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J. Phys. Chem. B 2009, 113, 14310–14314
Properties of the Diffuse Double Layer at High Electrolyte Concentrations W. Ronald Fawcett,* Peter J. Ryan, and Thomas G. Smagala Department of Chemistry, UniVersity of California, DaVis, California 95616 ReceiVed: June 22, 2009; ReVised Manuscript ReceiVed: September 1, 2009
The equations necessary to calculate the potential drop across the diffuse layer and its differential capacity are derived for 1:1 electrolytes on the basis of the Eigen and Wicke theory for concentrated electrolyte solutions. The results of this model are then compared with Monte Carlo data for more concentrated solutions and solutions with ions of large diameters. It is shown that the Eigen-Wicke model is inadequate because it fails to consider the change in potential at a given ion due to its surrounding atmosphere. Introduction
cM )
The well-known Debye-Hu¨ckel theory of electrolyte solutions attributes the nonideality of these systems to interactions of a given ion with the surrounding ionic atmosphere. The extended version of this theory describes the variation of the mean activity coefficient with the concentration of a 1:1 electrolyte quite well for concentrations up to 0.1 M. It is generally understood that if one wishes to extend the theory to higher concentrations one must consider the size of all the ions in the ionic atmosphere, not just the size of the central ion. Eigen and Wicke1 proposed such an extension using a lattice model for the electrolyte solution and Fermi statistics instead of Boltzmann statistics for the distribution of ions. In a recent feature article, Kornyshev2 discussed the effects of ion size on the properties of the diffuse double layer in concentrated solutions and ionic liquids. Most work3-6 in this area involving planar double layers is based on the description of concentrated electrolyte solutions given by Eigen and Wicke.1 As a result the equations for diffuse layer properties such as the potential drop φd and the differential capacity Cd are quite different from those obtained using the traditional GouyChapman (GC) theory. The Eigen-Wicke approach is not the only method for considering ion size effects for the diffuse double layer. Other recent contributions to this topic have been the Fawcett-Smagala model,7,8 which is based on the hypernettedchainintegralequation,andthemodifiedPoisson-Boltzmann theory of Outhwaite and Bhuiyan.9,10 The purpose of the present paper is to compare the Eigen-Wicke (EW) approach to the double layer problem with the results of Monte Carlo (MC) simulations for concentrated solutions and electrolytes with large ions. These are systems in which the fraction of the volume occupied by the ions is large. The discussion is limited to 1:1 electrolytes which are restricted, that is, which have ions of equal size. Theory Consider a restricted 1:1 electrolyte in which the ions have a radius ri and a volume Vi ) 4πri3/3. One now imagines the electrolyte solution to be a lattice of close packed hard ionic spheres which do not penetrate one another. The maximum number of ions of either type is equal to the reciprocal of the ionic volume.1 Thus, the hypothetical maximum concentration of either ion cM is given by
3 4πNLri3
(1)
where NL is the Avogadro constant, and electrolyte concentrations are measured in mol m-3. The ion density F for an actual electrolyte concentration of ce is
F ) 2NLce
(2)
and the fraction of the volume occupied by ions
η)
2ce 8πNLceri3 πFσi3 ) ) cM 3 6
(3)
where σi is the diameter of ion i. The final form of expressing the volume fraction is familiar from our previous work.7,8 The remainder of the volume is occupied by solvent molecules. Expressions are now written for the electrochemical potentials of the cation and anion using the EW approach.1 At any point in the diffuse layer, the electrochemical potential of the cation of the electrolyte is 0 µ˜ + ) µ+ + RT ln c+ - RT ln(cM - c+ - c-) + Fφ
(4) where µ0+ is the standard chemical potential of the cation and φ is the local electrostatic potential. The third term on the righthand side accounts for the change in free energy with degree of filling of the available sites in the diffuse layer. In the limit of dilute solutions where c+ and c- are very small with respect to cM, this term is a constant, and the electrochemical potential has its more usual form obtained by applying Boltzmann statistics. The corresponding equation for the anion is 0 µ˜ - ) µ+ RT ln c- - RT ln(cM - c+ - c-) - Fφ
(5) where µ0- is the standard chemical potential of the anion. The electrochemical potentials for these ions in the bulk of the solution are
10.1021/jp9058577 CCC: $40.75 2009 American Chemical Society Published on Web 10/07/2009
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J. Phys. Chem. B, Vol. 113, No. 43, 2009 14311
0 µ˜ +0 ) µ+ + RT ln ce - RT ln(cM - 2ce)
(6)
0 µ˜ -0 ) µ+ RT ln ce - RT ln(cM - 2ce)
(7)
and
This is equivalent to eq 14 obtained by Kornyshev.2 Now using Gauss’ law, the field at the outer Helmholtz plane (oHp) can be related to the electrode charge density in the absence of ionic specific adsorption:
( dφdx )
) d
Equating the electrochemical potentials of each ion in the diffuse layer and the bulk, one obtains
ci )
ce(cM - c+ - c-) exp(-ziφ) cM - 2ce
(8)
where ci is the concentration of the ion with charge zi, and φ ) Fφ/RT is the dimenionless potential at the given location in the diffuse layer. Adding eq 8 written for the cation to eq 8 for the anion and rearranging, one obtains
c+ + c- )
2cecM cosh φ 2ce cosh φ ) cM - 2ce + 2ce cosh φ 1 - η + η cosh φ (9)
Since c- ) c+ exp(2φ), one easily derives the equation
ci )
ce exp(-ziφ) 1 - η + η cosh φ
-Fz d2φ RT d2φ ) ) 2 2 ε0εs dx F dx
(11)
Here Fz is the charge density due to the ions, ε0 is the permittivity of free space, and εs is the relative permittivity of the ionic liquid. The charge density is easily found from the ionic concentrations so that
-2Fce sinh φ Fz ) Fc+ - Fc- ) 1 - η + η cosh φ
d
σm ε0εs
(15)
σm )
21/2RTε0εsκ Fη1/2
{ln[1 + 2η sinh2(φd /2)]}1/2
(16)
This may also be written as
E2 )
2 ln[1 + 2η sinh2(φd /2)] η
(17)
where AGC ) RTε0εsκ/F is the GC constant11 and E ) σm/AGC is the dimensionless electrode field. The corresponding equation in GC theory is d E2 ) 2 cosh φGC -2
(18)
The differential capacity of the diffuse layer Cd is given by
Cd )
dσm d
)
dφ
FAGC dE dE ) Cd0 d ) Cd0Yd RT dφd dφ
(19)
where Cd0 is the diffuse layer capacity at the point of zero charge (pzc), and Yd is the dimensionless differential capacity of the diffuse layer. Differentiating eq 18, one obtains
2E dE ) )
d
d
/2) cosh(φ /2) dφ ( η2 ) 2η 1sinh(φ + 2η sinh (φ /2) 2
d
d
2 sinh φd dφd 1 + 2η sinh2(φd /2)
(20)
Simplifying, one obtains the following expression for Cd:
(13)
where κ ) (2F2ce/ε0εsRT)1/2 is the Debye-Hu¨ckel reciprocal length. This equation is integrated to obtain an expression for the field in the diffuse layer in exactly the same way as in GC theory.11,12 The result is
dφ 21/2κ ) - 1/2 {ln[1 + 2η sinh2(φ/2)]}1/2 dx η
)-
(12)
Combining eqs 11 and 12, one obtains the Poisson-Fermi equation written in terms of the dimensionless potential:
d2φ sinh φ ) κ2 2 dx 1 + 2η sinh2(φ/2)
( )
where σm is the electrode charge density in C m-2. Combining eqs 14 and 15, one obtains the relationship between the electrode charge density and the dimensionless potential drop across the diffuse layer φd:
(10)
This is often called the Fermi equation for the ionic concentrations in the diffuse layer. When the solution is dilute and η goes to zero, it reduces to the Boltzmann equation. One now introduces the Poisson equation in one dimension:
RT dφ F dx
(14)
Cd )
Cd0 sinh φd E[1 + 2η sinh2(φd /2)]
(21)
In GC theory, the corresponding result is
Cd )
Cd0 d d ) Cd0 cosh(φGC /2) sinh φGC E
(22)
Substituting in eq 18 for E, one obtains eq 20 in Kornyshev.2 It is obvious from the above results that the equations derived for concentrated solutions using the EW approach are quite different from the classical GC equations. It should be noted that ion size effects have also been considered for systems with
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Fawcett et al.
spherical double layers. These results have been discussed recently by Bhuiyan and Outhwaite.13 In the following section, series expressions for φd and Cd are obtained using the EW model. Development of Series Expressions for the Diffuse Layer Potential Drop and Differential Capacity It was shown in earlier work14 that a very effective way of examining the properties of a model for ion size effects in the diffuse layer is to expand the function giving φd as an infinite series in the dimensionless field E. On the basis of eq 17, such an expansion leads to the result d d φEW - φGC ) c3E3 + c5E5 + ...
(23)
where the coefficients c3 and c5 can be specified. Now this series d : is converted to a series in φGC
d d φEW ) φGC +
d 3 d 5 η(φGC ) ) 5η2(φGC + + ... 8 384
Figure 1. Plots of estimates of the potential drop across the diffuse layer φd obtained by MC simulations ([) and estimated by the EW theory (b) for an ion diameter of 200 pm at concentrations of 1, 2, 5, and 10 M. For the sake of clarity the data at 2, 5, and 10 M have been shifted vertically by 0.1, 0.2, and 0.3 V, respectfully. The solid line shows the GC prediction.
(24)
It is immediately apparent that the estimate of φd by the EW model is greater than the GC estimate. Defining Yd,EW as the dimensionless differential capacity of the diffuse layer in the EW model, it may be obtained as an infinite series by differentiating eq 24 with respect to the dimensionless field E:
1 Yd,EW
d d 2 d 4 dφEW 3η(φGC 25η2(φGC ) ) 1 ) + + ) dE Yd,GC 8Yd,GC 384Yd,GC (25)
Here Yd,GC is the value of the dimensionless differential capacity according to GC theory. The value of Yd clearly depends on both the GC estimate and the GC value of φd. As shown earlier,8 the MC estimates of φd may be described d as follows: by a cubic equation in φGC d d d 3 φMC ) d1φGC + d3(φGC )
(26)
Alternatively, the MC estimate may expressed as an infinite series in the dimensionless field E: d φMC ) c1E + c3E3
(27)
Differentiation of eq 27 with respect to E directly gives the MC estimate of Yd. Thus
1 Yd,MC
)
d dφMC ) c1 + 3c3E2 dE
(28)
In the following section, the EW estimates of φd and Yd are compared with the values obtained from MC simulations. Results and Discussion In order to compare the predictions of the present model with results from MC simulations, the MC data obtained earlier were
Figure 2. As in Figure 1, but for an ion diameter of 400 pm.
extended to higher concentrations. Simulations were possible provided the value of the volume fraction η did not exceed 1/3. Estimates of the diffuse layer potential drop φd obtained at 1.0, 2.0, 5.0, and 10 M for an ion size of 200 pm are shown in Figure 1. At each concentration φd has been calculated for eight values of the electrode charge density in the range from 5 to 40 µC cm-2. Data from the EW model are approximately equal to the GC prediction, being slightly higher at higher charge densities. On the other hand, the MC results show significant deviation from the GC results, with the extent of the deviation increasing with increase in electrolyte concentration and electrode charge density. All of the MC simulations were carried out for a solvent relative permittivity of 78.5. Such an assumption can be seriously questioned at such high concentrations. On the other hand studies of the effect of solvent permittivity on the MC simulations show that it does not play the most important role in determining φd.15 Data from the EW model and MC simulations obtained for an ion diameter of 400 pm are shown in Figure 2. The results of the EW model lie above the GC prediction by an amount which increases with increase in electrode charge density. On the other hand the MC data all lie below the GC results and display a curvature which varies with electrolyte concentration. Unfortunately, an MC simulation could not be carried out for the 10 M solution in which the fraction of the volume occupied by the ions is 0.404. It seems clear from the earlier work7-10 and the results presented here that the reason the EW model fails is that it does
Diffuse Layer at High Electrolyte Concentrations
Figure 3. Plots of estimates of the potential drop across the diffuse layer φd obtained by MC simulations ([) and estimated by the EW theory (b) for an ion diameter of 500 pm at concentrations of 1 and 2 M. For the sake of clarity the data at 2 M have been shifted vertically by 0.1 V. The solid lines show the GC results.
J. Phys. Chem. B, Vol. 113, No. 43, 2009 14313
Figure 5. Plots of the dimensionless differential capacity of the diffuse layer Yd against the dimensionless field E for an ion diameter of 200 pm at electrolyte concentrations of 1, 2, 5, and 10 M. The solid curves were obtained from the GC theory; data designated by b were calculated using the EW model, and those designated by [ were estimated from the MC results. For clarity, results at 2, 5, and 10 M are shifted vertically by 1, 2, and 3 units, respectively. The dimensionless capacity Yd may be converted to Cd by multiplying by 228 µF cm-2; similarly, E may be converted to the charge density σm by multiplying by 5.87 µC cm-2.
Figure 4. As in Figure 3 but for an ion diameter of 600 pm.
not account for the change in local potential due to the electrostatic effect of surrounding ions. This can be seen by examining the effects of further increasing ion size. In previous work, MC simulations were carried out for ion diameters of 200, 300, and 400 pm. This range of ion sizes is appropriate for monatomic ions. However, polyatomic ions approximated by spheres can be much larger. Values of the diffuse layer potential drop φd obtained by MC simulations and estimated by the EW model are shown in Figures 3 and 4 for ion diameters of 500 and 600 pm. In all cases the EW results are greater than the GC estimates by an amount which increases with inrease in ion size. On the other hand, the MC data show much different characteristics when plotted against the GC value of φd. The initial slope is always smaller than unity, reflecting the effects of both finite ion size and the change in the ion atmosphere with respect to the Debye-Hu¨ckel description. In fact, at 2 M for an ion diameter of 600 pm, the initial slope is negative. In addition, the curvature increases with increase in ion diameter and electrolyte concentration. The latter result reflects the crowding of the large ions in the diffuse layer. It is interesting to compare values of the dimensionless diffuse layer capacity estimated from the EW theory and from the MC data. Values of Yd as a function of the dimensionless field are shown in Figure 5 for an ion diameter of 200 pm. These dimensionless results are easily converted to values of Cd in µF cm-2 by multiplying by 228. Firstly, the estimates of Yd according to the EW model are close to or slightly smaller than the values found using the GC theory. On the other hand the estimates of Yd obtained from the MC data using eq 28 are larger
Figure 6. Plots of the dimensionless differential capacity of the diffuse layer Yd against the dimensionless field E for an ion diameter of 400 pm at electrolyte concentrations of 1, 2, and 5 M. The solid curves were obtained from the GC theory; data designated by b were calculated using the EW model, and those designated by [, 2, and 1 were estimated from the MC results at 1, 2, and 5 M, respectively. For clarity, results at 2 and 5 M are shifted vertically by 1 and 2 units, respectively. The value of Yd at the pzc for the 5 M solution is 28.2. The dimensionless capacity Yd may be converted to Cd by multiplying by 228 µF cm-2; similarly, E may be converted to the charge density σm by multiplying by 5.87 µC cm-2.
than the GC results, the difference increasing with increase in electrolyte concentration. In all cases the differential capacity is a minimum at the pzc. Results obtained for an ion diameter of 400 pm are shown in Figure 6. The difference between the GC and EW estimates is now more pronounced. Values of Yd estimated from the MC data show a very different behavior. At 1 and 2 M, the estimates are much larger than the GC results. In addition, for an electrolyte concentration of 5 M, a pronounced maximum is found in Yd at the pzc, with the estimated value of Yd being 28.2. Even more complicated behavior is seen for an ion diameter of 600 pm (see Figure 7). Maxima are found in Yd at the pzc for electrolyte concentrations of 1 and 2 M. At 2 M, the value of Yd at the pzc is 38.3. The EW model also predicts a maximum in Yd at the pzc for a concentration of 5 M for an ion diameter of 400 pm, and at
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Fawcett et al. The Fawcett-Smagala model,7,8 which is based on the hypernetted chain approximation in the integral equation approach, treats both the ion size effect and the change in local potential. Alternatively, the modified Poisson-Boltzmann theory of Outhwaite and Bhuiyan9,10 achieves the same goal. Yet another effective way of treating ion size effects involves using density functional theory.17 The discussion here has been limited to restricted electrolytes in which both ions have the same diameter. In a more realistic model, the ions should be allowed to have different sizes.
Figure 7. Plots of the dimensionless differential capacity of the diffuse layer Yd against the dimensionless field E for an ion diameter of 600 pm at electrolyte concentrations of 1 and 2 M. The solid curves were obtained from the GC theory; data designated by b were calculated using the EW model, and those designated by 2 and 1 were estimated from the MC results at 1 and 2 M, respectively. Results at 2 M are shifted vertically by 1 unit for clarity. The value of Yd at the pzc for a 2 M solution is 38.3. The dimensionless capacity Yd may be converted to Cd by multiplying by 228 µF cm-2; similarly, E may be converted to the charge density σm by multiplying by 5.87 µC cm-2.
2 M for an ion diameter of 600 pm. Lamperski et al.16 found that the change in behavior from a minimum in Yd to a maximum occurs at approximately 3 M at 400 pm, whereas it occurs close to 0.8 M at 600 pm. Their results using modified PoissonBoltzmann theory agree well with the present MC results. As is apparent from the present results, the MC data are subject to some scatter. The estimate of the random error in φd is (2 mV. The error in Yd depends on the estimated error in the coefficients c1 and c3 defined in eq 27. This error depends on the curvature in the plot of φd against E. A rough estimate of the level of error is (5%. As expected for a quantity obtained by differentiation, this corresponds to approximately 10 times the level of error in φd or φd. In conclusion, the present study has shown clearly that the EW model is inadequate because it neglects the effect of ion size on the local potential experienced by an individual ion.
Acknowledgment. The authors are grateful to Dr. Dezso Boda of the University of Veszprem, Hungary, for making available his Monte Carlo program. The financial support of the National Science Foundation, Washington, DC (Grant CHE0906373), is gratefully acknowledged. References and Notes (1) Eigen, M.; Wicke, E. J. Phys. Chem. 1954, 58, 702. (2) Kornyshev, A. A. J. Phys Chem. B 2007, 111, 5545. (3) Kralj-Iglic, V.; Iglic, A. J. Phys. II 1996, 6, 447. (4) Borukhov, I.; Andelman, D.; Orland, H. Phys. ReV. Lett. 1997, 79, 435. (5) Borukhov, I.; Andelman, D.; Orland, H. Electrochim. Acta 2000, 46, 221. (6) Kilic, M. S.; Bazant, M. Z.; Ajdari, A. Phys. ReV. E 2007, 75, 021502. (7) Fawcett, W. R.; Smagala, T. G. Langmuir 2006, 22, 10635. (8) Fawcett, W. R. Electrochim. Acta 2009, 54, 4997. (9) Outhwaite, C. W.; Bhuiyan, L. B. J. Chem. Soc., Faraday Trans. 2 1983, 79, 707. (10) Bhuiyan, L. B.; Outhwaite, C. W. Phys. Chem. Chem. Phys. 2004, 6, 3467. (11) Fawcett, W. R. Liquids, Solutions, and Interfaces; Oxford University Press: New York, 2004; Chapter 10. (12) Girault, H. H. Analytical and Physical Electrochemistry; EPFL Press: Lausanne, 2004; Chapter 5. (13) Bhuiyan, L. B.; Outhwaite, C. W. J. Colloid Interface Sci. 2009, 331, 543. (14) Fawcett, W. R.; Smagala, T. G. J. Phys. Chem. B 2005, 109, 1930. (15) Fawcett, W. R.; Smagala, T. G. Electrochim. Acta 2008, 53, 5136. (16) Lamperski, S.; Outhwaite, C. W.; Bhuiyan, L. B. J. Phys. Chem. B 2009, 113, 8925. (17) Gillespie, D.; Valisko, M.; Boda, D. J. Phys.: Condens. Matter 2005, 17, 6609.
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