Editors' Choice—Review—Impedance Response of Porous Electrodes: Theoretical Framework, Physical Models and Applications

The electrolyte solution is assumed to be binary and symmetric. Therefore, under the premise of electroneutrality condition, the concentrations of cations and anions are equal, ${c}_{+}\,=\,{c}_{-}\,=\,{c}_{e}.$ The electroneutrality assumption is valid if the characteristic size of the volume containing the electrolyte solution is much larger than the Debye length which is proportional to ${\left({c}_{e}\right)}^{-0.5}$ and equal to 0.96 nm for an aqueous strong electrolyte of 0.1 M. ⁷⁵ Ion transport takes place via diffusion and migration, while the convection effect is neglected. Consideration of natural convection using a modified Poisson-Nernst-Planck theory can be found in Ref. 76. We further assume there is no chemical reaction in electrolyte. A general theoretical framework considering chemical reactions and mechanical effects has been developed by Dreyer et al. ^77,78

According to the lattice-gas model, the electrochemical potential of ions in electrolytic solution considering the ion size effect is expressed as ⁷⁹

$$\begin{eqnarray}&&{\mu }_{{\rm{i}}}\,=\,{B}_{{\rm{i}}}+RT\,\mathrm{ln}\,\displaystyle \frac{v{c}_{{\rm{i}}}}{\left(1-vc\right)},\,i=+,-,\end{eqnarray} \tag{ 14 }$$

where ${B}_{+}$ and ${B}_{-}$ are the molar enthalpy of cations (+) and anions (−), ${c}_{+}$ and ${c}_{-}$ are the concentration of cations (+) and anions (−), $c\,=\,{c}_{+}+{c}_{-}$ and $v$ refer to the total concentration and average molar volume of ions, respectively. $R$ and $T$ have their usual meanings. If we assume that electrostatic interactions described in a mean-field manner dominate over short-range forces in the electrolyte solution, we have ${B}_{i}\,=\,{\mu }_{i}^{0}+{z}_{i}F{\phi }_{e},$ with ${\mu }_{i}^{0}$ being an unspecified constant with trivial importance here, ${z}_{i}$ the charge number of ions, and ${z}_{+}\,=\,-{z}_{-}\,=\,z,$ $F$ Faraday constant, and ${\phi }_{e}$ the electrostatic potential in the electrolyte phase.

Microscopically, ion transport has been described as a vacancy-coupled ion transfer reaction. ^80,81 Within this picture, the mass conservation law of ion transport is written as

$$\begin{eqnarray}&&\displaystyle \frac{\partial {c}_{i}}{\partial t}\,=\,-{\rm{\nabla }}\cdot {N}_{i},\,{N}_{i}\,=\,-{D}_{i}{c}_{i}\left(1-vc\right){\rm{\nabla }}\left(\displaystyle \frac{{\mu }_{i}}{RT}\right),\end{eqnarray} \tag{ 15 }$$

where ${D}_{i}$ are the diffusion coefficients of cations ($i\,=\,+$) and anions ($i\,=\,-$), respectively. Compared to its usual form, Eq. 15 features the term $\left(1-vc\right),$ which is the direct consequence of the steric effect.

Substituting Eq. 14 into the flux term gives

$$\begin{eqnarray}&&{N}_{i}=-{D}_{i}\left(1-vc\right){\rm{\nabla }}{c}_{i}-\displaystyle \frac{{z}_{i}F{D}_{i}\left(1-vc\right){c}_{i}}{RT}{\rm{\nabla }}{\phi }_{e}.\end{eqnarray} \tag{ 16 }$$

The ionic conductivity is defined as, ${\kappa }_{i}\,=\,{c}_{i}{m}_{i}{\left({z}_{i}F\right)}^{2},$ where ${m}_{i}$ is the mobility. The relationship between diffusion coefficient and mobility is given by Einstein's relation, ${D}_{i}\,=\,{m}_{i}RT.$ Equation 16 is then rewritten as

$$\begin{eqnarray}&&{N}_{+}\,=\,-{D}_{+}\left(1-2v{c}_{e}\right){\rm{\nabla }}{c}_{e}-\displaystyle \frac{{\kappa }_{+}\left(1-2v{c}_{e}\right)}{{z}_{+}F}{\rm{\nabla }}{\phi }_{e},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{N}_{-}\,=\,-{D}_{-}\left(1-2v{c}_{e}\right){\rm{\nabla }}{c}_{e}-\displaystyle \frac{{\kappa }_{-}\left(1-2v{c}_{e}\right)}{{z}_{-}F}{\rm{\nabla }}{\phi }_{e},\end{eqnarray} \tag{ 18 }$$

where we have used the electroneutrality assumption which prescribes ${c}_{+}\,=\,{c}_{-}\,=\,{c}_{e}$ and $c\,=\,{c}_{+}+{c}_{-}\,=\,2{c}_{e}$ for a binary symmetric electrolyte.

The total conductivity of electrolyte ${\kappa }_{e}$ is the sum of cationic and anionic conductivity, ${\kappa }_{e}\,=\,{\kappa }_{+}+{\kappa }_{-}.$ The ion transference number, representing the contribution of a sort of ion to the total ionic conductivity, is given by, ${t}_{+}\,=\,\tfrac{{\kappa }_{+}}{{\kappa }_{e}};\,{t}_{-}\,=\,\tfrac{{\kappa }_{-}}{{\kappa }_{e}}\,=\,1-{t}_{+},$ respectively. The total current density in the electrolyte phase is carried by both cations and anions, given by, ${\boldsymbol{J}}\,=\,{{\boldsymbol{J}}}_{+}+{{\boldsymbol{J}}}_{-}\,=\,zF\left({N}_{+}-{N}_{-}\right),$ which is expanded as

$$\begin{eqnarray}&&{\boldsymbol{J}}\,=\,-{\kappa }_{e}\left(1-2v{c}_{e}\right){\rm{\nabla }}{\phi }_{e}-zF\left(1-2v{c}_{e}\right)\left({D}_{+}-{D}_{-}\right){\rm{\nabla }}{c}_{e}.\end{eqnarray} \tag{ 19 }$$

It is readily seen from the above formula that the current density in electrolyte is driven by the potential gradient via migration, and together by the concentration gradient via diffusion. For the case of ${D}_{+}\,=\,{D}_{-},$ the contribution driven by concentration gradient disappears. In addition, for the case of diluted solution, we have $v{c}_{e}\,\to \,0,$ and Eq. 19 reduces back to the Ohmic law used in the standard de Levie model. ⁶⁴

By some algebra manipulation, we can represent ion fluxes in terms of collective electrolyte variables, namely, ${\boldsymbol{J}}$ and ${c}_{e},$

$$\begin{eqnarray}&&{N}_{+}\,=\,\displaystyle \frac{{t}_{+}}{zF}{\boldsymbol{J}}-\left({t}_{-}{D}_{+}+{t}_{+}{D}_{-}\right)\left(1-2v{c}_{e}\right){\rm{\nabla }}{c}_{e},\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{N}_{-}\,=\,-\displaystyle \frac{{t}_{-}}{zF}{\boldsymbol{J}}-\left({t}_{-}{D}_{+}+{t}_{+}{D}_{-}\right)\left(1-2v{c}_{e}\right){\rm{\nabla }}{c}_{e},\end{eqnarray} \tag{ 21 }$$

Substituting this flux expression back into the mass conservation law of, for example, anions, we obtain

$$\begin{eqnarray}&&\displaystyle \frac{\partial {c}_{e}}{\partial t}\,=\,-{\rm{\nabla }}{N}_{-}\,=\,{\rm{\nabla }}\displaystyle \frac{{t}_{-}{\boldsymbol{J}}}{zF}+{\rm{\nabla }}\left[\left({t}_{-}{D}_{+}+{t}_{+}{D}_{-}\right)\left(1-2v{c}_{e}\right){\rm{\nabla }}{c}_{e}\right].\end{eqnarray} \tag{ 22 }$$

Equations 19 and 22 are the basic equations which describes the charge conservation and the transport of ions in a symmetric binary electrolyte. Generally, ${t}_{-}$ and ${t}_{+}$ can be considered as a constant. Using Fourier transform, one can rephrase Eqs. 19 and 22 in terms of perturbed variations marked with an over-tilde

$$\begin{eqnarray}&&\tilde{{\boldsymbol{J}}}\,=\,-{\kappa }_{e}\left(1-2v{c}_{e}\right){\rm{\nabla }}{\tilde{\phi }}_{e}-zF\left(1-2v{c}_{e}\right)\left({D}_{+}-{D}_{-}\right){\rm{\nabla }}{\tilde{c}}_{e},\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&j\omega {\tilde{c}}_{e}\,=\,{\rm{\nabla }}\displaystyle \frac{{t}_{-}\tilde{{\boldsymbol{J}}}}{zF}+{\rm{\nabla }}\left[\left({t}_{-}{D}_{+}+{t}_{+}{D}_{-}\right)\left(1-2v{c}_{e}\right){\rm{\nabla }}{\tilde{c}}_{e}\right].\end{eqnarray} \tag{ 24 }$$

The major difference between the current formulation and the Newman theory exists in the electrochemical potential of ions. ⁴ In the Newman theory, Eq. 14 is replaced with

$$\begin{eqnarray}&&{\mu }_{i}\,=\,{B}_{i}+RT\,\mathrm{ln}\left(v{f}_{i}{c}_{i}\right),\end{eqnarray} \tag{ 25 }$$

with a generic, unspecified activity coefficient, ${f}_{i},$ which in the present theory is explicitly given by, ${f}_{+}\,=\,{f}_{-}\,=\,{\left(1-2v{c}_{e}\right)}^{-1},$ considering the ion size effect.

The volume averaging method provides a means to account for the essential features of a porous electrode without going into exact geometric detail. ^82,83 In this section, we upscale the microscopic electrolyte equations expressed in Eqs. 23 and 24 to macroscopic electrolyte equations with an average consideration of the complex porous structure.

Representative volume element (RVE) is a key concept in the volume averaging method. ^84,85 The RVE is a characteristic volume ${V}_{0}$ that is sufficiently large to effectively sample all microscopic heterogeneities, and is sufficiently small to serve as a volume element of the continuum media. In this regard, the RVE has a characteristic scale of ${l}_{rve},$ satisfies

$$\begin{eqnarray}&&{l}_{mi}\ll {l}_{rve}\ll {l}_{ma},\end{eqnarray} \tag{ 26 }$$

where ${l}_{mi}$ is the characteristic scale of microscopic heterogeneities, and ${l}_{ma}$ is the characteristic scale of the porous electrode, as shown in Fig. 3a.

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The volume-averaged value of a property $a$ over the volume ${V}_{0}$ is defined as

$$\begin{eqnarray}&&\left\langle a\right\rangle \,=\,\displaystyle \frac{1}{{V}_{0}}\displaystyle {\int }_{{V}_{0}}adV.\end{eqnarray} \tag{ 27 }$$

If $a$ belongs only to the liquid volume ${V}_{e},$ we obtain

$$\begin{eqnarray}&&\left\langle a\right\rangle \,=\,\displaystyle \frac{{V}_{e}}{{V}_{0}}\left(\displaystyle \frac{1}{{V}_{e}}\displaystyle {\int }_{{V}_{e}}adV\right)\,=\,{\varepsilon }_{e}{\left\langle a\right\rangle }^{e},\end{eqnarray} \tag{ 28 }$$

where ${\varepsilon }_{e}\,=\,{V}_{e}/{V}_{0}$ is the liquid volume fraction and ${\left\langle a\right\rangle }^{e}$ is the intrinsic liquid-phase average of $a.$ The fluctuation of $a$ is given by

$$\begin{eqnarray}&&\hat{a}\,=\,a-{\left\langle a\right\rangle }^{e}.\end{eqnarray} \tag{ 29 }$$

By means of Leibniz's rule for differentiation under the integral sign, the volume averaged counterparts of the temporal derivative, gradient and divergence follow ^86,87

$$\begin{eqnarray}&&\left\langle \displaystyle \frac{\partial a}{\partial t}\right\rangle \,=\,\displaystyle \frac{\partial \left\langle a\right\rangle }{\partial t}-\displaystyle \frac{1}{{V}_{0}}\displaystyle {\int }_{S}a\vec{{\bf{v}}}\cdot \vec{{\boldsymbol{n}}}dS,\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray} &&\langle {\rm{\nabla }}a \rangle \,=\,{\rm{\nabla }} \langle a \rangle +\frac{1}{{V}_{0}}{\int }_{S}a\overrightarrow{{\boldsymbol{n}}}dS,\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray} &&\langle {\rm{\nabla }}\cdot {\boldsymbol{a}} \rangle \,=\,{\rm{\nabla }}\cdot \langle {\boldsymbol{a}} \rangle +\frac{1}{{V}_{0}}{\int }_{S}{\boldsymbol{a}}\cdot \overrightarrow{{\boldsymbol{n}}}dS,\end{eqnarray} \tag{ 32 }$$

where $\vec{{\bf{v}}}$ is the velocity of interface, $\mathop{n}\limits^{ \rightharpoonup }$ is a unit normal vector pointing outward from the phase under consideration. Eq.(30) is also termed Reynolds transport theorem in the field of fluid dynamics. If we neglect volume change, $\vec{{\bf{v}}}=0$ and the integral term on the right-hand side of Eq. 30 disappears.

The volume average of the product of two quantities is,

$$\begin{eqnarray}&&\left\langle a\cdot b\right\rangle \,=\,{\varepsilon }_{e}{\left\langle a\cdot b\right\rangle }^{e}={\varepsilon }_{e}{\left\langle a\right\rangle }^{e}\cdot {\left\langle b\right\rangle }^{e}+{\varepsilon }_{e}\left\langle \hat{a}\cdot \hat{b}\right\rangle \end{eqnarray} \tag{ 33 }$$

We neglect ${\varepsilon }_{e}\left\langle \hat{a}\cdot \hat{b}\right\rangle$ in subsequent development, based on the assumption that the order of magnitude of the fluctuation is smaller than that of the average.

By this point, we are fully-fledged to derive macroscopic volume-averaged equations for ion transport in the electrolyte phase. We assume that only cations exchange between the electrolyte and solid phase through the interface, while the anion flux at the interface is zero. Therefore, substituting ${\boldsymbol{a}}$ with the Fourier-transformed cation flux ${\tilde{{\boldsymbol{J}}}}_{+},$ and anion flux ${\tilde{{\boldsymbol{J}}}}_{-},$ we have,

$$\begin{eqnarray} &&\langle {\rm{\nabla }}\cdot {\tilde{{ \mathcal J }}}_{+} \rangle \,=\,{\rm{\nabla }}\cdot \langle {\tilde{{ \mathcal J }}}_{+} \rangle +\frac{1}{{V}_{0}}{\int }_{S}{\tilde{{ \mathcal J }}}_{+}\cdot \overrightarrow{{\boldsymbol{n}}}dS\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&\left\langle {\rm{\nabla }}\cdot {\tilde{{\boldsymbol{J}}}}_{-}\right\rangle ={\rm{\nabla }}\cdot \left\langle {\tilde{{\boldsymbol{J}}}}_{-}\right\rangle \end{eqnarray} \tag{ 35 }$$

in the frequency domain. The source flux in Eq. 34 is further given by,

$$\begin{eqnarray}&&\frac{1}{{V}_{0}}{\int }_{S}{\tilde{{ \mathcal J }}}_{+}\cdot \overrightarrow{{\boldsymbol{n}}}dS\,=-\,{a}_{v}\frac{{ \langle {\tilde{{\phi }}}_{s} \rangle }^{s}-{ \langle {\tilde{{\phi }}}_{e} \rangle }^{e}}{{Z}_{{loc}}}\end{eqnarray} \tag{ 36 }$$

where ${Z}_{loc}$ is the local impedance at the electrode/electrolyte interface (EEI), ${a}_{v}$ is the volumetric electrochemical surface area. If we assume that solid particles are spherical and all surfaces are electrochemically accessible, we have ${a}_{v}\,=\,3{\varepsilon }_{{\rm{s}}}/{R}_{a},$ with ${\varepsilon }_{{\rm{s}}}$ representing the volume fraction of solid active particles and ${R}_{a}$ their average radius.

Adding Eqs. 34 and 35, substituting ${\tilde{{\boldsymbol{J}}}}_{+}\,+\,{\tilde{{\boldsymbol{J}}}}_{-}$ with $\tilde{{\boldsymbol{J}}},$ and using Eq. 36, one can obtain

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot \left\langle \tilde{{\boldsymbol{J}}}\right\rangle \,=\,{a}_{v}\displaystyle \frac{{\left\langle {\tilde{\phi }}_{s}\right\rangle }^{s}-{\left\langle {\tilde{\phi }}_{e}\right\rangle }^{e}}{{Z}_{loc}}.\end{eqnarray} \tag{ 37 }$$

The volume averaging of both sides of Eq. 23 yields

$$\begin{eqnarray}&&\begin{array}{l}\left\langle \tilde{{\boldsymbol{J}}}\right\rangle \,=\,-{\varepsilon }_{e}{\left\langle {\sigma }_{e}\right\rangle }^{e}{\left\langle 1-2v{c}_{e}\right\rangle }^{e}{\left\langle {\rm{\nabla }}{\tilde{\phi }}_{e}\right\rangle }^{e}-z{\varepsilon }_{e}F{\left\langle 1-2v{c}_{e}\right\rangle }^{e}\\ \,{\left\langle {D}_{+}-{D}_{-}\right\rangle }^{e}{\left\langle {\rm{\nabla }}{\tilde{c}}_{e}\right\rangle }^{e}.\end{array}\end{eqnarray} \tag{ 38 }$$

In addition, according to Eq. 31, ${\left\langle {\rm{\nabla }}{\tilde{\phi }}_{e}\right\rangle }^{e}$ and ${\left\langle {\rm{\nabla }}{\tilde{c}}_{e}\right\rangle }^{e}$ can be written as

$$\begin{eqnarray}&&{ \langle {\rm{\nabla }}{\tilde{{\phi }}}_{e} \rangle }^{e}\,=\,{\rm{\nabla }}{ \langle {\tilde{{\phi }}}_{e} \rangle }^{e}+\frac{1}{{V}_{e}}{\int }_{S}{\tilde{{\phi }}}_{e}\overrightarrow{{\boldsymbol{n}}}dS\approx {\rm{\nabla }}{ \langle {\tilde{{\phi }}}_{e} \rangle }^{e},\end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray}&&{ \langle {\rm{\nabla }}{\widetilde{c}}_{e} \rangle }^{e}\,=\,{\rm{\nabla }}{ \langle {\widetilde{c}}_{e} \rangle }^{e}+\frac{1}{{V}_{e}}{\int }_{S}{\widetilde{c}}_{e}\overrightarrow{{\boldsymbol{n}}}dS\approx {\rm{\nabla }}{ \langle {\widetilde{c}}_{e} \rangle }^{e}.\end{eqnarray} \tag{ 40 }$$

Therefore, Eq. 38 can be rewritten as

$$\begin{eqnarray}&&\begin{array}{l}\left\langle \tilde{{\boldsymbol{J}}}\right\rangle \,=\,-{\varepsilon }_{e}{\left\langle {\kappa }_{e}\right\rangle }^{e}{\left\langle 1-2v{c}_{e}\right\rangle }^{e}{\rm{\nabla }}{\left\langle {\tilde{\phi }}_{e}\right\rangle }^{e}\\ \,\,\,-\,z{\varepsilon }_{e}F{\left\langle 1-2v{c}_{e}\right\rangle }^{e}{\left\langle {D}_{+}-{D}_{-}\right\rangle }^{e}{\rm{\nabla }}{\left\langle {\tilde{c}}_{e}\right\rangle }^{e}.\end{array}\end{eqnarray} \tag{ 41 }$$

Substituting Eq. 41 into Eq. 37, one can obtain the frequency-domain volume-averaged equation describing charge conservation in the electrolyte phase,

$$\begin{eqnarray}&&\begin{array}{l}{a}_{v}\displaystyle \frac{{\left\langle {\tilde{\phi }}_{s}\right\rangle }^{s}-{\left\langle {\tilde{\phi }}_{e}\right\rangle }^{e}}{{Z}_{loc}}\,=\,-{\varepsilon }_{e}{\rm{\nabla }}\cdot \left({\left\langle {\kappa }_{e}\right\rangle }^{e}{\left\langle 1-2v{c}_{e}\right\rangle }^{e}{\rm{\nabla }}{\left\langle {\tilde{\phi }}_{e}\right\rangle }^{e}\right)\\ \,-\,z{\varepsilon }_{e}F{\rm{\nabla }}\cdot \left({\left\langle {D}_{+}-{D}_{-}\right\rangle }^{e}{\left\langle 1-2v{c}_{e}\right\rangle }^{e}{\rm{\nabla }}{\left\langle {\tilde{c}}_{e}\right\rangle }^{e}\right).\end{array}\end{eqnarray} \tag{ 42 }$$

Similarly, we transform Eq. 24 to

$$\begin{eqnarray}&&\begin{array}{l}{\varepsilon }_{e}j\omega {\tilde{c}}_{e}\,=\,\displaystyle \frac{\left(1-{t}_{+}\right)}{F}{a}_{v}\displaystyle \frac{{\left\langle {\tilde{\phi }}_{s}\right\rangle }^{s}-{\left\langle {\tilde{\phi }}_{e}\right\rangle }^{e}}{{Z}_{loc}}\\ \,+\,{\varepsilon }_{e}{\rm{\nabla }}\cdot \left({\left\langle {t}_{-}{D}_{+}+{t}_{+}{D}_{-}\right\rangle }^{e}{\left\langle 1-2v{c}_{e}\right\rangle }^{e}{\left\langle {\rm{\nabla }}{\tilde{c}}_{e}\right\rangle }^{e}\right).\end{array}\end{eqnarray} \tag{ 43 }$$

It should be noted that the divergence and gradient operators in Eqs. 42 and 43 are conducted in the local torturous space, denoted $\left(\alpha ,\beta ,\gamma \right)$ and depicted in Fig. 3b. We now embark on transforming Eqs. 42 and 43 from the local coordinate $\left(\alpha ,\beta ,\gamma \right)$ to the regular Cartesian coordinate $(x,y,z)$ as depicted in Fig. 3b.

For a scalar quantity, say $a,$ the gradients in two coordinate systems are correlated as ⁶⁵

$$\begin{eqnarray}&&{{\rm{\nabla }}}_{\left(\alpha ,\beta ,\gamma \right)}a\,=\,\displaystyle \frac{1}{J}\displaystyle \frac{\partial }{\partial {x}_{i}}\left(Ja{\rm{\nabla }}{x}_{i}\right),\end{eqnarray} \tag{ 44 }$$

$$\begin{eqnarray}&&{{\rm{\nabla }}}_{\left(\alpha ,\beta ,\gamma \right)}^{2}a\,=\,\displaystyle \frac{1}{J}\displaystyle \frac{\partial }{\partial {x}_{i}}\left(J{g}_{i,j}\displaystyle \frac{\partial a}{\partial {x}_{j}}\right),\end{eqnarray} \tag{ 45 }$$

where we follow Einstein summation convention, and the coordinate transformation matrix determinant $J$ and coefficients ${g}_{i,j}$ are given by

$$\begin{eqnarray}\begin{array}{l}J\,=\,\displaystyle \frac{\partial \left(\alpha ,\beta ,\gamma \right)}{\partial \left(x,y,z\right)}\,=\,\left|\begin{array}{ccc}{\alpha }_{x} & {\alpha }_{y} & {\alpha }_{z}\\ {\beta }_{x} & {\beta }_{y} & {\beta }_{z}\\ {\gamma }_{x} & {\gamma }_{y} & {\gamma }_{z}\end{array}\right|,\\ {g}_{i,j}\,=\,{\rm{\nabla }}{x}_{i}\cdot {\rm{\nabla }}{x}_{j},\\ {\rm{\nabla }}{x}_{i}\,=\,\displaystyle \frac{\partial {x}_{i}}{\partial {\xi }_{j}}{e}_{j},\end{array}\end{eqnarray} \tag{ 46 }$$

where ${\xi }_{j}$ indexes over ($\alpha ,\beta ,\gamma$) and ${e}_{j}$ is the unit vector in the ($\alpha ,\beta ,\gamma$) coordinate. For the one-dimension case, we have

$$\begin{eqnarray}&&J\,=\,\displaystyle \frac{\partial \left(\gamma \right)}{\partial \left(z\right)}\,=\,\tau ,\,g\,=\,\displaystyle \frac{1}{{\tau }^{2}},\end{eqnarray} \tag{ 47 }$$

where $\tau$ is the local tortuosity.

Thereafter, Eqs. 42 and 43 in the regular Cartesian coordinate $(x,y,z)$ are rewritten as,

$$\begin{eqnarray}&&\begin{array}{l}{a}_{v}\displaystyle \frac{{\left\langle {\tilde{\phi }}_{s}\right\rangle }^{s}-{\left\langle {\tilde{\phi }}_{e}\right\rangle }^{e}}{{Z}_{loc}}\,=\,-{\varepsilon }_{e}\displaystyle \frac{1}{J}\displaystyle \frac{\partial }{\partial {x}_{i}}\left(\left({\rm{\nabla }}{x}_{i}\right){\left\langle {\kappa }_{e}\right\rangle }^{e}{\left\langle 1-2v{c}_{e}\right\rangle }^{e}\displaystyle \frac{\partial }{\partial {x}_{i}}\right.\\ \,\left.\times \,\left(J\left({\rm{\nabla }}{x}_{i}\right){\left\langle {\tilde{\phi }}_{e}\right\rangle }^{e}\right)\right)-zF{\varepsilon }_{e}\displaystyle \frac{1}{J}\displaystyle \frac{\partial }{\partial {x}_{i}}\cdot \left(\left({\rm{\nabla }}{x}_{i}\right){\left\langle {D}_{+}-{D}_{-}\right\rangle }^{e}{\left\langle 1-2v{c}_{e}\right\rangle }^{e}\right.\\ \,\left.\,\times \,\displaystyle \frac{\partial }{\partial {x}_{i}}\left(J\left({\rm{\nabla }}{x}_{i}\right){\rm{\nabla }}{\left\langle {c}_{e}\right\rangle }^{e}\right)\right)\end{array}\end{eqnarray} \tag{ 48 }$$

$$\begin{eqnarray}&&\begin{array}{l}{\varepsilon }_{e}j\omega {\tilde{c}}_{e}\,=\,\displaystyle \frac{\left(1-{t}_{+}\right)}{F}{a}_{v}\displaystyle \frac{{\left\langle {\tilde{\phi }}_{s}\right\rangle }^{s}-{\left\langle {\tilde{\phi }}_{e}\right\rangle }^{e}}{{Z}_{loc}}+{\varepsilon }_{e}\displaystyle \frac{1}{J}\displaystyle \frac{\partial }{\partial {x}_{i}}\\ \,\times \,\left(\left({\rm{\nabla }}{x}_{i}\right){\left\langle {t}_{-}{D}_{+}+{t}_{+}{D}_{-}\right\rangle }^{e}{\left\langle 1-2v{c}_{e}\right\rangle }^{e}\displaystyle \frac{\partial }{\partial {x}_{i}}\left(J\left({\rm{\nabla }}{x}_{i}\right){\rm{\nabla }}{\left\langle {c}_{e}\right\rangle }^{e}\right)\right)\end{array}\end{eqnarray} \tag{ 49 }$$

which are the general equations of ion transport in the electrolyte phase in porous electrodes obtained via volume averaging and coordinate transformation.

Provided assumptions and boundary conditions to be detailed below, we are able to obtain analytical impedance expressions from Eqs. 48 and 49. Firstly, we consider a non-dimensional case where $J$ and $g$ given in Eq. 47 are uniform. Secondly, we define effective transport properties,

$$\begin{eqnarray}&&{\kappa }_{e,e,eff}\,=\,\displaystyle \frac{{\varepsilon }_{e}}{{\tau }^{2}}{\left\langle {\kappa }_{e}\right\rangle }^{e}{\left\langle 1-2v{c}_{e}\right\rangle }^{e},\end{eqnarray} \tag{ 50 }$$

$$\begin{eqnarray}&&{D}_{e,eff}\,=\,\displaystyle \frac{{\varepsilon }_{e}}{{\tau }^{2}}{\left\langle {D}_{+}-{D}_{-}\right\rangle }^{e}{\left\langle 1-2v{c}_{e}\right\rangle }^{e},\end{eqnarray} \tag{ 51 }$$

$$\begin{eqnarray}&&{\left({D}_{+}-{D}_{-}\right)}_{e,eff}\,=\,\displaystyle \frac{{\varepsilon }_{e}}{{\tau }^{2}}{\left\langle {D}_{+}-{D}_{-}\right\rangle }^{e},\end{eqnarray} \tag{ 52 }$$

which embody the structure effect and ion size effect. Thirdly, we further neglect the spatial inhomogeneities of these effective transport properties. Then, we have

$$\begin{eqnarray}&&{a}_{v}{\tilde{j}}_{loc}\,=\,-{\kappa }_{e,e,eff}\displaystyle \frac{{\partial }^{2}{\tilde{\phi }}_{e}}{\partial {x}^{2}}-F{\left({D}_{+}-{D}_{-}\right)}_{e,eff}\displaystyle \frac{{\partial }^{2}{\tilde{c}}_{e}}{\partial {x}^{2}},\end{eqnarray} \tag{ 53 }$$

$$\begin{eqnarray}&&{\varepsilon }_{e}j\omega {\tilde{c}}_{e}\,=\,{D}_{e,eff}\displaystyle \frac{{\partial }^{2}{\tilde{c}}_{e}}{\partial {x}^{2}}+\displaystyle \frac{\left(1-{t}_{+}\right)}{F}{a}_{v}{\tilde{j}}_{loc},\end{eqnarray} \tag{ 54 }$$

where we have dropped the bracket and superscript "e" on the intrinsic variables for brevity. Thus, we have recovered the electrolyte-phase controlling equations in the Newman theory. As for electron transport in the solid phase, we use the Ohmic' law,

$$\begin{eqnarray}&&{a}_{v}{\tilde{j}}_{loc}\,=-\,{\kappa }_{s,eff}\frac{{\partial }^{2}{\tilde{\phi }}_{s}}{\partial {x}^{2}},\end{eqnarray} \tag{ 55 }$$

with an effective electron conductivity ${\kappa }_{s,eff}.$

The boundary conditions of Eqs. 53, 54 and 55 are as follow. As shown in Fig. 4, at $x\,=\,0,$ the ionic flux and electrical potential gradient in the electrolyte phase are zero, and a current stimulus, ${\tilde{j}}_{app},$ is imposed, that is

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\tilde{c}}_{e}}{\partial x}\,=\,0,\,\displaystyle \frac{\partial {\tilde{\phi }}_{e}}{\partial x}\,=\,0,\,-{\kappa }_{s,eff}\displaystyle \frac{\partial {\tilde{\phi }}_{s}}{\partial x}\,=\,{\tilde{j}}_{app}.\end{eqnarray} \tag{ 56 }$$

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At $x\,=\,{l}_{p},$ the ion concentration is constant, namely, the perturbation in the ion concentration is zero, and the total current flows through the electrolyte path only, that is

$$\begin{eqnarray}&&{\tilde{c}}_{e}\,=\,0,\,\displaystyle \frac{\partial {\tilde{\phi }}_{s}}{\partial x}\,=\,0,\,-{\kappa }_{e,eff}\displaystyle \frac{\partial {\tilde{\phi }}_{e}}{\partial x}-F{\left({D}_{+}-{D}_{-}\right)}_{e,eff}\displaystyle \frac{\partial {\tilde{c}}_{e}}{\partial x}\,=\,{\tilde{j}}_{app}.\end{eqnarray} \tag{ 57 }$$

Equations 53, 54 and 55 are put into a concise form,

$$\begin{eqnarray}&&\displaystyle \frac{{\partial }^{2}{\tilde{c}}_{e}}{\partial {x}^{2}}\,=\,{{\rm{\Theta }}}_{1}{\tilde{c}}_{e}+{{\rm{\Theta }}}_{2}\left({\tilde{\phi }}_{s}-{\tilde{\phi }}_{e}\right),\end{eqnarray} \tag{ 58 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\partial }^{2}\left({\tilde{\phi }}_{s}-{\tilde{\phi }}_{e}\right)}{\partial {x}^{2}}\,=\,{{\rm{\Theta }}}_{3}{\tilde{c}}_{e}+{{\rm{\Theta }}}_{4}\left({\tilde{\phi }}_{s}-{\tilde{\phi }}_{e}\right),\end{eqnarray} \tag{ 59 }$$

with the coefficients ${{\rm{\Theta }}}_{i}$ expressed as

$$\begin{eqnarray}&&{{\rm{\Theta }}}_{1}\,=\,\displaystyle \frac{{\rm{j}}\omega {\varepsilon }_{e}}{{D}_{e,eff}},\end{eqnarray} \tag{ 60 }$$

$$\begin{eqnarray}&&{{\rm{\Theta }}}_{2}\,=\,-\displaystyle \frac{\left(1-{t}_{+}\right)}{F{D}_{e,eff}}\displaystyle \frac{{a}_{v}}{{Z}_{loc}},\end{eqnarray} \tag{ 61 }$$

$$\begin{eqnarray}&&{{\rm{\Theta }}}_{3}\,=\,\displaystyle \frac{{\rm{j}}\omega {\varepsilon }_{e}{\left({D}_{+}-{D}_{-}\right)}_{e,eff}}{{D}_{e,eff}{\kappa }_{e,eff}},\end{eqnarray} \tag{ 62 }$$

$$\begin{eqnarray}&&{{\rm{\Theta }}}_{4}\,=\,\displaystyle \frac{{a}_{v}}{{Z}_{loc}}\left[\displaystyle \frac{{\left({D}_{+}-{D}_{-}\right)}_{e,eff}\left(1-{t}_{+}\right)}{F{D}_{e,eff}}+\left(\displaystyle \frac{1}{{\kappa }_{e,eff}}+\displaystyle \frac{1}{{\kappa }_{s,eff}}\right)\right].\end{eqnarray} \tag{ 63 }$$

In what follows, ${{\rm{\Theta }}}_{i}$ are taken as constant coefficients. For this end, ${Z}_{loc}$ is assumed to be uniform across the porous electrode. It is known that ${Z}_{loc}$ is determined by the local concentration of species, potential difference across and structural properties of the EEI. Therefore, the porous electrodes should be homogeneous and there is no concentration or potential gradient under the static state condition (no dc current), in order to justify the assumption of uniform ${Z}_{loc}.$ In the presence of dc current, numerical solution is usually resorted to.

With algebra manipulations detailed in the SI, ${Z}_{p}$ is analytically solved as,

$$\begin{eqnarray}&&\begin{array}{l}{Z}_{p}\,=\,\displaystyle \frac{{l}_{p}}{{\kappa }_{s,eff}}+\displaystyle \frac{{a}_{v}}{{Z}_{loc}{\kappa }_{s,eff}{\tilde{j}}_{app}}\left(\left(\displaystyle \frac{{\alpha }_{1}}{\sqrt{{\lambda }_{1}}}+\displaystyle \frac{{\beta }_{1}}{\sqrt{{\lambda }_{2}}}\right){l}_{p}+\left(\displaystyle \frac{{\alpha }_{2}}{{\lambda }_{1}}+\displaystyle \frac{{\beta }_{2}}{{\lambda }_{2}}\right)\right.\\ \,\,\left.-\,\left(\displaystyle \frac{1}{{\lambda }_{2}}-\displaystyle \frac{{v}_{2}}{{v}_{1}{\lambda }_{1}}\right){\rm{\Lambda }}\right)+\displaystyle \frac{1}{{\tilde{j}}_{app}}\left(1-\displaystyle \frac{{v}_{2}}{{v}_{1}}\right){\rm{\Lambda }},\end{array}\end{eqnarray} \tag{ 64 }$$

with the coefficient ${\rm{\Lambda }}$ denoting

$$\begin{eqnarray}&&{\rm{\Lambda }}\,=\,\displaystyle \frac{{\tilde{j}}_{app}}{{\kappa }_{e,eff}}\displaystyle \frac{{v}_{1}\left(1+\tfrac{{\kappa }_{e,eff}}{{\kappa }_{s,eff}}\tfrac{1}{{v}_{2}-{v}_{1}}\left(\displaystyle \frac{\left(1-\tfrac{{{\rm{\Theta }}}_{3}}{{{\rm{\Theta }}}_{1}}{v}_{1}\right){v}_{2}}{\cos \,{\rm{h}}\left(\sqrt{{\lambda }_{1}}{l}_{p}\right)}-\displaystyle \frac{\left(1-\tfrac{{{\rm{\Theta }}}_{3}}{{{\rm{\Theta }}}_{1}}{v}_{2}\right){v}_{1}}{\cos \,{\rm{h}}\left(\sqrt{{\lambda }_{2}}{l}_{p}\right)}\right)\right)}{\sqrt{{\lambda }_{2}}\left(1-\tfrac{{{\rm{\Theta }}}_{3}}{{{\rm{\Theta }}}_{1}}{v}_{2}\right){v}_{1}\,\tan \,{\rm{h}}\left(\sqrt{{\lambda }_{2}}{l}_{p}\right)-\sqrt{{\lambda }_{1}}\left(1-\tfrac{{{\rm{\Theta }}}_{3}}{{{\rm{\Theta }}}_{1}}{v}_{1}\right){v}_{2}\,\tan \,{\rm{h}}\left(\sqrt{{\lambda }_{1}}{l}_{p}\right)},\end{eqnarray} \tag{ 65 }$$

and the coefficients $\alpha ,$ $\beta ,$ ${\lambda }_{1},$ ${\lambda }_{2},$ ${v}_{1},$ ${v}_{2}$ are provided in the SI. Though ${\tilde{j}}_{app}$ is explicitly included in Eq. 64, ${Z}_{p}$ is independent of ${\tilde{j}}_{app},$ a consequence of linearity of the model, as ${\alpha }_{i},$ ${\beta }_{i}$ and ${\rm{\Lambda }}$ are proportional to ${\tilde{j}}_{app}$ and ${\tilde{j}}_{app}$ is cancelled out eventually.

The above impedance model can be simplified to a varying extent in following four cases. The first case assumes ${\kappa }_{e,eff}\ll {\kappa }_{s,eff},$ or equivalently, ${\kappa }_{e,eff}/{\kappa }_{s,eff}\,\approx \,0,$ namely, the electronic conduction is sufficiently fast compared to the ionic transport. This assumption is reasonable in many electrochemical systems, for example, carbon-based/supported electrodes for supercapacitors, LIBs and PEFCs. Under this assumption, we have ${\alpha }_{1}\,=\,0,$ ${\beta }_{1}\,=\,0,$ and much simplified expressions for ${\alpha }_{2},$ ${\beta }_{2}.$ The simplified impedance expression for this case 1 is

$$\begin{eqnarray}&&{Z}_{p}\,=\,\displaystyle \frac{1}{{\kappa }_{e,eff}}\displaystyle \frac{{v}_{1}-{v}_{2}}{\sqrt{{\lambda }_{2}}\left(1-\tfrac{{{\rm{\Theta }}}_{3}}{{{\rm{\Theta }}}_{1}}{v}_{2}\right){v}_{1}\,\tan \,{\rm{h}}\left(\sqrt{{\lambda }_{2}}{l}_{p}\right)-\sqrt{{\lambda }_{1}}\left(1-\tfrac{{{\rm{\Theta }}}_{3}}{{{\rm{\Theta }}}_{1}}{v}_{1}\right){v}_{2}\,\tan \,{\rm{h}}\left(\sqrt{{\lambda }_{1}}{l}_{p}\right)}.\end{eqnarray} \tag{ 66 }$$

In the second case, we neglect ionic diffusion in the electrolyte phase. Specifically, Eq. 58 is neglected and Eq. 59 is simplified to

$$\begin{eqnarray}&&\displaystyle \frac{{\partial }^{2}\left({\tilde{\phi }}_{s}-{\tilde{\phi }}_{e}\right)}{\partial {x}^{2}}\,=\,{{\rm{\Theta }}}_{4}\left({\tilde{\phi }}_{s}-{\tilde{\phi }}_{e}\right),\end{eqnarray} \tag{ 67 }$$

with the corrected expression of ${{\rm{\Theta }}}_{4}\,=\,\tfrac{{a}_{v}}{{Z}_{loc}}\left[\left(\tfrac{1}{{\kappa }_{e,eff}}+\tfrac{1}{{\kappa }_{s,eff}}\right)\right].$ Thus, the simplified impedance expression for the case 2 is

$$\begin{eqnarray}&&\begin{array}{l}{Z}_{p}\,=\,\displaystyle \frac{{l}_{p}}{{\kappa }_{s,eff}+{\kappa }_{e,eff}}\,+\,\displaystyle \frac{\coth \left(\sqrt{{{\rm{\Theta }}}_{4}}{l}_{p}\right)}{\sqrt{{{\rm{\Theta }}}_{4}}\left({\kappa }_{s,eff}+{\kappa }_{e,eff}\right)}\\ \,\,\times \,\left(\displaystyle \frac{{{\kappa }_{s,eff}}^{2}+{{\kappa }_{e,eff}}^{2}}{{\kappa }_{s,eff}{\kappa }_{e,eff}}+\displaystyle \frac{2}{\cosh \left(\sqrt{{{\rm{\Theta }}}_{4}}{l}_{p}\right)}\right),\end{array}\end{eqnarray} \tag{ 68 }$$

with the detailed derivation presented in the SI.

In the third case, we neglect both ionic diffusion and electronic conduction (${\kappa }_{s,eff}\,\to \,\infty ,$ a stronger assumption than ${\kappa }_{e,eff}\,\ll \,{\kappa }_{s,eff}$ used in the case 1. Therefore, Eq. 68 is further simplified as

$$\begin{eqnarray}&&{Z}_{p}\,=\,\sqrt{\displaystyle \frac{{Z}_{loc}}{{a}_{v}{\kappa }_{e,eff}}}\,\coth \left(\sqrt{\displaystyle \frac{{a}_{v}}{{\kappa }_{e,eff}{Z}_{loc}}}{l}_{p}\right),\end{eqnarray} \tag{ 69 }$$

which is the standard de Levie model.

In the fourth case, in addition to above assumptions made in the de Levie model, we further assume that ${\kappa }_{e,eff}$ is sufficiently high, satisfying

$$\begin{eqnarray}&&\displaystyle \frac{{a}_{v}\cdot {l}_{p}^{2}}{{\kappa }_{e,eff}\cdot {Z}_{loc}}\,\ll \,1.\end{eqnarray} \tag{ 70 }$$

In this case, the impedance is simply given by

$$\begin{eqnarray}&&{Z}_{pe}\,=\,\displaystyle \frac{{Z}_{loc}}{{a}_{v}\cdot {l}_{p}}.\end{eqnarray} \tag{ 71 }$$

Equation 71 shows that ${Z}_{pe}$ is proportional to ${Z}_{loc}$ with a trivial structural coefficient.

By this point, we have outlined a general theoretical framework of impedance response of uniform porous electrodes impregnated with a symmetric binary electrolyte under the electroneutrality approximation and the static state condition. The theory is expected to be applicable to model impedance response of porous electrodes in supercapacitors, various sorts of metal-ion batteries, and PEFCs. In what follows, we review chronologically literature studies introducing different transport processes and ${Z}_{loc}$ in the context of the above theoretical framework.

The impedance response of gas-diffusion porous electrodes was studied by Darby in 1960s, ⁸⁸ and by Keddam et al. in 1984. ⁸⁹ The transport process considered in their models is gas transport in the axial direction, which is coupled to a Faradic reaction, with first- or arbitrary order kinetics, at the pore wall. Double-layer charging current is neglected. The key controlling equation in their models is

$$\begin{eqnarray}&&\displaystyle \frac{\partial c}{\partial t}\,=\,D\displaystyle \frac{{\partial }^{2}c}{\partial {x}^{2}}-Kc,\end{eqnarray} \tag{ 72 }$$

where $D$ is the diffusion coefficient and $K$ a pseudo-homogeneous rate constant. The boundary conditions are: $c\left(x\right)={c}_{0}$ at $x=0$ with ${c}_{0}$ being the concentration in the solution bulk, and $dc\left(x\right)/dx=0$ at $x={l}_{p}$ as gas cannot penetrate the pore end. The gas concentration distribution at steady state is obtained as

$$\begin{eqnarray}&&c\left(x\right)\,=\,{c}_{0}\displaystyle \frac{\cosh \left[\left(x-{l}_{p}\right){\left(K/D\right)}^{1/2}\right]}{\cosh \left[{l}_{p}{\left(K/D\right)}^{1/2}\right]},\end{eqnarray} \tag{ 73 }$$

with ${l}_{p}$ being the thickness of the porous electrode.

When the gas-diffusion porous electrode is submitted to a small perturbing signal around its steady-state condition, the response, namely, the concentration change ${\rm{\Delta }}c,$ can be derived from the Taylor expansion of Eq. 72 limited to first-order terms,

$$\begin{eqnarray}&&\displaystyle \frac{d{\rm{\Delta }}c}{\partial t}=D\displaystyle \frac{{\partial }^{2}{\rm{\Delta }}c}{\partial {x}^{2}}-K{\rm{\Delta }}c-K^{\prime} c\left(x\right){\rm{\Delta }}E,\end{eqnarray} \tag{ 74 }$$

where $K^{\prime} \,=\,dK/dE,$ ${\rm{\Delta }}E\,=\,\left|{\rm{\Delta }}E\right|\exp \left(j\omega t\right)$ is a sinuous voltage perturbation, with $\left|{\rm{\Delta }}E\right|$ being the amplitude and $\omega$ the angular frequency. Via Fourier transform, Eq. 74 is written as

$$\begin{eqnarray}&&j\omega \tilde{c}=D\displaystyle \frac{{\partial }^{2}\tilde{c}}{\partial {x}^{2}}-K\tilde{c}-K^{\prime} c\left(x\right)\tilde{E}.\end{eqnarray} \tag{ 75 }$$

Different from the uniform static-state concentration distribution assumed in the above theoretical framework, Eq. 75 features a nonuniform $c\left(x\right)$ given in Eq. 73. For this reason, the model cannot be solved analytically, and numerical impedance simulation is in order.

Lasia extended the de Levie model by replacing ${Z}_{loc}\,=\,1/{\rm{j}}\omega {C}_{dl}$ with ${Z}_{loc}\,=\,1/T{\left({\rm{j}}\omega \right)}^{\phi },$ with $T$ being a constant in ${\rm{F}}\,{{\rm{cm}}}^{-2}{{\rm{s}}}^{\phi -1}$ and $\phi$ being the exponent of a constant-phase element (CPE). ⁹⁰ CPE was firstly proposed by Fricke in 1932 to describe the phenomenon of capacitance dispersion, ⁹¹ which may have many possible causes including surface roughness and specific adsorption, see a critical discussion by Pajkossy. ^92,93 Brug et al. ⁹⁴ proposed another CPE expression, ${Z}_{loc}\,=\,Q/{\left({\rm{j}}\omega \right)}^{1-\alpha },$ where $Q$ is a constant with the dimension of ${\rm{\Omega }}\,{{\rm{cm}}}^{-2}{{\rm{s}}}^{-\left(1-\alpha \right)}$ and $\left(1-\alpha \right)$ has the same meaning as $\phi$ in Lasia's expression. Zoltowski proposed two definitions of the CPE impedance, ${Z}_{loc}\,=\,1/{Q}_{a}{\left({\rm{j}}\omega \right)}^{\alpha }$ and ${Z}_{loc}\,=\,1/{\left({Q}_{b}{\rm{j}}\omega \right)}^{\alpha }.$ ⁹⁵ The dimensions of ${Q}_{a}$ and ${Q}_{b}$ are ${{\rm{\Omega }}}^{-1}{{\rm{cm}}}^{-2}{{\rm{s}}}^{\alpha }$ and ${\left({\rm{\Omega }}{{\rm{cm}}}^{2}\right)}^{-1/\alpha }{{\rm{s}}}^{\alpha },$ respectively. The CPE represents a capacitor for $\alpha \,=\,1,$ a resistor for $\alpha \,=\,0,$ and an inductor for $\alpha \,=\,-1.$ Although the CPE expression is different in these models, the transport process is the same as in the de Levie model, namely, ion transport in the electrolyte solution described by the Ohmic law. Replacing the pure capacitance with the CPE gives rise to an inclined tail in low frequency range.

In 1993, Paasch et al. ⁴⁷ derived the electrochemical impedance based on a macroscopically homogeneous porous electrodes theory. Their model describes ionic conduction in the electrolyte phase and electronic conduction in the solid phase, double layer charging and charge transfer reaction at the solid/liquid interface, and diffusion in the electrolyte-filled pores. They assumed that the electronic conductivity in the solid phase is much larger than the ionic conductivity in the electrolyte phase, so that the solid phase has a uniform electrical potential. The model of Paasch et al. ⁴⁷ corresponds to the case 1 in the above theoretical framework.

In 1999, Bisquert et al. ⁹⁶ studied impedance response of a disordered medium with anomalous transport effects that frequently occur in disordered materials where the charge transfer proceeds via discrete transitions between localized states. And they extended the model in the presence of redox reactions in 2000. ^97,98 Research in electrical properties of disordered materials has revealed a frequency-dependent ac conductivity $\kappa ^{\prime} \left(\omega \right),$ described as

$$\begin{eqnarray}&&\begin{array}{l}\kappa ^{\prime} \left(\omega \right)\,=\,{\kappa }_{0}\,\left(\omega \ll {\omega }_{m}\right),\\ \kappa ^{\prime} \left(\omega \right)\,\propto \,{\omega }^{p}\,\left(\omega \gg {\omega }_{m}\right),\end{array}\end{eqnarray} \tag{ 76 }$$

where ${\omega }_{m}$ is the crossover frequency, and $0\lt p\lt 1.$ For amorphous semiconductors, the values of $p$ is close to 0.8. ⁹⁶ The values of ${\omega }_{m}$ vary widely among different systems and depend on the temperature. ⁹⁶ Considering this anomalous transport effect, ${Z}_{loc}$ is now given by

$$\begin{eqnarray}&&{Z}_{loc}\,=\,\displaystyle \frac{{R}_{ct}}{1+{\left({\rm{j}}\omega /{\omega }_{m}\right)}^{p}},\end{eqnarray} \tag{ 77 }$$

which shares the same form with ${Z}_{loc}$ considering the CPE.

In 1999, Eikerling and Kornyshev developed the impedance theory of the catalyst layer of PEFCs based on the macrohomogeneous porous electrode theory. ⁴⁸ Following mass transport processes are included: (i) proton transport through the polymer-electrolyte porous network, (ii) oxygen supply through hydrophobized gas pores, and (iii) oxygen reduction reaction and double-layer charging at the electrochemical interface. In the case where oxygen transport is neglected, the local impedance is given by, ${Z}_{loc}\,=\,{R}_{ct}/\left({\rm{j}}\omega {C}_{dl}{R}_{ct}+1\right),$ where the chare transfer resistance is ${R}_{ct}\,=\,RT/nF{i}_{0},$ with an apparent Tafel slope $R{\rm{T}}/nF$ ($n$ is the number of electrons transferred) and the exchange current density ${i}_{0}.$ The electrode impedance is obtained by substituting ${Z}_{loc}\,=\,{R}_{ct}/\left({\rm{j}}\omega {C}_{dl}{R}_{ct}+1\right)$ into Eq. 68 corresponding to the case 2, with the following asymptotic behaviors,

$$\begin{eqnarray}&&{Z}_{p}\left(\omega \to 0\right)\,=\,\displaystyle \frac{{l}_{p}}{{\kappa }_{s,eff}+{\kappa }_{e,eff}}\left(1+\displaystyle \frac{{{\kappa }_{s,eff}}^{2}+{{\kappa }_{e,eff}}^{2}}{3{\kappa }_{s,eff}{\kappa }_{e,eff}}\right)+\displaystyle \frac{{R}_{ct}}{{a}_{v}{l}_{p}},\end{eqnarray} \tag{ 78 }$$

and

$$\begin{eqnarray}&&{Z}_{p}\left(\omega \to \infty \right)\,=\,\displaystyle \frac{{l}_{p}}{{\kappa }_{s,eff}+{\kappa }_{e,eff}}+\displaystyle \frac{{{\kappa }_{s,eff}}^{2}+{{\kappa }_{e,eff}}^{2}}{{\left({\kappa }_{s,eff}{\kappa }_{e,eff}\right)}^{0.5}{\left({\kappa }_{s,eff}+{\kappa }_{e,eff}\right)}^{1.5}}\sqrt{\displaystyle \frac{1}{{\rm{j}}\omega {C}_{dl}{a}_{v}}}.\end{eqnarray} \tag{ 79 }$$

When ${\kappa }_{s,eff}\gg {\kappa }_{e,eff},$ corresponding to the case 3, above equations are simplified further to

$$\begin{eqnarray}&&{Z}_{p}\left(\omega \to 0\right)\,=\,\displaystyle \frac{{l}_{p}}{3{\kappa }_{e,eff}}+\displaystyle \frac{{R}_{ct}}{{a}_{v}{l}_{p}},\end{eqnarray} \tag{ 80 }$$

and

$$\begin{eqnarray}&&{Z}_{p}\left(\omega \to \infty \right)\,=\,\sqrt{\displaystyle \frac{1}{{\rm{j}}\omega {C}_{dl}{a}_{v}{\kappa }_{e,eff}}}.\end{eqnarray} \tag{ 81 }$$

Extensions to the Eikerling-Kornyshev impedance model will be reviewed in the section 5.2.

In 2000, Meyers et al. modelled the impedance response for porous electrodes in lithium-ion batteries. ⁴⁹ As in the de Levie model, the Meyers model describes ion transport in the electrolyte phase and electron transport in the solid phase using the Ohmic law. The model features the expression of ${Z}_{loc},$ which describes the impedance response of a single intercalation particle, given by

$$\begin{eqnarray}&&{Z}_{loc}=\displaystyle \frac{1}{\left(\tfrac{1}{{R}_{ct}+\tfrac{\partial U}{\partial {c}_{s}}{Y}_{s}}\right)+{\rm{j}}\omega {C}_{dl}},\end{eqnarray} \tag{ 82 }$$

where ${R}_{ct}$ is the charge transfer resistance, $U$ is the equilibrium potential which depends on ${c}_{s},$ the Li⁺ concentration in the solid phase. ${Y}_{s}\,=\,\tfrac{{R}_{ap}}{F{D}_{s}}\tfrac{\tanh \left({{\rm{\Omega }}}_{s}\right)}{\tanh \left({{\rm{\Omega }}}_{s}\right)-{{\rm{\Omega }}}_{s}}$ is related to spherical diffusion, with ${{\rm{\Omega }}}_{s}\,=\,\sqrt{\tfrac{{\rm{j}}\omega {R}_{ap}^{2}}{{D}_{s}}}$ being the dimensionless frequency-dependent constant, ${R}_{ap}$ the radius of the active particles, and ${D}_{s}$ the solid-phase diffusion coefficient. ^65,99 When $\omega \to 0,$ Eq. 82 is asymptotic to

$$\begin{eqnarray}&&{Z}_{loc}\,=\,{R}_{ct}-\displaystyle \frac{3}{{\rm{j}}\omega F{R}_{ap}}\displaystyle \frac{\partial U}{\partial {c}_{s}}.\end{eqnarray} \tag{ 83 }$$

When $\omega \,\to \,\infty ,$ Eq. 82 is asymptotic to

$$\begin{eqnarray}&&{Z}_{loc}\,=\,\displaystyle \frac{1}{{\rm{j}}\omega {C}_{dl}}.\end{eqnarray} \tag{ 84 }$$

The impedance response of the porous electrode is then obtained by substituting Eqs. 83 and 84 into Eq. 68, corresponding to the case 2. When $\omega \,\to \,0,$ the electrode impedance is asymptotic to

$$\begin{eqnarray}&&\begin{array}{l}{Z}_{p}\,=\,\displaystyle \frac{{l}_{p}}{{\kappa }_{s,eff}+{\kappa }_{e,eff}}\left(1+\displaystyle \frac{{{\kappa }_{s,eff}}^{2}+{{\kappa }_{e,eff}}^{2}}{3{\kappa }_{s,eff}{\kappa }_{e,eff}}\right)+\displaystyle \frac{{R}_{ct}}{{a}_{v}{l}_{p}}\\ \,\,\,-\,\displaystyle \frac{3}{{\rm{j}}\omega F{a}_{v}{l}_{p}{R}_{ap}}\displaystyle \frac{\partial U}{\partial {c}_{s}}.\end{array}\end{eqnarray} \tag{ 85 }$$

And when $\omega \to \infty ,$ the electrode impedance is asymptotic to

$$\begin{eqnarray}&&\begin{array}{l}{Z}_{p}\,=\,\displaystyle \frac{{l}_{p}}{{\kappa }_{s,eff}+{\kappa }_{e,eff}}+\displaystyle \frac{{{\kappa }_{s,eff}}^{2}+{{\kappa }_{e,eff}}^{2}}{{\left({\kappa }_{s,eff}{\kappa }_{e,eff}\right)}^{0.5}{\left({\kappa }_{s,eff}+{\kappa }_{e,eff}\right)}^{1.5}}\\ \,\,\times \,\sqrt{\displaystyle \frac{1}{{\rm{j}}\omega {C}_{dl}{a}_{v}}}.\end{array}\end{eqnarray} \tag{ 86 }$$

Assuming ${\kappa }_{s,eff}\gg {\kappa }_{e,eff},$ as in the case 3, Eqs. 85 and 86 can be simplified further as follows,

$$\begin{eqnarray}&&{Z}_{p}\left(\omega \to 0\right)\,=\,\displaystyle \frac{{l}_{p}}{3{\kappa }_{e,eff}}+\displaystyle \frac{{R}_{ct}}{{a}_{v}{l}_{p}}-\displaystyle \frac{3}{j\omega F{a}_{v}{l}_{p}{R}_{ap}}\displaystyle \frac{\partial U}{\partial {c}_{s}},\end{eqnarray} \tag{ 87 }$$

$$\begin{eqnarray}&&{Z}_{p}\left(\omega \to \infty \right)\,=\,\sqrt{\displaystyle \frac{1}{{\rm{j}}\omega {C}_{dl}{a}_{v}{\kappa }_{e,eff}}},\end{eqnarray} \tag{ 88 }$$

Impedance models developed for lithium-ion batteries will be reviewed in greater detail in the section 5.1.

Explicit pores have a definite structure, viz., in a specific shape with a definite size, where the solid and electrolyte phases are explicitly separated. In the following, we start with uniform pores, then expand the treatment by introducing through-plane and in-plane distributions, successively, and touch upon fractal pores in the end.

Uniform pore structure

The standard de Levie model expresses the impedance response of a uniform cylinder pore as Eq. 9. Avoiding repeating the standard de Levie model that has been expounded previously, we herein focus on an additional structure effect, firstly pointed out by Lasia et al., concerning the contribution of the outer flat layer, for example, the pore end interfacing the electrolyte, to the total impedance of the porous electrode. ¹⁰³ This contribution becomes nontrivial in high frequency range where the penetration length is much shorter than the pore length. Under such scenarios, Lasia et al. took into account the capacitance of the top flat layer and modified the impedance response as, ¹⁰³

$$\begin{eqnarray}&&Z\,=\,{R}_{s}+\displaystyle \frac{1}{{\rm{j}}\omega {C}_{flat}+1/{Z}_{p}},\end{eqnarray} \tag{ 89 }$$

where ${R}_{s}$ is the solution resistance out of the pore, ${C}_{flat}$ is the capacitance of outer flat layer and ${Z}_{p}$ is impedance of the pore.

The simulation results (Fig. 5) show an inclined line with a phase angle larger than 45° in high frequency range, in agreement with experiment result (dots in Fig. 5c), and a vertical line in low frequency range. Besides, the phase angle of the inclined line in high frequency range grows as the ratio of the flat capacitance ${C}_{flat}$ to the pore capacitance ${C}_{pore}$ increases.

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Through-plane distribution of pore structure

By through-plane distribution we mean that properties, most often, the pore radius, change along the direction perpendicular to electrode surface. In a pioneering work, Keiser et al. ⁵¹ uniformly split the pore into $N$ discs with equivalent height $l/N$ but different radius ${r}_{i}$ $(1\leqslant i\leqslant N);$ each disc is then treated as a uniform cylinder pore with the solution resistance ${R}_{i}$ and capacitance ${C}_{i},$ as shown in Fig. 6a.

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Afterwards, considering the solution resistance out of the pore, denoted ${R}_{0},$ the total impedance as illustrated in Fig. 6b is given by

$$\begin{eqnarray}&&{Z}_{0}\,=\,{R}_{0}+\displaystyle \frac{1}{{\rm{j}}\omega {C}_{1}+\tfrac{1}{{Z}_{1}}},\end{eqnarray} \tag{ 90 }$$

with ${Z}_{1}$ determined by the recursion formula,

$$\begin{eqnarray}&&{Z}_{i}\,=\,{R}_{i}+\displaystyle \frac{1}{{\rm{j}}\omega {C}_{i+1}+\tfrac{1}{{Z}_{i+1}}}.\end{eqnarray} \tag{ 91 }$$

Dividing both sides of Eq. 91 by $R,$ one obtains

$$\begin{eqnarray}&&\displaystyle \frac{{Z}_{i}}{R}\,=\,{Z}_{i}^{* }\,=\,\displaystyle \frac{1}{N{g}_{i}^{2}}+\displaystyle \frac{1}{{\rm{j}}\tfrac{1}{2N}{\left(\tfrac{1}{\lambda }\right)}^{2}{g}_{i+1}+\tfrac{1}{{Z}_{i+1}^{* }}}\left(1\leqslant i\leqslant N\right),\end{eqnarray} \tag{ 92 }$$

where $R\,=\,l/\kappa \pi {r}^{2}$ with $r$ (m) being the mean pore radius and $l$ (m) the total length of the noncylindrical pore, ${g}_{i}={r}_{i}/r$ is the dimensionless shape factor with ${r}_{i}$ (${\rm{m}}$) being the radius of the ith disc, $\lambda \,=\,\tfrac{1}{2}\sqrt{\kappa r/\omega {C}_{dl}}$ (${\rm{m}}$) is the penetration depth of the a.c. signal with $\kappa$ $({\rm{S}}\,{\rm{m}})$being the ionic conductivity.

Since the pore end is assumed to be insulated $({Z}_{N+1}\to \infty ,\,{C}_{N+1}\,=\,0),$ the starting value of the recursion formula reads

$$\begin{eqnarray}&&\displaystyle \frac{1}{{Z}_{N+1}^{* }}\,=\,0,\,{g}_{N+1}\,=\,0,\end{eqnarray} \tag{ 93 }$$

The total impedance of the pore can be obtained via iterating Eq. 92 with the initial condition of Eq. 93. With the solution resistance out of the pore being zero (${R}_{0}\,=\,0$), Fig. 6c shows that the pore impedance significantly varies with the pore shape. Even for the case without any Faradaic reaction, a semicircle (usually interpreted as the impedance response of a Faradaic reaction in parallel with double layer charging) is manifested in high frequency range for the pear-shaped pore. Therefore, special care should be added in interpreting a semicircle of impedance curves. Lasia et al. extended the work of Keiser et al. to spherical and bi-spherical pores, and showed that the impedance curve depends on the pore opening size of the spherical pore and two overlapped semicircles exist in the impedance curve of the bi-spherical pore. ¹⁰⁴

Eloot et al. developed the matrix method to treat through-plane distributions, as shown in Fig. 7. ¹⁰² In the matrix method, the pore and the wall material are split up into $N$ discs with random height ${l}_{i}$ and radius ${r}_{i}$ ($1\leqslant i\leqslant N$), and each disc is described using the standard transmission line model with constant properties. The current through the electrolyte phase (${I}_{e}^{i}$) and solid phase (${I}_{s}^{i}$) as well as the interface potential (${\varphi }^{(i)}$) of the ith disc could be obtained as detailed in SI (S1 ∼ S21). The adjacent discs are correlated by a transfer matrix ${M}^{i},$

$$\begin{eqnarray}&&\left[\begin{array}{c}\begin{array}{c}{I}_{e}^{i}({a}_{i})\\ {\varphi }^{i}({a}_{i})\end{array}\\ I\end{array}\right]\,=\,{M}^{i}\left[\begin{array}{c}\begin{array}{c}{I}_{e}^{i-1}({a}_{i-1})\\ {\varphi }^{i-1}({a}_{i-1})\end{array}\\ I\end{array}\right],\end{eqnarray} \tag{ 94 }$$

where $I$ ($A$) is the total current.

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The matrix ${M}^{i}$ could be determined by invoking the constraint that the current and potential in both electrolyte and solid phases must be continuous at the interface between two adjacent discs, resulting in

$$\begin{eqnarray}{M}^{i}\,=\,\left[\begin{array}{ccc}{C}^{i} & -{S}^{i}\displaystyle \frac{{\chi }^{i}}{{Z}^{i}} & {D}^{i}(1-{C}^{i})\\ -{S}^{i}\displaystyle \frac{{Z}^{i}}{{\chi }^{i}} & {C}^{i} & {S}^{i}{\chi }^{i}{R}_{s}^{i}\\ 0 & 0 & 1\end{array}\right],\end{eqnarray} \tag{ 95 }$$

with ${\chi }^{i}\,=\,\sqrt{\left(\tfrac{{Z}^{i}}{{R}_{s}^{i}+{R}_{e}^{i}}\right)},$ ${D}^{i}\,=\,\tfrac{{R}_{s}^{i}}{{R}_{s}^{i}+{R}_{e}^{i}},$ ${S}^{i}\,=\,\sin \,{\rm{h}}\left(\tfrac{{l}_{i}}{{\chi }^{i}}\right),$ ${C}^{i}\,=\cos \,{\rm{h}}\left(\tfrac{{l}_{i}}{{\chi }^{i}}\right).$ Here, ${\varphi }_{e}^{i}$ (V) and ${\varphi }_{s}^{i}$ (V) are the potential in electrolyte and solid phases, respectively; ${R}_{s}^{i}$ (${\rm{\Omega }}\,{{\rm{m}}}^{-1}$), ${R}_{e}^{i}$ (${\rm{\Omega }}\,{{\rm{m}}}^{-1}$), and ${Z}^{i}$ (${\rm{\Omega }}\,{\rm{m}}$) are the specific resistances of the solid phase, the electrolyte phase, and the interface impedance per unit length, respectively. Provided the transfer matrix, we can determine the total impedance of the porous electrode, defined as, ${Z}_{p}\,=\,[{\varphi }_{s}^{N}-{\varphi }_{e}^{0}]/I,$ as (the corresponding solving process details are presented Eqs. (S45) to (S67) in the SI),

$$\begin{eqnarray}&&\begin{array}{l}{Z}_{p}=\displaystyle \frac{{\varphi }^{N}}{I}\\ \,+\,\displaystyle \sum _{i=1}^{N}\left(\displaystyle \frac{{R}_{e}^{i}}{{\gamma }_{i}^{\displaystyle \frac{1}{2}}{S}^{i}}\left({C}^{i}-1\right)\cdot \displaystyle \frac{{I}_{e}^{i}\left({a}_{i}\right)+{I}_{e}^{i-1}\left({a}_{i-1}\right)-2{D}^{i}I}{I}+{R}_{e}^{i}{D}^{i}{l}_{i}\right).\end{array}\end{eqnarray} \tag{ 96 }$$

Simulated EIS data are displayed in Nyquist plot in Fig. 7c, indicating that the impedance curves depend on the pore shape in high frequency range and converge to vertical lines in low frequency range. In high frequency range, the a.c. signal could only penetrate a part of the pore. Consequently, the total active surface area changes with the frequency and also with the pore shape. This explains why the EIS depends on the pore shape. In low frequency range, the a.c. signal penetrates the whole pore and the total active surface area does not change with the frequency any longer, resulting in an invariably vertical line.

As regards a porous electrode with a capacitive interface and an insulated pore base, Eloot et al. compared the recursion method and the matrix method. ¹⁰² The results show that the impedance obtained from the recursion method deviates from a 45° line and does not converge to the origin point of the coordinate system in high frequency range. This is because the recursion method involves serial resistances $1/N{g}_{i}^{2},$ which are zero only if $N$ is infinite. Eloot et al. concluded that the matrix method is more accurate than the recursion method in high frequency range with the same number of discs. In addition, Eloot et al. found that the real part of the impedance obtained from the matrix method depends on the pore geometry, which is in contrast with the results of Keiser et al. ¹⁰² Besides the pore shapes mentioned above, de Levie developed V-grooved pores to describe the influence of surface roughness on the electrode impedance. ⁴³ Later on, Gunning obtained an exact solution for the impedance response of the V-grooved pore. ¹⁰⁵

In-plane distribution of pore structure

A porous electrode can be considered as a parallel collection of pores with in-plane distributions of various properties, such as radius, length and so on. Accounting for these in-plane distributions represents a step closer to the reality. Song et al. have pioneered the introduction of pore size distribution into the impedance response of porous electrodes composed of cylinder pores of an identical pore length (Fig. 8). ^53,106

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The impedance of a single pore given by the de Levie model (Eq. 9) is rearranged in the following form,

$$\begin{eqnarray}&&\begin{array}{l}{Z}_{p}^{* }(\alpha )\\ =\,\alpha \left[\displaystyle \frac{\sinh \left(1/\alpha \right)-\,\sin (1/\alpha )}{\cosh \left(1/\alpha \right)-\,\cos (1/\alpha )}-{\rm{j}}\displaystyle \frac{\sinh \left(1/\alpha \right)+\,\sin (1/\alpha )}{\cosh \left(1/\alpha \right)-\,\cos (1/\alpha )}\right],\end{array}\end{eqnarray} \tag{ 97 }$$

where ${Z}_{p}^{* }(\alpha )\,=\,{Z}_{p}(\alpha )/{R}_{0}$ is a dimensionless impedance, with ${Z}_{p}(\alpha )$ (${\rm{\Omega }}$) being the impedance of a single pore, ${R}_{0}={l}_{p}/\kappa \pi {r}^{2}$ (${\rm{\Omega }}$) the total electrolyte resistance, ${l}_{p}$ (${\rm{m}}$) the pore length, $r$ (${\rm{m}}$) the pore radius and $\kappa$ (${\rm{S}}\,{{\rm{m}}}^{-1}$) the specific conductance. $\alpha \,=\,\lambda /{l}_{p}={\alpha }_{0}{\omega }^{-0.5}$ is the penetrability, with $\lambda$ (${\rm{m}}$) being the penetration depth. ${\alpha }_{0}=\tfrac{1}{2{l}_{p}}\tfrac{\kappa r}{{C}_{dl}}$ (${{\rm{s}}}^{-0.5}$) is the penetrability coefficient.

The total impedance of a porous electrode composed of an array of pores in parallel connection is expressed as

$$\begin{eqnarray}&&\displaystyle \frac{1}{{Z}_{tot}(\alpha :{a}_{1})}\,=\,\displaystyle {\int }_{0}^{+\infty }\displaystyle \frac{1}{{Z}_{p}\left(\alpha \right)}g\left({\alpha }_{0}:{a}_{1}\right)d{\alpha }_{0},\end{eqnarray} \tag{ 98 }$$

where $g\left({\alpha }_{0}:{a}_{1}\right)$ is a distribution density function of the penetrability coefficient (${\alpha }_{0}$), Hence, $g\left({\alpha }_{0}:{a}_{1}\right)d{\alpha }_{0}$ means the number of pores between ${\alpha }_{0}$ and ${\alpha }_{0}+d{\alpha }_{0}.$ ${a}_{1}$ is an array of parameters such as, pore size, pore length and so on.

If the structural parameter possessing a distribution is denoted as $x,$ one can conduct the transform, $g\left({\alpha }_{0}:{a}_{1}\right)d{\alpha }_{0}\,=\,f\left(x:{a}_{2}\right)dx,$ with $f\left(x:{a}_{2}\right)$ given by

$$\begin{eqnarray}&&f\left(x:{a}_{2}\right)\,=\,\displaystyle \frac{dn}{dx}\,=\,{\left(\pi {r}^{2}{l}_{p}\right)}^{-1}\left(\displaystyle \frac{dV}{dx}\right).\end{eqnarray} \tag{ 99 }$$

Here, ${\rm{d}}n$ is the pore number between $x$ and $x+dx,$ and $V$ is the pore volume. Through normalization, Eq. 99 is rewritten as, $f\left(x:{a}_{2}\right)\,=\,\left[{V}_{tot}/\pi {r}^{2}{l}_{p}\right]k(x:{a}_{2}),$ with $\displaystyle \int k\left(x:{a}_{2}\right)dx\,=\,1,$ and ${V}_{tot}$ being the total pore volume of the electrode.

Therefore, the total impedance is given by

$$\begin{eqnarray}&&{Z}_{tot}{\left(\omega :{a}_{2}\right)}^{-1}\,=\,\displaystyle \frac{\kappa {V}_{tot}}{{l}_{p}^{2}}\displaystyle \int {Z}_{p}^{* }{(\alpha )}^{-1}k\left(x:{a}_{2}\right)dx.\end{eqnarray} \tag{ 100 }$$

Parameterized with a lognormal distribution, $k\left(x:{a}_{2}\right)\,=\,\tfrac{1}{x\sqrt{2\pi }\sigma }\exp \left(-\tfrac{{\left(\mathrm{ln}(x)-\mu \right)}^{2}}{2{\sigma }^{2}}\right),$ with $\mu$ and $\sigma$ being the average and standard deviation of the structural parameter $x$ which is the pore radius $r$ in Ref. 53, respectively, Eq. 100 generates synthetic EIS data shown in Fig. 8b. Different from the usual case in Fig. 2, a much more inclined line appears in low frequency when $\sigma$ is larger, namely, when the pore size distribution is wider. This phenomenon is interpreted as follows. From the expression of the penetration length given by $\lambda \,=\,\tfrac{1}{2}\sqrt{\tfrac{\kappa r}{{C}_{dl}}}{\omega }^{-0.5},$ it is known that the penetration length becomes longer when the perturbation frequency is lower and when the pore radius is larger. Therefore, when the frequency is high enough that all pores act like a semi-infinite electrode, the overall impedance shows a 45° line. As the frequency decreases, the perturbation length of large pores exceeds the pore length, and the impedance response of these pores transitions to a 90° line, while the overall impedance response of the whole porous electrode shows an inclined line. If the pore distribution is wider, the transition from 45° to 90° lines will extend over a wider frequency range, as shown in Fig. 8b.

Recently, Musiani et al. ¹⁰⁷ reconsidered the influence of pore size distribution on the impedance response of a porous electrode. The pore radius r follows a lognormal distribution,

$$\begin{eqnarray}&&f\left(r:{a}_{2}\right)\,=\,\displaystyle \frac{dn}{dr}\,=\,\displaystyle \frac{1}{r\sqrt{2\pi }\sigma }\exp \left(-\displaystyle \frac{{\left(\mathrm{ln}(r)-\mu \right)}^{2}}{2{\sigma }^{2}}\right)\end{eqnarray} \tag{ 101 }$$

where $\mu \,=\,\mathrm{ln}({r}_{\mu })$ with ${r}_{\mu }$ being the median of the pore radius, at which the cumulative probability is 50%, and $\sigma$ is its standard deviation. And assuming the total pore volume to be half of the electrode (the pore volume fraction was arbitrarily chosen), the number of pores per unit surface is expressed as,

$$\begin{eqnarray}&&n\,=\,\displaystyle \frac{1}{2\pi \displaystyle {\int }_{0}^{\infty }{r}^{2}f(r:{a}_{2})dr}\end{eqnarray} \tag{ 102 }$$

Musiani et al. have calculated the impedance response of the porous electrode with a constant pore volume for two cases: (i) constant median of the pore radius (${r}_{\mu }$) and variable pore number ($n$); (ii) constant pore number ($n$) and variable median of the pore radius (${r}_{\mu }$). In contrast with the work of Song et al., Musiani et al. found the typical 45° line for porous electrode disappeared in high frequency range for the both case.

The main difference between the work of Song et al. and Musiani et al. is that Song et al. adopted pore volume density ($dV/dr$) while Musiani et al. used pore number density ($dn/dr$), to describe the pore distribution. Once a larger standard deviation ($\sigma$) is implanted into the distribution while the total pore volume is constrained, these two different distributions result in disparity. The volume distribution leads to an enormous increase of smaller pore number, thus a decrease of the median of the pore radius, in order to maintain the same total pore volume of pores on each side of the median pore. These smaller pores are accessible at very low frequency and therefore the typical 45° line extends to lower frequencies. In contrast, the pore number distribution retains the same number of pores on each side of the median pore size, which results in a large surface-area ratio of the electrode geometric surface to the pore inner surface when the standard deviation ($\sigma$) becomes larger. As a results, the porous electrode behaves more like a flat electrode and the frequency corresponding to the porous-to-capacitive transition shifts to a higher frequency or even disappears.

Fractal pore structure

A fractal pore possesses similar geometrical patterns at progressively smaller scales. Le Méhauté et al. ^108,109 made the first attempt to correlate the fractal dimension ${D}_{f}$ with the CPE exponent, ${\eta }_{CPE},$ as

$$\begin{eqnarray}&&{\eta }_{CPE}\,=\,1/{D}_{f}.\end{eqnarray} \tag{ 103 }$$

Following the work of Le Méhauté et al., a surge of research activities has sprung up to describe the effect of fractal geometry on the impedance response of interfaces and porous electrodes, among which three representative groups of models are discussed as below.

Liu ¹⁰⁰ and Kaplan et al. ¹⁰¹ have independently derived the same formula to describe the relation between the fractal dimension and the CPE exponent in low frequency range. The fractal interface is pictured as a hierarchical network, as depicted in Fig. 9b, in which a portion of the pore at the upper scale is split into two branches at the lower scale with a scaling length ratio of a. The impedance of this hierarchical network is mapped into an equivalent electrical circuit depicted in Fig. 9a, of which the impedance is expressed as,

$$\begin{eqnarray}&&Z\left(\omega \right)\,=\,R+\displaystyle \frac{1}{{\rm{j}}\omega C+\tfrac{2}{aR+\tfrac{1}{{\rm{j}}\omega C+\tfrac{2}{{a}^{2}R+\tfrac{1}{{\rm{j}}\omega C+\cdots }}}}},\end{eqnarray} \tag{ 104 }$$

where $a$ is the length ratio between two adjacent scales; $R$ and $C$ are the resistance and capacitance of primary pores, respectively. Substituting $\omega$ with $\omega /a$ in Eq. 104 and multiplying the fraction by $a$ yield

$$\begin{eqnarray}&&Z\left(\displaystyle \frac{\omega }{a}\right)\,=\,R+\displaystyle \frac{aZ(\omega )}{{\rm{j}}\omega CZ(\omega )+2}.\end{eqnarray} \tag{ 105 }$$

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Assuming that $Z(\omega )$ diverges much slower than $1/\omega ,$ namely, $\omega Z(\omega )$ goes to 0, as $\omega \to 0,$ the low-frequency limit of Eq. 105 becomes $Z(\omega /a)=\tfrac{a}{2}Z(\omega )$ (the ohmic resistance R has been safely neglected). Then, the recursion expression of the impedance response is given by

$$\begin{eqnarray}&&Z\left(\omega \right)\,=\,kR{\left({\rm{j}}RC\omega \right)}^{-{\eta }_{CPE}},\end{eqnarray} \tag{ 106 }$$

with,

$$\begin{eqnarray}&&{\eta }_{CPE}\,=\,1-\displaystyle \frac{\mathrm{ln}\,2}{\mathrm{ln}\,a}\,=\,1-{D}_{f},\end{eqnarray} \tag{ 107 }$$

where $k,$ ${\eta }_{CPE}$ and ${D}_{f}$ are the scaling factor, the CPE exponent and the fractal dimension of the interface, respectively. When $a\to \infty ,$ the fractal dimension of the interface converges to zero. In other words, when the interface is an ordinary smooth interface, we retrieve the classical result of ${\eta }_{CPE}=1.$

As for a finite number of generation of new branches, numerical calculation in Fig. 9c shows a frequency-independent regime in low frequency range and a frequency-dispersion regime in intermediate frequency range. Specifically, the frequency-dispersion regime is extended to lower frequencies for the interface with a larger number of generation of branches. This is because a larger generation number stands for a rougher interface with more small and deep pores, which are penetrated only by lower frequency signals.

Pajkossy et al. ^110,111 recognized that a practical electrode may contain many kinds of self-similar irregularities, such as grooves, cracks, pores and so on, and can be described, approximately and without going into the detailed irregularities, as self-similar fractals with an effective dimension (${D}_{f}$). By dividing the electrode into elementary pieces in parallel connection (Fig. 9c), Pajkossy et al. express the admittance of the electrode as

$$\begin{eqnarray}&&Y\left(1,\,\omega \right)\,=\,\displaystyle \displaystyle \sum _{j}\displaystyle \frac{{\rm{j}}\omega {C}_{j}}{1+{\rm{j}}\omega {R}_{j}{C}_{j}},\end{eqnarray} \tag{ 108 }$$

where ${C}_{j}$ and ${R}_{j}$ are the capacitance and resistance of elementary species, respectively.

The admittance $Y$ is a two-dimensional macroscopic quantity, which should be inversely correlated with the macroscopic surface area. The capacitance ${C}_{j},$ a microscopic quantity, should be proportional to the microscopic surface area. The resistance ${R}_{j}$ is of bulk nature which is inversely proportional to the scale of the object. Consequently, the following scaling laws are obtained when scaling up the macroscopically two-dimensional electrode by a factor of $r,$

$$\begin{eqnarray}&&\begin{array}{l}Y\left(r\cdot 1,\omega \right)\,=\,{r}^{2}Y\left(1,\omega \right),\,{C}_{j}\left(r\cdot 1\right)={r}^{{D}_{f}}{C}_{j}\left(1\right),\\ \,{R}_{j}\left(r\cdot 1\right)={r}^{-1}{R}_{j}\left(1\right),\end{array}\end{eqnarray} \tag{ 109 }$$

where 1 stands for the original system and $r\cdot 1$ the newly enlarged system.

Hence, the admittance of the new system is given by

$$\begin{eqnarray}&&Y\left(r\cdot 1,\omega \right)\,=\,\displaystyle \displaystyle \sum _{j}\displaystyle \frac{{\rm{j}}\omega {C}_{j}\left(r\cdot 1\right)}{1+{\rm{j}}\omega {R}_{j}\left(r\cdot 1\right){C}_{j}\left(r\cdot 1\right)}\,=\,rY\left(1,\omega {r}^{{D}_{f}-1}\right),\end{eqnarray} \tag{ 110 }$$

which is transformed to

$$\begin{eqnarray}&&\displaystyle \frac{Y(1,{r}^{{D}_{f}-1}\omega )}{Y(1,\omega )}\,=\,r.\end{eqnarray} \tag{ 111 }$$

We have for the CPE, $Y(\omega )\,=\,\sigma {(j\omega )}^{{\eta }_{CPE}},$ which gives the relationship $\tfrac{Y(k\omega )}{Y(\omega )}={k}^{{\eta }_{CPE}}.$ Equation 111 describes a CPE with an exponent,

$$\begin{eqnarray}&&{\eta }_{CPE}\,=\,{({D}_{f}-1)}^{-1}.\end{eqnarray} \tag{ 112 }$$

For a smooth surface $({D}_{f}=2),$ this correlation gives ${\eta }_{CPE}=1,$ as expected.

Fractal models introduced above treat a rough interface, rather than a porous electrode (a porous electrode has deep pores while a rough interface has shallow pores). Sapoval constructed a porous electrode model by combining the transmission line model and the fractal model as shown in Fig. 10b. ^112,113 The porous electrode is pictured as N square holes with the side length reduced by a factor $\alpha$ at the surroundings of the primary pore and the corresponding fractal dimension is then calculated as

$$\begin{eqnarray}&&{D}_{f}\,=\,\displaystyle \frac{\mathrm{log}\,N}{\mathrm{log}\,\alpha },\end{eqnarray} \tag{ 113 }$$

with $N\lt {\alpha }^{2}-1.$

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Each pore is described using the de Levie method. The admittance of a pore obtained at nth stage of the decimation is given by

$$\begin{eqnarray}&&{Y}_{n}\,=\,{\left(\displaystyle \frac{2a}{{\alpha }^{n}}\right)}^{\displaystyle \frac{3}{2}}{\left(C\omega \kappa \right)}^{\displaystyle \frac{1}{2}}(1+j)\tan \left[{\left(\displaystyle \frac{C\omega {l}^{2}{\alpha }^{n}}{2a\kappa }\right)}^{\displaystyle \frac{1}{2}}(1+j)\right],\end{eqnarray} \tag{ 114 }$$

where $a$ is the side length of the primary pore, $C$ is the capacitance per unit area, l is the pore length, $\kappa$ is the electrolyte conductance. Accordingly, the total impedance of the electrode is

$$\begin{eqnarray}&&{Z}^{-1}\,=\,\displaystyle \displaystyle \sum _{n}{N}^{(n-1)}{Y}_{n},\end{eqnarray} \tag{ 115 }$$

which converges when N $\lt \,{\alpha }^{3/2},$ meaning that the highest fractal dimension that can be described by this model is

$$\begin{eqnarray}&&{D}_{f}\,=\,1+\left(\displaystyle \frac{\mathrm{log}\,{\alpha }^{\displaystyle \frac{3}{2}}}{\mathrm{log}\,\alpha }\right)\,=\,5/2.\end{eqnarray} \tag{ 116 }$$

In case of ${D}_{f}\gt 5/2,$ the total impedance is dominated by small pores, the impedance thus diverges. In case of ${D}_{f}\lt 5/2,$ there is no explicit equation to correlate the fractal dimension and the exponent of CPE, but numerical results of Eq. 115 exhibits frequency dispersion in low frequency range, with a CPE exponent that is same as the value in the Liu model. ¹⁰⁰

Itagaki et al. ^115,114 have modelled a fractal structure with a three-scale pore structure (macropores, mesopores and micropores). The pores at the lower scale randomly locate on the pore wall at the upper scale as shown in Fig. 10d. Following the de Levie model, the impedance of a triangular pore is given by,

$$\begin{eqnarray}&&{Z}_{m}\,=\,\sqrt{{R}_{sol,m}^{* }{Z}_{m}^{* }}\,\cot {\rm{h}}\left({l}_{m}\sqrt{\displaystyle \frac{{R}_{sol,m}^{* }}{{Z}_{m}^{* }}}\right),\end{eqnarray} \tag{ 117 }$$

with ${R}_{sol,m}^{* }\,=\,\tfrac{\rho }{{A}_{m}}$ and ${Z}_{m}^{* }\,=\,\tfrac{{Z}_{m}^{* * }}{{X}_{m}}.$ Here, ${R}_{sol\,m}^{* }$ (${\rm{\Omega }}\,{{\rm{m}}}^{-1}$) is the solution resistance per unit pore length; ${Z}_{m}^{* * }$ (${\rm{\Omega }}\,{{\rm{m}}}^{2}$) is the area specific interfacial impedance; $\rho$ is the resistivity of electrolyte (${\rm{\Omega }}\,{\rm{m}}$); ${l}_{m}$ (${\rm{m}}$) is the pore length; ${Z}_{m}^{* }$ (${\rm{\Omega }}\,{\rm{m}}$) is the interfacial impedance per unit pore length; ${A}_{m}$ (${{\rm{m}}}^{2}$) is the cross-section area of pores; ${X}_{m}$ (${\rm{m}}$) is the circumference of cross-section of pores; and the subscript $m\,=\,ma,me,mi$ stands for macropores, mesopores and micropores, respectively.

There are ${n}_{mi}$ micropores per unit area of the mesopore wall. (1−$\xi$) of the mesopore wall is occupied by these micropores, and the remaining part with a fraction of $\xi$ by the capacitive interface. Therefore, the mesopore impedance is expressed as,

$$\begin{eqnarray}&&\displaystyle \frac{1}{{Z}_{me}^{* * }}\,=\,\displaystyle \frac{{n}_{mi}(1-\xi )}{{Z}_{mi}}+{\rm{j}}\omega \xi {C}_{dl}^{* }.\end{eqnarray} \tag{ 118 }$$

In the same way, the macropore impedance is derived from the mesopore impedance via,

$$\begin{eqnarray}&&\displaystyle \frac{1}{{Z}_{ma}^{* * }}\,=\,\displaystyle \frac{{n}_{me}\left(1-\xi \right)}{{Z}_{me}}+{\rm{j}}\omega \xi {C}_{dl}^{* },\end{eqnarray} \tag{ 119 }$$

where ${n}_{me}$ is the number of mesopores per unit area of the macropore wall.

Provided with the scaling coefficient $S$ for the pore depth and side length, the total impedance of the porous electrode is readily simulated using Eq. 119. The typical impedance spectra simulated using the model of Itagaki et al. exhibits a frequency dispersion region in high frequency range and a vertical line in low frequency range. If there is no mesopore on the macropore wall ($\xi \,=\,1$), the impedance response exhibits a 45° line in high frequency range, regressing to the standard behavior of the de Levie model. If the pore wall at the upper scale is totally occupied by sub-pores, the impedance response exhibit 22.5° and 11.25° lines in high frequency range for the double- and triple-scale pore structure, respectively. As $\xi$ decreases from 1 to 0, the typical 45° line in high frequency range decreases to a 11.25° line, indicating that less capacitive interface is detected by the a. c. signal. This is because the pore wall with a lower $\xi$ is dominated by more small pores on the upper scale, which can be penetrated only at much lower frequency. Readers are also referred to Lasia's book ¹⁵ for a comprehensive account of experimental and modelling works on fractals.

The explicit pore approach may be inadequate for a porous electrode with a large variety of pore structure. In such circumstances, the macrohomogenous porous electrode theory based on the volume averaging method is useful. ^4,6,40 In this implicit pore approach, the structure complexity is largely reduced by using average structural parameters, such as the void volume fraction (${\epsilon }$). Effective transport properties are used to reflect the impact of the porous structure on mass and charge transport, as expressed in Eqs. 50–52.

In macrohomogenous porous electrode theory, a widely studied structural effect is so-called gradient design for batteries and fuel cells. As for fuel cells, it have been found that 1) in the gas diffusion layer (GDL), increasing the porosity and PTFE content (hydrophobicity) from the catalyst layer (CL)/GDL interface toward the GDL/channel interface along the through-plane direction and from the channel inlet to the outlet in the in-plane direction improve the cell performance at high current densities ^116–126 ; 2) in the micro porous layer (MPL), increasing the PTFE content from the CL∣MPL interface to the MPL∣GDL interface helps to hydrate the membrane under low humidity and to remove liquid water from the electrode under high humidity condition ¹²⁷ ; 3) in the CL, a decrease in the ionomer content from the membrane/CL interface to the CL/GDL interface benefits the cell performance. ^128,129

There has also been a surge of interest in gradient design for LIBs, but controversial conclusions exist. For instance, the gradient porosity distribution designed by Subramanian et al. ^130,131 decreases from the separator to the current collector in the cathode electrode, which agrees with the conclusion of Maute et al. ¹³² On the contrary, Srinivasan et al. ¹³³ and Hosseinzadeh et al. ¹³⁴ show that the gradient porosity distribution leads to marginal improvement of the cell performance as compared with the uniform counterpart. However, these works did not discuss the impedance response, which can be readily calculated using the methods developed for nonuniform explicit pores.

An interesting structural effect frequently encountered in macrohomogenous porous electrode theory is agglomeration. An agglomerate is formed by holding together a group of particles on a smaller scale. As shown in Fig. 11, the agglomerate (∼200 nm) in fuel cell electrodes is composed of carbon black particles (∼20 nm) deposited with platinum nanoparticles. ¹³⁵ Intra-agglomerate mesopores (2 ∼ 20 nm) are partially penetrated with ionomer. In addition, the surface of the agglomerate is partially covered with an ionomer film. The oxygen dissolves into the ionomer thin film covering the surface of the agglomerate and then undergoes reaction-coupled diffusion inside the agglomerate.

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Gerteisen et al. modelled the case where ionomer cannot impregnate the inner pores of the agglomerate and therefore ORR does not occur on platinum nanoparticles inside the agglomerate due to the lacking of protons. ⁷⁴ In general cases where the ORR occurs inside the agglomerate, the agglomerate per se is a spherical porous electrode with ORR-coupled oxygen and charge transport. Consequently, the whole porous electrode actually possesses a three-scale structure, including the solid-electrolyte interphase (with Debye length as the characteristic length) where the ORR occurs at the bottom of the scale spectrum, the agglomerate scale, and the electrode scale. More impedance models of PEFC will be introduced in section 'Polymer electrolyte fuel cells'. The same circumstance is encountered in LIBs. Huang et al. have developed a three-scale impedance model for LIB electrodes, ^65,66 which will be introduced in section 'Batteries'.

The SEI film has a bilayer structure consisting of an inner compact layer and an outer porous layer. ^137–139 As shown in Fig. 15, $x=0$ locates the electrode surface, $0\lt x\lt {L}_{c}$ ("c" stands for "compact") the compact layer, ${L}_{c}\lt x\lt {L}_{c}+{L}_{p}$ ("p" stands for "porous") the porous layer, $x\gt {L}_{c}+{L}_{p}$ the bulk electrolyte, and $x={L}_{c}+{L}_{p}+{L}_{e}/2$ ("e" stands for "electrolyte") the mid-plane of the bulk electrolyte phase with ${L}_{e}$ being the thickness of the electrolyte bulk. It is safe to draw the approximation of ${L}_{e}\gg {L}_{c}+{L}_{p}.$

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The porous layer is impregnated with an electrolyte solution, allowing electrolyte-phase transport of solvated Li⁺ and anions. At the compact-porous interface, $x={L}_{c},$ Li⁺ ions have to undergo desolvation, which is proposed to be the rate-determining step by Ogumi. ^140–142 Secondary ion mass spectrometer analysis from Lu and Harris leads us to assume that anions cannot penetrate through this interface. ¹⁴³ In the compact layer, there may be multiple charge carriers, among which we assume excess interstitial Li⁺ ions are the major player. At the interface between the electrode and the compact layer, $x=0,$ Li⁺ ions intercalate into the electrode structure or capture an electron and deposit on the electrode surface. Electrical double layers are formed at the interface between the active particle and the compact layer, as well as the interface between the compact layer and the porous layer. There is no double-layer effect between the porous layer and the bulk electrolyte as ion transport occurs in the same electrolyte phase. In the literature, the relation between the electrical double layer and the SEI film is sometimes obscure and even confusing. Here, we note that the SEI film is a phase and the electrical double layer exsits between two phases.

As the thickness of the bulk electrolyte is usually several orders larger than the Debye length (ca. 0.1 nm for 1M electrolyte), the electroneutrality condition, implying that the concentration of Li⁺ is equal to that of anions, both denoted as $c,$ is satisfied. In bulk electrolyte, Li⁺ transport via diffusion and migration, which are described by the Nernst-Planck equation in the frequency domain (a phenomenological free energy based derivation can be found in Ref. 99),

$$\begin{eqnarray}&&{\rm{j}}\omega \tilde{c}\,=\,{\rm{\nabla }}\cdot \left({D}_{amb}{\rm{\nabla }}\tilde{c}\right),\end{eqnarray} \tag{ 153 }$$

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot \left({\sigma }_{e}{\rm{\nabla }}{\tilde{\phi }}_{e}+F{D}_{\pm }^{chem}{\rm{\nabla }}\tilde{c}\right)\,=\,0,\end{eqnarray} \tag{ 154 }$$

where the variables $\tilde{c}$ and ${\tilde{\phi }}_{e}$ are perturbed concentration and potential that are zero at steady state. Here ${D}_{amb}\,=({D}_{+}{D}_{-}^{chem}+{D}_{-}{D}_{+}^{chem})/({D}_{+}+{D}_{-})$ is the ambipolar diffusion coefficient, and ${D}_{+/-}^{chem}$ and ${D}_{+/-}$ are the chemical diffusion coefficient and the tracer diffusion coefficient in the dilute-solution limit, respectively, which are correlated as ${D}_{+/-}^{chem}\,=\,{D}_{+/-}\left(1+\partial \,\mathrm{ln}\,{\gamma }_{+/-}/\partial \,\mathrm{ln}\,c\right),$ with ${\gamma }_{+/-}$ being the activity coefficient. ${D}_{\pm }^{chem}\,=\,{D}_{+}^{chem}-{D}_{-}^{chem}$ is the differential diffusion coefficient. ${\sigma }_{e}\,=\,{F}^{2}\left({D}_{+}+{D}_{-}\right)c/RT$ is the ionic conductivity.

At the midplane of the bulk electrolyte, $x\,=\,{L}_{c}+{L}_{p}+{L}_{e}/2,$ a symmetric condition applies,

$$\begin{eqnarray}&&{\rm{\nabla }}\tilde{c}\,=\,0.\end{eqnarray} \tag{ 155 }$$

At the boundary between the porous layer and the bulk electrolyte, $x={L}_{c}+{L}_{p},$ the concentration and its gradient should be continuous,

$$\begin{eqnarray}&&{\tilde{c}}_{{2}^{-}}\,=\,{\tilde{c}}_{{2}^{+}},\,{\rm{\nabla }}{\tilde{c}}_{{2}^{-}}\,=\,{\rm{\nabla }}{\tilde{c}}_{{2}^{+}},\end{eqnarray} \tag{ 156 }$$

where ${\tilde{c}}_{{2}^{-}}$ and ${\rm{\nabla }}{\tilde{c}}_{{2}^{-}}$ are to be derived from the submodel of the porous layer. The subscript "2−" denotes the compact-layer side of the interface at $x={L}_{c}+{L}_{p}.$ The subscript "2+" denotes the bulk-electrolyte side of the interface at $x={L}_{c}+{L}_{p}.$

Solving Eq. 153 in the frequency domain with the boundary conditions in Eq. 155 and applying the approximation that ${L}_{e}\gg {L}_{c}+{L}_{p}$ result in,

$$\begin{eqnarray}&&\tilde{c}\,=\,{f}_{e}\exp \left(-\displaystyle \frac{x-{L}_{c}-{L}_{p}}{{\delta }_{e}}\right),\end{eqnarray} \tag{ 157 }$$

with ${\delta }_{e}\,=\,\sqrt{{D}_{amb}/{\rm{j}}\omega }$ being the characteristic length of diffusion in bulk electrolyte, and ${f}_{e}$ a coefficient to be determined later.

Ion transport in the porous layer is essentially the same as in the bulk electrolyte; the only difference is that the transport parameters should be replaced with effective values considering the effect of the porous structure,

$$\begin{eqnarray}&&j\omega {{\epsilon }}_{p}\tilde{c}\,=\,{\rm{\nabla }}\cdot \left({D}_{amb}^{eff}{{\epsilon }}_{p}{\rm{\nabla }}\tilde{c}\right),\end{eqnarray} \tag{ 158 }$$

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot \left({\sigma }_{e,eff}{\rm{\nabla }}{\tilde{\phi }}_{e}+F{D}_{\pm }^{chem,eff}{{\epsilon }}_{p}{\rm{\nabla }}\tilde{c}\right)\,=\,0,\end{eqnarray} \tag{ 159 }$$

where ${{\epsilon }}_{p}$ is the porosity of the porous layer, ${D}_{amb}^{eff}$ and ${D}_{\pm }^{chem,eff}$ are the effective ambipolar and chemical diffusion coefficient in the porous layer, respectively, and ${\sigma }_{e,eff}$ is the effective ionic conductivity.

The solution to Eq. 158 is given by

$$\begin{eqnarray}&&\tilde{c}\,=\,{f}_{p}^{1}\exp \left(\displaystyle \frac{x-{L}_{c}}{{\delta }_{p}}\right)+{f}_{p}^{2}\exp \left(-\displaystyle \frac{x-{L}_{c}}{{\delta }_{p}}\right),\end{eqnarray} \tag{ 160 }$$

with ${\delta }_{p}\,=\,\sqrt{{D}_{amb}^{eff}/{\rm{j}}\omega }$ being the characteristic length of diffusion in the porous layer. ${f}_{p}^{1}$ and ${f}_{p}^{2}$ are two coefficients to be determined by the boundary conditions. At the interface between the bulk electrolyte and the porous layer, $x\,=\,({L}_{c}+{L}_{p}),$ the continuity conditions expressed in Eq. 156 apply. At the compact-porous interface, $x\,=\,{L}_{c},$ the current flux is carried by cations only and the flux of anions is zero, as a result, ⁹⁹

$$\begin{eqnarray}&&-{D}_{amb}^{eff}\left(1+\displaystyle \frac{{D}_{+}^{eff}}{{D}_{-}^{eff}}\right)F{{\epsilon }}_{p}{\rm{\nabla }}\tilde{c}={\tilde{j}}_{app},\end{eqnarray} \tag{ 161 }$$

with ${\tilde{j}}_{app}$ being the applied current density. Applying the boundary conditions, we obtain

$$\begin{eqnarray}&&{f}_{p}^{1}\,=\,\displaystyle \frac{{\delta }_{p}{\tilde{j}}_{app}}{F{{\epsilon }}_{p}{D}_{amb}^{eff}\left(1+\tfrac{{D}_{+}^{eff}}{{D}_{-}^{eff}}\right)}\displaystyle \frac{\lambda \exp \left(-\tfrac{{L}_{p}}{{\delta }_{p}}\right)}{\exp \left(\tfrac{{L}_{p}}{{\delta }_{p}}\right)-\lambda \exp \left(-\tfrac{{L}_{p}}{{\delta }_{p}}\right)},\end{eqnarray} \tag{ 162 }$$

$$\begin{eqnarray}&&{f}_{p}^{2}=\displaystyle \frac{{\delta }_{p}{\tilde{j}}_{app}}{F{{\epsilon }}_{p}{D}_{amb}^{eff}\left(1+\tfrac{{D}_{+}^{eff}}{{D}_{-}^{eff}}\right)}\displaystyle \frac{\exp \left(\tfrac{{L}_{p}}{{\delta }_{p}}\right)}{\exp \left(\tfrac{{L}_{p}}{{\delta }_{p}}\right)-\lambda \exp \left(-\tfrac{{L}_{p}}{{\delta }_{p}}\right)},\end{eqnarray} \tag{ 163 }$$

with $\lambda \,=\,({\delta }_{e}-{\delta }_{p})/({\delta }_{e}+{\delta }_{p})$ being a frequency-dependent dimensionless parameter.

Provided the concentration distribution, we are to calculate the potential distribution. The current densities in the electrolyte solution and the porous layer are given by

$$\begin{eqnarray}&&{\sigma }_{e}{\rm{\nabla }}{\tilde{\phi }}_{e}+{D}_{\pm }^{chem}F{\rm{\nabla }}\tilde{c}\,=\,-{\tilde{j}}_{app},\end{eqnarray} \tag{ 164 }$$

$$\begin{eqnarray}&&{\sigma }_{e,eff}{\rm{\nabla }}{\tilde{\phi }}_{e}+{D}_{\pm }^{chem,eff}{{\epsilon }}_{p}F{\rm{\nabla }}\tilde{c}\,=\,-{\tilde{j}}_{app}.\end{eqnarray} \tag{ 165 }$$

Integrating Eqs. 164 and 165 and substituting concentration distribution, we solve the potential drop in the porous layer and bulk electrolyte, from which we arrive at the expression for the impedance corresponding to Li+ transport in the bulk solution and the porous layer,

$$\begin{eqnarray}&&{Z}_{p-bulk}\,=\,\displaystyle \frac{{L}_{e}}{2{\sigma }_{e}}+\displaystyle \frac{{L}_{p}}{{\sigma }_{e,eff}}-\displaystyle \frac{{\delta }_{p}{\gamma }_{d}}{{\sigma }_{e}},\end{eqnarray} \tag{ 166 }$$

with ${\gamma }_{d}$ being a dimensionless length coefficient related to ion transport, defined as

$$\begin{eqnarray}&&{\gamma }_{d}\,=\,\displaystyle \frac{{D}_{\pm }^{chem}}{{{\epsilon }}_{p}{D}_{amb}^{eff}\left(1+\tfrac{{D}_{+}^{eff}}{{D}_{-}^{eff}}\right)}\displaystyle \frac{\exp \left(\tfrac{{L}_{p}}{{\delta }_{p}}\right)+\lambda \exp \left(-\tfrac{{L}_{p}}{{\delta }_{p}}\right)}{\exp \left(\tfrac{{L}_{p}}{{\delta }_{p}}\right)-\lambda \exp \left(-\tfrac{{L}_{p}}{{\delta }_{p}}\right)}.\end{eqnarray} \tag{ 167 }$$

In Eq. 166, the term ${L}_{e}/2{\sigma }_{e}$ corresponds to migration in half of the bulk electrolyte, the term ${L}_{p}/{\sigma }_{e,eff}$ migration in the porous layer, and the term ${\delta }_{p}\gamma /{\sigma }_{e}$ diffusion in the porous layer and the bulk electrolyte.

At the compact-porous interface, $x={L}_{c},$ a solvated ${{\rm{Li}}}^{+}$ ion in solution, ${{\rm{Li}}}^{+}\cdots {\left({\rm{sol}}\right)}_{n},$ undergoes desolvation and then hops into the solid phase,

$$\begin{eqnarray}&&{{\rm{Li}}}^{+}\cdots {\left({\rm{sol}}\right)}_{n}\,\to \,{{\rm{Li}}}^{+}+n\cdot {\rm{sol}},\end{eqnarray} \tag{ 168 }$$

with $n$ being coordinate number of solvents. This ion transfer reaction can be described using the Butler-Volmer equation,

$$\begin{eqnarray}&&{j}_{it}\,=\,-{j}_{0,it}\left(\exp \left(-\displaystyle \frac{\alpha F\eta }{RT}\right)-\exp \left(\displaystyle \frac{\left(1-\alpha \right)F\eta }{RT}\right)\right),\end{eqnarray} \tag{ 169 }$$

where ${j}_{0,it}$ is the exchange current density, which is dictated by the activation energy of the desolvation process, and $\eta \,={\phi }_{1-}^{s}-{\phi }_{1+}^{s}-{E}_{it}^{0}$ the overpotential with ${E}_{it}^{0}$ being the equilibrium potential. ${E}_{it}^{0}\,=\,{E}_{it}^{\varnothing }+\tfrac{RT}{F}\,\mathrm{ln}\left(\tfrac{{c}_{1+}}{{c}_{1-}}\right)$ where ${E}_{it}^{\varnothing }$ is the standard value, ${c}_{1+}$ and ${c}_{1-}$ are taken at the diffuse-diffusion boundary. The subscript "1-" and "1+" denote the compact-layer and porous-layer side of the interface at $x={L}_{c},$ respectively, as shown in Fig. 15.

Under a small perturbation, Eq. 169 is written as

$$\begin{eqnarray}&&{\tilde{j}}_{it}\,=\,\displaystyle \frac{{j}_{0,it}F}{RT}\left({\tilde{\phi }}_{1-}^{s}-{\tilde{\phi }}_{1+}^{s}-\displaystyle \frac{RT}{F{\bar{c}}_{1+}}{\tilde{c}}_{1+}\right),\end{eqnarray} \tag{ 170 }$$

where ${\bar{c}}_{1+}$ denotes the concentration at the steady state. Here, we neglect variations in ${\tilde{c}}_{1-}$ exerted by the perturbation, for the fact that the compact layer is very thin, and the concentration in the compact layer is controlled by double layer effects.

In addition to the Faradaic current contributed by the ion transfer reaction, there is also double layer charging current at the interface between the compact layer and the porous layer. Since the compact layer is a semiconductor, the double layer is composed of two parts in series, a part belonging to the solution phase, and a part to the solid phase. ¹⁴⁴ As a result, the total double layer capacitance, ${C}_{c-p},$ is thus given by

$$\begin{eqnarray}&&\displaystyle \frac{1}{{C}_{c-p}}\,=\,\displaystyle \frac{1}{{C}_{c}}+\displaystyle \frac{1}{{C}_{p}},\end{eqnarray} \tag{ 171 }$$

where ${C}_{c}$ and ${C}_{p}$ refer to the double layer capacitance of the compact and porous layer, respectively. No attempt is made, herein, to relate ${C}_{c}$ and ${C}_{p}$ with physical properties of the compact and porous layer, because such practice involves many parameters at the microscopic scale of the interface that are difficult to obtain at the present.

The total current density is the sum of two parts: Faradaic and double layer charging, namely,

$$\begin{eqnarray}&&{\tilde{j}}_{app}\,=\,{\tilde{j}}_{it}+{\tilde{j}}_{dl},\end{eqnarray} \tag{ 172 }$$

with the double layer charging current density given by

$$\begin{eqnarray}&&{\tilde{j}}_{dl}\,=\,{\rm{j}}\omega {C}_{c-p}\left({\tilde{\phi }}_{1-}^{s}-{\tilde{\phi }}_{1+}^{s}\right).\end{eqnarray} \tag{ 173 }$$

Substituting Eqs. 170 and 173 into Eq. 172 with some mathematic manipulations, we obtain the impedance at the compact-porous interface as

$$\begin{eqnarray}&&{Z}_{c-p}\,=\,\displaystyle \frac{{R}_{it}\left(1+{\gamma }_{it}\right)}{{\rm{j}}\omega {R}_{it}{C}_{c-p}+1},\end{eqnarray} \tag{ 174 }$$

with ${R}_{it}=\tfrac{RT}{F{j}_{0,it}}$ being the ion-transfer resistance, and ${\gamma }_{it}\,=\tfrac{{j}_{0,it}{\delta }_{p}}{F{\bar{c}}_{1+}{D}_{\pm }^{chem}}{\gamma }_{d}$ embodying the effect of ion transport on the ion transfer reaction.

Next, we model the compact layer as a like solid electrolyte with only one ion moving, namely, composed of mobile Li⁺ in a rigid neutralizing background of negative charge. Li⁺ transport in the compact layer can be described using the PNP equation. Jamnik and Maier are able to convert the PNP equation into an equivalent circuit representation. ^145,146 Further neglect of spatial variation of circuit elements, which is valid beyond the diffuse layer, leads them to obtain analytical impedance expressions. ^145,146 However, the compact layer in our case, whose thickness is comparable with the Debye length, is governed by the double layer effects, and the assumption used in the model of Jamnik and Maier to obtain analytical results is invalid any longer. Nevertheless, the model of Jamnik and Maier reveals an important observation that the impedance of the compact layer manifests itself as a semicircle in the high-frequency range. Consequently, we employ a phenomenological model for the compact layer for the moment, which is a parallel connection of a capacitance, ${C}_{dl,c},$ and a resistance related with charge transport across the compact layer, ${R}_{c},$ namely,

$$\begin{eqnarray}&&{{\rm{Z}}}_{{\rm{c}}}^{int}\,=\,\displaystyle \frac{{R}_{c}}{1+{\rm{j}}\omega {R}_{c}{{\rm{C}}}_{{\rm{dl}},{\rm{c}}}}.\end{eqnarray} \tag{ 175 }$$

We are left with the task of relating the capacitance and the charge transport resistance with physical properties of the compact layer. This task is by no means trivial. The complexities are two-fold, to say the least. Regarding ${C}_{dl,c},$ it is a mixed capacitance composed of the differential double layer capacitance, the dielectric capacitance, and the chemical capacitance. These three types of capacitance are connected in a complex circuit, rendering a closer examination of ${C}_{dl,c}$ challenging for the moment. Regarding ${R}_{c},$ we could express it as

$$\begin{eqnarray}&&{R}_{c}\,=\,\displaystyle \frac{{L}_{c}}{{\sigma }_{c,eff}},\end{eqnarray} \tag{ 176 }$$

with ${\sigma }_{c,eff}$ being the effective conductivity of the compact layer. By effective we mean it is an overall quantity as the charge conductivity is spatially dependent, an immediate consequence of the distribution of the concentration of charge carriers, in the compact layer.

Combined, we express the impedance of the SEI shown in Fig. 15 as follows,

$$\begin{eqnarray}&&\begin{array}{l}{Z}_{SEI}\,=\,\displaystyle \frac{{R}_{c}}{1+{\rm{j}}\omega {R}_{c}{C}_{dl,c}}+\displaystyle \frac{{R}_{it}\left(1+{\gamma }_{it}\right)}{1+{\rm{j}}\omega {R}_{it}{C}_{c-p}}\\ \,\,\,+\,\left(\displaystyle \frac{{L}_{e}}{2{\sigma }_{e}}+\displaystyle \frac{{L}_{p}}{{\sigma }_{e,eff}}-{\gamma }_{d}\displaystyle \frac{{\delta }_{p}}{{\sigma }^{e}}\right),\end{array}\end{eqnarray} \tag{ 177 }$$

where the first, second and third term on the right-hand side denotes the impedance of solid-state ion transport in the compact layer, ion desolvation at the compact-porous layer interface, and electrolyte-phase ion transport in the porous layer and the solution bulk, respectively. Note that the charge transfer reaction at the electrode-compact layer interface is not included in ${Z}_{SEI},$ but will be included in the impedance of primary particles. Assigning model parameters with typical values as listed in the caption of Fig. 15, we simulate the EIS as shown in Fig. 15. In the Nyquist plot, we see a high-frequency semicircle corresponding to solid-state ion transport via the compact layer, a moderate-frequency semicircle corresponding to desolvation of Li⁺ ions at the compact-porous layer interface, and a 45° line in low-frequency range corresponding to transport of Li⁺ ions through the porous layer and the solution bulk.

A primary particle may have a SEI film covered on its surface or be bare without a SEI film. Huang et al. have developed an impedance model for a spherical primary particle without a SEI film, considering the complex coupling between intercalation and deintercalation of lithium ions at the interface, double-layer charging, and diffusion in solid phase. ⁶⁵ The impedance expression for the case without a SEI film is given by

$$\begin{eqnarray}&&{Z}_{pp}\,=\,\displaystyle \frac{1}{\tfrac{{\chi }_{pp}}{{R}_{ct}^{pp}+{Z}_{d}^{pp}}+{\rm{j}}\omega {C}_{dl}^{pp}},\end{eqnarray} \tag{ 178 }$$

where ${\chi }_{pp}=\tfrac{{\delta }_{RP}}{{\delta }_{RP}+{\lambda }_{pp}}$ is a structural parameter with ${\delta }_{RP}$ being the distance between the reaction plane (assigned as the outer Helmholtz plane in Ref. 65) and the electrode surface, and ${\lambda }_{pp}$ being the Debye length of the double layer of the primary particle. ${C}_{dl}^{pp}$ is the double layer capacitance at the electrode-electrolyte interface. ${R}_{ct}^{pp}=\tfrac{RT}{F{j}_{0}^{c}}$ is the charge transfer resistance with ${j}_{0}^{c}$ being the exchange current density corrected with double layer effects, ${Z}_{d}^{pp}=\tfrac{\partial U}{\partial {c}_{s}}{Y}_{s}$ is the diffusion impedance with $\tfrac{\partial U}{\partial {c}_{s}}$ being the local slope of the open circuit potential (U) curve as a function of the lithium ion concentration in solid (${c}_{s}$), ${Y}_{s}$ being a frequency-dependent factor of spherical diffusion, ^147,148

$$\begin{eqnarray}&&{Y}_{s}\,=\,\displaystyle \frac{{R}_{pp}}{F{D}_{pp}}\left(\displaystyle \frac{\tanh \left(\sqrt{{\rm{j}}\omega {\tau }_{d}}\right)}{\tanh \left(\sqrt{{\rm{j}}\omega {\tau }_{d}}\right)-\sqrt{{\rm{j}}\omega {\tau }_{d}}}\right).\end{eqnarray} \tag{ 179 }$$

Here ${\tau }_{d}\,=\,{R}_{pp}^{2}/{D}_{pp}$ is the time constant of solid phase diffusion with ${R}_{pp}$ being the radius and ${D}_{pp}$ the diffusion coefficient. It is worth noting that the diffusion impedance is in a form different with the oft-used Warburg impedance based on semi-infinite planar diffusion. Recently, Huang extended the diffusion impedance to different structures with different boundary conditions. ⁹⁹

Manifestation of double layer effects in Eq. 179 is two-fold. On the one hand, the exchange current density is corrected with double layer effects, namely, ${j}_{0}^{c}={j}_{0}\exp \left(-\tfrac{\alpha F{\rm{\Delta }}{\phi }_{dl}}{RT}\right)$ with ${\rm{\Delta }}{\phi }_{dl}$ being the potential drop in the diffuse layer and $\alpha$ the symmetry factor. On the other hand, ${\chi }_{pp}$ is introduced to consider the double layer structure on the charge transfer reaction.

As regards primary particles covered by a SEI film, we combine Eqs. 177 and 178 and obtain the following impedance expression, ⁶⁵

$$\begin{eqnarray}&&\begin{array}{l}{Z}_{pp}=\displaystyle \frac{1}{\tfrac{{\chi }_{pp}}{{R}_{ct}^{pp}+{Z}_{d}^{pp}}+{\rm{j}}\omega {C}_{dl}^{pp}}+\displaystyle \frac{{R}_{c}}{1+{\rm{j}}\omega {R}_{c}{C}_{dl,c}}+\displaystyle \frac{{R}_{it}\left(1+{\gamma }_{it}\right)}{1+{\rm{j}}\omega {R}_{it}{C}_{c-p}}\\ \,\,+\,\left(\displaystyle \frac{{L}_{p}}{{\sigma }_{e,eff}}-{\gamma }_{d}\displaystyle \frac{{\delta }_{p}}{{\sigma }^{e}}\right).\end{array}\end{eqnarray} \tag{ 180 }$$

In Eq. 180, we have neglected $\tfrac{{L}_{e}}{2{\sigma }_{e}}$ corresponding to ion transport in the bulk electrolyte phase, which will be considered on a higher scale. In addition, we have divided the interfacial charge transfer reaction into three steps: desolvation at the compact-porous layer, ion transport in the compact layer, and electron transfer at the electrode-compact layer interface, corresponding to the third, second, and first term on the right hand, respectively. The division brings many parameters which are unfortunately difficult, if not impossible, to obtain from impedance measurement on a cell. Model systems are required to determine them, instead.

An agglomerate (secondary particles impregnated with electrolyte) consisting of primary particles is actually a spherical porous electrode. Huang et al. developed a physical impedance model for a spherical agglomerate, starting with deriving basic equations of mass transport from a general free energy functional, continuing with applying the volume averaging method to describe mass transport in porous media, and ending with employing Fourier transform to solve the impedance response. ⁶⁵ An analytical impedance expression is obtained

$$\begin{eqnarray}&&\begin{array}{l}{Z}_{ag}\,=\\ \,\displaystyle \frac{RT{\tau }_{ag}}{F{{\epsilon }}_{ag}}\left(\displaystyle \frac{{\nu }_{1}-{\nu }_{2}}{\tfrac{RT{\sigma }_{ag}}{F{R}_{ag}}\left({\nu }_{1}L\left({{\rm{\lambda }}}_{1}\right)-{\nu }_{2}L\left({{\rm{\lambda }}}_{2}\right)\right)-\tfrac{F{D}_{\pm }^{chem}{c}_{ref}}{{R}_{ag}}\left(L\left({{\rm{\lambda }}}_{1}\right)-L\left({{\rm{\lambda }}}_{2}\right)\right)}\right),\end{array}\end{eqnarray} \tag{ 181 }$$

with $L\left(x\right)=\sqrt{x}\,\coth \left(\sqrt{x}\right)-1.$ Here, ${R}_{ag},$ ${{\epsilon }}_{ag}$ and ${\tau }_{ag}$ are the radius, electrolyte phase volume fraction and tortuosity of the agglomerate, respectively. ${\sigma }_{ag}$ is the electrolyte conductivity in the agglomerate. ${D}_{\pm }^{chem}$ is difference diffusion coefficient. ${\lambda }_{1},$ ${\lambda }_{2},$ ${v}_{1}$ and ${v}_{2}$ are functions of matrix elements Θ_i $({\rm{i}}\,=\,1,2,3,4),$ as expressed in Eqs. 60–63. The matrix elements Θ_i are functions of properties of primary particles, as given by Eq. (A3.3) in Ref. 65.

The impedance response of agglomerates can be measured using the single particle technique. Figure 16a compares the model simulation and the EIS data measured by Dokko et al. on a LiCoO₂ particle with a diameter of $15\,\mu {\rm{m}}$ at different storage time. ¹⁴⁹ The measured EIS data show a 45° line in the high frequency range, a clear fingerprint of porous electrodes, indicating that the single particle composed of primary particles has pores impregnated with electrolyte. The model expressed in Eq. 181 well captures this feature. On the contrary, the model neglecting electrolyte transport inside the single particle does not agree, qualitatively, with experimental data in high frequency range. In addition, the Warburg impedance cannot fit the impedance data in low frequency because solid diffusion in primary particles takes place in a spherical structure rather than the semi-infinite structure assumed in the Warburg model. However, the model expressed in Eq. 181 agrees well with the experimental data also in low frequency.

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In the section of extension in physics, we have obtained a general expression, Eq. 64, for the impedance response of porous electrodes considering mass transport in both electrode and electrolyte phases. This formula can be applied to model the impedance response of various kinds of battery electrodes, as detailed below. For three-scale battery electrodes made up of agglomerates, the agglomerate impedance, expressed in Eq. 181, is set as the local impedance element, ${Z}_{loc},$ in Eq. 64. For two-scale battery electrodes made up of primary particles, Eqs. 178 and 180 are set as ${Z}_{loc}$ for the case with and without a SEI film covering primary particles, respectively. In Ref. 66, Huang et al. developed a three-scale impedance model for a LiNi_1/3Mn_1/3Co_1/3O₂ porous cathode. The model was parameterized with experimental data at a series of state of charge (SOC) measured using a reference electrode. As shown in Fig. 16c, the model agrees well with experiments over a wide range of SOC, and close examination of the resultant parameters lend further credence to the validity of the model.

Published in 2000, the MDDN (shorthand for Meyers-Doyle-Darling-Newman) model describes the impedance response of a porous electrode consisting of intercalation particles on two scales. ⁴⁹ On the microscopic scale, the impedance of a single intercalation particle is phenomenologically described using an equivalent electric circuit of which some parameters are further derived from physical laws. On the macroscopic scale, the impedance of a porous electrode is described using the transmission line model taking into account both ionic conduction in the electrolyte phase and electronic conduction in the electrode phase. The MDDN model has following inadequacies. Firstly, the two-scale MDDN model cannot adequately describe porous electrodes that are composed of agglomerates and have a three-scale structure. Secondly, the physical meanings of the electric circuit and its elements describing the SEI film are elusive. Thirdly, ion transport in the electrolyte phase is described using the Ohm's law which considers migration only, while diffusion and the concentration gradient are neglected. The second limitation has been coped with by considering details of ion transport in the SEI film by Dees et al. ^150,151 and Single et al. ¹⁵² The third limitation has been amended by Doyle et al. who numerically solved the full Newman model, ¹⁵³ and by White et al. who provided analytical results. ^67,152,154

The White group at University of South Carolina has been striving to develop analytical solutions for the impedance response of battery electrodes. In 2004, Devan and White obtained an analytical expression for the porous electrode with intercalation reactions. ¹⁵⁴ The Devan-White model employs the concentrated solution theory for ion transport in the electrolyte phase and the Butler-Volmer equation for charge transfer reactions, but neglects diffusion in the solid phase. The synthetic impedance simulated using the model shows an arc in low frequency range corresponding to the electrolyte phase diffusion. Later on, Sikha and White complemented the Devan-White model by including solid-phase diffusion, presenting a comprehensive impedance model for two-scale battery electrodes. ^67,152

The first limitation of the MDDN model was resolved by Huang et al. ^65,66 They developed a three-scale impedance model for battery electrodes composed of agglomerates which are treated as spherical porous electrodes. A salient feature of the three-scale impedance model is that a 22.5° line exists in high frequency range, rather than a 45° in classical impedance models of porous electrodes. Of note, 22.5° and 11.25° lines in high frequency range have also been obtained by Itagaki et al. for the double- and triple-scale pore structures, ¹¹⁴ see a discussion in chapter 9.1.3 in Lasia's book. ¹⁵ In a more general sense, the impedance response of a n-scale porous electrode shows a $\pi /{2}^{n}$ line in high frequency range. In other words, the number of scales of the porous electrode may well be calculated as $n=\,\mathrm{ln}\left(\tfrac{\pi }{\theta }\right)/\,\mathrm{ln}(2)$ with $\theta$ being the angle of the inclined line in high frequency range. Generally, $n$ can be nonintergal. Besides the extension in the structure scale, the model developed by Huang et al. also features the coupling of double layer charging and charge transfer reactions and the detailed description of the SEI film. ⁶⁵ In a more recent study, Huang and Zhang developed a general model for impedance response of various kinds of porous electrodes. ⁵⁰ Applied to battery electrodes, the Huang-Zhang model is reduced to the earlier Sikha-White model. Furthermore, the Huang-Zhang model accounts for in-plane and through-plane pore size distribution, and obtains analytical solutions for several limiting cases, as discussed before. We note that in the field of time-domain modeling, Farrell et al. developed a three-scale model for primary alkaline battery cathode, ¹⁵⁵ and later extended it to a LiFePO₄ electrode. ¹⁵⁶

Other notable progress in theoretical modelling of impedance response of battery electrodes includes the treatment of phase transition by Ciucci and Lai, ¹⁵⁷ and Gambhire et al., ¹⁵⁸ the consideration of active particle morphology and electrode microstructure by Chen and Mukherjee, ¹⁵⁹ the description of dendrite growth by Talian et al., ¹⁶⁰ and the detailed discussion of the SOC-dependence by Gruet et al. ¹⁶¹ and Schönleber et al. ¹⁶² Specifically, Ciucci and Lai employed the Cahn–Hilliard framework to describe phase transition and the small signal perturbation approach to obtain impedance response, ¹⁵⁷ while Gambhire et al. employed the core–shell model to describe phase transition. ¹⁵⁸

Although EIS is routinely applied in battery investigations, interpretation of the results using a physical impedance model is relatively rare. Herein, instead of presenting a comprehensive survey of relevant activities, which has been attempted in recent reviews, ^22,163 we highlight several exemplary works filtrated through our eyes.

Lindbergh et al. employed EIS to investigate the aging mechanism of a Li_xNi_0.8Co_0.15Al_0.05O₂ porous electrode. ^164,165 In so doing, they developed a physical impedance model based on the Newman model, featuring consideration of the realistic multilayer structure, size distribution of active particles in two shapes (spheres and flakes). EIS measurements were conducted using a three-electrode setup with flexible temperature and pressure control. EIS data were interpreted using their physical impedance model. They found that the impedance evolution along aging can be attributed to loss of active particles, increased local contact resistance between the active material and the conductive carbon, and increased resistance of a layer on the current collector. Recently, Zhou et al. ¹⁶⁶ employed the physical impedance model developed by Huang et al. ⁶⁶ to identify the mechanism of power fade of lithium-ion batteries, revealing that the electrolyte phase transport is the major cause.

EIS has also been proven powerful in assessing ion transport properties in battery porous electrodes. For this purpose, special design of the experimental setup is usually required. Roling et al. ¹⁶⁷ and Latz et al. ¹⁶⁸ determined the transference number of Li⁺ in electrolyte from very-low frequency impedance measured on a symmetric Li∣electrolyte∣Li cell. Gasteiger et al. determined the tortuosity of a battery porous electrode from the impedance measured under a blocking electrode configuration, which is realized by using a non-intercalating electrolyte. ¹⁶⁹

Physics in PEFC porous electrodes

A PEFC consists of an anode where hydrogen is oxidized to protons, ${{\rm{H}}}_{2}\,\to \,{{\rm{H}}}^{+}+{{\rm{e}}}^{-},$ a proton-conducting membrane responsible for proton conduction, electrical insulation and water transport, and a cathode where oxygen from the gas diffusion layer reacts with protons and electrons forming water, namely, via the oxygen reduction reaction (ORR), ${{\rm{O}}}_{2}+4{{\rm{H}}}^{+}+4{{\rm{e}}}^{-}\,\to \,2{{\rm{H}}}_{2}{\rm{O}}.$ As the ORR constitutes the largest solo voltage losses in a PEFC, most attention has been paid to the cathode.

At the cathode, multi-dimension (thickness direction and along-channel direction), multi-media (CCL (cathode catalyst layer), GDL (gas diffusion layer) and gas flow channel), multi-component (oxygen, proton, electron and water) transport is coupled with the ORR. Schematic illustration of overpotential, proton, electron current density and gas concentration profiles at the cathode is depicted in Fig. 17. Water transport process is neglected and the effect of water is taken into account in the effective oxygen diffusion coefficient and the proton conductivity, assuming a uniform distribution of water saturation in pores and water content in ionomer.

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The full problem is simplified to a quasi-2D (1D + 1D) problem. The CCL is described along the through-plane direction, where the ionic, electronic, and faradaic current densities are correlated via the charge conservation law,

$$\begin{eqnarray}&&{C}_{dl}\displaystyle \frac{\partial \eta }{\partial t}+\displaystyle \frac{\partial {j}_{e}}{\partial x}\,=\,{S}_{1},\,{S}_{1}=-2{i}_{* }\left(\displaystyle \frac{c}{{c}_{ref}}\right)\sin \,{\rm{h}}\left(\displaystyle \frac{\eta }{b}\right).\end{eqnarray} \tag{ 182 }$$

Here, ${S}_{1}$ is a flux term from the ORR, ${C}_{dl}$ is the volumetric double-layer capacitance (F cm⁻³), $t$ is time, ${j}_{e}$ is the local proton current density (A cm⁻²), $x$ is the distance from the membrane (m), ${i}_{* }$ is the volumetric exchange current density (A cm⁻³), $c$ is the local oxygen concentration in the CCL (mol m⁻³), ${c}_{ref}$ is the reference oxygen concentration (mol m⁻³), $b$ is the Tafel slope (mV). $\eta$ is the local ORR overpotential given by $\eta \,=\,-({\phi }_{s}-{\phi }_{e}-{E}_{ORR}^{eq}),$ where ${\phi }_{e}$ is the potential in the ionic-conducting phase, ${\phi }_{s}$ is the potential in the electronic phase and ${E}_{ORR}^{eq}$ is the equilibrium ORR potential. A transference number of 0.5 is used in the Butler-Volmer equation. The model is generic for other values of the transference number, but some of the analytical solutions rely on the specific value of 0.5.

Based on the Ohmic law, ${j}_{e}$ is further given by,

$$\begin{eqnarray}&&{j}_{e}\,=\,-{\sigma }_{e}\displaystyle \frac{\partial {\phi }_{e}}{\partial x},\end{eqnarray} \tag{ 183 }$$

with ${\sigma }_{e}$ being the CCL proton conductivity. Note that there is no proton flux in the GDL. As the ORR is the proton-coupled electron transfer reaction, the generation/consumption of protons should be equal to that of electrons, namely,

$$\begin{eqnarray}&&\displaystyle \frac{\partial {j}_{e}}{\partial x}\,=\,-\displaystyle \frac{\partial {j}_{s}}{\partial x},\end{eqnarray} \tag{ 184 }$$

where ${j}_{s}\,=\,-{\sigma }_{s}\tfrac{\partial {\phi }_{s}}{\partial x}$ is electron current density described using the Ohmic's law with ${\sigma }_{s}$ being the electron conductivity of CCL. The sum of local proton and electron current density is equal to the cell current density ${j}_{0},$ namely, ${j}_{e}+{j}_{s}={j}_{0}.$ As there is no proton flux in the GDL, Eq. 184 indicates readily that $\tfrac{\partial {j}_{e}}{\partial x}\,=\,0$ in the GDL. As for oxygen transport in the CCL and GDL, only the through-plane direction is considersed,

$$\begin{eqnarray}&&\displaystyle \frac{\partial c}{\partial t}-D\displaystyle \frac{{\partial }^{2}c}{\partial {x}^{2}}\,=\,{S}_{2},\end{eqnarray} \tag{ 185 }$$

with $D$ being the effective oxygen diffusion coefficient, and ${S}_{2}$ a flux term. $D$ equals ${D}_{ox}$ in the CCL and ${D}_{b}$ in the GDL, and ${S}_{2}=\tfrac{{S}_{1}}{4{\rm{F}}}$ in the CCL and zero elsewhere.

The other dimension is the along-channel direction. Neglecting the rib/channel effect of flow field and assuming a plug gas flow with a constant velocity along the channel, the oxygen transport is described by,

$$\begin{eqnarray}&&\displaystyle \frac{\partial {c}_{h}}{\partial t}\,+\,v\displaystyle \frac{\partial {{\rm{c}}}_{{\rm{h}}}}{\partial z}\,=\,{\left.-\left(\displaystyle \frac{D}{h}\right)\displaystyle \frac{\partial c}{\partial x}\right|}_{x={l}_{t}+{l}_{b}}.\end{eqnarray} \tag{ 186 }$$

Here ${c}_{h}$ is the local oxygen concentration in the channel, $z$ is the distance from the channel inlet, $v$ is the flow velocity, $h$ is the channel depth, ${l}_{t}$ is the thickness of CCL and ${l}_{b}$ is the thickness of GDL.

Dimensionless equation set

The equation set consisting of Eqs. 182–186 is normalized to,

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\eta }^{nd}}{\partial {t}^{nd}}-{{\varepsilon }_{rpd}}^{2}\displaystyle \frac{{\partial }^{2}{\phi }_{e}^{nd}}{\partial {\left({x}^{nd}\right)}^{2}}=-{c}^{nd}\,\sinh \left({\eta }^{nd}\right),\,{x}^{nd}\in \left(0,1\right),\end{eqnarray} \tag{ 187 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\eta }^{nd}}{\partial {t}^{nd}}+{{\varepsilon }_{rpd}}^{2}{k}_{\sigma }\displaystyle \frac{{\partial }^{2}{\phi }_{s}^{nd}}{\partial {\left({x}^{nd}\right)}^{2}}=-{c}^{nd}\,\sinh \left({\eta }^{nd}\right),\,{x}^{nd}\in \left(0,1\right),\end{eqnarray} \tag{ 188 }$$

$$\begin{eqnarray}&&{{t}_{c}}^{2}\displaystyle \frac{\partial {c}^{nd}}{\partial {t}^{nd}}-{{\varepsilon }_{rpd}}^{2}{D}_{ox}^{nd}\displaystyle \frac{{\partial }^{2}{c}^{nd}}{\partial {\left({x}^{nd}\right)}^{2}}\,=\,-{c}^{nd}\,\sinh \left({\eta }^{nd}\right),\,{x}^{nd}\in \left(0,1\right),\end{eqnarray} \tag{ 189 }$$

$$\begin{eqnarray}&&{{t}_{c}}^{2}\displaystyle \frac{\partial {c}^{nd}}{\partial {t}^{nd}}-{{\varepsilon }_{rpd}}^{2}{D}_{b}^{nd}\displaystyle \frac{{\partial }^{2}{c}^{nd}}{\partial {\left({x}^{nd}\right)}^{2}}\,=\,0,\,{x}^{nd}\in \left(1,1+{l}_{b}^{nd}\right),\end{eqnarray} \tag{ 190 }$$

$$\begin{eqnarray}&&{{t}_{h}}^{2}\displaystyle \frac{\partial {c}_{h}^{nd}}{\partial {t}^{nd}}+{\lambda }_{ox}{j}^{nd}\displaystyle \frac{\partial {c}_{h}^{nd}}{\partial {z}^{nd}}\,=\,-{\left.{D}_{b}^{nd}\left(\displaystyle \frac{\partial {c}^{nd}}{\partial {x}^{nd}}\right)\right|}_{{x}^{nd}=1+{l}_{b}^{nd}},\end{eqnarray} \tag{ 191 }$$

$$\begin{eqnarray}&&{\eta }^{nd}\,=\,{\phi }_{e}^{nd}-{\phi }_{s}^{nd}+{E}_{ORR}^{eq,nd},\end{eqnarray} \tag{ 192 }$$

using dimensionless variables,

$$\begin{eqnarray}&&\begin{array}{l}{t}^{nd}=\displaystyle \frac{t}{{t}_{* }},\,{x}^{nd}=\displaystyle \frac{x}{{l}_{t}},\,{z}^{nd}=\displaystyle \frac{z}{L},\,{\eta }^{nd}=\displaystyle \frac{\eta }{b},\,{\phi }_{s}^{nd}=\displaystyle \frac{{\phi }_{s}}{b},\,{\phi }_{e}^{nd}\\ \,\,=\,\displaystyle \frac{{\phi }_{e}}{b},{j}^{nd}=\displaystyle \frac{j}{{j}_{* }},\,{c}^{nd}=\displaystyle \frac{c}{{c}_{ref}},\,{D}^{nd}=\displaystyle \frac{D}{{D}_{* }}.\end{array}\end{eqnarray} \tag{ 193 }$$

Here $L$ is the channel length. Additional care should be paid to different length scales used for ${x}^{nd}$ and ${z}^{nd}.$ In addition,

$$\begin{eqnarray}&&{t}_{* }\,=\,\displaystyle \frac{{C}_{dl}b}{2{i}_{* }},\,{j}_{* }\,=\,\displaystyle \frac{{\sigma }_{e}b}{{l}_{t}},\,{D}_{* }\,=\,\displaystyle \frac{{\sigma }_{e}b}{nF{c}_{ref}}\end{eqnarray} \tag{ 194 }$$

are the reference values for time, current density and diffusion coefficient, respectively. A dimensionless reaction penetration depth is given by,

$$\begin{eqnarray}&&{\varepsilon }_{rpd}\,=\,\sqrt{\displaystyle \frac{{\sigma }_{e}b}{2{i}_{* }{l}_{t}^{2}}}.\end{eqnarray} \tag{ 195 }$$

The characteristic time of oxygen concentration variation in the CCL and GDL is given by,

$$\begin{eqnarray}&&{t}_{c}\,=\,\sqrt{\displaystyle \frac{nF{c}_{ref}}{{C}_{dl}b}}.\end{eqnarray} \tag{ 196 }$$

The oxygen stoichiometry reads,

$$\begin{eqnarray}&&{\lambda }_{ox}\,=\,\displaystyle \frac{4Fhv{c}_{h}^{in}}{L{j}_{tot}},\end{eqnarray} \tag{ 197 }$$

where ${c}_{h}^{in}={c}_{ref}$ is the inlet oxygen concentration. The characteristic time of oxygen concentration in the channel is given by,

$$\begin{eqnarray}&&{t}_{h}\,=\,\sqrt{\displaystyle \frac{8Fh{c}_{h}^{in}{i}_{* }{l}_{t}}{{C}_{dl}{\sigma }_{e}{b}^{2}}},\end{eqnarray} \tag{ 198 }$$

and ${k}_{\sigma }$ is the ratio of electron to proton conductivity,

$$\begin{eqnarray}&&{k}_{\sigma }\,=\,\displaystyle \frac{{\sigma }_{s}}{{\sigma }_{e}}.\end{eqnarray} \tag{ 199 }$$

When analyzing the impedance response, we introduce the dimensionless impedance ${Z}^{nd},$ and the dimensionless angular frequency ${\omega }^{nd},$ as follows.

$$\begin{eqnarray}&&{Z}^{nd}=\displaystyle \frac{Z{\sigma }_{e}}{{l}_{t}},\,{\omega }^{nd}=\omega {t}_{* }.\end{eqnarray} \tag{ 200 }$$

Here $Z$ is the impedance and $\omega$ is the angular frequency.

The boundary conditions for the system ^61–66 are as follows. At the interface between the membrane and the CCL, ${x}^{nd}\,=\,0,$ the ionic-phase potential is zero and set as the reference potential, namely,

$$\begin{eqnarray}&&{\left.{\phi }_{e}^{nd}\right|}_{{x}^{nd}=0}\,=\,0,\end{eqnarray} \tag{ 201 }$$

and the electronic current is zero as electrons cannot go through the membrane, namely,

$$\begin{eqnarray}&&{\sigma }_{s}{\left.\displaystyle \frac{\partial {\phi }_{s}^{nd}}{\partial {x}^{nd}}\right|}_{{x}^{nd}=0}\,=\,0,\end{eqnarray} \tag{ 202 }$$

and the oxygen flux is zero since the oxygen cannot penetrate the membrane, namely,

$$\begin{eqnarray}&&{D}_{ox}{\left.\displaystyle \frac{\partial {c}^{nd}}{\partial {x}^{nd}}\right|}_{{x}^{nd}=0}\,=\,0.\end{eqnarray} \tag{ 203 }$$

At the CCL and the GDL interface, ${x}^{nd}\,=\,1,$ the proton current is zero as protons cannot penetrate the GDL, namely,

$$\begin{eqnarray}&&{\sigma }_{e}{\left.\displaystyle \frac{\partial {\phi }_{e}^{nd}}{\partial {x}^{nd}}\right|}_{{x}^{nd}=1}\,=\,0,\end{eqnarray} \tag{ 204 }$$

and the electronic phase potential is fixed, which is usually the independent variable controlled in EIS measurement,

$$\begin{eqnarray}&&{\left.{\phi }_{s}^{nd}\right|}_{{x}^{nd}=1}\,=\,{\phi }_{1}^{nd},\end{eqnarray} \tag{ 205 }$$

and the oxygen concentration distribution is continuous,

$$\begin{eqnarray}&&{\left.{c}^{nd}\right|}_{{x}^{nd}={1}^{-}}\,=\,{\left.{c}^{nd}\right|}_{{x}^{nd}={1}^{+}},\end{eqnarray} \tag{ 206 }$$

where ${1}^{-}$ and ${1}^{+}$ denote the left and right side of the CCL/GDL boundary, and the oxygen flux is equal on both sides of the interface,

$$\begin{eqnarray}&&{\left.{D}_{ox}^{nd}\displaystyle \frac{\partial {c}^{nd}}{\partial {x}^{nd}}\right|}_{{x}^{nd}={1}^{-}}\,=\,{\left.{D}_{b}^{nd}\displaystyle \frac{{c}^{nd}}{\partial {x}^{nd}}\right|}_{{x}^{nd}={1}^{+}}.\end{eqnarray} \tag{ 207 }$$

At the GDL and air channel interface, ${x}^{nd}=1+{l}_{b}^{nd},$ the oxygen concentration distribution is continuous,

$$\begin{eqnarray}&&{\left.{c}^{nd}\right|}_{{x}^{nd}=1+{l}_{b}^{nd}}\,=\,{c}_{h}^{nd}.\end{eqnarray} \tag{ 208 }$$

At the air channel inlet, ${z}^{nd}=0,$ the dimensionless oxygen concentration is fixed at 1,

$$\begin{eqnarray}&&{\left.{c}_{h}^{nd}\right|}_{{z}^{nd}=0}\,=\,1.\end{eqnarray} \tag{ 209 }$$

Impedance model

Impedance response can be obtained by applying a small-amplitude harmonic perturbation to steady-state conditions. Let ${\bar{\eta }}^{nd},$ ${\bar{\phi }}_{s}^{nd},$ ${\bar{\phi }}_{e}^{nd},$ ${\bar{c}}^{nd},$ ${\bar{c}}_{b}^{nd}$ and ${\bar{c}}_{h}^{nd}$ be the steady-state solutions to Eqs. 187–192 and ${\tilde{\eta }}^{{\rm{nd}}},$ ${\tilde{\phi }}_{s}^{{\rm{nd}}},$ ${\tilde{\phi }}_{e}^{{\rm{nd}}},$ ${\tilde{c}}^{nd},$ ${\tilde{c}}_{b}^{nd}$ and ${\tilde{c}}_{h}^{nd}$ be the corresponding perturbations. Performing Fourier transform to the control equations given in Eqs. 187–192 subtracting the steady-state parts, and linearizing the nonlinear terms, we get a system of linear equations for the perturbed variables,

$$\begin{eqnarray}&&\begin{array}{l}{{\varepsilon }_{rpd}}^{2}\displaystyle \frac{{\partial }^{2}{\tilde{\phi }}_{e}^{nd}}{\partial {({x}^{nd})}^{2}}\,=\,\sinh \left({\bar{\eta }}^{nd}\right){\tilde{c}}^{nd}+\left({\bar{c}}^{nd}\,\cosh ({\bar{\eta }}^{nd})+{\rm{j}}{\omega }^{nd}\right){\tilde{\eta }}^{nd},\\ \,{x}^{nd}\in \left(0,1\right),\end{array}\end{eqnarray} \tag{ 210 }$$

$$\begin{eqnarray}&&\begin{array}{l}-{{\varepsilon }_{rpd}}^{2}{k}_{\sigma }\displaystyle \frac{{\partial }^{2}{\tilde{\phi }}_{s}^{nd}}{\partial {({x}^{nd})}^{2}}\,=\,\sinh \left({\bar{\eta }}^{nd}\right){\tilde{c}}^{nd}+\left({\bar{c}}^{nd}\,\cosh \left({\bar{\eta }}^{nd}\right)+{\rm{j}}{\omega }^{nd}\right){\tilde{\eta }}^{nd},\\ \,{x}^{nd}\in \left(0,1\right),\end{array}\end{eqnarray} \tag{ 211 }$$

$$\begin{eqnarray}&&\begin{array}{l}{{\varepsilon }_{rpd}}^{2}{D}_{ox}^{nd}\displaystyle \frac{{\partial }^{2}{\tilde{c}}^{nd}}{\partial {({x}^{nd})}^{2}}\,=\,\left(\sinh \left({\bar{\eta }}^{nd}\right)+{\rm{j}}{\omega }^{nd}{{t}_{c}}^{2}\right){\tilde{c}}^{nd}+{\bar{c}}^{nd}\,\cos \,{\rm{h}}\left({\bar{\eta }}^{nd}\right){\tilde{\eta }}^{nd},\\ \,{x}^{nd}\in \left(0,1\right),\end{array}\end{eqnarray} \tag{ 212 }$$

$$\begin{eqnarray}&&{{\varepsilon }_{rpd}}^{2}{D}_{b}^{nd}\displaystyle \frac{{\partial }^{2}{\tilde{c}}^{nd}}{\partial {({x}^{nd})}^{2}}={\rm{j}}{\omega }^{nd}{{t}_{c}}^{2}{\tilde{c}}^{nd},\,{x}^{nd}\in \left(1,1+{l}_{b}^{nd}\right),\end{eqnarray} \tag{ 213 }$$

$$\begin{eqnarray}&&{\lambda }_{ox}{j}_{0}^{nd}\displaystyle \frac{\partial {\tilde{c}}_{h}^{nd}}{\partial {z}^{nd}}\,=\,-{\rm{j}}{\omega }^{nd}{{t}_{h}}^{2}{\tilde{c}}_{h}^{nd}-{\left.{D}_{b}^{nd}\displaystyle \frac{\partial {\tilde{c}}^{nd}}{\partial {x}^{nd}}\right|}_{{x}^{nd}=1+{l}_{b}^{nd}},\end{eqnarray} \tag{ 214 }$$

$$\begin{eqnarray}&&{\tilde{\eta }}^{nd}={\tilde{\phi }}_{e}^{nd}-{\tilde{\phi }}_{s}^{nd}.\end{eqnarray} \tag{ 215 }$$

The boundary conditions for the equation system Eqs. 210–215 are as follows. At the interface between the membrane and the CCL, ${x}^{nd}\,=\,0,$ where the reference potential is defined, the perturbation of ionic-phase potential is naturally zero,

$$\begin{eqnarray}&&{{\tilde{\phi }}_{e}}^{nd}\left(0\right)\,=\,0,\end{eqnarray} \tag{ 216 }$$

and according to Eqs. 202–209, we have,

$$\begin{eqnarray}&&{\left.\displaystyle \frac{\partial {{\tilde{\phi }}_{s}}^{nd}}{\partial {x}^{nd}}\right|}_{{x}^{nd}=0}\,=\,0,\end{eqnarray} \tag{ 217 }$$

$$\begin{eqnarray}&&{\left.\displaystyle \frac{\partial {\tilde{c}}^{nd}}{\partial {x}^{nd}}\right|}_{{x}^{nd}=0}\,=\,0,\end{eqnarray} \tag{ 218 }$$

$$\begin{eqnarray}&&{\left.\displaystyle \frac{\partial {{\tilde{\phi }}_{e}}^{nd}}{\partial {x}^{nd}}\right|}_{{x}^{nd}=1}\,=\,0,\end{eqnarray} \tag{ 219 }$$

$$\begin{eqnarray}&&{\left.{\tilde{\phi }}_{s}^{nd}\right|}_{{x}^{nd}=1}\,=\,{\tilde{\phi }}_{1}^{nd},\end{eqnarray} \tag{ 220 }$$

$$\begin{eqnarray}&&{\left.{\tilde{c}}^{nd}\right|}_{{x}^{nd}={1}^{-}}={\left.{\tilde{c}}^{nd}\right|}_{{x}^{nd}={1}^{+}},\end{eqnarray} \tag{ 221 }$$

$$\begin{eqnarray}&&{D}_{ox}^{nd}{\left.\displaystyle \frac{\partial {\tilde{c}}^{nd}}{\partial {x}^{nd}}\right|}_{{x}^{nd}={1}^{-}}={\left.{D}_{b}^{nd}\displaystyle \frac{\partial {\tilde{c}}^{nd}}{\partial {x}^{nd}}\right|}_{{x}^{nd}={1}^{+}},\end{eqnarray} \tag{ 222 }$$

$$\begin{eqnarray}&&{\left.{\tilde{c}}^{nd}\right|}_{{x}^{nd}=1+{l}_{b}^{nd}}={\tilde{c}}_{h}^{nd},\end{eqnarray} \tag{ 223 }$$

$$\begin{eqnarray}&&{\left.{\tilde{c}}_{h}^{nd}\right|}_{{z}^{nd}=0}=0.\end{eqnarray} \tag{ 224 }$$

Thus, the cell impedance can be obtained using ohmic law,

$$\begin{eqnarray}&&{Z}^{nd}={\left.\displaystyle \frac{{\tilde{\phi }}_{s}^{nd}}{{\tilde{j}}^{nd}}\right|}_{{x}^{nd}=1}\,=\,{\left.\displaystyle \frac{{\tilde{\phi }}_{s}^{nd}}{{k}_{\sigma }\partial {\tilde{\phi }}^{nd}/\partial {x}^{nd}}\right|}_{{x}^{nd}=1}.\end{eqnarray} \tag{ 225 }$$

The latter equality holds true as the current is contributed exclusively by electron conduction at ${x}^{nd}=1.$ If we neglect electron conduction resistance when ${k}_{\sigma }\gg 1,$ Eq. 211 is eliminated and Eq. 210 becomes

$$\begin{eqnarray}&&{{\varepsilon }_{rpd}}^{2}\displaystyle \frac{{\partial }^{2}{\tilde{\eta }}^{nd}}{\partial {({x}^{nd})}^{2}}\,=\,\sinh \left({\bar{\eta }}^{nd}\right){\tilde{c}}^{nd}+\left({\bar{c}}^{nd}\,\cosh \left({\bar{\eta }}^{nd}\right)+{\rm{j}}{\omega }^{nd}\right){\tilde{\eta }}^{nd},\end{eqnarray} \tag{ 226 }$$

and the boundary conditions Eqs. 219 and 220 are now,

$$\begin{eqnarray}&&{\left.{\tilde{\eta }}^{nd}\right|}_{{x}^{nd}=1}={\tilde{\eta }}_{1}^{nd},\,{\left.\displaystyle \frac{\partial {\tilde{\eta }}^{nd}}{\partial {x}^{nd}}\right|}_{{x}^{nd}=1}\,=\,0.\end{eqnarray} \tag{ 227 }$$

The cell impedance Eq. 225 now reads,

$$\begin{eqnarray}&&{Z}^{nd}={\left.\displaystyle \frac{{\tilde{\eta }}^{nd}}{{\tilde{j}}^{nd}}\right|}_{{x}^{nd}=0}\,=\,{\left.\displaystyle \frac{{\tilde{\eta }}^{nd}}{\displaystyle \frac{\partial {\tilde{\eta }}^{nd}}{\partial {x}^{nd}}}\right|}_{{x}^{nd}=0},\end{eqnarray} \tag{ 228 }$$

as the current is contributed exclusively by proton transport at ${x}^{nd}\,=\,0.$

PEFC enjoys a rich variety of operation conditions and cell configurations, e.g. the current density, gas stoichiometry, gas flow rate, properties of membrane electrode assembly (MEA), cell size and channel design etc. Under different circumstances, the theory expressed in Eqs. 187–192 can be simplified to different extent by neglecting one or several terms or even equations. In the following, we are to obtain analytical impedance expressions or numerically calculate the impedance for each limiting case and then to discuss its application to PEFC characterization and diagnosis. As shown in Table VI, six limiting cases are differentiated based on the transport phenomena described, and each case is further separately discussed under different reaction conditions.

Case 1:  Proton conduction in the CCL

Table VI. Definition of six limiting cases in terms of mass transport processes.

Case Number	Mass Transport in the CCL
	${{\rm{H}}}^{+}$ Conduction	${{\rm{e}}}^{-}$ Conduction	O2 Transport	O2 Transport in the GDL	O2 Transport in the Channel
1	√	x	x	x	x
2	√	√	x	x	x
3	√	x	√	x	x
4	x	x	x	√	x
5	√	x	√	√	x
6	√	x	√	√	√

This limiting case focuses on proton conduction, in absence or presence of ORR, in the CCL only, while neglecting gas transport in the channel, GDL and CCL and electron conduction in the CCL. A PEFC operating at H₂/N₂ atmosphere neatly fits this case. In order to meet this limiting case under H₂/O₂ atmosphere, the cell should be made as uniform as possible in in-plane direction, fed with a sufficiently large gas stoichiometry ${\lambda }_{ox}\gg 1,$ and operated at a current density ${j}_{0}^{nd}$ that is much smaller than the diffusion-limited current densities for oxygen transport in the CCL ${j}_{ox}^{nd}$ and GDL ${j}_{b}^{nd},$ namely, ${j}^{nd}\ll \,\min \left\{{j}_{ox}^{nd}={D}_{ox}^{nd}{c}_{1}^{nd},\,{j}_{b}^{nd}={l}_{b}^{nd}{D}_{b}^{nd}{c}_{1+{l}_{b}^{nd}}^{nd}\right\}.$ In addition, the electron conductivity ${\sigma }_{s}$ should be much larger than the proton conductivity ${\sigma }_{e},$ namely, ${k}_{\sigma }\gg 1$ or ${\sigma }_{s}\gg {\sigma }_{e},$ to justify the neglect of electron conduction. In the following, we will discuss the impedance response in these circumstances, separately.

Blocking electrode

When the cathode is fed with nitrogen and the electrode potential is adjusted to be within the double-layer region, there is no charge transfer reaction at the electrode/electrolyte interface (EEI). We are then left with proton transport in the CCL which is described using the standard de Levie model. The impedance expression of this simple case is Ref. 50

$$\begin{eqnarray}&&Z\,=\,\sqrt{\displaystyle \frac{{Z}_{loc}}{{a}_{v}{\sigma }_{e}}}\,\coth \left(\sqrt{\displaystyle \frac{{a}_{v}{l}_{t}^{2}}{{\sigma }_{e}{Z}_{loc}}}\right),\end{eqnarray} \tag{ 229 }$$

where ${a}_{v}$ is the volumetric area of the EEI, and ${Z}_{loc}={\left({\rm{j}}\omega {C}_{dl}\right)}^{-1}$ is the local impedance at the EEI.

In the limit of $\omega \to \infty ,$ Eq. 229 is asymptotic to

$$\begin{eqnarray}&&Z(\omega \to \infty )\,=\,\sqrt{\displaystyle \frac{1}{{\rm{j}}\omega {C}_{dl}{a}_{v}{\sigma }_{e}}},\end{eqnarray} \tag{ 230 }$$

which is represented by a 45° line in the Nyquist plot. In the limit of $\omega \to 0,$ Eq. 229 is asymptotic to

$$\begin{eqnarray}&&Z(\omega \to 0)\,=\,\displaystyle \frac{{l}_{t}}{3{\sigma }_{e}}-{\rm{j}}\displaystyle \frac{1}{{a}_{v}{l}_{t}{C}_{dl}\omega },\end{eqnarray} \tag{ 231 }$$

which is represented by a vertical line in the Nyquist plot. The full Nyquist plot of impedance response given by Eq. 229 is shown as the black line in Fig. 19. The proton conductivity is related with the real part of $Z(\omega \to 0)$ via

$$\begin{eqnarray}&&{\sigma }_{e}\,=\,\displaystyle \frac{{l}_{t}}{3}\displaystyle \frac{1}{Z^{\prime} (\omega \to 0)}.\end{eqnarray} \tag{ 232 }$$

By fitting the experimental impedance spectrum with Eq. 229, we can obtain ${\sigma }_{e},$ ${C}_{dl}$ and ${a}_{v}.$ Pickup et al. ¹⁷³ used this method to compare the proton conductivity of electrodes with and without ionomer in the CCL. Results show that the ionomer-containing CCL possesses a higher proton conductivity, bringing larger active area and thus better performance. Gasteiger et al. ^174,175 employed this method to measure ${\sigma }_{e}$ of the CCL with various ionomer/carbon weight ratios (I/C ratios) at different relative humidities (RHs). Using the same method, Zhang et al. ¹⁷⁶ investigated the evolution of ${\sigma }_{e}$ of PEFC during cold start.

Small current density

The assumption of fast oxygen transport in the channel, GDL and CCL leads to ${\tilde{c}}^{nd}=0$ and ${\bar{c}}^{nd}=1,$ simplifying Eq. 226 to

$$\begin{eqnarray}&&{{\varepsilon }_{rpd}}^{2}\displaystyle \frac{{\partial }^{2}{\tilde{\eta }}^{nd}}{\partial {({x}^{nd})}^{2}}\,=\,\left(\cos \,{\rm{h}}{\bar{\eta }}^{nd}+{\rm{j}}{\omega }^{nd}\right){\tilde{\eta }}^{nd}.\end{eqnarray} \tag{ 233 }$$

Furthermore, when the current density is much smaller than the characteristic current density for proton conduction, ${j}_{0}^{nd}\ll 1,$ the steady-state distribution of the overpotential, controlled by Tafel kinetics, is given by ¹⁷⁷

$$\begin{eqnarray}&&{\bar{\eta }}^{nd}\,=\,\arcsin {\rm{h}}\left(\displaystyle \frac{{{\varepsilon }_{rpd}}^{2}{\gamma }^{2}}{2}\left(1+{\tan }^{2}\left(\displaystyle \frac{\gamma }{2}\left(1-{x}^{nd}\right)\right)\right)\right),\end{eqnarray} \tag{ 234 }$$

where $\gamma$ is the solution of $\gamma \,\tan \left(\tfrac{\gamma }{2}\right)={j}^{nd}.$ Provided ${\bar{\eta }}^{nd}$ in Eq. 234, Eq. 233 becomes

$$\begin{eqnarray}&&{{\varepsilon }_{rpd}}^{2}\displaystyle \frac{{\partial }^{2}{\tilde{\eta }}^{nd}}{\partial {({x}^{nd})}^{2}}\,=\,\left({\rm{j}}{\omega }^{nd}+\sqrt{1+\displaystyle \frac{{{\varepsilon }_{rpd}}^{4}{\gamma }^{4}}{4{\cos }^{4}(\gamma \left(1-{x}^{nd}\right)/2)}}\right){\tilde{\eta }}^{nd}.\end{eqnarray} \tag{ 235 }$$

When the current density is sufficiently small, ${j}_{0}^{nd}\ll 1/{{\varepsilon }_{rpd}}^{2},$ which is realized when the voltage is close to OCV, the second term in the bracket approaches $1.$ Then, Eq. 235 closed with the boundary conditions given in Eq. 227 is analytically solved. resulting in the impedance expression of the real and imaginary parts given by ¹⁷⁸

$$\begin{eqnarray}&&{Z}^{nd}{\prime} \,=\,\displaystyle \frac{{{\varepsilon }_{rpd}}^{2}}{\sqrt{1+{({\omega }^{nd})}^{2}}}\left(\displaystyle \frac{\beta \,\sinh \left(2\beta \right)-\alpha \,\sin \left(2\alpha \right)}{\cosh \left(2\beta \right)-\,\cos \left(2\alpha \right)}\right),\end{eqnarray} \tag{ 236 }$$

$$\begin{eqnarray}&&{Z}^{nd}{\prime\prime} \,=\,-\displaystyle \frac{{{\varepsilon }_{rpd}}^{2}}{\sqrt{1+{({\omega }^{nd})}^{2}}}\left(\displaystyle \frac{\beta \,\sin \left(2\alpha \right)+\alpha \,\sinh \left(2\beta \right)}{\cosh \left(2\beta \right)-\,\cos \left(2\alpha \right)}\right),\end{eqnarray} \tag{ 237 }$$

where two coefficients are given by

$$\begin{eqnarray}&&\alpha \,=\,\sqrt{\displaystyle \frac{\sqrt{1+{({\omega }^{nd})}^{2}}-1}{2{{\varepsilon }_{rpd}}^{2}}},\,\beta \,=\,\sqrt{\displaystyle \frac{\sqrt{1+{({\omega }^{nd})}^{2}}+1}{2{{\varepsilon }_{rpd}}^{2}}}.\end{eqnarray} \tag{ 238 }$$

In the limit of $\omega \to \infty ,$ Eqs. 236 and 237 are asymptotic to $Z^{\prime} \left(\omega \to \infty \right)\,=\,-Z^{\prime\prime} \left(\omega \to \infty \right)\,=\,\tfrac{1}{\sqrt{2{\sigma }_{e}{C}_{dl}\omega }}.$ Thus, the high-frequency impedance is manifested as a 45° straight line in the Nyquist plot, associated with the proton conduction in the CCL. In the limit of $\omega \to 0,$ Eqs. 236 and 237 are asymptotic to $Z^{\prime} (\omega \to 0)\approx b/(2{i}_{* }\,{l}_{t}\,),$ and $Z^{\prime\prime} (\omega \to 0)=0.$ It implies that the impedance plot converges to a pure resistance on the real axis in the low-frequency limit. The low-frequency impedance is manifested as an arc related with the ORR in the CCL. The full Nyquist plot of impedance response is shown in Fig. 18a, and it well fits the numerical results. Two values of ${\varepsilon }_{rpd}$ are compared. When ${\varepsilon }_{rpd}$ grows from 1 to 866, meaning much faster proton conductivity, the 45° line diminishes while the arc size remains constant.

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Fitting the experimental EIS at OCV with Eqs. 236 and 237, one can obtain ${i}_{* }$ and $b$ of ORR, and, principally speaking, ${\sigma }_{e}$ in the CCL as well. However, the ORR usually obscures the 45° line corresponding to proton transport in the CCL, except under ultra-dry operation condition or for electrodes with poor proton conduction, ¹⁷⁹ making it difficult to extract the proton conductivity with high confidence. Consequently, Eqs. 236 and 237 are usually used to determine ORR parameters, such as charge-transfer resistance ¹⁸⁰ and the activation energy of ORR. ¹⁸¹

When the current density falls into the range of $1/{{\varepsilon }_{rpd}}^{2}$ ≪ ${j}_{0}^{nd}$ ≪ $\min \left\{{j}_{e}^{nd}=1,{j}_{ox}^{nd}={D}_{ox}^{nd}{c}_{1}^{nd},{j}_{b}^{nd}={l}_{b}^{nd}{D}_{b}^{nd}{c}_{1+{l}_{b}^{nd}}^{nd}\right\},$ we then neglect the unity under the square root of the second term in the bracket of Eq. 235 and replace the cos² term with 1, thus $A={{\varepsilon }_{rpd}}^{2}{\gamma }^{2}/2\,.$Then, the impedance expression becomes ^182–184

$$\begin{eqnarray}&&{Z}^{nd}{\prime} \,=\,\displaystyle \frac{{{\varepsilon }_{rpd}}^{2}}{\sqrt{\tfrac{{\gamma }^{4}{{\varepsilon }_{rpd}}^{4}}{4}+{\left({\omega }^{nd}\right)}^{2}}}\left(\displaystyle \frac{{\beta }_{* }\,\sinh \left(2{\beta }_{* }\right)-{\alpha }_{* }\,\sin \left(2{\alpha }_{* }\right)}{\cosh \left(2{\beta }_{* }\right)-\,\cos \left(2{\alpha }_{* }\right)}\right),\end{eqnarray} \tag{ 239 }$$

$$\begin{eqnarray}&&{Z}^{nd}{\prime\prime} =-\displaystyle \frac{{{\varepsilon }_{rpd}}^{2}}{\sqrt{\tfrac{{\gamma }^{4}{{\varepsilon }_{rpd}}^{4}}{4}+{\left({\omega }^{nd}\right)}^{2}}}\left(\displaystyle \frac{{\beta }_{* }\,\sin \left(2{\alpha }_{* }\right)+{\alpha }_{* }\,\sinh \left(2{\beta }_{* }\right)}{\cosh \left(2{\beta }_{* }\right)-\,\cos \left(2{\alpha }_{* }\right)}\right),\end{eqnarray} \tag{ 240 }$$

where two coefficients are

$$\begin{eqnarray}&&\begin{array}{l}{\alpha }_{* }=\sqrt{\displaystyle \frac{\sqrt{\tfrac{{\gamma }^{4}{{\varepsilon }_{rpd}}^{4}}{4}+{\left({\omega }^{nd}\right)}^{2}}-\tfrac{{\gamma }^{2}}{2}{{\varepsilon }_{rpd}}^{2}}{2{{\varepsilon }_{rpd}}^{2}}},\,{\beta }_{* }\\ \,\,=\,\sqrt{\displaystyle \frac{\sqrt{\tfrac{{\gamma }^{4}{{\varepsilon }_{rpd}}^{4}}{4}+{\left({\omega }^{nd}\right)}^{2}}+\tfrac{{\gamma }^{2}}{2}{{\varepsilon }_{rpd}}^{2}}{2{{\varepsilon }_{rpd}}^{2}}}.\end{array}\end{eqnarray} \tag{ 241 }$$

In the limit of $\omega \to \infty ,$ Eqs. 239 and 240 are asymptotic to $Z^{\prime} \left(\omega \to \infty \right)=-Z^{\prime\prime} \left(\omega \to \infty \right)=\tfrac{1}{\sqrt{2{\sigma }_{e}{C}_{dl}\omega }},$ as the same with Eqs. 236 and 237. In the limit of $\omega \to 0,$ Eqs. 239 and 240 are asymptotic to $Z^{\prime} (\omega \to 0)=b/j$ and $Z^{\prime\prime} \left(\omega \to 0\right)=0,$ implying that the impedance plot converges to a pure resistance associated with the charge transfer on the real axis in the low-frequency limit, and the charge transfer resistance monotonically decreases with increasing the cell current density. The full Nyquist plot of impedance response given by Eqs. 239 and 240 is shown in Fig. 18b, consisting of a 45° line in high-frequency range and an arc associated with the ORR in low-frequency range. When the cell current density increases, the low-frequency arc shrinks while the 45° straight line remains constant. The model developed here may well be used to interpret the impedance spectrum measured at small current density, for example, the data reported by Baker et al., ¹⁸⁵ Stamper et al. ¹⁸⁶ and Lindbergh et al. ¹⁸⁷

The proton conduction of CCL exhibits a 45° straight line in impedance spectrum, as discussed above. However, the straight line could deviates from 45° slope when considering ${\sigma }_{e}$ distribution through CCL depth, caused by non-uniform ionomer distribution. Reshetenko and Kulikovsky ¹⁸⁸ introduced a distributed electrolyte proton conductivity ${\sigma }_{e}(x)={\sigma }_{e}s(x)$ with $s\left(x\right)=\exp (-\beta x)$ into Eq. 183. Thus, in high-frequency region, Eq. 233 transfers to

$$\begin{eqnarray}&&{{\varepsilon }_{rpd}}^{2}\displaystyle \frac{\partial }{\partial {x}^{nd}}\left(s\left({x}^{nd}\right)\displaystyle \frac{\partial {\tilde{\eta }}^{nd}}{\partial {x}^{nd}}\right)\,=\,{\rm{j}}{\omega }^{nd}{\tilde{\eta }}^{nd}.\end{eqnarray} \tag{ 242 }$$

Solving Eq. 242 with the boundary conditions given in Eq. 227, they write the impedance expression in high-frequency region as

$$\begin{eqnarray}&&{Z}_{HF}^{nd}=\left(\sqrt{\displaystyle \frac{{\rm{j}}{{\varepsilon }_{rpd}}^{2}}{{\omega }^{nd}}}\right)\displaystyle \frac{{Y}_{0}\left(q\cdot \exp \left(\tfrac{\beta }{2}\right)\right){J}_{1}\left(q\right)-{Y}_{1}\left(q\right){J}_{0}\left(q\cdot \exp \left(\tfrac{\beta }{2}\right)\right)}{{Y}_{0}\left(q\right){J}_{0}\left(q\cdot \exp \left(\tfrac{\beta }{2}\right)\right)-{Y}_{0}\left(q\cdot \exp \left(\tfrac{\beta }{2}\right)\right){J}_{0}\left(q\right)},\end{eqnarray} \tag{ 243 }$$

where

$$\begin{eqnarray}&&q=\displaystyle \frac{2\sqrt{-{\rm{j}}\tfrac{{\omega }^{nd}}{{{\varepsilon }_{rpd}}^{2}}}}{\beta }.\end{eqnarray} \tag{ 244 }$$

and $J,$ $Y$ are the Bessel functions of the first and second kind, respectively. Then, they fitted Eq. 243 to the HF part of the experimental Nyquist spectra at the open–circuit conditions, ¹⁸⁹ and obtained the characteristic scale of ${\sigma }_{e}$ decay along $x.$

Case 2:  Electron and proton conduction in the CCL.

This limiting case focuses on proton and electron conduction, without or coupled with ORR, in the CCL only, while neglecting gas transport in the channel, GDL and CCL. Compared to the case 1, another transport process, namely electron conduction, comes into play in the case 2. Most often, the electron conductivity is usually one to two orders of magnitude larger than the proton conductivity in the CCL. ¹⁹⁰ However, in some occasions, e.g. Pt-free electrodes or degraded CCL due to carbon corrosion, the electron conductivity decreases greatly and electron conduction exerts a noticeable effect on the impedance spectrum, this limiting case deals with such occasions. The operation conditions for this case are nearly the same as those fulfilling the limiting case 1: a PEFC operating at H₂/N₂ atmosphere or H₂/O₂ atmosphere with a small current density, ${j}_{0}^{nd}\ll \,\min \left\{{j}_{ox}^{nd}={D}_{ox}^{nd}{c}_{1}^{nd},\,{j}_{b}^{nd}={l}_{b}^{nd}{D}_{b}^{nd}{c}_{1+{l}_{b}^{nd}}^{nd}\right\},$.

Blocking electrode

When the cathode is fed with nitrogen and the electrode potential is adjusted to be within the double-layer region, there is no charge transfer reaction at the EEI. We are then left with proton and electron transport in the CCL which can be described using transmission line (TML) model. This case corresponds to the case two in the section of "extension in physics," and the impedance expression reads ⁵⁰

$$\begin{eqnarray}&&Z\,=\,\displaystyle \frac{{l}_{t}}{{\sigma }_{e}+{\sigma }_{s}}+\displaystyle \frac{\coth \left(\sqrt{{\rm{\Theta }}}{l}_{t}\right)}{\sqrt{{\rm{\Theta }}}({\sigma }_{e}+{\sigma }_{s})}\times \left(\displaystyle \frac{{\sigma }_{e}^{2}+{\sigma }_{s}^{2}}{{\sigma }_{e}\cdot {\sigma }_{s}}+\displaystyle \frac{2}{\cosh \left(\sqrt{{\rm{\Theta }}}{l}_{t}\right)}\right),\end{eqnarray} \tag{ 245 }$$

with

$$\begin{eqnarray}&&{\rm{\Theta }}\,=\,\left(\displaystyle \frac{1}{{\sigma }_{e}}+\displaystyle \frac{1}{{\sigma }_{s}}\right)\displaystyle \frac{{a}_{v}}{{Z}_{loc}},\end{eqnarray} \tag{ 246 }$$

where ${Z}_{loc}\,=\,{\left({\rm{j}}\omega {C}_{dl}\right)}^{-1}$ is the local impedance at the EEI.

In the limit of $\omega \to \infty ,$ Eq. 245 is asymptotic to

$$\begin{eqnarray}&&Z\,=\,\displaystyle \frac{{l}_{t}}{{\sigma }_{e}+{\sigma }_{s}}+\displaystyle \frac{{\sigma }_{e}^{2}+{\sigma }_{s}^{2}}{{\left({\sigma }_{e}\cdot {\sigma }_{s}\right)}^{0.5}{\left({\sigma }_{e}+{\sigma }_{s}\right)}^{1.5}}\sqrt{\displaystyle \frac{1}{{\rm{j}}\omega {C}_{dl}{a}_{v}}},\end{eqnarray} \tag{ 247 }$$

which is represented by a 45° line in the Nyquist plot. When ${\sigma }_{s}\gg {\sigma }_{e},$ Eq. 247 simplifies to Eq. 230. In the limit of $\omega \to 0,$ Eq. 245 is asymptotic to

$$\begin{eqnarray}&&Z=\displaystyle \frac{{l}_{t}}{{\sigma }_{e}+{\sigma }_{s}}\left(1+\displaystyle \frac{{\sigma }_{e}^{2}+{\sigma }_{s}^{2}}{3{\sigma }_{e}\cdot {\sigma }_{s}}\right)+\displaystyle \frac{1}{{\rm{j}}\omega {C}_{dl}{a}_{v}{l}_{t}},\end{eqnarray} \tag{ 248 }$$

which is represented by a vertical line in the Nyquist plot. When ${\sigma }_{s}\gg {\sigma }_{e},$ Eq. 248 simplifies to Eq. 231. The length of the real part of the 45° straight line ${L}^{45^\circ }$ is obtained by subtracting Eq. 247 from Eq. 248,

$$\begin{eqnarray}&&{L}^{45^\circ }\,=\,\displaystyle \frac{{l}_{t}}{{\sigma }_{e}+{\sigma }_{s}}\displaystyle \frac{{\sigma }_{e}^{2}+{\sigma }_{s}^{2}}{3{\sigma }_{e}\cdot {\sigma }_{s}}=\displaystyle \frac{{l}_{t}}{3{\sigma }_{e}}\left(1+\displaystyle \frac{1}{{k}_{\sigma }}-\displaystyle \frac{2}{{k}_{\sigma }+1}\right),\end{eqnarray} \tag{ 249 }$$

which reaches the minimal value when the ratio of electron to proton conductivity ${k}_{\sigma }\,=\,1+\sqrt{2}.$ The full Nyquist plot of impedance response given by Eq. 245 is shown in Fig. 19. ${k}_{\sigma }$ dictates ${L}^{45^\circ }$ via Eq. 249, and ${\sigma }_{e},$ ${\sigma }_{s}$ can be extracted from high-frequency resistance via

$$\begin{eqnarray}&&{\sigma }_{e}\,=\,\displaystyle \frac{{l}_{t}}{1+{k}_{\sigma }}\displaystyle \frac{1}{{\rm{Re}}(Z(\omega \to \infty ))},\end{eqnarray} \tag{ 250 }$$

$$\begin{eqnarray}&&{\sigma }_{s}\,=\,\displaystyle \frac{{k}_{\sigma }{l}_{t}}{1+{k}_{\sigma }}\displaystyle \frac{1}{{\rm{Re}}(Z(\omega \to \infty ))}.\end{eqnarray} \tag{ 251 }$$

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In Fig. 19, three impedance spectra are compared to show the effect of ${k}_{\sigma }.$ When ${k}_{\sigma }$ reduces from infinite via 1 to 0.2, the spectrum shifts right towards and ${L}^{45^\circ }$ decreases first, and then increases much, which is consistent with the analysis above. Very recently, this model has been used to extract the electron and proton conductivity of a Fe–N–C electrode by Reshetenko et al. ¹⁹¹

Small current density

In the small current density regime, fast oxygen transport is assumed in the channel, GDL and CCL, namely, ${\tilde{c}}^{nd}=0$ and ${\bar{c}}^{nd}=1.$ We can transform Eqs. 210–211 to

$$\begin{eqnarray}&&\displaystyle \frac{{\partial }^{2}{{\tilde{\phi }}_{e}}^{nd}}{\partial {\left({x}^{nd}\right)}^{2}}\,=\,\left(\displaystyle \frac{\cosh \left({\bar{\eta }}^{nd}\right)+{\rm{j}}{\omega }^{nd}}{{{\varepsilon }_{rpd}}^{2}}\right)\left({{\tilde{\phi }}_{e}}^{nd}-{{\tilde{\phi }}_{s}}^{nd}\right),\end{eqnarray} \tag{ 252 }$$

$$\begin{eqnarray}&&-{k}_{\sigma }\displaystyle \frac{{\partial }^{2}{{\tilde{\phi }}_{s}}^{nd}}{\partial {\left({x}^{nd}\right)}^{2}}\,=\,\left(\displaystyle \frac{\cosh \left({\bar{\eta }}^{nd}\right)+{\rm{j}}{\omega }^{nd}}{{{\varepsilon }_{rpd}}^{2}}\right)\left({{\tilde{\phi }}_{e}}^{nd}-{{\tilde{\phi }}_{s}}^{nd}\right).\end{eqnarray} \tag{ 253 }$$

Furthermore, when the current density ${j}_{0}^{nd}$ is much smaller than the characteristic current density for proton (${j}_{e}^{nd}$) and electron (${j}_{s}^{nd}$) conduction in the CCL, ${j}_{0}^{nd}\ll \,\min \left\{{j}_{e}^{nd}=1,\,{j}_{s}^{nd}={k}_{\sigma }\right\},$ we can neglect the static overpotential distribution through the CCL depth, namely, ${\bar{\eta }}^{nd}\approx {\eta }_{0}^{nd}.$ Thus, Eqs. 252–253 closed with the boundary conditions Eqs. 219–220 can be analytically solved, resulting in the impedance expression given by ¹⁹²