3. Electrode effect on the capacitive performance

### 3.1. Electrode geometry optimization

where <sup>β</sup>�<sup>1</sup> <sup>¼</sup> kBT, <sup>R</sup> � ð Þ <sup>r</sup><sup>δ</sup>þ;r<sup>δ</sup>� represents two coordinates specifying the positions of two segments in each solvent molecule, μα is the chemical potential of an ionic species, μ<sup>M</sup> is the chemical potential of the solvent, Ψað Þr stands for the external potential for ions, ΨMð Þ R is the

is the total intrinsic Helmholtz energy. The number densities of the positive and negative

The intrinsic Helmholtz energy F includes an ideal-gas contribution and an excess contribution

Where V<sup>b</sup> stands for the bonding potential of the solvent molecule. The detailed expression for each contribution and the numerical details can be retrieved from Ref. [45, 48]. In evaluation of the Coulomb energy, we calculate the mean electrostatic potential (MEP) from the density

Vbð Þ <sup>R</sup> <sup>r</sup>Mð Þ <sup>R</sup> <sup>d</sup><sup>R</sup> <sup>þ</sup><sup>X</sup>

4πe ε

a

ð

ð

ð

<sup>i</sup>¼δþ, <sup>δ</sup>� <sup>φ</sup><sup>i</sup>

lnrað Þ� <sup>r</sup> <sup>1</sup> � �rað Þ<sup>r</sup> <sup>d</sup><sup>r</sup> <sup>þ</sup> <sup>β</sup>Fex (5)

rcð Þr (6)

ð Þþ <sup>r</sup>; <sup>t</sup> <sup>β</sup>Við Þ<sup>r</sup> � � � � (7)

ð Þ r; t is the local chemical potential and could be

ð Þ r; t , follows the generalized

dRδð Þ r � r<sup>δ</sup><sup>þ</sup> rMð Þ R (3)

dRδð Þ r � r<sup>δ</sup>� rMð Þ R (4)

ð Þ r<sup>i</sup> , and F

summation of the external potential for a solvent molecule, i.e. <sup>Ψ</sup>Mð Þ¼ <sup>R</sup> <sup>P</sup>

rδþð Þ¼ r<sup>δ</sup><sup>þ</sup>

rδ�ð Þ¼ r<sup>δ</sup>�

ð

∇2

the time evolution for the local density profiles of ionic species, r<sup>i</sup>

<sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>∇</sup> � Dir<sup>i</sup>

∂rið Þ r; t

ψð Þ¼� r

Eq. (9) can be integrated with the boundary conditions that defined by the operation potential. The surface charge density Q is obtained from the condition of overall charge neutrality. The differential capacitance C<sup>d</sup> of the EDLs could be calculated by a derivative of the surface charge

Time-dependent density functional theory (TDDFT) is an extension of the CDFT to describe dynamic or time-dependent processes based on the assumption of local thermodynamic equilibrium [54–62]. For ion diffusion in an electrolyte solution near electrodes, TDDFT asserts that

ð Þ r; t ∇ βμ<sup>i</sup>

Where D<sup>i</sup> stands for the self-diffusivity of ion i, β ¼ 1=ð Þ kBT , kB is the Boltzmann constant, T

obtained by a derivative of the intrinsic Helmholtz energy F with respect to the density, and Við Þr denotes the external potential arising from the electrodes. With TDDFT, we could the

segments of the solvent are calculated from

140 Supercapacitors - Theoretical and Practical Solutions

due to intermolecular interactions Fex.

½ � lnrMð Þ� R 1 rMð Þ R dR þ β

distributions of the ions by using the Poisson equation

density Q with respect to the surface potential.

stands for the absolute temperature, μ<sup>i</sup>

capture the ion dynamics inside the nanopores.

βF ¼ ð

diffusion equation

Whereas practical porous electrodes involve micropores with complicated morphology and pore size distributions [63, 64], theoretical modeling of EDLCs is mostly based on simplistic models to represent the pore geometry and the electrolyte-electrode interactions [65]. Specifically, three types of electrode structures are commonly used in theoretical investigations [66– 68]: (i) planar surfaces (e.g., a flat surface or slit pores); (ii) cylindrical pores with their concave inner surfaces or cylindrical particles with their convex outer surfaces (e.g., carbon nanotubes); and (iii) spherical surfaces (e.g., onion-like carbons). The slit and cylindrical pore models are conventionally used for porous materials characterization [69]. Despite the fact that a great variety of porous carbons have been utilized in EDLCs, the effects of the pore size and geometry on the EDL structure remain poorly understood [70]. At the heart of the issue is the question: What is the microscopic structure of porous electrodes and how does the capacitance of EDLCs depend on the electrode pore geometry and electrolyte composition? Recent simulations and experiments indicate that both the pore size and geometry play an important role in determining the capacitance of EDLCs [25, 68, 71–73]. An important question is whether this behavior is generally valid, given the slit-pore model or solid particles used in theoretical calculations and the diversity of pore structure for realistic carbon electrodes. Specially, how does the pore structure and curvature affect the capacitance dependence on the pore size? To address these questions, we propose in this work a generic model to represent both pore size and curvature of carbon electrodes using the CDFT. CDFT is an ideal computational tool for examining the pore size and geometry effects, as it is computationally efficient and applicable over a wide range of pore sizes ranging from that below the ionic dimensionality to mesoscopic scales.

Figure 1 shows the integral capacitance as a function of pore width D at different inner core radii. In all cases, the surface electrical potential is fixed at ψ<sup>0</sup> = 1.5 V. As observed in an previous work for an ionic liquid in slit pores [41], the EDL capacitance exhibits the oscillatory dependence on the pore size. The distance between neighboring peaks (or valleys) is approximately equal twice the ion diameter. The oscillatory variation of the integral capacitance is closely affiliated with the layering structures of ion distributions inside the nanopores. The layer-by-layer distributions of cations and anions are evident near the charged surfaces [41].

As inner radius R decreases, the capacitance increases significantly. This is also expected from the increased EDL capacitance at both the inner and outer surfaces. The oscillatory dependence of the capacitance on the pore size is consistent with those corresponding to individual EDLs. Our results show that the EDLs have a smaller influence on the overall ion distributions inside the pore as the pore size falls, leading to a diminishing difference in average counterion and coion densities. On the other hand, a smaller inner core radius results in more counterions in the pore thus a larger capacitance.

This work illustrates the curvature and pore size effect of realistic porous electrodes and suggests the significant role of convex surfaces for the synthesis of new porous electrodes to

pores when the electrodes are operated at sufficiently high voltages [74]. It has also been shown that the charging kinetics of an empty ionophobic nanopore is much faster than that of an ionophilic nanopore at similar conditions [75, 77]. Experimentally, the ionophobicity may be controlled by modifying the surface properties of nanoporous materials or by introducing

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We discussed the effects of non-electrostatic ion-surface interactions based on the classical density functional theory (CDFT) [54, 78]. δE stands for the resolvation energy, i.e. the energy cost to transfer an ion from the bulk to the slit pore. δE is used as an indication of the ionophobicity of nanopores: negative δE promotes adsorption of ions within the pore (viz., an

Figure 2. Theoretical predictions for the surface charge density (a), the differential capacitance (b), the average cation and anion densities inside the nanopore (c), and the energy stored per surface area (d) for an EDLC containing an organic electrolyte. Because of the system symmetry, the results are the same if the surface potential, ψ, is extended to the negative values. Here the concentration of the organic electrolyte is fixed at 1.0 M, the pore size is D = 0.6 nm. The ionophobicity of the nanopores is reflected in the ion resolvation energy, δE, which represents the ion transfer energy from the bulk reservoir into the nanopore. In (c), the solid lines are average densities inside the pore for the counterions, and the dashed lines are for the coions. Because of their small values, the curves for the average coion densities collapse into the same zero

line. Reproduced from Ref. [54] with permission. Copyright 2016 IOP Publishing.

special functional groups to the ionic species.

Figure 1. The overall integral capacitance versus the pore size for spherical shells of different inner radii. Reproduced from Ref. [41] with permission. Copyright 2016 American Chemical Society.

optimize EDLC performance. In particular, the spherical shell model provides a simple yet generic description of both pore size and curvature, opening up a new dimension to characterize nanoporous materials and quantify their performance for diverse applications including EDL capacitors.

#### 3.2. Electrode surface modification

Energy storage via electrosorption depends not only on the electrical potential and the geometric compatibilities of the electrode pores and ionic species but also on specific interactions between mobile ions and the surface properties of electrode materials. While existing theoretical reports are mostly devoted to analyzing electrostatic interactions and confinement effects, relatively little is known on how specific ion-surface associations may influence the EDLC performance. Very recently, Kondrat and Kornyshev found that the capacitive performance is sensitive to the ion affinity with nanopores: their theoretical results show electrodes with ionophobic nanopores may have slightly lower, the same, or even higher energy storage capacity than the ionophilic ones, all depending on the electrode voltage [74–76]. The capacitance voltage curve is shifted to substantially higher voltages as the pore ionophobicity increases. Within an ionophobic pore, the stored energy could be higher than for ionophilic pores when the electrodes are operated at sufficiently high voltages [74]. It has also been shown that the charging kinetics of an empty ionophobic nanopore is much faster than that of an ionophilic nanopore at similar conditions [75, 77]. Experimentally, the ionophobicity may be controlled by modifying the surface properties of nanoporous materials or by introducing special functional groups to the ionic species.

We discussed the effects of non-electrostatic ion-surface interactions based on the classical density functional theory (CDFT) [54, 78]. δE stands for the resolvation energy, i.e. the energy cost to transfer an ion from the bulk to the slit pore. δE is used as an indication of the ionophobicity of nanopores: negative δE promotes adsorption of ions within the pore (viz., an

optimize EDLC performance. In particular, the spherical shell model provides a simple yet generic description of both pore size and curvature, opening up a new dimension to characterize nanoporous materials and quantify their performance for diverse applications including

Figure 1. The overall integral capacitance versus the pore size for spherical shells of different inner radii. Reproduced

Energy storage via electrosorption depends not only on the electrical potential and the geometric compatibilities of the electrode pores and ionic species but also on specific interactions between mobile ions and the surface properties of electrode materials. While existing theoretical reports are mostly devoted to analyzing electrostatic interactions and confinement effects, relatively little is known on how specific ion-surface associations may influence the EDLC performance. Very recently, Kondrat and Kornyshev found that the capacitive performance is sensitive to the ion affinity with nanopores: their theoretical results show electrodes with ionophobic nanopores may have slightly lower, the same, or even higher energy storage capacity than the ionophilic ones, all depending on the electrode voltage [74–76]. The capacitance voltage curve is shifted to substantially higher voltages as the pore ionophobicity increases. Within an ionophobic pore, the stored energy could be higher than for ionophilic

EDL capacitors.

3.2. Electrode surface modification

142 Supercapacitors - Theoretical and Practical Solutions

from Ref. [41] with permission. Copyright 2016 American Chemical Society.

Figure 2. Theoretical predictions for the surface charge density (a), the differential capacitance (b), the average cation and anion densities inside the nanopore (c), and the energy stored per surface area (d) for an EDLC containing an organic electrolyte. Because of the system symmetry, the results are the same if the surface potential, ψ, is extended to the negative values. Here the concentration of the organic electrolyte is fixed at 1.0 M, the pore size is D = 0.6 nm. The ionophobicity of the nanopores is reflected in the ion resolvation energy, δE, which represents the ion transfer energy from the bulk reservoir into the nanopore. In (c), the solid lines are average densities inside the pore for the counterions, and the dashed lines are for the coions. Because of their small values, the curves for the average coion densities collapse into the same zero line. Reproduced from Ref. [54] with permission. Copyright 2016 IOP Publishing.

ionopobilic pore), while a positive δE means an ionophobic pore. For simplicity, we assume that δE is independent of the ion valence, the pore size, and the surface electrical potential.

dependence on the pore size for ionic liquid and organic electrolyte EDLCs. In the previous section, we discussed the capacitance dependence on pore size for an organic electrolyte with a moderate-polarity solvent (with a dipole moment of 3.4 Debye). But how would the solvent polarity affect the pore-size dependence of capacitance in a wide range of organic electrolyte

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Our CDFT calculations provide valuable insights into the effects of the dipole moment of the solvent in an organic electrolyte [43]. We found an optimal dipole moment that yields a maximum in the large-pore capacitance. These theoretical results further illustrate the rich behavior of the organic electrolytes inside porous electrodes. Moreover, it provides new considerations that can be taken into account when designing new experiments to select organic

RTIL has been used as electrolytes widely to increase the capacitive performance of electrochemical capacitors [82]. However, there are always some small amounts of impurity (e.g., water, alkali salts, and organic solvents) in RTIL, which may affect the electrochemical behavior of electrochemical devices. To understand the impurity effect, the RTIL with different impurities in the porous carbon electrodes is studied via CDFT. With a different type of binding energy with the surface or ionic species, the impurity shows a different influence on the EDL microstructures and contributes differently to the integral capacitance. It is noted that the impurity can be considered as either a contaminant or an additive to the ionic liquid, all depending on the interaction between the impurity and the electrode or ions [79, 80]. Meanwhile, the capacitance strongly oscillates with the variation of the pore size similar to that for the pure ionic liquid electrolyte. With strong binding of impurity to the ionic species, the RTIL/ impurity mixture may lead to an enhanced capacitance oscillation. In certain pores, a significant increase in the capacitance can be obtained. The theoretical results provide insights for further investigation of supercapacitors aiming at rational design of porous electrode materials

We also demonstrate that, under conditions favoring impurity accumulation in the nanopores of the electrode, impurity can change the EDL charging mechanism even at low bulk densities, shown in Figure 3. As the adhesion energy of impurity molecules with the electrode surface increases, the capacitance-potential curves can change from the bell shape to the two-hump camel shape, with the peak shifting toward a higher charging potential. Qualitatively the impurity effect on the charging behavior is similar to the solvent effect as studied by Rochester and coworkers [81]. Whereas the amount of impurity and solvent in the bulk is considerably different, the concentration effect can be compensated by the change in the transfer energy to yield a same charging behavior. As an ionophobic pore could be beneficiary for improving charge storage and charging dynamics, introduction of impurity molecules inside the pore makes it essentially more ionophobic thus enhances the energy storage. Our theoretical results suggest that special attention should be given to the nature of impurity and operation voltages when the surface properties nanoporous electrodes are modified to enhance the performance of EDLCs. It is worth noting that association between ions and impurity molecules, which

EDLC?

electrolytes for EDLCs.

4.2. Impurity or additive

and charge carriers.

In an ionopobic pore with a large ion resolvation energy, both counterions and coions are nearly excluded from the nanopore at low surface electrical potential. In this case, counterions are inserted into the empty pore only when the electrical potential is sufficiently large to overcome the surface repulsion. The differential capacitance increases with the surface potential until it reaches a maximum. A further increase of the surface electrical potential leads to saturation of counter ions inside the pore and thus a decline of the capacitance. In both ionopobic cases shown in Figure 2(b), the capacitance versus potential curve has a two-hump camel shape, and the capacitance value at the minimum is close to zero, in stark contrast to the maximum for an ionophilic pore. Figure 2(b) indicates that the peak capacitance shifts to a higher potential as the ionophobicity increases. In other words, we may find a bell shape to the two-hump camel shape transition in the capacitance-electrical potential curves by changing the surface ionophobicity. Figure 2(d) shows that, at low electrical potential, the energy density (Eð Þ¼ <sup>ψ</sup> <sup>Ð</sup> <sup>ψ</sup> <sup>0</sup> CDð Þx xdx) for an ionophilic pore is higher than that of an ionophobic pore, while the trend is opposite at high potentials. Because the peak of the camel shape differential capacitance shifts to higher potential, a more ionophobic pore offers higher storage energy. On the other hand, increasing ionophobicity prohibits counter ions from entering the pore, thus reduces the energy density at low potential. From the discussions above, we find that an ionophobic nanopore prevents counterion insertion and shifts the saturation point to a higher voltage. The energy stored in the EDLC can be promoted by the ionophobicity only when the electrode voltage is larger than a critical value [54].
