1 Introduction

As the utilization of large-scale batteries is currently increasing, so does the demand for more accurate battery diagnostics, thereby generating a rapid development of different battery management systems (BMSs) [1, 2]. Here, the state-of-charge (SoC) is one of the most critical parameters for indication of the remaining energy left in a cell. Developing efficient yet accurate SoC algorithms remains a challenge. Electrochemical impedance spectroscopy (EIS) is in this context considered a fast, non-destructive and reliable method. EIS can be used to determine important parameters such as charge-transfer resistance and double-layer capacitance, from which information on battery utilization can be derived [2, 3]. By extracting electrical parameters which vary as a function of the battery SoC, the impedance spectra can be used as a predictive tool. Generally, only the high-frequency response is used to estimate internal resistance and thereby SoC [3].

The measured total impedance constitutes the result of a large number of resistive components and reactive processes, which are usually assessed by an equivalent-circuit analogue. This circuit represents, for example, equilibrium potential, ohmic behaviour, charge transfer, double-layer effects at the electrode/electrolyte interfaces and mass transfer processes. In present battery diagnostics, both simple lumped-parameter circuits and more complex finite-transmission-line circuits are used [4,5,6]. However, most of the proposed electric equivalent models do not represent the electrochemical phenomena at each of the electrodes, but globally at cell level, thus generating an oversimplification. Moreover, porous-electrode effects, transient and non-linear responses, and additional artefacts of the battery current collectors, terminals, and other more peripheral components are not assessable by equivalent-circuit modelling. The contributions of these physically distinct processes are instead merged into one EIS response, rendering a meaningful interpretation less possible.

Since the impedance spectra of batteries strongly depend on the short-term charge/discharge history, the validity of equivalent-circuit modelling is restricted. To overcome this problem, we suggest an approach that combines EIS with a physics-based cell model. Finite element methodology (FEM) has been a valuable approach for modelling batteries with a range of cell types and geometries [7,8,9,10]. A FEM approach allows for a model based on physical properties of the system—such as thermodynamics, kinetics and diffusion—as opposed to simplified circuit elements representing these processes. In principle, such a model can be extended to three-dimensional models of complete cells or battery systems.

However, this approach has so far not been widely applied to exploring factors affecting SoC in Li-ion batteries (LiBs). In one recent example, such an approach is used for commercial LiB cells with reasonable accuracy [11], but is limited due to the restricted information on the electrode formulation and morphology. In this work, we attempt to explore the possibilities of correlating the impedance of lab-scale half-cells with controlled chemistry to their SoC, using FEM modelling and direct comparisons of the simulated EIS results with experimental analogues, also including electrode morphological effects.

The principal aim of this study is to investigate if this combination of FEM with impedance spectroscopy is a feasible approach for battery SOC diagnostics, and also to pinpoint some of the current limitations with such an approach.

2 Finite element studies

FEM studies have proven fruitful for battery modelling. Many of the mathematical models use porous-electrode theory [12], thereby treating the electrodes as a uniform mix of active material, binder, conducting additive and electrolyte. Wang and Sastry [13], however, implemented concentrated solution theory using FEM to break down the electrode material mix in order to study the influence of size distribution of the active material particles. Similarly, FEM has proven useful for simulating the impedance response of complex energy storage devices [14]. However, the physical processes and properties influencing the impedance response and its correlation to the SoC in a LiB are still to a large degree unexplored. In this work, we attempt to investigate the possibilities of correlating cell impedance and SoC using FEM modelling. To achieve this, we have used the COMSOL Multiphysics batteries and fuel cells module (BFC) to construct a 1D model of a LiB, which incorporates morphological properties such as porosity in the determination of the impedance spectra. Experimental data are used to validate the simulations.

2.1 Model background

A lithium half-cell with a lithium-iron phosphate (LiFePO\(_4\)) positive electrode has been modelled using the BFC module in COMSOL Multiphysics 4.4, simulating the battery charging profile. 1D half-cell lithium battery model consists of the cell cross section in one dimension, implying that edge effects in length and height are neglected. The cell is constituted in three regions (two electrodes and a separator) rendering four distinct boundaries at

  1. (a)

    \(0< x <\delta _\mathrm{cc}\) : Positive current collector (aluminium)

  2. (b)

    \(\delta _\mathrm{cc}< x < \delta _\mathrm{p}\) : Positive porous electrode (LiFePO\(_4\)). Thickness: 25 \(\upmu \mathrm{m}\)

  3. (c)

    \(\delta _\mathrm{p}< x < \delta _\mathrm{s}\): Separator (1 M LiPF\(_6\) salt in 1:1 EC:DEC as electrolyte). Thickness: 50 \(\upmu \mathrm{m}\)

  4. (d)

    \(\delta _\mathrm{s}< x < \delta _\mathrm{n}\): Counter-electrode (lithium foil). Thickness: 25 \(\upmu \mathrm{m}.\)

For the electronic current balance, a potential of 0 V is set on the counter electrode boundary (\(x = \delta _\mathrm{n}\)). At the working electrode current collector/feeder (\(x = \delta _\mathrm{cc} \)), the current density (harmonic perturbation) is specified. The inner boundaries facing the separator \(x = \delta _\mathrm{p} \) and \(x = \delta _\mathrm{s} \)) are insulating for electric currents. To maintain the ionic charge balance in the electrolyte, the current collector boundaries (\(x = \delta _\mathrm{cc} \) and \(x = \delta _\mathrm{n} \)) are ionically insulating. Insulating conditions also apply to the material balances.

2.2 Electrochemical description

The positive electrode is assumed porous. The region \(0< x <\delta _\mathrm{p}\) therefore contains both solid porous-electrode and liquid (electrolyte) phases. At the particle surface, the material flux is determined by the local electrochemical reaction rate given by linearized Butler–Volmer kinetics. The electrical potential in the electron conducting phase (\(\phi _\mathrm{s}\)) is calculated using a charge balance based on Ohm’s law, where the charge-transfer reactions result in a mathematical source or sink term. Taking porosity and tortuosity into account, the effective conductivities of the electrodes (\(\phi _\mathrm{eff}\)) is given by

$$ \phi _\mathrm{eff}=\phi _\mathrm{s} \epsilon ^{\gamma }, $$

where \(\gamma \) is the Bruggeman coefficient, set to 1.5 in this model, corresponding to a packed bed of porous particles. The ionic charge balances and material balances are modelled for binary 1:1 electrolytes. Fickian diffusion describes the mass transport of lithium ions to the particles (Eq. 2), whereby the resulting diffusion equation can be expressed from the concentration gradients of lithium ions in the porous electrode (Eq. 3). Butler–Volmer electrode kinetics describes the local charge-transfer current density in the electrodes and is introduced as source or sink terms in the charge balances and material balances (Eqs. 58).

Lithium diffusion in the spherical particle of radius “r” is described by Fick’s law according to [15]

$$\begin{aligned} \frac{\partial }{\partial t} \delta C_\mathrm{Li}= & {} \frac{D}{r^2}\frac{\partial }{\partial r} \left( r^2\frac{\partial C_\mathrm{Li}}{\partial r}\right) \end{aligned}$$
$$\begin{aligned} \frac{\partial (\epsilon _\mathrm{e} C_\mathrm{Li})}{\partial t}= & {} \frac{\partial }{\partial x} \left( D_\mathrm{Li}\frac{\partial C_\mathrm{Li}}{\partial x}\right) +\frac{1-t_\mathrm{Li}^0}{F}j \end{aligned}$$

with boundary conditions as

$$\begin{aligned} \frac{\partial C_\mathrm{Li}}{\partial x}\bigg |_{x=0}=\frac{\partial C_\mathrm{Li}}{\partial x}\bigg |_{x=L}=0 \end{aligned}.$$

Charge conservation in the solid phase of electrode is defined by Ohm’s law:

$$\begin{aligned} \frac{\partial }{\partial x}\left( \sigma _\mathrm{eff} \frac{\partial }{\partial x}\phi _\mathrm{s}\right) -j=0 \end{aligned}$$

with the following limit conditions at the current collectors

$$\begin{aligned} \sigma ^\mathrm{eff}_+\frac{\partial \phi _\mathrm{s}}{\partial x}\bigg |_{x=0}=\frac{I}{A} \end{aligned}$$

and the null current conditions at the separator:

$$\begin{aligned} \frac{\partial \phi _\mathrm{s}}{\partial x}\bigg |_{x=\delta _\mathrm{n}}=\frac{\partial \phi _\mathrm{s}}{\partial x}\bigg |_{x=\delta _\mathrm{n}+\delta _\mathrm{s}}=0 \end{aligned}.$$

If \(\phi _i(x,t)\) denotes the electrolyte potential, charge conservation in the electrolyte is defined by

$$\begin{aligned} \frac{\partial }{\partial x}\left( \kappa ^\mathrm{eff} \frac{\partial }{\partial x}\phi _i\right) +\frac{\partial }{\partial x}\left( \kappa ^\mathrm{eff} \frac{\partial }{\partial x}\mathrm{ln}(C_i)\right) +j=0 \end{aligned}$$


$$\begin{aligned} \frac{\partial \phi _i}{\partial x}\bigg |_{x=0}=\frac{\partial \phi _i}{\partial x}\bigg |_{x=L}=0 \end{aligned},$$

where \(\kappa ^\mathrm{eff}\) is the conductivity in the electrolyte. The four differential Eqs. 2, 5, 6 and 8 are linked by the Butler–Volmer equation

$$\begin{aligned} j/A =i_0 \left( e^{\frac{\alpha n F \eta }{RT}}-e^{\frac{(1-\alpha )n F \eta }{RT}}\right) \end{aligned}.$$

where A is the electrode area. In Eq. 10, the total current j is induced by overvoltage \(\eta \), defined by the potential difference between the solid phase and the electrolyte and equilibrium thermodynamic potential U:

$$\begin{aligned} \eta =\phi _\mathrm{s}+\phi _i-U \end{aligned},$$

where U is a function of the degree of lithiation in the active material particles, and thus dependent on the SoC.

The admittance response (the inverse of the impedance) of the device is then generated through the diffusive flux of the charges species, and is related to the current through the following equations:

$$\begin{aligned} Y(\omega )= & {} \frac{\iota \omega }{\eta _{0}} \int _{0}^{\infty } \mathrm{d}t\; e^{-\iota \omega t}\;j(t)\end{aligned}$$
$$\begin{aligned} SoC(t)= & {} \frac{1}{Q}\int _0^t j(t)\,\mathrm{d}t\end{aligned}$$
$$\begin{aligned} j(\omega )\propto \,& {} Y(\omega )=\frac{\iota \omega }{\eta _{0}} \int _{0}^{\infty } \mathrm{d}t\; e^{-\iota \omega t}\;j(t) \end{aligned}.$$

These relations can, in turn, be related to Eqs. 10 and  12, where the current passing through the battery is an indicator of the SoC. The relationship between impedance and SoC can thus be explored using this set of equations, and is here investigated by calculating the changing impedance spectra while incrementing the SoC with known intervals. Parameter values are listed in Table 1.

Table 1 List of symbols and parameters

2.3 Mesh density and solver settings

Equations 114 have been solved using the COMSOL Multiphysics 5.1 software. A mesh length of 3.35 \(\times 10^{-4}\) m was used, and the device was divided into 47 mesh elements in total, using 4 vertex elements. The element length ratio has an average mesh growth rate of 1.083. The same mesh was used in all simulations. A PARDISO solver with 0.01 relative and 0.001 absolute tolerances was used for the time-dependent matrix equations. A rectangular periodic pulse (inset in Fig. 1) with the same amplitude and period as employed in the experiments was used as the load for the applied current. By applying a prescribed load cycle, a time-dependent charging cycle was modelled. The flux was linearized at intervals of 30 mAh/g to yield the frequency domain response at different SoC levels using a harmonic perturbation of 15 mV.

3 Experimental

LiFePO\(_4\) (LFP, Phostech) was dispersed with carbon black (Super P, Imerys) and poly(vinylidene fluoride-co-hexafluoropropylene) (Kynar FLEX 2801, Arkema) in a ratio of 80:15:5 by weight into N-methylpyrrolidone (NMP, VWR) and mixed by planetary ball-milling for 1 h. The resulting slurry was bar-coated onto Al foil and dried at 70 \(^\circ \)C in air. The sheet was cut into 20-mm-diameter discs, transferred to an Ar-filled glove box and dried at 120 \(^\circ \)C under vacuum overnight. The LFP loading on the electrodes was 1.5 mg/cm\(^2\). Vacuum-sealed pouch-type half-cells were assembled with a 22-mm-diameter Li metal disc counter-electrode (Cyprus Foote Mineral, 125 \(\upmu \)m thickness) and a concentric ring of Li metal (22 mm inner diameter) as the reference electrode. The electrolyte was 1 M LiPF\(_6\) in 1:1 EC:DEC. Electrochemical experiments were performed using a VMP2 (Bio-Logic). Cells were first cycled under galvanostatic conditions at C/10 (17 mA/g) between 2.5 and 4.1 V versus Li/Li\(^+\) to confirm acceptable cell performance. Impedance spectra were then collected at successive intervals of 15 mAh/g (approximately 10% SoC, given a practical capacity of about 150 mAh/g for the electrodes). The cell was allowed to relax at OCV for 20 min before each impedance measurement. Impedance spectra were collected over the range of 200 kHz to 10 mHz with a peak-to-peak amplitude of 15 mV.

4 Results and discussions

The impedance spectra were generated at time intervals corresponding to specific states of lithiation/delithiation in the electrode, which in turn influence the transport properties and hence the electrochemical response. The inset in Fig. 1 shows the pulsed constant current charging profile. The same load cycle as utilized in the experiments was adapted in the simulations, where each pulse generates a 10 % increment in the SoC from 0 to 100%, and where the simulated impedance response is generated at every increment. Figure 1 shows the resulting Nyquist plot of the half-cell at various SoC levels, calculated using Eq. 14. The general features of the impedance spectra are similar, mainly displaying a large semicircle for medium frequencies and a finite diffusion branch for high frequencies. The most dominant contribution is assigned to the charge-transfer process between the LFP electrode and the electrolyte, corresponding to an increased value of \(C_\mathrm{Li}\) in Eq. 2 for increasing SoC values. The strong SoC dependency found for the degree of lithiation in the positive electrode has also been confirmed in several experimental studies [1, 12] . The identified charge-transfer processes there are in the same frequency range and show a similar SoC dependency as in the simulations, suggesting an agreement with the proposed interpretation. The impedance response can be characterized as different polarization processes in the LiFePO\(_4\) cathode: solid state diffusion or intercalation of Li ions (as described in Eqs. 2 and 3), charge transfer (cathode/electrolyte) described via Butler–Volmer kinetics (Eq. 6) and contact (particle) resistance described by Ohm’s law (Eqs. 59).

Fig. 1
figure 1

Simulated Nyquist plots for various state-of-charge levels. Inset showing load-cycling

Figure 2 shows the Bode modulus and phase plots, respectively, generated from the simulations and thereby the frequency dependence of the SoC. The resistance and frequency range of the different processes in the cathode can then be separated. The Bode phase plots display a suppression of the peak at \(\sim 10^{2.7}-10^3\) Hz, while the Bode modulus shows a lower resistance and a shift to higher frequencies for the diffusion onset for higher SoC values. This corresponds to higher charge-transfer and solid state diffusion resistances for higher cathode lithiation, equivalent to a higher \(C_\mathrm{Li}\) in Eq. 3. The different time constants for both these phenomena are responsible for a constant phase element (CPE)-type behaviour (i.e. phase angles \(>45^\circ \)). The CPE behaviour indicates the exact frequency of shifting power law behaviour in the Bode magnitude plots, seen as a shift in the slope of the linear curves at ca. \(10^{3.3}\) Hz. The crossover frequency, where the Bode plots display at plateau at lower frequencies, is higher for higher SoC levels, indicating a smaller diffusion length for these SoC levels inside the bulk of the porous electrode. Moreover, the transition from diffusion controlled to charge-transfer-controlled responses at different frequency ranges can be observed in the simulations, in agreement with previous studies [2, 3, 16, 17] where high SoC is indicated through lower charge-transfer resistance.

The Warburg-like behaviour seen at higher frequencies can be attributed to several factors, but primarily to the porosity of the electrodes. For a porous insertion electrode, the arc displays a characteristic shape previously described in the literature [18,19,20]. This type of porous-electrode interfacial impedance response was, for example, discussed by de Levie, and results from the increasing ohmic drop experienced as the ionic current penetrates into the electrode with decreasing signal frequency. This is also known as ambipolar diffusion of ions and electrons within a transmission line formed from a conductive pore of uniform diameter and a capacitive wall with a well-defined and constant area per unit length [21]. It is worth noting the similarity between this response and that of a diffusion process with a non-transmissive (blocking) interface. For a porous electrode, the current within the pore changes more sluggishly, and the penetration depth decreases with frequency. Therefore, after a very short time or at very high frequencies, primarily the capacitive effects of the more or less flat external electrode surface are measured, while the penetration distance of the sinusoidal varying current wave charging the walls of the structure is restricted at high frequencies by the resistance of the electrolyte in the pores.

Fig. 2
figure 2

Bode modulus and phase angle for various state-of-charge levels

Figures 3 and 4 show the simulated and experimental data in Nyquist and Bode phase diagrams at various SoC levels. The Nyquist plot displays remarkably good fits between simulated and experimental EIS data, showing the validity of the model used. In the Bode plot, it can be seen that the match is striking in the frequency region characteristic of the charge-transfer regime (\(10^{2.7} { - }10^3\) Hz), while the agreement is less good for higher frequencies. This is indicative of the importance of cell design parameters such as particle contacts, porosity and current collector/electrode contacts. These would appear in the high-frequency region, where the EIS response is considered to be dependent on porosity or particle–particle resistance [22,23,24].

Fig. 3
figure 3

Simulated and experimental Nyquist plots

Fig. 4
figure 4

Simulated and experimental Bode phase plots

To further explore this phenomenon, the effect of porosity directly correlated to the volume fraction of electrolyte in the electrode (\(\epsilon _\mathrm{l}\) in Eqs. 3 and 5) is investigated. Porosity is an important parameter, as it scales the effective diffusion coefficient and conductivity as described by the porous-electrode theory [25]. Figure 5 shows Nyquist plots for 20, 40 and 60% SoC levels and different electrode porosities. Generally, two features can be highlighted. First, the set of plots generated for the same SoC levels yield significantly different polarization for the different electrolyte volume fractions. More explicitly, lower porosity values yields a smaller semicircle in the Nyquist plot, corresponding to a lower charge-transfer resistance. A larger volume fraction of liquid electrolyte, on the other hand, leads to higher polarization. This can be explained by that a denser packing of the particles decreases the intra-particle resistance since the overall connectivity in the electrode provides a better interface for charge transfer, lowering the polarization resistance (see Eq. 6). Secondly, the real axis impedance (high-frequency intercept), which is a signature of the ohmic resistance, shows the reverse trend, i.e. it is greater for the lower electrolyte volume fraction in the porous electrode and lower for the more porous electrodes. This is due to the limited ionic conduction resulting from the low electrolyte content (see Eq. 8).

Fig. 5
figure 5

Nyquist plots for various electrolyte volume fractions (\(\epsilon _\mathrm{l}\)) and state-of-charge levels

Fig. 6
figure 6

Bode plots for various electrolyte volume fractions (\(\epsilon _\mathrm{l}\)) and state-of-charge levels

Figure 6 shows the phase plot of EIS data generated for different SoC levels and different porosities (\(\epsilon _\mathrm{l}\)), which directly corresponds to the volume fraction of electrolyte in the electrode composite. As stated above, the porosity yields the effective transport parameters through Eqs. 3, 5, 6 and 8. High porosities will yield good electrode/electrolyte contact, which renders high “effective” diffusivity and ionic conductivity through the electrode volume, as can be seen through the transport Eqs. 58. It can be seen from the simulation results that the high-frequency response is more or less independent of the SoC level, i.e. one cannot sense any deviation in the high-frequency region for the SoC variation. However, the change in phase angle is quite dramatic when altering the electrolyte volume fraction. This shift of the phase at higher frequencies can therefore be taken as an indication of porosity changes (or porosity in general) during the course of battery cycling. This region in the Bode phase plot can hence display the effect of experimental variables such as electrode calendering, volume changes or battery packing. If comparing the simulated phase angle data with the experimental equivalents in Fig. 4, the values are most similar for electrode porosities equivalent to 0.20 in electrolyte volume fraction. This can be considered somewhat of a low porosity estimate for an uncalandered LiFePO\(_4\) electrode. We estimate the electrodes used in the experimental part of this work as being in the range of 75–80% porous, based on the ratio of the density of the coating relative to the theoretical density of a perfectly compacted non-porous electrode. This is a reasonable estimate for an uncalendered electrode with a nanosized active material. However, this estimate does not include a consideration of the electrode tortuosity, which requires advanced techniques such as tomography to be quantified.

It is also clear from Fig. 6 that there are significant variations is this regime of the phase plots, also for constant porosities, indicating that more refined modelling is necessary to adequately capture all phenomena appearing in this frequency domain. Nevertheless, the current model clearly captures the qualitative features of the experimental curves, i.e. the plateau and the crossing of the curves when going from higher to lower SoC values (or vice versa). The porosity seems to vary substantially during cycling, in the range of 0.2–0.5, which certainly corresponds to realistic values.

5 Conclusions

The relationship between SoC and EIS data has been explored for LiFePO\(_4\) half-cells by employing a multiphysics approach. It can be seen that the charge-transfer resistance is dependent of the SoC during charging, and can be modelled in good agreement with experimental results. Cell design parameters, e.g. calendering, will influence the EIS response since they control the solid state conduction path in the electrode, and the simulations thus provide information on morphological parameters. Especially at higher frequencies, the simulations can qualitatively reproduce the features of porosity changes during battery cycling. With controlled variation of electrode porosity and model refinement, this relationship can most likely be explored to a higher degree in future studies. Such an approach will be necessary for SOC diagnostics using an electrochemical modelling tool when investigating batteries with porous electrodes. The simulations here evidently show clear shifts in porosity while cycling the battery, which can be quantified by simulation data.

In summary, the FEM approach provides insight into the impedance response without the need for equivalent-circuit modelling, thereby better capturing the interplay between cell chemistry, geometry and morphology. This may ultimately yield better input parameters for adaptive BMS algorithms such as ‘fuzzy logic’ or Kalman filtering.