Differential Capacitance of Room Temperature Ionic Liquids: The Role

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Differential Capacitance of Room Temperature Ionic Liquids: The Role of Dispersion Forces Martin Trulsson,† Jenny Algotsson,† Jan Forsman,*,† and Clifford E. Woodward‡ †

Theoretical Chemistry, Chemical Centre, P.O. Box 124, S-221 00 Lund, Sweden, and ‡School of Physical, Environmental and Physical Sciences, University College, University of New South Wales, ADFA Canberra ACT 2600, Australia

ABSTRACT We investigate theoretical models of room temperature ionic liquids, and find that the experimentally observed camel-shape of the differential capacitance is strongly related to dispersion interactions in these systems. At low surface charge densities, the loss of dispersion interactions in the vicinity of the electrodes generates depleted densities, with a concomitant drop of the differential capacitance. This behavior is not observed in models where dispersion interactions have been removed. SECTION Surfaces, Interfaces, Catalysis

oom temperature ionic liquids (RTILs) are currently attracting a great deal of attention and are often viewed as “green” solvents, due to their extremely low volatility. They also find application as charge carriers in many systems that would normally employ molten salts.1 The use of RTILs as electrolytes in electrochemical applications, including batteries and capacitors, is currently being explored.2 Despite this, the behavior of RTILs at charged interfaces is still not well understood. In RTILs the electrostatic coupling is much stronger than in usual electrolytes, and the Debye length is anomalously small. This means that steric as well as electrostatic effects are relevant. In a recent seminal review, Kornyshev3 outlined the importance of steric interactions in determining interfacial properties in RTILs. For example, the differential capacitance (CD) of ionic liquids decreases at large absolute values of the surface potential. This can be attributed to a saturation of the packing density of counterions at charged surfaces, caused primarily by steric interactions. Capacitance measurements on imidazoliumbased RTILs confirm not only this effect, but also show a minimum at around zero surface potential.4-7 This gives rise to the so-called “camel shape” for CD verus surface potential curves.3 Fedorov and Kornyshev8 simulated a charged repulsive sphere model, with asymmetric sizes, and obtained a capacitance curve weakly reminiscent of the experimental camel-shape. Their model also displayed the asymmetry in CD observed in experiments on RTILs. This asymmetry is clearly due to the different anion and cation sizes used in their model. However, it still remains unclear as to whether the simulated fluid in that study was a true liquid. The occurrence of a camelshaped differential capacitance is not unusual for ionic fluids at low density. For example, both computer simulations and modified Poisson-Boltzmann theory predict that dilute charged, symmetric hard spheres will display this effect.9 However, as the density increases, the capacitance curve quickly becomes bell-shaped. That is, CD, for the charged, symmetric hard sphere electrolyte, eventually only displays a

single maximum (near zero surface potential) at densities consistent with typical RTILs.9 Thus, the reason for the occurrence of a minimum in CD for RTILs remains uncertain. Lauw et al.10 used a mean-field theory to obtain a camelshaped capacitance for an RTIL fluid model at relatively high density. They used a hard-sphere oligomeric model for the RTIL, together with an anzats of a varying dielectric response in the vicinity of the surfaces. The mean-field approach they adopted will not predict a net energetic attraction in the bulk solution, i.e., the bulk is not a liquid. Recently, Fedorov et al.11 simulated a model RTIL, where the oligomers were modeled as dimers or trimers (with one monomer charged). They did observe a camel-shaped capacitance curve, but no dispersion interactions were included. It is not clear whether their model displays a liquid-gas transition (i.e., whether their model is a liquid). The theoretical work described above used simple, coarsegrained, models for RTILs. Other simulation studies have appeared in the literature, which have used all-atom approaches.12 These employ detailed molecular models for RTILs. However, the numerical demands of these approaches are not insignificant. Furthermore, there is a considerable risk that such approaches generate results that are difficult to interpret, since the complexity of the models make it hard to separate out the important physics from irrelevant details. To the best of our knowledge, an all-atom approach has not been used to systematically investigate the shape of the differential capacitance curve. Of course, it is numerically much simpler to use a coarse-grained model for such studies. Surprisingly, while steric interactions have received much attention in the recent literature, it appears that dispersion forces have not been included in coarse-grained models for RTILs. Generally,

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r 2010 American Chemical Society

Received Date: December 16, 2009 Accepted Date: March 17, 2010 Published on Web Date: March 22, 2010

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DOI: 10.1021/jz900412t |J. Phys. Chem. Lett. 2010, 1, 1191–1195

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polarizability responses to static fields and thus a value of ɛr ≈ 2 should reflect this.13-15 One should note that the dispersion force, although due to electronic polarizability, does not account for dielectric screening of static charge by the electrons. On the other hand, other ionic correlational effects should be present within the simulation. This notwithstanding, the dipole moment associated with more detailed intramolecular charge distributions within the ions are not present within our model and could plausibly give rise to an orientation polarization contribution to ɛr of around 0.6-1.4.14 To this end, we have performed simulations with ɛr = 2.3, 3.8, 7.6. The largest value was chosen for some of our systematic investigations of the nonelectrostatic potential contributions, in order to decrease the coupling in the system and allow faster convergence of the simulation and smaller statistical noise. It is essentially equivalent to performing calculations at higher temperature. Unlike the work of Lauw et al.,10 we will assume that ɛr is uniform throughout the fluid. The electrodes are modeled as planar surfaces, extending infinitely in the (x,y) directions. The surfaces carry uniform surface charge densities of opposite sign, with magnitude σel. We will consider both repulsive and attractive surface electrodes. Repulsive electrodes were modeled with a nonelectrostatic particle-surface interaction given, at a distance z from a surface, by

RTILs consist of large polarizable molecules at high density. Under these conditions, one cannot easily disregard the contribution of dispersion forces to the total energy. Clearly, these forces play a role in determining the liquid-vapor phase envelope of the bulk liquid. Furthermore, although generally weaker than steric forces, dispersion forces may still play an important role at interfaces. In addition, nonelectrostatic interactions with the electrode surface may also provide an additional mechanism for tuning double layer behavior. Here we report some preliminary results, which indicate that dispersion forces do indeed play an important role in determining the behavior of the RTIL double-layer. We use a simple (coarse-grained) model that mimics the molecular structure of imidazolium-based RTILs. Monte Carlo simulations are employed to investigate the phase behavior of the model, as well as the double-layer structure and differential capacitance at charged surfaces. We model the cationic species as a linear pentamer, consisting of bound Lennard-Jones (L-J) monomers. These are all identical, with the exception that one of them is positively charged. The L-J pair potential, ΦL-J, is given by " 6 # σ 12 σ -k ΦL-J ðrÞ ¼ 4ɛL-J ð1Þ r r

e - ðz - dÞ=t βΦref w ðzÞ ¼ A ððz - dÞ=tÞ3

where r is the separation between particles. We have here introduced a parameter k, which normally is unity. However, in order to illustrate the importance of dispersion interactions, we will also compare our results with those obtained when k = 0, representing an analogous fluid without dispersion interactions. We chose σ to also be the fixed bond length between monomers in the cationic oligomer. The bond potential, VB, contains no angular part. Specifically, e-βVB(rij) µ δ(rij - σ), where rij is the distance between connected monomers i and j, while β = 1/kT is the inverse thermal energy. This approximately models imidazolium cations, which consist of a charged (nitrogen-containing) ring, with attached hydrocarbon tails. Various lengths of the substituted hydrocarbons can thus easily be accommodated within this model. In principle, we could include a bond angle potential (chain stiffness) in our model. However, in an effort to reduce the number of parameters in this study, we have only used a flexible version. The anion is treated as a “free” L-J monomer, carrying a negative charge. All particles (monomers and anions) are described by the same L-J parameters. An illustration of our model is provided in the Supporting Information. In this preliminary study, we have limited ourselves to cases where the L-J strength parameter is fixed at ɛL-J = 0.5 kT. All simulations were performed at a temperature of 393 K (a typical experimental temperature7). The interaction between two charges, q1 and q2, separated by r is given by 1 q1 q2 ð2Þ Φel ðrÞ ¼ 4πɛ0 ɛr r

where z is the distance from the plane of charge. The choice τ = 1 Å ensures a short ranged repulsion. The amplitude factor, A, is set to 411. The parameter d was chosen such that the distance of closest approach to the surface place of charge became approximately σ, i.e., one monomer diameter. This is consistent with other studies on similar systems.11 Here, we will consider the values σ = 6 Å and 4 Å, which correspond to the choices d = 3.5 Å and 2.5 Å, respectively. Adsorbing electrodes will be modeled as L-J surfaces, and will be described later. The mean electrostatic potential, at the plane of charge, was calculated directly from the contributions due to all charges present, including the surfaces. As we assumed bulk conditions in the central part of the slit, the appropriate surface potential, ΨS, was measured relative to the potential at the midplane. The differential capacitance was then obtained as a numerical derivative of the surface charge density with respect to the surface potential, i.e., CD = δσel/δΨS. In order to ensure that the system we have investigated represents a true liquid, rather than a dense fluid, we performed canonical ensemble simulations within the coexistence region of the fluid between neutral surfaces. This was demonstrated via the formation of gas-liquid interfaces, parallel to the surfaces. The positions of these interfaces could be shifted by removing or adding salt pairs to the simulation box (not shown). The simulations reported below were performed at densities slightly higher than that of the observed coexisting liquid phase. Figure 1 shows the differential capacitance as a function of surface potential for three different sets of values for σ and nc. The latter refers to the position of the charged monomer on the pentamer chain modeling the cation. For example, nc= 3,

where ɛ0 is the permittivity of vacuum, and ɛr is due to the contributions to the low frequency dielectric response that are not captured by our model (the static response is strictly infinite, as the fluid is a conductor). There has been a great deal of study regarding this term. The model we use here lacks electronic


ð3Þ

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Figure 1. Differential capacitance (CD) versus surface potential (ΨS), for our RTIL models, with σ = 6 Å (diamonds, crosses) and 4 Å (circles, squares), respectively. Five point running averages are displayed. Two models are tested with σ = 4 Å. They differ in terms of the location of the charged monomer along the oligomeric salt component. Specifically, nc = 2 (circles) and 3 (squares), respectively. The bulk (mid-plane) salt densities are 0.00185 Å-3 and 0.00047 Å-3, for σ = 4 and 6 Å, respectively; ɛr was here set to 7.6.

Figure 2. Differential capacitance curves, with σ = 6 Å and nc = 3, as obtained with different choices of ɛr.

corresponds to the central monomer being charged. We see that, in all cases, there is a pronounced minimum in CD. That is, the experimentally observed6,7 camel-shape is reproduced by our theoretical model. Our model for the RTIL is crude, and relies on the assumption that the main physics is insensitive to its details. This is supported by these results. For instance, CD does not change at all, when the position of the charged monomer is shifted. Increasing σ from 4 to 6 Å, at constant volume fraction, does result in more significant changes, but the overall shape of the curve remains the same. We see that the capacitance is lower with the model using larger diameters. This is due to the less efficient screening of the surface charge by the larger fluid particles. Can we understand the general appearance of the CD curve and predict how it will respond to changes of model parameters or external conditions? Let us first consider the case when the surfaces are almost neutral. Then, the ion concentration will be depleted in the vicinity of the surfaces. With few responding “surface ions”, ΨS will change rapidly with σel, i.e., CD will be lower. The stronger the depletion, the lower CD will be (at low surface charge). If we increase the (absolute value of the) surface charge density, ions will accumulate at the surfaces. With more “surface ions” available to respond, the surface potential will change less rapidly with σel. This generates an increased differential capacitance. With the arguments presented thus far, we would anticipate a parabolic shape of CD(ΨS), with a minimum close to ΨS = 0. However, in reality, the surfaces will become saturated by ions at high surface charge, which limits the ability to produce a given shift of the ionic surface density, as a response to a change of σel. Thus, CD(ΨS) will display a maximum at relatively high absolute values of ΨS, resulting in a camel-hump shaped curve. The minimum in the differential capacitance is thus due to the depletion of liquid at the electrodes as the surface charge density approaches zero. The degree of depletion depends upon the cohesive properties of the liquid. Hence, dispersion interactions seem to play a crucial role in these system. In principle, one would therefore expect stronger depletion at higher liquid-wall surface tensions, but as the compressibility


Figure 3. Differential capacitance curves, as obtained with (k = 1 in eq 1) and without (k = 0) dispersion interactions between the spherical particles of our model. Comparisons are made for two different choices of ɛr.

of the liquid generally decreases at higher cohesion, predictions about the overall outcome can sometimes be subtle. It is interesting to relate our findings to the analysis provided by Kornyshev.3 In his mean-field model with excluded volume, the parameter γ (the ratio of the actual fluid density to its close packed limit) was used to measured the “compressibility” of the double layer. A camel-shaped capacitance in that model occurred for γ e 1/3, which is generally too low for a liquid. For higher fluid densities, the capacitance was bell-shaped. In our model, the double layer compressibility is enhanced by the depletion mechanism described above. That is, the fluid density in the vicinity of the surfaces at around zero surface charge is much less than the bulk value, which can be roughly interpreted as an effectively reduced γ value, even when the fluid has a liquid-like density in the bulk. We now switch focus to the role of ɛr. This is illustrated, for reasonable ɛr values, in Figure 2. We see how the capacitance drops with ɛr, and also attains a wider minimum. This is in agreement with our suggested mechanism, since depletion effects will be more pronounced at stronger electrostatic coupling. Still, even at high electrostatic coupling, dispersion interactions are of crucial importance. This is shown in Figure 3, where we we compare the predictions of the L-J model with those from a purely repulsive fluid, at the same bulk density, where the attractive part of the L-J potential has been

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SIMULATION METHODS In our simulations, we imposed periodic boundary conditions in the (x,y) directions parallel with the surfaces. The separation between the surfaces was assumed to be large enough, so that the fluid near the midplane was sufficiently “bulk-like”. We ensured that the local concentrations (around the midplane) were suitably flat, with local electroneutrality over a range of some 4σ. We also checked that orientational properties were isotropic in this regime. We used two different simulation approaches. One of these uses a grand canonical scheme, with the Rosenbluth insertion technique for the oligomeric cations. Long-range corrections were then handled by Ewald summation for the slab geometry.17,18 This approach is appealing in the sense that the correct bulk density is obtained “automatically”. However, the simulations also become time-consuming. In many cases, we have therefore performed simulations in the canonical ensemble. These do require some test runs in order to establish that the prescribed bulk density is obtained at the central part of the slit. Here, we used a mean-field correction19 to handle long-ranged interactions. We have verified that these two approaches produce the same results. Typical simulation box sizes are 90 90 90 Å3, when σ = 6 Å (59 59 90 Å3 with GC/Ewald), and 60 60 60 Å3, when σ = 4 Å. About 10-20 million configurations were devoted to equilibrate the system, followed by 10 times longer production runs.

Figure 4. Differential capacitance curves with nonadsorbing (reference, aw = 0)) and adsorbing (aw > 0) nonelectrostatic surface potentials, respectively.

removed. This amounts to setting k = 0 in eq 1. Results are provided for two choices of ɛr. With ɛr = 7.6, the repulsive fluid fails to reproduce any significant differential capacitance minimum. There is some tendency of a minimum, even for the repulsive fluid, at ɛr = 3.8, but it is not nearly as pronounced as for the liquid, where dispersion interactions are included. The capacitance minimum thus appears to be intimately connected to a loss of dispersion interactions at the electrodes, which effectively render surfaces with low charge density “solvophobic”, with a concomitant density depression. We also emphasize that, in the absence of dispersion (k = 0), the fluids are no longer proper liquids, i.e., there is no liquid-gas transition. Finally, we investigate the role played by nonelectrostatic surface/liquid interactions at the electrodes. In particular, we considered a L-J description of the (nonelectrostatic) surface potential: pffiffiffi" 3 # 3 3 σ 9 σ L-J ð4Þ - aw βΦw ðzÞ ¼ z z 2

SUPPORTING INFORMATION AVAILABLE An illustration

of our model, and its (crude) interpretation as an imidazolium chloride RTIL and differential capacitance curves in the absence of dispersion interactions. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author: *To whom correspondence should be addressed. E-mail: jan.forsman@ teokem.lu.se.

where the adsorption strength is regulated by aw. This potential has similar “excluding” properties as our reference repulsive potential, Φref w . With an L-J surface, monomers and ions will “automatically” be repelled at a distance of about σ. Given our arguments about desorption at low potentials being of fundamental importance to the capacitance minimum, we anticipate a more shallow minimum in the presence of a nonelectrostatic adsorption. This is corroborated in Figure 4, which highlights effects from nonelectrostatic liquid-electrode attraction. The results for the purely repulsive L-J wall agree with those obtained with our reference wall, which further motivates the chosen value of “d”, in the latter case. Ion adsorption in RTILs has been experimentally studied by Su et al.16 Our results indicate that attractive dispersion forces play a major role for the establishment of a camel-shaped differential capacitance in ionic liquids. The mechanism is intrinsically linked to the surface tension at the interface between the electrode and ionic liquid. The depth of the capacitance minimum near zero surface potential is also affected by the strength of dispersion attraction between the electrode surface and the ionic liquid, which in principle enables a way to tune the capacitance by an appropriate choice of electrode.


ACKNOWLEDGMENT J.F. acknowledges financial support by the Swedish Research Council.

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