Z. Phys. Chem. 220 (2006) 427–439 / DOI 10.1524/zpch.2006.220.4.427 by Oldenbourg Wissenschaftsverlag, München
An Empirical Approach to Modeling Ion Size Effects in Diffuse Double Layer Theory By Thomas G. Smagala and W. Ronald Fawcett ∗ Department of Chemistry, University of California, Davis, CA 95616, USA
Dedicated to Prof. Dr. Walther Jaenicke on the occasion of his 85 th birthday (Received November 30, 2005; accepted December 21, 2005)
Double Layer Theory / Diffuse Double Layer / Ion Size Effects A novel empirical model for the diffuse double layer is found by generalizing the simple analytical equations of Gouy–Chapman theory. Two adjustable parameters are introduced into the Boltzmann equation for the exponential dependence of the ion–wall correlation functions on the diffuse layer potential. Optimal parameter values and model validation for 1 : 1 and 2 : 1 electrolytes are provided by Monte Carlo simulations. Simple relationships are obtained between these empirical parameters and those commonly associated with the mean-spherical approximation. The new empiricism accurately models diffuse layer potential profiles and ion–wall correlation functions for a restricted 1 : 1 electrolyte in a primitive solvent.
1. Introduction For nearly a century, there have been mathematical models for the diffuse double layer in contact with a charged surface. The Gouy–Chapman theory provides the earliest and simplest model, governed by the nonlinear Poisson– Boltzmann differential equation. Despite its analytical simplicity, the model is flawed because it neglects the finite size of the ions and variable solvent permittivity. Such conclusions are apparent upon comparison with Monte Carlo simulations of a restricted electrolyte in a primitive solvent. Sophisticated theories have been devised using density functional theory [1–5] and the integral equation approach [6–8]. However most of these models are not analytical and require rather complicated numerical solutions. Thus they are rarely used by experimentalists. * Corresponding author. E-mail:
[email protected]
428
T. G. Smagala and W. R. Fawcett
One of the greatest attributes of GC theory is its simple analytical form, and this is why it is still currently used for comparisons with experimental diffuse layer data. This has fueled the development of more detailed analytical models. The authors have recently developed an analytical series approach for modeling the potential drop across the diffuse layer [9]. The coefficients of the series can be related to parameters commonly associated with the mean-spherical approximation. Within the past quarter-century, considerable success in modeling the diffuse layer has been achieved by further generalizing and extending the PB equation. This approach, known as modified PB theory, involves deriving and solving integro-differential equations [10–13]. Unless major simplifying assumptions are invoked, these models also lack explicit analytical solutions. The goal of the present work is to provide a complete analytical model for the diffuse double layer that yields the potential profiles and ion–wall correlation functions by generalizing GC theory in a novel empirical approach. Testing of the new empiricism for 1 : 1 and 2 : 1 electrolytes is provided by MC simulations for a restricted electrolyte in a primitive solvent.
2. The empirical model and its application to 1 : 1 electrolytes The key to our empirical approach is the generalization of the simple exponential dependence of the ion–wall correlation functions on the potential in the Boltzmann relation. The new model introduces two adjustable parameters, α and θ, into the Boltzmann equation to account for the effects of ion size g i (x) = θ exp[−z i αϕ(x)] − θ + 1
(1)
where g i (x) is the ion–wall correlation function and z i is the ionic charge. Throughout this work, all equations can be reduced to their GC form by setting α and θ equal to unity. The quantity ϕ(x) is the dimensionless potential at position x in the diffuse layer, and is related to the actual potential φ(x) by the equation ϕ(x) = Fφ(x)/RT .
(2)
The position x is made dimensionless by σ, the ionic diameter. The origin of the system is defined at the outer Helmholtz plane (oHp). Thus, the potential at x = 0 is the potential drop across the diffuse layer, ϕ 0 . When the potential ϕ(x) goes to zero, g i (x) must go to 1. This is the reason for the last two terms on the right-hand side of Eq. (1). The Poisson equation is the governing differential equation for the potential. In terms of dimensionless variables in one dimension, it is given by Fσ 2 ρ z d2 ϕ = − dx 2 RTε 0 ε s
(3)
An Empirical Approach to Modeling Ion Size Effects . . .
429
where ε 0 is the vacuum permittivity and ε s is the relative permittivity of the pure solvent. The average spatial charge density ρ z is generally given by z j c j g j (x) (4) ρz = F j
where c j is the concentration of ionic species j in SI units. For simplicity, the case of 1 : 1 electrolytes is explored first. In this case, ρ z has the form ρ z = Fc e {θ exp[−αϕ(x)] − θ + 1 − θ exp[αϕ(x)] + θ − 1} = − 2Fce θ sinh[αϕ(x)]
(5)
where ce is the electrolyte concentration in moles m −3 . Now the empirical PB equation can be written as d 2 ϕ(x) = κ 2 σ 2 θ sinh[αϕ(x)] dx 2
(6)
where κ is the Debye–Hückel reciprocal length. For 1 : 1 electrolytes, this is given by 1/2 2F 2 ce . (7) κ= RTε 0 ε s The method of solution of the nonlinear differential equation given in Eq. (6) is well known from GC theory [14]. Multiplying both sides by 2 dϕ yields 2 dϕ d = 2κ 2 σ 2 θ sinh(αϕ)dϕ . (8) dx Integrating from infinity to some general position x in the diffuse layer yields 2 2κ 2 σ 2 θ dϕ = (9) [cosh(αϕ) − 1] . dx α Further simplification is obtained by invoking the trigonometric identity cosh(αϕ) − 1 = 2 sinh 2 (αϕ/2) . Taking the negative square root of Eq. (9) produces 1/2 θ dϕ(x) sinh[αϕ(x)/2] . = −2κσ dx α
(10)
(11)
The above manipulations convert the second-order differential equation, Eq. (6), into a first-order differential equation.
430
T. G. Smagala and W. R. Fawcett
In order to solve the governing equations of any diffuse layer model, boundary conditions must be specified that satisfy important physical constraints. Electroneutrality requires that the total charge in the solution must be equal and opposite that of the wall. This can be expressed as Fσ
j
∞ g j (t)dt = σ m
z j cj
(12)
0
where σ m is the charge density of the wall, not to be confused with σ, the ionic diameter. This equation is derived by a single integration of Eq. (3) and application of Gauss’ Law Fσ dϕ(x) =− σm . (13) dx RTε 0 ε s x=0 For our purposes, Eq. (13) is sufficient to describe electroneutrality as the boundary condition of the slope of the potential profile. For the case of 1 : 1 electrolytes, the electroneutrality boundary condition has the simple dimensionless form dϕ(x) = −κσE (14) dx x=0 where E is the dimensionless field given by E = σ m /A GC
(15)
and the Gouy–Chapman constant is defined as A GC = (2RTε 0 ε s ce )1/2 .
(16)
Upon combining Eqs. (11) and (14), the result is E = 2(θ/α)1/2 sinh(αϕ0 /2) . This equation can be solved explicitly. αE αE 2 2 + +1 . ϕ0 = ln α θ 2 4θ
(17)
(18)
Thus, the potential drop across the diffuse layer in the empirical approach maintains the simple analytical form of GC theory, except with new adjustable parameters. Now Eq. (11) may be solved to obtain the potential profile in the diffuse layer. First one separates variables csch(αϕ/2)dϕ = −2κσ(θ/α)1/2 dx
(19)
An Empirical Approach to Modeling Ion Size Effects . . .
431
and integrates from the oHp to some position x in the diffuse layer, thus obtaining ln tanh[αϕ(x)/4] − ln tanh(αϕ 0 /4) = −κσ(αθ)1/2 x
(20)
or more simply tanh[αϕ(x)/4] = tanh(αϕ 0 /4) exp[−κσ(αθ)1/2 x] .
(21)
With this analytical equation for the potential profile, one can express the ion– wall correlation functions for the cation and anion using Eq. (1).
3. Empirical model for 2 : 1 electrolytes Most sophisticated models for 1 : 1 electrolytes offer little improvement over GC theory for describing many of the quantities of interest in the diffuse layer. GC theory overestimates MC values of the potential profiles more than the integral equation approach, though the difference is relatively small [7]. However, the differences between all these theories become much greater for asymmetric electrolytes. Simple methods such as GC theory and the Henderson–Blum approach [15–19], which work well for 1 : 1 electrolytes, deviate greatly from results of MC simulations for 2 : 1 electrolytes [20]. This suggests more work is necessary in developing sophisticated models for such systems. The empirical approach described so far is readily extended to 2 : 1 electrolytes. According to Eq. (4), the spatial charge density for this case is ρ z = Fc e [2θ exp(−2αϕ) − 2θ + 2 − 2θ exp(αϕ) + 2θ − 2] .
(22)
Substituting into Eq. (3), one obtains d2 ϕ κ 2 σ 2 θ = [exp(αϕ) − exp(−2αϕ)] dx 2 3
(23)
where κ is the Debye–Hückel reciprocal length, but now for 2 : 1 electrolytes. κ=
6F 2 ce RTε 0 ε s
1/2 (24)
Once again, it is desirable to convert this second-order differential equation into a first-order differential equation. Multiplying both sides of Eq. (23) by 2dϕ yields d
dϕ dx
2 =
2κ 2 σ 2 θ [exp(αϕ) − exp(−2αϕ)]dϕ . 3
(25)
432
T. G. Smagala and W. R. Fawcett
Integrating from infinity to some general position x in the diffuse layer, one obtains 2 κ 2σ 2 θ dϕ = [2 exp(αϕ) + exp(−2αϕ) − 3] . (26) dx 3 α For 2 : 1 electrolytes, the dimensionless form of Gauss’ law is dϕ κσ = −√ E . dx x=0 3
(27)
Combining the previous two equations yields E 2 = (θ/α)[2 exp(αϕ0 ) + exp(−2αϕ 0 ) − 3] .
(28)
From this equation, it is trivial to obtain E as a function of ϕ 0 , but the inverse is more complicated. Using the substitution X = exp(αϕ0 )
(29)
a cubic equation is obtained. 2X 3 − 3 + (α/θ)E 2 X 2 + 1 = 0
(30)
Only the first or principle root is physically valid. However, this solution is valid only for positive charge densities, because the inverse of Eq. (29) indicates ϕ0 = +α −1 ln X,
E > 0.
(31)
Thus by properties of logarithms, ϕ0 is always positive. The negative side is solved by means of a different substitution Y = exp(−2αϕ0 ) .
(32)
This results in a very different cubic equation 2 Y 3 − 2 3 + (α/θ)E 2 Y 2 + 3 + (α/θ)E 2 Y − 4 = 0 .
(33)
Ironically, the only physically valid root is the third one, the second of the complex conjugate pair. For negative charge densities this always yields the correct real and negative potentials ϕ0 = −(2α)−1 ln Y,
E < 0.
(34)
Using the powerful symbolic math software, Mathematica, it is trivial to write the full expressions for ϕ0 as a function of E for the positive and negative sides, though the formulas are far too long to show here.
An Empirical Approach to Modeling Ion Size Effects . . .
433
The differential equation, Eq. (26), may now be solved to obtain the potential profile in the diffuse layer. The solution of the equivalent equation in GC theory was outlined in detail by Fawcett and Henderson [20]. They indicate that it is convenient to make the substitution 2 exp(αϕ) = u 2 − 1 .
(35)
The differential element becomes αdϕ =
2udu . u2 − 1
(36)
After substituting into Eq. (26) and simplifying, the following differential equation is obtained √ κσ αθ du (37) = − √ (u 2 − 3) . dx 2 3 For the sake of brevity, we choose to solve it by means of Mathematica’s DSolve command. The resulting solution for the potential profile looks quite different in structure from Eq. (60) of Fawcett and Henderson [20]. However, a little algebra will show that when α and ϕ are both unity, the two expressions are exactly equivalent. √
κσ αθx 1 + 2 exp(αϕ0 ) 3 1 2 + arctanh ϕ(x) = ln 1 − sech α 2 2 3 (38) This expression resembles those derived by Grahame [21] for GC theory. Once again, the explicit solution of the potential profile allows one to express the ion–wall correlation functions for the cation and anion using Eq. (1).
4. Comparisons with Monte Carlo simulations The details of the Monte Carlo simulations are discussed in our previous work [9]. For 1 : 1 electrolytes, the MC calculations were performed for 5 different concentrations and 8 different positive charge densities. By symmetry, the potential profiles for the negative side are equal in magnitude, but opposite in sign to the positive side potentials. This is not true for asymmetric electrolytes. Thus, data for 8 more negative charge densities are included for 2 : 1 electrolytes. All calculations are carried out at room temperature, T = 298 K, and assume the solvent has the same relative permittivity as water, ε s = 78.46. Many previous comparisons of diffuse layer models [7, 8] with MC data [22, 23] used the rather large ion size of 425 pm. This is greater than the diameter of most unsolvated alkali metal and halide ions. Instead we choose
434
T. G. Smagala and W. R. Fawcett
Table 1. Best fit empirical parameters for 1 : 1 and 2 : 1 electrolytes for positive charge densities. ce /M
0.1
0.2
0.5
1.0
2.0
α (1 : 1) θ (1 : 1) α (2 : 1) θ (2 : 1)
1.45 0.445 1.53 0.336
1.43 0.567 1.45 0.513
1.47 0.621 1.48 0.614
1.49 0.789 1.43 0.993
1.60 0.937 1.44 1.55
Fig. 1. Dimensionless diffuse layer potential drop versus charge density for 1 : 1 electrolytes.
a diameter of 300 pm for all calculations. This seems to be a more physically realistic choice to represent the diameter of an average unsolvated monatomic ion. The potential drop across the diffuse layer as a function of charge density was tabulated from MC simulations. Using these data and analytical expressions for ϕ0 , the parameters α and θ were determined by nonlinear regression using Mathematica’s NonlinearFit package. The resulting best fit parameters for 1 : 1 and 2 : 1 electrolytes are given in Table 1. Note that for the 2 : 1 case, the least-squares routine converged only for data with positive charge densities. For 1 : 1 electrolytes, the empirical potential drops across the diffuse layer for all concentrations agree very well with the MC values, as shown in Fig. 1. This is a great improvement over GC theory, which consistently overestimates the
An Empirical Approach to Modeling Ion Size Effects . . .
435
Fig. 2. Dimensionless diffuse layer potential drop versus charge density for 2 : 1 electrolytes.
potential drops, especially at high charge densities. The same is true for 2 : 1 electrolytes, but only for the positive side, as shown in Fig. 2. The negative side potential drops exhibit minima and changes in slope. GC theory and GC-like models such as our empiricism fail to predict this behavior. The series approach of Fawcett and Smagala [9] provided similar improvements in the modeling the diffuse layer potential drop. However, it did not provide any means of modeling the potential profiles or ion–wall correlation functions. This is an additional advantage of the new empiricism. Fig. 3 shows the dimensionless potential profiles for a bulk concentration of 0.5 M and a wall charge density of 0.20 C m −2 for both 1 : 1 and 2 : 1 electrolytes. Note how GC theory always overestimates MC values to a greater extent than the empirical model. The potential behaves similarly for both 1 : 1 and 2 : 1 electrolytes with positive charge densities. The same is true for the ion–wall correlation functions, which are shown in Fig. 4 for the same parameters used in Fig. 3. The 2 : 1 correlation functions closely resemble those of the 1 : 1 case, and thus are omitted for brevity. The empirical model generally agrees better with MC data than GC theory under most circumstances. However, empirical values of the ion–wall correlation functions deviate more than GC values in the region near the wall, as shown in Fig. 4. Figs. 3 and 4 are generally representative of our calculations for charge densities ranging from 0.05 to 0.4 C m −2 and concentrations ranging from 0.1 M to 1 M.
436
T. G. Smagala and W. R. Fawcett
Fig. 3. Dimensionless diffuse layer potential profiles versus dimensionless distance, with bulk concentration = 0.5 M and charge density = 0.20 C m−2 .
Fig. 4. Ion–wall correlation functions versus dimensionless distance in diffuse layer for 1 : 1 electrolyte only and same parameters as in Fig. 3.
Monte Carlo simulations are tedious, and thus, it is desirable to derive explicit analytical expressions for α and θ that do not require the use of MC data. Fawcett and Smagala were able to relate the coefficients in their series approach to the MSA volume fraction, η, and reciprocal distance parameter, Γ . This last quantity is defined as Γ = (1 + 2κσ)1/2 /2 − 1/2 .
(39)
An Empirical Approach to Modeling Ion Size Effects . . .
437
Fig. 5. Relationship between empirical parameter θ and MSA reciprocal distance parameter Γ for 1 : 1 electrolytes.
We conjecture that α is a weak function of the compressibility factor and θ is a strong function of Γ . Table 1 indicates that α is roughly constant and can be approximated by 1.5 for both electrolytes considered here. The parameter θ varies in a monotonic way with Γ . Fawcett and Smagala deduced simple analytical expressions for coefficients by manipulating data in order to obtain a linear trend. This is the approach followed here. Fig. 5 demonstrates that θ obeys a simple relation for 1 : 1 electrolytes θ = Γ 1/2 + Γ/2 .
(40)
The corresponding relationship for the 2 : 1 case is not the same and has not yet been examined in detail.
5. Discussion and conclusion There have been many advances in recent years in modeling the diffuse double layer. The increased mathematical and computational complexity of these models hinders their application to experimental electrochemical systems. However, the simple analytical models available are usually not sophisticated enough to accurately describe reality. We hope the empirical model presented here can provide a compromise between these approaches. The new empiricism has proven to be more effective in modeling diffuse layer potential profiles and potential drops than Gouy–Chapman theory. All
438
T. G. Smagala and W. R. Fawcett
the analytical expressions derived here resemble the corresponding GC expressions, except that they contain empirical parameters. This resemblance is the reason why both models fail to describe the behavior for 2 : 1 electrolytes with negative charge densities. Complex phenomena such as ion pairing are not taken into account in such simple models. Thus, another level of sophistication is required for systems with ions having charges greater than unity. However, the empirical model was very successful for asymmetric electrolytes when the predominant ion in the diffuse layer has a charge of unity. The empirical model can be used to calculate many other quantities of interest, such as the ionic surface excesses and the diffuse layer differential capacity. In future work, we will examine such applications. In addition, a more sophisticated and accurate analytical model for asymmetric electrolytes remains to be developed.
Acknowledgement The authors are grateful to Dr. Dezs˝o 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 1. D. Boda, K.-Y. Chan, and D. Henderson, J. Chem. Phys. 109 (1998) 7362. 2. D. Boda, D. Henderson, and K.-Y. Chan, J. Chem. Phys. 110 (1999) 5346. 3. D. Boda, W. R. Fawcett, D. Henderson, and S. Sokołowski, J. Chem. Phys. 116 (2002) 7170. 4. D. Boda, D. Henderson, L. Mier y Teran, and S. Sokołowski, J. Phys. Condens. Matter 14 (2002) 11 945. 5. D. Boda, D. Henderson, P. Plaschko, and W. R. Fawcett, Mol. Simul. 30 (2004) 137. 6. S. L. Carnie, D. Y. C. Chan, D. J. Mitchell, and B. W. Ninham, J. Chem. Phys. 74 (1981) 1472. 7. M. Lozada-Cassou, R. Saavedra-Barrera, and D. Henderson, J. Chem. Phys. 77 (1982) 5150. 8. M. Lozada-Cassou and D. Henderson, J. Phys. Chem. 87 (1983) 2821. 9. W. R. Fawcett and T. G. Smagala, J. Phys. Chem. B 109 (2005) 1930. 10. S. Levine and C. W. Outhwaite, J. Chem. Soc. Faraday Trans. 2 74 (1978) 1670. 11. L. B. Bhuiyan, C. W. Outhwaite, and S. Levine, Chem. Phys. Lett. 66 (1979) 324. 12. C. W. Outhwaite, L. B. Bhuiyan, and S. Levine, J. Chem. Soc. Faraday Trans. 2 76 (1980) 1388. 13. S. Levine, C. W. Outhwaite, and L. B. Bhuiyan, J. Electroanal. Chem. 123 (1981) 105. 14. W. R. Fawcett, Liquids, Solutions, and Interfaces. Oxford University Press, New York (2004). 15. D. Henderson and L. Blum, J. Electroanal. Chem. 111 (1980) 217. 16. D. Henderson and L. Blum, Can. J. Chem. 59 (1981) 1906. 17. W. R. Fawcett and D. Henderson, J. Phys. Chem. B 104 (2000) 6837. 18. W. R. Fawcett, J. Electroanal. Chem. 500 (2001) 264.
An Empirical Approach to Modeling Ion Size Effects . . .
19. 20. 21. 22. 23.
R. Andreu and W. R. Fawcett, J. Electroanal. Chem. 552 (2003) 105. W. R. Fawcett and D. Henderson, J. Phys. Chem. B 109 (2005) 22 608. D. C. Grahame, J. Chem. Phys. 21 (1953) 1054. G. M. Torrie and J. P. Valleau, J. Chem. Phys. 73 (1980) 5807. G. M. Torrie and J. P. Valleau, J. Phys. Chem. 86 (1982) 3251.
439
