Microscopic Insights into the Electrochemical ... - ACS Publications

Mar 28, 2013 - Nonaqueous Electrolytes in Electric Double-Layer Capacitors. De-en Jiang*. ,† ... effects of the pore size on the structure of the el...
3 downloads 0 Views 391KB Size
Download PDF
Perspective pubs.acs.org/JPCL

Microscopic Insights into the Electrochemical Behavior of Nonaqueous Electrolytes in Electric Double-Layer Capacitors De-en Jiang*,† and Jianzhong Wu*,‡ †

Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States Department of Chemical and Environmental Engineering, University of California, Riverside, California 92521, United States

‡

ABSTRACT: Electric double-layer capacitors (EDLCs) are electrical devices that store energy by adsorption of ionic species at the inner surface of porous electrodes. Compared with aqueous electrolytes, ionic liquid and organic electrolytes have the advantage of larger potential windows, making them attractive for the next generation of EDLCs with superior energy and power densities. The performance of both ionic liquid and organic electrolyte EDLCs hinges on the judicious selection of the electrode pore size and the electrolyte composition, which requires a comprehension of the charging behavior from a microscopic view. In this Perspective, we discuss predictions from the classical density functional theory (CDFT) on the dependence of the capacitance on the pore size for ionic liquid and organic electrolyte EDLCs. CDFT is applicable to electrodes with the pore size ranging from that below the ionic dimensionality to mesoscopic scales, thus unique for investigating the electrochemical behavior of the confined electrolytes for EDLC applications.

A

number of theoretical and experimental investigations on the dependence of the capacitance on the pore size.9−13 Meanwhile, there has been a surge of theoretical efforts toward understanding the electrochemical behavior of ionic liquid/ electrode interfaces, catalyzed by two independent publications of analytical theories concerning variation of the differential capacitance with the surface charge density.14,15 These theoretical studies were mostly focused on how the differential capacitance (Cdiff) varies with the surface potential, in particular, transition of the curve from “camel shape” to “bell shape” as the ionic density increases.14,15 Although a main factor determining the shape of Cdiff is affiliated with the ionic size,14 recent theoretical work found that the Cdiff dependence on potential can also be influenced by the dielectric constant and ion polarizability,16 the dispersion interaction,17 or the electrode surface topography18 if the features of the surface roughness are comparable to the ionic diameter.19 Both the camel shape and the bell shape of the Cdiff dependence on the electric potential at the ionic liquid/electrode interface contrasts with the “U shape” curve predicted by the Gouy−Chapman (GC) theory for EDL in dilute aqueous solutions,20 suggesting that the conventional method is not applicable to EDL of ionic liquids.14 The focus of recent theoretical and simulation work on ionic liquids seems to be justified, given their unique characteristics as an electrolyte, including low vapor pressure, versatile chemical functionalities, and wide electrochemical windows. Especially, great efforts have been devoted to investigating the

n electric double-layer capacitor (EDLC), also known as a supercapacitor, is typically composed of two symmetric porous electrodes, a liquid electrolyte, and a separator, with a configuration similar to the setup of a rechargeable Li ion battery.1 Unlike a battery, however, an EDLC stores electrical energy not by electrochemical reactions but by adsorption of ionic species at the internal surfaces of porous electrodes of the opposite charge, thereby having faster response, more power, and longer cycle life than a battery as an energy storage device.2 For EDLCs, porous carbons have been the prime choice of the electrode material due to their versatile porosity, robust microscopic structure, ultrahigh surface area, and good electronic conductivity.3,4 The charge carriers in an EDLC can be an organic electrolyte, an ionic liquid, or an aqueous electrolyte solution. Normally, ionic liquids and organic electrolytes provide higher energy densities than aqueous electrolytes, owing to their larger electrochemical windows. A major effort for the ongoing EDLC research is to improve the energy and power densities by employing novel carbon architectures and optimizing the equilibrium and transport properties of the charge carriers.5,6 While porous carbons with a broad range of pore sizes have been used for the electrodes, the effects of the pore size on the structure of the electric double layer (EDL) and ion transport remain poorly understood. For example, a report published in 2006 by Chimola et al.7 surprised the electrochemistry community with the finding that the surface-area-normalized capacitance showed an anomalous increase as the average pore size became comparable to the dimensionality of the ionic species. It was hypothesized7 that ion desolvation within small pores was responsible for the anomalous increase in the capacitance and thus energy density. Similar behavior was later observed in the EDLC with an ionic liquid electrolyte.8 These experimental reports motivated a © 2013 American Chemical Society

Received: February 9, 2013 Accepted: March 28, 2013 Published: March 28, 2013 1260

dx.doi.org/10.1021/jz4002967 | J. Phys. Chem. Lett. 2013, 4, 1260−1267

The Journal of Physical Chemistry Letters

Perspective

expressing the system energy as a functional of the electronic density, CDFT for electrolytes starts with the free energy as a functional of the local densities of ions and solvent molecules. These density profiles are obtained by minimizing the grand potential, and subsequently, quantities such as the surface charge density and the mean electric potential can be derived from the converged density profiles. In our applications of CDFT to EDLs of ionic liquids and organic electrolytes, we adopted a simplistic model, as shown in Figure 1, where a

dependence of capacitance on the pore size in the subnanometer region for an ionic liquid electrolyte. Invoking a phenomenological model of charged ions inside of a metallic slit pore,21 Kornyshev and co-workers explained the anomalous increase as image forces exponentially screening out the repulsion of the same-charged counterions inside of the narrow pore. Later, they supported this model by Monte Carlo (MC) simulations of charged hard spheres inside of slit pores13 and further incorporated the consideration of pore size dispersion.22 All-atom molecular dynamics (MD) simulations were also used to address the anomalous increase for carbon tube electrodes12,23 and slit pores.24,25 In the latter case, the capacitance could increase up to 40% from that of flat surfaces.24,25 Recently, Qiao and co-workers analyzed in detail the mechanisms of charge storage and capacitance enhancement at different voltages for a subnanometer pore;26 Xing et al. further examined the mechanisms at different widths of conductive pores and found that the enhancement can be very different between the positive and the negative electrodes.19 For the organic electrolytes, the dependence of the capacitance on the pore size was studied with phenomenological methods.9,10 In addition, MD simulation has been reported on the structure and dynamic behavior of the EDL.27 On the experimental side, the anomaly in the capacitance in organic electrolytes was later challenged by Centeno et al.,11 who showed that the surface-area-normalized capacitances for 28 porous carbons of different pore size distributions were approximately invariant with the pore size. Centeno et al. argued that a different surface area model should be used for normalization of the capacitance.11 We addressed the microscopic behavior of the electrode/ electrolyte interface and the dependence of the capacitance on the pore size with the classical density functional theory (CDFT). Although CDFT had been used extensively to study the EDL structure in aqueous systems, it was never before used to examine the differential capacitance at the electrode/ nonaqueous electrolyte interface and the pore size dependence of an EDLC. We realized the great potential of the CDFT method in offering a microscopic view with minimal molecular details and computational cost, especially its ability to address a large pore size range from that comparable to the ionic size to macroscopic scales. Besides, it is straightforward to incorporate a polar solvent in the CDFT calculations so that the desolvation hypothesis could be tested and the contrast between ionic liquids and organic electrolytes could be revealed.

Figure 1. A slit pore model for the porous electrode of an EDLC: (a) ionic liquid electrolyte; (b) organic electrolyte. In both cases, the cations and the anions are represented by hard spheres of equal diameter (0.5 nm). For the organic electrolyte, each solvent molecule is represented by two touching hard spheres of equal diameter (0.3 nm) but opposite charges (0.29 e). The dipole orientation of the solvent molecule is indicated by an arrow.

nonporous electrode is represented by a planar hard wall, a porous electrode is represented by a slit pore consisting of two planar hard walls, and all ionic species and solvent molecules are composed of charged hard spheres. Before we embarked on the study of the EDLC, we tested the performance of CDFT by examining the differential capacitance versus the surface potential at different ionic densities.31,32 We were able to reproduce the transition from the camel shape to the bell shape as predicted by previous investigations.14,15 We also showed that an ionic liquid forms alternating layers of cations and anions near a highly charged surface,33,34 in excellent agreement with experiment.35 The CDFT predictions were supported by further theoretical and experimental studies.36,37 To elucidate the effect of pore size on the capacitance, we considered ionic liquids and organic electrolytes in slit pores (Figure 1) that mimic the microscopic environment within porous carbon electrodes. To make further simplifications, we assume that the cations and anions have equal size, and each solvent molecule consists of two touching spheres of equal size but opposite charges (Figure 1b).38,39 Approximately, these models represent those electrolytes studied in experiments, that is, tetraethylammonium tetrafluoroborate (TEA-BF4) dissolved in acetonitrile (ACN) at 1.5 M, and the ionic liquid model corresp onds to 1-ethyl-3-met hylimidazolium bis(trifluoromethylsulfonyl) imide (EMIM-TFSI). Whereas the simplistic models neglect atomic details, they capture the essential features of surface charging and enable a systematic investigation of the capacitance behavior over a broad parameter space such as pore size, ion and solvent densities, ionic and solvent excluded volume, ionic valences, and surface potential. The numerical results from the CDFT predictions

We realized the great potential of the CDFT method in offering a microscopic view with minimal molecular details and computational cost, especially its ability to address a large pore size range from that comparable to the ionic size to macroscopic scales. The basic ideas of CDFT and its applications to electrolyte systems have been discussed in recent reviews.28,29 Here, it suffices to say that the mathematical foundation of CDFT is the same as that of the well-known electronic DFT.30 Instead of 1261

dx.doi.org/10.1021/jz4002967 | J. Phys. Chem. Lett. 2013, 4, 1260−1267

The Journal of Physical Chemistry Letters

Perspective

profiles in the corresponding EDLs. In particular, we may affiliate the peak positions in Figure 2a with constructive interferences and the trough positions with destructive interferences. A constructive interference occurs when the density profile of cations from one surface overlaps most significantly with that from the other surface, the same being true for the anionic density profiles. At the first peak, the pore size is slightly larger than the ion diameter (0.5 nm); the constructive interference can be easily visualized as overlapping the anion density profiles against the two positively charged walls, while the cations are excluded. At larger pore sizes, the density profiles from two noninteracting walls may facilitate an understanding of the interference of two EDLs, as we explain below. Figure 3a shows the ionic density profiles at the pore size of 0.85 nm, which corresponds to the first trough in the ionic liquid curve shown in Figure 2a. We see no significant overlaps of the density profiles for the cations or the anions. Figure 3b shows the density profiles at the pore size of 1.23 nm, which corresponds to the second peak shown in Figure 2a. Apparently, the cation density from the left wall corroborates well that from the right, and likewise for the anion. Analysis of the overlapping of the density profiles from the noninteracting walls at larger separations further confirms that the peak positions coincide with good superposition of the ionic density profiles and the trough positions with bad superposition.40 Beyond a certain pore size (in our case, ∼4.0 nm or 8 times the ion diameter), the two EDLs are virtually independent, and the capacitance becomes constant. To further quantify the correlation between the density overlapping and the capacitance oscillation, we have examined the density overlapping probability, which, for a given pore size, is defined as the sum of the overlapped areas for both cation and anion density profiles weighted by the difference between the average counterion and co-ion densities. We plot both the density overlapping probability and the capacitance together against the pore size in Figure 4. One can see the almost exact correspondence between the capacitance oscillation and the variation in the density overlapping probability with the pore size. This plot firmly places the origin of the capacitance oscillation to the interference of the EDLs. So, why does the density profile overlapping or EDL interference lead to capacitance oscillation? Intuitively, this resembles a resonance

thus provide a comprehension of the EDLC design from a microscopic perspective. Figure 2 illustrates an important example concerning the dependence of the capacitance on the pore size for the

Figure 2. The dependence of the integral capacitance (at 1.5 V surface potential) of an EDLC on the pore size for the ionic liquid (a)40 and the organic electrolyte (b).38

electrolyte/electrode models. CDFT predicts that, for an ionic liquid EDLC, the capacitance oscillates with the pore size with a dampened magnitude around an average value of about 7.5 μF/ cm2; beyond 4.0 nm (or about 8 times the ion diameter), the capacitance becomes basically invariant with the pore dimensionality.40 The maximum capacitance occurs when the pore size is about the same as the ion diameter, which coincides with the anomalous increase of capacitance observed by Largeot et al.8 Whereas a similar peak appears for the organic electrolyte, there is no significant oscillation as the pore size increases, and the capacitance at the first peak is less dramatic in comparison with the asymptotic value.38 The theoretical results thus offer a reconciliatory picture of the capacitance dependence on the pore size for ionic liquid and organic electrolyte EDLCs. The simplicity of the theoretical model also allows us to readily pinpoint the physical origin of the capacitance oscillation in an ionic liquid electrolyte. As aforementioned, CDFT predicts strong oscillation of the ionic density profiles near a charged surface,31,33 and a small separation between two charged surfaces leads to the interference of the ionic density

Figure 3. Density profiles of counterions (anion) and co-ions (cation) inside of a slit pore of two noninteracting walls (in other words, the density profiles are from a single, charged planar electrode, the left wall, but are mirrored by the pore center plane to the right wall) at 1.5 V surface potential: (a) pore size at 0.85 nm; (b) pore size at 1.23 nm. The captions in (b) also apply to (a). 1262

dx.doi.org/10.1021/jz4002967 | J. Phys. Chem. Lett. 2013, 4, 1260−1267

The Journal of Physical Chemistry Letters

Perspective

becomes responsible for the capacitance increase; at even higher voltages and when the co-ions are completely removed from the pore, counterion insertion leads to the capacitance decrease.26 Later, Xing et al.19 showed from their MD simulations of the EMIM-TFSI ionic liquid insides slit pores that the shape of the differential capacitance versus the potential can change from the bell shape to the U shape, depending on the pore size and that the dominant charging process also varies with the pore size. Here we use CDFT to examine the charging mechanism at different surface potentials for pore sizes corresponding to the capacitance maxima and minima as investigated by MD simulations. To facilitate comparison, we have analyzed the surface-area-normalized ion accumulation inside of the pore as the surface potential increases. The results are shown together with both the integral (Cint) and differential (Cdiff) capacitances for 0.525 (Figure 5a) and 0.85 nm (Figure 5b) pore widths against the applied surface potential. The 0.525 nm pore corresponds to the first capacitance peak in Figure 2a, while the 0.85 nm pore corresponds to the first trough. For the 0.525 nm pore, the charging process can be divided into two processes: below 1.5 V, the dominant mechanism is the swapping of the co-ions (cation) in the pore with the counterions (anion) from the bulk, and above 1.5 V, the countion insertion dominates as the co-ions have been excluded from the pore. For the 0.85 nm pore, the charging mechanism is predominantly the counterion insertion, mainly because the 0.5 nm ions are not packed well into the 0.85 nm pore, leaving more empty space. In both cases, the capacitance versus potential curve has a bell shape; this is due to the simple ionic liquid model and high ionic density that we used in our CDFT.31 In fact, all of the pore sizes examined here have a bell-shaped curve for the capacitance dependence on the surface potential; this might be the reason why we did not find a dominant co-ion-expulsion process in the intermediate voltage range identified by Qiao and co-workers that gives rise to an increase of the capacitance with the potential.26 With a more realistic model for the ionic liquid, we expect that the CDFT is able to predict more complex shapes of the capacitance−potential curves and their associated charging mechanisms (including the co-ion-expulsion process) as well as the change of the curvature from the bell shape to the U shape with the pore size.19 Compared with ionic liquids, organic electrolytes are more widely employed in commercial EDLC devices. The original discovery of the anomalous increase in capacitance was also based on the organic electrolyte of 1.5 M TEA-BF4 in acetonitrile.7 With the microscopic theory at hand, we are now in a good position to dissect the role of the solvent in the

Figure 4. The density overlapping probability (right axis; see the text for its definition) and the integral capacitance at 1.5 V (left axis) with the pore size for the model ionic liquid.

effect: when frequencies match, the amplitude enhances. In the EDLC case, when the density profiles overlap the most, the response of the surface charge to the fixed surface potential is reinforced, leading to an above-average capacitance. To go beyond this simple argument, we are currently working to develop a quantitative model of the causal relationship between the density overlapping and the capacitance oscillation. CDFT not only predicts capacitance oscillation with the pore size for ionic liquid EDLCs, a novel phenomenon that awaits experimental confirmation, but also provides an intuitive microscopic understanding of the oscillatory behavior. We note that the capacitance oscillation of the ionic liquid EDLC was also revealed by two classical all-atom MD simulations published at about the same time;24,25 more interestingly, one of them further confirmed that the EDL interference is responsible for the capacitance oscillation.25 However, the entire oscillation profile was not reproduced by MD simulations because high computational cost might have prevented simulation of ionic liquids in a pore with the size beyond the second peak. A later all-atom MD simulation showed that the capacitance oscillation is asymmetric: the pattern on the positive electrode is very different from that on the negative electrode due to the different size and shape of the cation versus the anion of the EMIM-TFSI ionic liquid.19 Charging behavior of ionic liquids inside narrow pores was also explored by several groups. For example, a recent all-atom MD simulation of an ionic liquid confined into a subnanometer pore reveals a potential-dependent charging process: at low surface potential, charge storage is dominated by the swapping of the co-ions in the pore with the counterions in the bulk; as the voltage increases, expulsion of the co-ions from the pore

Figure 5. The differential (Cdiff) and integral (Cint) capacitances and the cumulative densities of cations and anions (that is, the integration of the density profile across the pore width) versus the electrode surface potential: (a) in a 0.525 nm slit pore; (b) in a 0.85 nm slit pore. 1263

dx.doi.org/10.1021/jz4002967 | J. Phys. Chem. Lett. 2013, 4, 1260−1267

The Journal of Physical Chemistry Letters

Perspective

performance of an organic electrolyte EDLC. But first, let us compare the CDFT prediction against the results from experimental measurements (Figure 6).7 There are two sets

The efficiency of CDFT calculations enables us to take into account the dependence of the capacitance on the pore size distribution. To illustrate, Figure 7 shows the capacitance

Figure 6. Comparison of the surface-area-normalized experimental capacitances with the CDFT prediction. The experimental capacitances were normalized by surface areas from two models (BET and DFT).7 The pore size is normalized by the pore size of the maximum capacitance, roughly the ion size.

Figure 7. The effect of pore size distribution on the capacitance of an organic electrolyte EDLC, as shown in Figure 2b. The Gaussian distribution was assumed for each average pore size, and the different curves correspond to a different FWHM.

of experimental data resulting from two different ways to get the surface areas of the carbide-derived carbons for normalization of the capacitance. The degree of the so-called anomalous increase is of course dependent on which surface area is used, a point that has been previously discussed in detail.11 Figure 6 shows that the CDFT results agree well with the experimental data from both qualitative and quantitative perspectives. Is desolvation responsible for the rapid increase of the capacitance with the pore size as previously suggested?7 We analyzed the density profiles of ions and solvent molecules at different pore sizes against a surface potential of 1.5 V and found that the average solvent density decreases by about 40% while the counterion density increases by about 9 times when the pore size changes from the trough position (at 1.4 in Figure 6) to the peak position (at 1.0 in Figure 6).38 In other words, when the pore size is comparable to the ion size, the desolvation indeed occurs, and the solvent plays almost no role in the capacitance. This explains why both the ionic liquid and organic electrolytes have the same capacitance for the first peak in Figure 2. What is interesting and important to notice is that, in an organic electrolyte, the capacitance recovers rather quickly from the trough (at 0.75 nm in Figure 2b) to the asymptotic value as the pore size increases. The CDFT predictions provide a complete picture of the capacitance as the pore varies from the subnanometer range to mesoscopic scales (>2 nm). When the pore size is larger than a few nanometers, the capacitance becomes virtually independent of the pore size. Previously, modeling of the EDLC behavior relies on a posteriori division of the pore size range (in other words, according to the experimental data) into several regions, such as below 1, 1−2, and 2−5 nm, and so forth.9,10 The applicability of CDFT to all of these regions makes the division unnecessary as it provides a continuous variation of the EDL structure and capacitance as the pore size changes. The broad applicability of CDFT also highlights the advantage of theoretical investigations where the parameter can be precisely controlled; it contrasts with experiment where a precise control of the pore size is difficult and usually a broad pore size distribution results.

behavior for a porous electrode with a Gaussian distribution for the pore size. For a specific material, we may use a more precise distribution function based on experimental characterizations. We divide the pore size into bins, use the capacitance for each pore size shown in Figure 2b, and then average the capacitance according to the weight of each bin in the Gaussian distribution. CDFT predicts that a broadened pore size distribution (that is, with increasing full width at half-maximum (FWHM)) leads to less dependence of the capacitance on the average pore size; in other words, the peak is averaged down while the trough is averaged up. When the FWHM is at 0.5 nm or above, we see a rather smooth or less varied capacitance versus average pore size curve within a narrow range of 8−10 μF/cm2. The theoretical prediction agrees well with recent reports that show a rather constant capacitance of 9.4 ± 1.1 μF/cm2 (the shaded region in Figure 7) from 0.7 to 15 nm based on their study of 28 porous carbons.11 Now that we have compared the CDFT predictions with the experiments regarding the capacitance versus pore size relationship, how do we explain the constant capacitance beyond a rather small pore size in the case of an organic electrolyte EDLC? Undoubtedly, here, the solvent plays a key role. By analyzing the EDL structure (that is, the density profiles of ions and solvent against the charged surface of porous electrodes), we found that beyond 1.0 nm, the EDL structure becomes almost independent of the pore size.38 This structure has a contact layer of counterions mixed with dipolar solvent molecules aligned against the charged wall. Figure 8 shows a schematic picture of the EDL structure responsible for the insensitivity of the capacitance in the case of an organic electrolyte for a pore size of about three times the ion size (or 1.5 nm). If one examines the density profiles at larger pore sizes such as 4.0 nm (Figure 9), the EDL structure remains about the same. In other words, unlike ionic liquids where a moderate surface potential causes strong oscillating density profiles of several ion layers away from the electrode surface, in the case of the organic electrolyte, the EDL is dominated by just one compact layer of counterions with the aligned dipoles. The crucial role played by the dipolar solvent is due to that at 1.5 M concentration (modeled here and also employed in the 1264

dx.doi.org/10.1021/jz4002967 | J. Phys. Chem. Lett. 2013, 4, 1260−1267

The Journal of Physical Chemistry Letters

Perspective

like a piezoelectric control device used in the scanning tunneling microscope. Graphene could be a good choice for such an experiment. In the case of organic electrolyte EDLCs, ion desolvation as predicted by CDFT could also be extremely interesting to investigate experimentally. There has been some effort to use solid-state NMR to probe the EDL structure during charging and discharging of an EDLC.41 It would also be interesting to look into the dipole alignment at different pore sizes, as we have seen from the CDFT density profiles. We may also discuss some implications for future modeling efforts given our progress. Although there have been quite a lot of MD simulations on the behavior of ionic liquid EDLCs,19,24−26,42 no such studies have been reported yet to address the capacitance dependence on the pore size for organic electrolyte EDLCs. It would be very interesting to see if all-atom MD simulations could confirm the trend shown in Figure 2b for an organic electrolyte EDLC and also the effect of desolvation predicted by CDFT. A recent coarse-grained MD simulation showed that indeed adding the solvent such as acetonitrile to an ionic liquid slightly increases the capacitance and suggested that the solvent molecules facilitate the separation of cations and ions under an applied potential.43 In our CDFT calculations, we used a rather generic representation of a solvent dipole; in principle, this could be applied to any polar solvent such as propylene carbonate or even water, not limited to acetonitrile. We hope that future MD simulations could also address such solvent systems and compare their capacitance versus pore size relationship and the EDL structure with our CDFT predictions. Finally, there are many unanswered questions for CDFT investigations; factors that could further enhance our understanding of organic electrolyte EDLCs include the size of solvent molecules, the solvent polarity, the concentration of ions, the charge of ions, the size disparity between the cations and the anions, and the shape of the electrode (for example, slit versus cylindrical versus spherical44 versus ink-bottle pores45). It would also be interesting to look into the effects of those factors on the comparison between the integral capacitance and the differential capacitance.46 Addressing those factors could help us answer questions such as why the asymptotic value of capacitance for the organic electrolyte is 30% higher than that for the ionic liquid (Figure 2). Irregular pore shapes are common in real porous carbons and provide both great challenges and opportunities in theory and modeling. Using an amorphous porous carbon model from reverse Monte Carlo, Merlet et al.47 showed very recently from MD simulations how the ions of an ionic liquid respond to charged disordered carbon pore structures, resulting in a higher capacitance than the simple pore geometries. This work suggests that the inherent randomness of pores in real carbon materials could also be a factor to enhance the capacitance, perhaps because strong local electrostatic fields generated around the rough edges promote ion segregations near the surface. As shown by Vatamanu et al., the surface topology of the electrode greatly affects the shape of differential capacitance versus the electrode potential.18 Using realistic pore structures will be a frontier in the theory and modeling of EDLC capacitors; CDFT could be a method of choice, as we further explain below. A great challenge from a modeling perspective is how we establish a predictive structure−capacitance relationship for a real porous material and specific electrolytes. We think that the CDFT approach could address such a challenge. For a porous

Figure 8. A cartoon of the EDL structure of an organic electrolyte inside of a slit pore.

Figure 9. Reduced density profiles of the anion, the cation, the negative pole of the dipolar solvent, and the positive pole of the dipolar solvent across the 4.0 nm slit pore at 1.5 V surface potential.

experiment7) the electrolyte is dominated by the solvent with a molar ratio of solvent/solute of about 10. So far, we have shown how CDFT provides a better understanding of the effect of pore size on the capacitance of porous electrodes in an ionic liquid or organic electrolyte. Not only does CDFT give quantitative predictions that can be directly compared with the experimental data, but it also offers an insightful microscopic view of what is responsible for the observed relationship of the capacitance versus the pore size. On the one hand, the good agreement with experiments reflects the fact that CDFT captures the most important physics in the ionic system. On the other hand, the simplicity of the theoretical model enables a full comprehension of all key parameters over a wide range of conditions.

Looking into the future, we are especially excited to see if experimentalists could confirm the CDFT predictions. Looking into the future, we are especially excited to see if experimentalists could confirm the CDFT predictions. In the case of ionic liquid EDLCs, the oscillation of capacitance with the pore size is certainly the most interesting theoretical prediction to be confirmed. Here, the experimental challenge is the design of a porous system that has not only a narrow pore size distribution but also a continuously tunable pore size at molecular levels. Ideally, this could be achieved by mechanically bringing together two flat, conducting surfaces via something 1265

dx.doi.org/10.1021/jz4002967 | J. Phys. Chem. Lett. 2013, 4, 1260−1267

The Journal of Physical Chemistry Letters

Perspective

development (J.W.) was supported by the DOE Grant (DEFG02-06ER46296).

carbon material, surface area and porosity are the most important characteristics for EDLC applications. The CDFT approach48,49 represents the state-of-the-art method used by experimentalists to characterize the surface area and the pore size distribution of porous materials. Combining the pore size distribution with the capacitance versus pore size relationship from CDFT, we can, in principle, predict the capacitance of realistic porous electrodes. In other words, CDFT modeling can cover the complete process of energy storage in EDLCs, from porosity characterization to EDL structures, capacitance estimation, and charging kinetics (a future topic not addressed in this Perspective). In the end, the power density and the energy density of an EDLC as measured by per unit of weight or volume are the targeted properties of the 3D porous carbon electrode materials together with the confined electrolytes.47 There is still a long way to go for the CDFT to fully incorporate the complexity of the porous electrodes and predict the best electrolyte and porous architecture to maximize the energy and power densities of practical EDLCs.

■

A great challenge from a modeling perspective is how we establish a predictive structure− capacitance relationship for a real porous material and specific electrolytes.

■

REFERENCES

(1) Winter, M.; Brodd, R. J. What Are Batteries, Fuel Cells, and Supercapacitors? Chem. Rev. 2004, 104, 4245−4270. (2) Miller, J. R.; Simon, P. Electrochemical Capacitors for Energy Management. Science 2008, 321, 651−652. (3) Simon, P.; Gogotsi, Y. Materials for Electrochemical Capacitors. Nat. Mater. 2008, 7, 845−854. (4) Simon, P.; Gogotsi, Y. Capacitive Energy Storage in Nanostructured Carbon−Electrolyte Systems. Acc. Chem. Res. 2013, 10.1021/ar200306b. (5) Pech, D.; Brunet, M.; Durou, H.; Huang, P. H.; Mochalin, V.; Gogotsi, Y.; Taberna, P. L.; Simon, P. Ultrahigh-power MicrometreSized Supercapacitorsbased on Onion-Like Carbon. Nat. Nanotechnol. 2010, 5, 651−654. (6) Zhu, Y. W.; Murali, S.; Stoller, M. D.; Ganesh, K. J.; Cai, W. W.; Ferreira, P. J.; Pirkle, A.; Wallace, R. M.; Cychosz, K. A.; Thommes, M.; et al. Carbon-Based Supercapacitors Produced by Activation of Graphene. Science 2011, 332, 1537−1541. (7) Chmiola, J.; Yushin, G.; Gogotsi, Y.; Portet, C.; Simon, P.; Taberna, P. L. Anomalous Increase in Carbon Capacitance at Pore Sizes Less than 1 Nanometer. Science 2006, 313, 1760−1763. (8) Largeot, C.; Portet, C.; Chmiola, J.; Taberna, P. L.; Gogotsi, Y.; Simon, P. Relation between the Ion Size and Pore Size for an Electric Double-Layer Capacitor. J. Am. Chem. Soc. 2008, 130, 2730−2731. (9) Huang, J. S.; Sumpter, B. G.; Meunier, V. A Universal Model for Nanoporous Carbon Supercapacitors Applicable to Diverse Pore Regimes, Carbon Materials, and Electrolytes. Chem.Eur. J. 2008, 14, 6614−6626. (10) Huang, J. S.; Sumpter, B. G.; Meunier, V. Theoretical Model for Nanoporous Carbon Supercapacitors. Angew. Chem., Int. Ed. 2008, 47, 520−524. (11) Centeno, T. A.; Sereda, O.; Stoeckli, F. Capacitance in Carbon Pores of 0.7 to 15 nm: A Regular Pattern. Phys. Chem. Chem. Phys. 2011, 13, 12403−12406. (12) Shim, Y.; Kim, H. J. Nanoporous Carbon Supercapacitors in an Ionic Liquid: A Computer Simulation Study. ACS Nano 2010, 4, 2345−2355. (13) Kondrat, S.; Georgi, N.; Fedorov, M. V.; Kornyshev, A. A. A Superionic State in Nano-Porous Double-Layer Capacitors: Insights from Monte Carlo Simulations. Phys. Chem. Chem. Phys. 2011, 13, 11359−11366. (14) Kornyshev, A. A. Double-Layer in Ionic Liquids: Paradigm Change? J. Phys. Chem. B 2007, 111, 5545−5557. (15) Kilic, M. S.; Bazant, M. Z.; Ajdari, A. Steric Effects in the Dynamics of Electrolytes at Large Applied Voltages. I. Double-Layer Charging. Phys. Rev. E 2007, 75, 021502. (16) Lauw, Y.; Horne, M. D.; Rodopoulos, T.; Leermakers, F. A. M. Room-Temperature Ionic Liquids: Excluded Volume and Ion Polarizability Effects in the Electrical Double-Layer Structure and Capacitance. Phys. Rev. Lett. 2009, 103, 117801. (17) Trulsson, M.; Algotsson, J.; Forsman, J.; Woodward, C. E. Differential Capacitance of Room Temperature Ionic Liquids: The Role of Dispersion Forces. J. Phys. Chem. Lett. 2010, 1, 1191−1195. (18) Vatamanu, J.; Cao, L.; Borodin, O.; Bedrov, D.; Smith, G. D. On the Influence of Surface Topography on the Electric Double Layer Structure and Differential Capacitance of Graphite/Ionic Liquid Interfaces. J. Phys. Chem. Lett. 2011, 2, 2267−2272. (19) Xing, L.; Vatamanu, J.; Borodin, O.; Bedrov, D. On the Atomistic Nature of Capacitance Enhancement Generated by Ionic Liquid Electrolyte Confined in Subnanometer Pores. J. Phys. Chem. Lett. 2013, 4, 132−140. (20) Fedorov, M. V.; Kornyshev, A. A. Towards Understanding the Structure and Capacitance of Electrical Double Layer in Ionic Liquids. Electrochim. Acta 2008, 53, 6835−6840.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (D.-e.J.); [email protected]. (J.W.). Notes

The authors declare no competing financial interest. Biographies De-en Jiang is a staff scientist at Oak Ridge National Laboratory. He received his B.S. degree in 1997 and M.S. degree in 2000 both from Peking University and a Ph.D. degree in 2005 from UCLA, all in chemistry. He joined Oak Ridge National Laboratory first as a postdoctoral research associate and then became a research staff member in 2006. His research is focused on applying state-of-the-art computational methods to important chemical systems and energyrelevant problems. Find out more about his research at http://sites. google.com/site/deenjiang. Jianzhong Wu is a professor of Chemical Engineering and a cooperating faculty member of Applied Mathematics at the University of California, Riverside. He received his Ph.D. degree in Chemical Engineering from the University of California, Berkeley, and M.S. and B.E. degrees in Chemical Engineering and a B.S.degree in Applied Mathematics from Tsinghua University, Beijing. His research is mainly concerned with development of statistical mechanical methods for describing the physiochemical properties of soft materials and biological systems. More information about his research group is available at http://www.cee.ucr.edu/jwu.

■

ACKNOWLEDGMENTS Calculations, data analysis, and writing (D.-e.J. and J.W.) were supported by the Fluid Interface Reactions, Structures, and Transport (FIRST) Center, an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences. Method 1266

dx.doi.org/10.1021/jz4002967 | J. Phys. Chem. Lett. 2013, 4, 1260−1267

The Journal of Physical Chemistry Letters

Perspective

Double Layer Capacitors. Electrochim. Acta 2013, 10.1016/j.electacta.2012.12.107. (44) Feng, G.; Jiang, D. E.; Cummings, P. T. Curvature Effect on the Capacitance of Electric Double Layers at Ionic Liquid/Onion-Like Carbon Interfaces. J. Chem. Theory Comput. 2012, 8, 1058−1063. (45) Nguyen, P. T. M.; Fan, C.; Do, D. D.; Nicholson, D. On the Cavitation-Like Pore Blocking in Ink-Bottle Pore: Evolution of Hysteresis Loop with Neck Size. J. Phys. Chem. C 2013, 117, 5475− 5484. (46) Pizio, O.; Sokolowski, S.; Sokolowska, Z. Electric Double Layer Capacitance of Restricted Primitive Model for an Ionic Fluid in SlitLike Nanopores: A Density Functional Approach. J. Chem. Phys. 2012, 137, 234705. (47) Merlet, C.; Rotenberg, B.; Madden, P. A.; Taberna, P.-L.; Simon, P.; Gogotsi, Y.; Salanne, M. On the Molecular Origin of Supercapacitance in Nanoporous Carbon Electrodes. Nat. Mater. 2012, 11, 306−310. (48) Ravikovitch, P. I.; Neimark, A. V. Density Functional Theory Model of Adsorption on Amorphous and Microporous Silica Materials. Langmuir 2006, 22, 11171−11179. (49) Gor, G. Y.; Thommes, M.; Cychosz, K. A.; Neimark, A. V. Quenched Solid Density Functional Theory Method for Characterization of Mesoporous Carbons by Nitrogen Adsorption. Carbon 2012, 50, 1583−1590.

(21) Kondrat, S.; Kornyshev, A. Superionic State in Double-Layer Capacitors with Nanoporous Electrodes. J. Phys.: Condens. Matter 2011, 23, 022201. (22) Kondrat, S.; Perez, C. R.; Presser, V.; Gogotsi, Y.; Kornyshev, A. A. Effect of Pore Size and its Dispersity on the Energy Storage in Nanoporous Supercapacitors. Energy Environ. Sci. 2012, 5, 6474−6479. (23) Yang, L.; Fishbine, B. H.; Migliori, A.; Pratt, L. R. Molecular Simulation of Electric Double-Layer Capacitors Based on Carbon Nanotube Forests. J. Am. Chem. Soc. 2009, 131, 12373−12376. (24) Wu, P.; Huang, J.; Meunier, V.; Sumpter, B. G.; Qiao, R. Complex Capacitance Scaling in Ionic Liquids-Filled Nanopores. ACS Nano 2011, 5, 9044−9051. (25) Feng, G.; Cummings, P. T. Supercapacitor Capacitance Exhibits Oscillatory Behavior as a Function of Nanopore Size. J. Phys. Chem. Lett. 2011, 2, 2859−2864. (26) Wu, P.; Huang, J.; Meunier, V.; Sumpter, B. G.; Qiao, R. Voltage Dependent Charge Storage Modes and Capacity in Subnanometer Pores. J. Phys. Chem. Lett. 2012, 3, 1732−1737. (27) Feng, G.; Huang, J.; Sumpter, B. G.; Meunier, V.; Qiao, R. Structure and Dynamics of Electrical Double Layers in Organic Electrolytes. Phys. Chem. Chem. Phys. 2010, 12, 5468−5479. (28) Wu, J. Z. Density Functional Theory for Chemical Engineering: From Capillarity to Soft Materials. AIChE J. 2006, 52, 1169−1193. (29) Wu, J. Z.; Li, Z. D. Density-Functional Theory for Complex Fluids. Annu. Rev. Phys. Chem. 2007, 58, 85−112. (30) Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. 1964, 136, B864−B871. (31) Jiang, D. E.; Meng, D.; Wu, J. Z. Density Functional Theory for Differential Capacitance of Planar Electric Double Layers in Ionic Liquids. Chem. Phys. Lett. 2011, 504, 153−158. (32) Henderson, D.; Lamperski, S.; Jin, Z. H.; Wu, J. Z. Density Functional Study of the Electric Double Layer Formed by a High Density Electrolyte. J. Phys. Chem. B 2011, 115, 12911−12914. (33) Wu, J. Z.; Jiang, T.; Jiang, D. E.; Jin, Z. H.; Henderson, D. A Classical Density Functional Theory for Interfacial Layering of Ionic Liquids. Soft Matter 2011, 7, 11222−11231. (34) Henderson, D.; Wu, J. Z. Electrochemical Properties of the Double Layer of an Ionic Liquid Using a Dimer Model Electrolyte and Density Functional Theory. J. Phys. Chem. B 2012, 116, 2520−2525. (35) Mezger, M.; Schroder, H.; Reichert, H.; Schramm, S.; Okasinski, J. S.; Schoder, S.; Honkimaki, V.; Deutsch, M.; Ocko, B. M.; Ralston, J.; et al. Molecular Layering of Fluorinated Ionic Liquids at a Charged Sapphire (0001). Surf. Sci. 2008, 322, 424−428. (36) Bhuiyan, L. B.; Lamperski, S.; Wu, J.; Henderson, D. Monte Carlo Simulation for the Double Layer Structure of an Ionic Liquid Using a Dimer Model: A Comparison with the Density Functional Theory. J. Phys. Chem. B 2012, 116, 10364−10370. (37) Zhou, H.; Rouha, M.; Feng, G.; Lee, S. S.; Docherty, H.; Fenter, P.; Cummings, P. T.; Fulvio, P. F.; Dai, S.; McDonough, J.; et al. Nanoscale Perturbations of Room Temperature Ionic Liquid Structure at Charged and Uncharged Interfaces. ACS Nano 2012, 6, 9818−9827. (38) Jiang, D. E.; Jin, Z. H.; Henderson, D.; Wu, J. Z. Solvent Effect on the Pore-Size Dependence of an Organic Electrolyte Supercapacitor. J. Phys. Chem. Lett. 2012, 3, 1727−1731. (39) Henderson, D.; Jiang, D. E.; Jin, Z. H.; Wu, J. Z. Application of Density Functional Theory To Study the Double Layer of an Electrolyte with an Explicit Dimer Model for the Solvent. J. Phys. Chem. B 2012, 116, 11356−11361. (40) Jiang, D. E.; Jin, Z. H.; Wu, J. Z. Oscillation of Capacitance inside Nanopores. Nano Lett. 2011, 11, 5373−5377. (41) Wang, H.; Koster, T. K. J.; Trease, N. M.; Segalini, J.; Taberna, P. L.; Simon, P.; Gogotsi, Y.; Grey, C. P. Real-Time NMR Studies of Electrochemical Double-Layer Capacitors. J. Am. Chem. Soc. 2011, 133, 19270−19273. (42) Rajput, N. N.; Monk, J.; Hung, F. R. Structure and Dynamics of an Ionic Liquid Confined Inside a Charged Slit Graphitic Nanopore. J. Phys. Chem. C 2012, 116, 14504−14513. (43) Merlet, C.; Salanne, M.; Rotenberg, B.; Madden, P. A. Influence of Solvation on the Structural and Capacitive Properties of Electrical 1267

dx.doi.org/10.1021/jz4002967 | J. Phys. Chem. Lett. 2013, 4, 1260−1267