Role of porosity and diffusion coefficient in porous electrode used in supercapacitors – Correlating theoretical and experimental studies

Abstract

Porous electrodes are fast emerging as essential components for next-generation supercapacitors. Using porous structures of Co₃O₄, Mn₃O₄, α-Fe₂O₃, and carbon, their advantages over the solid counterpart is unequivocally established. The improved performance in porous architecture is linked to the enhanced active specific surface and direct channels leading to improved electrolyte interaction with the redox-active sites. A theoretical model utilizing Fick's law is proposed, that can consistently explain the experimental data. The porous structures exhibit ∼50%–80% increment in specific capacitance, along with high rate capabilities and excellent cycling stability due to the higher diffusion coefficients.

1 INTRODUCTION

Electrochemical energy storage devices, for example, mobile-ion-based secondary batteries and supercapacitors, are amongst the most investigated technologies, which are associated with cleaner energy utilization. The massive progress in these systems is directly linked to the large-scale integration of nanomaterial in such devices.^{[1, 2]} Till a few years back, the main focus was to develop nanomaterials with higher specific surface area.^{[3, 4]} It was mostly believed that the higher is the surface area, the higher would be the specific capacity of these materials. Since the discovery of graphene and two-dimensional structures like MXenes, it is clear that there are other factors, which drive the enhancement in the storage capacity.^{[5, 6]} For example, layered rGO, gC₃N₄, GO, etc., which have lower specific surface area than most of the reported activated carbons, are able to deliver much high specific capacitance.^[7-9] Similar features have been observed in functional metal oxides based on 2D- and 3D materials. Here also, the exfoliated 2D structures have been able to deliver much higher specific capacitance than their 2D counterparts, which have higher surface area.^{[10, 11]} Therefore, it is clear that the role of a higher specific area of nanomaterials in driving enhanced electrochemical performance is overestimated. This is because, once a binder, conducting component, and the active material mixed film is fabricated, the effective surface area is appreciably suppressed. It has been shown that the major additional factors, that lead to enhanced performance are: porosity, ion-channel size, pore-volume, interlayer distance, termination elements, and so forth.^{[8, 12-15]} Hence, there has been a paradigm shift in focus, where research is mostly being directed toward the development of porous structures of the various particle morphologies that were being routinely used in electrode materials.

The major advantages of lighter but stable porous electrodes are linked to the higher interaction of electrolytes with the active material's surface. This leads to enhanced electrochemical reactions, shorter diffusion lengths, and reduced Ohmic, polarization, or concentration resistances.^[16] This brings appreciable improvement in the energy and power density values of the devices. Pseudocapacitive transition metal oxides serve as supercapacitor electrodes. Still, poor capacitance and low energy density impose restrictions on their practical applications.^[17] According to the working principle of pseudo-capacitors, electrode materials and electrolytes should synergistically combine. The electrolyte ions should transport fast in and out of the bulk of the electrode and at the interface between electrode and electrolyte in order to achieve excellent electrochemical properties.

In this paper, using a large number of metal oxides and carbon, the advantages of using porous electrodes in pseudocapacitors and electric double layer (EDL) are unequivocally established. Various morphologies of M-oxides (M = Co, Mn, Fe, Ni, Cu, etc.) are generally reported in the literature.^[17-23] But a comprehensive report dealing with their porous or hierarchical nanostructures remains elusive. Here, various synthesis protocols, that can be used to easily obtain porous structures, with tunable electrochemical responses, are discussed. The advantages of these morphologies are proven by comparing the performance with their corresponding solid structures. The electrochemical characterization of the synthesized porous materials shows superior performance, with ∼50%–80% increment in specific capacitance, in comparison to their solid morphologies. Using a mathematical model, it is also shown that the role of diffusion coefficient becomes critical in enhancing the response characteristics of porous structures. Therefore, the usefulness of porous materials for large-scale industrial use is proposed.

2 EXPERIMENTAL SECTION

2.1 Materials

2.1.1 Synthesis of M-oxides (M = Co, Mn, Fe) and carbon

Cobalt (II) chloride hexahydrate (CoCl₂.6H₂O), manganese (II) acetate (C₄H₆MnO₄), urea (CH₄N₂O), hexadecyltrimethyl-ammonium bromide (CTAB), iron (II) sulfate heptahydrate (FeSO₄.7H₂O), oxalic acid (C₂H₂O₄), sodium dihydrogen phosphate (NaH₂PO₄), and dimethylformamide (DMF) were purchased from Merck Industries Pvt. Ltd (India). All the precursors were used without further purification.

Solid and porous nanostructure of Co₃O₄

Porous nanostructures of Co₃O₄ were prepared using a low-temperature hydrothermal protocol.^[24] Initially, CoCl₂.6H₂O and urea were mixed in deionized (DI) water with a molar ratio of 2:1 and stirred for 30 min. The resultant solution was left undisturbed for ∼12 h. Subsequently, the solution was transferred to a sealed borosilicate glass bottle. It was heated in an oven at 120°C for 8 h, to obtain a pink precipitate. The precipitate was washed with DI water and ethanol several times to remove the spare urea. The resultant filtrate was vacuum dried at 80°C for 12 h. The dried powder was calcined at 500°C for 3 h in air, which resulted in the desired final product. The schematic of the synthesis protocol is given in Scheme S1a. The solid nanostructure of Co₃O₄ was synthesized by the co-precipitation method^[25] as described in the Supporting Information (SI).

Solid and porous nanostructure of Mn₃O₄

To synthesize porous structures of Mn₃O₄ using a hydrothermal method, 0.2 g of CTAB was dissolved in DMF. Subsequently, 0.01 M manganese acetate was added under constant stirring for 1 h. Note that, 0.6 g of urea was then added and the solution was poured into a 300 ml Teflon stainless steel autoclave. This was heated at 140°C for 4 h. The precipitate was washed several times with DI water and ethanol. Finally, the washed precipitate was air-dried at 60°C for 8 h. The schematic of the particle growth mechanism is presented in Scheme S1b. The solid nanostructure of Mn₃O₄ was synthesized by following a typical co-precipitation method. The details of the synthesis procedure are provided in the SI.

Solid and porous structure of Fe₂O₃

For obtaining porous Fe₂O₃ structures, 0.4 M FeSO₄.7H₂O was mixed with an equal amount of 0.4 M C₂H₂O₄ and NaH₂PO₄ solutions. The reaction was kept stirring at 70°C for 30 min. During the reaction, iron (II) oxalate complex formed, which could be confirmed from the change of colorless solution to yellow. The precipitate was washed with distilled water and ethanol, before being vacuum dried at 45°C for 12 h. The product was calcined at 400°C in the air for 12 h. This led to the formation of a porous microrod of Fe₂O₃ (see Scheme S2 for particle growth mechanism). The synthesis procedure of solid micro rods is also given in the SI.

Food waste-derived porous carbons

The raw food waste (FW) was used to prepare porous carbon. The collected FW was dried in a hot air oven for 24 h and was grounded using a blender to enhance the consistency and diffusion of sub-critical water into the FW during the hydrothermal container. The ground FW was sieved using 475-75μ sieves. The portions containing 300–75 μm (to maintain uniform particle size) sieved samples were used for hydrothermal carbonization (HTC) experimentation.

The FW was mixed with distilled water in a 1:10 solid/solvent ratio, and the mixture was eventually subjected to an HTC reactor for 6 h at 220°C before it naturally cooled to room temperature. The final solid product (hydrochar) was collected through vacuum filtration, washed with distilled water, and then oven-dried at 100°C for 12 h. The prepared hydrochar was further activated by pyrolyzing at 900°C for 2 h (Ants Prosys, India), with a slow heating rate of 5°C/min under the N₂ atmosphere.^[26] The growth mechanism is depicted in Scheme S3. For the solid structure, commercially available activated carbon was used.

2.2 Materials characterization

The crystalline phases of all the synthesized materials were analyzed by utilizing the X-ray diffraction (XRD) studies. The diffraction data were collected using a Rigaku analytical diffractometer with Cu-Kα incident radiation (λ = 0.15406 nm). The morphologies of the synthesized particles were investigated using scanning electron microscopy (SEM CARL ZEISS SUPRA 40) and transmission electron microscopy (TEM FEI-TECHNAI G220S-Twin). The pore size distribution and specific surface area of synthesized materials were tested by the Brunauer-Emmett-Teller (BET) method, using Quantachrome Novatouch surface area and a pore size analyzer.

2.3 Electrochemical characterization

The electrochemical tests, such as cyclic voltammetry (CV), galvanostatic charge-discharge (GCD) of the synthesized materials, were performed using a Metrohm Autolab (PGSTAT302N) potentiostat. The measurements were carried out in 3-electrode cells, using a compatible aqueous electrolyte. Additionally, the three-electrode cell comprised of an Ag/AgCl (in 3 M KCl) as the reference electrode, a platinum rod-based counter electrode, and a working electrode. The working electrodes were prepared by mixing 80 wt% of active material, 10 wt% of activated carbon as a conductive agent, and 10 wt% polyvinylidene fluoride as a binder, using acetone as a mixing media. The mixture was ultrasonicated and stirred at 60°C to obtain a homogeneous slurry. The obtained slurry was drop cast uniformly onto a graphite sheet so as to cover an area of 1 cm². The typical mass loading of the active electrode materials was kept at ∼1 mg/cm². Finally, electrode films were vacuum dried at 80°C for 12 h. Electrochemical impedance spectroscopy (EIS) measurements were also conducted using a Zehner impedance analyzer (ZEHNER ZENNIUM pro with THALES XT Software) in the frequency range of 5 mHz to 1 MHz.

2.4 Theoretical model

The flux of ionic substance (J) can be expressed by Fick's law. The concentration of electrolyte ions was evaluated from the expression of Fick's law and utilizing the Laplace transformation technique. Using Nernst's and two parameterized equations; one each for charging and discharging, respectively, diffusion coefficient could be estimated. C++ programming was used to write the codes to fit the two parameterized equations.

3 RESULTS AND DISCUSSIONS

3.1 Morphology and structural analysis

Powder XRD data were analyzed to confirm the phase formation of the synthesized materials. Figure 1a–d depicts the XRD patterns for Co₃O₄, Mn₃O₄, Fe₂O₃, and carbon-based solid and porous powders, respectively. No impurity peaks were observed in any of the profiles, indicating the formation of single-phase materials. The diffraction profiles could be indexed using the relevant Joint Committee on Powder Diffraction Standards cards (the respective number is mentioned in the figure). The calculated lattice parameters for each of the materials are tabulated in Table 1. These clearly indicated that the unit cell parameters or the unit cell, for a given metal oxide did not vary significantly with the particle morphology. Hence, if there are changes in electrochemical performance, they can be mostly linked to the variation in the particle shape and sizes.

FIGURE 1

X-ray diffraction (XRD) profiles of solid and porous particles of (a) Co₃O₄, (b) Mn₃O₄, (c) Fe₂O₃, and (d) carbon

TABLE 1. Calculated lattice parameters values from the X-ray diffraction (XRD) profile

		(Å)			(degree)
Material	System name	A	B	C	Α	Β	$${\bm{\gamma }}$$	Cell volume(Å^3)	Grain size(nm)
Co3O4 solid	Monoclinic	8.09	8.08	7.96	90	89.4	90	520.1	18.4
Co3O4 porous	Monoclinic	8.09	8.04	8.14	90	90.5	90	529.4	36.4
Mn3O4 solid	Monoclinic	6.06	5.06	9.18	90	79.4	90	276.9	13.9
Mn3O4 porous	Monoclinic	5.98	5.12	9.23	90	79.5	90	277.8	9.5
Fe2O3 solid	Triclinic	5.05	5.06	13.76	90.18	89.8	120.3	303.43	24.3
Fe2O3 porous	Triclinic	5.05	5.06	13.77	90.31	89.76	120.3	303.76	18.5

The SEM pictures are shown in Figure 2. SEM micrographs revealed that the particle size of the porous materials was smaller than their corresponding solid particles. For example, in porous Co₃O₄, particles of 50–75 nm were discernible (Figure 2b), while, in Co₃O₄ solid material, particles of > 200 nm were observed (Figure 2a). The SEM micrographs for other materials were also investigated and are depicted in Figure 2c–h. Porous structures of the materials are generally developed by the arrangements of small units. So, porous structures can grow up to a certain range of radius to maintain their stability. With respect to that, solid particles are more stable in nature and can grow bigger in size than their porous counterparts. The importance of these smaller and homogeneous particle size distributions in the porous materials will become evident after the electrochemical analysis. In porous structures, particles were loosely bound together, leading to the appearance of directed channels. The TEM micrographs further support this result as shown in Figure S1.

FIGURE 2

Scanning electron microscopy (SEM) micrographs of the synthesized solid and porous structure of (a, b) Co₃O₄, (c, d) Mn₃O₄, (e, f) Fe₂O₃, and (g, h) carbon

In addition to morphological features, the specific surface area plays an important role in determining the capacitive behavior of these materials. It was determined by collecting and analyzing the N₂ adsorption-desorption isotherms. Figure 3 depicts the isotherms for the porous and solid structures, respectively, of the investigated materials. The calculated BET-specific surface areas for these materials are listed in Table 2. Clearly, the specific surface area values of the porous materials were much higher than their solid counterparts.

FIGURE 3

(a–d) N₂ adsorption-desorption isotherms of the synthesized porous structured materials (Inset: corresponding solid structures)

TABLE 2. The calculated surface area and pore radius in solid and porous materials

Material	Morphology	BET surface area (m^2/g)	Pore radius (nm)
Co3O4	Solid	23	2.52
	Porous	49	7.20
Mn3O4	Solid	31	1.70
	Porous	91	1.82
Fe2O3	Solid	30	1.58
	Porous	61	1.61
Carbon	Solid	24	1.53
	Porous	220	1.57

The pore size and volume were determined by the Barrett-Joyner-Halenda (BJH) studies. Mesoporous and microporous structures were observed. The exact values are listed in Table 2.

3.2 Electrochemical performances of solid and porous transition metal oxides

Among the various kinds of aqueous electrolytes (acidic, alkaline, and neutral electrolyte), acidic electrolyte (H₂SO₄) has the highest ionic conductivity. But acidic electrolytes are commonly not suitable for the electrode of metal oxides due to their highest corrosive nature. After the acidic electrolyte, alkaline electrolyte (KOH, NaOH, and LiOH) has a greater conductivity than the neutral electrolyte (KCl, NaCl, K₂SO_4, and Na₂SO₄). So, alkaline electrolytes have been utilized extensively in the literature. Potassium hydroxide (KOH) is the most commonly used alkaline aqueous electrolyte, simply because of its higher ionic conductivity (0.54 S/cm, Table S1) and higher ionic mobility with a lower hydration radius (Table S2), which helps in enhancing the accumulation of charges at the electrode-electrolyte interface. Hence, all the measurements were performed in a 2 M KOH aqueous alkaline electrolyte. Figure 4a–c depicts the representative CV curves for solid and porous structures of Co₃O₄, Fe₂O_3, and Mn₃O₄, at a scan rate of 50 mV s^–1. The CV profiles of porous structure materials enveloped a larger area than their solid counterparts. This indicated improved electrochemical properties, as the value of the specific capacitance is directly proportional to the area under the CV curves. The CV profiles of solid and porous structures of transition metal oxides (TMOs), at various other scan rates, are given in Figure S2. The electrodes showed quasi-rectangular CV curves, with clear reduction-oxidation peaks, indicating the coexistence of pseudocapacitive and as well as the EDL-type nature in the materials. The redox peaks could be attributed to the following redox reactions occurring at the respective electrode-electrolyte interfaces:

$$$$\begin{equation}{\bm{C}}{{\bm{o}}_3}{{\bm{O}}_4}:\ C{o_3}{O_4} + {H_2}O + O{H^ - } \leftrightarrow 3CoOOH + {e^ - }\end{equation}$$$$(1)

$$$$\begin{equation}CoOOH + O{H^ - } \leftrightarrow Co{O_2} + {H_2}O + {e^ - }\end{equation}$$$$(2)

$$$$\begin{equation}{\bm{M}}{{\bm{n}}_3}{{\bm{O}}_4}:{\rm{\ }}M{n_3}{O_4} \to {K_\delta }Mn{O_x}.n{H_2}O\end{equation}$$$$(3)

$$$$\begin{equation}{K_\delta }Mn{O_x}.n{H_2}O \leftrightarrow Mn{O_x}.n{H_2}O + \delta {K^ + } + \delta {e^ - }\end{equation}$$$$(4)

$$$$\begin{equation}{\bm{F}}{{\bm{e}}_2}{{\bm{O}}_3}:\ F{e_2}{O_3} + 2{e^ - } + 3{H_2}O \leftrightarrow 2Fe{\left( {OH} \right)_2} + 2O{H^ - }\end{equation}$$$$(5)

$$$$\begin{equation}Fe{\left( {OH} \right)_2} + 2O{H^ - } \leftrightarrow FeOOH + {H_2}O + {e^ - }\end{equation}$$$$(6)

FIGURE 4

Cyclic voltammetry (CV) profiles of (a) Co₃O₄, (b) Mn₃O₄, and (c) Fe₂O₃ at 50 mV/s scan rate. (d) Comparison of specific capacitance between solid and porous morphology of TMOs at 50 mV/s scan rate

Interestingly, it was noticed that the redox peaks were more intense in the porous electrodes. As suggested by N₂ adsorption-desorption isotherms and microscopic analysis, the porous materials have a higher specific surface area and directed channels. These features could lead to increased diffusion and transmission of the electrolyte into the porous electrode, resulting in intense redox peaks and higher specific capacitance. Specific capacitance values obtained in these materials are given in Table 3. The formula (Equation [S1]) to calculate specific capacitance from the CV profile is mentioned in SI. At 50 mV/s scan rate, there was 51%, 38%, and 76% decrement in the specific capacitance value, on shifting from porous to solid morphologies of Co₃O₄, Mn₃O_4, and Fe₂O₃, respectively. This could be attributed to the much-reduced interaction between the electrolyte ions and electrode surface. With increasing scan rates, the specific capacitance value also gradually decreased, as shown in Figure S3. This happens to owe to the fact that the electrolyte ions, at faster scan rates, do not have sufficient time for diffusion into the electrode.

TABLE 3. Calculated specific capacitance in the solid and porous materials using the cyclic voltammetry (CV) and galvanostatic charge-discharge (GCD) profiles

Material	Morphology	Specific capacitance from CV profile (F/g)	Specific capacitance from GCD profile (F/g)
		at a fixed scan rate	at 1 A/g current density
Co3O4	Solid	121 F/g at 5 mV/s	134
	Porous	241 F/g at 5 mV/s	247
Mn3O4	Solid	32 F/g at 5 mV/s	37
	Porous	55 F/g at 5 mV/s	73
Fe2O3	Solid	12 F/g at 50 mV/s	13
	Porous	50 F/g at 50 mV/s	79
Carbon	Solid	84 F/g at 20 mV/s	78
	Porous	128 F/g at 20 mV/s	114

To further reaffirm the higher electrochemical behavior of porous electrodes, the materials were further characterized using GCD studies. Figure 5a–c presents the observed GCD profiles at 1 A/g current density, in both solid and porous particles. The results further proved the superiority of porous structures. It was observed that, for porous electrodes, the discharge curves had two distinct regions. A linear region is associated with the electrochemical double-layer capacitor (EDLC) nature and a non-linear region, which is normally linked with pseudocapacitance. The specific capacitances were estimated from the GCD profiles at 1 A/g current density using the formula given in the SI (Equation [S2]). The specific capacitance values are given in Table 3. The GCD profiles, at current densities ranging from 1 to 5 A/g, were further performed (Figure S4) to investigate the electrochemical rate capabilities at higher scan rates. Figure S5 gives the specific capacitance of these samples at various current densities. Clearly, the capacitance values decreased with increasing current densities. At higher specific currents, the electron transfer toward the electrode is faster, and hence the increment in potential is higher. Consequently, the electrode has reduced time to stay at a specified voltage and a lower specific capacitance value is observed. The electrochemical results at a higher current density of 5 A/g revealed that porous architectures of Co₃O₄, Mn₃O_4, and Fe₂O₃ showed excellent specific capacitance retention of 52%, 67%, and 58% of their maximum value at 1 A/g, whereas the solid architectures retained only 41%, 37%, and 50%, respectively.

FIGURE 5

Charge-discharge (CD) profile of (a) Co₃O₄, (b) Mn₃O₄, and (c) Fe₂O₃ at 1 A/g current density. (d) Comparison of specific capacitance between solid and porous morphologies at 1 A/g current density

The electrochemical properties of solid and porous structured carbon were also investigated and compared. The detailed CV and charge-discharge profiles are given in Figure S6. Carbon-based electrode showed ∼52% enhancement in the specific capacitance value on moving from solid to the porous structure, as given in Table 3.

To further understand the charge transport kinetics of the materials during the electrochemical response, EIS measurements were performed and the corresponding Nyquist plots are shown in Figure 6a–c. The Nyquist plot consisted of two portions, that is, a semi-circular region in the high-frequency range and a straight line at the low-frequency range. The semicircle in the high-frequency region is attributed to the interfacial charge transfer resistance (R_ct) formed between the electrolyte and the surface of the electrode, which can be estimated from the diameter of the semicircle (inset of Figure 6a–c). The magnitude of diameter of the semicircles suggested that the porous electrodes had a lower value of R_ct, compared to their solid counterparts. The straight line in the low-frequency region is attributed to the Warburg diffusion process. The linear curve at ∼45° confirmed their capacitive nature and it is noteworthy that porous structured electrodes displayed more vertical EIS curves. This indicated a reduced diffusion resistance. The obtained R_ct and equivalent series resistance (ESR) values are tabulated in Table 4.

FIGURE 6

Nyquist plot of porous and solid morphology of (a) Co₃O₄, (b) Mn₃O₄, and (c) Fe₂O₃. (d) Cycling stability test of porous Co₃O₄, Mn₃O₄, and Fe₂O₃

TABLE 4. Charge transfer (R_ct) and ESR values of the solid and porous structure of Co₃O₄, Mn₃O₄, and Fe₂O₃

Material	Solid structure		Porous structure
	ESR (Ω)	Rct (Ω)	ESR (Ω)	Rct (Ω)
Co3O4	2.92	4.52	2.03	2.22
Mn3O4	3.51	3.42	2.95	3.15
Fe2O3	4.25	7.72	2.91	7.06
Carbon	1.85	3.82	1.77	3.45

Finally, the electrochemical stability of the porous materials was evaluated by repetitive charging and discharging, at a higher current density, for 1000 cycles. The porous materials showed excellent cycling stability, with a capacity retention of ∼97%–98%, as shown in Figure 6d. This stability could be attributed to a limited volume change in porous structures, owing to the presence of the voids within the material.

The EIS characterization and cyclic stability of FW-derived porous carbon and the solid carbon are presented in SI (Figure S7).

The results obtained in the presented porous materials were compared with the published literature. This is elaborated in Table 5.

TABLE 5. Comparison of electrochemical results of the porous Co₃O₄, Mn₃O₄, Fe₂O₃, and carbon with some published results on the various morphologies and composites of similar materials

Material	Structure	Current density/scan rate	Specific capacitance (F/g)	Cycling stability	Ref.
Co3O4
Co3O4	Porous nanocubes	0.2 A/g	350	—-	[27]
Co3O4	Hollow boxes	0.5 A/g	278	—-	[28]
Co3O4/Ni-Co layered double hydroxide	Core-shell	0.5 A/g	318	92% after 5000 cycles	[29]
Co3O4	nanoparticles	0.2 A/g	179.7	73.5% after 1000 cycles	[30]
Needle-like Co3O4/graphene	Composite	0.1 A/g	157.7	70 % after 4000 cycles	[31]
Co3O4	Layered parallel folding	1 A/g	202.5	∼99% after 1000 cycles	[32]
Co3O4	Porous nanorod	1 A/g	247	∼97 % after 1000 cycles	Present work
Mn3O4
Mn3O4	Porous hollow microtube	1 mA/cm^2	51.7	57 % after 2000 cycles	[33]
Mn3O4	Thin film	50 mV/s	12.3	91 % after 2000 cycles	[34]
Mn3O4	Nanoparticle	1 A/g	55.7	81 % after 1200 cycles	[35]
MnOOH@Mn3O4	Composite	1 A/g	71	——	[36]
Mn3O4	Hollow nanofiber	0.3 A/g	155	99 % after 500 cycles	[37]
Mn3O4	Porous	1 A/g	73	97 % after 1000 cycles	Present work
Fe2O3
Fe2O3	Nanotube	2.5 mV/s	30	92 % after 2000 cycles	[38]
Fe2O3/MnO2	Core-shell	0.1 A/g	159	97.4 % after 5000 cycles	[39]
Fe2O3/Ni-foam	Sheets	0.36 A/g	147	86 % after 1000 cycles	[40]
Fe2O3	Bulk	1 A/g	40	—-	[41]
Fe2O3	Rod	1 A/g	80	98% after 1000 cycles	Present work
Carbon
Bare carbon	Porous	1 mA/cm^2	92 F/g	97.8 % after 1000 cycles	[42]
Porous carbon	nanofibers	25 mV/s	98.4 F/g	——-	[43]
CNF	nanofiber	1 mA/cm^2	60 F/g	——-	[44]
Rice husk derived carbon	Porous (RHPC)	1 mA/cm^2	60 F/g	78 % after 5000 cycles	[45]
Porous carbon aerogel	Hierarchical hole-like	1 A/g	132 F/g	93.9% after 5000 cycles	[46]
Food waste derived-carbon	Porous	1 A/g	114 F/g	96% after 2000 cycles	Present work

3.3 Theoretical interpretation of the performance enhancement in porous electrodes

Several factors affect the electrochemical performance of electrode material. Amongst them, electrolyte-electrode interaction is important in both EDLCs and pseudocapacitors. Porous structures significantly enhance the contact area between the electrode and electrolyte ions. Hence, they are expected to have higher electrochemical performances. The channels facilitate higher ion intercalations and deintercalation. Scheme 1 highlights the fact of the presence of higher electrochemically active sites in porous structures as compared to their solid counterparts.

SCHEME 1

Schematic representation of the transportation of electrolytes ions through the solid and porous structured electrode material

In a solid structure, the electrolyte ions are confined only at the surface of the electrode, whereas the presence of the larger pores, in the porous electrode, provides much higher access for the transmission of electrolyte ions through the material. Hence, the role of the diffusion process becomes important. This can be modeled using Fick's law^[47] of ionic diffusion. According to this law, flux J of an ionic substance, at a position x and time t, is given by:

$$$$\begin{equation}J = - D\frac{{\partial C}}{{\partial x\ }}\end{equation}$$$$(7)

where C represents the concentration of the electrolyte ions near the electrode surface and D is the diffusion coefficient of the electrolyte ions. By calculating the concentration of electrolyte ions C, during charging and discharging and substituting it into the Nernst's equation,^[48] the potential of the electrode surface can be obtained as:

$$$$\begin{equation}E\ = {E_0} + \frac{{RT}}{{nF\ }}\ln \left( {\frac{{{C_0}}}{{{C_R}}}} \right)\end{equation}$$$$(8)

where $${E_0}$$ denotes the reference potential of the electrochemical system. Hence, the substitution of analytical expression of $${C_0}( {0,t} ),\ {C_R}( {0,t} )$$, (C₀: concentration of electrolyte ions during oxidation, C_R is that for reduction) gives an expression for the time-varying electrode potential across the solid electrolyte interface. This leads to two time-dependent equations for the electrode surface (owing to charging and discharging). After parameterization, the two equations can be written:

$$$$\begin{equation}E = a_1^c + b_1^c{t^{\frac{1}{2}}} + c_{1\ }^ct\ {\text{for\ charging}}\end{equation}$$$$(9)

$$$$\begin{eqnarray} E &=& a_2^d + b_2^d({t^{\frac{1}{2}}} - e_2^d{\left( {t - {t_c}} \right)^{\frac{1}{2}}})\nonumber\\ &&+\ c_2^d{\left( {{t^{\frac{1}{2}}} - e_2^d\ {{\left( {t - {t_c}} \right)}^{\frac{1}{2}}}} \right)^2}\ {\text{for\ discharging}} \end{eqnarray}$$$$(10)

where $$a_1^c$$, $$b_1^c$$, $$c_1^c$$, and $$a_2^d$$, $$b_2^d$$, $$c_2^d$$ are the coefficients of charging and discharging, respectively. The system achieves the highest potential (E_max) during charging, corresponding to the time taken to fully charge (t_c). Hence, the charging time can be obtained from the following equation^[49]:

$$$$\begin{equation}{E_{max}} = a_1^c + b_1^ct_c^{\frac{1}{2}} + c_{1\ }^c{t_c}\end{equation}$$$$(11)

where $$a_1^c$$, $$b_1^c$$ and $$c_1^c$$ are the values obtained from the fitted data. Similarly, discharge time $${t_d}$$ was evaluated from the equation:

$$$$\begin{eqnarray} {{ \def\eqcellsep{&}\begin{array}{l} {E_{min}} = a_2^d + b_2^d({t_d}^{\displaystyle\frac{1}{2}} - e_2^d{\left( {{t_d} - {t_c}} \right)^{\displaystyle \frac{1}{2}}}) + c_2^d{\left( {{t_d}^{\displaystyle \frac{1}{2}} - e_2^d\ {{\left( {{t_d} - {t_c}} \right)}^{\displaystyle \frac{1}{2}}}} \right)^2} \end{array} }}\nonumber\\ \end{eqnarray}$$$$(12)

By subtracting $${t_c}$$ from the value of $${t_d}$$, one can estimate the time taken to discharge a supercapacitor. From the value of charging and discharging coefficients, the diffusion coefficient of electrolyte ions can be easily obtained. For current varying electrochemical cells, the potential across the electrode advances as a function of time. On accumulation of charges, the potential differences at the interface can change with time but this effect is neglected in the calculation, to maintain simplicity.^[49] This leads to:

$$$$\begin{equation}\frac{{\partial C}}{{\partial t}} = D\frac{{{\partial ^2}C}}{{\partial {x^2}}}\end{equation}$$$$(13)

As one can see, Equation (13) is a 2nd-order differential equation, involving time and position. The details of solving the equation have been already reported by our group.^[49] Here also, a similar protocol was followed. Solving Equation (13) and substituting it in Nernst's equation, electrode potential during charging and discharging can be written as:

$$$$\begin{eqnarray} E &\approx & {E_0} - \frac{{RT}}{{2nF}}\ln \left( {\frac{{{D_0}}}{{{D_R}}}} \right) - \frac{{RT}}{{nF}}\left( \left( {i\alpha } \right){t^{\frac{1}{2}}}\right.\nonumber\\ &&\left. +\ {{\left( {i\alpha } \right)}^2}\frac{t}{2} \right)\ {\text{for\ charging}} \end{eqnarray}$$$$(14)

and

$$$$\begin{eqnarray} E &\approx& {E_0} - \frac{{RT}}{{2nF}}\ln \left( {\frac{{{D_0}}}{{{D_R}}}} \right)\nonumber\\ &&+ \frac{{RT}}{{nF}}\left( - \left( {i\alpha } \right)\left( {{\rm{\ }}{t^{\frac{1}{2}}} - 2{\rm{\ }}{{\left( {t - {t_c}} \right)}^{\frac{1}{2}}}} \right)\right.\nonumber\\ &&\left.-\, \frac{{{{\left( {i\alpha } \right)}^2}}}{2}{{\left( {{\rm{\ }}{t^{\frac{1}{2}}} - {\rm{\ }}2{\rm{\ }}{{\left( {t - {t_c}} \right)}^{\frac{1}{2}}}} \right)}^2} \right)\ {\text{\ for\ discharging}}\nonumber\\ \end{eqnarray}$$$$(15)

where E represents the potential of the electrode surface both for charging and discharging. The terms $${E_0}$$, $$i$$,$$\ {D_O}$$ and $${D_R}$$ stand for the reference potential of the electrochemical system, current density, and the initial and final diffusion coefficient, respectively. $$F,\ R,\ T\ $$represent the Faraday number, the universal gas constant, and the temperature of the system, respectively, whereas, $$\alpha $$ is a constant parameter of the system which take the following value: $$\alpha \ = ( {\ \frac{2}{{nFAC_0^*D_0^{\frac{1}{2}}{\pi ^{\frac{1}{2}}}}}\ } )\ $$, where A, D₀, and $$C_0^*$$ represent the area of the electrode, the diffusion coefficient, and the concentration of ion at time t = 0. Further, a comparison of Equations (9), (10), (14), and (15) give the values^[25]:

$$$$\begin{equation*}b_1^c = - \frac{{RT}}{{nF}}\left( {i\alpha } \right),\ \ c_1^c = - \frac{{RT}}{{2nF}}{\left( {i\alpha } \right)^2}\ {\text{for\ charging}}\end{equation*}$$$$

$$$$\begin{equation*}b_1^d = - \frac{{RT}}{{nF}}\left( {i\alpha } \right),\ c_1^d = - \frac{{RT}}{{2nF}}{\left( {i\alpha } \right)^2}{\text{\ for\ discharging}}\end{equation*}$$$$

$$$$\begin{equation}{\rm{Therefore}},{\rm{\ }}2\frac{{c_1^c}}{{b_1^c}} = {\rm{\ }}\left( {i\alpha } \right)\end{equation}$$$$(16)

On solving Equation (16), one can obtain the value of i$$\alpha $$, which is required during the calculation of diffusion coefficients. This was obtained as:

$$$$\begin{equation}i\alpha = \left( {\ \frac{{2i}}{{nFAC_0^*D_0^{\frac{1}{2}}{\pi ^{\frac{1}{2}}}}}\ } \right)\end{equation}$$$$(17)

where i is the applied current density and n is the number of electrons passing the electrode during the reaction.

So, to establish the statement of higher electrolyte ions diffusion occurs through the porous electrodes, the values of the diffusion coefficients were calculated from the charging and discharging coefficients; obtained by fitting the CD profiles of the porous and solid particles. Figure 7a,c,e shows the comparison curve of theoretical and experimental data for porous Co₃O₄, Mn₃O₄, and Fe₂O₃, respectively. All of the GCD curves obtained at 1 A/g current density were used. Comparison curves for solid structures of metal oxides and solid and porous carbon are given in Figure S8a–e. The theoretical curves were in good agreement with the experimental data, with low relative error (<5%) [see Table 6]. Like the experimental results, the specific capacitance for each porous material achieved a greater value than their solid structures; the values are tabulated in Table 6. This can be explained by the diffusion behavior of the electrolyte ions. Figure 7b,d,f illustrates the comparison of diffusion coefficients of electrolyte ions in solid and porous structured Co₃O₄, Mn₃O₄, and Fe₂O₃, respectively. The variation of diffusion coefficient for solid and porous structured carbon is shown in Figure S8f. The diffusion coefficients of the electrolyte ions were calculated for solid and porous structures at 1 A/g current density. In each of these three metal oxides, and carbon, the diffusion coefficient increased when we moved from solid to porous structures. The diffusion coefficients in porous and solid structures are given in Table 6. The higher D values in porous nanostructures further proved their importance. It could be inferred that there exists a correlation between diffusion coefficient and specific capacitance. Diffusion coefficients depend upon different factors. Amongst them, the morphology of electrode material is critical. Usually, the electrochemical performance increases due to the increase in mobility of the electrolyte ions into porous structures.

FIGURE 7

Comparison between theoretical and experimental data at 1 A/g current density and diffusion coefficient of (a, b) Co₃O₄, (c, d) Mn₃O₄, and (e, f) Fe₂O₃, respectively

TABLE 6. Theoretical specific capacitance with relative errors and diffusion coefficients values of the solid and porous structure of the materials

Material	Morphology	Theoretical specific capacitance(F/g)	Relative error(%)	Diffusion coefficients(m^2/s)
Co3O4	Solid	135.9	1.42	1.8744
	Porous	243.0	1.62	4.244
Mn3O4	Solid	35.2	4.81	0.0453
	Porous	72.4	0.82	0.4824
Fe2O3	Solid	12.0	7.69	0.0535
	Porous	78.0	1.26	10.755
Carbon	Solid	75.4	3.33	0.01904
	Porous	104.1	3.42	0.1153

The kinetics of the electrolyte ion change with the change in porosity of the materials. Each of the porous materials discussed here had mesoporous or microporous nature, with a higher pore radius than their solid counterparts. Due to the increase in porosity, diffusion (of electrolyte ions) can occur in several directions. Also, effective diffusion length becomes larger and leads to an increase in the diffusion coefficient.^[50]

So, both the theoretical modeling and experimental analysis demonstrated the improved electrochemical performance of porous structure electrodes at a given rate compared to their solid bulk material. There was ∼46%, 49%, 83%, and 32% decrement in specific capacitance value when the porous structures of the Co₃O₄, Mn₃O_4, Fe₂O_3, and carbon were replaced by their corresponding solid counterpart. The BET results also suggested that the porous materials have greater specific surface area i.e. large area per unit mass, which in turn facilitates enhanced contact area of electrode-electrolyte interface. The larger contact area means that the current density per unit surface area decreases, which reduces the electrode polarization and drives higher charge transfer at the interface.

The pore in the porous Co₃O₄, Mn₃O_4, Fe₂O_3, and carbon was also large. These would act as an ‘‘ion reservoir'' and have ample room, to absorb lattice expansion during cycling. Therefore, the porous materials are able to deliver excellent cycling stability of > 97% after 1000 cycles with the improvement in charge-discharge efficiency (Table S3). The porous material also showed lower values of ESR and R_ct. This confirmed the facile charge transfer reactions and much lower diffusion resistance of the electrolyte ions at the interface.

4 CONCLUSIONS

It is clearly shown that the porous nanostructures of transition metal oxides and carbon have higher specific capacitance values than their solid counterparts. Porous structures of Co₃O₄ were able to deliver ∼84% increment in specific capacitance value, at 1 A/g current density, in comparison to its solid counterparts. Other investigated materials also showed similar results. The enhanced performance could be attributed to the higher surface adsorption sites and specific surface area in the porous structures. The porous network facilitates higher ion diffusion and improved electrochemical interactions. The porous materials also showed higher cyclic retention, with lower ESR values. All the experimentally observed data are consistently explained by a theoretical model, which uses Fick's law as the starting point to calculate the extent of ion diffusion within the material.

ACKNOWLEDGMENTS

The authors acknowledge the funding received from DST (India) under its MES scheme for the project entitled, “Hierarchically nanostructured energy materials for next-generation Na-ion based energy storage technologies and their use in renewable energy systems” (Grant No.: DST/TMD/MES/2k16/77).

CONFLICT OF INTEREST

The authors declare no conflict of interest.

Open Research

DATA AVAILABILITY STATEMENT

Data available on request from the authors.