Capacitance and surface of carbons in supercapacitors

Abstract This research is focused in the missing link between the specific surface area of carbons surface and their electrochemical capacitance. Current protocols used for the characterization of carbons applied in supercapacitors electrodes induce inconsistencies in the values of the interfacial capacitance (in F m −2 ), which is hindering the optimization of supercapacitors. The constraints of both the physisorption of N 2 at 77 K and the standard methods used for the isotherm analysis frequently lead to a misleading picture of the porosity. Moreover, the specific surface area of carbons loses their meaning when the supercapacitor operates with organic electrolytes and ionic liquids and the actual surface involved in charge storage has to be assessed by molecular probes suiting the critical dimensions of the ions. In the case of certain carbons such as graphene type-materials, the voltage-driven mechanism may facilitate the access of electrolyte ions to spaces between carbon layers, providing a larger area than that estimated by gas adsorption. Finally, the morphological and porous features of carbons can be extremely modified when they are processed in electrodes. Due to their impact, all these issues should not be neglected and the characterization protocols must be adapted for this specific application of carbons.


Introduction
The great technological implementation of nanoporous carbons stems from the opportunity to be produced with a tailored porosity in a variety of configurations. They are found as powders, fibers, foams, cloths, monoliths, etc. and as a result of their wide spectrum of porous features are able to suit multiple applications in very diverse industrial processes. For many years, they have acted as purification filters, adsorbents for emissions control, catalyst supports, host matrices for active materials, etc. [1]. In the last two decades, nanoporous carbons have also found a niche in electrical energy storage as electrodes in electrochemical double-layer capacitors, EDLC (also known as supercapacitors, SC) [2,3].
All these applications correspond to very different processes but their description essentially relies on porosity parameters such as pore volume, specific surface area and pore-size distribution. It follows that the standard textural characterization of carbons for adsorption or catalysis is also applied when they are used in EDLC systems.
The porosity of carbons is generally assessed by the determination of the adsorption isotherm (amount of gas adsorbed at increasing relative pressure and constant temperature) of small molecules such as N 2 , CO 2 , Ar, etc. There are different methods for analyzing the corresponding isotherms which provide the specific surface area as well as other porous features responsible for their performance [3][4][5][6].
In carbon-based SC, electrical energy storage takes place through the electroadsorption of electrolyte ions at the surface of charged carbon electrodes. This mechanism implies that the capacitance (C) is proportional to the extent of the electrochemical double-layer (S) and to the relative permittivity of the solution ( ) and reciprocally dependent on the charges separation (d) Following the simplest approach from Eq. 1, the capacitance is expected to be proportional to the specific surface area of the carbon used in the electrode.
In the case of carbons for supercapacitors, their characterization is carried out routinely with N 2 at 77 K and the most popular approach to evaluate the specific surface rests on the BET method (S BET ). The more recent availability of a variety of DFT based-methods in commercial adsorption equipments has boosted their use for the assessment of both the pore size distribution (PSD) and the specific surface area (S DFT ). Quite frequently, the average pore size from DFT is combined with S BET although they are based on completely different models and S DFT and S BET are often different [7][8][9].
The lack of a clear linear relationship between C and S BET or S DFT encouraged studies on the specific role of carbon porosity in the formation of the double layer and a variety of interpretations were proposed. Among others, it has been argued that porous carbons often have a significant proportion of narrow micropores inaccessible to ions of electrolyte and, therefore, the entire surface is not used for charge accumulation [10,11].
This hypothesis seems reasonable but it results puzzling that the capacitance tends to level off in microporous materials with pores mostly above 1.2-1.5 nm in width and negligible ultramicroporosity (< 0.7 nm). In this context, the limitation of C in highly porous carbons was ascribed to a space constriction for charge accommodation inside their very thin pore walls. Carbons, being semiconductors, do not feature high chargecarrier density and the space-charge capacitance would be more limited for those with larger surface area [12,13].
It was also suggested that the capacitance corresponds to separate contributions from the surface of the micropores walls (C mi ) and the external surface (C ext ) [14,15].
The fact that the sets of C mi and C ext values differ between types of carbons [14][15][16][17][18][19] reveals that this approach does not provide a general description of carbons in SC.
The finding that the double-layer capacitance of the edge orientation of graphite was one order of magnitude higher than that of the basal layer [20] led to conclude that carbons with similar specific surface area but higher ratio of edge/basal orientations would reach higher capacitance [21][22][23].
More recently, it was pointed out that pores with sizes matching the ions dimensions result more effective for charge storage, maximizing the capacitance normalized to the surface area (in F m -2 ) [8,9,24].
Finally, it was reported that the puzzling interfacial capacitance found for certain highly porous carbon electrode materials may be originated by a quantum capacitance component of graphene layers, in addition to surface area [25].
In order to address such outstanding questions here we summarize a comprehensive study on the correlations between surface and electrochemical capacitance of a wide variety of carbons tested as SC electrodes under the same conditions. It is found that the standard textural characterization of carbons may be very limited to evaluate their actual porosity involved in charge storage. The protocols must be adapted for eliminating possible inconsistencies between textural parameters and capacitance. In fact, a critical re-examination of current insights and theoretical models will certainly benefit the technological development of supercapacitors with optimal performance.

Carbons
This extensive study is based on a wide variety of carbons of different origins and obtained by diverse procedures.    [11, in 2M aqueous H 2 SO 4 and 1M Et 4 NBF 4 /acetonitrile with the specific surface area estimated by the BET equation (a and b) and the total surface area assessed by a combination of different methods (c and d) [5,6,38].

Specific surface area
In spite of the specific weaknesses of each of the methods available for the interpretation of physisorption isotherms [3,4,50], the systematic analysis carried out on a variety of carbons has found little discrepancies between areas derived by other methods such as comparison plot (S comp ), the Dubinin-Radushkevich equation (S DR ) and the adsorption of phenol from aqueous solutions (S phenol ) [5,28]. Some scatter of the specific surface area based on the density functional theory (S DFT ) with respect to the other determinations [5] derives from the complex nature of such advanced modeling approach with strict requirements [3,4,50] (see Section 3.2).
Instead of relying on a single method, it is more adequate the simultaneous use of several independent approaches in order to cross-check results and eliminate possible contradictions. It has been previously reported [5,28] that the average value of S DR , S comp , S DFT , S phenol leads to reasonably accurate evaluation of the specific surface area (S total ) of numerous carbons. The limitations of the BET equation for microporous carbons are clearly reflected by the quantitative correlation found between S total and being L o the average micropore width.
Eq. 2 reports that S BET underestimates the total surface area of carbons with a high proportion of pores below 0.8 nm and overestimates it in micropores above 1.1 nm [5]. simply reflects the fact that in classical aqueous electrolytes such as H 2 SO 4 and KOH, the energy storage mechanism does not correspond exclusively to an electrochemical interaction and certain surface functionalities also participate through Faradaic pseudocapacitive reactions [27,39,[51][52][53][54][55][56][57]. As an additional effect, surface chemistry may also influence the hydrophobic/hydrophilic character of carbons and, therefore, the electrode-electrolyte interaction.
Studies carried out over last decade have reported a pronounced capacitance enhancement by reversible oxidation/reduction of hydroquinone/quinone O-groups [51,52,54,56] and pseudo-capacitive interactions on negatively charged pyrrolic-and pyridinic-N functionalities [55,56]. Improved electron transfer through the carbon appears to derive from positive charge on quaternary-N and pyridinic-N-oxide [56].
Quantitative assessments of activated carbons in 2M aqueous H 2 SO 4 indicated that the fraction of the surface oxygen desorbed as CO in TPD experiments contributes by around 50-60 F mmol -1 [27,39,57]. The correction of the experimental capacitances by subtracting the corresponding pseudocapacitance from oxygen functionalities led to 0.105 F m -2 which virtually corresponds to the effective contribution from the doublelayer in this electrolyte [39]. It coincides with the lower bound displayed in Fig. 1c found for carbons with low oxygen content, independently of the pore width.
The participation of oxygen functionalities was further confirmed by the good agreement observed between the experimental capacitances in H 2 SO 4 and those calculated from the enthalpies of immersion of carbons into benzene and water [32].
Whereas the enthalpy of immersion into benzene is related to the porous structure, the enthalpy of immersion into H 2 O also reflects the oxygen content of the surface.
Recent studies by Wu et al. [74,75] challenged this interpretation by illustrating that the pseudocapacitance of functionalized carbons may be largely overestimated due to inadequate characterization of the porous structure. As previously reported [76], low N 2 adsorption capacity of carbons is not always indicative of the absence of pores. N 2 at 77 K is not accurate to assess very narrow porosity, its adsorption being too slow to reach the equilibrium under standard operation times. Such a limitation is usually sorted out by using CO 2 at 273 K. Although both molecules display similar dimensions, the higher adsorption temperature allows avoiding diffusional problems and CO 2 adsorption results more convenient to evaluate ultramicroporosity [76][77][78][79].
Textural characterization by CO 2 adsorption at 273 K revealed that the specific surface area of a seaweed-derived carbon was as high as 410 m 2 g -1 instead of < 2 m 2 g -1 probed by N 2 at 77 K [74]. Consequently, the surface capacitance drops from an extraordinary C/S N2 = 58.45 F m -2 to a value of C/S CO2 = 0.285 F m -2 within the standard ranges reported for functionalized carbons. Further assessments on a variety N-doped carbons [75] confirmed that pores below 0.7 nm participate actively in the formation of the double-layer in aqueous electrolyte and the initial values of 0.31-2.52 F m -2 based on S N2 (22-312 m 2 g -1 ) actually range between 0.16 and 0.28 F m -2 when one uses the more reliable 383-587 m 2 g -1 of S CO2 .
In summary, N 2 adsorption at 77 K provides reliable information on the surface available for charge storage in the case of typical porous carbons but it is not sufficiently accurate to probe materials with a high proportion of ultramicropores. CO 2 adsorption at 273 K has to be implemented for a better estimate of the effective doublelayer in aqueous electrolyte.

Surface area accessible to electrolyte ions
Whereas the total surface area of most porous carbons is virtually accessible to the ions of the aqueous electrolytes, the direct correlation between C and S total clearly fails for a large number of materials in the organic electrolyte Et 4 NBF 4 /AN (Fig. 1d).
It should be emphasized that the specific surface areas determined by physical adsorption of standard molecules (N 2 , Ar, CO 2 ) involve micropores of widths as low as 0.3-0.4 nm, which are not accessible to larger ions of organic electrolytes [8] and ionic liquids [80]. It follows that predictions based on the classical probes no longer apply and the technique must be adapted in order to provide an accurate assessment of the surface area available to larger ions.
Advanced analyses of the isotherms based on Density Functional Theory (DFT) [4,81,82] have been proposed to provide areas taking into account the ions dimensions [83][84][85]. With the help of the software of commercial adsorption equipments, the pore size distribution of nanoporous carbons is assessed and, subsequently, the surface area involved in charge storage is calculated by subtracting that in pores smaller than the ions size.
The uncertainties in the characterization of nanoporous carbons using the complex DFT-approaches have been pointed out in the literature [4,6,50,56,61,86]. and the pore size distributions if the selected kernel depending on the pores geometry is not consistent with the nanoporous system of the carbons under study.
Very importantly, since DFT-methods are based on a theoretical reference for a given adsorbent/adsorptive system, they must be applied on equilibrium data. Fig. 2 reveals that the results are extremely sensitive to the isotherm accuracy. It contains the combined analysis of CO 2 (NLDFT) and N 2 (QSDFT) adsorption for a microporous carbide derived-carbon and illustrates how relevant are the equilibrium conditions imposed on the system for taking data. Whereas the total surface area estimated from the N 2 isotherm obtained under standard experimental conditions (i.e. equilibrium interval = 40 s, absolute pressure tolerance =5 mmHg and relative pressure tolerance =5 %) is 1425 m 2 g -1 , the surface in pores above 0.7 nm and, therefore, available for the cation Et 4 N + (0.68 nm) accounts for 379 m 2 g -1 (Fig. 2a). The clear downward deviation of the low-pressure points observed in the corresponding Dubinin-Radushkevich representation of the isotherm (Fig. 2b) reveals that the N 2 adsorption equilibrium was not achieved and, consequently, the QSDFTresults are in error.
Much longer data acquisition time (details in Supplementary Material) allowed to reach comparable N 2 and CO 2 adsorption at similar relative pressures (Fig. 2d). The better assessment of N 2 isotherm reports a total area of 1761 m 2 g -1 whereas the surface available for Et 4 N + is around 759 m 2 g -1 , twice than that obtained previously from the routine N 2 test.
The need for severe equilibrium inputs to get reliable results from DFT approaches is also shown in Fig. S1 for   [38,47] excellent linear correlation between C in Et 4 NBF 4 /AN and S for different materials with average pore sizes between 0.66 and 15 nm (Fig. 3b). The deviation from the general pattern observed for carbons with narrow porosity (those of Fig. 3a) disappears when one uses the surface in pores above 0.63 nm. Such a linear relationship indicates that the performance of carbons in this organic electrolyte corresponds essentially to a doublelayer mechanism (as opposed to the behavior in aqueous media) with a virtually constant contribution from the surface of around 0.094 F m -2 (dotted line in Fig. 3b) [38,47].
The latter is in contradiction with the increase in surface capacitance claimed for pores smaller than 1 nm by Gogotsi et al. In a first study [8], such an enhancement was observed for C/S BET but in view of the low reliability of S BET to evaluate the surface in microporous carbons [5,6,38,47,50], the issue was recently revisited by using an approach based on the DFT-pore size distributions [84,85]. It consisted in the evaluation of the total surface area by adding the CO 2 -NLDFT-PSD up to a pore size of 0.9 nm to the N 2 -QSDFT-PSD for pores wider than 0.9 nm. In those cases in which the S-CO 2 is higher than S-N 2 , the offset is added to the PSD data obtained from N 2 isotherm. The surface involved in energy storage is evaluated by taking the surface in pores above the Et 4 N + size.   [47]. The latter study refuted the possibility that the presence of binder or wide pore size distributions may hide the variation of the surfacecapacitance with the pore size [88]. The constant pattern for surface-capacitance of carbons obtained experimentally from the effective surface area accessible to the ions agrees with modeling studies [39,41,42,[89][90][91][92] and would be consistent with a decrease in the relative permittivity r in narrow micropores [6,93]. Quantitative estimations of the latter by systematic analyses would provide its incidence on the different electrolytes.
The preparation of materials with a much larger surface and hence higher energy storage density is one of the greatest challenges faced by supercapacitors. In this context, graphene with an extremely high surface area of 2630 m 2 g -1 , fully accessible to the electrolytes ions, and an outstanding electrical conductivity has emerged as the best candidate for SC electrodes.

Extremely low surface area of certain graphene related-materials
Undoubtedly, graphene has great potential for supercapacitors but its actual application still presents many weaknesses [102][103][104][105]. Whereas the synthesis of singlelayer graphene is extremely costly to compete with activated carbons, industrial production provides a wide variety of multilayer-graphenes with chemical and structural imperfectness, which do not reach the expected outstanding performance [103,[105][106][107][108][109].
The testing of a variety of graphene materials such as chemically- [110] and thermally-reduced graphene oxide [111,112], microwave exfoliated graphite oxide [113] and chemically activated graphene [114]  electrolytes [114]. The origin of such low value is likely the unreliable surface of 3100 m 2 g -1 assessed for this supermicroporous material by the BET method (Section 3.1).
Under similar testing conditions, a chemically reduced graphene oxide with 705 m 2 g -1 displayed standard values of, respectively, 0.19 and 0.13 F m -2 [110]. At the other end, one finds the extraordinary 5.5-7.0 F m -2 achieved in aqueous, organic and ionic liquid electrolytes by lacey reduced graphene oxide nanoribbons [121]. As reported above, the latter would simply reflect the inability of N 2 adsorption to evaluate the surface operating in charge storage.
Our recent studies have revealed that the approaches described in Sections 3.1 and 3.2 cannot be applied in a straightforward manner for certain graphene relatedcarbons. As summarized in Table 2, expanded graphite oxide (S43 [45]), highly functionalized carbon nanoflakes (OF1, OF2 [48]) and thermally-and chemically (NaBH 4 )-reduced graphene oxides (series S77 and GO-B, respectively) display meaningless C/S N2 of 1.7-204 F m -2 in H 2 SO 4 . Such extremely high values of interfacial capacitance as those observed for GO2-B and GO3-B result from their negligible N 2 adsorption capacity (Fig. 4a) which reports a surface not exceeding 1 m 2 g -1 while its gravimetric capacitance is about 100 F g -1 . Being more accessible to very narrow cavities, the adsorption of CO 2 at 273 K is much more relevant (Fig. 4b) and provides much higher specific surface areas ( Table 2).    (C/S CO2 ) are also displayed in the organic electrolyte, in which the pseudocapacitance is negligible [29].
Although other factors may be contributing to increase the interfacial capacitance of graphene materials, these unusual values suggest that the specific surface measured by standard gas adsorption may not be a relevant parameter for the present cases. It should be noted that the preparation of the materials quoted in Table 2 involved the expansion of graphitic structures by intercalation of oxygen-containing groups between graphene layers. It is very likely that electrochemical cycling facilitates the access of electrolyte ions to the interlayer space (channels, cavities) by a voltage-driven mechanism [122][123][124][125], providing a surface area larger than that estimated by gas physisorption. Therefore, outstanding surface-capacitance of graphene materials is not excluded but it may be affected by the difficulty to estimate accurately the actual surface area involved in energy storage.
No less important, one must be aware that the majority of the results reported in the literature for graphene-based supercapacitors are obtained by using electrodes with extremely low weight. This may result in inaccurate calculations of the gravimetric capacitance and, consequently, of the interfacial capacitance.

Surface area of carbons when processed in electrodes
An important issue for the identification of the parameters determining the performance carbon based-SC concerns the integrity of the porous structure of carbons when are processed into electrodes. The effect of pore blocking due to the addition of binder is frequently argued to explain the absence of a proportionality between the specific surface area and the electrochemical capacitance [85,88].  The results of a comprehensive assessment by N 2 adsorption of the porosity of various carbon electrodes compiled in Fig. 6 illustrate that the processing with 5% of PTFE or even with excess of 20% of PVDF reduces the specific surface area of typical activated carbons by less than 10%. Furthermore, no significant change in PSD (not shown) has been detected. Unlike reported by Abbas et al. [126], Fig. S2 reveals no significant incidence of the binder type on the porosity loss, the differences being within the limits of experimental error of N 2 adsorption. It is important to note that even for the extremely ultramicroporous carbon monolith M1 with narrow pore size distribution around 0.73 nm, the surface area undergoes a reduction of only 7 % when the material is ground and compacted with 20% PVDF to form a film electrode. In the case of templated mesoporous carbons obtained by carbonization of mixtures of a thermoplastic precursor with magnesium citrate [31], the maximum decrease in surface is limited to 12%.
With such low differences, the correlation of the surface of the raw carbons (in powder) with electrochemical performance of the film electrodes ( Fig. 1 and 3b) does not yield a significant discrepancy. Besides the results previously obtained for binderfree carbon monoliths [47], the present findings reject that the binder hides an increase of C/S in ultramicropores [8,88] and support the reliability of constant surfacecapacitance in carbons [38,39].
Indeed, the limited incidence of the binder on the porosity loss observed for the variety of the above materials should not be extrapolated for all carbons and each case requires to be evaluated. Thus, further studies have shown that problems may arise for some advanced carbons. Recently, it has been pointed out the precautions which must be taken for the SC assessment of ordered mesoporous carbons (OMCs) obtained from 3-aminobenzoic acid with SBA-15 as template [49]. As illustrated by Fig. S3, their outstanding structure consisting in chains of particles with bean-like shape is not preserved after the fabrication of electrodes matching the thickness and the carbon loading of commercial devices, and their specific surface area can be reduced by almost half. The detection of such a porosity loss allowed to evidence the remarkable electrochemical activity of the surface complexes, contrary to what suggested when the specific capacitance was related to the surface area of the materials in powdered form [49].
Constraints also detected for graphene-based materials should not be underestimated. Fig. 7a illustrates the drop of surface area (up to 55%) when graphene materials obtained by wet oxidation of graphite are shaped into electrodes. It appears that the expanded structure experiences some re-stacking whereas the voids between graphene-type flakes GO5-D2 and rGO-75 ( Fig. 8) are also significantly reduced by following standard electrode processing. As stressed by Fig. 7b, totally different values for the interface capacitance are obtained depending on the surface taken for calculations. This should be considered in order to get reliable interpretation of the graphene performance in supercapacitors.
The much lower surface loss in the case of processing graphene nanoplatelets C750 derived by mechanical exfoliation of graphite (Fig. 8), suggests that there is no general pattern and the electrode preparation must attend to the specific features of each graphene material. This is another relevant challenge for the application of graphene as active material in supercapacitor electrodes.

Conclusions
Besides the S BET and S DFT based on N 2 adsorption which are mostly adopted for the characterization of carbons for supercapacitors, other approaches have to be considered.
Whereas the analysis of N 2 isotherm by a combination of different methods leads to excellent correlations between the total surface area and the specific capacitance of typical porous carbons in H 2 SO 4 aqueous electrolyte, CO 2 adsorption has to be implemented for a better estimate of the effective double-layer in carbons with a high contribution from ultramicropores.
The aqueous electrolyte virtually uses the entire surface area of porous carbons but the adsorption of larger ions of organic electrolyte Et 4 NBF 4 /acetonitrile is notably affected by the existence of a pore size distribution and/or constrictions at the entrance of the pores. Therefore, the specific surface area determined by standard gas adsorption (i.e. N 2 and CO 2 as adsorbates) may not be a relevant parameter and the surface available for charge storage has to be estimated by using molecular probes with similar dimensions to the ions.
The effective surface area accessible to the ions, which may differ considerably from the BET-surface and the more advanced DFT-PSD derived-surface, leads to a virtually constant surface-capacitance for carbons in both H 2 SO 4 and Et 4 NBF 4 /acetonitrile of around 0.1 F m -2 , independently of the pore size. This finding is of great importance for R&D of carbon supercapacitors as it suggests the existence of limits for their energy storage capacity. In the absence of pseudocapacitive contributions, the upper-bound for capacitance expressed in F g -1 would be essentially determined by the surface area available to ions that can be generated in carbons.
The determination of the surface area involved in the formation of the doublelayer in certain graphene-materials is an actual challenge. It appears that electrochemical operation (based on a voltage driven-process) facilitates the access of electrolyte ions to the expanded structure resulted from harsh oxidation of graphitic precursors. Therefore, standard gas adsorption may result useless for some graphene type-materials as it does not always match their electrochemically active area.
Another important concern is the significant textural changes experienced by some advanced carbons when processed in electrodes with features of commercial devices.
Due to the impact of all these issues on the knowledge of the supercapacitor performance, the characterization protocols must be adapted for this specific application of carbons.