 using Monte Carlo (MC) data. In the present paper, this is extended to 2:1 and 1:2 electrolytes, which is a much more difficult task. First of all, the GC estimate of the potential drop across the diffuse layer involves solving two cubic equations, one for positive charge densities and another for negative charge densities. Second, the plot of the potential drop against electrode charge density is asymmetric and therefore more difficult to fit to a polynomial equation. Once again, the coefficients of the series can be related to parameters associated with ion size effects in the MSA. A method is presented to estimate the diffuse layer potential drop and differential capacity for restricted electrolytes in a primitive solvent. Validation of the new model for 2:1 electrolytes is provided by MC simulations.
Grahame presented Gouy-Chapman theory for 2:1 and 1:2 electrolytes some time ago.5 It is worthwhile to review and expand upon his derivation. GC theory is governed by the Poisson-Boltzmann differential equation
dx2
)-
F2σ2 RT0s
∑j zjcj exp(-zjφ)
φ(x) ) Fφ(x)/RT
(1)
* Corresponding author. Phone: (530) 752-1105. Fax: (530) 752-8995. E-mail:
[email protected].
(2)
The remaining quantities are 0, the vacuum permittivity, s, the relative permittivity of the pure solvent, cj, the electrolyte concentration in SI units, zj, the ionic charge, and σ, the ionic diameter. The last quantity should not be confused with σm, the charge density of the wall. Equation 1 can be integrated to yield a more convenient form
( ) dφ dx
2
)
2F2cj
∑j RT [exp(-zjφ) - 1]
(3)
0 s
For 2:1 electrolytes, eq 3 becomes
dφ 2 κ2σ2 [exp(-2φ) + 2 exp(φ) - 3] ) dx 3
( )
(4)
The Debye-Hu¨ckel reciprocal length for both 2:1 and 1:2 electrolytes is given by
6F2ce κ2 ) RT0s
Gouy-Chapman Theory for 2:1 and 1:2 Electrolytes
d2φ
where φ is the dimensionless potential profile at dimensionless position x in the diffuse layer. The summation is carried out over all ionic species in solution. The dimensionless potential is related to the actual potential φ by the equation
(5)
Note that ce is the bulk concentration in SI units. The origin of the system, x ) 0, is defined at the edge of the diffuse layer, also known as the outer Helmholtz plane (oHp). Thus, the potential at x ) 0 is the potential drop across the diffuse layer, φd, and is our main quantity of interest. At this plane, Gauss’ law is obeyed and is expressed in dimensionless form for both electrolytes as
(dφdx )
)x)0
10.1021/jp073868l CCC: $37.00 © 2007 American Chemical Society Published on Web 11/08/2007
κσ E x3
(6)
13076 J. Phys. Chem. B, Vol. 111, No. 45, 2007
Smagala and Fawcett
The quantity E is the dimensionless charge density or field defined as
E ) σm/AGC
(7)
where AGC is the Gouy-Chapman constant.
AGC ) (2RT0sce)1/2
(8)
Combining eqs 4 and 6 yields the governing algebraic equations for φdGC, which for 2:1 electrolytes is
exp(-2φdGC) + 2 exp(φdGC) ) E2 + 3
(9)
From eq 9, it is trivial to obtain E as a function of φdGC, but the inverse is more complicated. Using the substitution X ) exp(φdGC), we obtain a cubic equation.
2 X3 - (3 + E2) X2 + 1 ) 0
(10)
Only the first or principle root is physically valid and only for positive charge densities. By properties of logarithms, φdGC is always positive.
φdGC ) + ln X,
E>0
(11)
The negative side is solved by means of a different substitution, Y ) exp(-φdGC). This results in a very different cubic equation:
Y3 - (3 + E2) Y + 2 ) 0
(12)
Again, the only first real root is physically valid. For negative charge densities, this always yields the correct real and negative potentials
φdGC ) - ln Y,
E 0
φdGC(2:1, E > 0) ) ln
[(
(E2 + 3)2 1 2 E +3+ + ∆+ 6 ∆+
)]
(14)
Figure 1. Dimensionless diffuse layer potential drops versus dimensionless charge density. Dashed curves show GC theory for 2:1 (long dashes) and 1:2 electrolytes (short dashes). Points indicate 2:1 MC data for concentrations of 0.1 (O), 0.2 (3), 0.5 (0), 1 ()), and 2 M (4) with an ion size of 300 pm.
Equations 14 and 16 are continuous and smooth when they meet at the origin, meaning that they both approach zero and have the same slope when E ) 0. Thus, the explicit solution of φdGC(E) requires two completely different expressions, but they are consistent with each other and match at the origin. A plot of φdGC(E) is shown in Figure 1. The agreement with MC data on the negative side for 2:1 electrolytes is poor because GC theory fails to predict the change in slope and the extrema. In the subsequent sections, an analytical model capable of matching the more exotic behavior of the MC data is presented. Note in Figure 1 that φdGC(1:2, E) is the mirror image of φdGC(2:1, E) across both the x and y axes. Series Expressions Obtained from the Integral Equation Approach Details of the integral equation approach (HNC/MSA theory) applied to 2:1 and 1:2 electrolytes are found in the work of Lozada-Cassou and Henderson.10 Numerical solution of these equations coupled with the Poisson equation requires iteration to solve for the ion-wall correlation functions, gi(x) and φ(x). At x ) 0, the HNC/MSA integral equation is
∑j Fj∫-∞[gj(t) - 1] Csrij (t) dt
where
ln gi(0) ) -zi φd + 2πσ3
∆+ ) [ E (E + 9E + 27) - 27 + 2
4
2
6x3x-E2(E4 + 9E2 + 27)]1/3 (15) For E < 0, the solution has a strikingly different form
)
(16)
∆- ) [-54 + 6x3x-E2(E4 + 9E2 + 27)]1/3
(17)
φdGC(2:1,
E < 0) ) - ln
(
18 + 6E2 + 21/3∆23‚22/3∆-
The solution is easily transformed to the 1:2 solution by a simple change of signs
(20)
where Fj is the number density of ion j, which can be calculated from the ion’s concentration in SI units by
Fi ) NLci
where
φd(1:2, E > 0) ) -φd(2:1, E < 0)
∞
(21)
and NL is Avogadro’s number. Csr ij is the short-range direct correlation function (SR DCF) between ions i and j in the bulk of the solution. In 2:1 electrolytes, species 1 is defined to be the cation with a charge of +2, and species 2 is the anion with charge -1. For convenience, the wall-sum and wall-difference correlation functions are introduced.
(18)
gws(x) ) [g1(x) + 2g2(x)]/3
(22)
gwd(x) ) [g2(x) - g1(x)]/3
(23)
and
and
φd(1:2, E < 0) ) -φd(2:1, E > 0)
(19)
Modeling Ion Size Effects for Asymmetric Electrolytes
J. Phys. Chem. B, Vol. 111, No. 45, 2007 13077
The wall-sum constraint for a 2:1 electrolyte, the condition for the contact value of the density profile for charged hard spheres near a charged wall, has the form
gws(0) ) a0 + E2/3
approximation that maintains both analytical simplicity and the ability to describe non-GC like behavior. Rather than directly using eq 29, an approximate expression for gwd(x) is introduced to obtain
(24)
where a0 is the compressibility factor. The currently accepted expression for a0 is
1 + η + η 2 - η3 Γ3 18η (1 - η)3
a0 )
(25)
The MSA volume fraction, η, and reciprocal distance parameter, Γ, are defined as
η ) πFσ3/6
(26)
Γ ) (1 + 2κσ)1/2/2 - 1/2
(27)
and
Note that Fσ3 is the dimensionless ion density for a 2:1 electrolyte where
F ) 3NLce
Ψ0 {exp[-2(φd + Θ0)] + 2 exp(φd + Θ0)} ) Ψ0 + E2/3 (34) This small change greatly simplifies the resulting series solution without any appreciable loss in accuracy. Both eq 29 and eq 34 can be solved exactly for φd and have three analytical solutions for E > 0 and for E < 0. The solutions resemble the GC solution of eqs 14-17 but with extra terms. Just as in GC theory, only one root on each side is physically valid. Numerical solution of eqs 9, 29, and 34 should be avoided, especially for E < 0, because the solution can converge to the wrong root and yield aphysical results. While GC theory fails to predict local extrema in φd(E), a simple cubic polynomial without a constant term can describe such behavior. Once again, a substitution is introduced to change the dependent variable from E to φdGC.
E ) ( [2 exp(φdGC) + exp(-2φdGC) - 3]1/2
(28)
When eq 28 is combined with HNC/MSA theory, the result is
Ψ0{exp[-2(φd + Θ0)] + 2 exp(φd + Θ0)} ) a0 + E2/3 (29) The quantities Ψ0 and Θ0 are defined as
∫-11(gws(t) - 1) Chs(t) dt]
Ψ0 ) exp[2πFσ3
(30)
and
∫01gwd(t)Ces(t) dt
Θ0 ) 4πFσ3
This conveniently removes some of the nonlinear curvature and allows MC data for φd to be fitted by a simple polynomial equation. In eq 35, the positive square root is used for E > 0, and the negative square root is used for E < 0. Upon expanding eq 34 as the Taylor series of the form φd(φdGC), one finds that a single series is capable of describing the whole range of positive and negative E. This is the great advantage of the series approach in modeling asymmetric electrolytes. Rather than dealing with two different expressions for φd for the positive and negative sides, a single continuous polynomial is capable of modeling the full range of E. Truncating to a cubic polynomial yields
(31)
These quantities are equivalent to eqs 12 and 13 of ref 4. In the bulk MSA, the SR DCF, Cijsr, is decomposed into a hard-sphere contribution, Chs(x), and an electrostatic contribution, Ces(x).8-9,12 The full mathematical definitions of these quantities are shown in previous work.8 In order to evaluate the integrals in eqs 30 and 31, the functions gws(x) and gwd(x) were taken from ion-wall correlation functions determined from MC simulations for a variety of different ion sizes, concentrations, and charge densities. This method is more accurate than the Henderson-Blum approach, especially for higher valence electrolytes.7 The results of the numerical integration indicate that the quantities Ψ0 and Θ0 are well-represented by simple analytical functions
Ψ0 ) (a0 - a2 E) sech(a1 E)
(32)
Θ0 ) b1E + b2E2 + b3E3
(33)
and
Note that each of these functions contain one additional term compared with their counterparts for symmetric electrolytes.4 It must be emphasized that this is not an analytical solution of the HNC/MSA equations. Instead, we seek the best
(35)
φd(φdGC) ) d1φdGC + d2(φdGC)2 + d3(φdGC)3
(36)
The Taylor series expansion generates the analytical functions for the {di} coefficients in terms of the parameters {ai} and {bi} from eqs 32 and 33
1 - x3 b1 a1/2 0
(37)
x3a2 b1 1 1 - 1/2 + 3/2 + - 3b2 6a0 6a 2a 2x3
(38)
d1 )
d2 )
0
0
and
d3 ) -
3a21 a2 1 1 1 + + + 3/2 1/2 1/2 18a 18a0 9a0 4a0 0 2x3a02 a2
+ 3/2
2x3a0
9a22 8a5/2 0
-
b1 3x3
+ b2 - 3x3b3 (39)
The remaining challenge is to find expressions for the unknown coefficients {ai} and {bi}. Least-squares regression of Ψ0 and Θ0 to eqs 32 and 33 yields the best fit estimates for {ai} and
13078 J. Phys. Chem. B, Vol. 111, No. 45, 2007
Smagala and Fawcett
{bi}. Relationships can then be found between these coefficients and the MSA parameters, η and Γ. Diffuse Layer Differential Capacity The previous analytical expressions for the potential drop across the diffuse layer can be used to calculate the differential capacity of the diffuse layer, Cd. The dimensionless diffuse layer capacity is defined as
C ˜d )
RT Cd F AGC
(40)
This quantity is computed by differentiation
C ˜d )
dE dφd
(41)
Using this definition, we find the GC differential capacity for a 2:1 electrolyte is
C ˜ d,GC(2:1) ) (
exp(φdGC) - exp(-2φdGC)
x exp(-2φdGC) + 2 exp(φdGC) - 3
Figure 2. Dimensionless diffuse layer potential drops versus their Gouy-Chapman estimates for a 2:1 electrolytes with ion size of 300 pm. Solid curves show cubic fits for concentrations of 0.1, 0.2, 0.5, 1, and 2 M. Dashed line shows GC theory. MC estimates are shown as points with same meaning as in Figure 1.
(42)
and for a 1:2 electrolyte
C ˜ d,GC(1:2) ) (
exp(2φdGC) - exp(-φdGC)
x2 exp(-φdGC) + exp(2φdGC) - 3
(43)
The positive sign is used for E > 0 and negative sign for E < 0. In the limit as E goes to zero, C ˜ d,GC at the point of zero charge (pzc) is x3 ≈ 1.73 for both electrolytes. Once again, it is possible to write simple transformations between the 2:1 and 1:2 values of Cd.
Cd(1:2, E > 0) ) Cd(2:1, E < 0)
(44)
and
Cd(1:2, E < 0) ) Cd(2:1, E > 0)
(45)
This indicates that the 2:1 and 1:2 values of Cd are mirror images of each other across the y axis on a plot of Cd(E) or Cd(φdGC). The Gouy-Chapman diffuse layer differential capacities are required for the calculation of C ˜ d in the series approach. For the case of asymmetric electrolytes, the estimate of C ˜ d by the series approach follows directly from eq 36
1 dφd 1 ) ) (d + 2d2(φdGC) + 3d3(φdGC)2) (46) C ˜ d dE C ˜ d,GC 1 For any electrolyte, the series approach provides a very simple result for calculating Cd at the pzc
˜ d,GC (pzc)/d1 C ˜ d (pzc) ) C
(47)
Because of the scatter and spacing of our MC data, accurate numerical differentiation is impossible. Consequently, a comparison with the Cd values estimated from MC data is not shown here. Results and Comparisons with Monte Carlo Simulations The specifics of the Monte Carlo simulations are discussed in earlier work.2 For each electrolyte, the MC calculations
Figure 3. Reciprocal dimensionless differential capacity of the diffuse layer versus φdGC. Dashed curves show GC theory for 2:1 (long dashes) and 1:2 electrolytes (short dashes). Solid curves are derived from the 2:1 cubic fits of Figure 2 and are for concentrations of 0.1, 0.2, 0.5, 1, and 2 M from top to bottom.
were performed for 5 different concentrations, 16 different positive and negative charge densities, and 3 different ion sizes. All calculations are carried out at room temperature, T ) 298 K, and assume a primitive solvent which has the same relative permittivity as water, s ) 78.46. Earlier comparisons of diffuse layer models10,13 with MC data14 used the rather large ion size of 425 pm. This is greater than the diameter of most unsolvated alkali metal and halide ions.5 Instead, we choose a diameter of 300 pm for most calculations. This is closer to the diameter of an average unsolvated monatomic ion. However, calculations at 200 and 400 pm are also shown for comparison. The potential drop across the diffuse layer as a function of charge density was tabulated from MC simulations. The MC values of φd were plotted as a function of φdGC and fitted to polynomials of the form of eq 36. The resulting best fit curves for 2:1 electrolytes are shown in Figure 2. The best fit coefficients are given in Table 1. The broken line with unit slope shows the GC results for reference. The MC data and the series approach deviate from this line by an amount which depends upon concentration and ion size. In all cases, GC theory becomes progressively less valid at higher charge densities, especially for negative charge densities. Fluctuations in the MC data
Modeling Ion Size Effects for Asymmetric Electrolytes
J. Phys. Chem. B, Vol. 111, No. 45, 2007 13079
TABLE 1: Best-Fit Coefficients from Least-Squares Regression of MC Diffuse Layer Potential Drop to the Corresponding GC Value (equation 46)a 200 pm 0.1 M 0.2 M 0.5 M 1M 2M a
300 pm
d2
d3
d1
d2
d3
d1
d2
d3
0.889 0.938 0.966 0.966 0.957
0.117 0.143 0.189 0.213 0.259
-0.0236 -0.0361 -0.0592 -0.0818 -0.1230
0.810 0.800 0.769 0.652 0.510
0.120 0.140 0.184 0.205 0.244
-0.0221 -0.0291 -0.0466 -0.0549 -0.0747
0.738 0.698 0.591 0.446 -0.0043
0.116 0.141 0.180 0.221 0.240
-0.0167 -0.0217 -0.0296 -0.0344 0.0025
Each set of data was fitted to a cubic polynomial.
TABLE 2: Dimensionless Differential Capacity of the Diffuse Layer at the Point of Zero Charge for 2:1 or 1:2 Electrolytes as Predicted by the Series Approach for Five Different Concentrations and an Ion Size of 300 pma
a
400 pm
d1
0.1 M
0.2 M
0.5 M
1M
2M
2.14
2.17
2.25
2.66
3.40
Gouy-Chapman value is 1.73 in all cases.
occurred because of the finite number of particles used in the simulations. Aside from these fluctuations, the series approach matches the behavior of φdMC rather successfully. The differential capacity of the diffuse layer is easily computed from the polynomial fits of Figure 2. Note that C ˜d approaches infinity whenever φd exhibits an extremum. Thus, it is easier to plot 1/C ˜ d as a function of φdGC, the results being shown in Figure 3. GC values are computed from eq 42. As stated previously, comparisons with MC data are not feasible. Once again, GC theory predicts a single curve, while the series approach shows branching dependent upon concentration and ion size. Table 2 shows how C ˜ d (pzc) varies over 5 different concentrations in the series approach, calculated by means of eq 47.
{di} given by eqs 37-39, the following expressions were obtained:
a1 ) 12η3/2
(48)
a2 ) 0.9η
(49)
b1 ) 0.45Γ2/3
(50)
b2 ) -(1/36)Γ0.5
(51)
b3 ) (3/50)Γ5 - (3/200)η0.05
(52)
These expressions are valid for all of the different concentrations and ion sizes used in a 2:1 electrolyte. Different expressions for the compressibility were considered, but the optimal a0 is given by eq 25. These expressions were determined sequentially by first considering the optimal fits for d1 via eq 37, then taking the known expressions and determining the unknown expression
Monte Carlo simulations are tedious, and thus, it is desirable to derive explicit analytical expressions for the coefficients di which do not require the use of MC data. As mentioned previously, one first must calculate the MC values of Ψ0 and Θ0. The best fit estimates for {ai} and {bi} are then obtained by least-squares regression to eqs 33 and 34. The resulting best fit curves compared to their MC values are shown in Figure 4 for a 2:1 electrolyte with an ion size of 300 pm. The excellent fits, which are also observed for the other sizes, demonstrate that complicated numerical quantities in the integral equation approach can be accurately described by simple analytical functions. Simple representations of coefficients {ai} and {bi} can be obtained in terms of the MSA parameters, η and Γ. We follow the same procedure for determining those relationships used in the most recent work on the series approach.4 Again, it was found that expressions obtained in this manner for {ai} and {bi} from accurate fits of both Ψ0 and Θ0 resulted in unsatisfactory representations of {di}. However, those expressions served as a useful guideline for the optimal functional form of the {di} coefficients in terms of the MSA parameters. For example, this analysis demonstrated that a1 and a2 are proportional to ηm and b1 and b2 are proportional to Γn, where n and m are simple fractions or integers. It must be emphasized that accurate analytical representations of Ψ0 and Θ0 are not the primary objective. From a practical sense, accurate expressions for {di} are more important for the analytical computation of φdGC. With a0 given by eq 25 and using the definition of
Figure 4. Plots of Ψ0 and Θ0 as a function of E for 2:1 electrolytes with ion size of 300 pm. MC estimates are shown as points with same meaning as in Figure 1. Solid curves show best fits of these MC estimates.
13080 J. Phys. Chem. B, Vol. 111, No. 45, 2007
Smagala and Fawcett It is interesting to examine the expressions for {di} in the limit that ionic diameter σ goes to zero. First of all, the MSA parameters η of Γ both go to zero in this limit (see eqs 26 and 27). At the same time, the parameter a0 goes to one (eq 25). All other parameters a1, a2, b1, b2, and b3 go to zero (eqs 4852). As a result, in the limit that ion size goes to zero, d1 goes to one, and d2 and d3 go to zero (eqs 37-39). Thus, as expected
φd (σ f 0) ) φdGC
(53)
Although eqs 37-39 and 48-52 have been written for 2:1 electrolytes, φd and Cd are readily transformed to their 1:2 forms. Equations 18-19 and 44-45 were written for discontinuous analytical solutions. Because the series approach is continuous at the pzc, it is more compact to write
φd(1:2, E) ) -φd(2:1, -E)
(54)
Cd(1:2, E) ) Cd(2:1, -E)
(55)
and
Interested users can now apply the analytical model directly for 2:1 and 1:2 electrolytes without performing MC simulations. Comparisons of each coefficient obtained from regression of the MC data with its theoretical estimate from the MSA parameters are shown in Figure 5. Discussion and Conclusion
Figure 5. Series coefficients obtained from regression of MC data as a function of their theoretical estimates. Data shown for d1, d2, and d3 coefficients for 2:1 electrolytes. Solid lines have unit slope. Points indicate coefficient data for ion sizes of 200 (2), 300 (b), and 400 pm (1).
contained in d2 via eq 38, and finally repeating the process for d3 using eq 39. Note how the {ai} coefficients are mostly functions of η, while the {bi} coefficients are mostly functions of Γ. This is not surprising because {ai} come from the analytical representation (eq 32) of the hard-sphere contribution (eq 30), while the {bi} come from the analytical representation (eq 33) of the electrostatic contribution (eq 31) to the MSA SR DCF. However, just as in the symmetric case, neither set of coefficients is purely a function of either η of Γ.4 Combining eqs 36-39, 46, and 48-52 yields a complete analytical model for the computation of potential drops and differential capacities of the diffuse layer.
The series approach has now been extended to more complicated asymmetric electrolytes. In 2:1 and 1:2 electrolytes, the MC values for φd differ greatly from their GC estimates. Until now, very little has been done to derive analytical models capable of describing such electrolytes. GC theory fails to predict the change in slope and local extrema of φd and Cd at higher charge densities. More recent methods for modeling these effects in such systems require more complicated numerical solution10,13,15 or long simulation times.14 This is the distinct advantage of the series approach, the only currently available analytical model to our knowledge capable of describing such behavior. The explicit solution of φd(E) in eqs 9, 29, and 34 requires two completely different expressions, one for E > 0 and another for E < 0. Fortunately, a single cubic series can be used to describe both the positive and the negative sides. This is another advantage of the series approach in the analytical modeling of asymmetric electrolytes. These electrolytes, such as 2:1 and 1:2 cases studied here, differ from symmetric electrolytes in that their Taylor series expansions require both even and odd terms. Thus, the analytical representations of integral quantities such as Ψ0 and Θ0 differ from the corresponding expressions for Ξ0 and Η0 in the symmetric case.4 However, the differences involved the addition of only one extra term, and the entire analysis for asymmetric electrolytes parallels the analysis for the symmetric case. The series expressions in this work for 2:1 electrolytes are readily transformed by substitution into the corresponding expressions for 1:2 electrolytes by eqs 54 and 55. Acknowledgment. The authors are grateful to Dr. Dezso´ Boda for making available his Monte Carlo program. Financial support was received from the National Science Foundation, Washington through a research grant (CHE 0451103). References and Notes (1) Fawcett, W. R.; Smagala, T. G. Condens. Matter Phys. 2004, 7, 709.
Modeling Ion Size Effects for Asymmetric Electrolytes (2) Fawcett, W. R.; Smagala, T. G. J. Phys. Chem. B 2005, 109, 1930. (3) Fawcett, W. R.; Smagala, T. G. Langmuir 2006, 22, 10635. (4) Smagala, T. G.; Fawcett, W. R. J. Phys. Chem. B 2007, 111, 1443. (5) Grahame, D. C. J. Chem. Phys. 1953, 21, 1054. (6) Fawcett, W. R. Liquids, Solutions, and Interfaces; Oxford University Press: New York, 2004; Chapters 3 and 10. (7) Fawcett, W. R.; Henderson, D. J. Phys. Chem. B 2005, 109, 22608. (8) Carnie, S. L.; Chan, D. Y. C.; Mitchell, D. J.; Ninham, B. W. J. Chem. Phys. 1981, 74, 1472. (9) Lozada-Cassou, M.; Saavedra-Barrera, R.; Henderson, D. J. Chem. Phys. 1982, 77, 5150.
J. Phys. Chem. B, Vol. 111, No. 45, 2007 13081 (10) Lozada-Cassou, M.; Henderson, D. J. Phys. Chem. 1983, 87, 2821. (11) Wolfram, S. The Mathematica Book, 4th ed.; Wolfram Research: Champaign, IL, 1999. (12) Waisman, E.; Lebowitz, J. L. J. Chem. Phys. 1972, 56, 3086; J. Chem. Phys. 1972, 56, 3093. (13) Outhwaite, C. W.; Bhuiyan, L. B. J. Chem. Soc., Faraday Trans. 2 1983, 79, 707. (14) Torrie, G. M.; Valleau, J. P. J. Phys. Chem. 1982, 86, 3251. (15) Boda, D.; Fawcett, W. R.; Henderson, D.; Sokołowski, S. J. Chem. Phys. 2002, 116, 7170.
