2. Classical density functional theory

energy by electrochemical reactions, while a supercapacitor stores energy through physicochemical adsorption, including formation of electric double layer and surface redox reactions. A battery has high energy density but low power density, while a supercapacitor boasts of high power density due to the fast surface physical and chemical processes. There are two types of supercapacitors: electric double-layer capacitors (EDLC) store electrical energy through formation of electric double layer at electrode/electrolyte interface, while pseudocapacitors store electrical energy by reversible redox reaction (including ion intercalation). Supercapacitors are becoming increasingly important for electrical energy storage due to their advantages on rapid charge/discharge rate, a virtually unlimited life, a wider range of working temperature, a very high efficiency, etc. However, the main drawback, a low energy density, has been limiting supercapacitors in the applications requiring many rapid charge/discharge cycles for short-term powder needs. It is the most urgent issue to improve the energy density of supercapacitors [1]. The EDLC performance is strongly correlated with electrosorption of

To design new materials and interfaces for EDLC with higher energy density, one requires a deeper understanding of the factors and contributions affecting the total capacitance of an EDLC. The most widely used electrode material for EDLC is porous carbon. Many types of carbon-based materials have been used for EDLC [3, 4], such as activated graphene oxide [5], activated carbon [6], carbide-derived carbon [7–9], carbon nanotube [10–12], onion-like carbon [13, 14] and graphene [15, 16]. The gravimetric capacitance of those carbon materials is quite sensitive to their structure, especially the porosity and specific surface area. The pore size can greatly affect the ion partitioning and packing inside the pore, which causes a large change on the capacitance. The relationship between the pore size and the capacitance of ionic liquids has been investigated by Simon and Gogotsi [1, 17, 18]. This important work reveals a clear physical insight into the pore size-dependent capacitance and suggests that the capacitance maximum can be achieved by optimally matching the pore size and ion size. Carbon nanotube has been reported as a novel EDLC electrode material [19, 20]. The reported capacitance of single-wall carbon nanotube is 180 F/g in aqueous electrolyte [11]. The onion-like carbon was also reported as a promising EDLC electrode and exhibit very large power density at discharging rate of up to 200 V/s [13, 21]. Moreover, graphene-based material also has been developed to be attractive EDLC material and its unique electronic structure could have large influence on the charge capacitive behavior [6, 15]. The pseudocapacitors and EDLC show distinctly different electrochemical behavior in cyclic voltammetry. Pseudocapacitance may

contribute more capacitance than double-layer capacitance for the same surface area.

In general, the electrode-electrolyte interface is the most key issue of supercapacitors, fundamental understanding on this should be crucial. Experimental tools such as Atomic Force Microscopy (AFM) and X-ray reflectometry were applied to study the structural properties of EDLs at the electrode surface [22–24]. However, it's difficult for experimental method to direct detect the nanoscale electrode-electrolyte interface. Computational methods, such as molecular dynamics (MD) simulations, were also used to investigate the distribution of ions near electrified interfaces [25–27]. Analytical methods [28–37] are computationally more efficient than MD simulations thereby suitable for a systematic investigation of the key parameters for relatively large systems. CDFT can be used to account for the ionic steric effects and electrostatic

ionic species at the inner surfaces of microporous electrode [2].

138 Supercapacitors - Theoretical and Practical Solutions

We use a non-primitive model to represent the ionic species, impurities, and solvent molecules in the electrolyte solution [48]. The model system consists of charged hard spheres for ionic species and a hard-sphere dimer for solvent molecules. The pair potential between two arbitrary spheres/segments in the system, i and j, is given by

$$u\_{i\bar{j}}(r) = \begin{cases} \rightsquigarrow & r < \frac{\sigma\_i + \sigma\_{\bar{j}}}{2} \\\\ \frac{Z\_i Z\_{\bar{j}} \varepsilon^2}{4 \pi \varepsilon\_0 \varepsilon r}, & r \ge \frac{\sigma\_i + \sigma\_{\bar{j}}}{2} \end{cases} \tag{1}$$

where r is the center-to-center distance, e is the elementary charge, ε<sup>0</sup> is the permittivity of free space, ε ¼ 1 is the local dielectric constant for the vacuum.

CDFT [44, 49–52] was used to obtain the EDL structure and capacitance for the carbon materials in contact with the electrolyte solution. The details of the CDFT calculations have been published before [42, 45, 47, 48, 53, 54]. Briefly, we obtained the surface charge densities at various electrical potentials. Given the number densities of ions and solvent molecules in the bulk and the system temperature, the pore size, the pore geometry, and the surface electrical potential, we solve for the density profiles of cations, anions and impurities, as well as the solvent segments inside the pore by minimization of the grand potential.

$$\begin{aligned} \left[\beta \Omega \left[\rho\_{\mathcal{M}}(\mathbf{R}), \left\{\rho\_{a}(\mathbf{r})\right\}\right] \right] &= \beta F \left[\rho\_{\mathcal{M}}(\mathbf{R}), \left\{\rho\_{a}(\mathbf{r})\right\}\right] + \int \left[\beta \Psi\_{\mathcal{M}}(\mathbf{R}) - \beta \mu\_{\mathcal{M}}\right] \rho\_{\mathcal{M}}(\mathbf{R}) d\mathbf{R} \\ &+ \sum\_{a} \int \left[\beta \Psi\_{a}(\mathbf{r}) - \beta \mu\_{a}\right] \rho\_{a}(\mathbf{r}) d\mathbf{r} \end{aligned} \tag{2}$$

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 summation of the external potential for a solvent molecule, i.e. <sup>Ψ</sup>Mð Þ¼ <sup>R</sup> <sup>P</sup> <sup>i</sup>¼δþ, <sup>δ</sup>� <sup>φ</sup><sup>i</sup> ð Þ r<sup>i</sup> , and F is the total intrinsic Helmholtz energy. The number densities of the positive and negative segments of the solvent are calculated from

$$
\rho\_{\delta+} (\mathbf{r}\_{\delta+}) = \int d\mathbf{R} \delta(\mathbf{r} - \mathbf{r}\_{\delta+}) \rho\_M(\mathbf{R}) \tag{3}
$$

3. Electrode effect on the capacitive performance

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

Classical Density Functional Theory Insights for Supercapacitors

http://dx.doi.org/10.5772/intechopen.76339

141

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

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

3.1. Electrode geometry optimization

mesoscopic scales.

the pore thus a larger capacitance.

$$
\rho\_{\delta-} (\mathbf{r}\_{\delta-}) = \int d\mathbf{R} \delta(\mathbf{r} - \mathbf{r}\_{\delta-}) \rho\_M(\mathbf{R}) \tag{4}
$$

The intrinsic Helmholtz energy F includes an ideal-gas contribution and an excess contribution due to intermolecular interactions Fex.

$$\beta \mathcal{F} = \int [\ln \rho\_M(\mathbf{R}) - 1] \rho\_M(\mathbf{R}) d\mathbf{R} + \beta \int V\_b(\mathbf{R}) \rho\_M(\mathbf{R}) d\mathbf{R} + \sum\_a \int \left[ \ln \rho\_a(\mathbf{r}) - 1 \right] \rho\_a(\mathbf{r}) d\mathbf{r} + \beta \mathcal{F}^{\rm ex} \tag{5}$$

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 distributions of the ions by using the Poisson equation

$$
\nabla^2 \psi(\mathbf{r}) = -\frac{4\pi e}{\varepsilon} \rho\_c(\mathbf{r}) \tag{6}
$$

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 density Q with respect to the surface potential.

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 the time evolution for the local density profiles of ionic species, r<sup>i</sup> ð Þ r; t , follows the generalized diffusion equation

$$\frac{\partial \rho\_i(\mathbf{r}, t)}{\partial t} = \nabla \cdot \left\{ D\_i \rho\_i(\mathbf{r}, t) \nabla \left[ \beta \mu\_i(\mathbf{r}, t) + \beta V\_i(\mathbf{r}) \right] \right\} \tag{7}$$

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 stands for the absolute temperature, μ<sup>i</sup> ð Þ r; t is the local chemical potential and could be 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 capture the ion dynamics inside the nanopores.
