2022 Volume 90 Issue 10 Pages 102007
Electrochemical impedance spectroscopy (EIS) enables the examination of the electrochemical nature of electrodes and electrochemical cells by applying an alternating voltage (or current) and measuring the resulting current (or voltage). The resistance and capacitance components of the electrode can be evaluated by applying an AC voltage and changing the frequency. In particular, analysis using the equivalent circuit can determine important parameters related to the electrochemical reaction of the electrode, such as the charge transfer resistance, electric double-layer capacitance, and Warburg impedance. Moreover, the internal resistance of the cell can be divided into resistances caused by the positive electrode, negative electrode, and electrolyte. Because of these advantages, EIS is a powerful technique used for basic research, such as in identifying the rate-determining step of an electrochemical reaction, and also for applied research, such as characterizing electrochemical devices (e.g., batteries and capacitors). In this paper, the concept of impedance, which represents the relationship between the AC voltage and current, is first explained; then, the AC characteristics of various circuit elements used in equivalent circuits, which are essential for understanding EIS, are described. Finally, treatments of more complex circuits based on transmission-line models (TLMs), which are used to represent equivalent circuits of porous electrodes, are presented. Analyses based on TLMs are the foundation for understanding electrodes for practical applications because porous electrodes are usually used in electrochemical devices.
Voltage E [V], current I [A], and time t [s] constitute the basic information in electrochemistry; thus, the purpose of electrochemical measurements is to control the voltage (or current) to an electrode and simultaneously measure the time variation of the resulting current (or voltage).1,2 For example, chronoamperometry measures the time change of the current I(t) when a constant voltage is applied to an electrode, and cyclic voltammetry measures the current response I(E, t) under various voltages with a constant scan rate (dE/dt = constant). Electrochemical impedance spectroscopy (EIS) is a technique for electrochemical measurement; it differs from other electrochemical measurement techniques in that alternating current (AC) voltage (or current) at various frequencies is applied to an electrode, and the resulting current (or voltage) is measured. In EIS, the relationship between voltage and current is expressed as impedance Z instead of resistance R in the direct current (DC) measurements.3,4 Similar to other electrochemical techniques, EIS obtains information consisting of voltage, current, and time because the unit of frequency has a reciprocal dimension of time (Hz = s−1), and the unit of impedance is Ω.
This paper describes the basics of the AC characteristics of circuit elements and their combined circuits to understand EIS. First, the AC characteristics of resistors, capacitors, and circuits combined in series and parallel are explained, with emphasis on the concept of impedance. Equivalent circuits, which are important in EIS analysis, are associated with circuit elements, such as charge transfer resistance, electric double-layer capacitance, and Warburg impedance. Next, the transmission-line model (TLM), which is used to understand EIS of porous electrodes, is presented. Porous electrodes are crucial for practical applications, such as lithium-ion batteries, fuel cells, and electric double-layer capacitors. Because porous electrodes have a large reaction area with many small pores, the electrochemical behavior of porous electrodes differs significantly from that of planar electrodes.
Figure 1 shows the current response of a resistor or capacitor when an AC voltage (E = E0 sin ωt) is applied. In the resistor, the current response obeys Ohm’s law (I = E/R); consequently, the ratio of the current to the applied voltage is always constant regardless of time, R [Ω] = E/I. By contrast, in a capacitor, the current is proportional to the time change of the voltage, I = C(dE/dt), where C [F] is the capacitance. As a result, the current flowing in the capacitor is proportional to the frequency of the applied voltage I = I0 cos(ωt) = E0ωC cos(ωt) = E0ωC sin(ωt − π/2). In AC measurements, the capacitor also exhibits similar behavior to the resistor, that is, a constant amplitude ratio of the voltage to the current, although the phase of the current is shifted by −90° with respect to the applied voltage. As shown in Fig. 1, the current response of the resistor and capacitor can be expressed using two parameters: the magnitude of impedance |Z| = E0/I0 corresponding to the amplitude ratio of voltage to current, and the phase difference between voltage and current, θ [°] or [rad]. For a resistor, |Z| is constant regardless of the frequency, and no phase difference occurs (0°), whereas for a capacitor, |Z| is inversely proportional to the frequency, and the phase difference is constant at −90°. By using the impedance expressed as Z = |Z|(cos θ + j sin θ), both circuit elements can be treated in the same manner in AC measurements.
Relationship between the applied sinusoidal voltage E(t) and resulting current I(t) to (a) resistor and (b) capacitor.
When an AC voltage is applied to a parallel RC circuit, the current is the sum of the currents flowing to the resistor IR and capacitor IC, corresponding to the sum of the vectors of IR and IC as shown in Fig. 2.
| \begin{align} I & = I_{\text{R}} + I_{\text{C}} = \frac{E_{0}}{R}\sin(\omega t) + E_{0}\omega C\cos(\omega t)\\ & = \frac{E_{0}\sqrt{1 + (RC\omega)^{2}}}{R}\sin(\omega t + \theta),\ \theta = \tan^{-1} (RC\omega) \end{align} | (1) |
| \begin{equation} |Z| = \frac{E_{0}}{I_{0}} = \frac{R}{\sqrt{1 + (RC\omega)^{2}}}, \ \theta = \tan^{-1} (RC\omega) \end{equation} | (2) |
Relationship between applied sinusoidal voltage E(t) and resulting current I(t) to a parallel RC circuit. The total current Itotal is the sum of the currents flowing through the resistor IR(t) and capacitor IC(t).
Conventionally, impedance is expressed using an imaginary unit j: Z = |Z|(cos θ + j sin θ). Because |Z| and θ of the resistor are equal to R and 0°, respectively, the impedance of the resistor is ZR = |Z| cos θ + j|Z| sin θ = R. However, the impedance of the capacitor becomes ZC = −j/(ωC) using |Z| = 1/(ωC) and θ = −90°. In addition to the calculation of the combined resistance of a series circuit, the impedance of a series RC circuit is the sum of impedances ZR and ZC.
| \begin{equation} Z = Z_{\text{R}} + Z_{\text{C}} = R - \frac{j}{\omega C} \end{equation} | (3) |
| \begin{align} &|Z| = \sqrt{Z'^{2} + Z''^{2}} = \sqrt{R^{2} + \frac{1}{\omega^{2}C^{2}}},\ \text{and}\notag\\ &\theta = \tan^{-1}\left(\frac{Z''}{Z'}\right) = \tan^{-1}\left(-\frac{1}{\omega RC} \right) \end{align} | (4) |
| \begin{align} &Z = |Z|(\cos\theta + j\sin\theta) = Z' + jZ'',\notag\\ &|Z|\cos\theta = Z'\ \text{and}\ |Z|\sin\theta = Z''. \end{align} | (5) |
| Z = Z′ + jZ′′ | Z′ = |Z| cos θ | Z′′ = |Z| sin θ | $|Z| = \sqrt{Z^{\prime 2} + Z^{\prime \prime 2}} $ | $\theta = \tan^{ - 1}\left( \dfrac{{Z''}}{{Z'}} \right)$ | |
|---|---|---|---|---|---|
| Series RC | $R - \dfrac{j}{\omega C}$ | R | $ - \dfrac{1}{\omega C}$ | $\sqrt{R^{2} + \dfrac{1}{\omega^{2}C^{2}}} $ | $\tan^{ - 1}\left( - \dfrac{1}{\omega RC} \right)$ |
| Parallel RC | $\left( \dfrac{1}{R} - \dfrac{\omega C}{j} \right)^{ - 1}$ | $\dfrac{R}{(\omega RC)^{2} + 1}$ | $\dfrac{ - \omega R^{2}C}{(\omega RC)^{2} + 1}$ | $\left( \dfrac{1}{R^{2}} + \omega^{2}C^{2} \right)^{ - 1/2}$ | tan−1(−ωRC) |
As described in the previous section, Z can be expressed using the absolute value of impedance |Z(ω)| and phase shift θ(ω), or the real part Z′(ω) and imaginary part Z′′(ω) of the impedance. Therefore, Z is illustrated in two manners: via a Bode plot, in which the logarithms of |Z(ω)| and θ(ω) are illustrated with respect to the logarithm of the frequency, or via a Nyquist plot, in which Z′(ω) and −Z′′(ω) are the horizontal and vertical axes, respectively. In the Nyquist plot, the vertical axis is taken as a negative value −Z′′(ω) to shift the locus of Z from the fourth quadrant to the first quadrant for easy viewing. The magnitude of the impedance |Z| corresponds to the distance from the origin, and the phase shift θ represents the angle with the real axis. The real and imaginary parts of impedance, Z′ and Z′′, are projected onto the real and imaginary axes, respectively. To show the features of the two representations, the impedances of the series and parallel RC circuits are depicted by a Bode (Fig. 3) and Nyquist plots (Fig. 4).
Bode plots of (a) series and (b) parallel RC circuits. Dashed lines indicate the frequency of f = 1/(2πRC) at θ = −45°.
Nyquist plots of (a) series and (b) parallel RC circuits. Dashed lines indicate the frequency of f = 1/(2πRC) at which θ = −45°.
Series RC circuit: Z of a series RC circuit is the sum of ZR and ZC. Therefore, Z of the circuit is equal to R in the high-frequency region because ZR ≫ ZC and equal to ZC in the low-frequency region because ZR ≪ ZC. As seen in the Bode plot, the series RC circuit behaves like a resistor in the high-frequency region (|Z| is independent of frequency, θ is 0°) and like a capacitor in the low-frequency region (|Z| is inversely proportional to frequency, θ is −90°). In the Nyquist plot, the locus of the impedance of the series RC circuit becomes a line extending vertically from the real axis, and the value of the intersection with the real axis is equal to R. The time constant of the circuit is RC, meaning that |ZR| = |ZC| and θ = −45° at a frequency ω = 1/(RC).
Parallel RC circuit: Z of a parallel RC circuit is calculated using the equation 1/Z = 1/ZR + 1/ZC. For a parallel RC circuit, the behavior of a circuit element with a smaller impedance appears. Therefore, Z of the parallel RC circuit is equal to ZC in the high-frequency region, whereas it is equal to ZR in the low-frequency region, which is the opposite of the series RC circuit. In the Nyquist plot of the parallel RC circuit, the locus Z is depicted as a semicircle, where the diameter of the arc is equal to R. The time constant of the parallel RC circuit is RC, which is the same as that of a series circuit. At a frequency of ω = 1/(RC), |ZR| = |ZC|, and θ = −45°, which corresponds to the vertex of the arc in the Nyquist plot.
Two types of EIS illustrations, namely, the Bode plot and Nyquist plot, represent the same meaning of Z; however, information obtained from the two illustrations contain certain differences. Figure 5 shows the Bode plots of a circuit with the same configuration but different capacitances (C = 10 mF and 100 mF, and 1 F). Because the time constant of the series RC circuit changes, the frequency at which |Z| changes in the low-frequency region varies depending on the capacitance C. The Bode plot allows us to discern the change in Z of the circuit against frequency because |Z| and θ are plotted as a function of frequency; moreover, the Nyquist plot is suitable for identifying the circuit configuration from the locus. The two semicircles in the Nyquist plot (Fig. 6a) indicate an equivalent circuit consisting of two parallel RC circuits with different time constants. To clarify the change in Z with respect to frequency, frequency information should be added to the figure. Another advantage of Nyquist plots is that the magnitude and change in Z can be intuitively understood particularly for minimal changes in Z because both the vertical Z′ and horizontal Z′′ axes are in linear scales instead of logarithmic scales in the Bode plot. When determining equivalent circuits by visual intuition in the Nyquist plots, particular attention should be given when analyzing the semicircles. Semicircles of two parallel RC circuits with similar time constants are difficult to distinguish, even when their resistances and capacitances are different. As shown in Fig. 6, a distorted semicircle resulting from the overlapping of two arcs is observed when the time constants of the two parallel RC circuits differ only 10 times. The two arcs can be clearly distinguished only when the time constants differ by at least 100 times. Therefore, when constructing the equivalent circuit, relying solely on the Nyquist plot is unwise. Verifying whether the equivalent circuit can rationally explain each elementary process in the electrochemical phenomena of interest is necessary.7
Bode plots of a circuit shown as the inset figure with varying capacitance C; (a) C = 10 mF, (b) 100 mF, and (c) 1 F.
Nyquist plots of a circuit shown as the inset figure with varying capacitances C; (a) C = 0.2 mF, (b) 1 mF, (c) 5 mF, and (d) 10 mF. The ratio of the time constants of the two parallel RC circuits is (a) 250, (b) 50, (c) 10, and (d) 5.
In EIS, the electrochemical behavior of an electrode is represented by an equivalent circuit comprising resistors and capacitors. A Randles circuit is generally used as the equivalent circuit of an electrode (Fig. 7).8
Nyquist plot of Randles circuit. Rs: solution resistance, Rct: charge transfer resistance, Cdl: electric double-layer capacitance, Zw: Warburg impedance.
Solution resistance9 Rs: The IR drop cannot be neglected, even when a reference electrode is used. The resistance involved in the conduction of an electrolyte is called solution resistance (Rs). Rs is obtained from the intersection with the real axis in the high-frequency limit in the Nyquist plot.
Charge transfer resistance10 Rct: The polarization behavior of conventional electrodes exhibits an exponential dependence between overvoltage and current when a large overvoltage is applied to an electrode. However, when the electrode potential approaches the equilibrium potential, the exponential relationship between overvoltage and current can be simplified as a linear relationship. Consequently, in the extremely low overvoltage region (<10 mV), the electrode behaves like a resistor, and its resistance is called the charge transfer resistance (Rct). Rct corresponds to the diameter of the semicircle in the Nyquist plots. Because the concept of charge transfer resistance can be utilized only in extremely small overvoltage regions where a linear relationship is obtained, the voltage amplitude in EIS is usually set to ≤10 mV.
Electric double-layer capacitance11 Cdl: The electric double layer at the interface between an electrode and an electrolyte is similar to a capacitor. Although the value of Cdl cannot be directly determined from the Nyquist plot, if Rct is determined, Cdl can be calculated from the time constant of the parallel circuit corresponding to the frequency of the apex of the arc using the following relation: τ = Rct × Cdl.
Warburg impedance12 ZW: Warburg impedance originates from the diffusion of active species in the electrolyte. In semi-infinite diffusion, ZW can be expressed by the following equation:
| \begin{equation} Z_{\text{W}} = \frac{RT}{n^{2}F^{2}Ac\sqrt{D}}\cdot \frac{1}{\sqrt \omega}\cdot \frac{1 - j}{\sqrt{2}} \end{equation} | (6) |
In the Bode plots shown in Fig. 8, the Warburg impedance has a magnitude slope of −0.5 decade per decade of frequency. When D increases, the active species are supplied rapidly to the electrode surface in response to a change in the concentration of active species there. Therefore, larger D values result in smaller Zw values. Conversely, Zw increases with decreasing D, thereby shifting the straight line to the low frequency side. As a result, in the Bode plot, the linear region of Zw overlaps with the region of the change of |Z| due to the parallel RC circuit. This phenomenon is more evident in the Nyquist plots shown in Fig. 9. When D is large, the straight line with a −45° slope and the arc are clearly distinguishable. However, the arc and line begin to overlap with as the value of D decreases. The overlapping symbolizes that the diffusion of active species is slower than the time constant of the charge transfer reaction. Essentially, the diffusion process is the rate-determining step.
Bode plots of the Randles circuit with varying values of RT/(n2F2AcD0.5): (a) 1, (b) 10, (c) 100, and (d) 300 Ω s−0.5.
Nyquist plots of the Randles circuit with varying values of RT/(n2F2AcD0.5): (a) 1, (b) 10, (c) 100, and (d) 300 Ω s−0.5.
In finite diffusion,14–16 the locus of ZW in the Nyquist plots significantly differs from that in semi-infinite diffusion. Finite diffusion can be divided into two cases depending on the boundary conditions: one is a “permeable boundary (PB),” in which the concentration at the boundary is constant, and the other is an “impermeable boundary (IPB),” in which no supply of species flows from outside the boundary. The ZW values for each case are shown in Fig. 10. The first boundary condition corresponds to a case in which the thickness of the diffusion layer is constant; examples thereof include a rotating disk electrode and a thin-film symmetrical electrode.14,15 Under the PB condition, ZW is expressed as follows:
| \begin{equation} Z_{\text{W,PB}} = \frac{RT}{n^{2}F^{2}Ac\sqrt{D}} \cdot \frac{1}{\sqrt \omega}\cdot \frac{1 - j}{\sqrt{2}}\cdot \tanh \sqrt{\frac{j\omega}{D}} \delta \end{equation} | (7) |
(a) Nyquist plots and (b) Bode plots of the impedance values of (A) semi-infinite diffusion, (B) finite diffusion with a PB condition, and (C) finite diffusion with an IPB condition. σeff is defined by Eq. A12 (see Appendix).
However, under the IPB condition, where no external species are supplied through the boundary, ZW is expressed as follows:
| \begin{equation} Z_{\text{W,IPB}} = \frac{RT}{n^{2}F^{2}Ac\sqrt{D}} \cdot \frac{1}{\sqrt \omega}\cdot \frac{1 - j}{\sqrt{2}}\cdot \coth \sqrt{\frac{j\omega}{D}} \delta \end{equation} | (8) |
The derivation of the Warburg impedance equation assumes only diffusion but not migration as the driving force of mass transfer. This indicates the presence of sufficient supporting electrolyte to mitigate migration. However, in some cases, for example, an energy device such as a Li-ion battery does not use a supporting electrolyte; instead, a binary electrolyte is typically used. In a binary electrolyte solution, the diffusion and migration of cations and anions are comparable in magnitude with one another. An exact description of the binary electrolyte solution is provided in Section 3.4.
TLM is a type of electronic circuit structure with a ladder-like feature. A basic example of a TLM in the field of electrochemistry is shown in Fig. 11. In this model, R and C microelements are used along and across lines, respectively. In a TLM, parameters have values per unit length, that is, [Ω cm−1] and [F cm−1] for the resistive microelement (rTLM) and capacitive microelement (cTLM), respectively, as shown in Fig. 11. Although Fig. 11a shows a circuit with discrete steps with a step length of Δx, a continuous model is typically assumed, as shown in Fig. 11b, because analytical solutions for continuous TLMs are available.17 In electrochemistry, TLMs are typically applied under two conditions: equivalent circuits for porous electrodes and diffusion (ZW). Although the diffusion condition is formulated in Eqs. 6–8, the corresponding TLMs help understand the physical meaning of these equations.
(a) TLM with R and C microelements (rTLM and cTLM) as a discrete circuit with a step length of Δx and (b) continuous model as the limit of Δx → 0. Here, the gray area indicates a bundle of infinitesimally small steps. ITLM and VTLM are the current and voltage, respectively, as functions of the position in the TLM.
Modeling of a porous electrode using TLM is schematically illustrated in Fig. 12. A porous electrode is a microscopic mixture of an electron-conducting phase (e.g., aggregation of particles of an electroactive material) and ion-conducting phase (impregnated electrolyte solution). Therefore, the potential and current distributions in the electron-conducting and ion-conducting phases should be considered separately. The TLM shown in Fig. 12b describes the potential and current distributions in the thickness direction using two lines for electronic and ionic currents. The interfacial impedance (zint) between the two lines represents the impedance at the electrode–electrolyte interface distributed in the porous electrode. The details are discussed in Section 3.2.
(a) Schematic of a porous electrode; (b) corresponding TLM.
Modeling of a Warburg element using TLM is illustrated in Fig. 13. A type of TLM can describe one-dimensional diffusion and can be used as a substitute for a Warburg element, as shown in Fig. 13. The details are discussed in Section 3.3. Notably, the TLM in Section 3.4 describing both diffusion and migration has more than two connecting terminals and requires more detailed consideration than that in Fig. 13.
(a) Randles-type equivalent circuit, (b) usage of a TLM as a Warburg element.
The mathematical solutions for the impedance values of typical TLMs are listed in Table 2, where L is the length of the model. The unit of the impedance values of the microelements along the lines (zA and zC) is [Ω cm−1], whereas that of values across the lines (zB) is [Ω cm]. Elements V, W, X, and Y in model (f) are set outside the ladder structure to describe the boundary conditions.
Instead of zA–zC, parameters can be combined with L as follows:
| \begin{equation} Z_{\text{A}} = z_{\text{A}}L \end{equation} | (9) |
| \begin{equation} Z_{\text{B}} = \frac{z_{\text{B}}}{L} \end{equation} | (10) |
| \begin{equation} Z_{\text{C}} = z_{\text{C}}L \end{equation} | (11) |
| \begin{equation} Z_{\text{TLM}} = \sqrt{Z_{\text{A}}Z_{\text{B}}}\coth \sqrt{\frac{Z_{\text{A}}}{Z_{\text{B}}}} \end{equation} | (12) |
| \begin{equation} Z_{\text{TLM}} = \sqrt{Z_{\text{A}}Z_{\text{B}}}\tanh \sqrt{\frac{Z_{\text{A}}}{Z_{\text{B}}}} \end{equation} | (13) |
In a porous electrode with a certain thickness, the condition of the electrode–electrolyte interface is locally distributed along the thickness direction owing to the ionic and/or electronic resistance inside the porous structure. As illustrated in Fig. 12, a TLM with two lines representing the ionic and electronic currents can be an equivalent circuit to describe this situation. This corresponds to model (d) shown in Table 2, in which microelements A and C represent the resistances for the ionic (rion [Ω cm−1]) and electronic (re [Ω cm−1]) conduction, respectively, and microelement B represents the interfacial impedance (zint), which may consist of the electric double layer (cdl [F cm−1]), charge-transfer resistance (rct [Ω cm]), and Warburg element (W), depending on the condition. For example, without an electrochemical reaction (blocking electrode), rion, 1/jωcdl, and re are substituted for zA, zB, and zC, respectively. In many cases, the electronic resistance is negligibly small; therefore, substituting zero for zC reduces the structure in model (a) in Table 2. It is noted that in some cases it might be convenience to use the total values of the parameters throughout the thickness direction, such as
| \begin{equation} R_{\text{ion}} = r_{\text{ion}}L \end{equation} | (14) |
| \begin{equation} C_{\text{dl}} = c_{\text{dl}}L, \end{equation} | (15) |
| \begin{equation} Z_{\text{A}} = R_{\text{ion}} \end{equation} | (16) |
| \begin{equation} Z_{\text{B}} = \frac{1}{j\omega C_{\text{dl}}} \end{equation} | (17) |
The effects of ionic resistance inside a porous electrode can be demonstrated by comparing a flat electrode (without TLM) and a porous electrode (with TLM), as shown in Fig. 14. In the Nyquist plots, the presence of a −45° line in the high-frequency region is the main effect of ionic resistance inside the porous electrode. In this frequency region, the penetration depth of the AC signal decreases with increasing frequency, that is, only the region adjacent to the boundary with the bulk solution senses the AC signal at the highest frequency. The magnitude of the −45° line region is approximately one-third of the ionic resistance throughout the thickness direction. Notably, rion, cdl, and rct are parameters with units of [Ω cm−1], [F cm−1], and [Ω cm], respectively, in accordance with the dimensions of the elements in the TLM.
Calculated impedance values of equivalent circuits without (a, c) and with (b, d) TLMs to reflect Rion, and those without (a, b) and with (c, d) Rct.
As shown in Fig. 15, zint may contain a Warburg element (W). However, attention must be focused on the applicability of this type of equivalent circuit. Because all the microelements for W are independent of each other, they cannot explain macroscopic diffusion in the thickness direction. Instead, a model for localized diffusion, typically inside each particle as a component of the solid phase of the porous electrode, can be used. Modeling of diffusion in the solution phase of a porous electrode is described in Section 3.4.
TLM with zint containing a Warburg element.
In this section, the length L of the TLM is assumed to be the thickness of the porous electrode. However, in some cases of a porous electrode made of a highly porous material, the ionic and/or electronic resistance of such materials may dominate the phenomena. In this case, a TLM can be an effective equivalent circuit for analyzing the EIS response. In this TLM, the physical meaning of L is the depth of the micropores in the material. Because the L value may be difficult to evaluate, the ZA–ZC values in Eqs. 9–11 can be used directly.
3.3 TLMs for Warburg impedanceDiffusion is a type of mass transfer driven by concentration gradient. During EIS measurements, sinusoidal periodic concentration changes in the active species are induced at the electrode surface and propagate into the bulk with attenuation. As shown in Fig. 13, one-dimensional diffusion can be electrically modeled using a TLM with R and C microelements by translating the physical quantities as:
| \begin{equation} V_{\text{TLM}} = \frac{\partial E^{\text{eq}}}{\partial c}\cdot \Delta c \left( = \frac{\partial \left(E^{\circ} + \cfrac{RT}{nF} \cdot \ln c\right)}{\partial c}\cdot \Delta c = \frac{RT}{nFc} \cdot \Delta c \right) \end{equation} | (18) |
| \begin{equation} I_{\text{TLM}} = nFA \cdot \Delta J \end{equation} | (19) |
| \begin{equation} r_{\text{TLM}} = r_{\text{diff}} = \cfrac{\cfrac{\partial E^{\text{eq}}}{\partial c}}{nFAD} \left(= \frac{RT}{n^{2}F^{2}ADc} \right) \end{equation} | (20) |
| \begin{equation} c_{\text{TLM}} = c_{\text{diff}} = \cfrac{nFA}{\cfrac{\partial E^{\text{eq}}}{\partial c}} \left(= \frac{n^{2}F^{2}Ac}{RT} \right) \end{equation} | (21) |
| \begin{equation} \Delta J = -D\frac{\partial c}{\partial x} \end{equation} | (22) |
| \begin{equation} \frac{\partial \Delta c}{\partial t} = -\frac{\partial \Delta J}{\partial x} \end{equation} | (23) |
| \begin{equation} I_{\text{TLM}} = - \frac{1}{r_{\text{TLM}}}\cdot \frac{\partial V_{\text{TLM}}}{\partial x} \end{equation} | (24) |
| \begin{equation} \frac{\partial V_{\text{TLM}}}{\partial t} = -\frac{1}{c_{\text{TLM}}}\cdot \frac{\partial I_{\text{TLM}}}{\partial x} \end{equation} | (25) |
If the assumption of the presence of sufficient supporting electrolyte is not guaranteed, diffusion and migration contribute to mass transfer. As the simplest case, a solution of a 1 : 1 binary electrolyte of M+ X− salt is considered. In a binary electrolyte solution, migration can be as large as diffusion for both cations and anions. The TLM shown in Fig. 16a is a one-dimensional diffusion–migration model for a binary electrolyte solution.22 In this model, the two lines correspond to the fluxes of the cation and anion:
| \begin{equation} I_{\text{M}^{+}} = +FA\cdot \Delta J_{\text{M}^{+}} \end{equation} | (26) |
| \begin{equation} I_{\text{X}^{-}} = -FA \cdot\Delta J_{\text{X}^{-}} \end{equation} | (27) |
(a) TLM for diffusion–migration in a binary electrolyte solution; (b) equivalent circuit of a symmetric cell of M | M+ X− solution | M.
As noted in Section 3.2, diffusion in the solution phase of a porous electrode cannot be modeled by simply using a Warburg element in zint. To account for concentration changes in a porous electrode, a combination of two different types of TLMs is required: one for a porous electrode (as shown in Fig. 12) and the other for diffusion–migration (Fig. 16). An example is shown in Fig. 17, which models diffusion–migration in a binary electrolyte solution impregnated in a porous electrode. The TLM is composed of three lines. Although they are not provided in Table 2, details are available in the literature.23
TLM for diffusion–migration in a binary electrolyte solution in a porous electrode. Here, only cdl is assumed as zint.
In this paper, the fundamentals of EIS, circuit elements corresponding to electrochemical processes, and transmission line models for expressing porous electrodes and Warburg impedance are briefly described. EIS provides useful information on the electrode kinetics at which various electrochemical processes of interest occur at the electrode, such as charge transfer reactions, ionic and electronic conduction, and diffusion. An understanding of the electrode kinetics is valuable because the power capability of electrochemical devices of practical importance can be predicted.
The study primarily focused on the equivalent circuits of the electrodes based on simulation data because electrodes show ideal electrochemical behavior. However, actual electrodes do not always obtain such spectra, which sometimes deviate from ideal electrodes, such as distorted arcs in the Nyquist plot. To further understand these concepts in more detail, readers should refer to Refs. 24–27. The application of EIS to practical electrochemical devices, such as batteries and capacitors, is described in the subsequent Part 2.
The data that support the findings of this study are openly available under the terms of the designated Creative Commons License in J-STAGE Data at https://doi.org/10.50892/data.electrochemistry.21100315. The authors' profiles of this paper can be found on the preface.28
Kingo Ariyoshi: Writing – original draft (Equal), Writing – review & editing (Equal)
Zyun Siroma: Writing – original draft (Equal), Writing – review & editing (Equal)
Atsushi Mineshige: Writing – review & editing (Supporting)
Mitsuhiro Takeno: Writing – review & editing (Supporting)
Tomokazu Fukutsuka: Writing – review & editing (Supporting)
Takeshi Abe: Writing – review & editing (Supporting)
Satoshi Uchida: Writing – review & editing (Supporting)
The authors declare no conflict of interest in the manuscript.
This paper constitutes a collection of papers edited as the proceedings of the 51st Electrochemistry Workshop organized by the Kansai Branch of the Electrochemical Society of Japan.
K. Ariyoshi and Z. Siroma: These authors contributed equally to this work.
K. Ariyoshi, Z. Siroma, A. Mineshige, M. Takeno, T. Fukutsuka, T. Abe, and S. Uchida: ECSJ Active Members
Faraday impedance (ZF) is the sum of charge transfer resistance (Rct) and Warburg impedance (ZW). ZW originates from sinusoidal periodic concentration changes of the active species at the surface of the electrode. Although the most familiar equation for ZW is
| \begin{equation} Z_{\text{W}} = \frac{RT}{n^{2}F^{2}Ac\sqrt D} \cdot \frac{1}{\sqrt \omega}\cdot \frac{1 - j}{\sqrt 2} \end{equation} | (6) |
| \begin{equation} Z_{\text{W}} = \frac{\partial E}{\partial c}\cdot \frac{\Delta c}{\Delta I} = \frac{\partial E}{\partial c}\cdot \frac{1}{nFA}\cdot \frac{\Delta c}{\Delta J} \end{equation} | (A1) |
| \begin{equation} \frac{\partial E}{\partial c} = \frac{\partial (E^{\circ} + RT/nF \cdot \ln c)}{\partial c} = \frac{RT}{nFc} \end{equation} | (A2) |
| \begin{equation} \frac{\Delta c}{\Delta J} = \frac{1}{\sqrt{j\omega D}} = \frac{1}{\sqrt{D}} \cdot \frac{1}{\sqrt \omega}\cdot \frac{1-j}{\sqrt 2} \end{equation} | (A3) |
| \begin{equation} c(x,t) - c_{\text{bulk}} = K \cdot \exp j\omega t \cdot \exp \left(- \sqrt{\frac{j\omega}{D}}x \right) \end{equation} | (A4) |
| \begin{equation} \Delta c(t) = K\cdot \exp j\omega t \end{equation} | (A5) |
| \begin{equation} \Delta J(t) = -D \cdot \left. \frac{\partial c}{\partial x}\right|_{x=0} {}= K \cdot \exp j\omega t \cdot \sqrt{j\omega D} \end{equation} | (A6) |
In addition to the semi-infinite diffusion described by Eq. 6, two types of finite diffusion, namely, those with “permeable boundary” (PB) and “impermeable boundary” (IPB), are typical conditions for EIS measurements. In these conditions, the thickness of the diffusion layer (δ) is an important parameter that is introduced. In the PB condition, the concentration at the boundary is fixed as
| \begin{equation} c(\delta,t) = c_{\text{bulk}}, \end{equation} | (A7) |
| \begin{align} &c(x,t) - c_{\text{bulk}} \notag\\ &\quad= K\cdot \exp j\omega t \cdot \left\{ \cosh\sqrt{\frac{j\omega}{D}} x - \coth \sqrt{\frac{j\omega}{D}} \delta \cdot \sinh \sqrt{\frac{j\omega}{D}} x \right\} \end{align} | (A8) |
| \begin{equation} Z_{\text{W,PB}} = \frac{RT}{n^{2}F^{2}Ac\sqrt{D}}\cdot \frac{1}{\sqrt{\omega}}\cdot \frac{1-j}{\sqrt{2}}\cdot \tanh \sqrt{\frac{j\omega}{D}} \delta \end{equation} | (7) |
| \begin{equation} \lim_{X \to \infty} \tanh X = 1 \end{equation} | (A9) |
| \begin{equation} \lim_{X \to 0}\tanh X \approx X \end{equation} | (A10) |
| \begin{equation} \lim_{\omega \to 0} Z_{\text{W,PB}} = \frac{\delta}{A\sigma_{\text{eff}}} \end{equation} | (A11) |
| \begin{equation} \sigma_{\text{eff}} = \frac{n^{2}F^{2}cD}{RT} \end{equation} | (A12) |
| \begin{equation} J(\delta,t) = 0 \end{equation} | (A13) |
| \begin{align} &c(x,t) - c_{\text{bulk}} \notag\\ &\quad= K \cdot \exp j\omega t \cdot \left\{ \cosh\sqrt{\frac{j\omega}{D}}x - \tanh \sqrt{\frac{j\omega}{D}}\delta \cdot \sinh \sqrt{\frac{j\omega}{D}}x \right\} \end{align} | (A14) |
| \begin{equation} Z_{\text{W,IPB}} = \frac{RT}{n^{2}F^{2}Ac\sqrt{D}} \cdot \frac{1}{\sqrt{\omega}} \cdot \frac{1-j}{\sqrt{2}}\cdot \coth \sqrt{\frac{j\omega}{D}} \delta \end{equation} | (8) |
| \begin{equation} \lim_{X \to \infty} \coth X = 1 \end{equation} | (A15) |
| \begin{equation} \lim_{X \to 0}\coth X \approx \frac{1}{X} + \frac{X}{3} \end{equation} | (A16) |
| \begin{equation} \lim_{\omega \to 0} Z_{\text{W,IPB}} = \frac{1}{3}\cdot \frac{\delta}{A\sigma_{\text{eff}}} + \frac{1}{j\omega C_{\text{lim}}} \end{equation} | (A17) |
| \begin{equation} C_{\text{lim}} = \frac{n^{2}F^{2}Ac\delta}{RT} \end{equation} | (A18) |
Each species involved in the redox reaction causes Warburg impedance. Therefore, Faraday impedance can be theoretically assumed as
| \begin{equation} Z_{\text{F}} = R_{\text{ct}} + Z_{\text{W,Red}} + Z_{\text{W,Ox}} \end{equation} | (A19) |
