Article pubs.acs.org/JPCC
Theoretical Prediction of the Capacitance of Ionic Liquid Films Ryan Szparaga,† Clifford E. Woodward,‡ and Jan Forsman*,† †
Theoretical Chemistry, Chemical Centre, Lund University, P.O. Box 124, S-221 00 Lund, Sweden School of Physical, Environmental and Mathematical Sciences, University College, University of New South Wales, ADFA Canberra ACT 2600, Australia
‡
ABSTRACT: We use classical density functional theory to investigate ionic liquid + solvent mixtures against smooth, model electrodes. We consider those mixtures that display a wetting transition at the electrodes. We find that a wetting film of an ionic liquid at the electrodes has similar properties to the neat liquid. A novel aspect of this transition is that, close to a surface critical point, we find an extraordinary increase in the capacitance, which is derived from the critical behavior.
I. INTRODUCTION Room temperature ionic liquids (RTILs) have very short electrostatic screening lengths. Furthermore, they do not possess a solvation shell, which enhances their ability to screen electrodes, even at the sub-nanometer level.1 These physical considerations, as well as their electrochemical stability, combine to make RTILs plausible replacements for conventional electrolytes in devices such as electric double layer capacitors (EDLCs). There are many applications that propose the use of RTIL, mixed with solvents. For example, the use of added solvent to reduce the viscosity of RTILs in electrochemical applications has often been suggested in the literature.2−4 This is due to the fact that high viscosity can adversely affect performance of devices, due to slow kinetics.2,4 There are approximately a million different RTILs. When combined with additional solvent, the number of RTIL + solvent combinations is nearly limitless, giving researchers the ability to tune these mixtures for particular applications. However, the addition of solvent would seem to obviate many of the advantages of using a pure RTIL. One might, for instance, question the ability of the RTIL + solvent mixture to effectively screen surface charges on electrodes. On the other hand, there is some evidence that the screening in a neat RTIL is already largely accomplished within a few monolayers of the liquid adjacent to electrode surfaces.5 An important difference between an RTIL + solvent mixture and a normal electrolyte solution is that the former may display a liquid−liquid miscibility gap upon saturation. This suggests the possibility of using RTIL + solvent combinations which manifest surface wetting transitions by the RTIL at electrodes.6 Even a wetting layer of a few monolayers should have a similar electrostatic screening length to the neat RTIL. It is certainly of significant academic interest to study the effect of surface phase transitions on the electrical properties of © 2012 American Chemical Society
ionic liquid solutions. There are a rich variety of surface wetting phenomena, and as far as we are aware, the implications of these on electrical properties, such as capacitance, have not yet been reported in the literature. We expect that wetting transitions can cause a profound effect on the electrolyte response (and the capacitance) via thermodynamic instabilities. To see this, we note that the dif ferential capacitance per unit area is a thermodynamic response function. Thus, in the vicinity of thermodynamic instabilities, such as surface transitions, spinodal and critical points, the differential capacitance may become large, or infinite. This raises the tantalizing possibility of a novel class of EDLCs with enhanced capacitance due to fluctuations induced by thermodynamic transitions. This notwithstanding, the experimental investigation of these systems may well be fraught with the same difficulties that accompany such attempts to study thin film wetting in other liquids.7 This may especially be the case for porous electrodes, wherein capillary condensation may well confound wetting studies, particularly in narrow pores. Here, we report a theoretical study, which will focus on the wetting phenomenon in RTIL + solvent mixtures, against model smooth surfaces. We use a classical density functional theory (DFT), to describe an RTIL model which manifests the wetting phenomenon.
II. THEORY A. The Model. This RTIL model has been described earlier,8,9 and a brief summary is given here. The cations are linear pentamers made up of freely jointed monomer spheres Received: May 31, 2012 Revised: July 2, 2012 Published: July 5, 2012 15946
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with a fixed bond length of 5 Å. A single positive charge is placed on the second monomer. The anions are simple negative monomeric spheres. This model broadly mimics an imidazolium-based RTIL. All the spherical monomers (cationic or anionic) are assumed to interact with a Lennard-Jones (LJ) potential given by ⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ ϕLJ(r ) = 4ϵLJ⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎝r⎠ ⎦ ⎣⎝ r ⎠
capacitance, are robust in the sense that they rely on wellestablished thermodynamic relations, which are independent of model details. Specifically, analogous behaviors can be established in a model comprising an explicit solvent representation. The semi-grand free energy,24 . , can be written as
∫ na(r)(ln[na(r)] − 1) dr + β -hs[nm ̅ (r), na̅ (r)] + β ∫ wLJ(r)(nm(r) + na(r))
id β . = β - cat [N (R)] +
(1)
where r is the distance between the centers of monomers. We chose σ = 5 Å (equal to the bond length between monomers in the cations). As described below, we employed an implicit solvent model for the mixture. Thus, ϵLJ is a solvent-mediated interaction strength. We have set ϵLJ = 0.625kT, where kT is the thermal energy unit. This is a typical value for hydrocarbons in a polar solvent. We note that the qualitative results we report here would not change significantly for other reasonable values of this parameter. All calculations were performed at a temperature of T = 300 K. The electrode is treated as a planar half-space, with a uniform charge per unit area. Nonelectrostatic, solvent mediated interactions between the fluid particles and the electrode surface were modeled with an integrated Lennard-Jones potential 3 ⎡ 2 ⎛ σ ⎞9 1⎛σ ⎞ ⎤ βwLJ(z) = 2π ϵLJ⎢ ⎜ w ⎟ − ⎜ w ⎟ ⎥ 3⎝ z ⎠ ⎦ ⎣ 45 ⎝ z ⎠
dr + β + 2 +
β 2
∬ ′ ∑ ∑ ng (r)nd(r′)ϕLJ(|r − r′|) dr dr′ g
d
∬ ∑ ∑ nγ (r)nδ(r′)Φelγδ(|r − r′|) dr dr′ γ
δ
∫ ∑ (βVel(r) + fDHH [nγ̅ (r)] − βμγ )nγ (r) dr γ
(4)
The sums extend over all particle types, except where Greek subscripts are used, whereby the sums only includes the charged monomers. The first two terms on the RHS of eq 4 represent the exact ideal contributions to the free energy due to cations and anions. As the cations are made up of chain-like molecules, their ideal free energy, - idcat[N(R], must reflect this. It can be exactly expressed as a functional of the density distribution, N(R), where R = (r1, ..., r5), with ri the coordinate of the ith cationic monomer.11
(2)
where z is the distance from the particle center and the plane of charge. For simplicity, the parameters in this surface potential are assumed the same as that for the fluid−fluid interactions. The Coulomb interaction between any two charged species, qα and qλ, in the model is 1 qαqλ Φelαλ(r ) = 4π ϵ0ϵr r (3)
id β - cat [N (R)] =
∫ N(R)(ln[N(R)] − 1) dR + β ∫ N (R)VB(R) dR
(5)
where VB(R) is the “bonding potential” that maintains the oligimeric structure. This is assumed to be a freely jointed chain in our model. The third term is the steric contribution, - hs[nm̅ (r), na̅ (r)], which describes the hard core interactions between the fluid species. As in other treatments,8 it is assumed to be a functional only of the total monomer densities of the cationic oligimer, nm(r), and the anion density, na(r). Note that these appear as weighted densities in the functional (denoted by the bar over densities). We choose the simple weighting scheme originally used by Nordholm,12 which gives a simple and reasonably accurate representation of excluded volume effects. For example, the weighted monomer density, nm ̅ (r), is given by
where ϵ0 is the permittivity of vacuum. The primary contribution to the relative dielectric constant, ϵr, is assumed to come from the solvent. Here, we chose ϵr = 30, which is typical of a number of nonaqueous solvents. The Coulomb interaction can be integrated over the uniform surface charge to give the interaction between charged particles and the electrode surface. B. Density Functional Theory. A detailed account of the functional describing the free energy (per unit electrode area), for our RTIL model, can be found in other references.8,9 In our present treatment, a solvent is also included, albeit implicitly, via an incompressibility constraint. That is, the total local volume fraction, which is the sum of the RTIL and solvent components, is constant. This assumption removes the explicit dependence on the solvent density, and the free energy is effectively only a functional of the densities of the ionic liquid components. In this way, the solvent renormalizes interactions so as to create an “effective” room temperature ionic liquid (eRTIL). In other words, we use McMillan−Mayer arguments10 to integrate out the solvent degrees of freedom, wherein the osmotic pressure (in an incompressible RTIL + solvent system) is directly related to the solvent chemical potential. This implicit solvent approach, generating solventmediated interactions, is reflected in our choice of parameter values, given earlier. We emphasize that our main findings in this study, with emphasis on a fluctuation-enhanced differential
nm ̅ (r) =
3 4πσ 3
∫|r−r | σ(c) + , the system temperature is above the surface critical temperature, Tsc. The presence of the critical points is related to the way in which the local excess chemical potential of the RTIL increases, near the electrode, as the magnitude of the surface charge density is increased. This excess chemical potential increase leads to a narrower film at the surface. Thus, the mutual attraction between the RTIL molecules in the film is effectively truncated, which causes a drop of the surface critical temperature. With a symmetric salt, Tsc would have its highest value at a neutral electrode, but the cooperative adsorption of the oligomeric cation shifts this maximum to positive surface charge densities. The change of Tsc with σs is pictorially illustrated in Figure 2. The structure of the RTIL films at neutral surfaces is shown in Figure 3. We see that thick films are only one or two monolayers wide. This is the case for all the surface charge densities investigated in our study, and indicates that almost complete screening of the electrode charges occurs across such a short distance. B. Capacitance. The differential capacitance, cD, is a thermodynamic response function, defined as
Figure 3. Concentration profiles of coexisting thick and thin anion and cation films adjacent to a neutral electrode. The dotted lines correspond to the thin profiles and the solid lines to the thick profiles.
c D = ∂σs/∂Ψs = {∂ 2./∂σs 2}−1 = {∂ 2Ω/∂σs 2}−1
(13)
Experimentally, this quantity is often measured as a function of the surface potential, Ψs. If the electrolyte undergoes a phase change, finite fluctuations in the surface charge density occur for a very small potential change (ideally zero), with a corresponding spike in the differential capacitance. We investigated the capacitance as a function of surface potential for a bulk concentration corresponding to the upper dotted line in Figure 1. The results are shown in Figure 4. Interestingly, we were able to obtain converged thick phase solutions at all potentials, even within the region where the thin phase is the more stable. This enables us to establish a thick phase capacitance across the full potential range. However, the thin phase becomes unstable if the absolute surface potential is too large. This is indicated by a huge increase in the thin phase capacitance, due to approaching spinodals, where the capacitance becomes infinite. Before the spinodals are reached, however, a transition to the thick phase would occur, accompanied by a sudden jump in σs, corresponding to a 15949
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of the neat ionic liquid, for negative surface potentials, where we expect the wetting film to be thicker. Remarkably, we see that close to zero surface potential there is a narrow range of surface potentials, where cD becomes huge. This behavior is clearly an “echo” of the nearby critical point, where the differential capacitance would be infinite. As the potential traverses this region, there would be a steep increase of the surface charge density on the electrodes.
IV. CONCLUSIONS The advantages to EDLCs of an RTIL + solvent system which wets electrodes with an essentially pure ionic liquid film are immediate. EDLC electrodes possess porosities over a range of length scales, Dp (where Dp is the pore diameter).21 At macroscopic scales (Dp > 50 nm), the capacitance is approximately inversely proportional to the screening length. In the RTIL film, this screening length is much smaller than the Debye length of the bulk electrolyte. For mesoporous carbon materials (2 nm < Dp < 50 nm), often obtained via templating methods, the capacitance is adversly affected by the need to desolvate ions, as the pore size decreases. This leads to a decrease in the capacitance. The ions in the RTIL film do not possess a solvation shell, and therefore, this effect is avoided. It should be noted that, in some cases, wetting may be supplanted by capillary condensation, depending upon the equilibrium thickness of the film and the pore diameter. Even in microporous materials (Dp < 2 nm), where large increases in capacitance have been observed, the loss of the solvation shell is an intrinsic free energy penalty that lowers the capacitance.22 This free energy cost is circumvented in our system, which will lead to an enhanced capacitance, as suggested in recent experiments with neat RTILs and microporous electrodes.21,23 Finally, the large charge fluctuations that occur in these systems close to critical conditions may also be exploited to further increase the capacitance. The natural inhomogeniety of surfaces in EDLCs will likely guarantee that a subclass of surface environments will have fluctuation enhanced capacitance as the system traverses a range of potentials. Indeed, the possibility that surfaces can be engineered to display enhanced capacitances over a specific potential window is an exciting possibility.
Figure 4. Differential capacitance, cD, as a function of the electrode potential, Ψs, for a bulk density (thin (black), thick (blue)) corresponding to the upper dotted line in Figure 1. At each surface transition point, the capacitance would display an “infinite” spike. In practice, a phase transition occurs slightly before the spinodal point is reached. The red curve shows the differential capacitance for the neat RTIL.
spike in cD. We compared the capacitance of the thick and thin films, with that of the neat RTIL. The latter was obtained from a wetting film of infinite thickness. The capacitance of the thin phase is generally below that of the neat RTIL (except toward the spinodal regions). It is worth pointing out, in this context, that our model would likely overestimate the capacitance of the thin film, as it does not account for the solvation shell around the ions. On the other hand, we find that the thick phase has a capacitance, which is almost identical to that of the neat liquid, consistent with our earlier conjecture that screening occurs in the first few monolayers of the neat RTIL. The dynamic properties of a narrow film are expected to be superior with respect to performance in electrical devices, compared to the neat ionic liquid. For example, a wetting film will readily dissipate upon discharging of electrodes in EDLCs. We also explored the behavior of wetting films above the surface critical temperature, where the wetting transition becomes second-order. We expected to find large capacitances at or around the surface critical point and investigated this possibility, by scanning the electrolyte behavior at the bulk density described by the lower dotted line in Figure 1. This line traverses a “supercritical” region close to the critical point. The corresponding capacitance is shown in Figure 5. Here we note that the capacitance has characteristics of both thick and thin films in Figure 4. The capacitance seems to closely follow that
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AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
REFERENCES
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Figure 5. Differential capacitance, cD, as a function of the electrode potential, Ψs, for a bulk density corresponding to the lower dotted line in Figure 1. Here the system is supercritical, though kept close to the surface critical point (as indicated by the terminus of the phase coexistence line). The dotted (red) line corresponds to the capacitance of the neat RTIL (as in Figure 4). 15950
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