# Mapping kinetic limitations in porous electrodes using impedance spectroscopy: a route to optimize electrochemical characterization

Justin Bouvet, Eric Woillez, Marion Chandesris

► To cite this version:

Justin Bouvet, Eric Woillez, Marion Chandesris. Mapping kinetic limitations in porous electrodes using impedance spectroscopy: a route to optimize electrochemical characterization. Electrochimica Acta, 2025, 538, pp.146924. 10.1016/j.electacta.2025.146924. cea-05200720

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Submitted on 5 Aug 2025

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# Mapping Kinetic Limitations in Porous Electrodes using Impedance Spectroscopy : a Route to Optimize Electrochemical Characterization

Justin Bouvet¹, Eric Woillez¹, Marion Chanderis¹,*

¹ University Grenoble Alpes, CEA-Liten, F-38054 Grenoble, France
* corresponding author: [marion.chandesris@cea.fr](mailto:marion.chandesris@cea.fr)

## Abstract

Porous electrode models of lithium-ion batteries (LIB) are becoming increasingly popular, while the proper determination of the material and electrode properties remains challenging and costly. In this work, we develop a methodology based on the conventional Electrochemical Impedance Spectroscopy (EIS) to extract most of the kinetics model parameters from a limited number of experiments. Building on our recent work [Woillez & Chandesris 2023], where we derived the EIS analytical solution of the linearized P2D-model, our methodology is based on the careful comparison of the various non-dimensional numbers and characteristics frequencies to discriminate the dominant mechanisms, especially in the low-frequency regime. Applied to original EIS measurements conducted in symmetric coin cell configuration on NMC 811 and graphite electrodes with standard loading at several temperatures, we show that electrolyte properties can be extracted from the NMC electrode measurements. While lithium diffusion in NMC is not accessible with the standard electrode loading and chosen frequency range, lithium diffusion in graphite can be measured, with increasing confidence at high temperature. We further provide experimental recommendations on cell design and operating conditions to improve the discrimination of low frequency phenomena.
# 1. Introduction

The advent of portable consumer electronics in the 1970s has boosted the research on lithium batteries, leading in 1991 to the first commercialization by Sony and Asahi Kasei of a rechargeable lithium-ion battery (LIB) with graphite at the negative electrode and LiCoO2 at the positive electrode [1] [2]. With their high energy density, long cycle life, and minimal self-discharge rates, LiBs have become the preeminent energy storage technology, gaining traction in EVs and considered as a technology of interest for stationary energy storage [3]. Predicting battery performances as a function of its components properties remains a major challenge to accelerate the deployment of these technologies but remains complex, as LiBs are inherently multiscale and multi-physics systems [4]. To account for the interplay between the different physics, mainly electrochemistry and charge and species transport at both electrode and particle levels, the so-called pseudo-2D (P2D) model has become the reference to predict the electrochemical performances of LiBs. Developed by Newman, Doyle and Fuller in the 90's [5] [6] [7] [8], its success comes from its ability to include all the leading order transport phenomena and electrochemical reactions in the cell, while remaining simple enough to be numerically simulated with very few computational power. Another of its strengths comes from its ability to link the electrochemical performances of the cell to the physical and geometrical properties of its components. However, determining these properties (16 physical and geometrical parameters in the simplest version of the P2D model: 10 parameters per electrode and 6 for the electrolyte [9]), remains a complex task, given that the various physical mechanisms are strongly coupled and might evolve with aging.

Electrochemical impedance spectroscopy (EIS) is extensively used to determine these transport and electrochemical properties, because measurements are made over a wide frequency range, which makes it easier to separate the different physical phenomena according to their characteristic timescales [10] [11] [12]. Typically, high frequency phenomena ($f>1$Hz) are related to solid-electrolyte interface kinetics and electrolyte conductivity, while low frequency phenomena ($f<1$Hz) are mainly related to diffusion of lithium in both the electrolyte and the active particles. Interpretation of these spectra with the help of physical models allows the determination of the different properties. In the high-frequency domain, the transmission line model (TLM) first introduced by de Levie [13] is still the reference model to interpret impedance spectra. For a long time, the low frequency domain was interpreted with the so-called "Warburg expressions" [14] that correspond to a simplified version of the P2D model considering either electrolyte diffusion in the pores or solid diffusion in the active particles, but not the coupling between both [15]. This approximation has been overcome more recently with the derivation of an equivalent electric circuit based on a discrete formulation of the P2D model
[16] [17] [18] . While this upgraded “TLM” model requires a numerical resolution contrary to the analytical formula, it is fully equivalent to the P2D model and has the advantage to reduce the P2D complexity to a few numbers of electrical elements that can be related to physical transport parameters [19]. In particular, the interplay between diffusive phenomena can be more easily studied and the effective electrolyte diffusion coefficient can be extracted from the low-frequency impedances of NMC electrodes [20] [21].

Another route has been initiated recently, when the analytical resolution of the complete P2D set of equations has been achieved by Sikha & White [22] [23] as well as Huang et al. [24] [25]. It provides a rather complex expression but still a comprehensive theoretical framework to interpret the impedance spectra while accounting for all the different coupled phenomena present in the P2D model, including coupled diffusion in both the electrolyte and the active particles. Even mastering the exact analytical solution of the P2D, the different asymptotic behaviours where one transport mechanism gives the overwhelming signature in the impedance spectrum, are difficult to derive. A possible systematic workflow to fit impedance spectra depending on their transport limiting regime has been proposed recently [26], as well as a fast numerical method to solve numerically the P2D model in the frequency domain [27].

Following a similar theoretical framework, we have recently shown [9] that the impedance expression corresponding to the P2D electrode model is fully characterized by only three non-dimensional numbers, three characteristic frequencies, and one resistance. Compared to equivalent electric circuits [17], the use of non-dimensional numbers allows a classification of the low frequency EIS phenomenology depending on the porous electrode kinetic limitations. This need to introduce non-dimensional numbers, to classify the various limitations as a function of component properties and geometric parameters, takes importance in the battery modelling community [28] [29] [30] [31] and seems crucial to ease comparisons between different technological solutions.

This is theory, but in practice, how many independent parameters can be extracted from a given experimental impedance spectra using the analytical resolution of the complete P2D model? The reduction from the initial 16 physical and geometrical parameters of the P2D model to 7 independent parameters is a crucial step to weight the contribution of the different physical phenomena and classify the different EIS phenomenologies. Nevertheless, for a given system, the electrode design (thickness, porosity, tortuosity, active material particle size), physical properties of the components, and operating conditions of the experiment (temperature, state of lithiation) lead to conditions where it might be not possible to extract the 7 parameters at once. This is particularly the case if the physical phenomena have close characteristic times and the contribution of one is dominant. The challenge then is to anticipate the values of these
7 parameters, in order to anticipate the measurement conditions and electrode design to
extract maximum information from the impedance measurements.

The aim of this work is to illustrate this approach using original EIS measurements conducted
in symmetric coin cell configuration on both $LiNi_{0.8}Mn_{0.1}Co_{0.1}O_2$ (NMC 811) and graphite
electrodes at several states of lithiation. The electrodes have a typical commercial design, with
a relatively high loading, low porosity and high active material content. The measurements are
performed at different temperatures to highlight its impact on the different limitations. The
physical parameters are first extracted using the complete analytical solution of the P2D model,
while discussing the validity range of the methodology. For this type of electrode design, the
parameters extraction can only be done over a limited range of experimental conditions
compared to electrochemical experiments dedicated to the measurement of one particular
parameter. The scope of the present work is not to focus on one transport mechanism, but to
show how to extract a maximum of information from a single electrode design. By switching
the measurement conditions, we show that the electrode enters different regimes where one
or the other transport mechanism are alternatingly dominant. It is also shown that the simplified
expressions of the P2D, in the form of analytical Warburg expressions, can be used to extract
the physical parameters at high frequency but fail to capture the diffusion parameters at low
frequency. Finally, the classification of porous electrodes based on the non-dimensional
numbers is detailed for the experimental results of this paper and extended to other electrode
designs that could be realized in experiments.

**2. Methods**

**2.1 Experiments**

**2.1.1 Electrodes**

The NMC-based positive electrode contains 96 wt% of $Li-Ni_{0.8}Mn_{0.1}Co_{0.1}O_2$ (mean particle radius of 5,4 µm), 1.5 wt% of super P, 0.5wt% of VGCF and 2 wt% of PolyVinyliDene Fluoride (PVDF) coated on an aluminum foil. The graphite-based negative electrode is made of 96.5 wt% of graphite (mean particle radius of 6.0 µm) with 1.3 wt% of CMC (sodium carboxymethyl cellulose), 0.5 wt% of super P and 1.7 wt% of latex coated on a copper foil. The electrodes are manufactured considering a loading of 22.92 $mg cm^{-2}$ for the positive and 14.13 $mg cm^{-2}$ for the negative with a porosity of 27% and 28%, respectively. The porosity values are estimated using mass and thickness measurements performed on pristine electrodes while knowing the density of the coating.
### 2.1.2 Cell design and impedance measurements

Electrodes are first mounted in half-cell configuration to perform the formation step, reach the desired state of lithiation and measure the corresponding OCV slope. These half-cells are manufactured using the studied electrode, with a 14mm diameter (positive or negative) paired with a lithium foil counter electrode of 16 mm diameter. They are assembled in an argon-filled glove box using a CR2032-type coin cell format. The separator is a 20 µm thick tri-layer (PP/PE/PP) Celgard 2320 with a porosity $ε = 0.4$ and a tortuosity $τ ≈ 2.6$. The electrolyte consists of 1 M of lithium hexafluorophosphate (LiPF6) in 1:1:1 weight proportion of ethylene carbonate (EC), ethyl methyl carbonate (EMC), and dimethyl carbonate (DMC). The formation step was performed at 45°C and consisted of a C/20 cycle between 4.3V and 2.7V for the NMC and 10mV and 1V for the graphite with a floating until the current drops below C/50 at 4.3V for the NMC and 10mV for graphite electrodes respectively.

At the end of this formation step, the cells are set to a potential of interest called $U_{target}$. The OCV slope at this potential, is measured by first setting the cell potential to $U_{target} - 5mV$ and then $U_{target} + 5mV$, before setting back the cell to $U_{target}$. For this step, 3 hours floating period followed by 1 hour rest are used at each tested potential to ensure good quality of the slope measurement. The targeted potentials for the NMC electrodes are 4.25V and 3.85V, corresponding roughly to $x_{Li} ≈ 0.16$ and $x_{Li} ≈ 0.52$, whereas for the graphite electrodes, they are set to 300mV, 95mV and 15mV, corresponding roughly to $x_{Li} ≈ 0.04$, $x_{Li} ≈ 0.5$ and $x_{Li} ≈ 0.99$ respectively. The graphite potentials were selected to avoid being on a diphasic plateau, while testing extremal lithiation states. The cell potential formation curves, as well as the potential vs stoichiometry increments used to determine the OCV slope are presented in Appendix E.

The half-cells are then disassembled to retrieve the electrodes at these potentials of interest. The electrodes are rinsed twice with DMC to remove any potential LiPF₆ salt remaining of their surface. Then electrodes at the same potential are paired for the manufacturing of symmetrical coin cells containing the same separator and electrolyte as used for the formation process. The electrochemical impedance spectra (EIS) measurements of the symmetrical cells are conducted in a climatic chamber using a Biologic VMP3 system. The tested temperatures range from -5°C to 45°C in the following sequence 25°C, 45°C, 35°C, 25°C, 15°C, 5°C, 0°C, -5°C, 25°C. After each temperature change, the cells are rested for at least 2h before carrying out the EIS measurement to ensure thermal stability. Potentiostatic EIS measurements were performed between 200 kHz and 2 mHz with a 10mV amplitude. The low frequency limit of the study has been set to 2mHz in order to keep a reasonable measuring time of roughly 3h for
each impedance spectra. Two cells were assembled at each stoichiometry for the NMC and graphite electrodes to check the reproducibility.

## 2.2 Interpretation of impedance spectra

The present section gives a summary of the previous work done in [9]. It is intended as a guideline for the interpretation of experimental impedance spectra. The analytical theory of porous electrode impedance is briefly recalled in Appendix B. The impedance is typically composed of a high-frequency domain and a low-frequency domain, which relative sizes are quantified by the three non-dimensional numbers and the three characteristic frequencies given in the **Table 1**. The parameters used in the P2D model are precisely defined and their values in the present work are gathered in Appendix A.

<table><thead><tr><th></th><th>Solid diffusion</th><th>Electrolyte diffusion</th><th>Electrochemical (fast) kinetics</th></tr></thead><tbody><tr><th>Characteristic frequencies</th><td>$f_s = \frac{D_s}{r^2}$</td><td>$f_{el} = 2\alpha_t \left(1 + \frac{\partial \ln \gamma}{\partial \ln c}\right) \frac{j_0 A t^+(1-t^+)}{2\pi S L F c_0 \varepsilon}$</td><td>$f_{capa} = \frac{F j_0}{2\pi R T C_{dbl}}$</td></tr><tr><th>Non-dimensional numbers</th><td>$N_s = \frac{j_0 r |\phi'_{OCV}|}{RT D_s c_s^{max}}$</td><td>$N_{el} = 1 + \frac{1-t^+}{\alpha_t t^+}$</td><td>$N_\sigma = \frac{\lambda}{L}$</td></tr><tr><th>Characteristic electrode resistance</th><td></td><td>$Z = \frac{\lambda}{\sigma_t^{\text{eff}}} [\Omega \cdot m^2]$</td><td></td></tr></tbody></table>

**Table 1** : The three non-dimensional numbers, three characteristic frequencies, and the characteristic resistance associated to the EIS of a porous electrode (reproduced from the table 1 of [9]). The parameters definitions are given in Appendix A. Throughout the article, a colour code is used to help the reader identifying the parameters. The high frequency parameters are displayed in green, the electrolyte diffusion parameters in blue, and the solid diffusion parameters in red.

Each transport mechanism across the porous electrode can be characterized by a couple $(f, N)$ where $1/f$ is the characteristic timescale at which the transport mechanism takes place, and $N$ is the non-dimensional number that scales the weight of the transport mechanism in the impedance spectrum. We briefly discuss in the following the physical interpretation of those numbers.

The electrochemical fast kinetics includes the electrolyte conduction, and the lithium exchange at the solid-electrolyte interface combining the double layer charging and the insertion/desinsertion kinetics. Thus, its characteristic frequency $f_{capa}$ can be interpreted as the double-layer charging frequency. The non-dimensional number $N_\sigma$ gives the ratio between the electrode penetration length $\lambda$ [32] [33] [34] [35] given by
$$ \lambda = \sqrt{\frac{RT\sigma_l^{\text{eff}SL}}{Fj_0A}} \qquad (1) $$

and the electrode thickness L. A large value of $N_σ$ means that the lithium insertion is homogeneous across the electrode, while a small value means that only a small electrode fraction close to the surface is electrochemically active.

The solid diffusion is the transport mechanism that drives the lithium from the surface to the core of an active particle. Its characteristic frequency $D_s/r^2$ can be well identified as the frequency of a simple diffusion process along the particle radius r. The number $N_s$ combines different parameters that give the balance between the lithium insertion flux and the diffusive flux.

The electrolyte diffusion contributes with the electrolyte conduction to the lithium transport in the electrode depth, but on a much longer timescale. Its characteristic frequency $f_{el}$ can be interpreted as a “pumping” frequency because the electrolyte diffusion process is driven by lithium insertion or desinsertion that act as a source term in the transport equation. Interestingly, the non-dimensional number $N_{el}$ only depends on two electrolyte parameters, the transport number $t^+$ and the Nernst-Einstein correction to the diffusion coefficient $\alpha_l$ defined as [9]

$$ D_{\pm} = \alpha_l \left( 1 + \frac{\partial \ln \gamma}{\partial \ln c} \right) t^+ (1 - t^+) \frac{2RT\sigma_l}{F^2 c_0} \qquad (2) $$

The values of the frequencies and non-dimensional numbers defined in **Table 1** impact the shape of the impedance spectra. Various phenomenologies can be observed depending whether one transport mechanism is overwhelming the other [20] [18] [21]. In some limit cases, the full solution of the P2D-Newman model reduces to simple analytic expressions that can be used to fit the EIS and identify some of the model kinetic parameters. We recall and discuss these expressions in the following.

The EIS high-frequency domain is usually observed above 1Hz for most active materials, provided that the characteristic frequency for double-layer charging $f_{capa}$ is large enough. If the electronic resistance can be neglected, the high-frequency EIS is well approximated by the TLM expression [13] [36].
$$Z_{TLM} = Z \frac{1}{\sqrt{1 + \frac{i\omega}{2\pi f_{capa}} \tanh\left(\frac{1}{\sqrt{1 + \frac{i\omega}{2\pi f_{capa}}}}\right)}} \quad (3)$$

where the couple ($f_{capa}$, $N_σ$) and the characteristic resistance $Z$ are given in **Table 1**. The shape of the Nyquist diagram in the high-frequency domain depends on the non-dimensional number $N_σ$ and is represented by the green curves in **Figure 1**. If $N_σ \gg 1$, the Nyquist diagram comprises a bottom 45° line and a sharp transition to a semi-circle shape, which exact location depends on the value of $N_σ$. On the reverse if $N_σ \ll 1$, the semi-circle is flattened and the EIS is in an asymptotic regime that do not depend on the precise value of $N_σ$. The tortuosity can thus be deduced from the value of $N_σ$ only if $N_σ \gg 1$. The characteristic frequency $f_{capa}$ is located on top of the green TLM spectrum. If the characteristic frequency $f_{capa}$ is too small, the high-frequency domain and the low-frequency domain are not clearly separated because the lithium insertion kinetics (green curve) comes in competition with the solid diffusion (red curve) as displayed in the lower half of **Figure 1**. The resulting EIS shape, displayed in black, shows that the high-frequency bump is damped by the mixing with the diffusive part. On contrary, when the frequency $f_{capa}$ is high enough, the separation between the insertion kinetics and the diffusive kinetics is marked by a deep inflexion in the EIS shape, as shown by the black curves in the upper half of **Figure 1**. The inflexion corresponding to the transition between the high and low-frequency domains is very important to properly measure the characteristic resistance $Z = \frac{\lambda}{\sigma_l^{eff}}$ from which the value of the exchange current density $j_0$ can be obtained. The measurement is thus much more precise when the separation between lithium insertion and diffusion is clear. For standard battery active materials, the characteristic frequency of solid diffusion hardly goes over 1 Hz, even at high temperature. The separation between a high frequency domain and a low frequency domain can be obtained with the criterion $f_{capa} \gg 1$Hz.

The low-frequency domain gives a signature of both the electrolyte and the solid diffusion. The relative magnitude of those phenomena in the EIS is controlled by the three non-dimensional numbers $N_s$, $N_{el}$ and $N_σ$, given in the **Table 1**. The dependency of the low frequency domain on $N_σ$ is highly non-linear. For high values of $N_σ$ (that can be reached for example with thin electrodes, or low temperatures), the low-frequency impedance depends almost linearly on $N_σ$. But its impact becomes much smaller when its value decreases. For low or moderate values of $N_σ$, it has been shown in [9] that the solid diffusion overwhelms the electrolyte diffusion in the case $N_s \gg N_{el}$. The low-frequency EIS is then close to a blocking Warburg that
starts with a 45° line and has a transition to a vertical line at the characteristic frequency $f_s$. In that case, the EIS is well approximated by the distributed particle impedance given by the expression [24] [9]

$$Z_{DP} = Z \frac{1}{\sqrt{\frac{1}{1+Z_s} + \frac{i\omega}{2\pi f_{capa}} \tanh\left(\frac{1}{N_{\sigma}\sqrt{\frac{1}{1+Z_s} + \frac{i\omega}{2\pi f_{capa}}}}\right)}} \quad (4)$$

$$Z_s = N_s \frac{\tanh(\sqrt{i\omega/f_s})}{\sqrt{i\omega/f_s} - \tanh(\sqrt{i\omega/f_s})} \quad (5)$$

In the reverse case where $N_{el} \gg N_s$, and for low or moderate values of $N_σ$ the electrolyte diffusion overwhelms the solid diffusion. In that case, the low-frequency EIS is close to a semi-circle controlled by the characteristic frequency $f_{el}$, and an analytic capacitive approximation with an RC model is given by [9]

$$Z_{RC} = Z \frac{\left( \frac{\sqrt{N_{el}}}{\tanh(\sqrt{N_{el}}/N_{\sigma})} - \frac{1}{\tanh(1/N_{\sigma})} \right)}{1 + \frac{i\omega}{2\pi f_{el}}} \quad (6)$$

It should be noticed that the magnitude of the electrolyte impedance depends on the non-dimensional number $N_{el}$, but also on the characteristic resistance $z$ and on $N_σ$. This will be important when discussing the impact of the electrode thickness, $L$, as $N_{el}$ depends only on electrolyte parameters, while $z$ and $N_σ$ do depend on $L$. In practice, the experimental EIS always deviates from the analytic expression Eq.(6) at low frequency when the blocking part of the solid impedance becomes important.

The competition between the solid and the electrolyte diffusion processes is displayed in the right part of **Figure 1** with the solid diffusion signature in red and the electrolyte diffusion signature in blue. The resulting EIS spectra obtained with the P2D model coupling the solid and electrolyte diffusions are displayed with black curves. In the left part where the solid diffusion is dominant, the EIS shape is close to the solid diffusion signature, with the electrolyte acting as a perturbation. In particular, the transient solid diffusion signature close to 45° can be identified on the spectra. In the right part where the electrolyte diffusion is dominant, the EIS shape shows an obvious negative curvature that is characteristic of the electrolyte diffusion. In the upper part of the graph, the electrolyte signature is the first to appear in the EIS because $f_{el} \gg f_s$, while the frequency $f_s$ lies out of the black spectra. The situation is reversed in the lower part of the graph where the solid diffusion is the first to appear because $f_s \gg f_{el}$ and the frequency $f_{el}$ lies out of the black spectra.
In the overwhelming solid diffusion case, the two parameters $D_s$ and $\phi'_{ocv}$ can be deduced from $N_s$ and $f_s$. In the overwhelming electrolyte diffusion case, the electrolyte diffusion coefficient $D_{\pm}$ and the thermodynamic factor $1 + \frac{\partial \ln \gamma}{\partial \ln c}$ can be deduced from $N_{el}$ and $f_{el}$.

<figure><img src="image_2.png" /><figcaption>**Figure 1** : Schematic representation of the different impedance regimes determined by the non-dimensional numbers and the characteristic frequencies. The left panel illustrates the competition between lithium insertion in the active material (green curves) and the solid diffusion (red curve) in the EIS high frequency range. The right panel illustrates the competition between the solid diffusion (red curve) and the electrolyte diffusion (blue curve) in the EIS low frequency range. The resulting spectra obtained with the simulation of the P2D model are displayed with black curves. The frequency 1Hz is displayed in the left panel to illustrate the frequency mixing between the insertion kinetics and the diffusion kinetics. In the right panel, the characteristic frequency of the first visible diffusion phenomenon is displayed, that is, the electrolyte diffusion in the upper part and the solid diffusion in the lower part.</figcaption></figure>

Note that the analytic expressions given by Eqs. (3)-(6) are given for the porous electrode alone and should be corrected in the presence of a non-negligible separator resistance. In the simplified limit where the electrolyte and the solid diffusion are decoupled, and using a capacitive approximation for the electrolyte diffusion, the impedance of a half-cell reduces to the simple equivalent electric circuit [9]

$$Z = Z_{DP} + Z_{RC} + \frac{R_{sep}}{2} \left( 1 + \frac{N_{el} - 1}{1 + \frac{i\omega}{2\pi f_{el}}} \right) \quad (7)$$

where the separator resistance is given by

$$R_{sep} = \frac{L_{sep}}{\sigma_{sep}^{eff}} \quad (8)$$

In the present work, the separator is 20µm thick, and its Mc Mullin number given by the tortuosity over porosity ratio is close to $\epsilon/\tau \approx 6,5$. The separator characteristic resistance can be evaluated between $R_{sep} \approx 2,2 \cdot 10^{-4} \Omega \cdot m^2$ at -5°C and $R_{sep} \approx 0,8 \cdot 10^{-4} \Omega \cdot m^2$ at 45°C. Those resistances are small compared to the characteristic resistances of NMC electrodes (see
**Figure 3)** and graphite electrodes (see **Figure 7**). Therefore, the presence of the separator does not overwhelms the dominant transport phenomenon (electrolyte or solid diffusion) to be seen at low frequency, and is not included in the list of characteristic numbers proposed in **Table 1**. However, we emphasize that the separator is not neglected in the analysis : the parameters extraction done in sections 3.2 and 3.3 with the P2D model do include the electrolyte diffusion in the separator. Simulations with and without separator are displayed in appendix C.

The expression Eq.(7) is only an approximation of the exact impedance solution of the P2D model. The P2D model includes the full coupling between all the transport mechanisms, and no such simple and compact expression exists in the general case (see also [23] [24] [25]). An extensive characterization of the electrode properties using an EIS fit with the approximated expression of Eq.(7) is done in section 4. The results are then compared with the characterization done using the full P2D model and show to which extent the equivalent electric circuit given by Eq.(7) can be used to characterize transport properties.

To sum up, the porous electrode impedance depends on the seven characteristic numbers given in the **Table 1**, that are given by a combination of the transport parameters of the P2D model. From those seven numbers, seven physical transport parameters can in principle be deduced. One possible (but not the only one) choice can be

* $\tau, C_{dbt}, j_0$ can be deduced from the high-frequency parameters $N_\sigma, f_{capa}$ and the characteristic resistance $z$

* $D_s, \phi'_{ocv}$ can be deduced from $N_s, f_s$ for the solid phase and $D_\pm, (1 + \frac{\partial \ln \gamma}{\partial \ln c})$ can be deduced from $N_{el}, f_{el}$ for the electrolyte phase, at low frequency.

The **Figure 1** illustrates the major difficulty of the parameter characterization. All the seven parameters might not be extracted with a single electrode design because one transport phenomenon might hide the other. In the present paper, we illustrate the methodology of transport characterization with the LiNi$_{0.8}$Mn$_{0.1}$Co$_{0.1}$O$_2$ (NMC 811) and graphite electrode designs presented in section 2.1. For each electrode design, we measure the non-dimensional numbers and their variation with temperature to assess the transport regime in the electrode. We then extract the relevant transport parameters of the electrode.
# 3. Results

## 3.1 Impact of temperature on the EIS spectra

Typical impedance spectra of the NMC//NMC and Gr//Gr cells recorded at different temperatures between -5°C and 45°C are presented **Figure 2**. A shift is applied on the x-axis, such that the minimum of the imaginary part between high-frequency and low-frequency contributions is located at $Re(Z) = 0$ for all the temperatures. This representation makes it possible to visualize the impact of temperature, while distinguishing between high- and low-frequency contributions.

<figure><img src="image_3.png" /></figure>

**Figure 2:** Illustration of graphite (top) and NMC (bottom) EIS spectra measured in symmetric configuration, between -5°C and 45°C, for an EIS frequency range between 200kHz and 2mHz. The state of lithiation is $x_{Li} \approx 0.50$ for the graphite and $x_{Li} \approx 0.52$ for the NMC. A shift of the different spectra is applied on the x-axis, such that the minimum of the imaginary part between high-frequency and low-frequency contributions is located at zero. This representation highlights the impact of temperature on both the low-frequency and high-frequency parts of the spectra. Left: whole spectra, right: zoom on the low frequency part.

In the high frequency range, i.e. $Re(Z) < 0$ with the chosen representation, the Nyquist diagram looks like a flattened semi-circle starting with a 45° bottom line. For NMC electrodes,
the bottom 45° line and sharp transition to the semi-circle shape is clearly visible at low
temperature, which is typical of high values of N<sub>σ</sub>. This sharp transition fades with increasing
temperatures. For the graphite electrode, the transition is never sharp, which seems to indicate
relatively low values of N<sub>σ</sub> at all temperatures.

In the low frequency range, i.e. Re(Z) > 0 with the chosen representation, the Nyquist diagram is more complex, but typical of the different phenomenologies that can be encountered depending on the value of N<sub>el</sub>, N<sub>s</sub>, f<sub>el</sub> and f<sub>s</sub> [9].

For the graphite electrode, the spectrum is almost a straight line, with a temperature dependent slope. As the temperature increases, the slope comes closer to a 45° line characteristics of the solid diffusion in the transient regime. This corresponds to an increasing contribution of the solid diffusion, that is N<sub>s</sub> > N<sub>el</sub>, with an electrolyte contribution occurring on shorter time scale f<sub>el</sub> > f<sub>s</sub>. The deviation from 45° to steeper slope when the temperature decreases can be attributed to an increasing contribution of the electrolyte diffusion.

For the NMC electrode, the spectra are never close to a straight line, highlighting the importance of the electrolyte contribution in the tested frequency range. At low temperature, the spectrum is close but slightly above the 45° line, which is characteristics of the solid diffusion in the transient regime, while at higher temperature, the vertical contribution, typical of the solid diffusion in the blocking limit, highlighting a possible decrease of the solid diffusion compared to the liquid one. The temperature has therefore a clear impact on the overall shape of the impedance spectra for both NMC and graphite configurations. A more detailed analysis of the different contributions will be tackled in the next section while extracting the physical parameters using the analytical impedance solution of the P2D model.

**3.2 Physical parameters determination for the NMC electrode**

We present here the determination of the NMC electrode parameters using the fit of
experimental EIS with the full P2D model. The choice of the free fitting parameters is done with
a preliminary estimation of the non-dimensional numbers for the NMC electrode, using orders
of magnitude for the active material and the electrolyte parameters. The literature indicates
that NMC active materials have a quite high solid diffusion coefficient, with typical values D<sub>s</sub> >
10<sup>-13</sup> m<sup>2</sup>s<sup>-1</sup> at 25°C for potentials larger than 3,9 V [37] [34]. With an OCV derivative φ' ≈
-1 V and an exchange current density j<sub>0</sub> ≈ 1 A m<sup>-2</sup>, this leads to a non-dimensional number
N<sub>s</sub> ≈ 0,5. The transport number of the electrolyte is usually very close to t<sup>+</sup> ≈ 0,3 [38] [39] [40].
With a Nernst-Einstein deviation coefficient α<sub>l</sub> ≈ 1, this leads to a non-dimensional number
$N_{el} \approx 3$. The NMC electrode should thus be in a regime where the electrolyte diffusion is overwhelming the solid diffusion at low frequency. The self-consistency of this latter assumption will be checked a posteriori on the experimental measurements.

The fit with the P2D model is done with the three parameters $N_σ, f_{capa}, Z$ at high frequency and the two parameters $N_{el}, f_{el}$ at low frequency. The electrolyte bulk conductivity has been measured independently and is given by

$$ \sigma_l(T) = 1.15e^{-\frac{E_a^{\sigma_l}}{R(\frac{1}{T}-\frac{1}{298})}} S m^{-1}, \quad E_a^{\sigma_l} = 14 kJ mol^{-1} \qquad (9) $$

with the temperature $T$ given in K. An Arrhenius law is postulated for the temperature dependency of the solid diffusion coefficient as

$$ D_s = D_0 e^{-\frac{E_a^{D_s}}{R(\frac{1}{T}-\frac{1}{298})}} m^2 s^{-1} \qquad (10) $$

The exact value of $D_s$ cannot be extracted from our experimental measurements because $N_s$ is too small compared to $N_{el}$, and the solid diffusion has thus little impact on the impedance spectra. The activation energies for solid diffusion in NMC can hardly be found in the literature, and the reported values strongly depend on the measurement technique [41]. We postulate an activation energy $E_a^{D_s} = 60 kJ mol^{-1}$ and typical values at 25°C $D_0 = 2.10^{-13} m^2 s^{-1}$ for the electrodes at 4,25 V and $D_0 = 1.10^{-13} m^2 s^{-1}$ for the electrodes at 3,85 V [37]. The appendix D discusses another possible value $E_a^{D_s} = 30 kJ mol^{-1}$, value reported in the recent work of Günther et al. [42] for stoichiometries below 0.8. It is shown that, while the temperature dependency of $N_s$ is then reversed, the fundamental assumption $N_s \ll N_{el}$ remains valid in the operating temperature range and the extracted electrolyte parameters are very similar. The OCV local derivatives are directly measured with small changes in the electrode state of charge as described in section 2.1.2. The obtained values are gathered in **Table 2**. All other parameters are known from the electrode manufacturing process and are also summarized in **Table 2**. A typical result of the fitting algorithm is displayed in the **Figure 3** at the three temperatures 45°C, 25°C and -5°C. The position of the three characteristic frequencies $f_{capa}, f_s$ and $f_{el}$ are displayed by arrows. The quality of the fit with the P2D model can be considered as excellent.
<figure><img src="image_4.png" /><figcaption>**Figure 3:** Comparison between the experimental EIS curves (in blue) and the theoretical curves fitted with the P2D model (in red) for the NMC symmetric coin cell assembled with electrodes at 3.85 V. The location of the three characteristic frequencies is displayed on the Nyquist curves (top) and on the curves of the imaginary part (bottom). The frequencies range from 200 kHz to 2 mHz. The diffusion frequencies ($f_s, f_{el}$) drop below 1 mHz at -5°C and do not appear in the Nyquist diagram.</figcaption></figure>

The high-frequency part is clearly separated from the low-frequency part by an inflexion point close to 1 Hz at 45°C, and to 30 mHz at -5°C. In particular, the diffusion phenomena are not perturbing the high-frequency part which allows a precise measurement of the exchange current density. The high-frequency loop is flattened at high temperature and becomes closer to a semi-circle when the temperature decreases. This shows that $N_σ$ increases when the temperature decreases and indicates a more homogeneous lithium transport in the electrode at low temperature. It can be seen that the model slightly deviates from the experiment above 10 kHz. This can be attributed to additional capacitive phenomena such as imperfect electrical contact between the current collector and the active material, or between the active particles themselves. The present work focuses on transport phenomena at intermediate and low frequencies that's why such effects are not included in the model.

The low-frequency part begins with a negative curvature that can be interpreted as the beginning of the electrolyte diffusive semi-circle. When the frequency decreases, the curvature reverses because the EIS enters in the solid diffusion blocking regime. We note that the blocking regime cannot be observed at low temperature (-5°C) because it would happen at a lower frequency than the minimal frequency (2 mHz) of the experiment. Probing the electrode
at lower frequencies would lead to unreasonable experimental times and was not done in the present work. The frequencies $f_s$ and $f_{el}$ have similar values, which indicates that both diffusion phenomena are mixed in the EIS although the electrolyte diffusion has the overwhelming contribution.

The **Figure 4** displays the parameters obtained from the fit. The upper panel displays the characteristic resistance $Z$, the two middle panels display the non-dimensional numbers and the bottom left panel displays the characteristic diffusion frequencies in logarithmic scale, as function of the temperature The bottom right panel displays the reduced electrolyte frequency defined by

$$ \tilde{f}_{el} = \frac{f_{el}}{(j_0 S_a / (2\pi F c_0 \varepsilon))} = 2\alpha_l \left(1 + \frac{\partial \ln \gamma}{\partial \ln c}\right) t^+(1 - t^+) \quad (11) $$

where the value of $j_0$ is deduced from the fit of the high-frequency part of the EIS. The major interest to introduce the reduced electrolyte frequency is that it only depends on the electrolyte parameters. The two parameters ($N_{el}$, $\tilde{f}_{el}$) can then be directly used to deduce the two transport parameters $D_\pm$ and $(1 + \frac{\partial \ln \gamma}{\partial \ln c})$. We recall that the electrolyte diffusivity is related to the Nernst-Einstein deviation parameters $\alpha_l$ by Eq. (2).

The characteristic resistance $Z$ and the non-dimensional number $N_\sigma$ have similar temperature variations, they both decrease when the temperature increases. This trend happens because the temperature decrease of $j_0$ is much steeper ($E_a^{j_0} \approx 56 \text{ kJ mol}^{-1}$, see Eq. (14)) than that of the electrolyte conductivity $\sigma_l$ ($E_a^{\sigma_l} \approx 14 \text{ kJ mol}^{-1}$, see Eq.(9)) such that the variations of $Z$ and $N_\sigma$ are governed by the variations of $j_0$. A high value of $N_\sigma$ is correlated to the observation that the high-frequency EIS becomes closer to a semi-circle with a transition to a 45° line at very high frequency. The configuration where the values of $N_\sigma$ are high is suited to the measurement of the tortuosity. The **Figure 4** displays an area where the measured tortuosity value is reliable, with the criterion $N_\sigma > 1$. As the tortuosity should not depend on temperature, the value measured at -5°C is the most reliable one. The value $\tau = 3,5$ gives the best EIS fit at -5°C for the four coin cells. This value is kept constant for all the fits at other temperatures.
<figure><img src="image_5.png" /><figcaption>**Figure 4** : The characteristic resistance (top), the three non-dimensional numbers (middle) and the two characteristic frequencies (bottom left) measured with the EIS of the four NMC symmetric coin cells. The bottom right panel displays the reduced electrolyte frequency given by Eq. (11). The data from the two coin cells at the lower lithiation state are displayed with dashed lines and the two at the higher lithiation state are displayed with full lines. The two red curves in the right panels are a fit of the electrolyte non-dimensional number (top) and reduced frequency (bottom) respectively.</figcaption></figure>

In the low-frequency range, the value of the solid diffusion coefficient is fixed (see Eq. (10)),
as its precise value is hidden by the electrolyte diffusion. The curves Ns and fs (orange curves
in **Figure 4**) should not be considered as a real measurement. Their values are only displayed to check the self-consistency of the underlying assumption that the electrolyte diffusion is overwhelming the solid diffusion for the NMC electrodes. It is not possible to measure the solid diffusion coefficient with the NMC electrodes, but the precise value does not impact significantly the EIS curve. The P2D model used for the fit does not include the particle size distribution, and only considers the solid diffusion in a representative particle of average radius $r_p \approx 5,4 \ \mu m$ (see the parameters list in **Table 2**). The non-dimensional number $N_s$ is proportional to $r_p$, which means that the presence of a distribution with large particles could increase the solid diffusion impact. For example, doubling the particle radius would set $N_s$ roughly at the same value as $N_{el}$. The particle size distribution of the present NMC electrode measured by laser diffractometer has only 10% of particle volumic fraction with a particle radius $r_p > 19 \ \mu m$, such that the impact of large particles is still assumed to be very small.

The fit result is the measurement of the two electrolyte parameters $N_{el}$ and $f_{el}$ (blue curves in **Figure 4**). The value of $N_{el}$ should be the same for all the samples, as the electrolyte properties do not depend on the electrode state of charge. The **Figure 4** shows that this assertion is satisfied for high temperatures $T>10^\circ C$. For lower temperatures, the $N_{el}$ values are much more dispersed and do not give a reliable result. This very likely comes from the fact that the characteristic frequency $f_{el}$ is smaller than 1 mHz under $10^\circ C$, which is smaller than the cut-off frequency 2 mHz of the measurement. The EIS curve thus only covers a small fraction of the electrolyte signal and the algorithm has not enough data to fit the electrolyte parameters. Another – but complementary – reason is that the value of $N_\sigma$ increases at low temperature and magnifies the impact of solid diffusion to the detriment of electrolyte diffusion. We define the conditions in which the measured electrolyte parameters are reliable with the present electrode design and frequency range with the criterion $T>10^\circ C$. Then a linear fit is done over the four curves $N_{el}(T)$ and a quadratic fit over the four curves $\tilde{f}_{el}(T)$. The result is displayed in red in the two right panels.

The measurements of the characteristic resistance $Z$ and the characteristic frequency $f_{capa}$ give independently access to the exchange current density $j_0$ (left panel of **Figure 5**) and the double layer capacity $C_{dbl}$ (right panel of **Figure 5**) using the expressions

$$j_0 = \frac{RTSL}{FA\sigma_{el}^{\text{eff}}Z^2} \qquad (12)$$

$$C_{dbl} = \frac{Fj_0}{2\pi RT f_{capa}} \qquad (13)$$
<figure><img src="image_6.png" /><figcaption>**Figure 5** : The exchange current density (left) and the double-layer capacity (right) as deduced respectively from the characteristic resistance Z and the characteristic frequency f_capa. The data from the two coin cells at the lower lithiation state are displayed with dashed lines and the two at the higher lithiation state are displayed with full lines.</figcaption></figure>

The values of $j_0$ for the two electrodes at the highest lithiation state ($x_{Li} \approx 0.5$, full lines) are very close together, and higher than the two electrodes at the lower lithiation state ($x_{Li} \approx 0.15$, dashed lines). In particular, the values of $j_0$ for the lower lithiation state differ by almost a factor two. This can be explained by the dependency of the exchange current density to the lithiation state. For NMC materials, the exchange current density displays a typical inverse parabola as function of the lithiation state, with a maximum around $x_{Li} \approx 0.5$ and drops down to zero at low and high lithiation state [43] [37]. Therefore, any difference in the real lithiation state of the two electrodes close to $x_{Li} \approx 0.15$ would result in a large variation of $j_0$ whereas a difference for the two electrodes close to $x_{Li} \approx 0.5$ has almost no impact because $j_0(x_{Li})$ is close to its maximum. The temperature dependency of $j_0$ is very close for all the electrodes, with an activation energy ranging from 50 kJ mol$^{-1}$ to 61 kJ mol$^{-1}$ and an average activation energy

$$E_a^{j_0} \approx 56 \text{ kJ mol}^{-1} \text{ (NMC)} \qquad (14)$$

consistent with literature results [44] [45] [46] [47]. The double-layer capacity (right panel of **Figure 5**) very weakly depends on the temperature and on the lithiation state. For the NMC material, the typical value is $C_{dbl} \approx 1 \text{ F m}^{-2}$, which is one order of magnitude larger than the typical value for the graphite (see **Figure 10**).

From the measurement of $N_{el}$ and $\tilde{f}_{el}$ displayed in **Figure 4**, two independent transport parameters can be deduced. The electrolyte conductivity is an input model parameter and is measured by independent electrochemical experiments. This leaves the three electrolyte
parameters $t^+$, $D_\pm$ and the thermodynamic factor $(1 + \frac{\partial \ln \gamma}{\partial \ln c})$ to measure. The measurement of the transport number has been done for a long time since the pioneer work [48] done with polymers, and afterwards with carbonate liquid electrolytes [39] [38] [49] [40]. Even if the values still display some dispersion, there seems to be an agreement that $t^+ \approx 0,3$ for most of the carbonate electrolytes, and that this value very weakly depends on concentration or temperature. The transport number can be interpreted in the dilute solution theory as a ratio between the mobility of the $Li^+$ and the sum of mobilities of both species in the binary electrolyte. In concentrated solutions, it should also depend on the ionic interactions and of the interactions with the solvant, and such a simple interpretation does no longer hold. However, it is a non-dimensional number that should thus vary much slower with temperature than the diffusion coefficient, that quantitatively measure an absolute effective kinetics in the electrolyte.

The reported values of $D_\pm$ and $(1 + \frac{\partial \ln \gamma}{\partial \ln c})$ are much more sparse, and there is still no agreement on the temperature dependency of those two parameters. In the present paper, the value of the transport number is fixed at the value $t^+ = 0,3$, independently of the temperature. The two remaining parameters are obtained from

$$D_{\pm} = \tilde{f}_{el} \frac{RT\sigma_l}{F^2 c_0} \qquad (15)$$

$$(1 + \frac{\partial \ln \gamma}{\partial \ln c}) = \frac{\tilde{f}_{el}(N_{el} - 1)}{2(1 - t^+)^2} \qquad (16)$$

The electrolyte diffusion coefficient and the thermodynamic factor are displayed in **Figure 6** in the trust conditions $T>10^\circ C$. The errorbars on the measurement are quite large, and come from the propagation of the measurements dispersion of $N_{el}$ and $\tilde{f}_{el}$. In particular, the error on $\tilde{f}_{el}$ has a very large contribution. This comes from the fact that the parameter $\tilde{f}_{el}$ is a non-dimensional parameter of order one, that is deduced from the dimensional parameter $f_{el}$ that follows an Arrhenius law with temperature. A small relative error (in logarithmic scale) on $f_{el}$ has thus a large impact on $\tilde{f}_{el}$. A comparison of our result with the literature [39] and [40] is also displayed in **Figure 6**. The measurement of Valøen and Reimers [39] is done with LiPF6 in PC:EC:DMC (10:27:63 w:w:w) and the measurement of Landesfeind and Gasteiger [40] with LiPF6 in EC:DMC (1:1 w:w), which is close to our composition 1M LiPF6 in EC:DMC:EMC (1:1:1 w:w:w)+2% VC. The literature results are obtained following a classical electrochemical protocol for electrolyte characterization, using a concentration cell for the thermodynamic factor and a pulse experiment in Li-metal coin cell for $t^+$ and $D_\pm$ [49] [40] [39]. Variations of this electrochemical technique are used either to reduce the number of experiments [50] or to account for convective effects [51], but still rely on lithium metal electrodes. The orders of
magnitude found for the electrolyte transport properties at 25°C are fully consistent with our results. Initiated by [20] [21], our work gives a new validation of the methods of electrolyte transport properties extraction using porous electrodes of NMC. The **Figure 6** shows that, while the experimental protocol is very different, our results are in close agreement with the literature. The best agreement is obtained for the diffusion coefficient (right panel). We propose the polynomial expression

$$D_{\pm} = (1,97 - 1,68 \cdot 10^{-2}T + 2,37 \cdot 10^{-3}T^2) \cdot 10^{-10} \text{ m}^2 \text{ s}^{-1} \quad (17)$$

with T in °C in the range 10°C - 45°C.

<figure><img src="image_7.png" /><figcaption>**Figure 6** : The electrolyte thermodynamic factor (left) and diffusion coefficient (right) as deduced from the fit of the non-dimensional number and reduced frequency displayed in **Figure 4**. The vertical lines display the error bars on the fit (blue curve).</figcaption></figure>

Although the reported values in the literature for the thermodynamic factor match at 25°C, there is no agreement relative to its variation with the temperature. It can be seen in the left panel of **Figure 6** that our measurement is in closer agreement to that of Landesfeind and Gasteiger with a thermodynamic factor increasing with the temperature. We propose the quadratic expression

$$\left(1 + \frac{\partial \ln \gamma}{\partial \ln c}\right) = 2,3 - 2,2 \cdot 10^{-2} T + 7,2 \cdot 10^{-4} T^2 \quad (18)$$

with T in °C in the range 10°C - 45°C.
Let's conclude this section with a brief discussion on the advantages and drawbacks of our new characterization methodology. First, it can be seen that the electrolyte characterization using EIS in symmetric coin cells composed of porous electrodes allows to measure two out of the four electrolyte transport parameters. It should thus be used as a technique complementary to other characterization techniques such as the conductivity cell and the concentration cell or the pulse experiment. Second, the temperature (and concentration) range in which this method can be applied strongly depends on the active material properties. A high solid diffusion coefficient and small active particles are required to avoid solid diffusion limitations. However, the very few numbers of extensive electrolyte characterization reported in the literature prove that the classical electrochemical techniques are not at all straightforward. One strong limitation is the instability of the lithium metal interfaces that perturb the electrochemical measurements in Li-metal cells on the same timescales that are required to measure the electrolyte diffusivity. One advantage of our methodology is the perfect stability of active material interfaces to avoid the problems inherent to the Li-metal electrodes. The second advantage – and this is the main statement of this paper – is that the electrolyte characterization is shared with other parameters characterization and thus the total amount of work to characterize the full lithium kinetics in the porous electrode is strongly reduced.

## 3.3 Physical parameters determination for the graphite electrode

We focus now on the determination of the graphite electrode parameters, still using the fit of experimental EIS with the full P2D model. The electrolyte used for the EIS with the graphite electrodes is the same as for the NMC electrodes. The electrolyte parameters are considered as input parameters for the P2D model with the values obtained in section 3.2. Even if the electrolyte parameters are known, the uncertainty on their value could be large enough to perturb the measurement of solid diffusion properties. It should be checked that the graphite electrode is indeed in the overwhelming solid diffusion regime. Following the results of section 3.2, we have $N_{el} \approx 3$. The literature indicates that the graphite diffusion coefficient (measured outside the phase transition plateau) lies between $10^{-15} - 10^{-13} m^2 s^{-1}$ [52] [53] [41] [54]. With an OCV derivative $\phi' \approx -1 V$ and an exchange current density $j_0 \approx 1 A m^{-2}$, this leads to a non-dimensional number $N_s$ between 0,8 and 80, depending on the value of $D_s$. It is therefore difficult to predict if the graphite electrode will also be in the overwhelming electrolyte regime. Fortunately, we will see in the results that the difference between the thermal activation energies of the parameters still gives some conditions where the diffusion coefficient can be accurately measured.
The fit with the P2D model is done with the two parameters $C_{dbl}$ and $j_0$ as high-frequency parameters, and $D_s$ (or equivalently $f_s$) as low-frequency parameters. All other parameters are known from independent measurements, which only leaves three fitting parameters. A typical result of the fitting algorithm is displayed in the **Figure 7** at the three temperatures 45°C, 25°C and -5°C. The position of the three characteristic frequencies $f_{capa}$, $f_s$ and $f_{el}$ are displayed by arrows. The quality of the fit with the P2D model is quite good given the very few numbers of fitting parameters, but not as accurate as for the NMC electrodes.

<figure><img src="image_8.png" /><figcaption>**Figure 7**: Comparison between the experimental EIS curves (in blue) and the theoretical curves fitted with the P2D model (in red) for the graphite symmetric coin cell assembled with electrodes at $x_{Li} = 0.5$. The location of the three characteristic frequencies is displayed on the Nyquist curves (top) and on the curves of the imaginary part (bottom). The frequencies range from 200 kHz to 2 mHz. The diffusion frequencies ($f_s$, $f_{el}$) drop below 1 mHz at -5°C and do not appear in the Nyquist diagram.</figcaption></figure>

The high-frequency part is clearly separated from the low-frequency part by an inflexion point close to 10 Hz at 45°C, and to 100 mHz at -5°C. In particular, the diffusion phenomena are not perturbing the high-frequency part which allows a precise measurement of the exchange current density. The high-frequency loop has the shape of a flattened semi-circle at all temperatures, which indicates that the number $N_σ$ is much lower than in NMC electrodes. This means that the tortuosity cannot be accurately measured with this experiment. An independent experiment of EIS in blocking conditions [55] with the same electrodes gives the value $τ ≈ 7$, which was the value used for the fit of **Figure 7**. Yet, it can be seen that the P2D model still deviates significantly from the experiment in the high-frequency part, because the semi-circle
is much flatter in the experiment than in the P2D model. This observation leads some
experimental works to introduce an additional empirical parameter in the TLM expression Eq.
(3) using a constant phase element (CPE) [33] [34]. This approach was not considered in the
present work as this empirical correction cannot be simply related to modified equations of the
P2D model. To account for this flattened semi-circle, another approach could be to add a RC
element associated to the SEI on the graphite surface [22]. However, to fit experimental data,
CPE for both the SEI and the graphite are often required [32] [56]. The need for more complex
approaches including a combination of the resistive and capacitive effect of the SEI with the
distributed particle impedance [23] or diffusive effect in the SEI [57] [58] can also emerge.
Implementing the physics of the SEI could be an interesting perspective to increase the
predictability of the model at high frequency, while keeping a direct link to physical equations.

The low-frequency part is very close to a 45° line that corresponds to the solid diffusion regime. The EIS down to 2mHz does not reach the characteristic frequencies $f_s$ and $f_{el}$ at low temperature, meaning that the frequency cut-off is too high to properly characterize diffusion phenomena at low temperature. Probing the electrode at lower frequencies would lead to unreasonable experimental times and was not done in the present work. However, the analysis of non-dimensional numbers has shown that the electrolyte diffusion has a non-negligible contribution in this part of the EIS. In the P2D model, the full interaction between solid and electrolyte diffusion is taken into account, and still results in a shape very close to 45°. In principle, two parameters can be identified from the fit of the low-frequency EIS. For the present experiment, the OCV derivative has been directly measured in the experiment as described in section 2.1.2 (the obtained values are gathered in **Table 2**), and is thus an input parameter. An exception is done for the measurement with the two coin cells at very low state of charge. As the OCV derivative is very high for $x_{Li} < 0.05$, the P2D model is very sensitive to this parameter and it is thus left as a fitting parameter for these two coin cells. For the four coin cells at intermediate and high state of charge, the solid diffusion coefficient is the only remaining fitting parameter at low-frequency.

The **Figure 8** displays the parameters obtained from the fit. The top left panel displays the characteristic resistance, the top right and bottom left panels display the non-dimensional numbers, and the bottom right panel displays the characteristic diffusion frequencies in logarithmic scale, as function of the temperature. The data from the six coin cells at three different state of charge are superimposed on the graphs.
<figure><img src="image_9.png" /><figcaption>**Figure 8:** The characteristic resistance (top left), the three non-dimensional numbers (top right and bottom left) and the two characteristic frequencies (bottom right) measured with the EIS of the six graphite symmetric coin cells. The electrolyte parameters are input parameters in the fit and are thus identical for all the EIS. The data from the two coin cells at the lowest lithiation state are displayed with dashed lines, two at the intermediate lithiation state with dotted lines, and two at the highest lithiation state with full lines.</figcaption></figure>

As expected from the Nyquist plots of **Figure 7**, the value of $N_σ$ displayed in the left panel of **Figure 8** is very low for the graphite electrode. Only at temperatures lower than 0°C does the number $N_σ$ become larger than one. However, the fit done with the tortuosity as a fitting parameter leads to inconsistently large values $τ ≈ 14$, even with the experimental data at -5°C. This means that much larger values of $N_σ$ may be required to properly measure the tortuosity in graphite electrodes. The tortuosity was independently measured using EIS in symmetric coin cells composed of delithiated graphite electrodes and the value $τ = 7$ was kept for the P2D simulations.

The non-dimensional numbers $N_s$ and $N_{el}$ (middle panel of **Figure 8**) are of the same order of magnitude at 25°C. Whereas $N_{el}$ weakly depends on temperature, $N_s$ steeply increases with the temperature and becomes dominant in the range $T > 20°C$, for all states of charge. The dashed black lines indicate the condition where the measurement of the solid diffusion
coefficient becomes possible based on the criterion $N_s > N_{el}$. The value of $N_s$ is proportional to the ratio $\frac{j_0}{D_s}$. The steep increase of $N_s$ with the temperature shows that the activation energy of $j_0$ is much larger than that of $D_s$ for graphite. At high temperature, the lithium flux at the particle boundary becomes much larger and the solid diffusion is no longer able to carry the lithium to the particle core such that solid diffusion becomes the limiting phenomenon.

Finally, the characteristic diffusion frequencies are displayed in the right panel of **Figure 8**. It can be seen that both frequencies are of the same order of magnitude at low temperature and that $f_{el}$ becomes much larger than $f_s$ at high temperature. Both diffusion phenomena are strongly coupled for the graphite electrode because they act over similar timescales. The fact that the solid diffusion can be measured only comes from the high value of $N_s$ and not from a timescale separation between the phenomena. From the expression $f_s = D_s/r^2$, the variations of $f_s$ with SOC and temperature correspond to the variations of $D_s$.

The value of $D_s$ in the reliable temperature range is displayed in **Figure 9**. At very low lithiation states, before the first phase transition plateau, $x_{Li} \approx 0.04$ (dashed lines), the value of $D_s$ is one order of magnitude larger than at higher lithiation states (dotted lines and full lines), a result that is consistent with previous measurements reported in the literature [54]. The solid diffusion coefficient has a very small variation with the temperature. Despite the small temperature range over which $D_s$ measurement is relevant, the activation energy is estimated, so that its order of magnitude can be compared with other kinetic parameters. The corresponding average activation energy is

$$E_a^{D_s} \approx 38 \text{ kJ mol}^{-1} \text{ (Graphite)} \qquad (19)$$

This value lies close between $E_a^{D_s} \approx 35 \text{ kJ mol}^{-1}$ reported in [59] and $E_a^{D_s} \approx 41 \text{ kJ mol}^{-1}$ reported in [41]. The measurements of the characteristic resistance $Z$ and the characteristic frequency $f_{capa}$ give independently access to the exchange current density $j_0$ (left panel of **Figure 10**) and the double layer capacity $C_{dwl}$ (right panel of **Figure 10**) using the equations (12)-(13). The graphite exchange current density has very small variations with the state of charge. All the samples at intermediate (dotted lines) and high lithiation states (full lines) have almost the same value of $j_0$, while $j_0$ drops by roughly a factor 2 for the samples at low state of charge. This latter observation is consistent with the fact that the exchange current density should vanish for a fully delithiated material, but this limit is hardly reached for graphite. The temperature dependency of $j_0$ is very close for all the electrodes, with an activation energy ranging from $50 \text{ kJ mol}^{-1}$ to $73 \text{ kJ mol}^{-1}$ and an average activation energy
$$E_a^{j_0} \approx 63 \text{ kJ mol}^{-1} \text{ (Graphite)} \qquad (20)$$

The double-layer capacity (right panel of **Figure 10**) very weakly depends on the temperature and on the lithiation state. For the graphite material, the typical value is $C_{abl} \approx 0.1 \text{ F m}^{-2}$, which is one order of magnitude smaller than the typical value for the NMC (see **Figure 5**).

<figure><img src="image_10.png" /><figcaption>**Figure 9:** The solid diffusion coefficient of graphite measured in the trust temperature range T > 25°C from the characteristic solid diffusion frequency displayed in **Figure 8**. The two coin cells at the lowest lithiation state have the higher diffusion coefficient (dashed lines3) whereas the coin cells at the intermediate and higher lithiation states have very similar diffusion coefficients (dotted and full lines).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 10:** The exchange current density (left) and the double-layer capacity (right) as deduced respectively from the characteristic resistance $Z$ and the characteristic frequency $f_{capa}$. The data from the two coin cells at the lowest lithiation state are displayed with dashed lines, two at the intermediate lithiation state with dotted lines, and two at the highest lithiation state with full lines.</figcaption></figure>
## 4. Comparison between P2D model and a simple equivalent circuit approximation

The extraction of electrode transport parameters from electrochemical titration is usually done using two approaches. The first, and the simplest one, is to fit the experimental data with a simple equivalent electrical circuit approximation (EEC). The second one, considered as more involved, is to fit the experimental data with the full analytical solution of the P2D model or the complex equivalent electric circuit based on the discrete formulation of the P2D model [16] [17] [18]. In the present paper, the full analytical solution of the linearized P2D model (Appendix B) is used. The first approach is based on a simplification of the P2D analytical equations, resulting in the simple equivalent electric circuit (EEC) given by Eq. (7). These two approaches are compared to illustrate the limit of the simple EEC.

Considering the case of NMC, the **Figure 11** compares the experimental EIS (in blue), the EIS reconstructed with the analytical equations using the parameters determined by fitting the experimental data with the P2D model (in red), the EIS reconstructed with the EEC model (Eq. (7)) using the parameters determined by fitting the experimental data with the EEC (in green), and the EIS reconstructed with the EEC model (Eq. (7)) using the same parameters as in the P2D model (in yellow).

<figure><img src="image_12.png" /><figcaption>**Figure 11** : Comparison between the experimental EIS curves (in blue), the theoretical curves fitted with P2D model (in red), the EEC curves fitted with EEC equations (in green) and the EEC curves using the fitting results from P2D model (in yellow) for the NMC symmetric coin cell assemble with electrodes at 3,85V. The frequencies range from 200 kHz to 2 mHz.</figcaption></figure>

First, fitting the low frequency part of the EIS spectrum using the P2D model results in better precision than using the EEC model. However, the high frequency part of the spectrum is well fitted by both approaches. Plotting the EIS curves (in yellow) with the EEC equations using the parameters determined with the P2D model (see section 3) confirms both the concordance of
the high frequency part of the spectra and the major differences observed at low frequencies. This result is consistent with the previous work [9] showing that, although there are no differences between the P2D model without the electrolyte contribution and its approximation by the distributed particle impedance described in Eq. (4), the approximation of the electrolyte impedance by a simple RC model from Eq. (6) does not match the P2D model.

Thereby, the use of the simplified Eq. (4) is adequate for the determination of $D_s$ and $j_0$ but does not allow a proper determination of $D_±$. Indeed, the use of a single RC electric circuit to simplify the electrolyte transport phenomenon is not enough to properly identify the electrolyte parameters. If the use of an equivalent electrical circuit expression is important, the challenge remains to find an expression that can better describe the electrolyte diffusivity than the single RC approximation. Our recommendation to capture electrolyte phenomena remains to use the full analytic expression presented in this paper or the equivalent complex equivalent electric circuit based on the discrete formulation of the P2D model [16] [17] [18].

# 5. Discussion

The experimental measurements obtained from the NMC electrodes and from the graphite electrodes are gathered in the **Figure 12** using a compact graphical representation. The left panel represents $\log(f_{capa}/1[Hz])$ as a function of $\log(N_σ)$. The right panel represents $\log(f_{el}/f_s)$ as a function of $\log(N_{el}/N_s)$. In both graphics, the experimental values are displayed with triangles for NMC and circles for graphite, as function of the state of charge and temperature. All the points correspond to the data given in **Figure 4** and **Figure 8**. Following the representation of **Figure 1**, the kinetic limitations can be illustrated using the two quadrant figures of **Figure 12**. They are emphasized with a particular colour code. The green background (zone 2 in the left panel) represents the conditions where tortuosity can be measured, the red background (zones 1 and 3 in the right panel) where the solid diffusion coefficient can be measured, and the blue background (zones 2 and 4 in the right panel) where the electrolyte diffusion can be measured.

The **Figure 12** (left) represents the kinetic limitations occurring at high frequencies. Points located in the green area where $\log N_σ > 1$ represent the experimental conditions for which the electrode is homogeneously active in the depth. In this zone the penetration length is larger than the electrode thickness leading to an EIS signal with a 45° linear shape at very high frequencies followed by an almost perfect semi-circle for charge transfer. This typical shape allows the distinction in the EIS spectrum between the contribution from the ionic conductivity in the electrode porosity and the lithium insertion in the active material. The magnitude and
localisation of the slope breaking at very high frequency gives the possibility to extract the
tortuosity value. It can be seen that the experimental points are dispersed in the graph following
almost a straight line going from the bottom right corner at low temperature to the upper left
corner at high temperature. This is due to the scaling of $f_{capa}$ and $N_σ$ with the parameter $j_0$,
that has a much higher activation energy than $σ_l$ and the other parameters. The data from
graphite electrodes are shifted up and left compared to the NMC electrodes, and cover only a
small fraction of zone 2. This shows that it is much more difficult to measure the tortuosity of
graphite electrodes than NMC electrodes.

The **Figure 12** (right) represents the kinetic limitations at low frequency and gives the conditions in which the electrolyte and solid diffusion coefficients can be measured. In zone 1, solid diffusion is dominant over electrolyte diffusion and occurs at lower frequencies than electrolyte diffusion. The low frequencies exhibit both solid and electrolyte diffusion phenomena. Similarly, in zone 4, electrolyte diffusion is dominant over solid diffusion but solid diffusion occurs at higher frequencies so that solid and electrolyte diffusion are both observed at low frequencies. In zone 2, electrolyte diffusion is dominant and solid diffusion occurs at lower frequencies meaning that there is a frequency window in which diffusion in the electrolyte can be observed with little perturbation, allowing the characterization of the electrolyte parameters. In zone 3, solid diffusion is dominant and electrolyte diffusion occurs at lower frequencies meaning that there is a frequency window in which solid diffusion can be observed with little perturbation, allowing the characterization of the solid diffusion parameter. It can be seen that the experimental values for graphite are dispersed following three straight lines going from zone 4 at low temperature to zone 1 at high temperature, but preferentially cover the solid diffusion area at high temperature. The variation with temperature is mainly due to the ratio $j_0/D_s$ in which the activation energy of $j_0$ dominates over that of $D_s$. By contrast, the experimental values for NMC electrodes are gathered in the electrolyte diffusion area, with little temperature variation. This is due to our model hypothesis that activations energies of $j_0$ and $D_s$ are the same, but the real value of the $D_s$ activation energy could not be measured with our electrode design. The graph emphasizes that only electrolyte transport properties could be measured with our NMC electrodes.

The chosen representation of $\log(f_{el}/f_s)$ as a function of $\log(N_{el}/N_s)$ provides a quantitative criterion for the decorrelation of the different phenomena. From an experimental perspective, it is important to understand what parameters can be changed to navigate inside **Figure 12**.

First, experimental conditions at which the EIS are performed for a given electrode design are
considered. In particular, temperature and lithiation state of the different electrodes were
examined because these are quite simple experimental conditions to modify. The experimental
data obtained on both NMC and graphite electrodes show that the green region where
tortuosity can be measured can be selected by performing EIS at low temperatures. This
property can be reached thanks to the very different activation energies of the electrolyte
conductivity and of the exchange current density both in NMC (Eq.(14)) and in graphite (Eq.
(20)). At low frequencies, the global behaviour strongly depends on the active material
properties. The NMC electrode is in the overwhelming electrolyte diffusion regime (blue region)
for the entire temperature range tested [-5°C to 45°C]. Moreover, the ratio N<sub>el</sub>/N<sub>s</sub> does not
strongly change with temperature but is affected by a change in stoichiometry. As the solid
diffusion is expected to drop significantly at high stoichiometries, NMC at 3.55V, corresponding
roughly to x<sub>Li</sub> ≈ 0.91 was tested to emphasize solid diffusion. The obtained EIS spectra are
presented in appendix F, **Figure 16**, but do not enable the observation of features related to
solid diffusion. Indeed even if a significant decrease of the solid diffusion coefficient is expected
at this high stoichiometry, the exchange current density j<sub>0</sub> also decreases significantly. It is
rather uncertain that the combination of both variations leads to a higher value of Ns.
Furthermore, the strong decrease of j0 results in a shift of the characteristic frequency f<sub>capa</sub> to
lower values. This makes the separation between low and high frequencies obsolete and it is
impossible to properly decorrelate charge transfer and diffusive phenomena as can be seen
**Figure 16**.

By contrast, the graphite electrodes experience overwhelming solid diffusion when the
temperature increases. This is due to the large difference in the activation energies of the
exchange current density, 63 kJ mol<sup>-1</sup>, and of the solid diffusion coefficient, 38 kJ mol<sup>-1</sup>.
Moreover, probing different lithiation states affect the f<sub>el</sub>/f<sub>s</sub> ratio. The low stoichiometry helps
observing the solid diffusion at a higher frequency than at other lithiation states due to the
higher value of the solid diffusion coefficient.

The evaluation of characteristic frequencies and non-dimensional numbers allows to anticipate
on the impact of other design parameters, in particular the particle size, the electrolyte
concentration and the electrode thickness.
<figure><img src="image_13.png" /><figcaption>**Figure 12** : (left) Representation of the high frequency impedance regimes using the non-dimensional number $N_σ$ and the characteristic frequency $f_{capa}/1[Hz]$; (right) representation of the low frequency impedance regimes using the ratio of the non-dimensional numbers $N_{el}/N_s$ and the ratio of the characteristic frequencies $f_{el}/f_s$. The experimental data from NMC electrodes are displayed with triangles, and the experimental data from graphite electrodes are displayed with circles. The colour scale represents the variation with temperature going from red for 45°C to dark blue for -5°C. The dotted lines indicate the state of charge with the same convention as in **Figure 4** and **Figure 8**.</figcaption></figure>

The particle size has a major impact on the relative magnitude between solid and liquid diffusion, and therefore on the position in the low-frequency quadrant of **Figure 12**. The solid diffusion frequency has the obvious scaling $f_s \sim \frac{1}{r^2}$ and the related non-dimensional number scales as $N_s \sim r$. The electrolyte non-dimensional number $N_{el}$ is independent of $r$. Finally, the electrolyte frequency $f_{el}$ depends on the exchange surface $A$. A simple spherical approximation gives the scaling $A \sim \frac{1}{r}$ (see **Table 2**) and thus $f_{el} \sim \frac{1}{r}$. We obtain

$$ \left\{ \begin{aligned} \frac{N_{el}}{N_s} &\sim \frac{1}{r} \\ \frac{f_{el}}{f_s} &\sim r \end{aligned} \right. \qquad (21) $$

The scaling given by Eq. (21) leads to a displacement with slope -1, represented by the black arrow in zone 1 in the diagram of **Figure 12**. As could be intuited, larger particles increase the magnitude of solid diffusion and decrease its characteristic frequency. This in particular
explains why the solid diffusion is easier to measure with graphite. Not only is the solid diffusion coefficient smaller in graphite than in NMC, but particles are also much larger.

The electrolyte concentration is quite easy to modulate in lab experiments, and can thus be used to modulate the diffusion phenomena. Its impact on electrolyte conductivity and diffusivity is important [39] [40]. However, the electrolyte properties that impact $N_{el}$ and $f_{el}$, are not the conductivity and diffusivity, but $t^+$, $\alpha_l$ and the thermodynamic factor $1 + \frac{\partial \ln \gamma}{\partial \ln c}$. The dependence of those properties with the electrolyte concentration is difficult to predict, because these transport parameters are difficult to precisely evaluate, especially at high concentrations. We therefore focus on these parameters variation at low electrolyte concentrations where dilute solution theory provides estimations. The solid diffusion frequency $f_s$ clearly does not depend on the electrolyte concentration. In the electrolyte, $t^+$ weakly depends on the concentration [39] [40], and the dilute solution theory gives the limit $\alpha_l \to 1$ and $1 + \frac{\partial \ln \gamma}{\partial \ln c} \to 1$ at low concentration. The ideal Butler-Volmer theory also states that $j_0$ should depend on the electrolyte concentration with the scaling $j_0 \sim \sqrt{c_l}$. We obtain at low concentration

$$ \left\{ \begin{array}{l} \frac{N_{el}}{N_s} \sim \frac{1}{\sqrt{c_l}} \\ \\ \frac{f_{el}}{f_s} \sim \frac{1}{\sqrt{c_l}} \end{array} \right. \qquad (22) $$

The scaling given by Eq. (22) leads to a displacement with slope +1, represented by the black arrow in zone 2 in the diagram of Figure 12. Decreasing the electrolyte concentration should thus lead to a larger signature of the electrolyte diffusion. However, the square root scaling gives a very small displacement in the diagram, except reaching very low electrolyte concentrations. This last estimation is performed in the context of low or moderate value of $N_\sigma$, for which the magnitude of the electrolyte impedance scale with $\sqrt{N_{el}}$ (See Eq. (6) of section 2.2). For thin electrode ($N_\sigma \gg 1$), the estimation would be much more complex given the dependence of the low-frequency spectrum on the three non-dimensional numbers $N_s$, $N_{el}$ and $N_\sigma$, including the highly non-linear contribution of $N_\sigma$.

We finally discuss the impact of the electrode loading, or electrode thickness. This parameter has an impact over all frequencies in the impedance spectrum via the non-dimensional number $N_\sigma$. As already discussed in section 2.2, its impact is highly non-monotonic. For low or moderate values of $N_\sigma$, the electrode is in an asymptotic regime, that we could call the “thick electrode limit”. In this limit, the electrode thickness has little impact on the impedance spectrum, because the electrode is only electrochemically active over the penetration length $\lambda \ll L$. The low-frequency spectrum is then determined mainly by the diffusion non-
dimensional numbers and characteristic frequencies. For high values of N<sub>σ</sub>, i.e. in the “thin electrode limit”, the low-frequency spectrum varies with all the non-dimensional numbers N<sub>σ</sub>, N<sub>el</sub> and N<sub>s</sub>. More precisely, a high value of N<sub>σ</sub> magnifies the solid diffusion, which scales as N<sub>σ</sub> and reduces the electrolyte diffusion, whose magnitude scales with (N<sub>el</sub> − 1)/N<sub>σ</sub>. Reducing the electrode thickness therefore increases the solid diffusion contribution compared to the liquid one, as can be observed in the impedance spectra of Günther et al. [42] measured for 3 different electrodes thickness. This explains why solid diffusion coefficients are preferentially measured with thin electrodes, and why the electrolyte signal at low frequency disappears with thin electrodes. Qualitatively, an increasing value of N<sub>σ</sub> would shift to the right the crossover between the solid diffusion domain (orange) and the electrolyte diffusion domain (blue) in **Figure 12**.

The results from this work also give recommendations to select the right experimental condition to better characterize different parameters of interest. Exchange current density and tortuosity are preferentially observed at low temperatures. Solid diffusion is preferentially observed at high temperatures and with thin electrodes. Electrolyte diffusion could be observed at all temperatures, preferentially using thick electrodes. However, this phenomenon can only be measured using particular active materials with relatively high diffusion coefficients and/or small particles, such as NMC. Changing the electrode design in experiments is not always an easy matter, that's why experimental conditions should be preferred whenever possible to choose the optimal condition to isolate one or the other transport phenomenon.

**6. Conclusion**

This work provides a methodology to extract most of the kinetics parameters of the P2D-
Newman model from conventional EIS measurements. The methodology relies on the
analytical solution of the linearized P2D-Newman model in the frequency domain, together
with a classification of the limiting transport regimes using the non-dimensional numbers and
characteristic frequencies introduced in **Table 1**. The parameter values found with the present
methodology are systematically compared to literature values and show an excellent
agreement, while they can only be obtained over a limited range of experimental conditions.
This shows that a lot of information on transport properties can be obtained with a single
commercial electrode design that is not particularly dedicated to characterization experiments.

In the high-frequency domain ($f > 1$ Hz), the EIS is driven by electrolyte conduction and the lithium exchange at the solid-electrolyte interface, whose ratio is associated to the non-dimensional number $N_\sigma$. In the low-frequency domain ($f < 1$ Hz), the EIS is dominated by
lithium solid diffusion in active materials and lithium diffusion in electrolyte, the two phenomena being difficult to characterize independently [21]. They are respectively associated to two non-dimensional numbers ($N_s, N_{el}$) and two characteristic frequencies ($f_s, f_{el}$).

For both the graphite and NMC electrodes, the separation between the low and high frequency phenomena is very well defined for all the tested temperatures and stoichiometries. This makes it possible to determine, the exchange current densities of active materials and their temperature dependence ($E_a^{j_0} \approx 56 \text{ kJ mol}^{-1}$ for NMC and $E_a^{j_0} \approx 63 \text{ kJ mol}^{-1}$ for graphite) with a high degree of confidence. Regarding the value of $N_σ$, as expected, its value increases with decreasing temperature. This yields a tortuosity value of 3.5 for the NMC electrode at -5°C, where confidence in the measurement is highest. For the tested graphite electrode, the value of $N_σ$ is never high enough to allow for tortuosity measurement without resorting to an additional experiment in blocking configuration.

In the low-frequency range, for the NMC electrode, the impedance spectra exhibit the characteristic features of the liquid diffusion process. The fit with the analytical solution shows that the electrolyte diffusion dominates solid diffusion in the active materials ($N_{el} > N_s$) for all the tested temperatures and stoichiometry. The solid diffusion cannot be measured with this electrode design, and would require the use of thinner electrodes. Regarding, the electrolyte properties, we were able to determine two out of the four electrolyte transport parameters. More specifically, we obtained quadratic expressions for the temperature dependence of the diffusion coefficient and thermodynamic factor, in the range [10°C – 45°C], without resorting to dedicated experiments and avoiding the problems inherent to the Li-metal electrodes.

For the graphite electrode, a 45°C slope, characteristic of the solid diffusion phenomena is visible, especially at high temperature. Building upon the electrolyte measurements performed with the NMC electrode, we can extract the solid diffusion and its dependence with temperature with a good degree of confidence for temperatures above 20°C. The steep increase of $N_s$ with the temperature shows that the activation energy of $j_0$ is much larger than that of $D_s$ for graphite, for which we obtain a value of $E_a^{D_s} \approx 38 \text{ kJ mol}^{-1}$.

We evaluate the pertinence of resorting to equivalent electrical circuit approximation to perform these parameters extraction instead of the full analytical model. As expected, while the TLM model is a very good approximation in the high frequency domain, simplified and decoupled expressions in the low frequency domain can lead to large error, especially in the electrolyte properties identification.
These results show that it is not always possible to use EIS data to extract the solid diffusion. More significantly, we have also proposed a simple methodology, based on the evaluation of the system's dimensionless numbers, to determine a degree of confidence associated with the properties measurements. Whenever the values of these dimensionless numbers can be anticipated, our methodology provides the measurement conditions and electrode design to be used to extract maximum of information from conventional EIS measurements.

# Appendix A: Notations

The **Table 2** and **Table 3** gather the P2D parameters and their respective values used in the present work. We briefly define below the parameters whose interpretation is not explicit.

The maximal lithium concentration is defined as the maximal theoretical concentration value in lithium for a perfect active material at SOC=0%, neglecting its volumic change during lithiation. For example, the chemical expression for the fully lithiated graphite is $LiC_6$, which means that the maximal concentration is given by

$$c_s^{\max} = \frac{\rho_{am}}{6M_{Gr}} = \frac{2,25 \cdot 10^6 [g \ m^{-3}]}{6 \cdot 12 [g \ mol^{-1}]} = 31250 [mol \ m^{-3}]$$

The particle mean radius $r$ is given by half of the D50 obtained from the active material manufacturer datasheet or by laser diffractometry.

The tortuosity $\tau$ is by definition given by $\tau = \varepsilon \frac{\sigma_l}{\sigma_l^{\text{eff}}}$ where $\sigma_l$ is the bulk electrolyte conductivity and $\sigma_l^{\text{eff}}$ is the effective ionic conductivity of the porous medium. Note that the tortuosity values given in the **Table 2** are measured by EIS and are much higher than the theoretical values predicted by the Bruggeman formula $\tau = \frac{1}{\sqrt{\varepsilon}}$.

The exchange current density $j_0$ is the prefactor of the well-known Butler-Volmer law for charge transfer at the electrolyte/solid interface, given by Eq. (29). The parameter $j_0$ is supposed to depend on the lithium solid concentration at the interface and on the temperature.

The double-layer capacity is the capacity of the electrolyte/solid interface modeled by an equivalent electric circuit, to account for charge storage at the interface. The mathematical expression involving $C_{abl}$ in the P2D model is given by Eq. (28).
<table><caption>**Table 2:** Notation of the electrode parameters and their respective values in the present work.</caption><thead><tr><th>Active material and electrode parameter</th><th>Notation [unit]</th><th>NMC electrode</th><th>Graphite electrode</th></tr></thead><tbody><tr><td>Maximal lithium concentration</td><td>c<sub>s</sub><sup>max</sup> [mol m<sup>-3</sup>]</td><td>48700</td><td>31250</td></tr><tr><td>Particle mean radius</td><td>r [µm]</td><td>5.4</td><td>6</td></tr><tr><td>Sample surface</td><td>S [cm<sup>2</sup>]</td><td>1.54</td><td>1.54</td></tr><tr><td>Loading</td><td>E [mg cm<sup>-2</sup>]</td><td>22.92</td><td>14.13</td></tr><tr><td>Thickness</td><td>L [µm]</td><td>70</td><td>100</td></tr><tr><td>Active material fraction</td><td>x<sub>am</sub> [%]</td><td>96</td><td>97</td></tr><tr><td>Porosity</td><td>ε [%]</td><td>28.6</td><td>33</td></tr><tr><td>Tortuosity</td><td>τ</td><td>3.5</td><td>7</td></tr><tr><td>Active material density</td><td>ρ<sub>am</sub> [g cm<sup>-3</sup>]</td><td>4.75</td><td>2.25</td></tr><tr><td>Active surface</td><td>A [cm<sup>2</sup>]</td><td>(3SE x<sub>am</sub>)/(rρ<sub>am</sub>) = 39</td><td>(3SE x<sub>am</sub>)/(rρ<sub>am</sub>) = 50</td></tr><tr><td>OCV derivative vs stoechiometry</td><td>φ'<sub>ocv</sub> [V]</td><td>-1.14 x<sub>Li</sub> ≈ 0.52<br>(±0.01)<br>-1.8 x<sub>Li</sub> ≈ 0.16<br>(±0.05)</td><td>-5.24 x<sub>Li</sub> ≈ 0.04<br>(±0.2)<br>-1.95 x<sub>Li</sub> ≈ 0.50<br>(±0.05)<br>-0.90 x<sub>Li</sub> ≈ 0.99<br>(±0.05)</td></tr><tr><td>Solid diffusion coefficient</td><td>D<sub>s</sub> [m<sup>2</sup> s<sup>-1</sup>]</td><td>Measured with EIS</td><td>Measured with EIS</td></tr><tr><td>Exchange current density</td><td>j<sub>0</sub> [A m<sup>-2</sup>]</td><td>Measured with EIS</td><td>Measured with EIS</td></tr><tr><td>Double-layer capacity</td><td>C<sub>dbl</sub> [F m<sup>-2</sup>]</td><td>Measured with EIS</td><td>Measured with EIS</td></tr></tbody></table>

<table><caption>**Table 3:** Notation of the electrolyte parameters and their respective values in the present work.</caption><thead><tr><th>Electrolyte parameter</th><th>Notation [unit]</th><th>Value</th></tr></thead><tbody><tr><td>Average concentration</td><td>c<sub>0</sub> [mol m<sup>-3</sup>]</td><td>1000</td></tr><tr><td>Conductivity</td><td>σ<sub>l</sub> [S m<sup>-1</sup>]</td><td>Eq. (9)</td></tr><tr><td>Transport number</td><td>t<sup>+</sup></td><td>0,3</td></tr><tr><td>Diffusion coefficient</td><td>D<sub>±</sub> [m<sup>2</sup> s<sup>-1</sup>]</td><td>Eq. (17)</td></tr><tr><td>Thermodynamic factor</td><td>1 + (∂ ln γ / ∂ ln c)</td><td>Eq. (18)</td></tr><tr><td>Effective conductivity</td><td>σ<sub>l</sub><sup>eff</sup> [S m<sup>-1</sup>]</td><td>σ<sub>l</sub> ε / τ</td></tr></tbody></table>
# Appendix B: Equations of the P2D model

The transport equation in the solid phase is given by the Fick's law

$$ \partial_t c_s = \nabla \cdot D_s \nabla c_s \qquad (23) $$

with the boundary condition at the particle surface

$$ -D_s \nabla c_s \cdot \mathbf{n}_{sl} = \frac{j_{out}}{F} \qquad (24) $$

In Eqs. (23) and (24), $c_s$ [mol m$^{-3}$] is the lithium concentration in the active material, $\mathbf{n}_{sl}$ is the unit vector normal to the particle surface pointing outwards, and $j_{out}$ [A m$^{-2}$] is the current density flowing out of the particle. The other parameters are given in **Table 2**.

The transport equations in the liquid phase are given by

$$ \partial_t c_l = \nabla \cdot \left( D_{\perp}^{\text{eff}} \nabla c_l - t^{+} \frac{i_l}{F} \right) + \frac{\nabla \cdot i_l}{F} \qquad (25) $$

$$ i_l = -\sigma_l^{\text{eff}} \nabla \varphi_l + \frac{2RT\sigma_l^{\text{eff}}(1-t^+)}{F} \left(1 + \frac{\partial \ln \gamma}{\partial \ln c}\right) \frac{\nabla c_l}{c_l} \qquad (26) $$

$$ \nabla \cdot i_l = \frac{A}{SL} (j_{out} + j_{dbl}) \qquad (27) $$

In Eqs. (25), (26), (27), $c_l$ [mol m$^{-3}$] is the electrolyte concentration, $i_l$ [A m$^{-2}$] is the charge current density, $\varphi_l$ [V] is the electrochemical potential and $j_{dbl}$ [A m$^{-2}$] is the current density in the double-layer. The other parameters are given in **Table 2** and **Table 3**.

The double-layer electric current density is given by

$$ j_{dbl} = C_{dbl} \partial_t (\Phi - \varphi_l) \qquad (28) $$

where $\Phi$ [V] is the electric potential in the solid phase.

Finally, the current outward the solid particle is given by the Butler-Volmer law

$$ j_{out} = j_0 \sqrt{\frac{c_l}{c_0}} \left( e^{\frac{F}{2RT}\eta} - e^{-\frac{F}{2RT}\eta} \right) \qquad (29) $$

with the overpotential given by

$$ \eta = \Phi - \Phi_{ocv} - \varphi_l \qquad (30) $$

In a previous work [9], it has been shown that the P2D model can be linearized and solved using a Fourier transform. The analytical solution can be expressed in terms of the non-dimensional numbers and characteristic frequencies given in the **Table 1**. The particle impedance should first be computed to find the relation between the particle outward current and the potential difference at the particle surface

$$ j_{out}(1 + Z_s) = \tilde{\Phi} - \tilde{\Phi}_l \qquad (31) $$
with $\tilde{j}_{out} = \frac{j_{out}}{j_0}$ the non-dimensional current fluctuations, $\tilde{\phi} - \tilde{\phi}_l = \frac{F}{RT}(\phi - \phi_l)$ the non-dimensional potential fluctuations, and $Z_s$ the solid diffusion impedance (Eq. (5)) which depends on $N_s$ and $f_s$ (see Eq. (32) in ref [9] for more details).

The double layer Eq. (28) can also be solved to obtain

$$ \tilde{j}_{dbl} \frac{2\pi f_{capa}}{i\omega} = \tilde{\phi} - \tilde{\phi}_l \qquad (32) $$

with $\tilde{j}_{dbl} = \frac{j_{dbl}}{j_0}$.

Finally, the set of non-dimensional transport equations in the electrolyte can be written (see Eqs. (44) and (45) in ref [9])

$$ \frac{i\omega}{2\pi f_{el}} \tilde{c}_{el} = \tilde{c}''_{el} + (1 - t^{+}) \tilde{i}'_{l} \qquad (33) $$

$$ \tilde{i}_{l} = \left( \frac{1}{\frac{1}{1+Z_s} + \frac{i\omega}{2\pi f_{capa}}} \right) \tilde{i}''_{l} + \frac{N_{el} - 1}{1 - t^{+}} \tilde{c}'_{el} \qquad (34) $$

where $\tilde{c}_{el} = \frac{c_{el}}{c_0}$ is the non-dimensional concentration fluctuation and $\tilde{i}_{l} = \frac{\lambda i_{l}}{F c_0 D_{\pm}^{\text{eff}}}$ is the non-dimensional current fluctuation in the electrolyte. The derivatives are given with respect of the non-dimensional length $\tilde{x} = \frac{x}{\lambda}$. It can be seen in Eqs. (33) and (34) that the non-dimensional set of equations still contains the second non-dimensional number $t^+$ in addition to $N_{el}$. However, the eigenvalues equation is

$$ k^4 - \left( \frac{i\omega}{2\pi f_{el}} + N_{el} \left( \frac{1}{1+Z_s} + \frac{i\omega}{2\pi f_{capa}} \right) \right) k^2 + \frac{i\omega}{2\pi f_{el}} \left( \frac{1}{1+Z_s} + \frac{i\omega}{2\pi f_{capa}} \right) = 0 \qquad (35) $$

This means that the eigenvalues do not explicitly depend on $t^+$ but rather on the non-dimensional number $N_{el}$. The transport number only appears in the eigenvector expression. According to the littérature [38] [39] [40], the transport number $t^+$ has very small variations as function of the concentration. This means that its precise value can be ignored to estimate the relative influence of the solid diffusion and of the electrolyte diffusion in the electrode impedance. The relative weight of both diffusion phenomena is thus only given by the comparison of $N_s$ and $N_{el}$.

## Appendix C : Impact of the separator

The present paper proposes a classification of dominant transport phenomena based on the non-dimensional numbers of porous electrodes only, and do not consider the characteristic numbers of the separator. The analysis is particularly relevant for commercial cells, where the electrodes have a high loading and are thick (see typical values in Table 2), whereas the separators are as thin as possible to increase the cell energy density. Typical commercial
separator thickness range between 10µm and 25µm, which is much thinner than the electrodes. The separator used in the present work is 20µm thick. The **Figure 13** shows that the electrolyte diffusion in the separator has a small impact on the low-frequency impedance spectra of half-cells. Therefore, the electrolyte diffusion in the separator is taken into account in the P2D model to extract the transport properties of porous electrodes, but is not considered to classify the low-frequency transport regimes as done in the right panel of **Figure 12**.

<figure><img src="image_14.png" /><figcaption>**Figure 13** : simulation of impedance spectra of the NMC half-cell (left) and graphite half-cell (right) at 25°C. The simulation including the separator is displayed in red and without the separator in black. The spectra have been horizontally shifted to highlight the impact of the separator on the low-frequency part.</figcaption></figure>

# Appendix D : Impact of the solid diffusion coefficient on NMC spectra

As discussed in section 3.2, the NMC solid diffusion coefficient cannot be measured with our NMC electrodes because those are in a configuration where the electrolyte diffusion is overwhelming the low-frequency part of the EIS. For the fit optimization, a reasonable value for $D_s$ and for its activation energy should thus be assumed as input parameters in the P2D model. We were not able to find a widely accepted value for the activation energy. The value $E_a^{D_s} = 60 \text{ kJ mol}^{-1}$ is assumed in the present article, but a value closer to $E_a^{D_s} = 30 \text{ kJ mol}^{-1}$ is also possible.

A lower value $E_a^{D_s} = 30 \text{ kJ mol}^{-1}$ would mean that the activation energy for the solid diffusion coefficient becomes smaller than that of the exchange current density (see Eq.(14)). This reverses the variation of $N_s$ with the temperature, with increasing its value at high temperature.
The impact of solid diffusion could thus become non-negligible at 45°C and perturb the extraction of electrolyte properties.

To check this point, we do a new fit with the P2D model of the EIS of the two coin cells at the higher lithiation state $x_{Li} \approx 0,5$. All the values of the high-frequency parameters $j_0, \tau, C_{dbl}$ do not change (not displayed). The impact of the solid diffusion activation energy can be seen in the low-frequency part. The **Figure 14** displays the non-dimensional numbers and characteristic frequencies obtained from the fit with the P2D model for the two possible activation energies. It can be seen that the fundamental assumption for NMC electrodes $N_s \ll N_{el}$ remains valid in the operating temperature range and the extracted electrolyte parameters are very similar. The new values assuming $E_a^{Ds} = 30 \text{ kJ mol}^{-1}$ are within the errorbars of measured electrolyte parameters.

<figure><img src="image_15.png" /><figcaption>**Figure 14** : The non-dimensional numbers (left) and the characteristic frequencies (right) measured with the EIS of the two NMC symmetric coin cells at $xLi \approx 0,5$. The thick curves with circles show the results with the hypothesis $E_a^{Ds} = 60 \text{ kJ mol}^{-1}$ (same curves as **Figure 4**). The thin curves with the diamonds show the results with the hypothesis $E_a^{Ds} = 30 \text{ kJ mol}^{-1}$.</figcaption></figure>

# Appendix E : Formation and OCV curves

The formation and the local OCV determination protocol are described in section 2.1.2. **Figure 15** presents the corresponding results, with the cell potential as function of time during the formation and potential setting for both NMC (left) and graphite (center) electrodes. The potential increment is presented as function of the stoichiometry increment (right) for the different cells. The resulting OCV slopes are presented in **Table 2**.
<figure><img src="image_16.png" /><figcaption>**Figure 15** : Formation cycles and potential setting for half-cells containing NMC (left) and graphite (center) performed at 45°C for the different potential targets. The local OCV slope is deduced from the potential vs stoichiometry increments (right).</figcaption></figure>

**Appendix F : EIS of NMC at high lithiation state**

In an attempt to observe the specific feature associated to solid diffusion in NMC in the EIS spectra, NMC at 3.55V, corresponding roughly to $x_{Li} \approx 0.91$ was tested. The obtained EIS spectra displayed in **Figure 16** do not present the specific solid diffusion features. The deep inflexion in the EIS spectra, marking the separation between the insertion kinetics and the diffusive kinetics is not present of this high stoichiometry. Performing an analysis based on the non-dimentional numbers and frequencies of **Table 1** can help to anticipate this result. Indeed, for NMC at high stoichiometries, a strong decrease of $D_s$ is expected, leading to a decrease of $f_s$. However, a strong decrease of $j_0$ is also expected, leading to a decrease of $f_{el}$ and $f_{capa}$, as well as an increase of $Z$ and $N_σ$. For $N_s$, it is rather uncertain that the combination of both variations would lead to a higher value of $N_s$. Furthermore, as observed in **Figure 16**, the decreasing $f_{capa}$ makes it impossible to decorrelate the charge transfer and solid diffusion phenomena, and thus prevent the extraction of the different kinetics parameters.
<figure><img src="image_17.png" /><figcaption>**Figure 16** : Illustration of NMC EIS spectra measured in symmetric configuration, between -5°C and 45°C, for an EIS frequency range between 200 kHz and 2 mHz. The state of lithiation is $x_{Li} \approx 0.91$. A shift of the different spectra is applied on the x-axis, such that the first point is located at zero. This representation highlights the impact of temperature on both the low-frequency and high-frequency parts of the spectra. Top: whole spectra, bottom: zoom for high temperatures.</figcaption></figure>

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