J. Phys. Chem. B 2007, 111, 5545-5557
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FEATURE ARTICLE Double-Layer in Ionic Liquids: Paradigm Change? Alexei A. Kornyshev Section of Theoretical and Experimental Physical Chemistry, Department of Chemistry, Faculty of Natural Sciences, Imperial College London, SW7 2AZ London, U.K. ReceiVed: NoVember 27, 2006; In Final Form: January 29, 2007
Applications of ionic liquids at electrified interfaces to energy-storage systems, electrowetting devices, or nanojunction gating media cannot proceed without a deep understanding of the structure and properties of the interfacial double layer. This article provides a detailed critique of the present work on this problem. It promotes the point of view that future considerations of ionic liquids should be based on the modern statistical mechanics of dense Coulomb systems, or density-functional theory, rather than classical electrochemical theories which hinge on a dilute-solution approximation. The article will, however, contain more questions than answers. To trigger the discussion, it starts with a simplified original result. A new analytical formula is derived to rationalize the potential dependence of double-layer capacitance at a planar metal-ionic liquid interface. The theory behind it has a mean-field character, based on the Poisson-Boltzmann lattice-gas model, with a modification to account for the finite volume occupied by ions. When the volume of liquid excluded by the ions is taken to be zero (that is, if ions are extremely sparsely packed in the liquid), the expression reduces to the nonlinear Gouy-Chapman law, the canonical result typically used to describe the potential dependence of capacitance in electrochemical double layers. If ionic volume exclusion takes more realistic values, the formula shows that capacitance-potential curves for an ionic liquid may differ dramatically from the GouyChapman law. Capacitance has a maximum close to the potential of zero charge, rather than the familiar minimum. At large potenials, capacitance decreases with the square root of potential, rather than increases exponentially. The reported formula does not take into account the specific adsorption of ions, which, if present, can complicate the analysis of experimental data. Since electrochemists use to think about the capacitance data in terms of the classical Gouy-Chapman theory, which, as we know, should be good only for electrolytes of moderate concentration, the question of which result is “better” arises. Experimental data are sparse, but a quick look at them suggests that the new formula seems to be closer to reality. Opinions here could, however, split. Indeed, a comparison with Monte Carlo simulations has shown that incorporation of restricted-volume effects in the mean-field theory of electrolyte solutions may give results that are worse than the simple Gouy-Chapman theory. Generally, should the simple mean-field theory work for such highly concentrated ionic systems, where the so-called ion-correlation effects must be strong? It may not, as it does not incorporate a possibility of charge-density oscillations. Somehow, to answer this question definitely, one should do further work. This could be based on density-functional theory (and possibly not on what is referred to as local density approximation but rather “weighted density approximation”), field theory methods for the account of fluctuations in the calculation of partition function, heuristic integral equation theory extended to the nonlinear response, systematic force-field computer simulations, and, most importantly, experiments with independently determined potentials of zero charge, as discussed in the paper.
Introduction Room-temperature ionic liquids have existed for almost a century. The first report about them goes back to 1914,1 but today, this area is experiencing a renaissance due to a multitude of potential applications. Ionic liquids are used as solvents for organic synthesis and catalysis, as electrolytes for practical electrochemical systems like supercapacitors and batteries, as extraction liquids for the purification of metals and proteins, as chromatography media or microfluidic phases for analytical
chemistry, as electronic gating media for nanosystems, and, finally, as simple lubricants. In comparison to ordinary electrolytes, ionic liquids have a number of advantages. They can incorporate various combinations of reactants and can replace the neutral-oil phase in immiscible-electrolytic systems. They eliminate concerns of electrolyte-solvent loss and have very low vapor pressure, and some have low freezing points. Being solvent-free, ionic liquids achieve maximal electrolyte concentration; unlike conventional electrolytes, there is no risk of salt precipitation. Having high
10.1021/jp067857o CCC: $37.00 © 2007 American Chemical Society Published on Web 05/01/2007
5546 J. Phys. Chem. B, Vol. 111, No. 20, 2007
Alexei A. Kornyshev, MS, PhD, DSci, is a Professor of Chemical Physics at Imperial College of Science, Technology and Medicine, London. His interests span widely in theoretical condensed matter chemical physics and its applications to electrochemistry, nanoscience, biological physics, and sustainable energy. He is an author of 180 original publications and 25 monographic and review articles, co-editor of 5 multiauthor books and number of special issues, and a member of the Editorial Board of J. Phys.: Condens. Matter. He is known for his contributions to the theory of solid liquid and liquid−liquid electrochemical interfaces (including functionalized interfaces), hydration, electron and proton transfer in complex environments including water, membranes and complex electrodes, and physical theory of modern fuel cells; his most recent works are in the field of novel electrowetting systems, single-molecule-based electronic devices, and molecular machines. A special area of his research interest is interaction and recognition between biological macromolecules. In a series of recent papers he and his colleagues developed a theory for describing the role of the helical structure of DNA, collagen and other biomolecules in their aggregates. Alexei Kornyshev is an elected Fellow of IUPAC, Institute of Physics, and ISE. He was a recipient of the Humboldt Prize of 1991 in Physical Chemistry and Electrochemistry, 2003 ChristianFriedrich Scho¨nbein medalist of the European Fuel Cell Forum, and a co-chairman of Division of Interfacial Electrochemistry for two successful terms (1995-1999). He joined Imperial in 2002 as a 2001 Royal Society Wolfson Awardee and a Chair of Chemical Physics, after 10 years of work at the Research Center Juelich (where he was Head of a Section of Theoretical Physical Chemistry in the Institute of Energy Processing and Materials, combined later with a post of a Professor of Theoretical Physics at the Heinrich-Heine University of Duesseldorf), preceded by 20 years of work at the Theoretical Department of the Frumkin Institute of Electrochemistry in Moscow. His research group cooperates with partners in USA, Russia, Israel, Germany, Canada, and Denmark. Ionic liquids are a new area of interest for him, caused by their possible applications as media for gating and energy storage. The article is intended to trigger discussions about understanding the “electrochemical performance” of these puzzling solvent-free electrolytes.
viscosity, ionic liquids may be used in various applications that require reduced fluidity of the active material (although they are not appropriate if high conductivity is needed). They are often “fatty”, with only a small tendency to corrode. Last, but not least, ionic liquids comprise a curious object of physical chemistry: an example of a concentrated plasma existent at ambient temperatures. Ionic-liquid research is “hot” indeed; symptomatically, Tom Welton’s 1999 review article has gathered 2200 citations!2 In a number of their features, ionic liquids are similar to molten salts, but they are composed of a different kind of ions. They typically contain at least one organic, fattyion, the radius of which is substantially larger than that of an alkaline or halide. Most are qualified as “room-temperature” ionic liquids, as they do not require temperatures higher than 100 °C to liquefy. At low temperatures, the viscosity of ionic liquids is substantially increased, which kinetically hinders crystallization. As a result, ionic liquids undergo a glass transition to an amorphous solid phase.3 The glassy phase is not crystalline, and cannot perform
Kornyshev any liquid functions either, although it can be considered a solid electrolyte with poor conductivity. As mentioned, ionic liquids find application in supercapacitors, where they may replace aqueous or organic electrolytic solutions near high-surface-area electrodes.3,4 The motivation is a typically large (up to 5 V) electrochemical window, within which ionic oxidation and reduction are insignificant. However, to design such capacitors rationally, one must understand first the properties of the double layer at a flat metal-ionic liquid interface. Measurement of ionic-liquid capacitance at plane electrodes has a practical value per se. It provides an inexpensive method to investigate how ionic liquids respond to an electric field, which can provide structural information. Last but not least, applications of ionic liquids as gating media cannot proceed “blind”, without understanding first the structure of, and the potential distribution in, the double layer, in linear- and nonlinear-response regimes. These characteristics all underlie the voltage dependence of interfacial capacitance. Controversy surrounds the physical interpretation of experimental capacitance-potential curves for ionic liquids. Overall, the character of such curves is puzzling. It defies rationalization by the Gouy-Chapman theory: the potential of zero charge obtained from an electrocapillary maximum (via a droppingmercury electrode) does not correspond to the capacitance minimum.5 This is unsurprising, because it is unjustified to apply the dilute-solution approximation, on which Gouy-Chapman theory relies, to an ionic liquid. The main point of this article is that the theory of the double layer in ionic liquids must be built differently, and the resulting expression for the double-layer capacitance could be quite dissimilar in general, while reproducing the Gouy-Chapman result as a limiting case. Below, we formulate an alternative model, that takes into account restrictions on the minimal ionic volume in the double layer. This model has a similar kind of relation to the Gouy-Chapman theory as the Eigen and Wicke model does to the Debye-Hu¨ckel theory. The model is somewhat crude, as it does not allow for the charge-density waves that are likely to occur in a system with such high ionic density. However, the new theory does result in a simple analytical expression for the double-layer capacitance, providing the first advance beyond the Gouy-Chapman formula. Several graphs will show features of the capacitance curves yielded by the modified theory. Interestingly, we will see that room-temperature ionic liquids and dilute electrolytes may represent two extremes, between which the capacitance of a molten salt lies. Can these results provide a new framework for the treatment of experimental data? The treatment given in this paper looks encouraging. Obviously, such a simple framework would be useful, if the theory was adequate, but the sufficiency of the present attempt is uncertain. However, the discussion here is intended to stimulate development of a more detailed theory, as well as systematic computer simulations. Further work must attack the atomistic aspects of ionic-liquid structure and charge response, as discussed below. Background The principles of double layers with restrictions on ion volume go back to the 1950s, beginning with the Eigen and Wicke theory of concentrated electrolytes.6 In the late 1970s, similar ideas were used by Kornyshev and Vorotyntsev7,8 and Gurevich and Kharkats9 to build a theory of the double layer in solid electrolytes with single charge-carrier species. The main
Feature Article effect in those works, which led to changes in the potential distribution near the interface, and, consequently, in the capacitance, was the limitation on the local concentration that ions could achieve in the double layer. In 1997, Boruchov, Andelman, and Orland10 published a theory of the double layer in binary electrolytes with maximal charge-density constraints (to account for the finite size of ions). That theory was motivated by the double layer which forms near the surfaces of charged lipids in Langmuir-Blodgett films, where large interfacial surface-charge density can be achieved, unaccompanied by Faradaic processes. In such systems, the consequences of charge-density constraints may, in principle, be seen, but they have not been detected in ordinary electrochemical systems. Actually, a year earlier, in a lesser known paper, Kralj-Iglic and Iglic11 published a similar but less involved theory, oriented toward highly charged surfaces of ionic lipid bilayers, a subject of high relevance in bioelectrochemistry. However, neither the papers of Kralj-Iglic and Iglic nor those of Boruchov et al. fully reflected some 10 years of previous activity in the theory of dense ionic systems, where the mere applicability of the local density approximation (LDA), used by both mentioned groups, was disputed. Instead, the so-called weighted density approximation (WDA) was proposed by Nordholm, Tarazona, Rosenfeld, and others12-14 and, in the context of the double-layer theory near charged colloid surfaces, explored in detail in seminal papers by Penfold et al.15 and Groot.16 This question was further analyzed in very clear studies by the Barbosa-Levin-Holm groups.17,18 This research found, in particular, that the restricted-volume correction of the meanfield theory does not make a better comparison with the results of Monte Carlo simulations than Gouy-Chapman theory, whereas WDA does!18 It was further shown that the emergence of a finer packing effect, such as a decaying charge-density wave, is a critical feature of high-surface-charge-density states, or of systems containing multivalent ions. The Barbosa-LevinHolm study, however, was not extended to highly concentrated systems, partially because the WDA variants used were not developed to describe them. (For such extensions, these authors refer to the schemes suggested by Gillespy and Eisenberg19 and Yu and Wu.20) The phenomena of ionic correlations in concentrated electrolytes, and Coulomb systems in general, have been a subject of intensive study in bulk electrolytes (for a review, see refs 21-23). One of the first studies of the effect of ionic correlations in the double layer of electrolytic solutions was performed in the pioneering papers of Kjellander and Marcelja, which used a heuristic integral equation approach, based on certain effective closure schemes for the chain of equations on correlation functions.24,25 Being focused on the problems of colloid science, but not electrochemistry, none of these groups reported numerical results or analytical expressions for the double-layer capacitance. Neither did Borukhov et al. obtain them, as the metalelectrolyte interface was not their focus either. Thus, before criticizing the mean-field LDA result for the double-layer capacitance, one must first get it! We derive this expression here in the context of ionic liquids and molten salts. Although formally applicable to the doublelayer capacitance in liquid electrolytes, it will not be of any value there: within the usual potential window of conventional electrolytes of moderate and even high concentrations, the deviations from Gouy-Chapman theory are hardly noticeable because the number of intervening solvent ions is so vast.
J. Phys. Chem. B, Vol. 111, No. 20, 2007 5547 However, the principles underlying the theory will be the same for moderately dilute electrolytes, as used and described by Boruchov et al.10 To save some space, we could have referred to some intermediate results of that paper. However, it will be instructive to derive the expression from scratch. First, the derivation itself will be clearer, as we start immediately from the mean-field approximation. Next, the derivation will reveal in which directions this primitive theory could be extended to describe finer effects that may be important for ionic liquids. Once more, the reader should be warned that the derivation will be entirely mean-field-like. In this simplified form, it will not incorporate fluctuations, ionic correlations, or charge-density waves; hence, it cannot predict the oscillations in ionicconcentration profiles which have been observed in numerical simulations.26 Notably, the more general, but still LDA-based, approach of Borukhov, Andelman, and Orland10 is amenable to such an extension. However, due to the so-called “saddlepoint approximation” that they used in the end, their model gives nothing other than a mean-field result in its present state. It is wrong to think that the Borukhov-Andelman-Orland theory cannot give anything other than mean-field predictions. Indeed, within the general theory, and in opposition to the meanfield (saddle-point) approximation, lies the strong coupling approximation, explored with statistical field theory methods by Netz and Orland for the case without ion-size restriction.27-29 The strong coupling approximation accounts for ion correlations, gives rise to charge-density waves, and even predicts the possible attraction of similarly charged surfaces due to these nonlinear effects. These were not explored in the model of Borukhhov et al.; it is work that remains to be done. However, this does not mean that these effects do not fall within their general theoretical construct. Note also that the above-mentioned WDA-based works explored, in fact, the saddle-point solution within the renormalized density functional designed to manifest correlation effects, whereas the model of Orland and Netz gets this result without an a priori renormalization of the density functional (action, in their case), by treating the strong coupling limit.30,31 No attempts have been reported yet about doing anything with this level of detail for ionic liquids. However, as usual, before going into a complicated analysis, one should first explore the simplest mean-field theory and compare it, at least qualitatively, with experiments. Note that a pioneering work by Dogonadze and Chizmadjev32 on the double layer in molten salts incorporated ionic correlations in a semi-phenomenological fashion, by expressing the capacitance through ion-distribution correlation functions. On the basis of the linear-response theory, it could predict only a linear capacitance. There is no analytical theory that can extend their approach to a potential dependent (i.e., nonlinear) capacitance, and such a theory should rather be sought in (i) explorations of the Borukhov et al. functional-integral approach, (ii) mean-field WDA-based solutions, (iii) efficient integral equation schemes, or (iv) computer simulations. The latter two approaches have been widely used in the past for a calculation of the linear-response capacitance of molten salts (for a review, see refs 33 and 34). A legitimate question arises: will a mean-field theory of the double layer in ionic liquids, that neglects ion-correlation effects, capture the main features of the capacitance-voltage dependence? From a rigorous point of view, the careful answer is “probably not”. If we compare the Coulomb interactions between the neighboring ions in ionic liquids, considering even
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the largest species, they will be of the order of 5-10 kBT, and there is no satisfactory reason why the saddle-point approximation should work. Liquid aqueous electrolytic solutions benefit from the high dielectric constant of intervening water; whenever the average distance between the ions is smaller than the Bjerrum length, LB ) e2/kBT ≈ 7 Å, the energy of interionic interaction is smaller than the thermal energy, and the saddlepoint approximation is justified. This is not the case for ionic liquids. However, you never know with the mean field: sometimes it works where it should not. A final judgment can be made only after comparing the results with (i) experiments, (ii) a theory which incorporates correlations, and (iii) numerical simulations. The experimental data are currently sparse and to a high degree indecisive in view of the poor knowledge of potentials of zero charge; the theory and simulations do not yet exist. Under these circumstances, it makes sense to report and discuss the results of the LDA-based mean-field theory, as they are, compare them with the existing experimental data, and outline possibilites for a further development of the theory. The “Fermi” Distributions The mean-field lattice-gas model of concentrated electrolytes starts with the free energy of the system:
F ) eΦ(N+ - N-) + B+N+2 + B-N-2 + CN+N- kBT ln W (1) Here, N+ and N- are the number of cations and anions; Φ is the electrostatic potential (hereafter, we assume that the valences of cations and anions are 1 and -1, respectively, true for all known ionic liquids); and B+, B-, and C are the constants that characterize short-range interactions between the cations, between the anions, and between the cations and anions, respectively. Note that the coefficients B+, B-, and C reflect energetics in excess of the Coulomb energy that owes to the mean field that charged species create, for instance, the specific energetics of short-range steric interactions, or “discreteness-of-charge” effects. It is understood that B+ and B- must be positive, as they describe steric repulsions and discrete Coulomb forces between the ions of the same science. However, C could be positive (due to steric repulsion) or slightly negative, if steric interaction is overcompensated by the short-range interactions of oppositely charged ions. Note that when analyzing the simplest variant of the theory, we will put B+ ) B- ) C ) 0, but the more general form (1) will be needed for a subsequent discussion. The last term describes the entropy of the distribution of ions over available sites (i.e., free energy of mixing). Denoting the total number of available sites that could be occupied by these ions as N, we may write W as a product of three terms: W ) W1‚W2‚W3. Here, W1 ) (N - N-)!/(N - N+ - N-)!N+! is the number of combinations for the distribution of N+ cations in the “hole sites” in the lattice after the occupation of the available sights by N- anions. Similarly, the number of combinations for the distribution of N- anions in the holes left after the occupation of the available sights by N+ cations is given by W2 ) (N - N+)!/(N - N+ - N-)!N-!. Finally, W3 ) N!(N - N+ - N-)!/(N - N+)!(N - N-)! is the number of ways these holes could be arranged. For the product W1‚W2‚W3, we thus get
W)
N! (N - N+ - N-)!N+!N-!
(2)
The Stirling formula (ln M! ≈ M ln M + M at M . 1) may be used to evaluate the mixing entropy, ln W. As a result, one obtains expressions for the electrochemical potentials of cations and anions:
µ+ )
µ- )
N+ ∂F ) eΦ + 2B+N+ + CN- - kBT ln ∂N+ N - N+ - N (3) N∂F ) eΦ + 2B-N- + CN+ - kBT ln ∂NN - N + - N(4)
As declared, we now neglect the terms of short-range interactions and equalize the electrochemical potentials of each sort of ion to the corresponding value of its chemical potentials in the bulk, where the electric field of the electrode is fully screened and the electrostatic potential is taken to be zero. We furthermore assume that the total volume of the system does not change with variation of potential (which in fact can break down, if there is “electrostriction” in the double layers or cations and anions have different sizes, so that in the bulk the smaller ions can pack between the larger ones). Rewriting the result in terms of the corresponding ionic concentrations, c+ and c-, and taking into account that in the bulk c+ ) c- ) c0 ) cj/2, where cj is the average density of the salt, and omitting all the intermediate algebra, we obtain
( ) ( ) ( ) ( )
exp c+ ) c0
1 - γ + γ cosh exp
c - ) c0
eΦ kBT
eΦ kBT
eΦ kBT
1 - γ + γ cosh
eΦ kBT
,
(5)
Here,
γ)N h /N
(6)
where N h is the total number of cations and anions in the bulk and N, again, is the total number of sites available for them. We can rewrite γ in terms of concentrations (which is straightforward, if the total volume of the system does not change)
γ ) 2c0/cmax
(7)
where cmax is the maximal possible local concentration of ions (both cations and anions). As noted in ref 8, density distributions like those in eq 5 can hardly be called Boltzmann distributions. They look rather like Fermi distributions. This is not surprising, as these distributions reflect the so-called “lattice-saturation effect”,7-9 c+(Φ f -∞) ) cmax and c-(Φ f +∞) ) cmax, which means that however the system is polarized, it cannot exceed the maximum allowed local concentration of ions. Boruchov et al.10 called it a “steric effect”. Why may γ be smaller than 1? Consider, for instance, negative fields which draw cations to the electrode and expel anions from it. The density of cations in the double layer may be somewhat higher, as the “plasma” of anions will be compressed, call it, again, electrostriction. Indeed, there is evidence that ions are not this “close-packed” within an ionic
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liquid. Experiments show that ionic liquids swell some 15% upon melting, without a corresponding increase in the average interionic distances measured by spectroscopy.2 Also, molecular simulations have shown the presence of vacuum cavities within ionic liquids,35 justifying the assertion that some potential “lattice sites” for ions may be unoccupied. The expressions in eq 5 have a remarkable limiting behavior. If Φ f -∞, c+ f (2/γ)c0 ) cmax and c- f 0. Thus, even if γ ) 1; that is, there are no free voids in the liquid and it is completely incompressible, the maximal local concentration of cations alone at large negative potentials can increase by a factor of 2, because the cations coming to the electrode replace the anions, which tend toward regions with more positive potential. By the same token, Φ f ∞, c- f (2/γ)c0 ) cmax, and c+ f 0. Such expressions were first obtained by Eigen and Wicke,6 who studied their consequences in the context of bulk-electrolyte properties; the same expressions were obtained in the meanfield limit of the more general Borukhov, Andelman, and Orland theory,10 for double layers near a charged surface. The simple solutions (eq 5) are meaningful when the sizes of cations and anionssor, more generally, the potential of their short-range interactionsare the same. Otherwise, it would be critical to retain the 2B+N+ and 2B-N- terms in eqs 3 and 4, taking into account the fact that B+ * B-. Keeping these terms does not allow a simple closed form solution, and will be considered elsewhere (a numerical study along these lines has been published by Burak and Andelman36 in the context of account of hydration interactions in the electrolytic double layer). Starting from eq 5, we concentrate below on ionic liquids with similar anion and cation sizes. After getting the result, we discuss how it can be semiempirically extended to the case of an asymmetric salt and show the simplest consequences of such asymmetry. Double-Layer Theory
ate-frequency” dielectric constant of the ionic liquid, extrapolated from the height of the semicircle on a Bode impedance plot. The latter is an important parameter which sets the scale of the electrostatic interactions and capacitance in ionic liquids. It is, therefore, sensible to clarify its microscopic significance in some detail. In ionic liquids, the value of is associated with contributions of electronic, vibrational, and even degrees of freedom owing to “rotation of ionic pairs”.39-41 This quantity is often called the static dielectric constant39 to distinguish it from the high-frequency dielectric constant, whose value is close to 2 and which is determined exclusively by the electronic polarizability of ions. However, the term “static” is inaccurate: the true zero-frequency or static dielectric constant of ionic liquid is equal to infinity, because it is a conducting medium. A better term is the intermediate dielectric constant. For a number of ionic liquids, the value of has been measured using the methods of dielectric spectroscopy. It was found to lie typically in the range 9-15.39,40 To keep notations and algebra compact, we use hereafter a dimensionless potential and distance from the electrode:
F ) e(c+ - c-) ) -2ec0
( ) ( )
1 + 2γ sinh2
eΦ 2kBT
(8)
Such equations are called “Poisson-Boltzmann equations” when ions are Boltzmann distributed, γ ) 0 in eq 8. These are known to be the “saddle-point” solutions of the functional integrals for the partition function, obtained when all fluctuations are neglected (see, e.g., the discussions in refs 10, 37, and 38). In our particular problem, the resulting equation should rather be called the “Poisson-Fermi” equation, because of the Fermi distributions in eq 5. Indeed, d2Φ/dx2 ) -4πF/ in this case reads
( ) ( )
eΦ sinh kBT
d2Φ 8πec0 ) eΦ dx2 1 + 2γ sinh2 2kBT
where Φ is the potential calculated relative to the potential in the bulk of the solution (taken for zero) and is the “intermedi-
X ) x/LD
(11)
where
x
4πe2cj kBT
1 ≡κ) LD
(12)
In other words, the potential will be measured in units of kBT/e (which is 25 mV at room temperature), whereas the distance from the electrode, in units of Debye length, LD. Then,
sinh u d2u ) dX2 1 + 2γ sinh2(u/2)
(13)
The boundary with the electrode will be set at x ) X ) 0, and the bulk of the liquid will correspond to x f ∞, that is, X . 1. Integrating this equation and taking into account that the electric field, that is, the negative gradient of the potential, is zero in the bulk, one obtains
du ) dX -
( )) |
xx ( 2 γ
ln 1 + 2γ sinh2
u 2
(-) for positively charged electrode
(+) for negatively charged electrode
(14)
This equation determines the potential distribution in the form of an expression for X(u):
X ) f(u,u0,γ) ≡ sgn(u0)
(9)
(10)
and
The density distributions given by eq 5 are useful as long as the potential distribution near the interface is known. This can be obtained from Poisson’s equation for potential, with the charge density taken to be
eΦ sinh kBT
u ) eΦ/kBT
xγ2‚∫
du
u0
u
x(
(15)
u ln 1 + 2γ sinh 2
( ))
2
where sgn stands for the sign function. Since f(u,u0,γ) can easily be calculated numerically for any given values of u, u0 and γ, eq 15 is an implicit statement of u(X), giving instead the inverse function, X(u). Since f(u,u0,γ) is a single-valued function of u, one can rotate the X(u) graph by 90° to get u(X).
5550 J. Phys. Chem. B, Vol. 111, No. 20, 2007
Kornyshev electrolytic solutions, although not in the theory of molten salts or ionic liquids. The expression for capacitance given below to our knowledge has never been reported, and this can only be explained by the fact that the papers related to the double layer in highly concentrated electrolytes at high electrode polarizations were not focused on classical electrochemical systems. The specific differential capacitance of the double layer is defined as
C)
dσ dU
(17)
where σ is the interfacial surface-charge density and U the overall potential drop between the bulk of electrolyte and the electrode surface. Recalling Gauss law
|
dΦ ) 4πσ dx x)0
D ) -
for the electrostatic displacement at the surface, we get
C)
| }
{
d dΦ 4π dU dx x)0
In our dimensionless units, the latter reads
{ | }
(18)
4πLD
(19)
du d C ) C0 du0 dX X)0 where
C0 ) Figure 1. Profiles of electric potential near a charged surface in the lattice-gas model, plotted for indicated values of the parameter γ, compared to those of the Gouy-Chapman theory (γ ) 0): u0 ) (a) 2; (b) 20.
Plots of this kind are shown in Figure 1. As is well-known, the Gouy-Chapman theory gives an invertible form for X(u) for 1-1 electrolytes, and u(X) can be written explicitly as
{
() ()
u0 1 + tanh exp(-X) 4 u ) 2 ln u0 1 - tanh exp(-X) 4
}
(16)
Thus, in Figure 1, we also compare these plots with the plots given by eq 16. For small γ, there is practically no difference: the finite-size effect does not matter. Also, for small polarizations, the Gouy-Chapman distribution and the one limited by the finite size of the ions lie very close to each other. This is expected because the effects of the “Fermi-like” denominators in eq 5 are negligible at low potentials. For large γ, the potential profile near the interface is far less steep than that predicted by the Gouy-Chapman theory. The reason is “lattice saturation”: since the ions cannot pack densely near the surface to screen very high surface charge, they instead occupy several layers in front of it. Thus, the effective thickness of the double layer grows with potential. As we will see below, this can entirely change the potential dependence of the doublelayer capacitance. Double-Layer Capacitance The equations presented so far in this or that form were known earlier in the context of the properties of concentrated
is the linear Gouy-Chapman or “Debye” capacitance. The substitution of eq 14 into eq 18 finally gives a new result
() ()
cosh C ) C0 ‚
u0 2
u0 1 + 2γ sinh 2 2
‚
x[
2γ sinh2
() ( )] u0 2
u0 ln 1 + 2γ sinh 2
(20)
2
With γ f 0, 1 + 2γ sinh2(u0/2) ≈ 1, ln[1 + 2γ sinh2(u0/2)] ≈ 2γ sinh2(u0/2), so that eq 20 gives the Gouy-Chapman law
C f CGC ) C0 cosh
() u0 2
(21)
γf0
However, for any finite value of γ, there will be sufficiently large values of u0 for which the Boltzmann distribution of ions no longer applies, and results will differ from the GouyChapman formula. The main difference is that instead of growing exponentially at large potentials, as the GouyChapman capacitance does, the capacitance decreases at the high-bias “wings”, and this is already seen for quite small γ. The limiting behavior of the decrease is obtained by replacing cosh(u0/2) by exp(|u0|/2)/2 and sinh2(u0/2) by exp(|u0|)/4; neglecting the terms that are not exponentially large, we find that at the wings the capacitance decreases as
C ≈ C0
x
1 2γ|u0|
(22)
This inverse-square-root decay is a classical signature of lattice saturation and can be rationalized without the algebra
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given above, as first shown in ref 7. It is a result of the effective double-layer thickness growing with electrode potential as
Leff ≈ LDx2γ|u0|
(23)
The latter follows from the charge conservation law. Indeed, when the surface-charge density of the electrode, σ, is large, its absolute value is approximately equal to the ionic counter charge density integrated over the thickness of the double layer: |σ| ∼ Leffecmax. On the other hand, the Gauss law for electric induction tells us that
|σ| )
Why such “universality”? This remains as one of the “mysteries” of the model. Can This Model Be Used to Describe Ions of Different Size? One way to describe liquids with asymmetric ion size would be to keep the 2B+N+ and 2B-N- terms in eqs 3 and 4, with B+ * B-. If the cations are smaller than anions, then B+ < B-, effectively allowing a higher maximal concentration of cations
| | |
(kBT/e)|u0| dΦ ≈ 4π dx x)0 4π Leff
Equalizing these two expressions gives us
Leff2 ≈
kBT 2c0|u0| (kBT/e)|u0| ) ) LD2γ|u0| 2 c 4π ecmax 8πc e max 0
that is, to the accuracy of the factor x2, this crude estimate reproduces eq 23. As we have mentioned, in ionic liquids with ions of the same size, even for γ ) 1, one may locally increase the concentration of each sort of ions by a factor of 2, expelling cations from the anode to the cathode and anions from the cathode to the anode (cf. eq 5). However, generally, γ < 1, as these densities near electrodes can be slightly increased, relative to the bulk, where the ions may be more “sparsely packed”. However, it is difficult to expect γ to be much smaller than 1, so that the capacitance will never have the classical Gouy-Chapman character. Figure 2 shows the family of capacitance curves, plotted via eq 20. What we notice is that the capacitance (i) has, for reasonable values of γ, the maximum at the potential of zero charge, but not the minimum, as prescribed by the Gouy-Chapman theory, and (ii) it, of course, decreases at the wings as prescribed by eq 22. For a densely packed ionic liquid, γ ) 1, and42
| ( )|
u0 x2 sinh u0 2 C ) C0 cosh 2 cosh u ln[cosh u ] 0x 0
()
(24)
If the above-discussed model is correct, and the capacitance is described more or less by one of the curves shown in Figure 2, one could obtain C ≈ C0 from the maximum of the capacitance if the curve is bell-shaped or from the local minimum between the two humps if the curve is “camel”shaped. The parameter γ can be obtained then from the slope of the curve C-2 versus |u0| which would appear as a straight line starting from zero, C-2 ) C0-22γ|u0|. Having obtained these two parameters, one may try to plot the whole curve and compare it with the experimental one. Interestingly, since at very small potentials
(
C ≈ C0 1 +
1 - 3γ 2 u0 8
Figure 2. Double-layer capacitance as a function of electrode potential for indicated values of the lattice-saturation parameter, γ: bell-shape versus camel-shape behavior.
)
the demarcation line between the bell shape and camel shape of the capacitance is γ ) 1/3. For any γ > 1/3, the curve will be bell-shaped. It means, for an ionic liquid, curves will practically always be bell-shaped. However, what is special about γ ) 1/3?
Figure 3. How the difference in the size of ions may affect the capacitance curve. Red (dotted) curve: γ+ ) 0.2, γ- ) 1; blue (dashed) curve: γ- ) 0.2, γ+ ) 1.
than anions in the double layer; the opposite effect would hold for B- < B+. Incorporation of ion-size asymmetry can be made using a perturbation theory which assumes B+ and B- to be small; this is predestined to give small effects and will not be too interesting. Alternatively, ion-size asymmetry could be explored numerically. Instead of this more complicated procedure, we do something very crude, which cannot be qualified as a rigorous “theory” but rather as a semiempirical modification to show how the result for capacitance may change. The idea is as follows. Imagine that anions are substantially smaller than cations. Then, the maximal local concentration of anions in an anion-rich region will be larger than the maximal local concentration of cations. If the ratio of ion diameters is only 2, then the ratio of their maximal concentrations could be
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close to 8 (the volume being proportional to the cube of the diameter). At positive polarizations, the double layer will be rich in anions, and the character of the charge distribution must be characterized, effectively, by smaller values of γ; in the mentioned example close to 1/9, whereas for the case of negative polarizations, when the double layer is rich in cations, γ will be close to 8/9. If this is so, just looking at the wings of the capacitance curve, one may expect that the wing at positive polarizations will be steeper with respect to potential, and therefore, the capacitance maximum will shift away from zero, toward positive potentials. In the reverse case, the shift will be to the negative side. To parametrize this model, we may brutally allocate a potential dependence to γ, which will interpolate it between the value characteristic for the cation-rich layer and that of the anion-rich layer, for instance, by a simple sigmoidal curve, such as
γ(u) ) γ- +
γ + - γ1 + exp(u)
(25)
To avoid introduction of extra parameters (this expression already contains two parameters, γ+ and γ-, instead of one, γ), we did not allocate any special voltage-scale parameter that controls the rate of the transition between γ+ and γ- as a function of u. Plotting the capacitance curve via eq 20 with potential dependent γ (eq 25), we can straightaway observe the related asymmetry (Figure 3). Now, the maximum of capacitance will no longer coincide with u ) 0 but may be shifted to the negative or positive potentials depending on whether the cations are larger or smaller than anions, respectively. We must not forget a speculative character of this conclusion. It may not be reproduced by simulations or a more involved theory, if some effect, less visible a priori, is missed. Thus, the content of this section may be oversimplified. An experimental disagreement with its predictions should not call into question the theoretical basis established in the preceding sections. However, there could be other reasons, as discussed below. Is Linear Capacitance Properly Described? For any value of γ, the linear double-layer capacitance, C|u0)0 ) C0, prescribed by eq 20, is given by the Debye capacitance (eq 19). This is different from the case of a solid electrolyte with a single charge-carrier species. Indeed, if γ ) 1 with a single charge carrier, the local concentration of mobile ions cannot increase anywhere, let alone in the double layer. As a result, the expression for the double-layer capacitance8 gives
C0 f 0 γf1
that is, the interface cannot be polarized. With both cations and anions being mobile, like in ionic liquids and molten salts, the situation is different. Even if γ ) 1, the cations and anions can swap places: cations leaving the double layer and moving toward the counter electrode will be replaced by anions leaving the double layer at the counter electrode and vice versa. As a result, one can polarize the interface, so that
C0 * 0 γf1
However, will C0 for such a concentrated system be described by the Debye capacitance? How good could such an approximation be?
The idea proposed by Dogonadze and Chizmadjev32 was that in a molten salt, as one moves toward the bulk solution from the electrode surface, the position of the maximum of the first peak of binary charge-density correlations plays the role of an effective double-layer “capacitor plate”. This kind of idea was independently explored in the theory of semi-infinite charged hard spheres near a charged wall by Blum43 and Painter et al.44 Tosi33,34 discussed in detail its various aspects in the context of molten salts, and the subject was later revisited by Boda, Henderson, and Chan.45 At the time of those studies, the most exciting feature to be explained was the anomalous temperature dependence of the linear-response double-layer capacitance of molten salts. Let us consider the properties of the linear-response doublelayer capacitance, using the formalism of Kornyshev, Schmickler, and Vorotyntsev46 for a description of the capacitance of a semi-infinite plasma-like media. Within certain approximations, it suggests a beguiling expression for the linear-response capacitance through the nonlocal dielectric function of the bulk:
1 4πL
(26)
∫0∞k2dk (k)
(27)
C0 ) L)
2 π
where (k) is the wavenumber dependent longitudinal static nonlocal dielectric function of the conducting liquid. This expression is not exact, and may even be untrustworthy, as it neglects the possible changes of the structure of the liquid near the surface, but is “handy”, as it gives a “fast track” connection between the capacitance and the structure of the plasma-like medium. For a medium with a purely Debye-like screening, for example, a diluted solution of strong electrolyte,
( )
(k) ) 1 +
κ2 k2
(28)
the integral eq 27 is easily calculated to give
L)
1 LD ) κ
(29)
which brings us back to the Debye capacitance (eq 19). However, generally, in a dense plasma, eq 28 must not be fulfilled. To write an equation more general than eq 29, we may use the fluctuation-dissipation theorem33,47
4π S(k) 1 )1kBT k2 (k)
(30)
which expreses (k) through the charge-density structure factor
S(k) ) 〈Fˆ kFˆ /k〉
(31)
where Fˆ k is the Fourier transform of the charge-density fluctuation, * denotes complex conjugate, and 〈 〉 means ensemble averaging. For a binary 1-1 ionic liquid or molten salt, 2 - ˆ+ ˆ k cˆ k *〉 S(k) ) e2{f+(k)2 〈cˆ + kc k *〉 + f-(k) 〈c
ˆˆˆ+ f+(k) f-(k)[〈cˆ + kc k *〉 + 〈c kc k *〉]} (32) Here, the indices + and - refer to cations and anions, cˆ ( k )
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J. Phys. Chem. B, Vol. 111, No. 20, 2007 5553
(1/xN()∑j eikrˆ j is the Fourier transform of the corresponding spontaneous ion-density distribution, and f((k) are the chargedensity form factors of individual ions. For instance, in an approximation in which the charge distribution within each ion may be approximated by a Born sphere of radius a, the form factors can be written as48,49
f((k) )
sin ka( ka(
(33)
For pointlike charge distributions, a( ) 0 and f((k) ) 1. The structure factors can be measured via X-ray or neutron scattering,50 “computer-simulated”51 or calculated using one of the models of the integral equation theory;33 the form factors can be measured via electron or X-ray scattering in the gaseous state, or obtained from quantum chemistry calculations.52 Given these measurements, one may calculate this integral to get L. This would be a lot of work to do, to get one number, even if it went smoothly with the wavenumber! Lacking the data for ionic liquids, we are not going to pursue this here. However, the goal of this section is twofold: (i) to prepare the soil when such data are systematically available and (ii) to warn the reader that L must not be necessarily equal to LD/, as it must contain information about particle sizes and the structure of the liquid. This has been known since Blum43 obtained L, using a somewhat different method, by employing the values of structure factors for positively and negatively charged hard spheres and performing calculations within the mean spherical approximation (MSA).43 For cations and anions of the same size, that approach resulted in an simple formula (see its discussion and also of other methods of calculation of the linear-response capacitance in refs 33 and 34), which is equivalent to
1
L)
d
(34)
x
d 1+2 -1 LD
where d is the sphere diameter. Abstaining from a debate about the applicability of MSA to molten salts or ionic liquids (this plasma may be too concentrated for MSA to be accurate), we just notice what this formula actually predicts. When the Debye length is large compared to the sphere diameter, one may expand the square root to get
L≈
LD + d/2 , LD . 2d
(35)
this suggests that the behavior of capacitance for small polarizations cannot be accurately captured by eq 20. At the same time, the capacitance at the wings may have a more universal character, and it would not be surprising if eq 20 reproduced the basic form of the observed potential dependence of the double-layer capacitance. One final point. As repeatedly stressed by Tosi and coworkers33,34,44 in the context of molten salts, the effective position of the metal plate relative to the center of mass of the countercharge in the liquid may also affect the capacitance. The role of this effect in liquid electrolytic solutions was the center of attention in the 1980s (for a review, see refs 56 and 57). We do know that the electron spillover and its response to electron charging or the position of closest approach of the liquid to the metal skeleton are very important for understanding the potential dependence of the “compact-layer” capacitance, as extracted from Parson-Zobel plots (for a review, see, e.g., ref 58). However, all of this is important when the diffuse-layer part gives a negligible contribution to the total capacitance, and the latter was true for the Gouy-Chapman type behavior at large electrode polarizations. In our model of an ionic liquid, this situation is reversed: rather than rising, the wings of capacitance that describe the space charge in the liquid (eq 22) instead fall, making fine interfacial details unimportant at large polarizations. However, how important are these details close to the potential of zero charge (pzc)? The Role of the Compact Layer The diffuse-layer theory relates the potential at the boundary (calculated relative to the potential in the bulk) to the capacitance of a half-space in the electrolytic medium, and the density profiles of cations and anions there. However, this potential, u0, in our notations, is not the potential of the metal! Really, the centers of the last layer of cations and anions neighboring the electrode are shifted from the metal surface by at least half of their diameters. All in all, the total capacitance of this interface, in the most elementary consideration (in the absence of specific adsorption, and when the diameters of anions and cations are similar), will be composed of the compact- and diffuse-layer parts in series, like in the Gouy-Chapman-Stern theory59
1 1 1 ) + Ctot C Cc
Here, C may be described by eq 20, if we accept the diffuselayer model described above, whereas
whereas in the opposite limit
L≈
x
1
d LD , 2
Cc ) LD , 2d
(36)
Thus, in both cases, L > LD and the capacitance is thereby smaller than the Debye one; that is, the MSA result giVes a different expression for capacitance than the mean-field theory discussed aboVe! Must the linear-response capacitance of a “structured” ionic liquid always be smaller than the Debye capacitance? This is not clear. Equations 26 and 27, if they have any relation to reality, may give a different answer. Indeed, we know that the dielectric function of molten salts contains a substantial domain of k where (k) < 033,34 (thermodynamic stability forbids only the range 0 < (k) < 153,54). The presence of such a domain suggests that it may happen that L < LD, and the capacitance will be larger than the Debye one.55 One way or another, all of
(37)
˜ 1 4π ξ(d /2) - zσ
(38)
where ˜ is the effective dielectric constant, which is unlikely much larger than the optical one; d is the diameter of an ion; the factor ξ takes into account the distance of closest approach of an ion to the ionic skeleton, which must not exactly equal d/2; and zσ is the position of the center of mass of the excess charge on the electrode relative to the edge of the metal ionic skeleton.56,57 zσ increases with negative charging of the electrode and decreases with the positive charging.56,57,60 However, the resulting value of Cc will depend also on what happens with the distance of closest approach ξ(d/2), as ξ also varies with the electrode charge, σ. There are two known trends: (i) this distance moves together with the overall shift of the surface electronic profile, repeating more or less the motion of zσ,61 which is, as mentioned, asymmetric in the sign of the charge,
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and (ii) this distance tends to decrease, as any flexible capacitor tends to shorten its electrode separation, independently of the sign of the charge.62 The two trends, reviewed in detail, for example, in refs 56 and 57, partially compensate each other with negative charging and amplify each other with positive charging. Altogether, with negative charging ξ(d/2) - zσ decreases, but at positive charging the decrease may slow, or even reverse, leading to an increase. The strength of these effects is unknown for ionic liquids, because no one has ever tried to calculate them. The electrode charge dependence of the compact-layer capacitance would be of importance only if the compact-layer contribution in eq 37 was outweighed by the diffuse one. In the Gouy-Chapman theory, it appears at the wings, when the capacitance of the diffuse double layer rises exponentially. In the present model, with wings decreasing, this effect will not be seen. What happens in between the wings? Here, without going to any speculations about the compactlayer charge dependence, which for ionic liquids will unlikely vary Cc by more than 100%, let us make simple estimates. We need to compare Cc with C at the maxima of the humps, in the case of a camel shape, or just with C0 in the case of a bell shape. Let us treat the latter. In the case of a bell shape, we do not need to bother about the compact-layer contribution when
d ξ - zσ L 2 D , ˜
(39)
A “mild” estimate, with ξ ) 1, d ) 5 Å, zσ ) 1 Å, ˜ ) 4, LD ) 3 Å, and ) 10, gives then 0.125 < 0.3. However, had we assumed ˜ ) 2 and d ) 6 Å, the inequality would change to 1 > 0.3. So, a priori, the winner is unknown in this competition to dominate capacitance close to the capacitance maximum. If the compact layer wins here, it will cut the maximum, but it must not necessarily make it blunt, because of the mentioned variation of Cc with electrode potential. Is the Observed Double-Layer Capacitance Parabolic or Bell-Shaped? There are not many data for capacitance curves in ionic liquids. The most pertinent results were reported by Nanjundiah, McDevitt, and Koch,5 who studied the interface between mercury and a series of liquids with 1-ethyl-3-methyl imidazolium cation (EMI+) and X- anions, where X- ) (CF3SO2)C(methide, Me-), (CF3SO2)2N- (imide, Im), CF3SO3- (triflate, Tf -), and BF4-; they also showed one curve for glassy carbon. The most spectacular for us are their mercury electrode curves, which we reproduce here (Figure 4). Unfortunately, the pzc is not unequivocally determined here (generally, there are a number of nontrivial difficulties in its measurement for ionic liquds63), but still, the data are gathered in their Table 3. Nanjundiah et al. attempted to attribute the poorly visible minima of capacitance to the pzc, in the spirit of GouyChapman theory. These disagree significantly with the pzc values obtained by electrocapillary curves: electrocapillary maxima were found to lie much closer to the maximum of capacitance for BF4-! (This observation is in harmony with the present model, which shows that, in opposition to the GouyChapman theory, the capacitance of an ionic liquid can reach a maximum at pzc). As we stressed above, the Gouy-Chapman theory does not apply to ionic liquids. It is much more likely that the long decaying wings of the capacitance for Tf-, BF4-, Im-, and Me-
Figure 4. Double-layer capacitance of ionic liquids with EMI cation and indicated anions as a function of electrode potential vs saturated calomel electrode. Reproduced from ref 5 with permission. Copyright 1997 The Electrochemical Society.
at negative potentials (i.e., in the range that these authors assume to be free from Faraday processes) are “lattice-saturation” wings. We may even try to make a stronger statement that the clearly seen peak of the BF4- curve is not an artifact, caused by the breakdown of ideal polarizability of the electrode, or some parasitic processes, but is what one should expect for the maximum of capacitance, prescribed by eq 20. To be fair, however, we should be equally ready to something completely different, which we currently fail to see. The sequence of curves has no obvious correlation with the size of the anions and cations, (CF3SO2)2N- > (CF3SO2)C- > CF3SO3- > BF4-, even if we take into account the fact that each curve should be replotted (shifted) relative to the pzc for each particular liquid. This could mean several things. Either the correction discussed above is too crude to describe the size difference of cations and anions adequately or the pzc for each liquid may not be accurately determined. Also, if the compact layer “interferes”, it will introduce its own ion-size dependence, although this must be important only close to the maximum. Last but not least, if specific adsorption or pseudocapacitance play any role here, the interpretation of these curves may be substantially confused. Under these circumstances, all that we can conclude is that the overall character of the curves is consistent with the present model, although one can never exclude the possibility that very different physics is taking place. There are more data for molten salts.64 Most of the reported data look parabolic, around a capacitance minimum. It should be noted, however, that upon salt melting the melt acquires a large number of free voids (holes) and that their net volume grows with temperature65 as ∝ T3/2. If we literally apply the theory suggested above, this would imply γ ∝ T -3/2. Substantially smaller γ together with larger values of kBT/e, to which the potential dependence is scaled, may result in a parabolic curve within the capacitance window.
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Figure 5. Some ions used in room-temperature ionic liquids (courtesy of Tom Welton).
Interestingly, Booth and Haymet, in their paper on the comparison of “singlet”-integral equation theory of molten salts66 to MC simulations,67 obtain not only the profiles of ionic concentrations near a hard charged wall, exhibiting decaying oscillations, but also the nonlinear capacitance. This was an important achievement, because even more recent works on concentrated ionic systems continued to focus on system behavior with small electrode surface charges.68 Remarkably, the Booth and Haymet capacitance plot (their Figure 7) looks parabolic in the (1 V potential range with a minimum at pzc, but if we look at this plot attentively, we will see that the range of capacitance variation is less than 1 µF/m2; that is, it is practically constant! Thus, it is far from being Gouy-Chapmanlike and may, again, indicate the possible subsequent crossover to the falling wings. This is, of course, not much of an argument, as long as the experimental data show explicit parabolic dependence. They often do but with a flatter area around the capacitance minimum than Gouy-Chapman theory suggests. Note also that there are models that try to describe the capacitance response of molten salts, introducing a complicated interplay between the compact- and diffuse-layer parts of the double layer69 or speculating about the compressibility of voids70 (for a review, see ref 64); this lies beyond the simple consideration here. Electrode Charges and Countercharges: Overscreening or Lattice Saturation? The last important question is which type of screening should we actually expect for electrode charges within the capacitance window? Here, the following calculation will be useful: -1
J. Phys. Chem. B, Vol. 111, No. 20, 2007 5555 µC/cm2 of surface-charge density corresponds to an in-plane 2d Wigner-Seitz radius (radius per one electron) of rs ) 20 Å. For Q µC/cm2, this radius will be a factor of xQ smaller. If we take for orientation the BF4- curve of Figure 4 of ref 5, very roughly, we could expect that negative charges of up to -20 µC/cm2 (rs ) 4.5 Å) could be achieved. At pzc, all layers in front of the electrodes will be neutral; that is, cations and ions will alternate in the plane parallel to the electrode surface. At -1 µC/cm2, electroneutrality must break down, and in several layers, this one-electron-per-20 Å circle must be compensated by a net positive countercharge. In principle, charge compensation could be achieved in several ways. If we adopt Tosi’s overscreening concept, we may admit the Wigner-Seitz radius per net positive countercharge in the first layer to be smaller than 20 Å, and that would mean oVerscreening, because the next layer must bear a net negative charge to compensate the extra, “nonrequested” positive charge of the first layer. And off the overcompensation goes, oscillating several times before it decays completely after a few layers: four to five for molten salts but possibly larger for room-temperature ionic liquids. However, once the electrode is charged with -20 µC/cm2 (rs ) 4.5 Å), it will be practically impossible not only to overscreen this electron charge in the first layer but also to screen it within one layer, if some in-plane alternation of positive and negative charges is preserved. If the first layer can be occupied entirely by positive charges (it is a question if this is physical), total screening by the first layer may be possible for cations with radii smaller than rs < 4.5 Å. If the intermediate dielectric constant of the liquid is taken to be of the order 10, at the distance of closest approach, two such ions may interact, as mentioned earlier, with an energy of 8 kBT, so that such an arrangement of charges may take place only as a collective effect, balanced by the next layer of negative charges. However, such a configuration will have an excessively low entropy and may not be physically realized. If so, such values of surface charges might cause a trend toward lattice saturation, which would only strengthen for higher charges. How the crossover between the regimes of oVerscreening and lattice saturation will be exactly realized, we do not know. However, we will learn a lot from a future microscopic theory or computer simulations, if we pose proper questions. This is one of the projects in the author’s research group, but it is natural to expect that a number of groups worldwide would be interested in finding the answer. A healthy competition will only be beneficial for this promising research area.
Figure 6. Speculative examples of structures (clusters) of asymmetric ions and their counterions in the bulk of an ionic liquid (a) and near a strongly charged interface (b). In the bulk, “micelles” with their counterion shields aggregate with cavities in between; the patterns are dynamics opening, closing, and rearranging all the time. Near an electrified interface, the structures may be much more ordered if the electric field is strong; otherwise, some transient structures may be realized (not sketched).
5556 J. Phys. Chem. B, Vol. 111, No. 20, 2007 Can Ions in a Room-Temperature Ionic Liquid Be Modeled as Charged Spheres? In a democratic society, you can model anything the way you want, but will your model be close to reality? Among the examples of ions shown in Figure 5, probably only one of them, the BF4- anion, may be considered as spherical in the first approximation, whereas the others certainly cannot. Often, the unique properties of these liquids owe to the dramatic asymmetry in the structure of anions and cations. Thus, conclusions coming from the theory of liquids of hard charged spheres, with all those ideas of overscreening, chargedensity oscillations, and so forth, may not necessarily show up in the results of the simulations. There are also ions with even larger fatty tails, and their packing structures are not yet clear. They may exhibit a higher degree of softness, increasing our γ parameter. One can imagine rather tricky “quaternary” structures built by such cations, particularly in combination with more compact anions, and vice versa, sketched in Figure 6. The packing patterns of such ions near a charged surface are also unknown, but one might expect a putative micellar structure in the bulk (a) and a layered structure near the surface (b), which is stabilized by the electric field. Agglomerates of structures will inevitably be porous, but it is quite natural to expect that in strong electric fields the highly porous structures will be replaced by more compact ones. Although this is speculative, it is analogous to the thinking typically applied to charged lipid selforganization in liquid crystals. Probably some of the experimental methods that matured that area could elucidate the structures of ionic liquids as well. Concluding Remarks These notes should not be considered as a new theory of the double layer in ionic liquids. They rather set few directions in which such a theory may be developed, taking into account atomistic details and the fine packing structure of ions in such liquids. Nevertheless, they present a new formula (eq 20) for the diffuse double-layer capacitance. It contains the GouyChapman law as a particular case, although this limit is unlikely to apply to ionic liquids. This area is ripe for Monte Carlo simulations. Hanke, Price, and Lynden Bell worked out handy intermolecular potentials for the simulations of liquid imidazolium salts;71 sensitivity of the resulting structure to the specific values of parameters of these potentials was analyzed in a recent paper of Lynden Bell and Young.72 In several papers of Lynden-Bell and co-authors the interfacial properties of ionic liquids were simulated on the basis of this kind of potentials: for the gas-ionic liquid interface73 and for the interfaces of ionic liquids with usual liquids.74 Several advanced quantum chemistry studies of interacting pairs or clusters of ions that form ionic liquids were studied in a series of papers of Hung et al.,75-78 which is an important contribution toward the understanding of nearest neighbor arrangements possible in ionic liquids in the condensed phase. It is only left to extend all of these investigations to electrified interfaces to answer the questions discussed above. The new formula looks relatively simple, and is perhaps too simple. The potential dependence of capacitance is scaled to thermal energy and is governed by a single parameter, called γ, the ratio of the bulk density of the ions in the liquid to the maximal possible density in the double layer. If this ratio is equal to 1, that is, the result is parameter-free, in that case, where have all the fine details of dense packing in this plasma gone? This is also discussed in the paper, but in brief, it is not excluded
Kornyshev that the potential or ionic density profiles that should exhibit oscillations in a more involved theory may be characterized in reality by the “envelopes” described above. Then, the wings of the capacitance will represent less model dependent behavior, associated with the conservation laws on bound charge, and eq 20 will have the character of an interpolation formula. As such, it may catch the main features of capacitance curves at potential extremes, being inaccurate in between. However, this view is optimistic. On a careful note, the model lacks structural details, and the linear-response limit of the basic formula does not reproduce approximate microscopic statistical mechanical calculations, such as those obtained in the beguiling mean spherical approximation (MSA). The role of the compact-layer contribution should not be ignored, as discussed in the paper. This condition is less stringent in microscopic theories or simulations, which incorporate it automatically. Further investigations of the theory, computer simulations, and systematic experimental measurement of the capacitance dependence on electrode potential will show whether our basic formula is as adequate as it is handsome. Theory and simulations cannot rest only on the models of ions as hard spheres. For many systems, they are not hard and are not spheres. Long and challenging investigations are still ahead. The goal of this article is to trigger such investigations. Acknowledgment. Special thanks are due to Charles Monroe for a critical reading of the manuscripts and numerous comments and suggestions. The author also wishes to thank Tim Albrecht, Steve Feldberg, Alexander Kuznetsov, Dominic Lee, and Tom Welton for useful discussions or valuable information. The 2001 Royal Society Wolfson Award and the Leverhulme grant are greatly appreciated. References and Notes (1) Walden, P. Bull. Acad. Imp. Sci. St. Petersbourg 1914, 1800. (2) Welton, T. Chem. ReV. 1999, 99, 2071. (3) Galinski, M.; Lewandowski, A.; Stepnyak, I. Electrochim. Acta 2006, 51, 5567. (4) Buzzeo, M. C.; Evans, R. G.; Compton, R. G. ChemPhysChem 2004, 5, 1106. (5) Nanjundiach, C.; Mcdevitt, S. F.; Koch, V. R. J. Electrochem. Soc. 1997, 144, 3392. (6) Eigen, M.; Wicke, E. J. Phys. Chem. 1954, 58, 702. (7) Vorotyntsev, M. A.; Kornyshev A. A. Dokl. Acad. Nauk SSSR 1976, 230, 631 (Engl. Transl. in Proc. Acad. Sci. USSR, ser. Phys. Chem.Consultants Bureau)). (8) Kornyshev, A. A.; Vorotyntsev, M. A. Electrochim. Acta 1981, 26, 303. (9) Gurevich, Yu. Ya.; Kharkats Yu. I. Dokl. Acad. Nauk SSSR 1976, 229, 367 (Engl. Transl. in Proc. Acad. Sci. USSR, ser. Phys. Chem.Consultants Bureau)). (10) Borukhov, I; Andelman, D.; Orland, H. Phys. ReV. Lett. 1997, 79, 435. (11) Kralj-Iglic, V.; Iglic, A. J. Phys. II 1996, 6, 447. (12) Nordholm, S.; Johnson, M; Freasier, B. C. Aust. J. Chem. 1980, 33, 2139. (13) Tarazona, P. Mol. Phys. 1984, 52, 81. Tarazona, P.; Evans, R.; Marconi, U. Mol. Phys. 1985, 54, 1357. (14) Rosenfeld, Y. Phys. ReV. Lett. 1989, 63, 980; J. Chem. Phys. 1993, 98, 8126. (15) Penfold, R.; Nordholm, S; Jo¨nsson, B.; Woodward, C. E. J. Chem. Phys. 1990, 92, 1915. (16) Groot, R. D. J. Chem. Phys. 1991, 95, 9191. (17) Diehl, A.; Tamashiro, M. N.; Barbosa, M. C.; Levin, Y. Physica A 1999, 274, 433. (18) Antypov, D.; Barbosa, M. C.; Holm, C. Phys. ReV. E 2005, 71, 061106. (19) Gillespie, W. N. D.; Eisenberg, R. S. Phys. ReV. E 2003, 68, 031503. (20) Yu, Y.-X.; Wu, J. J. Chem. Phys. 2002, 117, 10156. (21) Fisher, M. E. J. Stat. Phys. 1994, 75, 1. (22) Weinga¨rtner, H.; Schro¨er, W. AdV. Chem. Phys. 2001, 116, 1. (23) Levin, Y. Rep. Prog. Phys. 2002, 65, 1577.
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