Jie Yang,Alejandro Gallegos,Cheng Lian,Shengwei Deng,Honglai Liu,Jianzhong Wu*

1 State Key Laboratory of Chemical Engineering,and School of Chemistry and Molecular Engineering,East China University of Science and Technology,Shanghai 200237,China

2 Department of Chemical and Environmental Engineering,University of California,Riverside,CA 92521,USA

3 College of Chemical Engineering,Zhejiang University of Technology,Hangzhou 310014,China

Keywords:Electric double layer Electrodes/electrolyte interface Curvature effects Classical density functional theory Machine learning

ABSTRACT Understanding the microscopic structure and thermodynamic properties of electrode/electrolyte interfaces is central to the rational design of electric-double-layer capacitors(EDLCs).Whereas practical applications often entail electrodes with complicated pore structures,theoretical studies are mostly restricted to EDLCs of simple geometry such as planar or slit pores ignoring the curvature effects of the electrode surface.Significant gaps exist regarding the EDLC performance and the interfacial structure.Herein the classical density functional theory (CDFT) is used to study the capacitance and interfacial behavior of spherical electric double layers within a coarse-grained model.The capacitive performance is associated with electrode curvature,surface potential,and electrolyte concentration and can be correlated with a regression-tree(RT)model.The combination of CDFT with machine-learning methods provides a promising quantitative framework useful for the computational screening of porous electrodes and novel electrolytes.

1.Introduction

Electric-double-layer capacitors (EDLCs),a.k.a.supercapacitors,have been widely used in transportation,consumer electronics and renewable energy industry.In comparison with batteries,EDLCs have the advantage of high power densities,fast charging ability,and long cycling lifespans for electrical energy storage[1,2].Recently,understanding the fundamental mechanisms of EDLC performance has received considerable attention.In contrast to batteries,which store energy by electrochemical reactions,EDLCs harness electrical energy by adsorption of ionic species at the surfaces and in the micropores of electrodes[3–8].The performance of EDLCs correlates well with the specific surface areas and the properties of electrode/electrolyte interfaces in amorphous porous electrodes [5,9,10].Thus,improving the performance of EDLCs hinges on the optimization of both electrolytes and electrode materials.

Over the past few decades,several attempts have been made to tune the pore size,surface area,morphology,architecture,and functionality of the electrode materials for their use in EDLCs[1,8,11–14].Seminal studies reveal an ‘‘anomalous”increase in capacitance when the pore size is decreased to 1 nm[10,15].However,the validity of this claim was challenged by a following experiment using organic electrolytes ([TEA][BF4]/ACN).It was found that the specific capacitance is nearly independent of pore width in the range of 0.7–15 nm[16,17].The abnormal capacitive behavior has been subject to controversy because its physical origins remain elusive.

Numerous theoretical studies have been devoted to investigating the capacitance of porous electrodes with simple pore geometries such as slits,cylinders and spheres[18].Huang et al.proposed an electric wire-in-cylinder capacitor(EWCC)model for cylindrical pores [19,20]and a sandwich model [21]for slit-shaped pores to rationalize the anomalous increase of capacitance as the pore size falls.Good agreement with experimental data was demonstrated for the pore size in the unexpected increasing regime.Based on the mean-field theory (MFT) and Monte Carlo (MC) simulations,Kondrat and Kornyshev et al.[22,23]suggest that the anomalous increase is ascribed to the ‘‘superionic state”of ions inside micropores.The spuerionic state arises from the image forces that exponentially screen out the electrostatic interactions among ions inside the pore.Later studies by classical molecular dynamic(CMD) determined the microscopic structure of ions near the charged electrode surface and observed the similar EDL behaviour[24].For instance,Wu and Qiao [25]et al.investigated EDLs in nanopores filled with a room-temperature ionic liquid(RTIL)using MD simulation.By considering the polarizability of pore walls and the geometrical anisotropy of ions,they demonstrated that the capacitance represents a U-shaped scaling behavior if the pore width is in the range of 0.75–1.26 nm.The simulation results verified the anomalous capacitance increase in sub-nanometer pores observed experimentally.Cummings et al.[26]further expanded the MD simulation to slit-shaped micropores ranging in size from 0.67 to 1.8 nm rather than only the anomalous capacitance region.They found two peaks in the capacitance curve versus pore size,located at 0.7 nm and 1.4 nm,respectively.This multiple peak behavior of the capacitance scaling curve suggested that the nearly independent capacitance with pore size [16]was owing to the broad pore-size distribution in microporous electrodes.Moreover,Feng et al.revealed the influence of the electrode curvature on the capacitance–potential curve on onion-like carbons[24]and singlewalled carbon nanotube [27].

Classical density functional theory(CDFT)has been used before to describe the EDL structure and the zeta potential near planar surfaces [28,29],cylinders [30–32]and spheres [33,34].The theoretical results show that the curvature effects are significantly different on convex and concave EDLs.Near surfaces with the same convex curvature,the contact values of counterion density profiles and the zeta potentials are smaller than those near a planar surface and follow the order of spherical convex [33]＜cylindrical convex[30,31].However,an opposite order is found for the concave case:planar ＜cylindrical pore [32]＜spherical concave [34].Although much progress has been made on the curvature effects on the EDL structure,the relationship between the capacitive performance and the curvatures of amorphous electrodes remains poorly understood.

In this paper,we tackle the curvature effects on the capacitive performances of spherical EDLCs using CDFT.It has been demonstrated that CDFT is an ideal computational tool for investigating the geometry effects with high efficiency [35–38].To examine the rich behavior of spherical electrodes at different electrolyte concentrations,we use the regression-tree(RT)model to correlate the contributions of the electrode and electrolyte properties (i.e.,curvature,electrode potential,and electrolyte concentrations) to the capacitive performance.It is hoped that a combination of CDFT and RT would yield new insights into the rational design of porous electrodes and novel electrolyte materials.

Fig.1.Schematic representation of ionic species near a planar(left)and a spherical(right) surface of uniform charge,respectively.

2.Models and Methods

2.1.Electrode and electrolyte models

Fig.1 illustrates a schematic setup for a planar and a spherical electrode submerged in an aqueous electrolyte solution.Approximately,the spherical system mimics a supercapacitor with the electrodes made of onion-like carbons.Predicting the spherical capacitor as a function of the electrode radius and ion concentrations would allow us to determine the curvature effect on the EDL structure and the capacitive performance.

To capture the essential physics of EDL charging,we use the restricted primitive model (RPM) for the electrolyte solution.A similar model was employed in our previous work to represent ion distributions outside the spherical electrode [33,39].In this model,ionic species are described as charged hard spheres of the same diameter whereas the solvent (e.g.,water) is accounted for implicitly through a dielectric background.Despite neglecting the chemical details,the primitive model accounts for ionic excluded volume effects and electrostatic correlations important for EDLC performance.Such effects are often neglected by conventional EDL theories (i.e.,the Poisson-Boltzmann and Poisson-Nernst-Planck equations).

Throughout this work,cations and anions are assumed as charged hard spheres of the same diameter(σ=0.5 nm)and equal but opposite valence (Zi=±1),and the dielectric constant for liquid water is εr=78.Based on these assumptions,the pair potential between ionic species can be expressed as

where r represents the center-to-center distance,e is the elementary charge,and ε0denotes to the permittivity of the free space.

Near a spherical electrode of radius R,the non-electrostatic component of the external potential on each ion is represented by

where r is the radial distance between the spherical center of the electrode and the ion center.When the spherical electrode is subject to a fixed surface electrical potential,the grand canonical ensemble can be used to describe the distributions of cations and anions near the spherical surface.As shown below,all equilibrium properties of the EDL can be obtained from the ion distributions.

2.2.Classical density functional theory

We use the classical density functional theory (CDFT) to calculate the ion distributions near the spherical electrode and the capacitance of EDLCs.The theoretical details can be found in previous publications [40–43].For the ionic systems considered in this work,the grand potential can be formally represented as a functional of the ionic density profiles,

where ρi（r）and μirepresent the local densities and the chemical potentials of ionic species,respectively;Vext,i（r）stands for the external potential,and F denotes the intrinsic Helmholtz energy.The intrinsic Helmholtz energy F consists of an ideal-gas contribution and an excess due to the intermolecular interactions,where β=1/（kBT）and kBrepresents the Boltzmann constant,Λ is the thermal de Broglie wavelength for species i.

With an appropriate expression for the excess Helmholtz energy [44,45],minimizing the grand potential Ω as shown in Eq.(3) yields a set of Euler-Lagrange equations for the ion distributions,

2.3.Regression-tree (RT) model

To examine the multifactor influence on the electrical double layer capacitance (EDLC),we analyzed the theoretical results with a nonlinear regression method named M5 decision tree(M5P).The integral capacitance predicted by CDFT is used as the dependent variable.The independent variables involve the electrolytes concentrations and electrode properties such as surface curvature and the surface charging potential.

M5P is one of the RT models in which each branch represents a choice between various alternatives and each node represents a decision.M5P generalizes the concept of RT by splitting the input data space into sub-spaces and subsequently applies a local specialized linear regression model to each sub-space.The RT model is a transparent approach that does not need optimization of parameters [47,48].Additionally,it has advantages in interpretation,low memory usage,and fast fitting and prediction.More details for the model can be found in a previous study [48].

We implement the M5P models using an open-source machinelearning package WEKA.The correlation coefficient(R),mean absolute error (MAE),and root mean square error (RMSE) are introduced to evaluate the accuracy of the M5P models in correlating the EDL capacitance data predicted by CDFT:

3.Results and Discussion

In this work,we mainly focus on the curvature effects on the capacitance of spherical electrodes using 1:1 electrolyte as an example.An isolated particle is used to represent how the radius of a spherical electrode influences the ion distributions,mean electrostatic potential,surface charge density and the capacitive performance.In general,CDFT is also capable of describing the curvature effects on EDL structure and capacitance for multivalent systems [33,53].Therefore,we expect that combining machine learning and CDFT would potentially be useful to predict the performance of EDLCs for all electrolyte systems.

3.1.The electric-double-layer structure

As stated above,different spheres are used to describe the curvature effects in porous carbon electrodes.We first investigated the local ionic densities near spherical electrodes of different radii,with R ranging from a small size comparable to the ion diameter(1 nm)to infinitely large(i.e.,the planar limit).The surface charge density was set as 0.8 C·m-2for all cases.Fig.2 shows the density profiles and the mean electrostatic potentials near different spherical particles.Similar to the EDL structure around an isolated colloidal particle [33],the normalized ionic densities display a strong accumulation of counterions (i.e.,ions in opposite charge of the surface)near the charged surface to compensate for the high surface charge of the spherical electrode.On the other hand,coions(i.e.,ions with the same charge of the surface) are depleted from the first few ion layers next to the surface.The coion depletion results from the strong electrostatic repulsion from the surface.Interestingly,significant accumulation of counterions is also observed in the second layer for the flat case.The layering effect is nearly absent for spherical electrodes of small size (e.g.,R=1 nm).Even the radius increasing from R=1 nm to 2 nm shows a noticeable change in the density profiles for both counterions and coions.

Fig.2.Ionic density profiles for (a) counterions and (b) coions,and (c) the mean electrostatic potentials for an aqueous electrolyte of 1.0 mol·L–1 around spherical electrodes of various radii.In all cases,the surface charge density of the electrode is fixed at 0.8 C·m-2.

The difference in the ionic distributions among the electrodes of various radii can be understood as a balance between the packing of ions near the electrode to compensate the surface charge and the attractive or repulsive interactions between ions of opposite and same charges,respectively.One factor is that there is less room to pack ions around an electrode with a smaller radius because the surface area for the electrode is given by 4πR2.On the other hand,at a fixed charge density,the total charge of ions within a distance from the electrode scales quadratically with the radius of the electrode.Thus,while there is less room to pack the ions,the necessary ions to compensate the electrode charge is also decreased as the electrode becomes smaller.Since the ions that compensate the electrode charge are located only outside the electrode,the surface area available to the ions at any position r is larger than that of the electrode.This difference in surface area is more noticeable for ions near an electrode with a small radius because it is related to r2/R2.The EDL that forms near the electrode extends only a few nanometers from the surface,thus the normalized surface area approaches unity for large R since r～R.As a result,spherical electrodes with various radii,but at a fixed charge density,will exhibit an increase in the average counterion density within the formed EDL as the electrode radius increases.Because of packing limitations near the surface,a second layer of counterions must be formed to compensate the surface charge.Such phenomenon cannot be captured in a theory that does not account for the ionic volume exclusion,e.g.,the Poisson-Boltzmann equation.

The mean electrostatic potentials of the spherical electrodes with various radii are presented in Fig.2(c) at conditions corresponding to Fig.2(a) and 2(b).The electrode potential surges as the radius increases but approaches the a symptotic limit by 10 nm.This follows from the previous discussion because there are more ions at the interface for large electrodes.All four cases show a similar trend with a rapid decrease of the electrostatic potential near the surface as the surface charge is fixed to 0.8 C·m-2.The more counterion presence for electrodes with larger radius leads to a stronger decrease in the electrostatic potential.The distance at which the bulk solution condition is reached (i.e.,outside the EDL) is almost independent of the electrode radius showing only a small increase when the radius initially decreases.

3.2.The surface charge density

Next,we examine the curvature effects on EDL capacitance over the entire voltage range of the electrode as typically considered in practical applications.As shown in Fig.3(a)and 3(b),CDFT predicts a monotonic increase of the surface charge density as the surface electrical potential increases for a spherical electrode of an arbitrary radius in an electrolyte solution at 1.0 mol·L–1and 2.0 mol·L–1,respectively.The surface charge density increases more significantly with the potential when the radius of the electrode is decreased.The small particles are more responsive than large ones because a smaller surface area means less total charge(i.e.,C,not C·m-2).As discussed in Section 3.1,the higher electrode potential,the slower the rate of the surface charge density increases.This trend is more noticeable for large particles because they require a low counterion density within the EDL to balance the surface charge.

Fig.3.Various physical quantities versus the electrical potential for spherical electrodes with radii ranging from 1.0 nm to infinite.(a)and(b):The surface charge density(Q).(c)and(d):Excess adsorptions of ions at the surface.(e)and(f):The differential capacitance.The bulk electrolyte concentrations are 1.0 mol·L–1(left)and 2.0 mol·L–1(right),respectively.

3.3.The ionic adsorption on the electrodes

To understand the Q-ψ curves,we investigated the excess ion adsorptions at the surface of the electrodes as a function of the surface potential.When the electrode is uncharged(i.e.,at zero potential),the ion concentration at the surface is similar to that in the bulk solution.When the electrode is applied with a potential,counterions are enriched at the electrode surface concomitant with the depletion of coions.As the radius of the spherical electrode becomes larger,less counterions are adsorbed as can be expected from the surface charge density plots in Fig.3(c) and 3(d).While the surface charge density was only slightly dependent on the electrolyte concentration at a large applied potential,the excess adsorption of the ionic species is more noticeable due to the higher presence of ions in the solution.As the coions are depleted from the surface,the excess adsorption of the coions only slightly decreases with the increase in the electrode potential.The depletion of coions is more noticeable at high electrolyte concentration because of the large concentration of coions at the surface at the zero applied potential.The decline in the slope of the surface charge density versus the applied potential results from the slight decrease in the excess adsorption of coions.

3.4.The differential capacitance

To evaluate the effects of pore curvature on the performance of EDLs,we have calculated the differential capacitance Cdas a function of the surface potential at two representative electrolyte concentrations,1.0 mol·L–1and 2.0 mol·L–1[Fig.3(e)and 3(f)].In both cases,the radius of the electrode is changed from 1 nm to the planar limit.It can be seen that the differential capacitance is noticeably larger for the electrode with a smaller radius.As discussed earlier,this results from the fact that the charge of a small particle can be more easily compensated by counterions than that of a large particle.While the differential capacitance exhibits a plateau at large applied potentials at both electrolyte concentrations,the behavior is quite different at low applied potentials.For the case of 1.0 mol·L–1electrolyte concentration,the differential capacitance shows maximum values around ± 0.2 V,leading to a‘camel-shaped’curve.By contrast,at high electrolyte concentration(2.0 mol·L–1),the differential capacitance is maximized at zero potential,i.e.,it exhibits a ‘bell-shaped’ curve.The effect of curvature has only a minor influence on the shape of the capacitance curve.The reason for the different shapes of the capacitance curves at low and high electrolyte concentrations has been discussed in previous studies[49–52].The shape of the differential capacitance can be understood by the ionic densities near the electrode surface.When the electrolyte concentration is high,the surface is saturated with ions,leading to little room for any additional ions.As a result,any charging of the surface occurs by ion-exchange of the counterions with the coions.Thus,the differential capacitance is maximized at a zero applied potential.When the electrolyte concentration is reduced,the surface is not saturated with ions so the counterions can adsorb to the surface without much disruption of the coion distribution.This leads to a lower differential capacitance at zero applied potential and a higher differential capacitance when the applied potential is increased.

3.5.The integral capacitance

We have also calculated the integral capacitance from the ratio of the surface charge respect to the surface electric potential.As illustrated in Fig.4,the relationship between integral capacitance with spherical radius and operating potential window under different bulk concentrations is similar to that for the differential capacitance.At high electrolyte concentration (2.0 mol·L–1),the integral capacitance versus the surface potential exhibits a bell-shape curve which is obviously modified by the curvature effects when the surface potential is nearly neutral.At low electrolyte concentrations(0.6 mol·L–1and 1.0 mol·L–1),the integral capacitance versus surface potential exhibits a camel-shape curve which is noticeably modified by the curvature effects at the maximum capacitance.The transition from the ‘camel’ shape to the ‘bell’ shape occurs at an intermediate concentration between 1.0 mol·L–1and 2.0 mol·L–1.

3.6.Multivariate analysis for integral capacitance

We used the regression-tree (RT) method to correlate the integral capacitance predicted by CDFT.As illustrated in Fig.5(a),the integral capacitance predicted by the M5P model is compared with the input integral capacitance predicted by CDFT.To prove the accuracy of this model,we have listed the estimated R,MAE and RMSE values in Fig.5(a).The M5P model works reasonably well for predicting the capacitance performance with correlation efficiency R up to 0.9138 and the RMSE is as small as 0.1361.

The RT model does not need to optimize the parameters by trial and error.Besides,it is easy to generate the relationship between the integral capacitance and the spherical radius under different electrolyte concentrations in the bulk (c):

Generally,the surface charge density is uniquely determined by the surface potential.Because the latter is commonly used in experiments to control the performance of the EDLC,we represent the integral capacitance as a function of the surface potential.The variables of the spherical electrode:radius R,operating potential window and electrolyte concentration c shown in Eq.(14)indicate that the integral capacitance could easily be influenced by changing these parameters.From the equation above,we can also see that the integral capacitance will be enhanced by increasing the electrolyte conccentration and the curvature,while it declines by raising the surface potential.

3.7.Effective EDL thickness

Fig.4.Relation between the integral EDL capacitance and the spherical radius under different operating potential window and bulk concentrations:(a) 0.6 mol·L–1,(b)1.0 mol·L–1,and (c) 2.0 mol·L–1.

Fig.5.(a) Comparison of the predicted capacitance from the M5P model (y-axis) and the capacitance predicted by CDFT (x-axis).(b) Relation between the effective EDL thickness and the spherical radius under different bulk concentrations at the surface electrical potential of 0.1 V.

In this work,we also delineated the effective EDL thickness near a charged spherical electrode.Experimentally,the performance of EDLCs is often correlated with the EDL thickness as predicted by the Helmholtz model.By fitting the integral capacitance with Eq.(14),we can get the effective EDL thickness as follows:

Fig.5(b) illustrates the relationship between effective EDL thickness and the spherical radius under different bulk concentrations.As shown,the effective EDL thickness declines with the electrolyte conccentration.At a given voltage,the curvature effect on the EDL thickness becomes less significant.

4.Conclusions

In summary,we report CDFT predictions of the ion distributions and mean electrostatic potentials of spherical electric double layer(EDL)using the restricted primitive model of electrolyte solutions.The EDL capacitance exhibits a positive correlation with the electrolyte concentration and the particle curvature.At low electrolyte concentrations (e.g.,＜1.0 mol·L–1),both differential capacitance and integral capacitance represent ‘‘camel”shape curves when it is plotted as a function of the surface potential,while they transferred to ‘‘bell”shape curves at higher electrolyte concentration due to the strong excluded volume effects.

In addition,we used the regression-tree model to correlate the capacitive performance with the electrode and electrolyte properties (curvature effect,electrode potential,electrolyte concentration).The M5P model works well for predicting the capacitance performance with a high correlation efficiency and a small RMSE.Among all factors considered in this work,both the curvature and the electrolyte concentration have positive correlations with the EDL capacitance.

Overall,the combination of CDFT and M5P models provides a quantitative description of the curvature effects and electrolyte properties on EDL performance (and the effective EDL thickness).The method is expected to be applicable to amorphous electrodes in general with different pore geometries such as cylindrical electrodes.These findings may open an avenue for quick screening of porous electrodes and electrolytes materials for diverse applications of EDL capacitors.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was sponsored by the National Natural Science Foundation of China(Nos.91834301,21908053,and 21808055),Shanghai Sailing Program (19YF1411700).J.W.and A.G.thank the financial support from the Fluid Interface Reactions,Structures and Transport (FIRST) Center,an Energy Frontier Research Center funded by the U.S.Department of Energy,Office of Basic Energy Sciences.We also acknowledge helpful discussions with Dr.K.Liu.