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ARTICLESPUBLISHED ONLINE: 21 MARCH 2014 | DOI: 10.1038/NMAT3916
Accelerating charging dynamics insubnanometre poresSvyatoslav Kondrat1,2*, PengWu3, Rui Qiao3* and Alexei A. Kornyshev2*
Supercapacitors have exceptional power density and cyclability but smaller energy density than batteries. Their energy densitycan be increased using ionic liquids and electrodes with subnanometre pores, but this tends to reduce their power density andcompromise the key advantage of supercapacitors. To help address this issue through material optimization, here we unravelthe mechanisms of charging subnanometre pores with ionic liquids using molecular dynamics simulations, navigated by aphenomenological model. We show that charging of ionophilic pores is a di�usive process, often accompanied by overfillingfollowed by de-filling. In sharp contrast to conventional expectations, charging is fast because ion di�usion during chargingcan be an order of magnitude faster than in the bulk, and charging itself is accelerated by the onset of collective modes.Further acceleration can be achieved using ionophobic pores by eliminating overfilling/de-filling and thus leading to chargingbehaviour qualitatively di�erent from that in conventional, ionophilic pores.
Supercapacitors offer unique advantages of high powerdensity and extraordinary cyclability but provide moderateenergy density1. Enhancing their energy density without
compromising the mentioned advantages would enable theirwidespread applications2. The current surge of interest in super-capacitors is driven by recent breakthroughs in developing novelelectrode materials and electrolytes3. In particular, electrodesfeaturing subnanometre pores and room-temperature ionic liquids(RTILs) are among the most promising materials for next-generation supercapacitors: The former affords a large specificsurface area and may also enhance the specific capacitance4–6and energy density7; the latter allows increasing the operationvoltage beyond that of conventional electrolytes8. These materialshave enabled impressive improvements in energy density9–13,and the thermodynamics of charge storage in these materials arenow understood reasonably well14–24. An emerging issue of thesematerials, however, is that they tend to lower the power density ofsupercapacitors25. For example, ion transport in RTILs is slow inthe bulk and could be even slower in nanoconfinement26–28, whichmay lead to sluggish charging dynamics, thus lowering the powerdensity. Resolving these issues, for example, by judicious selectionof pores and RTILs, necessitates the fundamental understanding ofcharging dynamics in subnanometre pores with RTILs.
The latter is, however, complicated by unique features emergingin subnanometre pores. In these pores, all ions of RTIL are inclose contact with each other. Consequently, charging dynamics insingle nanopores are affected by a multitude of collective effectsthat cannot be described by conventional theories proved validfor mesoporous electrodes. For instance, in single nanopores, theclassical transmission line model is valid only when the pore size ismuch larger than the double-layer thickness (in some cases it maycapture the shape of charging curves correctly also in the case ofextremely narrow pores—this is, however, rather by ‘coincidence,’as we shall see). Furthermore, conventional ion transport theories,rigorous in the limit of weak ion–ion correlations, cannot bedirectly used to predict the transport of RTILs in nanopores29,30.
To obtain a comprehensive picture of charging dynamics ofnanoporous supercapacitors with RTILs, the following questionsmust be addressed. First, whether there are ‘universal features’ ofcharging in such systems. Second, whether slow ion transport inbulk RTILs necessarily implies slow ion transport during charging.Finally, whether it is feasible to accelerate charging by tailoringthe size, geometry, and surface properties of nanopores. Resolvingthese issues can shape and guide the development of novel materialsfor supercapacitors.
Here we use molecular dynamics (MD) simulations and arecently developed31 phenomenological mean-field type (MFT)model to study the dynamics of charging nanoporous electrodeswith RTILs. We investigate the charging of a pair of slit nanoporesin two metallic electrodes. Figure 1 shows our nanopore systemtogether with a few snapshots of the time evolution during charging.In MD simulations, we consider ions as charged van der Waalsparticles of identical size (Methods). Such an approach does not takeinto account the electronic structure of carbon electrodes, neithergoes into the details of the atomistic structure of real ions. Withthese deliberate simplifications we aim at revealing the essentialphysics responsible for generic features of the charging dynamics,unobscured by the chemical complexity of RTILs and real carbonmaterials. Regarding the pore shape, the results of this study shallbe directly applicable to an emerging class of new materials, suchas graphene- and MXene-based nanoporous electrodes10,12,13, wherethe slit pores are well-aligned, comparable in width to the iondiameter, and have very large length-to-width aspect ratios. Ina more general context, understanding the charging dynamics ofindividual simple-shaped nanoporeswill help to build up an analysisof chargingmore complicated electrodematerials with, for example,interconnected nanopores of variable size and shape.
Charging of nanopores pre-wetted by RTILsMFT predictions. The flux (along the pore) of monovalent ionsconfined inside a metallic pore of width comparable to the ion
1Forschungszentrum Jülich, IBG-1: Biotechnology, 52425 Jülich, Germany, 2Department of Chemistry, Faculty of Natural Sciences, Imperial College London,SW7 2AZ, UK, 3Department of Mechanical Engineering, Clemson University, Clemson, South Carolina 29634-0921, USA.* e-mail: [email protected]; [email protected]; [email protected]
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ARTICLES NATUREMATERIALS DOI: 10.1038/NMAT3916
xL
V
V−H/2 H/2
±(equ) (V = 0)
t = 0 ns t = 3 ns t = 6 ns
a b
c d e
ρ ±(equ) (V)ρ ±
(equ) (V)ρ
Figure 1 | Charging of narrow electrode pores with room-temperature ionic liquids. a, Side-view snapshot of one half of the molecular dynamics systemfeaturing an electrode pore (width: 0.53 nm) and part of the room-temperature ionic liquid reservoirs connected to it. b, Schematic of a continuumphenomenological model. The equilibrium ion densities at zero electrode polarization (corresponding to the potential of zero charge) are set inside the poreof width L and length H. The ion densities corresponding to a non-zero voltage are set close to the pore entrances, and are kept fixed as the system isallowed to evolve in time. c–e, Top-view snapshots of the ionic structure inside the negative electrode pore when a voltage of 3 V is imposed impulsivelybetween the positive and the negative electrode pores (for a 3 ns video, see Supplementary Movie M2). The blue and orange spheres represent cations andanions, respectively. Wall atoms are not shown for clarity.
diameter can be written as31
J±=−D±∇ρ±∓D±ρ±G∇c−D±ρ±
ρmax−ρ6∇ρ6 (1)
where D± is the ion’s diffusion coefficient (for simplicity we shalluse the same D≡D±), ρ± is the ion density, c = ρ+ − ρ− is thecharge (in units of the elementary charge) and ρ6 the total iondensity, and ρmax is the total ion density at close packing; we neverreach ρmax in our calculations. G is a parameter characterizingthe screening of the ion–ion electrostatic interactions due to theelectronic polarizability of metallic pore walls; when the pore ismade narrower, the screening becomes stronger, and the ion–ioninteractions and hence G decrease (Methods). The first term in theion flux is simply diffusion. The second term comes from the ion‘migration’. It is due to the screened electrostatic interactions and iscollective in nature. The last term has entropic origin and representsthe transport of ions due to the gradient of total ion density along thepore. Equation (1) together with the local conservation law definethe MFT model for the dynamics of pore charging.
The RTIL reservoir is not explicitly accounted for in the MFTmodel. Rather, the ion densities close to the pore entrance areset to the equilibrium densities corresponding to some non-zerovoltage (Methods), and the ion densities inside the pore are allowedto evolve from their equilibrium values at the potential of zerocharge (PZC).
The numerical solution reveals that charging of pores wettedby RTILs at the PZC is a diffusive process. This can also be seenanalytically by noting that the time/space variation of total iondensity is small compared with the variation of charge density (thisis true up to times ≈15 in dimensionless units, see SupplementaryMovie M1). Then the last term in equation (1) can be ignored andone easily arrives at the diffusion equation for the charge density,∂tc = ∂xDeff∂xc(x , t), where Deff(ρ6)=D(1+ ρ6G) is the effectivediffusion coefficient. Interestingly, the diffusion-type equationarises also in the transmission line model (TLM). This may explainthe success of the TLM in interpreting impedance spectroscopydata for nanoporous electrodes4,32. We emphasize, however, that thephysical meanings of the above equation and the TLM model aredifferent: in TLM, the ‘effective diffusion coefficient’ comprises thedouble-layer capacitance and the bulk resistance of RTILs. Here,it expresses the collective behaviour of ions due to the inter-ioninteractions in ultra-narrow pores.
From the analytical solution of the diffusion equation33 onereadily finds a square-root behaviour at short times (Fig. 2a)
Q/Q∞≈4(Deff/πH 2)1/2√t (2a)
and an exponential saturation at long times (Fig. 2b)
Q/Q∞≈1−8π2
e−t/τ (2b)
with the relaxation time τ=H 2/π2Deff, where H is the pore length.The diffusive nature of charging originates from the fact that
the ion migration is proportional to the charge density gradient(the second term in equation (1), which follows from the solutionof the Poisson equation for the electrostatic potential inside thepore). This contribution enhances the ion transport, as comparedto the ion’s self-diffusion, and leads to Deff/D�1. By narrowing thepore, the ion–ion interactions become more screened, thus G andDeff∼ 1+ρ6G decrease; this means that wider pores charge faster(Fig. 2d). Interestingly, a similar diffusion slow-down is observed inmicellar systems,where the ‘apparent’ diffusion coefficient decreaseson adding salt34. Similarly to our case, where the screening is dueto metallic pore walls, the salt screens the electrostatic interactionsbetween the micelles and reduces their collective diffusivity.
The pore occupancy (that is, the total number of ions inside thepore) increases in the course of charging and reaches values higherthan the final, equilibrium occupancy. This overfilling is moredistinct for narrow pores (Fig. 2c) and disappears for sufficientlywide pores (not shown). Interestingly, de-filling extends over timescales much longer than overfilling and is accompanied by a third‘super-slow’ regime (see the long tail in Fig. 2b). This super-slowregime, however, seems to be of little practical importance in thepresent system as the pore is approximately 99% charged at its onset.
Ion diffusion in charged nanopores. Although the ions’ self-diffusion coefficient is frequently assumed constant at a giventemperature29–31,35, it depends on ion densities, pore size and otherfactors. In bulk and in mesopores such a dependence is relativelyweak or moderate27,36, and can be neglected in many relevantsituations. As we shall see, however, this is not the case forsubnanometre pores, where the ion diffusivity dependsmarkedly onion concentrations or the degree of pore charging.
For other parameters kept fixed, the self-diffusion coefficient(D±) becomes a complicated function of total (ρ6) and charge (c)
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NATUREMATERIALS DOI: 10.1038/NMAT3916 ARTICLES
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Figure 2 | Charging of nanopores predicted by the mean field model. a,b, Until a pore is nearly fully charged, Q(t)=Q∞=Q(t=∞), charging is a di�usiveprocess that follows a square-root law at short times (a) and an exponential law at larger times (b). c, Charging is accompanied by overfilling and isfollowed by de-filling, during which charging is slower. The thin dashed lines in a and b show the solution of the di�usion equation with De�/D≈26,corresponding to zero voltage, in a, and De�≈35, corresponding to the final state (≈0.78 V) in b. A long tail in b, deviating from the exponential regime,marks a super-slow charging regime (Fig. 4d). d, The speed of charging can be improved by making the pore wider or ionophobic. In all plots time ismeasured in units of d2/D (D is the ion’s di�usion coe�cient and d its diameter), which amounts to approximately 87 ps for the di�usion coe�cientD=Dbulk=2.32× 10−9m2 s−1. Note, however, that D varies with the density and degree of charging (Figs 3b and 5b).
densities. For simplicity, therefore, we look at D± along certain‘paths’ on the (ρ6 , c) plane, closely related to the actual chargingconditions; Fig. 3a shows such paths. We find that the averagetotal ion density during charging, ρ̄6(t), does not drop below theequilibrium density, ρ(equ)6 (c), at the same degree of charging—thatis, for c = c̄(t). Therefore, we calculate (Methods) the in-planeself-diffusion coefficient along the equilibrium path, as a limitingcase, and compare it with D± along the average ρ̄6(c) at c= c̄(t)corresponding to the impulsive charging at 3 V. We focus on thediffusion coefficient of cations (D+) and note thatD− shows similarbehaviour (Supplementary Figs 4–6).
The ion diffusion coefficient varies non-monotonically with thecharge density inside the pores: when the pore is neutral, the ion’sself-diffusion is nearly two orders of magnitude slower than in thebulk; as the charge inside the pore increases, the ion’s self-diffusionaccelerates and can become ten times faster than in the bulk; it slowsdown only when the pore become highly charged (c'2.6e nm−2).These phenomena seem general and are observed in equilibriumand during ‘impulsive’ charging, for wider pores and for morerealistic RTILs (Supplementary Figs 4–6).
The non-monotonic variation of the diffusion coefficientoriginates from the different structure of an ionic liquid inside thepore at different states of charging (Fig. 3b insets). At the PZC,ions form a two-dimensional lattice with counter- and co-ionsinterlocked with each other like in an ionic crystal. Diffusion ofions in such an environment requires a large activation energy tounbind counter/co-ion pairs or to cleave their ‘bonds’37, and thusthe ion diffusion is slow. As more counter ions are introduced,the perfect inter-locked counter/co-ion lattice gradually disappearsand ions diffuse more freely. Such accelerated self-diffusion has
also been observed near charged planar surfaces27, but the effect ismoderate. This is because counter ions near charged surfaces arestill bound to many co-ions in adjacent ionic layers. When the ionsformamonolayer inside a narrowpore, such binding disappears andthe acceleration of ion diffusion is much more dramatic. At largecounter ion density, counter ions form a quasi-Wigner crystal witha small number of co-ions as impurities, and the diffusion coefficientdecreases. In this case, however, there is a mostly steric contributionto the activation energy, which is much lower than at the PZC, andhence the diffusion in highly charged pores is much faster than atthe PZC.
Thus, a careful examination of RTILs inside nanopores undercharging conditions is necessary to select an optimal electrode/RTILpair, rather than a simple ‘extrapolation’ of the bulk properties ofRTILs. Although this renders the design of RTILs more complex, italso opens up exciting opportunities for tailoring RTILs for specificpores and degrees of charging.
Charging dynamics from MD simulations. Let us now returnto the dynamics of charging. We impose ‘impulsively’ a potentialdifference of 3 V between the negative and positive electrodes, butanalyse the charging of only one electrode pore (negative, to bespecific, see Fig. 1a), as our system is fully symmetric; we shall alsorestrict our considerations to pores of two different widths.
In line with the MFT predictions, the pore occupancy behavesnon-monotonically with time. Initially, the incoming flux of counterions overweights the outgoing flux of co-ions, leading to a slightoverfilling (Fig. 4a,b). Although overfilling is similar for both pores,the subsequent de-filling differs significantly. In the case of a widerpore (0.66 nm), the pore occupancies at the PZC and in the final
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ARTICLES NATUREMATERIALS DOI: 10.1038/NMAT3916
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Figure 3 | Ion self-di�usion during charging of ionophilic pores. a, A map showing the average total (ρ6) and charge (c) densities during ‘impulsive’charging (blue solid line) and in equilibrium (black dash-double-dot line); the green dash-dot line corresponds to an ionophobic pore (compare with Fig. 5).The average total and charge densities during charging lie within the shaded (grey) area. b, Cation’s self-di�usion coe�cient along the equilibrium path andalong ρ6(c) corresponding to impulsive charging at 3 V. The di�usion coe�cient is expressed in terms of the di�usion coe�cient of a neutral bulk system(Dbulk). For the 3 V charging, the data only up to 6 ns is shown. Blue and orange spheres in the inset denote cations and anions, respectively. The pore sizeis 0.53 nm.
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Figure 4 | Charging of ionophilic pores obtained frommolecular dynamics simulations. a,b, A voltage of 3 V is imposed impulsively between the negativeand positive electrodes at t=0. The average cation and anion densities are shown in a and the total pore occupancy in b. c,d, The evolution of the netcharge inside the pores exhibits a di�usive behaviour: the initial stage of charging follows a square-root law (c) and the late stage of charging follows anexponential saturation law (d).
state (corresponding to 3 V) are comparable, and de-filling has littleeffect on charging. Indeed, we find that the accumulated chargereaches nearly 98% of the final charge at the onset of de-filling,which is thus mainly characterized by ‘removal’ of both co- andcounter ions from the pore. This is followed by a ‘super-slow’ regimesimilar to the one predicted by the MFT (Fig. 2b). In the narrowerpore (0.53 nm), the difference between the initial (at the PZC) andfinal occupancies is considerable, and the charging at later times isessentially due to de-filling (SupplementaryMovieM2), which leadsto a significant slow-down of the charging process.
The evolution of net charge inside the pore, Q(t), exhibits thesquare-root and exponential saturation regimes revealed by theMFT model (compare Figs 2a,b and Figs 4c,d). Motivated by this,we use equations (2) to fit Q(t) and extract the effective diffusioncoefficients, Deff; note that Deff characterizes the whole system ina given time frame. In the square-root regime, equation (2a), weget Deff = 3.09± 0.39× 10−8 m2 s−1 for pore width L= 0.53 nm
and Deff=7.17±0.88×10−8 m2 s−1 for pore width L=0.66 nm; inthe exponential regime, equation (2b), we obtain Deff= 0.4± 0.08×10−8 m2 s−1 and Deff=5.91±0.71×10−8 m2 s−1, respectively. Theextracted values of Deff show a decrease with reducing pore width,manifesting slower charging in narrower pores.
It is instructive to compareDeff with the self-diffusion coefficient(D±). However, this is impeded by the fact that D± varies withRTIL density and composition (however, recall that D+≈D−≈D,Supplementary Figs 4–6). To be on the safe side, in mostcases we take the highest value of D at relevant conditions(Supplementary Section IV.B). For the 0.66 nm pore we get Deff/
D≈ 30 in both square-root and exponential regimes. This is inqualitative agreement with the MFT, which predicts a considerableenhancement of ion transport due to collective effects (the secondterm in equation (1)).
Similar enhancement is obtained in the square-root regime forthe narrower pore (0.53 nm), Deff/D≈10. At later times, however,
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Figure 5 | Self-di�usion and charging in ionophobic pores. a, Comparison of charging ionophobic and ionophilic pores of the same width (0.53 nm). Theionophobic pore is free of ions at the potential of zero charge. Local defects protruding into the pore centre slow down charging but do not change thecharging dynamics qualitatively (dotted line; only an ionophobic pore with defects was considered). b, Cation’s self-di�usion coe�cient inside a negativelycharged ionophobic pore at di�erent cation densities (no anions present in the pore, and pore walls are free of defects). The dashed line denotes theKnudsen di�usion coe�cient. For comparison, the short dash line shows the bulk di�usion coe�cient (Fig. 3).
Deff becomes comparable to the self-diffusion coefficients, withDeff/D≈0.5 to 1.0. This is closely related to the de-filling characterof charging discussed above. In this case, the first and third terms inequation (1) dominate, and charging becomes subdominant to de-filling. Physically, such a slow-down can be understood by notingthat low co-ion concentrations and strong screening of ion–ioninteractions in nanopores reduce collective effects. In other words,the co-ions have to diffuse on their own in the sea of counter ions tofind a way out of the pore, and henceDeff becomes comparable toD.
Fast charging of nanopores filled with RTILs is a result of bothcollective effects, due to the ion–ion interactions, and acceleratedions’ self-diffusion during charging (Fig. 3b). The collective effectsalso play an important role in charging such narrownanopores filledwith conventional organic electrolytes, as ion–ion interactions arestrong enough due to partial ion de-solvation3. However, we donot expect ions’ self-diffusion to vary significantly in the course ofcharging. This is because the ions, although partially desolvated, aresurrounded by solvent molecules throughout the charging process,and thus ions’ diffusion is to a large extent controlled by the ion–solvent, rather than ion–ion interactions.
Acceleration by ionophobicityOur results suggest that charging of narrow ionophilic pores isnearly always accompanied by overfilling, which itself is a fastprocess. The price one has to pay, however, is de-filling, which turnsout to slow down charging significantly. It thus seems beneficialfrom the practical point of view to use electrodes with widepores, where overfilling and hence de-filling are reduced or vanish.Unfortunately, however, in most cases increasing pore size reducesthe capacitance and stored energy density5,7.
Motivated by the MFT results (Fig. 2c,d; ref. 31), we explorehere a different possibility of accelerating charging, by makingthe surface of nanopores ionophobic. Pore ionophobicity can beachieved, for instance, by using mixtures38 of different RTILs or byadding surfactants39. In this workwemimic it by tuning the ion–wallvan der Waals interactions, so that the pores are free of RTILs at thePZC (Methods).
Ionophobic pores charge initially in a front-like fashion, withcounter ions spreading quickly throughout the pore (SupplementaryMovie M3); this is followed by a slower ‘diffusive’-like charging,much like for wide ionophilic pores. Importantly, however, we findthat ionophobic pores charge an order of magnitude faster thanionophilic pores at the same conditions. For instance, in ionophilicpores 90% of charging is achieved in 4 ns, whereas only ≈0.2 ns isneeded in the case of ionophobic pores.
A distinct feature of ionophobic pores is the behaviour of theself-diffusion coefficient (D+ in our case). At an early stage ofcharging, the ion density inside the pore is low and the ion–ionseparation is much larger than the ion–wall separation, hence theion diffusion is limited by collisions with the pore walls. In thiscase, the self-diffusion coefficient is very large and approachesthe Knudsen limit (Fig. 5b), where the ion’s mean free path islarge compared to the pore size, and hence the collisions of ionswith the pore walls, rather than between the ions, contribute themost to the diffusion coefficient (see Supplementary Section III.1).As more counter ions enter the pore, the diffusion coefficientgradually reduces. Importantly, the pore becomes highly chargedbefore the diffusion coefficients decreases significantly. For instance,when charging reaches 90%, the self-diffusion coefficient,D+≈2.57× 10−8 m2 s−1, is higher than in an ionophilic pore and in thebulk at comparable conditions. Incidentally, the strong variation ofD+ explains why the MFT, where we assumed a constant diffusioncoefficient, underestimates the acceleration of charging due toionophobicity of pore walls.
So far we have focused on ‘smooth’ walls, but local defects areubiquitous in practical electrode materials and may slow downmolecular transport significantly, as, for example, in the case offluid transport in carbon nanotubes40. To assess the effect ofsurface defects on charging, we performed simulations where thepore-wall atoms protrude into the pore centre and locally impedethe ion transport (see Supplementary Section II). Although chargingof the pore with defects is slower (Fig. 5a), the defects do not seemto affect the charging dynamics qualitatively, and in particular itsacceleration in ionophobic pores.
Finally, it is interesting to note that we observe a transitionbetween collective Fickian diffusion and (near) self-diffusion inboth ionophobic and ionophilic pores. Its effect on charging isdifferent, however. For ionophilic pores the charging undergoes atransition from collective to self-diffusion, and this slows downcharging. In contrast, for ionophobic pores a transition fromKnudsen-type self-diffusion to collective diffusion is observed, andthe onset of collective modes slows down the dynamics.
In summary, a phenomenological model and molecular dyna-mics simulations show that charging of ionophilic pores, of widthcomparable to the ion diameter, follows an effective diffusion law.Such charging is a complex process, complicated by a myriad offactors, such as extreme confinement and ion crowding, imageforces and screened interactions, and so on. Thus, the ‘law of effect-ive diffusion’ is not only remarkable but also of practical importance.Indeed, it can, for instance, help simplify the development of ‘whole
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porous-electrode’ models, and thus open doors for optimizingelectrode materials beyond the single-pore level.
Ion’s self-diffusion in subnanometre pores shows an interestingdependence on ion densities and composition. The self-diffusioncoefficient varies during charging over a few orders of magni-tude, and can exceed a few times the ion diffusion in the bulk (undersimilar conditions). This suggests that fast charging can in principlebe achieved if an ionic liquid is optimized specifically for selectedporous materials and the required degree of charging.
We have found that charging is often accompanied byoverfilling. Although overfilling can in fact accelerate charging, asdemonstrated by high effective collective diffusivity, the subsequentde-filling slows down charging significantly, and should be avoidedin practical applications. One way to achieve this is to make poresionophobic. We have shown that ionophobic pores can acceleratecharging by an order of magnitude. The calculations in the presentpaper, as well as those performed for an exactly solvable model ofcharge storage in cylindrical pores (A.A. Lee, S.K. and A.A.K., tobe published) show that pore ionophobicity leads to comparablevalues of capacitance and enhanced energy density at moderatelyhigh voltages. We therefore believe that ionophobic pores presentan exciting opportunity for increasing both the power and energydensity of nanoporous supercapacitors.
MethodsMean-field model. We consider a single layer of ionic liquid confined in a slitnanopore formed by two parallel metal walls. The free energy of the system canbe written as31 F[ρ±]=Eel [ρ±]−TS[ρ±]+
∫(h+ρ+(x)+h−ρ−(x))dx , where T is
temperature and ρ± are the ion densities. In the first term we take into accountexplicitly the pore-induced exponential screening of the ion–ion electrostaticinteractions17. To account for excluded volumes, we adopt theBorukhov–Andelman–Orland expression41 for the entropy, S[ρ±]. Thevoltage-dependent ‘external fields,’ h±, consist of the ions’ electro-chemicalpotentials, resolvation energy, and the van der Waals and image-force17interactions of ions with the pore walls. The h± control the equilibrium iondensities inside the pores and do not participate in the dynamics other thanthrough initial and boundary conditions.
The dynamics is defined by the continuity equation ∂tρ±=−∂x J±. For thecurrent we postulate J±=−Γρ±∂x(δF/δρ±), where Γ ≡Γ±=D±/kBT is aphenomenological mobility parameter and D± the diffusion constant, which weassumed independent of pore width, voltage and density; kB is the Boltzmannconstant. Plugging the free energy F[ρ±] into the continuity equation results inequation (1): the first and third terms follow from the entropy, and the secondterm is due to Eel , where31
G=4LBRc
∞∑n=1
sin2(πn/2)n
K1 (πnRc/L) (3)
is a parameter characterizing screening of the electrostatic interactions by themetal pore walls; it depends on the pore width, L, and decreases with narrowingof the pore. Here K1 is the modified Bessel function of the second kind of thefirst order, Rc is the cut-off radius31 and LB=e2/εpkBT (in Gaussian units) is theBjerrum length, where e is the elementary charge and εp the dielectricpermittivity inside the pore. It has been shown42 that the pore-width-dependentpermittivity, εp(L), may have a substantial effect on the surface-specificcapacitance, and so can also influence the dynamics of charging. For electrolytesolutions, εp varies with the pore width owing to the effects of spatial dispersion(see, for example, ref. 43) and varying concentration of the solvent inside thepore. However, for neat RTILs, which we consider here, these effects arenegligible and we assume a constant value of εp.
Equations (1) were solved numerically using the GSL library44. The solutionprovides the ion densities, and thus the accumulated charge and pore occupancyat any given time.
MD simulations. The MD system consists of a pair of identical slit pores andtwo reservoirs separating the pores. The access widths of pores were 0.53 nm and0.66 nm, the pore length was 12.09 nm, and periodic boundary conditions wereapplied in all directions. Each pore wall was made of a square lattice ofLennard-Jones (LJ) particles, and cations and anions were modelled as charged LJparticles. The ionophobicity of the pore wall was varied by tuning the LJparameters of the ion–wall interactions. A schematic picture of the MD system,the number of electrode atoms and ions inside the system, and the force field
parameters are provided in Supplementary Section II. To compute theequilibrium charge at a given applied voltage, a separate system with 50% shorterpores was set up and run for 20 ns. We determined the integral capacitance of theionophilic pores at 3 V to be 12.6 and 8.1 µF cm−2 for the 0.53 and 0.66 nm widepores, respectively; this is comparable to the experimental values5,6. Thecorresponding capacitance of the 0.53 nm-wide ionophobic pore is 11.4 µF cm−2.
MD simulations were performed using a customized Gromacs code45. Porewalls were maintained as equi-potential surfaces with their image planescoinciding with the geometric plane of the wall atoms. In the method46 we used,the electronic polarizability of pore walls is taken into account at the continuumelectrostatics level. This method is in good agreement19,47 with other models ofpolarizable electrodes22,23. Further MD details, such as time step, thermostating,and calculation of forces are provided in Supplementary Section II.
The system was first equilibrated for 2 ns at PZC at 400 K. After the systemreached equilibrium, a voltage difference of 3 V was impulsively imposed betweenthe negative and positive electrodes, and the system was allowed to evolve in theNVT ensemble for 6 ns. Note that the voltage difference between an electrodeand the bulk RTILs was used in the MFT, which thus corresponds to the halfpotential applied in MD simulations. Each charging case was repeated 50 (5)times for ionophilic (ionophobic) pores, with independent initial configurations,to obtain reliable statistics.
It is known that the temperature of an ionic liquid inside nanopores tends torise above the thermostat temperature during charging (see, for example, ref. 48),as a result of the so-called Joule effect. Indeed, in our simulations the RTILtemperature inside the 0.66 nm-wide pore rises from 400 K to a maximum of422 K after 250 ps, but returns to 400 K by t≈550 ps, and remains close to 400 Kthereafter. To assess how this ‘over-heating’ affects charging, we performed extrasimulations in which the RTILs’ temperature inside the pore is controlled by acanonical velocity rescaling method49. We verified that this method eliminatesoverheating, and that the extracted diffusion coefficients agree within thestatistical error with those obtained by the ‘conventional’ simulations usedthroughout this study.
To study self-diffusion of ions inside nanopores, we set up MD systemswhich consist of a single pore only (and the ions in it), with periodic boundaryconditions in all directions. We tuned the number of cations and anions to matchthe desired total and charge densities inside the pore. Because pore walls aremodelled as an equi-potential surface, they form a Faraday cage around ions, andelectroneutrality is automatically satisfied. The diffusion coefficient of ions wascomputed using the Einstein–Helfand relation50; the ion trajectories wereobtained from at least 5 ns equilibrium runs.
Received 22 October 2013; accepted 10 February 2014;published online 21 March 2014
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AcknowledgementsWe thank the Clemson-CCIT office and E. Duffy for providing computer facilities.R.Q. acknowledges the support of the NSF (CBET-1264578). S.K. and A.K. weresupported by the Engineering and Physical Science Research Council via GrantEP/H004319/1. We are grateful to Y. Gogotsi, P. Simon, C. Pérez, J. Griffin, G. Oshaninand F. Stoeckli for fruitful discussions, and X. Jiang for technical assistance.
Author contributionsA.A.K., R.Q. and S.K. designed the research. P.W. performed MD simulations and S.K.performed MFT calculations. The results were analysed jointly by R.Q., S.K., A.A.K. andP.W., and all participated in writing the paper.
Additional informationSupplementary information is available in the online version of the paper. Reprints andpermissions information is available online at www.nature.com/reprints.Correspondence and requests for materials should be addressed to S.K., R.Q. or A.A.K.
Competing financial interestsThe authors declare no competing financial interests.
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