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October 21, 2011

C 2011 American Chemical Society

Complex Capacitance Scaling in IonicLiquids-Filled NanoporesPeng Wu,† Jingsong Huang,‡ Vincent Meunier,§ Bobby G. Sumpter,‡ and Rui Qiao†,*

†Department of Mechanical Engineering, Clemson University, Clemson, South Carolina 29634-0921, United States, ‡Center for Nanophase Materials Sciences andComputer Science and Mathematics Division, Oak Ridge National Laboratory, Bethel Valley Road, Oak Ridge, Tennessee 37831-6367, United States, and§Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York 12180-3590, United States

Supercapacitors store electrical energyphysically in electrical double layers (EDLs)at the electrode/electrolyte interface.1

Despite their high power density and extra-ordinary cyclability, the widespread deploy-ment of supercapacitors is limited by theirmoderate energy density. The current surgein interest in supercapacitors is driven byrecent breakthroughs in developing novelelectrode materials and electrolytes thatpromise to significantly improve their en-ergy density.2�6 In particular, a group ofexperiments showed that the area-normal-ized capacitance (or specific capacitance,hereafter referred to as capacitance) of sub-nanometer pores filled with organic orroom-temperature ionic liquid (RTIL) elec-trolytes increases anomalously as the porewidth decreases.6�8 These findings chal-lenge the long-held presumption that sub-nanometer pores are too small to contributeto energy storage, and suggest that non-classical behaviors of EDLs emerge in sub-nanometer pores.The validity of the anomalous capaci-

tance scaling in subnanometer pores hasbeen challenged by recent experimentalstudies,9 in which the capacitances werefound to be nearly pore-width independentfor pores from 0.7 to 15 nm. The sharpcontrast in the observed capacitance scal-ings was ascribed to the discrepancy in thespecific surface area (used for determiningthe specific capacitance) probed by differ-entmethods.9 The debate about anomalouscapacitance is in part fueled by the absenceof a complete theoretical description of allthe intricate processes taking place at thenanoscale. Along that front, the anomalouscapacitance enhancement in subnanometerpores was first described using a heuristicmodel that assumes that counterions forma wire in the center of a cylindrical pore (ora single layer in the center of a slit-shapedpore).10,11 Among the recent atomistic

simulations of EDLs in nanopores,12�15

molecular dynamics (MD) simulations ofcarbon nanotubes (CNTs) filled with RTILshave shown a capacitance increase withdecreasing CNT diameter in the range of2.0 to 0.9 nm.16 However, the polarizabilityof the nanotube wall was neglected and thecomputed capacitances were ∼10 foldsmaller than experimental values. Recently, amodel that takes into account thepolarizability

* Address correspondence torqiao@clemson.edu.

Received for review August 24, 2011and accepted October 21, 2011.

Published online10.1021/nn203260w

ABSTRACT

Recent experiments have shown that the capacitance of subnanometer pores increases

anomalously as the pore width decreases, thereby opening a new avenue for developing

supercapacitors with enhanced energy density. However, this behavior is still subject to some

controversy since its physical origins are not well understood. Using atomistic simulations, we

show that the capacitance of slit-shaped nanopores in contact with room-temperature ionic

liquids exhibits a U-shaped scaling behavior in pores with widths from 0.75 to 1.26 nm. The left

branch of the capacitance scaling curve directly corresponds to the anomalous capacitance

increase and thus reproduces the experimental observations. The right branch of the curve

indirectly agrees with experimental findings that so far have received little attention. The

overall U-shaped scaling behavior provides insights on the origins of the difficulty in

experimentally observing the pore-width-dependent capacitance. We establish a theoretical

framework for understanding the capacitance of electrical double layers in nanopores and

provide mechanistic details into the origins of the observed scaling behavior. The framework

highlights the critical role of “ion solvation” in controlling pore capacitance and the

importance of choosing anion/cation couples carefully for optimal energy storage in a given

pore system.

KEYWORDS: supercapacitor . electrical double layer . room-temperature ionicliquids . anomalous enhancement . nanopores . transmission line model

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of a pore wall showed that the screening of electro-static interactions and the negative self-energy of ionsin pores lead to a superionic state.17 This model andsubsequent Monte Carlo (MC) simulations18 also re-produced the anomalous capacitance enhancementin subnanometer pores observed experimentally. De-spite this agreement, an important difference exists:comparison was made between the experimentallymeasured integral capacitance Cint at a potential dropof φw from an electrode to bulk electrolyte and thetheoretically predicted differential capacitance Cdiff atφw = 0 V. Using the data presented in the MC simu-lations,18 one can show that, for a φw = 1.5 V (which canbe justified, assuming a symmetric capacitor, for anexperimental potential drop of 3 V between the twoelectrodes7), Cint increases as the pore width decreasesfrom 1.2 to 1.0 nm but decreases as the pore widthfurther decreases, which disagrees with the experi-mental trend. The experimental trend of Cint is recov-ered if φw < 1.0 V.In this study we use MD simulations to investigate

EDLs in slit-shaped nanopores with polarizable walls inequilibrium with RTILs. Our simulations reproduce theanomalous capacitance enhancement when the porewidth reduces below a critical value, but also revealthat capacitance increases as the pore width increasesbeyond another critical value. We provide mechanisticinsights into these scaling behaviors by delineating therelation between the structure and the capacitance ofthe EDLs in these pores. We show that the EDLs insidesubnanometer pores filled with RTILs represent a newcapacitor regime in which the ubiquitously used trans-mission-line model for supercapacitors fails qualitatively.

RESULTS AND DISCUSSION

Figure 1a shows a schematic of the MD system thatconsists of a slit-shaped pore in contact with a reservoirof RTILs. The entire electrode surface (the horizontalpore walls and the vertical walls blocking the RTILreservoir) is modeled as an equi-potential surface witha net charge. The magnitude of surface charge andelectrical potential of the pore walls was not directlycontrolled, but determined from the ion distributionobtained in the MD simulations (see Methods). Ionswere modeled using semicoarse grained models that(1) resolve their geometrical anisotropy and van der

Waals interactions, which are important in determiningEDL capacitance,19 and (2) assign the charge of eachion to one of its atoms (Figure 1b and Methods). Thesemodels were adopted in the spirit of the hard spheremodel18 which has been shown to capture key featuresof EDLs in nanopores. Simulations in which the chargeof each ion was distributed on its multiple atomsyielded qualitatively similar results, in agreement withour prior finding that charge delocalization plays a rela-tively unimportant role in determining the thermo-dynamic properties of EDLs.19 We note that, however,charge delocalization plays a key role in determiningdynamics of RTILs. Hence caution must be used if thepresent RTIL model is to be used to study EDL dynamics.To compute the integral capacitance, two systems

were simulated for each pore width: in a quasi-poten-tial of zero charge (qPZC) system, the net charge of theentire electrode was zero; in a charged (CHG) system,the net charge of the entire electrodewas positive, andthe anion sketched in Figure 1b serves as the counter-ion. The potentials of the pore wall with respect to theRTIL reservoir in these two systemswere determined tobe φqPZC and φCHG, respectively. The surface chargedensities of the pore wall were determined as σqPZCand σCHG, respectively. φqPZC, φCHG, σqPZC, and σCHG forthe various pores studied are summarized in Table 1. Inthe qPZC systems, σqPZC deviates slightly from zero dueto the different affinity of ions to the horizontal porewall and the open vertical walls. The integral capaci-tance of each porewas computed using Cint =Δσ/Δφ,

Figure 1. (a) A schematic of the simulation system featuring a slit-shaped nanopore in equilibrium with a RTIL reservoir. Theelectrode features an equi-potential surface that carries a net charge that is balanced by ions inside the simulation box(denoted by the red dashed line). (b) Schematics of the semicoarse grainedmodel for the cation and the anion. The red spherein the cation and the green sphere in the anion denote the atoms which carry charges of þe and �e, respectively.

TABLE 1. Surface Charge Density (σ) and Electrical

Potential (O) in the qPZC and CHG Systems for Various

Nanopores. The Electrical Potential in Bulk RTILs Is Taken

as Zero

qPZC system CHG system

pore width W (nm) σqPZC (C/m2) φqPZC (V) σCHG (C/m

2) φCHG (V) Δφ (V)

0.75 �0.007 0.117 0.172 1.693 1.5760.78 �0.005 0.010 0.150 1.602 1.5920.84 �0.003 �0.035 0.118 1.735 1.7710.91 �0.001 �0.138 0.108 1.584 1.7211.04 0.006 �0.266 0.115 1.445 1.7111.12 0.007 �0.283 0.121 1.373 1.6561.17 �0.002 �0.084 0.148 1.672 1.7571.26 �0.005 �0.028 0.157 1.621 1.649

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whereΔσ= σCHG� σqPZC is the increase of pore surfacecharge density as the pore is electrified byΔφ= φCHG�φqPZC volts. Δφ of all pores studied is within 0.1 V fromthe target value of 1.68 V (cf. Table 1), which is similar tothe experimental potential.7 Since σCHG .σqPZC andσqPZC ≈ 0 in all pores, the computed capacitanceapproximates well the integral capacitance definedby Cint = σCHG/(φCHG � φPZC),

21 where φPZC is thesurface potential of a neutral pore.

Capacitance Scaling. Figure 2a shows the variation ofcapacitance (Cint) with the center-to-centerwidth (W) ofthe pores, where a U-shaped Cint�W curve is observed.The trend revealed by the left branch of the Cint�W

curve agrees with that reported experimentally (seeFigure 2b): it simultaneously captures the sharp in-crease of Cint asW decreases from 0.91 to 0.75 nm andits slight change asW decreases from 1.12 to 0.91 nm;the latter was not found by a prior analytical model.10

We note that the pore width reported in ref 7 is mostlikely the “accessible width”, which is about one vanderWaals diameter (∼0.3 nm) smaller than the “center-to-center width” used in Figure 2a. If Cint in Figure 2a isplotted against the “accessible width”, it would shifttoward a smaller pore width by ∼0.3 nm, suggestingthat the capacitance enhancement would appear in anarrower pore width range (0.45�0.61 nm) than ex-periments (0.7�0.8 nm).7 Such a difference can beattributed to the fact that the counterion in oursimulation (∼0.52 nm in diameter) is smaller than thatused in the experiments (the largest dimension of bothanions and cations is ∼0.8 nm).7 Nevertheless, the trendof the capacitance scaling, the focus of the present study,bears a close resemblance to the experimental one.

The sharp increase of Cint withW ranging from 1.12to 1.26 nmhas not been directly reported experimentallybut is indirectly supported by measurements. Figure 2b

shows that the experimental capacitance of a 1.1 nmpore is ∼7 μF/cm2,7 whereas the experimental capaci-tance of planar electrodes made of glassy carbon in thesame electrolyte is ∼12 μF/cm2.20 This difference im-plies that the capacitance of pores larger than 1.1 nmshould increase toward the planar surface value withincreasingW. Figure 2a does show that the capacitanceincreases with increasing pore width but it does notapproachC¥ in anasymptoticway since the capacitancein a 1.26 nmpore is higher thanC¥. This suggests that asthe pore width further increases, capacitance will de-crease again and eventually reachC¥. HowC¥ is reachedasW increases, for example, by a damped oscillation orby a smooth decay, is an interesting question, and maybe pursued in future studies. However, we expect thecapacitance will not vary significantly in wider poresbased on the insights obtained from the analysis below.Specifically, as will be shown later, the pore capacitanceis controlled primarily by the changeof “ion solvation” inpores as the pore becomes electrified (the “ion solva-tion” in RTILs refers to the electrostatic and van derWaals interactions between a cation and its surroundinganions (and vice versa). This is qualitatively similar toorganic electrolytes consisting of ions solvated by sol-vent molecules. The quotation marks are removed forlater usage of solvation for simplicity). When pore widthW increases, the solvation experienced by ions in differ-ent pores becomes increasingly similar. Hence thechange of their solvation upon pore electrification willnot differ significantly as W increases, subsequentlyresulting in a weaker dependence of capacitance onW. The capacitance scaling trend shown in Figure 2 wasalso observed in simulations in which the charge ofcation and anion was distributed among all their atoms,although the capacitance increases less sharply; e.g., thecapacitance enhancement is 20% less sharp comparedto data herein as pore size varies from 0.91 to 0.75nm.

The above results suggest that the dependence ofthe nanopore capacitance on the pore width is morecomplex than previously expected in experimental andtheoretical studies. In view of the ongoing debate onthe anomalous capacitance enhancement in subnan-ometer pores, our results strongly support the exis-tence of the anomalous capacitance enhancement.Additionally, the sharp increase of capacitance oververy narrow pore widths (0.75�0.91 nm and 1.12�1.26 nm) and theU-shapedCint�W curve forW<1.26nm(and the possibly oscillatory Cint�W curve for W >1.26 nm) indicate that, for the pore-width dependentcapacitance to be observed experimentally, it is essen-tial that the electrode materials used in experimentalcharacterization feature unimodal and narrow pore-width distribution. When electrodes with multimodaland/or wide pore-width distribution are used, the capa-citance variation in different pores can be averagedout. In this regard, we note that most of the electrodesused in ref 9, where the capacitance was found to be

Figure 2. (a) Variation of EDL capacitance from MD simula-tions in pores with center-to-center width from W = 0.75�1.26 nm. C¥ denotes the capacitance of an open planarelectrode. (b) Variation of EDL capacitance in pores filledwith EMI-TFSI electrolye reported in ref 7. The experimentalcapacitance, C¥,exp, was measured on planar glassy carbonelectrodes in the same electrolyte.20 The statistical errorfor the 1.17 nm pore is relatively large because ion diffusionin this pore is very slow (due to the nearly crystal-like packinginside the neutral pore), which renders simulation difficult.

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nearly pore-width independent, are based on activatedcarbons. Since such electrodes are prone to a relativelywide pore-width distribution, it is possible that theindependence of capacitance on pore width observedin those experiments is caused by the averaging effect.

EDL Structure Inside Pores. The capacitance scalingshown in Figure 2a is ultimately controlled by the diffe-rent EDL structures in various pores. Figure 3 shows thecation and anion density profiles across pores withW =0.75, 0.91, 1.12, and 1.26 nm in the qPZC system and inthe CHG system (density profiles for other pores aregiven in the Supporting Information). We observe that,when the pore is neutral or nearly neutral, a largeamount of ions accumulate inside the pore. In poreswith W = 0.75 and 0.78 nm, cations and anions form amonolayer in the pore center, due to the wall confine-ment. In wider pores, anions always form one layernear each wall, which is an expected behavior formolecules confined in pores wider than twice theirdiameter. The cation distribution, however, showsmore complex behavior: in pores with W = 0.84 and0.91 nm, they reside rather uniformly across the acces-sible width of the pore; in pores withW = 1.12 nm, theyaccumulate in a narrow band near the pore centralplane; in pores withW = 1.17 and 1.26 nm, they form a

distinct layer near each pore wall. The different cationdistributions in these pores are mainly driven by thetendency of cations to maximize their solvation byanions. We computed the solvation number of cationsacross these pores (i.e., the number of anions withinr1 from a cation, where r1 is the first local minimum ofthe cation�anion radial distribution function) andfound that its distribution profile resembles the cationdensity profile. The ion distribution profiles in electrifiedpores are simpler. In pores with W = 0.75 and 0.78 nm,limited amount of cations (co-ions) and mostly anions(counterions) form a monolayer in the pore center. Inwider pores, counterions form two distinct layers, whileco-ions accumulate mainly in the pore center.

The observed ion distributions are closely related tothe geometry of the ions used here. For example, inneutral pores withW = 1.17 and 1.26 nm, contact pairsof cations and anions are positioned side-by-sideacross the pore.22 Such crystal-like packing of cationsand anions results in strong solvation of these ions,which in turn leads to a distinct layer of cation/anionnear each pore wall and significantly slowed ion diffu-sion inside the pores (the diffusion coefficients of bothions are less than 1/10 of their bulk value). In neutralpores with W = 1.12 nm, however, the pores are too

Figure 3. The upper pannels show the ion density in the qPZC systems, and the lower panels show the ion density in the CHGsystems. The porewall surface charge density and surface potential for each system are shown in Table 1. The position of ionsis based on the location of their charged atoms.

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narrow to accommodate a contact pair of cation andanion positioned across the pore, and cations mustreside in the pore center to maximize their solvation.Since ions used in this study are representative of RTILsin which the cation shape is strongly anisotropic andthe anion is quasi-spherical, the results obtained hereare relevant to this broad class of RTILs.

Physical Origins of Capacitance Scaling in Nanopores. Togain mechanistic insights into the capacitance scalingbehavior shown in Figure 2a and its relation with theEDL structure shown in Figure 3, we derive a model forthe EDL capacitance. Consider a pore in equilibriumwith an RTIL reservoir. Under an electrifying potentialφ,thepore surfacesacquireachargedensityofσ (throughoutthis work, the electrical potential in reservoir is taken aszero). The electrochemical potential of an ion i insidethe pore, μi,pore(φ,σ), is the free energy cost for insertingthe ion into the pore, which is equal to its electro-chemical potential in the reservoir (μi,¥):

μi,pore(φ, σ) ¼ Ei f all(σ)þΔEall others(σ) � TΔS(σ)

¼ μi,¥ (1)

where the first term on the right-hand side is theinteraction energy between the inserted ion and allspecies in the pore including the walls, and (σ) is usedto emphasize that the configuration of all species(including ion i) used to determine Eifall correspondsto the state at which the pore has a surface chargedensity of σ; the second term is the change of interac-tion energy between all other species as the ion isinserted into the pore, and it includes the cavitationenergy cost when the ions around the inserted ion areseparated from each other to generate a cavity toaccommodate the inserted ion; the third term standsfor entropic effects. Splitting the energy between ion i

and all species into an electrostatic part Eifallelec (σ) and a

non-electrostatic part Eifallnon-elec(σ), we have

Eeleci f all(σ)þ Enon-eleci f all (σ)þΔEall others(σ) � TΔS(σ)¼ μi,¥ (2)

With the use of the superposition principle in electro-statics, the energy of an ion i due to its electrostaticinteractions with all species inside a polarizable slit-shaped pore with a surface potential of φ is

Eeleci f all(σ) ¼ qiφþ ∑j 6¼i

Eeleci f j(σ)þ Ei, self (σ) (3)

where qi is the charge of ion i, j is the index of all ions,Eifjelec(σ) is the electrostatic interaction energy betweenions i and j given by17

Eeleci f j(σ) ¼qiqj

πε0εrW∑¥

n¼ 1sin

nπziW

� �sin

nπzjW

� �K0

nπRijW

� �

(4)

where ε0 and εr are the vacuum permittivity and back-ground dielectric constant, respectively; zi and zj are

the distance of ion i and j from the lower pore wall,respectively; K0 is the zeroth order modified Besselfunction of the second kind, which decreases sharplyas its argument increases; Rij is the lateral ion�ionseparation, that is, separation between ions i and j indirection parallel to the pore wall; Ei,self is the self-energy of ion i, which depends on the pore width W

and ion's position inside the pore.17,23 Using eqs 2 and3, we obtain

φ ¼ [ μi,¥ � ∑j 6¼i

Eeleci f j(σ) � Ei, self (σ) � Enon-eleci f all (σ)

� ETS(σ)]=qi (5)

where ETS(σ) =ΔEall others(σ)� TΔS(σ). Applying eq 5 toan electrified pore (surface potential φ; surface chargedensity σ) and a neutral pore (surface potential φPZC)with the same width, and using Cint = σ/(φ � φPZC), weobtain

Cint ¼ �qiσ=[ΔEeleci f ions(0 f σ)þΔEnon-eleci f all (0 f σ)

þΔEi, self (0 f σ)þ (ETS(σ) � ETS(0))] (6)

where ΔEifallelec (0fσ) = ∑j6¼iEifj

elec(σ) � ∑j6¼iEifjelec(0) is the

change in electrostatic interaction energy of an ion i

with other ions as the pore surfaces acquire a chargedensityσ (note that the number of counter/co-ions andtheir positions inside the pore change accordinglyduring this process). ΔEifall

non-elec(0fσ) = Eifallnon-elec(σ) �

Eifallnon-elec(0) and ΔEi,self(0fσ) = Ei,self(σ) � Ei,self(0) arethe changes in non-electrostatic energy and in self-energy of an ion i as the pore surfaces acquire a chargedensity σ, respectively. The first three terms in thedenominator of eq 6 are expected to dominate overthe remaining terms that represent higher order inter-actions and entropic effects. ΔEi,self(0fσ) is small andcan be neglected. Consequently, for the scaling ofcapacitance, we have

Cint∼qi=[�ΔEeleci f ions(0 f σ)=σ �ΔEnon-eleci f all (0 f σ)=σ]

∼qi=[�ΔEtoti f all(0 f σ)=σ] (7)

where ΔEifalltot = ΔEifions

elec þ ΔEifallnon�elec. Equation 7

correlates the capacitance of EDLs in a pore with thevariation of ion energy (and ultimately the variation ofEDL structure) as the pore is electrified.

To understand the physical meaning of eq 7, weconsider a pore electrified by a positive potential φ.Here, each cation inside the pore is affected by twofactors when the pore becomes electrified: (1) itsenergy due to the interaction with the pore wallsincreases by qiφ, and (2) its energy due to the interac-tion with others species becomes more negative (andstabilized) when the pore acquires net negative spacecharge, that is,ΔEifall

tot < 0. While the first factor tends todrive the cations out of the pore, the second factortends to hold them inside. Consequently, for each unitnegative space charge the pore gains (or, for each unit

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positive surface charge the pore walls gain), if theenergy gain of cations due to their interactions withother species inside the pore is significant, that is,�ΔEifall

tot /σ is large, it will be difficult to remove themfrom the pore. This tends to lead to a small net negativespace charge in the pore and accordingly to a smallcapacitance, as predicted by eq 7. Below we delineatethe physical origins of the capacitance scaling shown inFigure 2a by analyzing the energy change of cationsinside each pore as it becomes positively electrified. Asimilar analysis has also been performed for anions butis not shown for brevity.

Figure 4 shows the variation ofΔEifalltot (0fσ)/σ and

its two components for cations. We observe that (1) thescaling of capacitance in these pores can be explainedby the scaling of ΔEifall

tot (0f σ)/σ using eq 7, and (2)ΔEifall

tot (0fσ)/σ is dominated by ΔEifionselec (0f σ).

The latter observation indicates that, as a pore iselectrified, the change of a cation's energy due to thechange in its electrostatic interactions with other ions,ΔEifions

elec (0fσ), is more important than the change inits non-electrostatic interactions with other species.These observations indicate that ΔEifions

elec (0fσ) þΔEifall

non-elec(0fσ) in eq 6 indeed dominates over theterm ETS(σ) � ETS(0), and eq 7 is justified. They alsoindicate that we can understand the capacitance scal-ing in Figure 2a by elucidating the physical origins ofthe scaling of ΔEifions

elec (0fσ)/σ shown in Figure 4.In pores within 0.75eWe 0.91 nm,ΔEifions

elec (0fσ)/σbecomes less negative as W decreases, and this iscaused by two factors. First, the screening of electro-static interactions increases as W decreases. As firstpointed out in ref 17 and shown in eq 4, the electro-static interaction between two ions confined in apolarizable pore decreases sharply as Rij/W increases.Consequently, as the same amount of net negativespace charge is introduced into a pore by addinganions or removing cations, the energy gain of a cationinside a narrower pore tends to be smaller. Second, the

degree of a two-dimensional nature of ion solvationincreases asW decreases. AsW decreases from 0.91 to0.75 nm, cations/anions inside the pores are increas-ingly solvated by their counterparts located near theirequator (the line connecting the north and south polesof an ion is assumed to be normal to the pore walls)because of wall confinement. As such, anions intro-duced into (and the cations removed from) narrowerpores during pore electrification tend to have alarger lateral distance Rij from the cations remaininginside the pores, which leads to a less negativeΔEifions

elec (0fσ)/σ in narrower pores according to eq 4.In pores within 0.91 <We 1.12 nm,ΔEifions

elec (0fσ)/σchanges slightly asW changes, and this can be under-stood as follows. AsW decreases from 1.12 to 0.91 nm,the increased screening of electrostatic interactionstends to make ΔEifions

elec (0f σ)/σ less negative. How-ever, as these pores become positively electrified, thenewly added anions can distribute near the north/south pole of the remaining cations in the pore dueto two effects: (1) the pores are wide enough to allowthree-dimensional solvation of ions, and (2) duringpore electrification, cations originally located nearneutral walls (interfacial cations) are removed andleave more space for the anions to be added into thepore. Of these effects, the second one is more impor-tant in the 0.91 nm pore as there are more interfacialcations in the neutral 0.91 nm pores (see Figure 3).Since a displacement of interfacial cations would de-crease the lateral anion�cation distance Rij, the secondeffect tends to make ΔEifions

elec (0f σ)/σ more negativeas W decreases from 1.12 to 0.91 nm. This effectcounteracts the increased screening of electrostaticinteractions in narrower pores and the result is thatΔEifions

elec (0fσ)/σ changes slightly asW decreases from1.12 to 0.91 nm.

In pores within 1.12 <We 1.26 nm,ΔEifionselec (0fσ)/σ

becomes less negative asW increases, and this trend ismainly caused by the qualitative change of ion solva-tion structure asW changes. As shown in Figure 3, in aneutral 1.12 nm pore, cations are located mostly in thecenter of the pore to maximize their solvation byanions located near the two walls. As the pore wallsbecome positively electrified, the number of anionssolvating each cation increases but the solvationstructure of these cations remains largely unchangedbecause they are still located at the pore center. Incomparison, the 1.26 nm pore follows a qualitativelydifferent scenario. As shown in Figure 3 and describedearlier, when these pores are neutral, cations and anionsform contact pairs and are positioned side by sideacross the pore. Such a solvation structure leads tosmall lateral separation Rij between a cation and itssolvation anions, which in turn leads to strong electro-static interactions according to eq 4. As the pore wallsbecome electrified, the solvation structure of cationchanges drastically: cations are displaced into pore

Figure 4. Variation of ΔEifalltot (0fσ)/σ and its two compo-

nents in the pores studied. The ion energy in the qPZCsystem is used to approximate the ion energy in neutralpores.

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center and can no longer be positioned side-by-sidewith anions across the pore. Such a change in cationsolvation structure increases the lateral separation Rijbetween a cation and some of its solvation anions, andthus weakens their electrostatic interactions. Conse-quently, as the pore walls of a 1.26 nm pore becomeelectrified, the energy gain of each cation due to itselectrostatic interaction with anions, is reduced by theabove weakening mechanism. Since this weakeningmechanism exists in the 1.26 nm pore but not in the1.12 nm pore, ΔEifions

elec (0f σ)/σ tends to be less nega-tive in the 1.26 nm pore.

The above discussions demonstrate that the scalingof EDL capacitance in pores is controlled by the screen-ing of electrostatic interactions in different nanopores,and, more importantly, by the ion solvation structureinside nanopores and how it changes as nanopores areelectrified. In particular, large capacitance can be ex-pected if the ion solvation is weak (as in the 0.75 nmpore) or if the electrification of a pore disrupts thestrong ion solvation structure inside it (as in the1.26 nm pore). These insights have rich ramificationsin the design of electrodematerials and electrolytes foroptimizing the capacitance of supercapacitors. Speci-fically, they suggest that one can optimize the capaci-tance of a nanopore through the manipulation of ionsolvation inside the pore by tailoring the size, shape,and other chemical details of ion pairs for the nano-pore. Such an approach goes beyond the state-of-the-art practice in which the capacitance of a nanopore isoptimized by matching the size of the counterion withthe nanopore width7 and provides richer opportunitiesfor enhancing the capacitance of supercapacitors byenabling one to exploit the vast chemical diversity ofRTILs.24

A New Capacitor Regime. To understand the collectivebehavior of EDLs in supercapacitors (e.g., their chargingkinetics), each pore in supercapacitors is often repre-sented by an equivalent circuit with capacitors con-nected in parallel and with resistors in serial.27 Such atransmission line model has been extensively used insupercapacitor research, and it implicitly requires that,at equilibrium, the potential drop between the porewall and the bulk electrolyte occurs entirely inside eachpore, that is, the potential drop between the porecenter and the bulk electrolyte is zero. However, wefound that the electrical potential at the pore centerincreases from�1.10 to 0.44 V as the pore width varies

from 0.75 to 1.26 nm while the total potential dropfrom the electrode surface to the bulk RTIL is close to1.68 V in all pores (see Supporting Information for thepotential distribution profiles in these pores). Theseresults suggest that the transmission linemodel breaksdown in pores studied here, and a capacitor regimequalitatively different from that inwide pores emerges.Specifically, although the charges on the pore wall arebalanced by ions enclosed between the pore wall andthe pore central plane, there exists a significant poten-tial drop between the pore center and the bulk elec-trolyte, which has no analogy in classical capacitors.Consequently, although the pore walls and the EDLstogether can still be viewed as capacitors, they can nolonger be described by the transmission line model.The transmission line model will be accurate whenpore width becomes large enough so that the EDLsnear opposing pore walls no longer interact with eachother. Since the thickness of EDL in RTILs is usually1�3 nm, we expect that the transmission line modelwill be accurate in pores wider than 2�6 nm.

CONCLUSION

In summary, we studied the EDLs in nanopores filledwith RTILs using MD simulations that account for thegeometrical anisotropy of ions and the polarizability ofpore walls. The results show that the scaling of porecapacitance exhibits a more complex behavior thanpreviously reported, and the rich scaling behaviorrevealed helps settle the debate on the validity ofanomalous capacitance enhancement in subnan-ometer pores and points to opportunities for newexperimental studies. Using a newly developed me-chanistic model for nanopore capacitance, we rationa-lized the observed scaling behaviors and showed thation solvation structure in nanopores and its responseto pore electrification play a critical role in controllingthe capacitance. This insight lays the theoretical basisfor optimizing pore capacitance through the manip-ulation of ion solvation inside nanopores by tailoringthe chemical details of ion pairs for a given pore.Finally, we showed that the EDLs in nanopores filledwith RTILs represent a new capacitor regime in whichthe transmission line model breaks down qualitatively.This result suggests that predictions of the transmis-sion line model, which is used ubiquitously in super-capacitor research, must be carefully reassessed in thefuture.

METHODSSimulationswere performed using a customizedMDpackage

Gromacs.25 Themethods for enforcing electrical potential on anelectrode surface and for computing electrostatic interactionsare the same as that used in ref 26. Using the Faraday cageeffect, the total charge on the electrode surface was controlledby fixing a corresponding amount of charges inside the electrode

(denoted by the shaded regions in Figure 1) and by adjustingthe number of ions inside the system to balance these charges.The length of the pores was g6.0 nm but varied slightly fordifferent pores. The RTIL layers adjacent to each vertical plateare thicker than 3 nm so that bulk-like RTILs are found atpositions away from the plate. The electrode was cut from acubic lattice (lattice constant: 0.17 nm) and its atoms were

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frozen. A united atom coarse grain model for the cation con-sisted of a 7-atom ion featuring a ring with the geometry of thenative imidazolium. The anion was modeled as a 5-atom ionwith the geometry of BF4

�. The charge of each ion was locatedon one of its atoms as shown in Figure 1b. A backgrounddielectric constant of 2.0 was used to account for the electronicpolarization of the ions. The Lennard-Jones parameters for allatoms were the same as those of carbon.26

Simulations were performed in the NVT ensemble (T = 400 K).Each system was simulated three times using different initialconfigurations and consisted of an equilibrium run of 10 ns anda production run of 10 ns. The charge density of pore walls wasobtained by computing the net ion charge in themiddle sectionof each pore;entrance effects associated with pore mouthwere found to be negligible at a distance greater than 1.2 nmfrom the mouth. The potential drop from the electrode surfaceto bulk RTILs was determined by solving the Poisson equationinside the system using the space charge density from thesimulations.

Acknowledgment. The authors thank the Clemson-CCIT of-fice for providing computer time. The Clemson authors ac-knowledge support from NSF under Grant No. CBET-0756496.R.Q. and V.M. were partially supported by an appointment to theHERE program for faculty at the Oak Ridge National Laboratory(ORNL) administered by ORISE. The authors at ORNL acknowl-edge the support from the Center for Nanophase MaterialsSciences, which is sponsored at ORNL by the Office of BasicEnergy Sciences, U.S. Department of Energy.

Supporting Information Available: Ion density distributionprofiles in the 0.78, 0.84, 1.04, and 1.17 nm pores, the elec-trical potential distribution profiles in pores studied in thiswork, and the method for modeling polarizable electrodes.This material is available free of charge via the Internet athttp://pubs.acs.org.

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