Role of the pore size on the capacitance of EDLCs

As mentioned in section 1.4.2, Chimola et al. observed an anomalous increase of surface area normalized capacitance for subnanometer pores with the size of the bare electrolyte ion (around 0.7 nm).^[152] The validity of the confinement effect has been subject of discussion in recent years. Some researchers assume the capacitance is independent of the pore size, which is described by a “regular pat-tern”.[184,185,190,191] Regarding these studies, results from various scientific papers were taken into account and compared, in which the carbons were used as electrode materials in organic electrolytes. The normalized capacitance values were calcu-lated using the BET specific surface area and the electrochemical active surface area generated by pores larger than 0.63 nm, which reflects the usable surface area of the carbon electrodes where capacitive charge storage is possible. The

capaci-35 tance values were plotted against the average micropore size calculated by the Du-binin-Radushkevich (DR) method as shown in Figure 18. The capacitance values normalized by the BET specific surface area show an anomalous increase of capac-itance for the smallest accessible pores around 0.7 nm, which would confirm previ-ous observations (Figure 18 a). The capacitance values normalized by the electro-chemically active surface area on the other hand, show no clear trends, for the entire pore range from 0.7 to 2 nm, hence behaving in a “regular pattern”. The normalized capacitance values scatter around 0.094 ± 0.011 F·m⁻² (Figure 18 b). Therefore the authors assume the anomalous increase of capacitance for pores around 0.7 nm is a result of the capacitance normalization with the inaccurate BET surface area. What is also important to note is that these studies represent the dependence of the capac-itance of the volume-weighted average micropore size. Even microporous carbons do not have a narrow and monomodal pore size distribution, instead they are rather multimodal. Therefore, the pore size specific influences on the capacitance cannot be obtained from the studies. This requires different approaches.

Figure 18: Variation of normalized capacitance in TEABF4/ACN vs. average mi-cropore size for a variety of porous carbons: (a) normalized to BET surface area;

(b) normalized to the electrochemical active surface area S>0.63 nm for pores larger than 0.63 nm.^[120] Copyright 2018, Elsevier.

The introduction of the electrochemical active surface area is an important factor for the proper determination of the influence on pore size and the capacitance. How-ever, one problem of the previous investigations was that the dependence of the capacitance was determined only against an average pore size, which does not re-flect the actual pore size distribution of most carbons as it only indicates an average

36

value. Chemically activated carbons usually have additional mesopores and there-fore a multimodal pore size distribution, which is why the relation between the av-erage pore size and the specific capacitance only is very inaccurate.^[189,192]

Although the subject is widely discussed, it was experimentally proven that par-tially of fully desolvated ions participate in the formation of the electric double layer.^[193] The confinement effect of partial or full removal of the solvation shell of ions, entering subnanometer pores is therefore possible. The Helmholtz model of the rigid double layer can be used for flat plate configurations and can describe the pore size-dependent thickness of the double layer in perfectly ordered slit pores, but is inaccurate for nanopores with different degrees of curvature caused by different pore geometries. Therefore, new models of capacitors have been proposed to de-scribe the double layer formation in a curved and highly nanoporous carbon net-work in porous carbons. Depending on the synthesis, nanoporous carbons can have various pore shapes, such as slit, cylindrical and sometimes spherical types.^[52]

Huang et al. adapted the classic Helmholtz model to develop a concept based on cylindrical mesopores (2 to 50 nm) and micropores (< 2 nm). In the mesopore re-gime, solvated ions enter the pores and approach the pore walls to form electric double-cylinder capacitors as shown in Figure 19 a.^[194,195] The surface area normal-ized double-cylinder capacitance is given by the following equation:

𝐶

𝐴 = 𝜀₀𝜀_r

𝑏ln[𝑏/(𝑏 − 𝑑)] (7)

where b is the radius of the cylinder pore and d the double-layer thickness. In such case, the effect of the pore size and pore curvature becomes prominent compared to the double-layer thickness d. In the micropore regime, partially or fully desolv-ated ions enter the pores and line up to form electric wire-in-cylinder capacitors, as shown in Figure 19 b and its capacitance is given by the following equation:

𝐶

𝐴 = 𝜀₀𝜀_r

𝑏ln(𝑏/𝑎₀) (8)

where the key quantity is a0, the effective size of the ion (the extent of electron density around the ions).[103,194,195] These models can be combined to simulate the

37 total capacitance of a carbon electrode material with a multimodal pore size distri-bution using equation (8) for micropores, equation (7) for mesopores and (4) for macropores with their respective surface areas.

Figure 19: Scheme of the (a) electric double-cylinder capacitor model with the pore radius b and the double layer thickness d and the (b) electric wire-in-cylinder ca-pacitor model with the effective size of the ion a0 or ion radius.

The electric double cylinder-capacitor and electric wire-in-cylinder capacitor els assume a cylindrical pore shape for micro- and mesopores and alternative mod-els have been proposed for other pore geometries. If the mesopores have a spherical shape, the geometric configuration of adsorbed electrolyte ions would differ, lead-ing to the followlead-ing equation derived by Wang et al.:^[196]

𝐶

𝐴 = 𝜀₀𝜀_r

𝑑 (𝑏 − 𝑑

𝑏 ) (9)

Although the electric double-cylinder capacitor model assumes a cylindrical pore geometry for micropores, it is supposed that micropores in carbon materials are rather slit pores, which can also show a curvature of pore walls.^[197] Feng et al. sug-gested that micropores area too narrow for electrolyte ions to experience the effect of pore wall curvature on the electric double layer. They adjusted the classic flat plate model, assuming slit-shaped micropores, to account for the confined double layer leading to a sandwich-type capacitor model, as described in equation (10) and

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shown in Figure 20.^[198] Furthermore, the effective double-layer thickness deff was introduced, which is the distance between the pore wall and the outer electron shell of the electrolyte ion.

𝐶

𝐴 = 𝜀₀𝜀_r

𝑏 − 𝑎₀ (10)

Figure 20: Sandwich-type capacitor model for confined slit-shaped micropores with the pore diameter of the electrode material 2b, the effective ion radius a0 and the effective double-layer thickness deff.

With the effective double-layer thickness, it is possible to describe the anomalous increase of capacitance for subnanometer pores very well. Hsieh et al. used the new capacitor model for slit pores (Figure 20) in combination with the electric double-cylinder capacitor model for mesopores (Figure 19 a) to simulate the capacitance of activated carbons based on data derived from physisorption measurements.^[199]

The relative permittivity of the solvent also plays a decisive role for determining the influence of the pore size on the capacitance. It is assumed that in confined spaces, e.g. in small pores, this permittivity is significantly smaller compared to bulk space. This is due to the restricted mobility of the solvate molecules, which leads to the fact that the electric field between electrolyte ions and electrode surface is maintained to a lesser extent due to the limited orientation polarization of the solvent molecules. Consequently, they proposed a pore size dependent dielectric permittivity, which increases linearly up to a pore size of 2 nm. According to equa-tion (10), the capacitance increases with increasing pore size, considering the in-crease of relative permittivity. However, since the term (b − a0) becomes larger with increasing pore size, the capacitance is reduced. These two counteracting effects

39 lead to a constant capacitance for the entire range of micropores. Despite this ap-proach, which contradicts the anomalous increase of capacitance at subnanometer pores, experimental and simulated capacitance values matched reasonably for or-ganic and aqueous electrolytes. Just recently, Zuliani et al. evaluated different com-binations of double layer capacitor models. They used the electrical wire and sand-wich type pore model for micropores and the cylinder and spherical model for mes-opores for various activated carbons, and compared the simulated capacitance with the measured capacitance.^[200] Using high resolution scanning electron microscopy (SEM) for investigation of the pore shape it was found that the activated carbon materials had curved mesopores. Furthermore, they observed the best match of sim-ulated and measured capacitance for a combined model of slit-shaped pores for mi-cropores and cylindrical shaped pores for mesopores, confirming the approach of Hsieh et al.