The M element in EIS: Impedance under restricted linear diffusion
Latest updated: February 4, 2025Introduction
Many applications involve electrochemical reactions taking place on electrodes consisting of a thin layer, usually solid, deposited on an electronically conductive substrate. Examples include electrochemichromism, electroinsertion and grafted polymer films. Each of these reactions involves material transport in a finite-dimensional space (the host material film), coupled with electronic and/or ionic charge transfer at the film boundaries. It is assumed that the host material behaves like an electronic conductor of constant thickness, that there is no phase change during the reaction and that the substrate is impermeable to the inserted species $\text M$ [1].
The reaction of inserting a monovalent $\text M^+$ cation into an electronically conductive host structure, for example when discharging the positive electrode of a battery or recharging its negative electrode, is written as:
\begin{equation}\text M^+ + \langle\;\;\rangle + \text e^- \overset{K_{\text r}}{\underset{K_{\text o}}{\longleftrightarrow}} \langle \text M \rangle \label{II.6.1} \tag{1}\end{equation}
where the notation $\langle\;\;\rangle$ indicates a free site for insertion and $\langle \text M \rangle$, the inserted species in the host material. The $\text M^+$ cation is present in the electrolyte solution (Fig. 1).
The principle of electrode impedance calculation is shown in Fig. 2.
Figure 1: Schematic diagram of the insertion reaction $\text M^+ + \langle\;\;\rangle + \text e^- \overset{K_{\text r}}{\underset{K_{\text o}}{\longleftrightarrow}} \langle \text M \rangle$.
Figure 2: Flow diagram for calculating the faradaic impedance and electrode impedance of an electrochemical reaction involving only species in solution or host material. $s = j 2 π f$.
If, due to faster transport of material in the electrolyte, the interfacial concentration $M^+(0,t) $ in metal cation remains close to that of the solution $M^{+*}$, the rate of reaction is given by:
\begin{equation} v(t) = K_{\text r}(t) M^{+*}\langle\;\;\rangle(0,t) – K_{\text o}(t) \langle M \rangle(0,t) \tag{2} \end{equation}
With
\begin{equation} K_{\text o}(t) = K_{\text o} \exp(\alpha_{\text o} f_\text N E(t)), K_{\text r}(t) = K_{\text r} \exp(- \alpha_{\text r} f_\text N E (t)), \alpha_{\text o} + \alpha_{\text r} = 1 \tag{3} \end{equation}
If we assume that there are no interactions between the inserted species and the host material or between the inserted atoms of $\text M$, and that the insertion isotherm is Langmuir.
The system behavior is described by the equations:
\begin{equation} i(t) =i_\text f(t) + C_\text{dl} \frac{\text d E(t)}{\text d t},\;i_\text f(t) = – F\, v(t) \tag{4} \end{equation}
The concentration of species $\text M$ in the host material is a solution of the Fick’s second law:
\begin{equation} \frac{\partial\langle M \rangle(x,t)}{\partial t}= D_{\langle \text M \rangle} \frac{\partial^2\langle M \rangle(x,t)}{\partial x^2} \tag{5} \end{equation}
with boundary conditions, using Fick’s first law:
- at external interface ($x = 0$) where the electron transfer reaction takes place, $\forall\; t \geq 0 $:
\begin{equation} J_{\langle \text M \rangle}(0,t) = – D_{\langle \text M \rangle}\frac{\partial\langle M \rangle(x,t)}{\partial x}|_{x=0} = -\frac{i_\text f(t)}{F} \tag{6} \end{equation}
- at internal interface ($x = l$), since the substrate is impermeable to the species $\text M$, $\forall\; t \geq 0$:
\begin{equation} J_{\langle \text M \rangle}(l,t) = 0 \tag{7} \end{equation}
Steady-state
Equation (5) becomes in steady-state:
\begin{equation} D_{\langle \text M \rangle} \frac{\text d^2 \langle M \rangle(x)}{\text d x^2} = 0 \tag{8} \end{equation}
The boundary conditions (6) and (7) are those of linear diffusion with an impermeable boundary. Integration of (8), given (6) and (7), leads to a horizontal concentration profile of the species inserted in the host material:
\begin{equation} \langle M \rangle(x) = \langle M \rangle,\; 0 \leq x \leq l \tag{9} \end{equation}
From (6) we then derive:
\begin{equation} i_\text f= – F\, v = 0 \tag{10} \end{equation}
The steady-state current density is zero, whatever the value of the electrode potential, and all electrode potentials are steady-state equilibrium potentials. From (2) and (10) we deduce:
\begin{equation} \frac{\langle M \rangle}{\langle\;\;\rangle} = \frac{K_{\text r} M^{+*}}{K_{\text o}} = \frac{K_{\text r} M^{+*}}{K_{\text o}} \exp (- f_\text N E) \tag{11} \end{equation}
Noting $\langle M \rangle_{\text{max}}$ is the maximum number of insertion sites in the host material per unit volume, $y_{\langle \text M \rangle}= {\langle M \rangle}/{\langle M \rangle_{\text{max}}}$ the filling level of these sites by the $\text M$ species and $y_{\langle\; \;\rangle} = 1 – y_{\langle \text M \rangle}$ the level of empty insertion sites, we deduce the expression of the insertion isotherm and the value of the steady-state insertion level:
\begin{equation} \frac{y_{\langle \text M \rangle}}{1 – y_{\langle \text M \rangle}} = \frac{K_{\text r} M^{+*}}{K_{\text o}} \exp (-f_\text N E) \Rightarrow y_{\langle \text M \rangle}= \frac{(K_{\text r} M^{+*}/K_{\text o}) \exp (- f_\text N E)}{ 1 + (K_{\text r} M^{+*}/K_{\text o}) \exp (- f_\text N E)}, y_{\langle\; \;\rangle} = 1 – y_{\langle \text M \rangle} \tag{12} \end{equation}
The evolution of the $\text M$ species insertion level with electrode potential is shown in Fig 3.
Figure 3: Animation. Change in $\text M$ species insertion level with steady-state electrode potential. Case of the positive electrode of a discharging battery, or the negative electrode of a recharging battery.
Electrochemical impedance spectroscopy
Insertion reaction impedance
The kinetic equations describing the dynamic behavior of the insertion reaction are given in Eqs. (2) – (7).
In Eq. (4), given (2), the faradaic current density is a function of time through the electrode potential $E$ and the concentration of inserted species $\langle M \rangle$.
The first-order Taylor series expansion of the current density is written:
\begin{equation} \Delta i_\text f(t) =\frac{\partial i_\text f}{\partial E}\,\Delta E(t) + \frac{\partial i_\text f}{\partial\langle M \rangle}\, \Delta \langle M \rangle(0,t) \Rightarrow Z_{\text f}(s) = \frac{\Delta E(s)}{\Delta i_\text f(s)} = R_{\text{ct}} + Z_{\langle \text M \rangle}(s) \;\text{with}\; s = j 2 π f \tag{13} \end{equation}
Partial derivatives are calculated from Eqs. (2) and (4), then by replacing the insertion level with its steady-state value. For example it is obtained:
\begin{equation} \frac{\partial i_\text f}{\partial E} = f_\text N F (\alpha_\text o K_\text o \langle M \rangle + (\alpha_\text r K_\text r ( \langle M \rangle_\text{max} – \langle M \rangle) M^{+*})= \frac{ f_\text N \,F\,K_{\text o}\, K_{\text r}\, M^{+*}\,\langle M \rangle_{\text{max}}}{K_{\text o} + K_{\text r}\, M^{+*}} \tag{14} \end{equation}
The faradaic impedance $Z_{\text f}(s)$, obtained after Laplace transformation of Eq. (13) then division by $\Delta i_\text f(s)$ is the sum of the transfer resistance and the impedance of concentration of the species inserted with:
\begin{equation} R_{\text{ct}} = \frac{1}{\partial i_\text f/\partial_E } = \frac{K_{\text o} + K_{\text r}\, M^{+*}}{ f_\text N \,F\,K_{\text o}\, K_{\text r}\, M^{+*}\,\langle M \rangle_{\text{max}}} \tag{15} \end{equation}
\begin{equation}
Z_{\langle \text M \rangle}(s) = – \frac{\partial i_\text f/\partial\langle M \rangle }{\partial i_\text f/\partial_E }\,\frac{\Delta \langle M \rangle(0,s)}{\Delta i_\text f(s)} = – R_{\text{ct}}\, F\,(K_{\text o} + K_{\text r}\, M^{+*}) \,\frac{\Delta \langle M \rangle(0,s)}{\Delta i_\text f(s)}\label{(III.5.14)} \tag{16}
\end{equation}
The diffusion equation (5) is linear and the concentration of the inserted species can be replaced by its variation with respect to the initial steady-state:
\begin{equation} \frac{\partial \Delta \langle M \rangle(x,t)}{\partial t} = D_{\langle \text M \rangle}\, \frac{\partial^2 \Delta \langle M \rangle(x,t)}{\partial x^2}\tag{17}\end{equation}
with boundary conditions:
- at external interface:
\begin{equation} \Delta J_{\langle \text M \rangle}(0,t) = – D_{\langle \text M \rangle}\,\left. \partial_x\Delta{\langle \text M \rangle}(x,t)\right|_{x=0} = \Delta v(t) = – \frac{\Delta i_\text f(t)}{F}\tag{18} \end{equation}
- at internal interface:
\begin{equation}\Delta J_{\langle \text M \rangle}(L,t) = – D_{\langle \text M \rangle}\, \left. \partial_x\Delta {\langle \text M \rangle}(x,t)\right|_{x=L} = 0 \tag{19} \end{equation}
Eqs. (17)-(19) transformed in the Laplace plane lead to (see Appendix):
\begin{equation} \Delta \langle M \rangle(0,s) = M_{\langle \text M \rangle}(s)\, \Delta J_{\langle \text M \rangle}(0,s) = – \frac{M_{\langle \text M \rangle}(s) \Delta i_\text f(s)}{F} \tag{20} \end{equation}
where $M_{\langle \text M \rangle}(s)$ is the restricted diffusion operator:
\begin{equation} M_{\langle \text M \rangle}(s) = \frac{1}{m_{\langle \text M \rangle}}\,\frac{\coth \sqrt{\tau_{\text d\langle \text M \rangle}\,s}}{\sqrt{\tau_{\text d\langle \text M \rangle}\,s}},\;\tau_{\text d\langle \text M \rangle}= \frac{L^2}{D_{\langle \text M \rangle}},\; m_{\langle \text M \rangle} = \frac{D_{\langle \text M \rangle}}{L} \tag{21}\end{equation}
By dividing (20) by $\Delta i_\text f(s)$, we get:
\begin{equation} \frac{\Delta {\langle M \rangle}(0,s)}{\Delta i_\text f(s)} = – \frac{M_{\langle \text M \rangle}(s)}{F} \tag{22} \end{equation}
and the expression for the concentration impedance of the inserted species is given, according to (16), by:
\begin{equation}Z_{\langle \text M \rangle}(s) =R_{\langle \text M \rangle}\, \frac{\coth \sqrt{\tau_{\text d\langle \text M \rangle}\,s}}{\sqrt{\tau_{\text d\langle \text M \rangle}\,s}},\;R_{\langle \text M \rangle} = \frac{R_{\text{ct}}\, (K_{\text o} + K_{\text r}\, M^{+*})}{m_{\langle \text M \rangle}}\tag{23} \end{equation}
by noting $R_{\langle \text M \rangle}$ the resistance of the concentration resistance of species $\text M$.
Insertion reaction impedance diagrams
The faradaic impedance of the insertion reaction is written,
\begin{equation} Z_{\text f}(s) = R_{\text{ct}} + R_{\langle \text M \rangle}\,\frac{\coth \sqrt{\tau_{\text d\langle \text M \rangle}\,s}}{\sqrt{\tau_{\text d\langle \text M \rangle}\,s}} \tag{24} \end{equation}
The electrode impedance is obtained by taking into account the double layer capacitance connected in parallel with the faradaic impedance and its electrical circuit is shown in Fig. 4. It corresponds to the Randles electrical circuit in which the Warburg impedance is replaced by a restricted linear diffusion impedance.
Figure 4: Equivalent electrical circuit for the impedance of the insertion reaction when the interfacial variation of the M$^+$ cation concentration in solution is negligible. C1 = C$_\text{dl}$, R1 = R$_\text{ct}$, M1 = $Z_{\langle \text M \rangle}$.
Change in inserted species concentration profile with electrode impedance measurement frequency is shown in Fig. 6. The behavior of the restricted linear diffusion impedance is equivalent to the Warburg impedance at high frequencies and to a series RC circuit at low frequencies.
The low-frequency limit of the real part is finite and equal to the insertion resistance given by:
\begin{equation} \lim_{f \rightarrow 0} \textrm{Re}\;Z_{\langle \text M \rangle}(s) = {R_{\langle \text M \rangle}}/{3}\tag{25} \end{equation}
Transfer and insertion resistances vary with electrode potential. The shape of the faradaic impedance diagram and that of the electrode impedance depend on the ratio $R_{\text{ct}}/R_{\langle \text M \rangle}$.
The various possible forms of impedance graph for the circuit shown in Fig. 4 are presented in Fig. 5, where impedances are normalized according to:
\begin{equation}Z_{\text f}^* = {Z_{\text f}}/({R_{\text{ct}} + {R_{\langle \text M \rangle}}/{3})},\;Z^* = {Z}/{(R_{\text{ct}} + {R_{\langle \text M \rangle}}/{3})}\tag{26} \end{equation}
The linear part making an angle of $-\pi/4$ with the axis of the reals (Warburg straight line) is observable on the graphs in Fig. 5 only when the value of the double-layer capacitance is sufficiently low and when $ R_{\langle \text M \rangle}/{3}$ is not negligible compared to $R_\text{ct}$.
At low frequencies the impedance $Z_{\langle \text M \rangle}$ is that of an $R_\text{lf}$+$C_\text{lf}$ circuit with $R_\text{lf}=R_\text{d}/3$ and $C_\text{lf}=\tau_\text d/R_\text d$.
Figure 5: Impedance diagram array for the Randles circuit with restricted diffusion (Fig. 4). Green points: $u_\text{c1}=3.88$, black points: $u_\text{c2}=1/T$. $u=\tau_\text d \omega$, $\rho={R_{\text{ct}}}/{R_{\text d}},\; T=\tau_{\text f}/\tau_{\text d},\;\tau_{\text f}=R_{\text{ct}}\,C_{\text{dl}}$ [2].
Figure 6: Animation. Change in inserted species concentration profile with electrode impedance measurement frequency [3,4]. a) Ultra low frequency, b) low frequency, c) middle frequency, d) high frequency, e) very high frequency. For ultra low frequencies, the concentration profiles of inserted species are horizontal.
Conclusion
The impedance of element M of ZSim and ZFit is given by the relation (22). The impedances of Ma and Mg elements of ZSim and ZFit are given in [5].
Complications can, of course, arise in all the electrochemical reactions that can take place at battery electrodes.
- Geometry of the host material, which can be spherical or cylindrical [2],
- Influence of particle size distribution on insertion processes in composite electrodes [6],
- Diffusion-trapping impedance under restricted linear diffusion [7]
- Indirect, (two- step) insertion reaction [8]:
\begin{equation} \text M^+ + \text s+ \text e^- \overset{K_{\text r}}{\underset{K_{\text o}}{\longleftrightarrow}} \text{M,s} \tag{27} \end{equation}
\begin{equation} \text{M,s} + \langle\;\;\rangle \overset{k_{\text a}}{\underset{k_{\text d}}{\longleftrightarrow}} \langle \text M \rangle+ \text s \tag{28} \end{equation}
Figure 7: Equivalent circuit for the electrode impedance related to the indirect (two-step) insertion reaction. C1 = C$_\text{dl}$, R1 = R$_\text{ct}$, R2 = R$_\text{ab}$, C2 = C$_\text{ads}$, M3 = $Z_{\langle \text M \rangle}$ [8].
Appendix
The diffusion equation (17) is written
\begin{equation} y′′(x) – a y(x) = 0,\; \text{ with } y(x) = \Delta \langle M \rangle(x,s), \;\text{ and }\;a = s/D_{\langle \text M \rangle} \tag{29} \end{equation}
This is an ordinary second-order differential equation without a second member. Its general solution is:
\begin{equation} y(x) = A\, \exp (\sqrt a\, x) + B\, \exp (- \sqrt a\, x) \tag{30} \end{equation}
with
\begin{equation} y(0)= A+B\tag{31} \end{equation}
The expressions of the constants $A$ and $B$ depend on the boundary conditions. The boundary condition at $x = L$ refers to the flux of the species $\langle \text M \rangle$ and we have:
\begin{equation} J_{\langle \text M \rangle}(L,t) = 0 \tag{32} \end{equation}
which means that the concentration profile of the species $\langle \text M \rangle$ is horizontal for $x = L$. Therefore
\begin{equation} \Delta J_{{\langle \text M \rangle}}(L,s) = – \left. \frac{D_{{\langle \text M \rangle}}\, \text d y(x)}{ \text d x}\right|_{x=L} = – D_{{\langle \text M \rangle}} \sqrt a\,(A\, \exp (\sqrt a\,L) – B\, \exp (- \sqrt a\, L)) = 0 \tag{33} \end{equation}
and:
\begin{equation} A\, \exp (\sqrt a\, L) – B\, \exp (- \sqrt a\, L) = 0 \tag{34} \end{equation}
From (31) and (34) it is obtain:
\begin{equation} A = \frac{y(0)}{1 + \exp (2\, \sqrt a\, L)},\;B = – \frac{y(0) \exp (2\, \sqrt a\, L)}{1 + \exp (2\, \sqrt a\, L)} \tag{35} \end{equation}
Using these expressions in (30) it is obtain:
\begin{equation} y(x) = \frac{y(0)\, \left(\exp (\sqrt a\, x) + \exp (2\, \sqrt a\, L) \exp (\sqrt a\, x) \right) } { 1 + \exp (2\, \sqrt a\, L)} \tag{36} \end{equation}
The variation in interfacial flux $\Delta J_{\text Xi}(0,s)$ is written:
\begin{equation} \Delta J_{{\langle \text M \rangle}}(0,s) = – D_{{\langle \text M \rangle}} \sqrt a\, y(0) \, \frac{1 – \exp (2\, \sqrt a\, L) }{1 + \exp (2\, \sqrt a\, L) } = y(0)\, \frac{D_{{\langle \text M \rangle}}}{L}\,\sqrt{a\,L^2}\, \tanh\sqrt{a\,L^2} \tag{37} \end{equation}
or taking into account (29):
\begin{equation} \Delta J_{{\langle \text M \rangle}} (0,s) = \Delta {\langle M \rangle}(0,s)\,\frac{D_{\langle \text M \rangle}}{L}\, \sqrt{\frac{s L^2}{D_{\langle \text M \rangle}}} \tanh{\sqrt{a\, L^2}} \tag{38} \end{equation}
which can also be written as:
\begin{equation} \Delta \langle M \rangle (0,s) = M_{\langle \text M \rangle}(s) \Delta J _{\langle \text M \rangle}(0,s) \tag{39} \end{equation}
with:
\begin{equation} M_{\langle \text M \rangle}(s) = \frac{1}{m_{\langle \text M \rangle}}\,
\frac{\coth \sqrt{\tau_{\langle \text M \rangle}\,s}}{\sqrt{\tau_{\langle \text M \rangle}\,s}},\;
m_{\langle \text M \rangle} = \frac{D_{\langle \text M \rangle}}{L},\;\tau_{\langle \text M \rangle} = \frac{L^2}{D_{\langle \text M \rangle}} \tag{40} \end{equation}
References
[1] J.-P. Diard, B. Le Gorrec, and C. Montella, Cinétique électrochimique. Paris : Hermann, 1996.
[2] Handbook of EIS – Diffusion impedances, DOI : 10.13140/RG.2.2.27472.33288.
[3] C. Montella, J.-P. Diard, and B. Le Gorrec, Exercices de cinétique électrochimique. II. Méthode d’impédance. Paris : Hermann, 2005.
[4] Claude Montella, Jean-Paul Diard (2017), “Concentration Profiles in Electrochemical Impedance Spectroscopy (EIS)” Wolfram Demonstrations Project. demonstrations.wolfram.com/ConcentrationProfilesInElectrochemicalImpedanceSpectroscopyE/
[5] Application note #61. How to interpret lower frequencies impedance in batteries ? http ://www.bio-logic.net/en/application-notes/.
[6] J.-P. Diard, B. Le Gorrec, and C. Montella, “Influence of particle size distribution on insertion processes in composite electrodes. potential step and eis theory : Part i. linear diffusion,” Journal of Electroanalytical Chemistry, vol. 499, no. 1, pp. 67–77, 2001.
[7] J.-P. Diard and C. Montella, “Diffusion-trapping impedance under restricted linear diffusion,” 2003. Electroanal. Chem., vol. 557, pp. 19–36, 2003.
[8] C. Montella, “EIS study of hydrogen insertion under restricted diffusion conditions. I. Twostep insertion reaction,” J. Electroanal. Chem., vol. 497, pp. 3–17, 2001.
