3. Understanding electrochemical kinetics of charge transfer process using numerical analysis

Modeling electrodes or full-cells of batteries or supercapacitors has been extensively studied [12–15]. For example, Popov and coworkers have developed a one-dimensional model to analyze the performance of a hybrid system comprised of battery and supercapacitor (based on a Sony 18650 battery and a Maxwell's 10F supercapacitor) under pulse discharge currents [15]. The proposed model has not only successfully predicted the power-energy relationship compared with the practical experimental conditions, but also reveals the capability of the hybrid system to deliver higher energy density than the battery-alone system while operating at high power density. However, little has been done on simulating the transition of the electrochemical behavior between the battery-type and capacitor-type charge storage mechanisms. In this book chapter, we present numerical solutions for a simple model of an electrode material and discuss in detail the interplay between redox reaction and diffusion of charge carriers, as well as the effect of dependence of open-circuit-voltage on chemical composition on the overall cyclic voltammetry behavior of the electrode in a half-cell setting. We are able to show the transition from a diffusion-limited charge transfer process (battery-like electrochemical behavior) to kinetic-limited charge transfer process (capacitor-like behavior) by changing the structural and experimental conditions.

#### 3.1. The description of the mathematical model

Figure 5 shows a schematic of a single spherical electrode material model. The following assumptions are made during the analysis:


Therefore, the flux of charge carriers (N) within the particle can be described using Eq. (9), and mass conservation of the charge carriers in the particle can be described using Eq. (10)

$$\mathbf{N} = -D\nabla \mathbf{c} \tag{9}$$

particle) processes can be described by a partial differential equation (Eq. (11)) with a set of initial condition (Eq. (12)) and boundary conditions at the surface of the particle (Eqs. (13) and

Figure 5. Schematic of a single spherical electrode material model in (a) a regular parameter denotation and (b) a

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i

<sup>F</sup> ¼ �<sup>D</sup> <sup>∂</sup><sup>c</sup>

∂c ∂r � � � � r¼0

where i is the current density at the electrode surface and r<sup>0</sup> is the radius of the particle.

ð Þ cs <sup>β</sup> exp

It is notable that the current density at the surface is also governed by the Butler-Volmer

where k is a reaction rate constant, C<sup>l</sup> is the concentration of the charge carrier in the liquid phase (assumed to be a constant in the calculation), c<sup>θ</sup> is the surface concentration of vacant sites ready for cation intercalation, cs is the concentration of cation on the surface of the electrode, and ct is the concentration of total sites for seating cations (c<sup>θ</sup> = ct � cs). R is the gas constant, T is the temperature, and β is the symmetry factor, representing the fraction of the electrical potential used to promote the cathode reaction (β is usually considered to be 0.5). The overpotential η is defined as the difference between the applied potential (Uapp) and the opencircuit potential of the particle (UÞ as shown in Eq. (16). It is assumed that Uapp is uniform

<sup>1</sup> � <sup>β</sup> � �F<sup>η</sup>

RT � � � exp � <sup>β</sup>F<sup>η</sup>

RT � � � � (15)

∂r � � � � r¼r<sup>0</sup>

!

c rð Þ¼ ; t ¼ 0 c<sup>0</sup> (12)

¼ 0 (14)

(13)

95

(17)) and at the center of the particle (Eq. (14)):

dimensionless parameter denotation (when c<sup>0</sup> = ct).

i <sup>F</sup> <sup>¼</sup> k cð Þ<sup>l</sup>

throughout the particle.

1�β ð Þ c<sup>θ</sup> 1�β

equation:

$$\frac{\partial \mathbf{c}}{\partial t} = -\nabla \cdot \mathbf{N} \tag{10}$$

where c is the concentration of charge carriers (such as Li+ or Na+ ), D is the diffusion coefficient during the ionic transport within the particle (assumed to be a constant).

Combining Eqs. (9) and (10) gives the conservation of the charge carriers in spherical coordinates:

$$\frac{\partial c}{\partial t} = D \left( \frac{\partial^2 c}{\partial r^2} + \frac{2}{r} \frac{\partial c}{\partial r} \right) \tag{11}$$

Starting from certain discharged state with a homogenous concentration of the charge carriers (c0Þ in the particle, the electrokinetics during charging (oxidation or extraction of charge carriers out of the particle) and discharging (reduction or insertion of charge carriers into the

Enhancing Pseudocapacitive Process for Energy Storage Devices: Analyzing the Charge Transport… http://dx.doi.org/10.5772/intechopen.73680 95

[15]. The proposed model has not only successfully predicted the power-energy relationship compared with the practical experimental conditions, but also reveals the capability of the hybrid system to deliver higher energy density than the battery-alone system while operating at high power density. However, little has been done on simulating the transition of the electrochemical behavior between the battery-type and capacitor-type charge storage mechanisms. In this book chapter, we present numerical solutions for a simple model of an electrode material and discuss in detail the interplay between redox reaction and diffusion of charge carriers, as well as the effect of dependence of open-circuit-voltage on chemical composition on the overall cyclic voltammetry behavior of the electrode in a half-cell setting. We are able to show the transition from a diffusion-limited charge transfer process (battery-like electrochemical behavior) to kinetic-limited charge transfer process (capacitor-like behavior) by changing

Figure 5 shows a schematic of a single spherical electrode material model. The following

ii. The transport of charge carriers (cations) within the solid particle is only limited by diffusion, and only radial diffusion has been considered. It is also assumed that the potential gradient within the particle is negligible, and thus migration of cations does

iii. The charge-transfer reaction is governed by the Butler-Volmer electrokinetic expression. Therefore, the flux of charge carriers (N) within the particle can be described using Eq. (9), and

N ¼ �D∇c (9)

¼ �∇∙N (10)

), D is the diffusion coefficient

(11)

mass conservation of the charge carriers in the particle can be described using Eq. (10)

∂c ∂t

Combining Eqs. (9) and (10) gives the conservation of the charge carriers in spherical coordi-

Starting from certain discharged state with a homogenous concentration of the charge carriers (c0Þ in the particle, the electrokinetics during charging (oxidation or extraction of charge carriers out of the particle) and discharging (reduction or insertion of charge carriers into the

<sup>¼</sup> <sup>D</sup> <sup>∂</sup><sup>2</sup> c ∂r<sup>2</sup> þ 2 r ∂c ∂r

where c is the concentration of charge carriers (such as Li+ or Na+

during the ionic transport within the particle (assumed to be a constant).

∂c ∂t

the structural and experimental conditions.

94 Supercapacitors - Theoretical and Practical Solutions

assumptions are made during the analysis: i. The particle is a perfect solid sphere.

not occur.

nates:

3.1. The description of the mathematical model

Figure 5. Schematic of a single spherical electrode material model in (a) a regular parameter denotation and (b) a dimensionless parameter denotation (when c<sup>0</sup> = ct).

particle) processes can be described by a partial differential equation (Eq. (11)) with a set of initial condition (Eq. (12)) and boundary conditions at the surface of the particle (Eqs. (13) and (17)) and at the center of the particle (Eq. (14)):

$$\mathcal{c}(r, t = 0) = \mathfrak{c}\_0 \tag{12}$$

$$\frac{\dot{I}}{F} = -D \left( \frac{\partial c}{\partial r} \bigg|\_{r=r\_0} \right) \tag{13}$$

$$\left.\frac{\partial c}{\partial r}\right|\_{r=0} = 0\tag{14}$$

where i is the current density at the electrode surface and r<sup>0</sup> is the radius of the particle.

It is notable that the current density at the surface is also governed by the Butler-Volmer equation:

$$\frac{\dot{q}}{F} = k(c\_l)^{1-\beta} (c\_\theta)^{1-\beta} (c\_s)^\beta \left\{ \exp\left(\frac{(1-\beta)F\eta}{RT}\right) - \exp\left(-\frac{\beta F\eta}{RT}\right) \right\} \tag{15}$$

where k is a reaction rate constant, C<sup>l</sup> is the concentration of the charge carrier in the liquid phase (assumed to be a constant in the calculation), c<sup>θ</sup> is the surface concentration of vacant sites ready for cation intercalation, cs is the concentration of cation on the surface of the electrode, and ct is the concentration of total sites for seating cations (c<sup>θ</sup> = ct � cs). R is the gas constant, T is the temperature, and β is the symmetry factor, representing the fraction of the electrical potential used to promote the cathode reaction (β is usually considered to be 0.5). The overpotential η is defined as the difference between the applied potential (Uapp) and the opencircuit potential of the particle (UÞ as shown in Eq. (16). It is assumed that Uapp is uniform throughout the particle.

$$
\eta = \mathcal{U}\_{app} - \mathcal{U} \tag{16}
$$

Eqs. (18)–(23) can be solved with a partial differential equation solver PDE2D using the

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97

From the numerical analysis, we have found that the value of the OCV of the particles at different concentration of the charge carriers strongly affects the shape of the CV, and therefore the entire charge storage mechanism. In this study, two different open-circuit voltage (OCV) expressions (as a function of the concentration of the charge carrier in the particle) are used as shown in Figure 6. They include (i) the simplified OCV of a capacitor material (Figure 6a), where the OCV linearly decreases with increasing concentration of charge carrier within the particle; (ii) the simplified OCV of a Li-ion battery material during phase transition (Figure 6b), where OCV of the particle is nearly independent of the charge carrier concentration within the

Considering one-electron transfer and the insertion of the charge carrier into the solid spherical

We assume that the conversion of MOX into MCOX is a one-phase reaction, where MOX and MCOX have a similar solid-solution type structure, analogous to proton intercalation into

where F is the degree of freedom, which is the number of thermodynamic parameters necessary for defining a system, C is the number of components, and P is the number of phases. For the reaction shown in Eq. (24), the system has two components (C = 2) including the charge

Beside the two intensive parameters, usually pressure and temperature, there is one additional degree of freedom that needs to be specified for the system. Thus, the chemical potential of the electrode (or OCV) has to be a function of temperature, pressure, and composition (the concentration of the charge carriers in the particle). Once composition changes (as the last degree

) and the host particle (MOX), the degree of freedom is equal to 3 (F = 2 � 1 + 2 = 3).

RuO2 materials. The degrees of freedom can be calculated by the Gibbs phase rule:

) 10�<sup>9</sup>

) 1.0 <sup>r</sup><sup>0</sup> (cm) <sup>5</sup> � <sup>10</sup>�<sup>4</sup>

) 10�<sup>4</sup>

�<sup>1</sup> mol�1/2) 6.3 � <sup>10</sup>�<sup>5</sup>

β 0.5

a (dimensionless) 1

Table 1. Parameters set for the calculation.

MO<sup>X</sup> þ C<sup>þ</sup> þ e� \$ MCO<sup>X</sup> (24)

F ¼ C � P þ 2 (25)

3.2. The influence of open-circuit voltage on the electrochemical behavior

parameters listed in Table 1.

particle for most of the composition range.

particles occurs as shown in Eq. (24).

carriers (C+

D (cm<sup>2</sup> s �1

cl (mol dm3

K (cm5/2 s

V (V s�<sup>1</sup>

Combination of Eqs. (13) and (15) gives the following equation:

$$-D\left(\frac{\partial c}{\partial r}\Big|\_{r=r\_0}\right) = k(c\_l)^{1-\beta}(c\_\theta)^{1-\beta}(c\_s)^\beta \left\{ \exp\left(\frac{(1-\beta)F\eta}{RT}\right) - \exp\left(-\frac{\beta F\eta}{RT}\right) \right\} \tag{17}$$

To facilitate the numerical analysis, the partial differential equation with its initial condition and boundary conditions as shown from Eqs. (11)–(17) can be transformed into dimensionless form with the following dimensionless variables:


Eqs. (11)–(14) then become following expressions accordingly:

$$\frac{\partial y}{\partial \tau} = \frac{\partial^2 y}{\partial x^2} + \frac{2}{x} \left. \frac{\partial y}{\partial x} \right|\_{,} \tag{18}$$

$$y(\mathbf{x},0) = \mathbf{1}(\text{when } \mathbf{c}\_0 = \mathbf{c}\_t) \tag{19}$$

$$j = -\left(\frac{\partial y}{\partial \mathbf{x}}\Big|\_{x=1}\right) \tag{20}$$

$$\left.\frac{\partial y}{\partial x}\right|\_{x=0} = 0\tag{21}$$

Eq. (19) is applicable only when (c<sup>0</sup> ¼ ct), while here we assume c<sup>0</sup> is lower but near ct (y < 1) and the electrochemical process starts from the oxidation of the particles (extraction of charge carriers from the solid particles as shown in the forward direction in Eq. (1)), Eq. (17) then becomes:

$$\frac{\partial y}{\partial x}\Big|\_{x=1} + a \big(y|\_{x=1}\big)^{\beta} (1-y|\_{x=1})^{1-\beta} \left\{ \exp\left(\frac{(1-\beta)F\{\mathcal{U}\_{app}-\mathcal{U}\}}{RT}\right) - \exp\left(-\frac{\beta F\{\mathcal{U}\_{app}-\mathcal{U}\}}{RT}\right) \right\} = 0 \tag{22}$$

where yj <sup>x</sup>¼<sup>1</sup> <sup>¼</sup> cs=ct and <sup>a</sup> <sup>¼</sup> kr0<sup>c</sup> 1�β l <sup>D</sup> .

Under potentiodynamic simulation, the applied potential changes linearly with time

$$\mathcal{U}\_{app} = \mathcal{U}\_0 + \mathfrak{v}t \tag{23}$$

where U<sup>0</sup> is the initial applied potential, v is the potential sweep rate.

Eqs. (18)–(23) can be solved with a partial differential equation solver PDE2D using the parameters listed in Table 1.

#### 3.2. The influence of open-circuit voltage on the electrochemical behavior

η ¼ Uapp � U (16)

� � � �

� exp � <sup>β</sup>F<sup>η</sup>

<sup>∂</sup><sup>x</sup> (18)

¼ 0 (21)

� exp � <sup>β</sup>F Uapp � <sup>U</sup> � �

Uapp ¼ U<sup>0</sup> þ vt (23)

RT

y xð Þ¼ ; 0 1ðwhen c<sup>0</sup> ¼ ctÞ (19)

RT

(17)

(20)

¼ 0 (22)

<sup>1</sup> � <sup>β</sup> � �F<sup>η</sup> RT � �

r0

Combination of Eqs. (13) and (15) gives the following equation:

1�β ð Þ c<sup>θ</sup> 1�β

ð Þ cs <sup>β</sup> exp

To facilitate the numerical analysis, the partial differential equation with its initial condition and boundary conditions as shown from Eqs. (11)–(17) can be transformed into dimensionless

¼ k cð Þ<sup>l</sup>

r2 0 ii. dimensionless distance from the particle center <sup>x</sup> <sup>¼</sup> <sup>r</sup>

Eqs. (11)–(14) then become following expressions accordingly:

x¼1 � �<sup>1</sup>�<sup>β</sup> exp

> 1�β l <sup>D</sup> .

where U<sup>0</sup> is the initial applied potential, v is the potential sweep rate.

<sup>x</sup>¼<sup>1</sup> <sup>¼</sup> cs=ct and <sup>a</sup> <sup>¼</sup> kr0<sup>c</sup>

ct

FDct

∂y <sup>∂</sup><sup>τ</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup> y ∂x<sup>2</sup> þ 2 x ∂y

<sup>j</sup> ¼ � <sup>∂</sup><sup>y</sup> ∂x � � � � x¼1

> ∂y ∂x � � � � x¼0

Eq. (19) is applicable only when (c<sup>0</sup> ¼ ct), while here we assume c<sup>0</sup> is lower but near ct (y < 1) and the electrochemical process starts from the oxidation of the particles (extraction of charge carriers from the solid particles as shown in the forward direction in Eq. (1)), Eq. (17) then

> <sup>1</sup> � <sup>β</sup> � �F Uapp � <sup>U</sup> � � RT � �

Under potentiodynamic simulation, the applied potential changes linearly with time

� � � �

� �

form with the following dimensionless variables:

�<sup>D</sup> <sup>∂</sup><sup>c</sup> ∂r � � � � r¼r<sup>0</sup>

i. dimensionless time <sup>τ</sup> <sup>¼</sup> tD

becomes:

where yj

þ a yj x¼1 � �<sup>β</sup> <sup>1</sup> � <sup>y</sup><sup>j</sup>

∂y ∂x � � � � x¼1

iii. dimensionless concentration <sup>y</sup> <sup>¼</sup> <sup>c</sup>

iv. dimensionless current density <sup>j</sup> <sup>¼</sup> ir<sup>0</sup>

!

96 Supercapacitors - Theoretical and Practical Solutions

From the numerical analysis, we have found that the value of the OCV of the particles at different concentration of the charge carriers strongly affects the shape of the CV, and therefore the entire charge storage mechanism. In this study, two different open-circuit voltage (OCV) expressions (as a function of the concentration of the charge carrier in the particle) are used as shown in Figure 6. They include (i) the simplified OCV of a capacitor material (Figure 6a), where the OCV linearly decreases with increasing concentration of charge carrier within the particle; (ii) the simplified OCV of a Li-ion battery material during phase transition (Figure 6b), where OCV of the particle is nearly independent of the charge carrier concentration within the particle for most of the composition range.

Considering one-electron transfer and the insertion of the charge carrier into the solid spherical particles occurs as shown in Eq. (24).

$$\text{MO}\_X + \text{C}^+ + \text{e}^- \leftrightarrow \text{MCO}\_X \tag{24}$$

We assume that the conversion of MOX into MCOX is a one-phase reaction, where MOX and MCOX have a similar solid-solution type structure, analogous to proton intercalation into RuO2 materials. The degrees of freedom can be calculated by the Gibbs phase rule:

$$\mathbf{F} = \mathbf{C} - \mathbf{P} + \mathbf{2} \tag{25}$$

where F is the degree of freedom, which is the number of thermodynamic parameters necessary for defining a system, C is the number of components, and P is the number of phases. For the reaction shown in Eq. (24), the system has two components (C = 2) including the charge carriers (C+ ) and the host particle (MOX), the degree of freedom is equal to 3 (F = 2 � 1 + 2 = 3). Beside the two intensive parameters, usually pressure and temperature, there is one additional degree of freedom that needs to be specified for the system. Thus, the chemical potential of the electrode (or OCV) has to be a function of temperature, pressure, and composition (the concentration of the charge carriers in the particle). Once composition changes (as the last degree


Table 1. Parameters set for the calculation.

shape is often observed in classic EDL capacitor that uses carbon as electrode material or pseudocapacitor that uses RuO2 as electrodes and H2SO4 as electrolyte. This simulation again highlights the fundamental difference between a battery material and a capacitor material: the former undergoes a phase transition upon the interaction with cation, while the latter can

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Parameter a is the dimensionless factor (shown in Eq. (22)) involving the concentration of the charge carriers in the liquid phase, and can be considered as the ratio of interfacial reaction rate

> 1�β l

A large value of a indicates a faster interfacial charge-transfer kinetics and a slower diffusion of charge carrier, usually resulting from a large rate constant (k), a large particle radius (r0) or a small diffusion coefficient (D). Figure 7 shows the simulated CVs for various values of a at the scan rate of 0.1 mV/s, in which all the parameters described in the definition of a are fixed except the value of k. It is clear that the shapes of the CV are strongly dependent on the value of a. When the values of a increases, the peak potential of the anodic scan (oxidation reaction) shifts to lower potential values with a relatively narrower peak. Similarly, the peak potential of

) to the diffusion rate of the charge carriers on the surface of the particle (Dcl/r0) as shown

<sup>D</sup> <sup>¼</sup> kc<sup>2</sup>�<sup>β</sup> l Dcl=r<sup>0</sup>

(26)

99

interact with cation without generation of a new crystalline phase.

electrochemical behavior

Figure 7. Simulated CV curves as a function of parameter a.

(kcl 2�β

in Eq. (26).

3.3. The influence of interfacial reaction and diffusion of charge carriers on the

<sup>a</sup> <sup>¼</sup> kr0<sup>c</sup>

Figure 6. Simulated open-circuit-voltage (OCV) as a function of concentration of charge carrier (Li<sup>+</sup> ) in (a) one-phase reaction and (b) two-phase reaction, and (c and d) their corresponding CVs curves.

of freedom), the OCV of the host particle changes accordingly at fixed temperature and pressure, as shown in Figure 6b. Therefore, in the one-phase discharge reaction (Eq. (24)), the voltage changes linearly with the concentration of the charge carriers.

As C+ continuously inserts into MCOX, the further reduction of M cation eventually leads to the formation of new M-containing species (now it becomes a two-phase reaction), resulting in a new degree of freedom of 2 (F = 2 2 + 2 = 2). For a fixed pressure and temperature, there is no more independent degree of freedom left and all the thermodynamic functions including OCV should remain constant once the composition changes. Therefore, as shown in Figure 6a in the two-phase discharge reaction, the OCV is constant with the concentration of the charge carrier.

It is clear that in a two-phase reaction (Figure 6d), the CV curves show the distinct redox features that represent conventional battery material behavior. This is also congruent with the fact that a typical battery intercalation/deintercalation reaction is accompanied with a phase transition. On the other hand, in a one-phase reaction (Figure 6c), the CV curve shows a square-shaped current versus potential plot without distinct redox features. This unique CV shape is often observed in classic EDL capacitor that uses carbon as electrode material or pseudocapacitor that uses RuO2 as electrodes and H2SO4 as electrolyte. This simulation again highlights the fundamental difference between a battery material and a capacitor material: the former undergoes a phase transition upon the interaction with cation, while the latter can interact with cation without generation of a new crystalline phase.

#### 3.3. The influence of interfacial reaction and diffusion of charge carriers on the electrochemical behavior

Parameter a is the dimensionless factor (shown in Eq. (22)) involving the concentration of the charge carriers in the liquid phase, and can be considered as the ratio of interfacial reaction rate (kcl 2�β ) to the diffusion rate of the charge carriers on the surface of the particle (Dcl/r0) as shown in Eq. (26).

$$a = \frac{k r\_0 c\_l^{1-\beta}}{D} = \frac{k c\_l^{2-\beta}}{D c\_l / r\_0} \tag{26}$$

A large value of a indicates a faster interfacial charge-transfer kinetics and a slower diffusion of charge carrier, usually resulting from a large rate constant (k), a large particle radius (r0) or a small diffusion coefficient (D). Figure 7 shows the simulated CVs for various values of a at the scan rate of 0.1 mV/s, in which all the parameters described in the definition of a are fixed except the value of k. It is clear that the shapes of the CV are strongly dependent on the value of a. When the values of a increases, the peak potential of the anodic scan (oxidation reaction) shifts to lower potential values with a relatively narrower peak. Similarly, the peak potential of

Figure 7. Simulated CV curves as a function of parameter a.

of freedom), the OCV of the host particle changes accordingly at fixed temperature and pressure, as shown in Figure 6b. Therefore, in the one-phase discharge reaction (Eq. (24)), the

) in (a) one-phase

As C+ continuously inserts into MCOX, the further reduction of M cation eventually leads to the formation of new M-containing species (now it becomes a two-phase reaction), resulting in a new degree of freedom of 2 (F = 2 2 + 2 = 2). For a fixed pressure and temperature, there is no more independent degree of freedom left and all the thermodynamic functions including OCV should remain constant once the composition changes. Therefore, as shown in Figure 6a in the two-phase discharge reaction, the OCV is constant with the concentration of the charge carrier. It is clear that in a two-phase reaction (Figure 6d), the CV curves show the distinct redox features that represent conventional battery material behavior. This is also congruent with the fact that a typical battery intercalation/deintercalation reaction is accompanied with a phase transition. On the other hand, in a one-phase reaction (Figure 6c), the CV curve shows a square-shaped current versus potential plot without distinct redox features. This unique CV

voltage changes linearly with the concentration of the charge carriers.

reaction and (b) two-phase reaction, and (c and d) their corresponding CVs curves.

98 Supercapacitors - Theoretical and Practical Solutions

Figure 6. Simulated open-circuit-voltage (OCV) as a function of concentration of charge carrier (Li<sup>+</sup>

the cathodic scan (reduction reaction) shifts to higher potential values with a narrower peak. These changes strongly suggest that a large value of a reflects a fast interfacial reaction and/or a slow diffusion. Therefore, the extraction (anodic scan) and insertion (cathodic scan) of the charge carriers are limited by diffusion of charge carriers in the solid spherical particle, probably attributed to a relatively low diffusion coefficient (D) and/or a large particles size (r0). Accordingly, at a large value of a, the solid particle behavior is closer to a battery electrode material, where diffusion-limited charge transport is the slowest step during the electrochemical process. In this context, to increase the capacitive charge transfer contribution, a smaller sized electrode and an electrolyte with cations having a higher diffusivity in the host electrode material is needed. The former points to the synthesis of nanostructured electrode material and the latter indicates Li-containing electrolyte. It is also notable that since the value of β is between 0 and 1, the value of (1 β) is always higher than zero. Therefore, it indicates that a low electrolyte concentration favors the contribution of capacitive process, though the overall charge storage might decrease.

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