A Generic Model for Electric Double Layers in ... - ACS Publications

Apr 6, 2016 - Department of Chemical and Environmental Engineering and. §. Department of Chemistry, University of California, Riverside,. California ...
1 downloads 0 Views 1MB Size
Download PDF
Article pubs.acs.org/JPCC

A Generic Model for Electric Double Layers in Porous Electrodes Cheng Lian,†,‡ De-en Jiang,§ Honglai Liu,† and Jianzhong Wu*,‡ †

State Key Laboratory of Chemical Engineering, East China University of Science and Technology, Shanghai 200237, People’s Republic of China ‡ Department of Chemical and Environmental Engineering and §Department of Chemistry, University of California, Riverside, California 92521, United States ABSTRACT: The performance of electric-double-layer capacitors (EDLCs) hinges on microscopic charge distributions near the electrode/electrolyte interfaces. Whereas practical EDLCs consist of electrodes made of amorphous porous materials, theoretical understanding of EDLCs is mostly based on EDL structure near a planar surface or on simplistic models that have little relevance to realistic systems. In this work, we propose a spherical shell model to account for both pore size and curvature effects of amorphous porous materials. The EDL structure in spherical shells has been investigated over a broad range of pore sizes and curvatures by use of classic density functional theory. Theoretical results reveal that the curvature effects on convex and concave EDLs are drastically different and that materials with extensive convex surfaces will lead to maximized capacitance. Like a slit pore, the spherical shell model also predicts oscillatory variation of capacitance with pore size, but the oscillatory behavior is magnified as the curvature increases. The joint effects of pore size and curvature identified in this work give new insight into materials design for porous electrodes with optimal EDLC performance.

1. INTRODUCTION

Whereas practical porous electrodes contain micropores with complicated morphology and pore-size distributions,12,13 theoretical modeling of EDLCs is mostly based on simplistic models to represent pore geometry and electrolyte−electrode interactions.3,14 Specifically, three types of electrode structures are commonly used in theoretical investigations:15−17 (i) planar surfaces (e.g., a flat surface or slit pores), (ii) cylindrical pores with their concave inner surfaces or cylindrical particles with their convex outer surfaces (e.g., carbon nanotubes), and (iii) spherical surfaces (e.g., onionlike carbons). The slit and cylindrical pore models are commonly used for porous material characterization.18 Despite the fact that a great variety of porous carbons have been utilized for EDLCs, the joint effects of pore size and geometry on the EDL structure remain poorly understood.5 At the heart of the issue is the question: What is the microscopic structure of porous electrodes and how does the capacitance of EDLCs depend on the electrode pore geometry and electrolyte composition? Recent simulations and experiments both indicate that pore size and geometry play important roles in determining the capacitance of EDLCs.17,19−22 For example, it has been shown that the surface-area-normalized capacitance displays an anomalous increase as the average pore size becomes comparable to the dimensionality of the ionic species.15,23−29 The curvature effect was also studied in terms of EDL formation at the outer surfaces of spherical or cylindrical particles.16 An important

Electric-double-layer capacitors (EDLCs), also known as supercapacitors, store electrical energy not by electrochemical reactions but by physical adsorption of ionic species at the internal surfaces of porous electrodes. In comparison to batteries, EDLCs have the advantages of faster charging kinetics, higher power densities, and longer cycling lifespans.1 While electrochemical reactions play a central role in batteries, the performance of supercapacitors is strongly correlated with ionic behavior inside the micropores of electrodes.2−8 A better understanding of the effects of pore size and geometry is essential for rational design of novel electrode materials with improved energy and power densities. Over the last few decades, a wide variety of electrode materials, with tunable pore-size distributions, morphology, architecture, and functionality, have been proposed for EDLCs. Meanwhile, diverse electrolytes have been tested to improve EDLC performance. Most porous electrodes are made of carbon materials including activated carbon, carbon nanotubes, templated carbons, carbon aerogels, and graphene.9 Porous carbons are attractive for practical applications owing to a number of significant advantages in comparison to other electrode materials, including easy processability, nontoxicity, high chemical stability, low density, large electrical conductivity, large surface area, and relatively low cost. Among various electrolytes for EDLCs, room-temperature ionic liquids (RTILs) are widely adopted as working electrolytes because of their large electrochemical windows, excellent thermal stability, and nonvolatility.10,11 © XXXX American Chemical Society

Received: January 28, 2016 Revised: April 5, 2016

A

DOI: 10.1021/acs.jpcc.6b00964 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

tigations in this work are focused on a generic pore model that intends to capture the essential features of realistic porous electrodes shown as Figure 1b. Specifically, we consider EDL behavior in individual spherical shells over broad ranges of inner radius R and thickness D (Figure 1c). To investigate the joint effects of pore size and curvature, we apply a positive potential of 1.5 V to the confining walls of the spherical shell. The symmetric setup for our model EDLC system implies that the negative electrode is at −1.5 V relative to the bulk electrolyte, leading to an overall voltage window of 3.0 V for the entire supercapacitor. This value is close to commonly applied voltages for EDLCs with RTILs.32,33 As in previous work,34,35 we use a restricted primitive model to represent RTILs inside nanopores. Whereas the primitive model ignores the chemical details, it accounts for electrostatic correlations and ionic excluded volume effects important for EDLC performance but neglected in conventional EDL theories. In this model, cations and anions are charged hard spheres of the same size (σ = 0.5 nm) with equal but opposite valence (Zi = ±1). Model parameters are selected to approximately match the charge and the molar volume of 1ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide (EMIM-TFSI), an ionic liquid often used in electrochemical devices. At 298 K and 1 bar, the molar volume of EMI-TFSI is 259 cm3/mol, corresponding to a reduced number density ρi0σ3 = 0.29 for both cations and anions.23 With these assumptions, the pair potential between ionic species is given by

question is whether this behavior is generally valid, given the slit-pore model or solid particles used in theoretical calculations and the diversity of pore structure for realistic carbon electrodes. Specially, would the shape of the C−V curves be different if one considered concurrently pore size as well as curvature of the electrolyte−electrode interfaces? How does the pore structure affect capacitance dependence on pore size? To address these and other questions, we propose in this work a generic model to represent both pore size and curvature of carbon electrodes by use of classical density functional theory (CDFT).30,31 CDFT is an ideal computational tool for examining pore size and geometry effects, as it is computationally efficient and applicable over a wide range of pore sizes, ranging from that below ionic dimensionality to mesoscopic scales.

2. MODELS AND METHODS 2.1. Electrode and Electrolyte Models. In this work, we use spherical shells of different radii and thicknesses to describe both pore size and curvature effects in porous carbon electrodes. The spherical shell model intends to capture the essential features of pore size and curvature effects on ionic distributions in porous electrodes at equilibrium. Unlike slit or cylindrical pore models conventionally used to represent porous materials, the spherical shell model allows us to account for not only pore size and curvature effects but also both convex and concave shapes of the electrolyte/electrode interfaces that are commonplace for realistic electrode materials. To our knowledge, the joint effects of pore size and curvature have not been systematically studied before. Figure 1a shows a schematic setup for a supercapacitor whereby porous carbons are used for both positive and negative

⎧ for r < σ ⎪∞ uij(r ) = ⎨ 2 ⎪ ⎩ ZiZje /4πε0r for r ≥ σ

(1)

where r is the center-to-center distance between ions, e is the elementary charge, and ε0 is the permittivity of free space. For ionic species in a spherical shell of inner radius R and thickness D, the confinement effect is represented by an external potential: σ σ ⎧ ⎪∞ for r < R + 2 or r > R + D − 2 Vi (r ) = ⎨ ⎪ 0 for R + σ ≤ r ≤ R + D − σ ⎩ 2 2

(2)

where r is radial distance from the spherical center. In the presence of an electrode potential, the exchange of cations and anions with the reservoir electrolyte can be described with the grand canonical ensemble. As explained below, the equilibrium properties of the EDL can be calculated from the ion distributions. 2.2. Classical Density Functional Theory. In this work, we are interested in understanding EDL behavior in realistic carbon electrodes for which a broad range of length scales are needed yet atomistic details are not available. Classical density functional theory (CDFT) is used to calculate ion distributions within the spherical shell and subsequently the capacitance of EDLCs.30,31 In comparison to alternative methods, here CDFT is a particularly good choice, not only because it is computationally much more efficient than simulation methods, such as molecular dynamics, but also because CDFT has been a standard tool for characterization of pore-size distribution for porous materials. Given temperature T and a set of ion concentrations in the bulk, ρi0, CDFT predicts ion distributions inside the pore:

Figure 1. (a) Schematic setup for a supercapacitor with two identical porous electrodes and symmetric positive and negative ions. (b) Representative microscopic structure for electrode. (c) Spherical shell model for nanopores.

electrodes. The electrodes are submerged in a room-temperature ionic liquid (RTIL) and separated by a permeable membrane that prevents electrical short circuit while permitting ion transport. For simplicity, we assume that the supercapacitor is perfectly symmetric in terms of both electrode and electrolyte; that is, cathode and anode are made of the same porous material, and cations and anions are identical other than the opposite signs for their electric charges. In addition, we assume that the carbon materials are ideally polarizable and that the confined electrolytes are in equilibrium with a bulk electrolyte away from the electrodes. Whereas disordered porous carbons have a complex poresize distribution and diverse pore shapes, theoretical invesB

DOI: 10.1021/acs.jpcc.6b00964 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C ρi (r ) = ρi 0 exp[−βVi (r ) − βZi eψ (r ) − β Δμi ex (r )]

(3)

where β = 1/kBT and kB is the Boltzmann constant. Electrical potential, ψ(r), is related to local charge density by the Poisson equation: 1 ∂ ⎡ 2 ∂ψ (r ) ⎤ e ⎢r ⎥=− 2 ∂r ⎦ ε0εbulk r ∂r ⎣

∑ Zi ρi (r) i

(4)

Equations 3 and 4 are solved self-consistently with the following boundary conditions for the electrical potential: ψ (R ) = ψ (R + D) = ψ0

(5)

The electrode potential, ψ0, is assumed to be constant at both surfaces of the spherical shell. Theoretical details in CDFT calculations are reflected in the last term on the right side of eq 3, Δμiex(r), which accounts for electrostatic correlations and ionic excluded volume effects. In this work, the former is represented by a quadratic expansion of excess Helmholtz energy with respect to that of a bulk system,36 and the latter is described by modified fundamental measure theory.37 Electronic polarization of ionic liquids was not accounted for in this work but can be incorporated by use of direct correlation functions for polarizable ions. Because EDL capacitance is mainly affiliated with inhomogeneous ionic distributions due to the electrostatic charge, we expect that inclusion of ion polarizability will not change the major conclusions. If Δμiex(r) = 0, eqs 3 and 4 reduce to the conventional Poisson−Boltzmann equations. To solve eqs 3 and 4 numerically, we start with an initial guess of the ionic density profiles, ρi(r). Local excess chemical potential for each ionic species, Δμiex(r), and local electrical potential, ψ(r), are then calculated from the initial density profiles. Subsequently, a new set of ionic density profiles are obtained from eq 3, and the procedure repeats until convergence. From the ion distributions and electrical potential profiles, we can readily calculate the surface charge density (Q) for both sides of the spherical shell and the individual EDL capacitances from Gauss’ law: Q R = −ε0εr

d ψ (r ) dr

Q R + D = ε0εr

d ψ (r ) dr

r=R

r=R+D

Figure 2. EDL capacitance vs pore thickness (D) at convex and concave interfaces of a spherical shell. Panels a−d correspond to spherical shells of different inner radii (R). When R is sufficiently large, the EDLs in a spherical shell become identical to those in a slit pore.

50% for the smallest pore at R = 1 nm, in comparison to that of a slit pore of the same width. As discussed below, a smaller surface area results in a stronger electric field and leads to integral capacitance at the inner surface (convex) systematically larger than that at the outer surface of the spherical shell (concave). Such differences vanish as the inner core radius approaches infinity. Oscillatory variation of the integral capacitance is closely affiliated with the layering structures of ion distributions inside the nanopores. To illustrate, we show in Figure 3 the local number densities of cations and anions in spherical shells of different radii while the pore size is fixed at D = 4 nm. Layer-bylayer distributions of cations and anions are evident near the charged surfaces. As for ionic liquids in slit pores, shown in Figure 3a, the interference of charged layers is responsible for

(6)

(7)

The integral capacitance at each surface is calculated from the ratio of surface charge to surface electric potential: C = Q/ψ0.

3. RESULTS AND DISCUSSION We consider first the effect of pore size and curvature on integral capacitance (C) of EDLs in spherical shells. Figure 2 shows integral capacitance as a function of pore width D at different inner core radii (R). In all cases, the surface electrical potential is fixed at ψ0 = 1.5 V. As observed in previous work for an ionic liquid in slit pores,34 the EDLs at both convex and concave surfaces exhibit an oscillatory dependence of integral capacitance on pore size when the pore thickness is comparable to the ion diameter in Figure 2a. The distance between neighboring peaks (or valleys) is approximately twice the ion diameter. While the curvature has little effect on EDL capacitance at the concave interface shown in Figure 2b−d, we see a significant increase in integral capacitance, up to about

Figure 3. Density profiles of cations and anions inside a spherical shell with thickness D = 4.0 nm and surface electrical potential ψ0 = 1.5 V. (a) R = ∞, that is, slit pore; (b) R = 5.0 nm; (c) R = 2.5 nm; (d) R = 1.0 nm. The positively charged surfaces are positioned at r − R = 0 and r − R = 4 nm, respectively. Here reduced number density is defined as ρiσ3, where ρi and σ are the number density and diameter of ion i. C

DOI: 10.1021/acs.jpcc.6b00964 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C the oscillatory dependence of capacitance on pore size.34 Figure 3b−d shows that the contact densities of counterions at convex surfaces are slightly larger than these at concave surfaces, but both increase as the inner radius R is reduced from infinity to 1.0 nm. The difference in contact densities is also related to electrical field or equivalently charge densities at inner and outer surfaces. To provide further insight into the capacitance difference between convex and concave EDLs, we evaluate charge densities at the surface regions numerically: q in ≈ −

1 4πR2

qout ≈ −

∫R

R+δ

4πr 2ρe (r ) dr

1 4π (R + D)2

(8)

R+D

∫R+D−δ 4πr 2ρe (r) dr

(9)

where δ = 0.5 nm and ρe(r) is local charge density. Figure 4 shows surface charge densities for spherical shells of different

Figure 5. Integral capacitance for (a) convex and (b) concave interface of a spherical shell at a fixed surface electric potential, ψ0 = 1.5 V.

Figure 4. Curvature efffect on the surface charge density of a spherical shell with fixed pore thickness D = 4.0 nm and surface electrical potential ψ0 = 1.5 V.

inner radii but the same pore size, D = 4.0 nm. Clearly, charge densities at both convex and concave surfaces increase with curvature (reducing inner radius R), suggesting more counterions near the surface. Increased surface charge density also explains the curvature effects as shown in Figure 2. Because the charge density of the convex interface increases greatly as the inner radius of the spherical shell become smaller, we expect larger EDL capacitance at the inner surface and subsequently a stronger oscillatory dependence on pore size (Figure 5a). Conversely, the charge density of the concave interface is relatively insensitive to curvature, yielding a capacitance almost unchanged with the inner core radius of the spherical shell (Figure 5b). Figure 6a presents the overall integral capacitance of the spherical shell, calculated from those corresponding to both convex and concave surfaces: CT =

C i R2 + Co(R + D)2 R2 + (R + D)2

(10)

where subscripts i and o denote the inner and outer surface of the spherical shell, respectively. Equation 10 is equally applicable to the overall differential capacitance. As inner radius R decreases, overall capacitance increases significantly. This is also expected from the increased EDL capacitance at both inner and outer surfaces. The oscillatory dependence of overall capacitance on pore size is consistent with those corresponding to individual EDLs. Figure 6b shows average

Figure 6. (a) Overall integral capacitance vs pore size for spherical shells of different inner radii. (b) Average densities of counterions (solid lines) and co-ions (dashed lines) inside the spherical shell at different pore thicknesses. The surface electrical potential is ψ0 = 1.5 V.

densities of counterions and anions inside the spherical shells. As pore size falls, the EDLs have a smaller influence on overall ion distributions inside the pore, leading to a diminishing D

DOI: 10.1021/acs.jpcc.6b00964 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C difference in average counterion and co-ion densities. On the other hand, a smaller inner core radius results in more couterions in the pore and thus a larger capacitance. We have also calculated the differential capacitance Cd of the EDLs by a derivative of surface charge density with respect to surface potential. Figure 7 shows both positive and negative

Figure 8. (a) Overall differential capacitance vs surface potential for spherical shells of different radii. (b) Average densities of counterions (solid lines) and co-ions (dashed lines) inside the spherical shell at different surface potentials. In all cases, pore size is D = 4.0 nm.

relative densities of both cations and anions versus surface electrical potential. As electrode potential increases from 0 to 1.5 V, the average density of counterions increases greatly inside the micropore, whereas that of co-ions declines monotonically. At the same surface potential, the average coion density rises as the inner radius of the spherical shell increases. However, the curvature effect is slightly more complicated for the average density of counterions. At low electrical potential, ion distributions are strongly influenced by ionic excluded volume effects, leading to a slight increase in average counterion density as the inner radius of the spherical shell increases, similar to that for co-ions. When the surface electrical potential is sufficiently large, however, the counterion distribution is mostly influenced by surface charge density. As a result, average counterion density increases with pore curvature. It should be noted that the curvature effects discussed in this work are qualitatively different from those identified in earlier work for EDLs at the outer surfaces of spherical or cylindrical particles.16 According to our spherical shell model, the curvature effect is manifested primarily at the convex interface of electrolyte and electrode, not much on the EDL capacitance at the concave interface. The theoretical prediction implies that the cylindrical pore model commonly used in materials characterization is not able to capture the curvature effects of realistic porous electrodes, which have extensive edges and rough surfaces in addition to diverse distributions of pore size and geometry. While an increase in EDL capacitance with curvature holds true for both the convex interface of a spherical shell and spherical/cylindrical particle, its response to the electrical potential is distinctively different. As shown in Figures 7 and 8, the spherical shell model predicts a bell-shaped curve for EDL capacitance versus electrical potential, regardless of curvature. However, the particle model predicts a flat

Figure 7. Curvature effect on differential capacitance of (a) convex and (b) concave interface of a spherical shell. Pore size is D = 4 nm.

branches of the differential capacitance of convex (Figure 7a) and concave (Figure 7b) surfaces versus surface potential. In both cases, the inner core radius is changed from 1 nm to the planar limit while pore size is fixed at D = 4 nm. Due to strong excluded volume effects, the differential capacitance shows a maximum at zero surface electrical potential for both convex and concave EDLs, leading to the so-called “bell-shaped” Cd−V curve.38−41 In comparison to that for the slit pore, capacitances of the convex interfaces are noticeably increased with curvature, in particular near the potential of zero charge. For instance, Cd for R = 1 nm is almost twice the slit pore value. While the increase in capacitance with curvature has been reported before, the shape of Cd−V curves shown in Figure 7 is distinctively different from those for a rough planar surface or onionlike carbon electrodes.17,42 Because previous work did not consider the confinement and pore-size effects or the overlap of EDLs, a nearly flat shape of differential capacitance versus potential was predicted. In stark contrast to curvature effect on differential capacitance of the convex EDL, curvature has negligible effects on differential capacitance of concave surfaces. Figure 8a shows overall differential capacitance versus surface potential for spherical shells of different inner radii. As discussed above, the increase in overall differential capacitance with curvature can be attributed to higher surface charge density and subsequently an increased number density of counterions near curved surfaces. To examine the curvature effect for the entire voltage range, we plot in Figure 8b average E

DOI: 10.1021/acs.jpcc.6b00964 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

(2) Burt, R.; Birkett, G.; Zhao, X. A Review of Molecular Modelling of Electric Double Layer Capacitors. Phys. Chem. Chem. Phys. 2014, 16, 6519−6538. (3) Zhong, C.; Deng, Y.; Hu, W.; Qiao, J.; Zhang, L.; Zhang, J. A Review of Electrolyte Materials and Compositions for Electrochemical Supercapacitors. Chem. Soc. Rev. 2015, 44, 7484−7539. (4) Shao, Y.; El-Kady, M. F.; Wang, L. J.; Zhang, Q.; Li, Y.; Wang, H.; Mousavi, M. F.; Kaner, R. B. Graphene-Based Materials for Flexible Supercapacitors. Chem. Soc. Rev. 2015, 44, 3639−3665. (5) Vatamanu, J.; Bedrov, D. Capacitive Energy Storage: Current and Future Challenges. J. Phys. Chem. Lett. 2015, 6, 3594−3609. (6) Wu, P.; Huang, J. S.; Meunier, V.; Sumpter, B. G.; Qiao, R. Complex Capacitance Scaling in Ionic Liquids-Filled Nanopores. ACS Nano 2011, 5, 9044−9051. (7) Kondrat, S.; Wu, P.; Qiao, R.; Kornyshev, A. A. Accelerating Charging Dynamics in Subnanometre Pores. Nat. Mater. 2014, 13, 387−393. (8) He, Y.; Huang, J.; Sumpter, B. G.; Kornyshev, A. A.; Qiao, R. Dynamic Charge Storage in Ionic Liquids-Filled Nanopores: Insight from a Computational Cyclic Voltammetry Study. J. Phys. Chem. Lett. 2015, 6, 22−30. (9) Frackowiak, E.; Abbas, Q.; Beguin, F. Carbon/Carbon Supercapacitors. J. Energy Chem. 2013, 22, 226−240. (10) Fedorov, M. V.; Kornyshev, A. A. Ionic Liquids at Electrified Interfaces. Chem. Rev. 2014, 114, 2978−3036. (11) Hayes, R.; Warr, G. G.; Atkin, R. Structure and Nanostructure in Ionic Liquids. Chem. Rev. 2015, 115, 6357−6426. (12) 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. (13) Palmer, J. C.; Llobet, A.; Yeon, S. H.; Fischer, J. E.; Shi, Y.; Gogotsi, Y.; Gubbins, K. E. Modeling the Structural Evolution of Carbide-Derived Carbons Using Quenched Molecular Dynamics. Carbon 2010, 48, 1116−1123. (14) Meunier, V.; Huang, J. S.; Feng, G.; Qiao, R.; Sumpter, B. G. Modern Theories of Carbon-Based Electrochemical Capacitors: A Short Review. Proceedings of the Asme International Mechanical Proc. ASME Int. Mech. Eng. Congr. Expo.–2010 2010, 11, 21−30. (15) Huang, J.; Sumpter, B. G.; Meunier, V. Theoretical Model for Nanoporous Carbon Supercapacitors. Angew. Chem., Int. Ed. 2008, 47, 520−524. (16) Huang, J.; Sumpter, B. G.; Meunier, V.; Yushin, G.; Portet, C.; Gogotsi, Y. Curvature Effects in Carbon Nanomaterials: Exohedral Versus Endohedral Supercapacitors. J. Mater. Res. 2010, 25, 1525− 1531. (17) Feng, G.; Li, S.; Presser, V.; Cummings, P. T. Molecular Insights into Carbon Supercapacitors Based on Room-Temperature Ionic Liquids. J. Phys. Chem. Lett. 2013, 4, 3367−3376. (18) Landers, J.; Gor, G. Y.; Neimark, A. V. Density Functional Theory Methods for Characterization of Porous Materials. Colloids Surf., A 2013, 437, 3−32. (19) Li, S.; Han, K. S.; Feng, G.; Hagaman, E. W.; Vlcek, L.; Cummings, P. T. Dynamic and Structural Properties of RoomTemperature Ionic Liquids near Silica and Carbon Surfaces. Langmuir 2013, 29, 9744−9749. (20) Van Aken, K. L.; McDonough, J. K.; Li, S.; Feng, G.; Chathoth, S. M.; Mamontov, E.; Fulvio, P. F.; Cummings, P. T.; Dai, S.; Gogotsi, Y. Effect of Cation on Diffusion Coefficient of Ionic Liquids at OnionLike Carbon Electrodes. J. Phys.: Condens. Matter 2014, 26, No. 284104. (21) Wang, H.; Pilon, L. Mesoscale Modeling of Electric Double Layer Capacitors with Three-Dimensional Ordered Structures. J. Power Sources 2013, 221, 252−260. (22) 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.

capacitance−potential curve when the particle size is comparable to ion diameter (i.e., when the curvature is large).42

4. CONCLUSIONS We have introduced a spherical shell model to represent the essential feature of micropores of carbon electrodes that have been extensively used for electric-double-layer capacitors (EDLCs). In comparison to conventional models of pore characterization, the spherical shell model allows us to describe pore size and curvature effects together. To the best of our knowledge, this work represents the first theoretical study of joint effects of pore size and curvature that are important for understanding the performance of realistic EDLCs. We investigated systematically the influences of pore size and curvature on EDL structures and capacitances using classical density functional theory (CDFT) within the primitive model of room-temperature ionic liquids (RTILs). Theoretical results indicate that overall capacitances of spherical shells noticeably increase as pore curvature rises, and the curvature effect is mainly manifested in the convex interface. EDL capacitance at the concave interface is little influenced by pore size. As observed for RTILs in slit pores, CDFT predicts oscillatory variation of capacitance with pore thickness when it is comparable to ion diameter. The pore thickness effect mainly arises from the interference of EDLs and is further enhanced when the pore curvature increases. In summary, this work illustrates curvature and pore-size effects of realistic porous electrodes and suggests the significant role of convex surfaces for the synthesis of new porous electrodes to optimize EDLC performance. In particular, the spherical shell model provides a simple yet generic description of both pore size and curvature, opening up a new dimension to characterize nanoporous materials and quantify their performance for diverse applications including EDL capacitors. Although the theoretical method has been validated with simulation results in previous publications, it will be interesting to see whether specific predictions from this work can be reproduced by molecular dynamics simulation or experiments.

■

AUTHOR INFORMATION

Corresponding Author

*Telephone (951) 827-2413; e-mail [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS C.L. is grateful to the Chinese Scholarship Council for a visiting fellowship. This research was sponsored by the Fluid Interface Reactions, Structures and Transport (FIRST) Center, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences. C.L. and H.L. also appreciate financial support by the National Natural Science Foundation of China (91334203, 21376074) and the 111 Project of China (B08021). Numerical calculations were performed at the National Energy Research Scientific Computing Center (NERSC).

■

REFERENCES

(1) Simon, P.; Gogotsi, Y. Capacitive Energy Storage in Nanostructured Carbon-Electrolyte Systems. Acc. Chem. Res. 2013, 46, 1094−1103. F

DOI: 10.1021/acs.jpcc.6b00964 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C (23) 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. (24) 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. (25) 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. (26) 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. (27) Shim, Y.; Kim, H. J. Nanoporous Carbon Supercapacitors in an Ionic Liquid: A Computer Simulation Study. ACS Nano 2010, 4, 2345−2355. (28) 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. (29) Wang, D.; Li, Y.; Li Puma, G.; Wang, C.; Wang, P.; Zhang, W.; Wang, Q. Dye-Sensitized Photoelectrochemical Cell on Plasmonic Ag/ Agcl@ Chiral Tio 2 Nanofibers for Treatment of Urban Wastewater Effluents, with Simultaneous Production of Hydrogen and Electricity. Appl. Catal., B 2015, 168-169, 25−32. (30) Wu, J. Z. Density Functional Theory for Chemical Engineering: From Capillarity to Soft Materials. AIChE J. 2006, 52, 1169−1193. (31) Wu, J. Z.; Li, Z. D. Density-Functional Theory for Complex Fluids. Annu. Rev. Phys. Chem. 2007, 58, 85−112. (32) Jiang, D.-e.; Wu, J. Unusual Effects of Solvent Polarity on Capacitance for Organic Electrolytes in a Nanoporous Electrode. Nanoscale 2014, 6, 5545−5550. (33) Van Aken, K. L.; Beidaghi, M.; Gogotsi, Y. Formulation of IonicLiquid Electrolyte to Expand the Voltage Window of Supercapacitors. Angew. Chem., Int. Ed. 2015, 54, 4806−4809. (34) Jiang, D.-e.; Jin, Z.; Wu, J. Oscillation of Capacitance inside Nanopores. Nano Lett. 2011, 11, 5373−5377. (35) Jiang, D.-e.; Wu, J. Microscopic Insights into the Electrochemical Behavior of Nonaqueous Electrolytes in Electric DoubleLayer Capacitors. J. Phys. Chem. Lett. 2013, 4, 1260−1267. (36) Li, Z. D.; Wu, J. Z. Density Functional Theory for Planar Electric Double Layers: Closing the Gap between Simple and Polyelectrolytes. J. Phys. Chem. B 2006, 110, 7473−7484. (37) Yu, Y.-X.; Wu, J. Structures of Hard-Sphere Fluids from a Modified Fundamental-Measure Theory. J. Chem. Phys. 2002, 117, 10156−10164. (38) Kornyshev, A. A. Double-Layer in Ionic Liquids: Paradigm Change? J. Phys. Chem. B 2007, 111, 5545−5557. (39) Bazant, M. Z.; Storey, B. D.; Kornyshev, A. A. Double Layer in Ionic Liquids: Overscreening Versus Crowding. Phys. Rev. Lett. 2011, 106, No. 046102. (40) 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. (41) Jiang, D.-e.; Meng, D.; Wu, J. Density Functional Theory for Differential Capacitance of Planar Electric Double Layers in Ionic Liquids. Chem. Phys. Lett. 2011, 504, 153−158. (42) 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.

G

DOI: 10.1021/acs.jpcc.6b00964 J. Phys. Chem. C XXXX, XXX, XXX−XXX