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Langmuir 1997, 13, 5470-5475
Limitations of Multistate Solvent Models at an Electrode/ Electrolyte Interface: A Numerical Study Using the Bethe Approximation R. Saradha and M. V. Sangaranarayanan* Department of Chemistry, Indian Institute of Technology, Madras 600 036, India Received March 20, 1997. In Final Form: June 26, 1997X The consequences of employing the Bethe approximation for dipolar interactions in the inner part of the electrical double layer are examined. Apart from providing explicit results for order parameters, dipolar potential, and differential capacitance for a two-state polarizable point dipole model, a nonlinear regression analysis using experimental data is carried out in order to estimate interfacial constants such as permanent dipole moment, polarizability, effective coordination number, etc. This indicates that the simultaneous agreement of dipole potential and differential capacitance variations with charge density on the metal surface cannot be obtained using conventional discrete multistate orientational state models, thereby pointing to the need for explicit introduction of a metal surface into the double layer theories.
1. Introduction The analysis of the solvent structure at a metal/ electrolyte interface has been the scene of intense activity ever since the postulate of a two-state orientational model by Watts-Tobin1 (WT) about 3 decades ago. Several modifications and refinements have subsequently been suggested in the mathematical and physical approximations, motivated by the desire to provide a quantitative and realistic description of the experimental behavior, in particular the variation of differential capacitance with potential and temperature.2 These improvements pertain to increasing the number of orientational states, viz., (i) inclusion of parallel state,3 (ii) postulate of monomers and clusters for solvent dipoles,4-6 and (iii) analysis of infinite states,7,8 etc. In addition, the role of hydrogen bonding9 and the contribution of image charges10 have also been investigated [cf. refs 11 -15 for reviews]. One of the earliest approaches for describing electrostatic interactions between discrete solvent dipoles in the inner layer is due to Levine et al.10 and was subsequently generalized to higher orientational states by Fawcett3,6 and Nikitas.16 These models, however, employ the molecular field approximation (MFA) in calculating the effective electric field due to dipolar orientations. ReX
Abstract published in Advance ACS Abstracts, August 15, 1997.
(1) Watts-Tobin, R. J. Philos. Mag. 1961, 6, 133. (2) Rangarajan, S. K. In Specialist Periodical Reports: Electrochemistry; Thrisk, H. R., Ed.; The Chemical Society: London, 1980. (3) Fawcett, W. R. J. Phys. Chem. 1978, 82, 1385. (4) Damaskin, B. B. J. Electroanal. Chem. 1977, 75, 359. (5) Parsons, R. J. Electroanal. Chem. 1975, 59, 229. (6) Fawcett, W. R.; Levine, S. A.; de Nobriga, R. M.; McDonald, A. C. J. Electroanal. Chem. 1980, 111, 163. (7) Macdonald, J. R.; Barlow, C. A. J. Chem. Phys. 1962, 36, 3062. (8) Fawcett, W. R.; de Nobriga, R. M. J. Phys. Chem. 1982, 86, 371. (9) Guidelli, R. J. Electroanal. Chem. 1981, 123, 59; 1983, 197, 77. (10) Levine, S.; Bell, G. M.; Smith, A. L. J. Phys. Chem. 1969, 73, 3534. (11) Reeves, R. M. In Modern Aspects of Electrochemistry; Bockris, J. O’M., Conway, B. E., Eds.; Plenum Press: New York, 1974; Vol. 9, p 239. (12) Habib, M. A. In Modern Aspects of Electrochemistry; Bockris, J. O’M., Conway, B. E., Eds.; Plenum Press: New York, 1977; Vol. 12, p 131. (13) Trasatti, S. In Modern Aspects of Electrochemistry; Bockris, J. O’M., Conway, B. E., Eds.; Plenum Press: New York, 1979; Vol. 13, p 8. (14) Parsons, R. In Trends in interfacial Electrochemistry; Silva, A. F., Ed.; Reidel: Dordrecht, The Netherlands, 1986. (15) Guidelli, R. In Trends in interfacial Electrochemistry; Silva, A. F., Ed.; Reidel: Dordrecht, The Netherlands, 1986. (16) Nikitas, P. Can. J. Chem. 1986, 64, 1286.
S0743-7463(97)00301-6 CCC: $14.00
cently, the effect of incorporating finite dipole length under MFA has also been analyzed.17 The inclusion of local order in this context was first studied by Guidelli9 on the basis of the quasi-chemical approximation. Marshall and Conway,18 Fawcett,19 and Schmickler20 performed more refined calculations of the interactions between the molecules gathered into small clusters. Further, Lamperski21,22 studied the effect of short-range correlation for point dipoles in detail using the Monte Carlo (MC) method. Parsons23 developed a method which combines the freedom of the MC method with an optimization technique to obtain the minimum-energy configurations of the solvent molecules using the Stillinger and Rahman24 model and showed that the molecules are oriented in the plane of a two-dimensional array at the potential of zero charge (PZC). Almost invariably in the above orientational models, the metal is assumed as a planar structureless wall incapable of maintaining an electric field beyond the surface. This weakness, first pointed out by Rice,25 has subsequently been brought to light due to the pioneering work of Schmickler,26 Badiali,27 Kornyshev,28 and Amokrane.29 The essential feature in all these approaches lies in incorporating the electronic structure of the metal explicitly into the formalism, using jellium models30,31 and thereby quantitatively accounting for the influence of different metal surfaces (and single crystals) for a given solvent. The capacitance of a simple metal modeled by jellium has also been studied analytically in the presence of an applied electric field, and the results indicate the important role played by metal in the interfacial struc(17) Gao, X.; White, H. S. J. Electroanal. Chem. 1995, 389, 13. (18) Marshall, S. L.; Conway, B. E. J. Electroanal. Chem. 1992, 337, 1. (19) Fawcett, W. R. J. Chem. Phys. 1990, 93, 6813. (20) Schmickler, W. J. Electroanal. Chem. 1983, 149, 15. (21) Lamperski, S. J. Electroanal. Chem. 1991, 318, 39. (22) Lamperski, S. J. Electroanal. Chem. 1994, 373, 211. (23) Parsons, R.; Reeves, R. M. J. Electroanal. Chem. 1981, 123, 141. (24) Stillinger, F. H.; Rahman, A. J. Chem. Phys. 1974, 60, 1545. (25) Rice, O. K. J. Phys. Chem. 1926, 30, 1501. (26) Schmickler, W.; Henderson, D. J. Chem. Phys. 1984, 80, 3381. (27) Badiali, J. P.; Rosinberg, M. L.; Goodisman, J. J. Electroanal. Chem. 1983, 143, 73; 1983, 150, 25. (28) Kornyshev, A.; Schmickler, W.; Vorotyntshev, M. Phys. Rev. 1982, B25, 5244. (29) Amokrane, S.; Badiali, J. P. J. Electroanal. Chem. 1989, 266, 21; 1991, 297, 377. (30) Smith, J. R. Phys. Rev. 1969, 181, 522. (31) Lang, N. D.; Kohn, W. Phys. Rev. 1970, B1, 4555; 1973, B8, 6010.
© 1997 American Chemical Society
Limitations of Multistate Solvent Models
ture.32 In contrast to the orientational models of WT genre, these approaches26-29,32 do not explicitly assume the division of an electrical double layer into inner and diffuse layers but provide a unified picture of the entire interfacial region. In these models, the electrolyte is visualized as a system of an ion-dipole mixture and analyzed using the mean spherical approximation (MSA),33,34 the reference hypernetted chain approximation (RHNC),35-37 etc. (cf. refs 38-43 for recent reviews in this field). In a series of recent papers, Price and Halley44-47 have considered a hierarchy of models to describe the interaction of metal electrons with solvent molecules. The variation of differential capacitance with σM, in the case of cadmium, exhibits a satisfactory agreement45 with experimental results, with one adjustable parameter for pseudopotential. Halley et al.48 have reported explicit calculations of differential capacitance variation with crystal faces for copper and silver, using jellium models for the metal surface. A combination of molecular dynamics and density functional methods to describe the structure of the nonelectrolyte interface is also recently studied.47 An alternate model of the metal/electrolyte interface on the basis of a multilayer lattice gas model was suggested by Macdonald, which enables the calculation of charge and potential distribution within the double layer.49 Here we may mention a recent study by Nazmutdinov et al.,50 for calculating the metal capacitance in the case of the Pt(100)/H2O interface as a function of electrode charge using molecular dynamics simulations. In this quantum mechanical approach, the metal surface is described using a cluster formalism (cf. ref 51 for a critical review on microscopic approaches to the theory of the metal/ electrolyte interface). Although the need for the above sophisticated treatments is unquestionable, a careful study of modeling dipolar interactions within the inner part of the electrical double layer using approximations superior to MFA has not yet been systematically carried out. This, if and when accomplished, will not only throw light on the current status of the orientational state models but also highlight the superiority of the models based on density functional theory of the metal/solution interface. With this objective, we analyze in this paper (i) a two-state polarizable point (32) Holmstron, S.; Apel, P. Surf. Sci. 1986, 165, 337. (33) Carnie, S. L.; Chan, D. Y. C. J. Chem. Phys. 1980, 73, 2949. (34) Henderson, D.; Blum, L. J. Chem. Phys. 1981, 74, 1902. (35) Berard, D. R.; Patey, G. N. J. Chem. Phys. 1991, 95, 5281. (36) Wei, D.; Torrie, G. M.; Patey, G. N. J. Chem. Phys. 1993, 99, 3990 and references therein. (37) Kinoshita, M.; Harada, M. Mol. Phys. 1994, 81, 1473. (38) Schmickler, W. In Trends in Interfacial Electrochemistry; Silva, A. F., Ed.; Reidel: Dordrecht, The Netherlands, 1986; p 453. (39) Schmickler, W.; Henderson, D. Prog. Surf. Sci. 1986, 22, 323. Schmickler, W. In Structure of Electrified Interfaces; Lipkowski, J., Ross, P. N., Eds.; VCH Publishers: New York, 1993; p 201. (40) Kornyshev, A. A. Electrochim. Acta 1989, 34, 1829. Kornyshev, A. A. In Condensed Matter Physics Aspects of Electrochemistry; Tosi, M. P., Kornyshev, A. A., Eds.; World Scientific: Singapore, 1991; p 7. (41) Amokrane, S.; Badiali, J. P. In Modern Aspects of Electrochemistry; Conway, B. E., White, R. E., Bockris, J. O’M., Eds.; Plenum Press: New York, 1992; Vol. 22, p 1. Badiali, J. P.; Amokrane, S. In Condensed matter Physics Aspects of Electrochemistry; Tosi, M. P., Kornyshev, A. A., Eds.; World Scientific: Singapore, 1991; p 157. (42) Henderson, D. In Trends in Interfacial Electrochemistry; Silva, A. F., Ed.; Reidel: Dordrecht, The Netherlands, 1986. (43) Berard, D. R.; Kinoshita, M.; Ye, X.; Patey, G. N. J. Chem. Phys. 1994, 101, 6271. (44) Price, D.; Halley, J. W. J. Electroanal. Chem. 1983, 150, 347. (45) Halley, J. W.; Price, D. Phys. Rev. 1987, B35, 9095. (46) Price, D. L.; Halley, J. W. Phys. Rev. 1988, B38, 9357. (47) Price, D. L.; Halley, J. W. J. Chem. Phys. 1995, 102, 6603. (48) Halley, J. W.; Johnson, B.; Price, D.; Schwalm, M. Phys. Rev. 1985, B31, 7695. (49) Macdonald, J. R. Surf. Sci. 1982, 116, 135. (50) Nazmutdinov, R. R.; Probst, M.; Heinzinger, K. Chem. Phys. Lett. 1994, 221, 224. (51) Schmickler, W. Chem. Rev. 1996, 96, 3177.
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dipole model using the Bethe approximation for dipolar interactions and investigate its validity (or otherwise) in reproducing the experimental behavior of differential capacitance, (ii) the prediction of nonlinear regression analysis for obtaining the “best” set of interfacial molecular constants such as permanent dipole moments, polarizabilities, etc., (iii) the interpretation of “effective coordination number” customarily employed in all multistate models hitherto known, and (iv) the dichotomy between differential capacitance vs potential data and dipole potential vs surface charge density analysis. 2. Analysis of a Simple Two-State Polarizable Point Dipole Model Using the Bethe Approximation 2.1. Salient Features of the Model. A two-dimensional hexagonal lattice with the lattice sites occupied by polarizable point dipoles in two orientational states is considered. ps1 and ps2 denote the normal component of the permanent dipole moments in the two states, Rs is the isotropic polarizability, and Us1 and Us2 represent the nonelectrostatic interaction energies of the two states with the metal surface (cf. ref 52) . In the Bethe approximation, a central dipole is chosen and its interaction with all its nearest-neighbor dipoles is treated exactly, whereas the interactions with nonnearest-neighbor dipoles are treated as in the conventional MFA. Hence, we require three different electrical fields, viz., ξo, ξj, and ξi representing respectively the field at the central dipole “o”, the field at the nearest-neighbor dipoles “j”, and the field at the non-nearest-neighbor dipoles “i”.52 ξo is given by the sum of the external field (E) due to σM, the charge density on the metal surface, and the reaction field (J) due to the interaction of the central dipole with solvent dipoles at site “j”, viz.
ξo ) E + J
(1)
E ) 4πσM/B
(2)
where
B)1+
∑fijRs + ∑fij∑j fojRs2〈si〉 - ∑j foj2Rs2〈so〉 〈ij〉
(3)
〈ij〉
and
J)
{∑ (
∆p 2B
foj
j
∆p 2
)
- Rs(4πσM) -
∑j
}
foj2Rs (∆p〈sj〉 + ptot) 2 (4)
where ptot ) ps1 + ps2 and ∆p ) ps1 - ps2. fij is the potential due to the pair of dipoles at i and j and ∑〈ij〉fij is proportional to 1/d3, with d being the nearest-neighbor distance. The proportionality constant defined as the “effective coordination number” Ce is ascribed a value greater than the geometric coordination number due to infinite imaging of the dipole in the two interfaces. foj is the potential due to a pair of dipoles at o and j and obviously ∑jfoj ) 1/zd3. 〈si〉, 〈so〉, and 〈sj〉 represent the average orientations of solvent dipoles at sites i, o, and j, respectively, and can be derived as shown below: 〈ξj〉 is given by the sum of the external field (E) and the fields F and J due to interactions with dipoles at sites i and o, respectively, (52) Saradha, R.; Sangaranarayanan, M. V. J. Chem. Phys. 1996, 105, 4284.
5472 Langmuir, Vol. 13, No. 20, 1997
Saradha and Sangaranarayanan
〈ξj〉 ) E + F + J
(5)
adsorbed dipoles as
where F is given by
F)
1
(
2B
∆φ )
∑j fojptot + ∑fij(ptot + ∆p〈si〉) + 〈ij〉
∑fij∑j fojRs〈si〉(ptot + ∆p〈sj〉))
(6)
〈ij〉
(7)
The limit to MFA is obtained by letting the coordination number z f ∞; thereby, in eqs 3, 4, and 6 ∑jfoj f 0. The total Hamiltonian incorporating coulombic and nonelectrostatic contributions is now given by (cf. ref 52) z
HT ) (∆U + ∆pE)
sj
∑ j)0 2
z
+ ∆pF
sj
∑ j)1 2
z
+J
sosj ∑ j)1
(8)
where ∆U ) Us1 - Us2. For the Hamiltonian given by eq 8, the long-range order parameter R denoting the net orientation of solvent dipoles in the two states (R ) (Nv - NV)/(Nv + NV), where Nv and NV are the number densities in the two states), is
〈so〉 ) R ) tanh[k1 + k2 + k3z/(z - 1)]
(9)
where k1, k2, and k3 are
k1 ) -
∆U , 2kT
k2 ) -
∆pE , 2kT
k3 ) -
∆pF 2kT
(10)
and the short-range order parameter Q defining the number of pairs of dipoles i.e., 2[(Nvv + NVV) - 2NvV]/zN is given by
〈sj〉 ) Q ) [tanh(k1 + k2 + k3 + γ) + exp[-2(k1 + k2 + k3z/(z - 1))] tanh(k1 + k2 + k3 + γ)]/{1 + exp[-2(k1 + k2 + k3z/(z - 1))]} (11) where γ ) -J/kT and 〈si〉 ) (R + Q)/2. When z f ∞, R ) Q ) RMFA. The rationale behind our providing the above detailed algebra is to bring the result pertaining to the Bethe approximation in conformity with the MFA result in the context of multistate models (cf. eq 9). The analysis leading to eqs 9-11 has been extensively discussed earlier.53,54 The dipole potential for this model follows as
gdip ) 4πN[(ptot + ∆pR)/2 - Rs(ξo + z〈ξj〉)] (12) where the first term is due to permanent dipole moments while the second term arises from the induced dipole moment contributions. Under the mean-field approximation, the sum ξo + z〈ξj〉 becomes equal to 〈ξi〉 and is given by eq 7 with the corresponding MFA expressions for E and F. When the condition z f ∞ is imposed upon eq 12, it yields the MFA expression as expected. The total potential drop across the interface is then given by the sum of the potential drop due to the charge density on the metal surface σM and that due to the (53) Kubo, R. Statistical Mechanics; Elsevier Science Publishers BV: Amsterdam, The Netherlands, 1965; Chapter 5. (54) Plischke, M.; Bergensen, B. Equilibrium statistical physics; Prentice Hall: Englewood Cliffs, NJ, 1989; Chapter 3.
(13)
where Kion is the integral capacitance.55 On differentiating the total potential drop as given by eq 13, with respect to σM, we obtain
Since, the interaction of the dipole at site i with the remaining dipoles is treated in MFA 〈ξi〉 follows as:
〈ξi〉 ) E + F
σM + gdip Kion
∂gdip 1 1 ) + M Ci Kion ∂σ
(
(14)
)
∂ξo z ∂〈ξj〉 1 1 ∂R ) + 2πN∆p M - 4πNRs M + Ci Kion ∂σ ∂σ ∂σM
(15)
The above expression enables the evaluation of differential capacitance by numerically differentiating the order parameters with respect to σM and will be analyzed further subsequently. 2.2. Nonlinear Regression Analysis for the Charge Density Dependence of Dipole Potential. The data given by Trasatti56 for the dipole potential, at the Hg/ aqueous solution interface for different σM values, are taken as the starting point for our numerical analysis. In view of the simplicity of the dipole potential variation, the nonlinear regression analysis was performed for these data (see below). The following numerical procedure is followed for the calculation. The fields given by eqs 2, 4, and 6 depend on both R and Q; hence, the two nonlinear eqs 9 and 11 were solved simultaneously for each value of σM to get the corresponding order parameters. Of the seven parameters required to fit the above model with Trasatti’s data, the dipole moment in one of the orientational states has been fixed as ps1 ) -6.13 × 10-30 C m. The remaining set of parameters ps2, Rs, Ce, d, and N were obtained using the Gauss-Newton method57 of nonlinear regression. In this method, one searches for the minimum in the sum of the squares of the residuals between the given data and the fitted eq, viz. n
Err )
(gi(exp) - gi(theory))2 ∑ i)1
(16)
where n is the number of data points. Since it is common to obtain unrealistic physical parameters in nonlinear regression analysis while employing a large number of variables, we kept the number of regression parameters to be five and assumed a value of 3.2 for k1 at 298 K. The best fit was obtained (cf. Figure 1), for the following set of parameters: ps2 ) 4.39 × 10-30 C m, Rs ) 1.66 × 10-31 m3, Ce ) 15.62, d ) 3.4 × 10-10 m, N ) 3.42 × 1017 molecules m-2. From Figure 1, it can be seen that a qualitative agreement with Trasatti’s data is obtained for σM < 0.05 C m-2. (The computer program developed in MATLAB for this purpose is available from the authors.) Our analysis predicts the saturation limit at a much lower value of gdip than the experimental ones (cf. section 2.3). Here mention must be made of the previous monomercluster model of Bockris and Habib58 and Damaskin.4 Bockris and Habib have calculated the dipole potential values for various σM values, but the theoretically com(55) Trasatti, S. J. Electroanal. Chem. 1977, 82, 391. (56) Trasatti, S. J. Electroanal. Chem. 1978, 91, 293. (57) Chapra, S. C.; Canale, R. P. Numerical Methods for Engineers, 2nd ed.; McGraw Hill International Editions, Applied Mathematics Series; McGraw-Hill: New York, 1989. (58) Bockris, J. O’M.; Habib, M. A. Electrochim. Acta 1977, 22, 41.
Limitations of Multistate Solvent Models
Figure 1. Variation of dipole potential with charge density on the metal surface (σM). The parameters employed are ps1 ) -6.13 × 10-30 C m, ps2 ) 4.39 × 10-30 C m, Rs ) 1.66 × 10-31 m3, Ce ) 15.62, d ) 3.4 × 10-10 m, N ) 3.42 × 1017 molecules m-2, z ) 6, and k1 ) 3.2 at 298 K. (O) Experiment. (s) This work.
puted behavior is at complete variance with the experimental behavior (cf. Figure 2 of ref 58). Damaskin using a three-state model has reported the surface potential dependence on charge density, but the absolute values obtained therein are much smaller than Trasatti’s derived data. The foregoing analysis implies that a two-state polarizable dipole model using the Bethe approximation for dipolar interactions is capable of yielding a qualitative agreement for the behavior of gdip, provided the molecular constants are estimated through a regression analysis (see later). (Although the fit becomes poorer beyond 0.05 C m-2 for this set of values, a dramatic improvement can be achieved albeit at the cost of obtaining unrealistic estimates for various parameters.) 2.3. Temperature Coefficient of the Dipole Potential at σM ) 0. The analysis of the solvent structure at the electrode/electrolyte interface will not be complete without a study of the temperature dependence of double layer parameters. We have taken Trasatti’s data56 for studying the effect of temperature on the dipole potential. None of the MFA-based models of the inner layer solvent structure hitherto developed (with the exception of the four-state model of Fawcett6) were able to predict the correct sign of the temperature coefficient of the dipole potential at σM ) 0. Fawcett et al.6 with the four-state orientational model predicted the correct sign for the temperature coefficient of the potential drop at PZC but could not obtain the absolute estimates for gdip values. Further, Guidelli’s analysis9 using the Quasi-chemical approximation is also unable to obtain the correct temperature dependence of the dipole potential. The predicted temperature dependence using our model is shown in Figure 2 for the same set of parameters as in Figure 1. Though the absolute values obtained are not in agreement with experimental values, the correct sign of the temperature coefficient of the dipole potential at PZC is obtained. The disagreement between computed and experimental estimates can be attributed to the noninclusion of polarizability differences between various dipolar states in our analysis. A rigorous methodology should, in addition, include a proper strategy for describing metal/solvent electrostatic interaction energies (cf. section 3). The effect of temperature on the dipole potential as well as the variation of 1/Ci with gdip at PZC has been further discussed in ref 59 for ad hoc values of interfacial parameters in the case of nonpolarizable point dipoles
Langmuir, Vol. 13, No. 20, 1997 5473
Figure 2. Variation of dipole potential with temperature at σM ) 0 for the same set of parameters employed in Figure 1. (O) Experiment. (s) This work.
Figure 3. Variation of inner layer capacitance with charge for k1 ) 15.5 at 298 K. The other parameters as in Figure 1. (O) Experiment. (-‚-) Fawcett four-state model. (s) The present model.
with equal permanent dipole moments in the two orientational states. 2.4. Charge Dependence of Inner Layer Differential Capacitance: Comparison with Grahame’s Data. Using the same set of interfacial parameters, the differential capacitance vs charge (σM) values were computed assuming Kion ) 19 F/m2 and the agreement with the experimental data60 is entirely unsatisfactory. Even if Kion is varied in a reasonable manner, this discrepancy persists. Before we discuss the origin of this behavior, let us investigate the effect of varying k1 arbitrarily. A good agreement with Grahame’s data is obtained if k1 is assumed a value of 15.5 at 298 K, with other parameters being kept as before (cf. Figure 3). Furthermore, the fit herein is found to be superior compared to the four-state model of Fawcett et al.6 which employs MFA. k1 ) 15.5 at 298 K implies that Us2 - Us1 ) 1.27 × 10-19 J. Using this value of ∆U, differential capacitance vs σM plots were obtained at other temperatures too and the agreement with experimental data is still satisfactory. In a related analysis by Guidelli9 using the quasi-chemical approximation, explicit results for gdip or Ci have not been reported. Furthermore, variation of the inner layer thickness with σM and variation of the (59) Saradha, R.; Sangaranarayanan, M. V. J. Colloid Interface Sci. 1996, 183, 610. (60) Grahame, D. C. J. Am. Chem. Soc. 1957, 79, 2093.
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Figure 4. Variation of inner layer capacitance with σM at various temperatures for this model: (- - -) 273 K, (-‚-) 318 K, (s) 338 K.
dielectric constant with σM were assumed therein to obtain satisfactory Ci vs σM behavior. Apart from this, Ci vs σM plots show an abrupt decrease at positive charges. We reiterate here that an assumed value of k1, though able to give a reasonable description for dipole potential variation with σM, fails to reproduce the capacitance behavior and this needs some comments. First, the multistate orientational models analyzed to date yield partial agreement for differential capacitance behavior, whereas all of them fail in obtaining the dipole potential variation with respect to σM or vice versa. Note that, in our analysis, a mere change in the k1 value employed is sufficient to obtain a satisfactory agreement for both gdip and Ci variation with σM. Recalling that k1 is the term which represents the so-called nonelectrostatic interaction energy of the metal with solvent states, this method of introducing the influence of metal surface in discrete multistate statistical mechanical models seems clearly invalid. The basis behind the introduction of this term in earlier treatments3,6,10 lies in obtaining a nonvanishing dipole potential at σM ) 0. Second, the above-mentioned inconsistency has been noticed in previous studies too and a phenomenological method of overcoming this limitation has also been suggested.61 Third, the present numerical analysis provides a methodology for obtaining the best choice of molecular parameters for reproducing differential capacitance data of Grahame, within a twostate point dipole version. This reinforces the dictum2 that if interfacial interactions are realistically incorporated, the need to introduce higher orientational state models can be obviated at least partially. 2.5. Analysis of the Temperature Coefficient of the Inner Layer Capacitance. It is of interest to investigate the behavior of the inner layer capacitance as a function of temperature. Damaskin4 and Parsons5 have pointed out that any theory which predicts correctly the temperature dependence of differential capacitance should be able to give the behavior of entropy of formation with charge on the metal surface, as well. Further, the behavior of experimental observables with respect to temperature is considered as a sensitive test for any theoretical model. The results obtained from the present analysis are given in Figure 4. It is clear that, except at extreme positive charges, the experimental and theoretical results give a satisfactory agreement. In Figure 5, the temperature coefficient of the inner layer capacity is plotted against (61) Damaskin, B. B.; Safonov, V. A.; Petrii, O. A. J. Electroanal. Chem. 1989, 258, 13.
Saradha and Sangaranarayanan
Figure 5. Variation of the temperature coefficient of inner layer capacitance with σM for k1 ) 15.5 and other parameters as in Figure 1. (O) Experiment. (-‚-) Fawcett four-state model. (s) The present model.
Figure 6. Variation of the long-range order parameter with σM. k1 ) 3.2 (a) and k1 ) 15.5 (b). Other parameters as in Figure 1.
charge density (σM) on the metal. The model predicts a maximum in ∂(1/Ci)/∂T at negative charge values in accordance with the observed experimental data. The model proposed by Damaskin and Frumkin,4 though found to give satisfactory agreement for the temperature coefficient of the inner layer capacity, fails to reproduce the correct sign of the temperature coefficient of the dipole potential at PZC as noted earlier. Their model treats the dipole-dipole interactions arising from polarizability effects in an approximate manner through the introduction of effective dielectric constant which is independent of electrode charge. Later, Parsons5 developed a four-state model which, though it gives agreement in the region of the hump and at positive charges, does not reproduce the experimental behavior at negative charges. In the fourstate model of Fawcett et al.6 the maximum in ∂(1/Ci)/∂T occurs close to PZC, which is in disagreement with experimental observations; further, in this monolayer model, the cluster dipole moment and the residual energy of the cluster in the “up” orientational state were varied with respect to σM in order to obtain a satisfactory agreement with the experimental behavior. 2.6. Variation of Short-Range and Long-Range Order Parameters. The variation of R and Q with respect to charge density σM on the metal surface is given in Figures 6 and 7 for k1 ) 3.2 and for k1 ) 15.5. From these two sets of figures, it is inferred that the short-range order parameter which gives the difference between the number
Limitations of Multistate Solvent Models
Langmuir, Vol. 13, No. 20, 1997 5475
parameters obtained from the regression procedure and for k1 ) 15.5, it turns out that the value required for z is 650 at PZC! Furthermore, z depends sensitively on the charge on the metal surface. Hence, it follows that the MFA results become meaningful only if an extremely large value for the (effective) coordination number is employed. This may be the prescription as noted by Borkowska64 wherein Ce > 14.5 does not give negative capacitance values and avoids the phenomenon of the CooperHarrison catastrophe.65
Figure 7. Variation of the short-range order parameter with σM. k1 ) 3.2 (a) and k1 ) 15.5 (b). Other parameters as in Figure 1.
densities of pairs of dipoles is less sensitive toward the variation of k1. But the variation of k1 has a significant influence upon the behavior of R as a function of σM. When k1 ) 3.2, R is always negative for the set of system parameters derived using regression analysis. For k1 ) 15.5, R takes positive and negative values and, in hindsight, this seems to be the origin of the inconsistency observed between gdip vs σM and Ci vs σM pointed out above (cf. section 2.4). The positive value of a short-range order parameter at PZC shows that the pair of dipoles which are in the same configurations (either up orientational state or down orientational state) is higher than the pair in which the dipoles are in opposite configurational states (i.e., with one dipole in up orientational state and the other in down orientational state). 2.7. Quantitative Limitations of MFA in the Analysis of the Interfacial Solvent Structure. In the Bethe approximation, the interactions between the dipoles inside the chosen central cluster are treated exactly and hence the geometrical coordination number z of the lattice arises. For the dipoles outside the cluster, the interactions are treated using the familiar MFA. According to Topping’s calculations,62 in order to account for the interactions due to next-nearest-neighbor dipoles under MFA, an increased value of 11 has to be ascribed for the geometrical coordination number in the case of a hexagonal lattice and is customarily designated as the effective coordination number. Levine et al.10 used the value of 15.2 instead of 11, which includes effects due to infinite imaging both on the metal and on the solution side. In our analysis, we have both the geometrical and the effective coordination numbers in contrast to the previous models. As is well-known,63 when z f ∞, the long-range and short-range order parameters become equal and the results of the Bethe approximation lead to those of MFA. It is imperative to enquire at what particular z value the MFA limit is obtained starting from the Bethe approximation for a chosen set of values. For the set of
3. Perspectives The foregoing analysis indicates that despite the plethora of models hitherto proposed for the solvent structure at the electrode/electrolyte interface, reproducing the experimentally observed differential capacitance and dipole potential variations continues to be a formidable exercise. The origin of the difficulties consists of an inadequate treatment of both dipole/dipole interactions (permanent and induced) and metal/solvent interaction energies (electrostatic, nonelectrostatic). The former, viz., dipolar interactions when studied using the Bethe approximation for a two-state polarizable model give an explanation of (i) the linear correlation of 1/Ci vs gdip at PZC,59 (ii) temperature dependence of gdip at σM ) 0,59 and (iii) differential capacitance vs σM behavior at a range of temperature for set of molecular parameters, etc. (in contrast to the conventional MFA models). Furthermore, the weakness of the MFA can only be gauged in a hierarchial manner, by analyzing approximations superior to it, and we have taken the first step in this direction by studying the simplest multistate model. However, it appears that a satisfactory description is unlikely to emerge unless a proper treatment of the metal/solvent interaction energy term is simultaneously carried out. The rationale behind our above speculation lies in the crucial role played by the (σM-independent!) parameter k1 which represents the nonelectrostatic interaction of solvent dipoles with the metal surface. Consequently, we propose to incorporate metal/solvent interaction energies in eq 8 and solve for order parameters for various metal surfaces explicitly using jellium and related models. 4. Summary The efficacy of employing the Bethe approximation in handling dipolar interactions for discrete solvent molecules at the electrode/electrolyte interface is studied using a two-state polarizable point dipole model as a case study. The best set of interfacial molecular constants was obtained using a nonlinear regression analysis of gdip vs σM data. Though a fairly satisfactory agreement can be obtained in this instance with realistic molecular constants, the same parameters lead to a total variance in so far as the differential capacitance vs charge data are concerned. This inconsistency is ascribed to the ad hoc introduction of metal/solvent interaction energy terms hitherto employed in multistate orientational models of Watts-Tobin type and clearly indicates the need for explicit introduction of a metal surface into the double layer theories. LA970301V
(62) Topping, J. Proc. R. Soc. London 1927, A113, 67. (63) Fowler, R. H. Statistical Mechanics, 2nd ed.; Cambridge Press: Cambridge, U.K., 1929; Chapter 21.
(64) Borkowska, Z.; Stafiej, J. J. Electroanal. Chem. 1985, 182, 253. (65) Cooper, I. L.; Harrison, J. A. J. Electroanal. Chem. 1975, 66, 85.
