Enhanced Porous Electrode Theory Based Electrochemical Model for Higher Fidelity Modelling and Deciphering of the EIS Spectra

Journal of The Electrochemical Society OPEN ACCESS Enhanced Porous Electrode Theory Based Electrochemical Model for Higher Fidelity Modelling and Deciphering of the EIS Spectra To cite this article: Igor Mele et al 2024 J. Electrochem. Soc. 171 080537 View the article online for updates and enhancements. You may also like Secure voice cryptography based on Diffie-Hellman algorithm Sura F. Yousif - Image ciphering algorithm based on novel ICMM and modified differential encoding Narbda Rani - Deciphering the Role of p-Type ZnCo2O4 Semiconductor Nanoflakes for Selective Enhancement of Voltammetric Responses Toward Redox Species System: Interfacial Electron-Transfer Kinetics and Adsorption Capacity Xuan Dinh Ngo, Ngoc Huyen Nguyen, Thi Lan Huong Phung et al. - This content was downloaded from IP address 92.172.2.43 on 18/11/2025 at 11:22 Enhanced Porous Electrode Theory Based Electrochemical Model for Higher Fidelity Modelling and Deciphering of the EIS Spectra Igor Mele, 1 Klemen Zelič, 1 Marko Firm, 2 JožeMoškon, 2 Miran Gaberšček, 2,3 and TomažKatrašnik 1,z 1 University of Ljubljana, Faculty of Mechanical Engineering, SI-1000 Ljubljana, Slovenia 2 National institute of Chemistry, Department of Materials Chemistry, SI-1000 Ljubljana, Slovenia 3 University of Ljubljana, Faculty of Chemistry and Chemical Technology, SI-1000 Ljubljana, Slovenia Electrochemical impedance spectroscopy (EIS) is essential for non-invasive battery characterization. This paper addresses the challenge of adequate interpretation of EIS spectra, which are often complicated by overlapping internal phenomena occurring on similar time scales. We present, for the first time, a high-fidelity numerical time-domain electrochemical model that can virtually replicate experimental EIS spectra with three superimposed high-frequency semicircles, a transition to the diffusion tail at elevated imaginary values, and a tilted diffusion tail at low frequencies. These advanced features were made possible by extending state-ofthe-art porous electrode model with innovative sub-models for the double layer phenomenon at the carbon black/electrolyte and metal Li-anode/electrolyte interfaces, and transport phenomena of charged species through the solid electrolyte interphase at the Lianode interface. Additionally, we modelled the diffusion tail inclination by introducing representative active particles of varying sizes. Results from custom-made half-cells confirm the model’s ability to decipher EIS spectra more accurately compared to existing models. Moreover, innovative physics-based battery model that is capable of accurately modelling intra-cell phenomena can reveal internal states and physical parameters of batteries using measured EIS spectra. The model, therefore, also enables functionality of an advanced virtual sensor, which is an important diagnostics feature in next-generation battery management systems. © 2024 The Author(s). Published on behalf of The Electrochemical Society by IOP Publishing Limited. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 License (CC BY, http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted reuse of the work in any medium, provided the original work is properly cited. [DOI: 10.1149/ 1945-7111/ad6eb9] Manuscript submitted May 30, 2024; revised manuscript received August 4, 2024. Published August 27, 2024. Supplementary material for this article is available online Greek letters αCharge transfer coefficient [−] ϵPorosity [−] κLiquid phase conductivity [S/m] μChemical potential [J/mol] ΦPotential [V] φRatio [−] ρDensity [kg/m 3 ] σSolid phase conductivity [S/m] Latin letters ¯ c Average concentration [mol/m 3 ] aSpecific surface [1/m] CCapacitance [F/m 2 ] cConcentration [mol/m 3 ] DDiffusion constant [m 2 /s] ECell voltage [V] FFaraday constant [As/mol] fActivity coefficient [−] ICurrent density [A/m 2 ] iCurrent density [mol/m 2 s] jReaction rate per unit area [mol/m 2 s] LLength [m] MMolar mass [kg/mol] mMass [kg] RGas constant [J/mol K] rCoordinate in particle’s radial direction [m] SElectrode surface [m 2 ] TTemperature [K] tTransference number [−] VVolume [m 3 ] xCoordinate across the cell length [m] zCharge of the species [−] Q Capacity [Ah] x Lithiation level [mol] Z Impedance [Ω] Subscripts or superscripts ± Positively or negatively charged species +Positively charged species a Anodic am Active material app Applied b+cb Binder and carbon black brugg Bruggeman BV Butler-Volmer c Cathodic cat Cathode DL Double layer e Electrolyte ecb Electrolyte/carbon black interface eLi Electrolyte/Li-anode interface es Electrolyte/solid interface GITT Galvanostatic intermittent titration technique Li Lithium p Particle ref Reference s Solid sep Separator th Theoretical V Volume Modelling plays a crucial role in the development process of advanced Li-ion batteries. On the one hand, it is crucial for a better understanding of the fundamental physico-chemical processes. On the other hand, if models have a sufficiently high consistency with the real underlying phenomena taking place in batteries and thus a sufficiently high predictive capability, they are an important building block for the virtualisation of the development process and enable efficient exploration of the design space and virtual prototyping. Battery materials, such z E-mail: [email protected] Journal of The Electrochemical Society, 2024 171 080537 as the Nickel-Manganese-Cobaltoxide (NMC) used as a demonstration material in this study, exhibit significantly different material properties and are susceptible to a variety of degradation phenomena such as oxygen release 1 and the dissolution of transition metals, 2 formation of rock salt structures, 3 formation of cathode-electrolyte interphases (CEI), 4 cracking of aggregates/secondary particles 5,6 and others. The family of NMC materials also exhibits several intriguing phenomena related to their strong dependence of the diffusion constant on the lithium content in the active material. The group of C. Grey has recently revealed the emergence of the so-called kinetic phase separation, i.e. a spatial non-equilibrium variation of the SOC within single particles 7 and proposed that this phenomenon is caused by the SOCdependent Li-ion diffusion constant and is also linked to the Li mobility in the bulk material. 8 Moreover, the change in the diffusion constant is particularly remarkable for the highly lithiated particles, 9,10 where the kinetics of Li diffusion becomes much slower. All these phenomena and the properties of other materials, as well as the design and morphology of the cells and their operating conditions, influence the performance and safety of batteries. Therefore, their adequate performance, durability and safety critically depend on appropriate non-invasive monitoring tools that allow the detection of various in-cell phenomena related to battery performance and degradation. Indeed, such tools can significantly improve battery control and thus safety in real-world applications through a variety of measures such as limiting battery performance, avoiding certain operating conditions, triggering self-healing mechanisms 11 or the battery replacement or other countermeasures. In addition, non-invasive monitoring tools that enable the detection of intra-cell phenomena are also crucial in the battery development process to improve cell design and to provide a feedback loop to material design. One such powerful method for non-invasive monitoring of intracell phenomena is electrochemical impedance spectroscopy (EIS), which is a widely used non-destructive method for characterising electrochemical devices such as Li-ion batteries. 12,13 The main idea of this method is to apply a small potential (potentiostatic EIS or PEIS) or a small current (galvanostatic EIS or GEIS) excitation to the electrochemical device in the form of a sinusoidal perturbation with frequencies ranging across several orders of magnitude 13,14 (typically from kHz to mHz) and measure its response, i.e. amplitude and phase shift. The result is an impedance spectrum presented with a Nyquist diagram that characterises the phenomena in the battery that occur on the different time scales. Since a typical battery consists of several components and each component leaves its fingerprint in the EIS spectrum, 15,16 deconvoluting the effects of each individual process within the battery cell is a challenging task when interpreting the EIS spectrum. There are still many nonaligned interpretations of the EIS spectrum in the literature, 13,17 especially in the medium to high frequency range, while for the low frequencies it is generally recognised that the EIS spectrum provides profound information about transport phenomena in the active material. The requirement for a correct interpretation is crucially linked to the ability to decipher the contribution of specific performance and degradation related phenomena to the overall EIS spectrum. This requirement is, therefore, inherently interlinked with an adequate model used to interpret the spectra. This requires an adequate physical basis of the model for more detailed analyses and, above all, the determination of realistic values for the model parameters. There are many publications in the literature 18–20 that analyze the EIS spectra with the electrical circuit representation of the battery, i.e. with the so-called equivalent circuit models (ECMs), which consist of electrical components such as resistors and capacitors and in many cases also of constant phase elements (CPE) 21,22 or finite space Warburg elements. 23–25 Despite the fact that there exist ECMs which feature physicochemical consistency, e.g. 14,26 most of ECMs, e.g. 18–20 do not feature physicochemical consistency and they feature relatively simple model topology. These models have the advantage of being easy to implement and parameterise and require little computational effort. However, they generally do not provide a direct correlation between the model parameters, featuring realistic values, and physicochemical processes in batteries and may also be degenerate. 27 In particular, CPEs are notorious for lacking a physical basis, 28 although they are widely used for fitting the tilted low-frequency parts of the EIS spectrum corresponding to the transport of Li within the electrode material. Only recently, Lasia 29 has indicated potential explanation for CPE phenomena involving the electrosorption diffusion of anions. With the emergence of the artificial intelligence (AI) based approaches, there are also several publications that apply these approaches for extraction of parameters or states of the electrochemical devices. Babaeiyazdi et al. 30 estimated SoC using machine learning algorithms from the EIS measurements. Pradyumna et al. 31 combined EIS measurements with convolutional neural network to estimate the remaining capacity. Recently, Liu et al. 32 proposed a new method for automatic feature extraction from EIS measurements that combines variational autoencoders and bidirectional gated recurrent unit. The shortcomings of AI-based approaches are usually attributed to the dependence on large, high-quality data sets for training and the difficult interpretation of the underlying physical processes due to the nature of these models. On the other side of the battery modelling spectrum, there are the physics-based continuum models inspired by the pioneering work of Newman and co-workers 33,34 followed by a number of research models, e.g. 35–40 that are based predominantly on the pseudo-two-dimensional (P2D) approach 41 and validated with the discharge curves. Simplified variants of these models are so-called linearised electrochemical models 42–45 and represent a common approach for modelling EIS spectra. The advantage of these models lies in the fact that the impedance can be derived analytically and the fitting can be done directly in the frequency domain 42,43 with some additional assumptions, e.g. using constant material properties such as solid and ionic conductivity and diffusion constants. However, this is not fully consistent with real experiments, where measurements are conducted in the time domain and Nyquist plots only represent post-processed data. Measurements have to also be conducted under favourable signalto-noise ratios with sufficiently high amplitude of the excitation signals. Therefore, until now, frequency domain models were subjected to several limitations when interpreting EIS spectra of NMC materials, in particular at higher lithiation levels, where the diffusion constant exhibits strong dependency on the level of lithiation. 9,10 This becomes even clearer when it comes to deciphering the contributions of various phenomena in the low-frequency EIS spectrum, as a non-negligible amount of charge, e.g. in GEIS measurements, is transferred at low frequencies. In addition to the diffusion constant, the surface concentration and thus also the chemical potential and OCV of the material changes during such experiments. Previous studies have already reported that variations in solid-phase diffusion constant and OCV have a non-negligible influence on the EIS spectra. 46 More advanced models, of which there are not many in the literature, model the EIS spectra with the full electrochemical models in the time domain corresponding to real experiments. One such work is by R-Smith et al. 47 in which they modelled the EIS spectra with COMSOL Multiphysics of the full commercial cell at different degradation levels and extracted several ageing parameters, supported by advanced microscopy techniques. In addition, Xie et al. 48 studied effects of cycling on the impedance response of the Liion battery using multiphysics-based EIS simulation. Reformulated P2D has also been used for the simulation of the EIS spectra, e.g. Pathak et al. 49 presented a hybrid analytical-collocation approach for the fast simulation of the impedance response for a Li-ion battery using the P2D model. Teo et al. 50 simulated the dynamic electrochemical impedance response of full Li-ion cell during charging and discharging using the P2D model using a computationally efficient approach and Murbach et al. 51 extended the P2D model to evaluate the linear and nonlinear response of the full Li-ion cell. These models have in common that they model the EIS spectra of the full-cell configurations with the standard set of Newman-based governing equations, which include standard transport and potential equations, a charge transfer model and a corresponding double-layer model for the electrode material. 37,48,52 The results generated with Journal of The Electrochemical Society, 2024 171 080537 these models feature a single high-frequency semicircle and a tilted low-frequency tail. Such behavior can indeed, in general, be qualitatively observed in many experimentally measured EIS spectra, e.g. 53–55 These spectra appear, at a first glance, as simplistic Nyquist plots, but the collective behavior of both electrodes in the full-cell systems results in an EIS spectrum that is a superposition of a number of intra-cell phenomena occurring on similar time scales. Resultantly, these apparently simplistic Nyquist spectra indeed mask several phenomena occurring at similar time scales, hence, making their model supported deciphering very challenging. These phenomena range from solid-phase diffusion in the electrode particles, diffusion in the electrolyte phase (porous electrodes and separator), interfacial electrochemical reactions and contact resistances in both electrodes 16 to various impedance contributions due to battery degradation. 47 This challenge can be circumvented by using custom-made halfcells in this work. We have designed these half-cells (details of which can be found in appendix Section Experimental) in a way that specific underlying phenomena in such a half-cell occur at different time scales, thus allowing the detection of phenomena that are often hidden or superimposed in the full-cell experiments, but are important for the operation of the cell and thus important to be modelled. In summary, the currently published Newman-based models are not capable to virtually replicate relevant intra-cell phenomena with physically meaningful values of model parameters compared to the model proposed in this paper because, as will be shown later in the paper, they lack submodels of specific phenomena that are crucial for modelling multiple semicircles characteristic of specific intra-cell phenomena. In order to overcome the above challenges related to deciphering the phenomena contributing to the overall shape of the EIS spectrum, we present a first time-domain based electrochemical model, validated on the tailor made half-cell over a wide range of frequencies. This was made possible by originating from an advanced modelling framework presented in previous references, 56–58 which has been significantly extended to comply with the objective to virtually replicate EIS results with high fidelity and physically meaningful parameters. This means that it is capable of modelling three superimposed high-frequency semicircles, which are inherent to specific intra-cell phenomena, as well as tilted diffusion tail at low-frequencies. These advanced modelling capabilities include significant extensions i.e. sub-models, namely the double layer phenomenon related to the carbon black/ electrolyte interface, the double layer phenomenon at the metal-Li anode/electrolyte interface, and transport of charged species through the discretised porous solid electrolyte interphase (SEI) layer placed between the separator and the Li-anode. The results of the model demonstrate that additional double-layer phenomena at the carbon black/electrolyte interface, the effect of introducing the particle size distribution to the slope of the EIS spectrum in the low-frequencies (one of the possible explanations/compensations for the CPE elements) and the effect of diffusivity of charged species through the SEI layer on the EIS spectra are crucial to properly decipher the underlying phenomena and to predict physically meaningful values of the model parameters. Results, in addition, reveal a significant impact of the need for a sufficiently high discretisation resolution, which significantly impacts results of the EIS spectra and, thus, impact values of the material properties when extracted from the EIS spectra. We also provide detailed and consistent root-cause explanations. Supported by the non-invasive EIS measurements, the presented high fidelity model can be considered as a tool for advanced characterization of the intra-cell parameters and states of the battery and be used as a virtual sensor in next-generation battery management systems. Figure 1. (a) A schematic representation of the modelling domain, i.e. half-cell configuration including the cathode with electrode particles and carbon black, the porous separator, the porous SEI layer and the Li counter anode; (b) a detailed scheme of a single representative particle surrounded by the electrolyte (light blue color) and the carbon black and emergence of the double layer at the interface between the solid (particle and carbon black) and the liquid phase, denoted by pink and gray circles with +/−symbols; (c) indicated contributions of individual phenomena to the full EIS spectra namely the three double layer models, the transport of Li within the particle and the transport of Li + through the porous SEI layer, the color coding of the individual phenomena refers to the subfigures (b) and (d); (d) representation of the porous SEI layer and the double layer phenomena at the Li-anode interface denoted by red circle with +/−symbols. Journal of The Electrochemical Society, 2024 171 080537 The following Sections are organised as follows. Section Battery Model presents the governing equations of the developed model and Section Results presents the validation of the calculated EIS spectra in the time-domain versus the experimentally measured ones and impact of certain sub-models or parameters on the spectrum. The appendix presents the experimental measurements supporting the simulation results, the numerical approaches, the parameters used in the simulations and the corresponding parameter sensitivity analysis. Battery Model This Section presents the governing equations of the model and the corresponding boundary conditions, which are based on our previous publications 56–58 and have been extended by several submodels discussed in this section. To complement the equations, a schematic representation of the modelling domain and the most important models can be seen in Fig. 1. Governing equations.—The following assumptions were made in the presented model when dealing with low current and voltage amplitudes during the EIS procedure: the active electrode material, i.e. secondary particles, is modelled as a spherical particle, which is a reasonable approximation of the NMC material, 59 the electrode material is ideally immersed in the electrolyte, which is the case for dilute electrodes (Fig. A·1), the volume change of the electrode material during (de)intercalation is neglected. This is a common assumption for batteries with liquid electrolyte, while this simplification can be considered valid for the diluted electrodes analysed in this work. The electrode material exhibits a monotonous dependence of the chemical potential on the stoichiometry of the particles, while parameters such as temperature, transference number, activity coefficient, diffusion constant and conductivity of the electrolyte were kept constant due to the negligible temperature, concentration and potential gradients and their respective absolute fluctuations over the half-cell. Although these dependencies are available in the model 56–58 they were switched-off reduce the influence of parameters with negligible effect. The transport and mass balance of the charged species (Li-ions in this case) in the electrolyte is governed by the following equation ϵ∂( ) ∂=∇· ∇ − − ( − )+ [] + + ⎜⎟ ⎛ ⎝⎞ ⎠ c tDc t zF iajj aj, 1 ee e eff eees es es DL ecb ecb DL where the first term ∇ D c e eff erepresents the diffusion flux of the charged species with the effective diffusion constant governed by the Bruggeman relation ϵ=() D Dc e eff eee brug g 60,61 with ϵ e representing porosity of the electrode, the second term + +i t zFerepresents the migration term 62 which is dependent on the liquid-phase current density i e and is defined in the Eq. 3,t + and z + represent the transference number and the charge of the species, respectively, a es and a ecb represent specific surface of the electrode material and the carbon black in the electrode, respectively, and finally j es represents reaction rate of the electrochemical reaction at the electrode material/electrolyte interface (Eq. 9), jes DL represents reaction rate due to double-layer effects at the electrode material/electrolyte interface (Eq. 13) and jecb DL represents reaction rate due to doublelayer effects at the carbon black/electrolyte interface (Eq. 14). Additional terms from double-layer phenomena ensure mass conservation in the system. The last two terms in the Eq. 1are not relevant for the separator region of the modelling domain. Under assumption of electro-neutrality 36 over the modelled halfcell (Fig. 1a), one can write the following equation ∇· + ∇· = [ ] ii0, 2 es where besides previously mentioned i e , the i s represents solid phase current density. Liquid-phase potential Φ e is governed by charge leaving or entering the liquid phase, and can be written as κ κ ∇· = ∇· − ∇Φ + ( − ) + ∂ ∂∇ =− ( ( − )− ) [] + ± ⎜⎟ ⎜⎟ ⎛ ⎝ ⎛ ⎝⎞ ⎠ ⎞ ⎠ iRT Ftf cc Fa j j a j 211 ln ln ln , 3 ee eff ee eff e e es tot es DL ecb ecb DL where κe eff represents the effective liquid-phase conductivity evaluated from the Bruggeman relation κκϵ=()c e eff eebrug g ,f ± represents the activity coefficient 63 and finally, F,Rand Tare the Faraday’s constant, the gas constant and the temperature, respectively. Similarly to Eq. 3, the solid phase potential Φ s is governed by the charge leaving or entering the solid phase, and can be written in the form of Ohmic law as σ∇· = ∇·( ∇Φ ) = ( ( − ) − ) [ ]iFajjaj.4 ss eff ses tot es DL ecb ecb DL The effective solid phase conductivity σ eff is calculated as σ eff =σ(1 −ϵ e −φ V,b+cb ), where φ V,b+cb represents the combined volume fraction of binder and carbon black. The transport and mass balance of Li in the electrode material is determined by the general form of the diffusion equation where the Li distribution is determined by the gradient of the chemical potential μand is written as follows μ ∂ ∂=∇· () ∇() [] ⎛ ⎝⎞ ⎠ c t Dcc RT c,5 s sss s where c s and D s represent the solid-phase concentration and the diffusion constant of Li in the electrode material, respectively. Assuming spherical electrode particles, the Eq. 5can be rewritten to the following form μ μ ∂ ∂=∂ ∂ () ∂() ∂ =∂ ∂ () ∂() ∂ ∂ ∂[] ⎜⎟ ⎛ ⎝⎞ ⎠ ⎛ ⎝⎞ ⎠ c trr rDcc RT c r rr rDcc RT c c c r 1 1,6 s 2 2sss s 2 2sss s s s where rrepresents the radial coordinate in the electrode material domain. For the concentration dependent chemical potential, the following expression was used μ()= [] ⎜⎟ ⎛ ⎝ ⎞ ⎠ cRT c c ln , 7 ss s,ref where c s,ref represents a reference solid-phase concentration. Inserting expression for chemical potential from Eq. 7into the Eq. 5leads to a well known Fick’s diffusion equation for the transport of Li in spherical particles ∂ ∂=∂ ∂() ∂ ∂[] ⎛ ⎝⎞ ⎠ c trr Dcr c r 1.8 s 2ss2s However, the modelling framework is designed to allow the inclusion of arbitrary chemical potential expressions by directly solving the Eq. 6, including non-monotonic chemical potentials characteristic of phase-separating materials. 56 The charge transfer reaction on the surface of the electrode particles (electrolyte/solid interface) is modelled with the widely used Butler-Volmer equation 36,64,65 αηαη=−− (− ) [] ⎡ ⎣ ⎢⎛ ⎝⎞ ⎠⎛ ⎝⎞ ⎠⎤ ⎦ ⎥ ji F F RT F RT exp exp 1,9 es 0,es c es a es Journal of The Electrochemical Society, 2024 171 080537 where i 0,es is the stoichiometry-dependent exchange current density, αis the charge transfer coefficient for the cathodic and anodic reaction, and η es is the overpotential for the electrochemical reaction and is calculated using the following equation η=(Φ −Φ)− [ ]U.10 es se OCV The last term in Eq. 10 represents the stoichiometry-dependent open circuit voltage (OCV). In this work, the OCV of the NMC811 cathode material vs. Li/Li + was determined using the GITT experiment (Section Electrochemical measurements). Similar to the Eq. 9, the electrochemical reaction at the electrolyte/Li-anode interface (Fig. 1b) is also modelled with the Butler-Volmer equation 66 αηαη=−− (− ) [] ⎡ ⎣ ⎢⎛ ⎝⎞ ⎠⎛ ⎝⎞ ⎠⎤ ⎦ ⎥ ji F F RT F RT exp exp 1,11 eLi 0,eLi c eLi a eLi with i 0,eLi and η eLi are the exchange current density and the overpotential at the electrolyte/Li-anode interface. The overpotential η eLi is defined as η=(Φ −Φ)− [ ]U,12 eLi se Li OCV where the open circuit voltage of the Li-anode is set as constant =U0.0 V Li OCV . 67 As already announced in the discussion of the Eq. 1, the modelling framework also contains several models of double-layer phenomena at different locations in the modelling domain and follows the widely adopted formulation from Ong et al. 34 The reaction rate jes DL due to the double-layer effect on the surface of the electrode particles (electrolyte/solid interface, Fig. 1b) is calculated as =∂(Φ − Φ ) ∂[]jC t,13 es DL es DL se where C es DL represents the capacitance of the double-layer on the surface of the electrode particles. Another double-layer effect is modelled at the surface of the carbon black (Fig. 1b) which is in contact with the electrolyte =∂(Φ − Φ ) ∂[]jC t,14 ecb DL ecb DL se where C ecb DL represents the capacitance of the double-layer of the carbon black. The last double-layer phenomena is modelled at the electrolyte/Li-anode interface (Fig. 1d) and is evaluated as =− ∂Φ ∂[]jC t,15 eLi DL eLi DL e where C eLi DL represents the capacitance of the double-layer on the Lianode foil in contact with the electrolyte and solid-phase potential Φ s is set to be 0.0 V. The Eq. 15 enters into the Eqs. 21 and 25 as a boundary condition at the interface. Finally, the porous SEI layer that forms on the surface of the Lianode was also modelled. The thin layer was spatially discretised, similar to the electrode an the separator. To account for the additional contribution of the diffusion impedance of the charged species through the porous SEI, the transport and mass balance equation can be written as a simplified version of the Eq. 1 ϵ∂( ) ∂=∇· ∇ − [ ] + + ⎜⎟ ⎛ ⎝⎞ ⎠ c tDc t zF i,16 e SEI e e eff,SEI ee where ϵ e SEI is the porosity of the porous SEI layer and D e eff,SEI represents the effective diffusion constant of the charged species through the SEI layer which is calculated again by the Bruggeman relation ϵ=() D D e eff,SEI e SEI e SEI brug g . 60,61 The Eq. 16 does not contain any source/sink terms, as the experiments were carried out with fresh cells without long-term cycling, as ageing effects are not within the scope of this work. In addition to the concentration, the liquid-phase potential Φ e in the porous SEI layer was also modelled using the following equation [] κ κ ∇· =∇· − ∇Φ + ( − ) + ∂ ∂∇= + ± ⎜⎟ ⎜⎟ ⎛ ⎝⎛ ⎝⎞ ⎠⎞ ⎠ 17 i RT Ftf cc211 ln ln ln 0, e e eff,SEI ee eff,SEI e e where κe eff,SEI represents the conductivity of the effective liquidphase in the porous SEI layer and is evaluated from the Bruggeman relation κκϵ=( ) e eff,SEI ee SEI brugg. Boundary conditions.—This Section presents the boundary conditions used in the computational domain analysed in this paper. They are organised according to the corresponding governing equations presented above. The coordinates of the individual boundary conditions are shown schematically in Fig. 1. Electrolyte concentration.—The following boundary conditions are imposed for the transport and mass balance of the charged species in the electrolyte governed by Eq. 1. The zero-flux boundary conditions are imposed at the current collector/cathode boundary: −∇∣= [] = Dc 0. 18 x e eff e0 Continuous flux boundary conditions are imposed inside the cell: −∇∣ =−∇∣ [] == −+ Dc Dc ,19 xL xL e eff ee eff e cat, cat, −∇∣ =−∇∣ [ ] == +− ++ Dc Dc .20 xL xL e eff ee eff e cat sep, cat sep, The reaction rate j eLi , which acts as a source of Li-ions into the computational domain, and reaction rate jeLi DL due to the double-layer phenomena are imposed at the Li-anode boundary: −∇∣ = −+ [] =++ Dc jj F.21 ex L e eff,SEI eLi eLi DL cat sep SEI Electrolyte potential.—The following boundary conditions are imposed for the Eq. 3that governs liquid-phase potential Φ e of the charged species in the electrolyte. Similar to the Section Electrolyte concentration, Zero-flux boundary conditions are imposed at the current collector/cathode boundary: κ−∇Φ∣= [ ] =0. 22 x e eff e0 Continuous flux boundary conditions are imposed inside the cell: κκ−∇Φ∣ =−∇Φ∣ [] == −+ ,23 xL xL e eff ee eff e cat, cat, κκ−∇Φ∣ =−∇Φ∣ [ ] == +− ++ .24 xL xL e eff ee eff e cat sep, cat sep, Table I. Lithiation level and corresponding open circuit voltage (OCV) of the NMC811 material obtained from measurements shown in Fig. A·2. x [mol] OCV [V] 0.338 4.055 0.551 3.805 0.747 3.665 Journal of The Electrochemical Society, 2024 171 080537 The reaction rates j eLi and jeLi DL are again imposed at the Li-anode boundary: κ−∇Φ∣ =−+ [] =++ jj.25 xL e eff eeLi eLi DL cat sep SEI Solid phase potential.—The following boundary conditions are imposed for the governing Eq. 4that governs the solid phase potential Φ s . The applied (dis)charge current density I app is imposed at the current collector/cathode boundary condition: σ−∇Φ∣ = [ ] =I.26 sx eff 0app The zero-flux boundary condition is imposed at the cathode/ separator boundary since no electric current is supposed to flow into the separator domain: σ−∇Φ∣ = [] =0. 27 xL s eff scat Electrode particle concentration.—Assuming spherical particles, the solution of the Eq. 6requires two boundary conditions for the solid-phase concentration c s in the electrode particle. The boundary condition of zero flux is applied in the centre of the electrode particle: −∇∣ = [ ] = Dc 0, 28 rss0 whereas on the surface of the particle the reaction rate is predetermined due to the electrochemical reaction −∇∣ = [ ] = Dc j.29 rRss es p Results In this section, the results of the proposed modelling framework validated with the experimental measurements are presented, followed by the variation of additional parameters to investigate their effects on the EIS spectra. Experimental validation.—The model was validated with the experimentally measured EIS spectra on the laboratory built half-cell with the electrode material NMC811. Details on the experimental electrode preparation, cell assembly and the electrochemical measurements can be found in the appendix Section Experimental. The starting point for the model validation with the experiments were raw measurement data points consisting of frequency, real and complex part of the impedance which were imported into the model. Three EIS curves were measured at three different voltage levels (see Table I). For each frequency, galvanostatic sinusoidal excitation was used as a boundary condition at the current collector/cathode boundary (Eq. 26) with an amplitude of C/10 4 to ensure a linear response without higher harmonics 13 and a duration of 20 periods with 10 4 time-steps within each period for accurate results. From the last period, the amplitude of the voltage response and the phase shift between the sinusoidal current and voltage time traces were extracted. Based on this information, the real and complex parts of the impedance for a single frequency were calculated. After completing the simulations for each frequency, the rootmean-square deviation (RMSD) between the experimental measured values and the simulated values was determined. The RMSD value was then entered into to the self-implemented evolutionary algorithm called Differential evolution proposed by Storn and Price 68 which performed an optimisation of the problem by minimising the total RMSD value (i.e. the sum of the individual RMSD values for all three cases and all frequencies) by iteratively improving set of fitting parameters. The parameters that were fitted are as follows: exchange current density i 0,es and i 0,eLi from Eqs. 9and 11, respectively, double layer capacitances C es DL, C ecb DL and C eLi DL from the Eqs. 13,14 and 15, solid-phase diffusion constant D s from Eq. 8, specific surface of the carbon black a ecb from Eqs. 1,3and 4, the diffusion constant of the Li-ions through the porous SEI layer at the Li-anode D e SEI from Eq. 16 and finally the number of representative electrode particles was fitted. The size of the electrode particles was determined using a Gaussian distribution according to the manufacturer’s size specification in Table A·I and SEM images of a pristine electrode material. The specific surface of the carbon black a ecb was Figure 2. A comparison between experimentally measured and simulated EIS spectra at different degrees of lithiation corresponding to the following open-circuit voltages: (a) 4.055 V; (b) 3.805 V; and (c) 3.665 V. To improve comparison, the positions of five selected frequencies over several orders of magnitude in the spectra are marked with the different coloured dot symbols. Journal of The Electrochemical Society, 2024 171 080537 fitted due to the fact that not all the surface of the carbon black is exposed to the electrolyte. Of the above fitting parameters, only the exchange current density i 0,es and solid phase diffusion constant D s were fitted separately for each case to account for the dependency of the i 0,es and D s to the different degrees of lithiation in the NMC material, 10,16 whereas the other fitting parameters were kept constant for all three cases in Fig. 2. For this validation, we discarded the commonly used parabolic ansatz before the exponential parts of the Butler-Volmer equation. Therefore, the entire pre-exponential term was fitted as a single parameter due to the fact that the ansatz qualitatively captures the dependence of the exchange current density to the lithiation level at the particle’s surface. 16 Furthermore, we assumed constant solid-phase diffusion constant due to negligible change in transferred charge during the galvanostatic EIS excitation for each EIS measurement (Table I). Figure 2shows the comparison between measured and simulated EIS spectra (calculated with the minimisation of the RMSD value) at three different cell voltages corresponding to three different degrees of lithiation, i.e. stoichiometries, of the NMC material (see Table I). In addition, five selected frequencies are labelled in each of the two EIS curves (measured and simulated) to illustrate their relative position on both spectra, as the Nyquist plot does not explicitly present the frequency information. The resulting simulated EIS spectra show good agreement with the experimentally measured spectra over a wide frequency range from 100 kHz to 1 mHz. Some important aspects from Fig. 2that support the model’s capabilities to model EIS spectra with a good accuracy, that exceeds that of other models analysed in the Introduction, are worth mentioning. Firstly, it is important to emphasise that the model is capable to model smaller semi circle at the highest frequencies, i.e. Re(Z) approximately between 3 Ωand 250 Ω. Its occurrence is due to the electrochemical reaction and the double layer effects at the separator/SEI/Li-anode interface 16 (Fig. 1d and Eqs. 11 and 15). This statement is supported by the fact that the impedance of the semi circle remains nearly constant for all three case which further confirms the adequacy and capability of the included sub-models at the Li-anode interface. The second semi circle, also known as the charge transfer (CT) semi circle, in the medium frequency range is due to electrochemical reaction and double layer effects at the particle’s surface 16,28 (Fig. 1b and Eqs. 9and 13). The impedance of this semi circle changes significantly between the three cases, with the largest change occurring at the lowest voltage (Fig. 2c), i.e. for the most lithiated NMC material of the three cases. The change in impedance correlated directly with the change in exchange current density i 0,es in Eq. 9, which was fitted separately for each case, as i 0,es depends on the surface stoichiometry of the electrode particles. The transition from the charge transfer semi circle to the diffusion tail occurs in the analysed case at -Im(Z)>0 for all three experimental EIS spectra. The same phenomenon can also be observed in the literature. 53–55 In special cases, as for example single electrode particle and micro-electrode technique 69 without the use of carbon black, this transition occurs near the Re(Z) axis. However, we identified this transition to be a consequence of the double layer phenomena at the electrolyte/carbon black interface, which required inclusion of the novel carbon black double layer model in the electrochemical model (Fig. 1and Eq. 14). This model is missing in the standard set of Newman-based governing equation and, hence, in the models from the literature. 35–40 The obtained value from the minimisation algorithm of the carbon black double layer capacitance = C 5.74 ecb DL μF/cm 2 (or 0.0574 F/m 2 in the Table A·I) shows a very good agreement with the experimental determination of specific double-layer capacitance due to carbon black conductive additive presented in appendix Section Experimental determination of specific double-layer capacitance due to carbon black conductive additive and in the Fig. A·4, which confirms the addition of the novel model. As shown later in the Fig. 4, the elevated transition can indeed still be modelled with reasonable accuracy, but at the cost that the values of other parameters have to be compensated for and thus become unphysical, confirming once again the need to include the carbon black double layer model. Such addition (Section Impact of novel submodels to the EIS spectra), which consistently represents the effects at the electrolyte/carbon black additive interface, makes possible simulating Figure 3. A comparison between experimental and full model EIS results from Fig. 2with the modelling results of the model without the double layer model at the carbon black/electrolyte interface and at the metal-Li anode/ electrolyte at different degrees of lithiation corresponding to the following open-circuit voltages: (a) 4.055 V; (b) 3.805 V; and (c) 3.665 V. Journal of The Electrochemical Society, 2024 171 080537 experimentally observed transitions from the charge transfer semi circle to the diffusion tail at -Im(Z)>0 and the obtaining of realistic model parameters, and thus, represents a meaningful and innovative extension of the standard set of Newman-based governing equation. The diffusion tail, which is due to the to the transport of Li within the electrode particles, corresponds to the low frequency diffusion impedance. 16,28 In this frequency range there are two important aspects, that need to be emphasised. The first is the challenge of modelling the tilt in the diffusion tail and the second is the agreement between measured and simulated values of the complex impedance at the lowest frequencies. The tilt proved to be challenging to model without the use of the CPE elements, e.g. 16 and with a single representative averaged sized particle which most of the P2D electrochemical models in the literature still use, e.g. 37,48,52 In this work, we propose the inclusion of number of differently sized representative particles in the electrode, which further enhances consistency between the modelled and experimental system. This functionality leads to combining different eigen diffusion time constants to achieve a reasonable tilt in the diffusion tail, that is characteristic of the measured spectra. This approach is also supported by the experimental work of Wu et al. 70 in which they demonstrated impact of the different particle size distributions on the EIS spectra. As discussed in the Introduction, the particle size distribution might be one of the possible explanations for the tilt, but we do not claim that is the only one. Diffusion impedance of charged species through the SEI layer at the Li-anode interface might also contribute to the mid to low frequency parts of the EIS spectra, i.e. transitioning from the charge transfer semi circle to the diffusion tail as indicated in Fig. 1. Later in the Fig. 5we demonstrate that a single particle of average size cannot replicate the tilt in the diffusion tail and additionally, in Figs. 6and A·5 we demonstrate the potential impact of the hindered diffusivity of charged species through the SEI layer on the increase of impedance in the EIS spectra. Knowledge of the size distribution of the electrode materials used in the cell (see size distribution data in Table A·I) plays another crucial role in physically based models. The third aspect regarding the diffusion tail is the agreement between measured and simulated values of the complex impedance at the lowest frequencies, which in all cases show a very good agreement with very small deviations. According to the analytical derivations the impedance of the single particle model impedance by Meyers et al. 46 the “height”of the imaginary/ complex impedance at the lowest frequencies depends on the local gradient of the OCV with respect to the actual degree of lithiation. It is therefore important to emphasise here that an accurate OCV measurement (Fig. A·2) is crucial for plausible modelling of the EIS spectra. The presented results show that the proposed model is able to capture the most important features from the experimentally measured EIS spectra, i.e. two distinguishable semi circles, an elevated transition to the low frequency diffusion tail and the tilt of the aforementioned diffusion tail. Based on these facts, the proposed model can be characterised as advancement over the existing models in the field of modelling EIS spectra in the timedomain. Impact of novel submodels to the EIS spectra.—To further underpin importance of the newly added modelling features, this Section presents the results of a full model and several variants of this model, in which these features were systematically switched off to demonstrate their effects on the EIS spectra. Figure 3shows the comparison of the experimental and simulated EIS spectra (from now referred as the “Full model”) with the EIS spectrum obtained without the double layer model at the electrolyte/ carbon black interface and with the EIS spectrum obtained without the double layer model at the Li-anode interface for the three different OCV voltages as in the previous Fig. 2. The other parameters were kept constant as they were calculated for the Full model. Switching off the double layer model at the electrolyte/carbon black interface (Fig. 1b and Eq. 14) and retaining the remaining modelling parameters the same, causes the spectrum to approach Re (Z) axis at the transition to the low frequency solid-diffusion impedance tail for all three cases, which is not in agreement with the measurements where this transition is substantially elevated in terms of -Im(Z) values. Furthermore, the high frequency semicircle Figure 4. A comparison between experimental and full model EIS results from Fig. 2with the fitted results of the model without the double layer model at the carbon black/electrolyte interface and with the model with the redistributed carbon black capacitance to the capacitance of the double layer at the Li-anode. The simulations were performed at different degrees of lithiation corresponding to the following open-circuit voltages: (a) 4.055 V; (b) 3.805 V; and (c) 3.665 V. Journal of The Electrochemical Society, 2024 171 080537 migration resistance contribution of the carbon black electrode and diffusional resistance of electrolyte in separator, finally transitioning into vertical capacitive impedance tail at about 0.1 Hz. The obtained EIS spectra were analysed by calculating effective capacitance of measured impedance tail by using the Brug type of relation for non-ideally polarised electrode. 73,74 For comparison we also did simple analysis where we applied relation for an ideal capacitor specifically for only the measurements at 1 mHz. Specific (area) capacitance was calculated by normalising the value of capacitance by the surface area of carbon black; the latter was calculated from the data about the mass and BET surface area of carbon black (62 m 2 g −1 ). The obtained results are shown in Fig. A·4 where it can be seen that simple ideal-capacitor approximation matches quite well with the results obtained by using the Brug relation, whereby the ideal-capacitor approximation overestimates capacitance values by about 5%–10%. It is interesting to note in Fig. A·4 that double-layer capacitance exhibits local minima at about 2.8 V (vs. Li + /Li 0 ). At the voltages most relevant for the present study the corresponding values of specific capacitance are estimated to be: 5.9 μFcm −2 (3.665 V), 6.2 μFcm −2 (3.805 V), and 6.4 μF cm −2 (4.055 V), respectively. SEI diffusion impedance analysis.—To complement the Fig. 6 from the main text where the absence of the porous SEI layer at the Li-anode interface did not significantly change the EIS curves compared to the full model result, a sensitivity analysis was performed to analyze the impact of varying the value of the diffusion constant D e SEI of the Li-ions through the discretised SEI layer on the EIS spectra. This analysis is shown in Fig. A·5 and is applied to all three EIS curves that correspond to different levels of lithiation of the NMC811 material (Table I). The diffusion constant D e SEI was varied from 4 · 10 −10 m 2 /s to 4 · 10 −15 m 2 /s. By decreasing D e SEI one order of magnitude at time, until =· − D 410 e SEI 13 m 2 /s there is no major impact on the EIS curves in all three cases. With the =· − D 410 e SEI 14 m 2 /s a more pronounced increase in the real part of the impedance can be observed due to an additional diffusion impedance contribution. Further decreasing the D e SEI by another order of magnitude has an even more profound effect on the EIS curves. The real part of the impedance increases significantly, especially the middle frequency range in Figs. A·5aandA·5b, while the values of the complex part of the impedance remain the same. In addition, Fig. A·5c shows an emergence of the third semicircle, which is more clearly defined due to the increased SEI diffusion impedance. Numerical approach.—The set of coupled partial differential equations presented in Section Battery Model was numerically discretised in the two domains, i.e. the electrode domain and the particle domain, using the Finite Volume Method (FVM) 75 approach, resulting in a set of coupled Differential algebraic equations (DAE). DAE system was solved numerically with the Implicit Differential-Algebraic solver (IDA), a general purpose solver for the initial value problem (IVP) for systems of DAEs from the SUNDIALS software suite. 76,77 Numerical meshes of the computational domain consisting of cathode and separator were generated using the open source 3D finite element method (FEM) mesher Gmsh. 78 In this study, a 1D mesh was used, although the modelling framework also supports unstructured 2D meshes as shown in our previous work. 57 An in-house implemented library then computes all the necessary geometric and connectivity information needed for the FVM method. The particles, on the other hand, were discretised spherically in the 1D radial dimension. The numerical convergence related to the discretisaton of the particles has already been shown in the main text in Fig. 7. In addition to Fig. 7, where the importance of sufficient particle discretisation was highlighted, Fig. A·6 shows the convergence of the EIS spectra when changing the number of control volumes in the cathode and separator. To ensure comparable results between the different numerical meshes, a single particle of average size was selected for each control volume in the cathode domain. The SEI layer at the Li-anode was spatially discretised with 4 CVs in all cases. The Fig. A·6 shows that even the lowest discretisation Figure A·6. Numerical convergence of EIS spectra when changing the discretisation number along the half-cell, i.e. the number of control volumes in cathode and separator. The parameters were taken from the full model case at 3.805 V (Fig. 2b), with exception of using a single particle per each control volume of the cathode. Table A·I. Model parameters used in the full model simulations shown on Fig. 2. Remarks: † measured, ‡ assumed from the literature, ⋆ fitted, • calculated. Parameter Unit Value brugg —1.5 ‡79 ϵ e,cat —0.845 • ϵ e,sep —0.880 • ϵ e SEI —0.02 ‡72 D e m 2 /s 4 · 10 −10 ‡80 D e SEI m 2 /s 1.573 · 10 −13 ⋆ κ e A/V m 0.335 ⋆ T°C 24.0 † t + —0.363 ‡79 a ecb 1/m 9.700 · 10 6⋆ z + —1 ( ) +( ) ∂ ∂ ±cT1, f c ln ln e e —1.0 ‡79 σA/V m 1.0 ‡81 φ V,b+cb —0.141 • D p =2R p μm D10: ≈5μm; D50: 9-15 μm; D90: <30 μm † ; μmR 2 p,max:<45 μm; 2R p,mean =6.46μm † i 0,es A/m 2 4.445 ⋆ (Fig. 2a); 5.790 ⋆ (Fig. 2b); 1.574 ⋆ (Fig. 2c) D s m 2 /s 4.723 · 10 −14⋆ (Fig. 2a); 5.243 · 10 −14⋆ (Fig. 2b); 2.186 · 10 −14⋆ (Fig. 2c) α c —0.5 ‡79 α a —0.5 ‡79 U OCV V see Fig. A·2 † i 0,eLi A/m 2 0.766 ⋆ U Li OCV V 0.0 ‡67 C es D L F/m 2 0.831 ⋆ C ecb D L F/m 2 0.0574 ⋆ C eLi D L F/m 2 0.0134 ⋆ L cat μm 8.0 † L sep μm 200.0 † L SEI nm 20.0 ‡72 M am /V M,am kg/m 3 4757.0 • Journal of The Electrochemical Society, 2024 171 080537 mesh with 1 CV in the cathode and 10 CVs in the separator sufficiently models the EIS spectra. The reason for this result lies in the fact that the cathode in the half-cell itself is very thin and measures only 8 μm across and in combination with the high porosity of both the cathode and the separator favours the transport of Li-ions across the half-cell. Based on this result, the mesh with 15 control volumes was selected (1 CV in the cathode, 10 CVs in the separator and 4 CVs for the SEI layer) for the experimental validation with the Differential evolution algorithm. Model parameters.—This Section presents model parameters (Table A·I) used in the modelling framework in the simulation results presented in this work. Sensitivity analysis of the fitted parameters.—This Section contains an analysis of the fitted parameters for all three cases. The value of each individual fitted parameter (Table A·I) was relatively perturbed by +10%. The resulting EIS spectrum was compared with the unperturbed spectrum by calculating the RMSD in the units of Ωusing the following equation = ( () − ()) +( () − ()) [·] N RMSD 1ReZReZ ImZImZ, A1 points P2P2 where the letter P indicates the perturbed value of the real ( ( ) Re Z )or complex part ( ( )I mZ) of the impedance Z in the EIS spectrum, which consists of N points points. Table A·II shows the results of the sensitivity analysis. Both exchange current densities (i 0,es and i 0,eLi ), especially the i 0,es , exhibit higher sensitivity for all three EIS cases. This is to be expected as the impedance of the charge transfer is inversely proportional to the exchange current density. 46 Double-layer capacitances show a slightly lower sensitivity on average. The exception is the capacitance of the double layer at the carbon black interface at 3.665 V in combination with the specific surface area a ecb of the carbon black that has also higher sensitivity in the latter case. 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