Advanced Porous Electrode Modelling Framework Based on More Consistent Virtual Representation of the Electrode Topology

The paper proposes an advanced continuum level modelling framework characterized by a more consistent virtual representation of electrode topology to enhance prediction capability and generality of porous electrode theory based models. The proposed modelling framework, therefore, establishes the missing link between the mesoscopic scale with a detailed 3D representation of electrode topology and the continuum single cell scale, where interrelation to the real electrode topology was missing. This link is established by elaborating a uni ﬁ ed approach for modelling materials with signi ﬁ cantly different topologies of active material by virtually creating agglomerates, representing secondary particles, from primary particles. Proposed approach relies on multi-particle size distribution of primary particles and particle-to-particle connectivity. Generality of the proposed modelling framework is demonstrated by simulating LFP and NMC materials featuring signi ﬁ cantly different electrode topologies by the same modelling framework while adapting only virtual representation of electrode topologies and intrinsic material properties. Credibility of the proposed modelling framework is con ﬁ rmed through good agreement with experimental results for various discharge tests. Insightful simulation results also reveal background of the topologically driven low Li utilization at high current densities of the LFP material and topologically driven voltage response difference during the memory effect of different LFP materials.

The paper proposes an advanced continuum level modelling framework characterized by a more consistent virtual representation of electrode topology to enhance prediction capability and generality of porous electrode theory based models. The proposed modelling framework, therefore, establishes the missing link between the mesoscopic scale with a detailed 3D representation of electrode topology and the continuum single cell scale, where interrelation to the real electrode topology was missing. This link is established by elaborating a unified approach for modelling materials with significantly different topologies of active material by virtually creating agglomerates, representing secondary particles, from primary particles. Proposed approach relies on multiparticle size distribution of primary particles and particle-to-particle connectivity. Generality of the proposed modelling framework is demonstrated by simulating LFP and NMC materials featuring significantly different electrode topologies by the same modelling framework while adapting only virtual representation of electrode topologies and intrinsic material properties. Credibility of the proposed modelling framework is confirmed through good agreement with experimental results for various discharge tests. Insightful simulation results also reveal background of the topologically driven low Li utilization at high current densities of the LFP material and topologically driven voltage response difference during the memory effect of different LFP materials.  Li-ion batteries are one of the most widespread energy storage devices, covering a very broad range of mobile and increasingly also stationary applications. All these applications share common high level requirements, namely, batteries with higher energy and power density as well as a prolonged lifetime and moreover increased safety at reduced battery manufacturing costs are anticipated. Additionally, the broad range of applications also imposes several application specific objectives which cover operating temperatures, specific load profiles and specific durability criteria. Therefore, a single design cannot fulfil all requirements in an optimal way and thus tailoring battery design to a specific application with the aim to approach engineering limits, represents a significant challenge.
On the global market there are namely several different types of electrode active materials used in the production of Li-ion batteries. Commonly used cathode materials are for example Lithium Iron Phosphate (LFP), Lithium Manganese Oxide (LMO), Lithium Cobalt Oxide (LCO), Lithium Nickel Cobalt Aluminium oxide (NCA) and Lithium Nickel Cobalt Manganese oxide (NCM), whereas for the anode, Graphite is usually used with an optional addition of Silicon. Inevitably, such a variety of materials also inherently reflects their different Li (de)intercalation dynamics during the (dis)charging process and their different equilibrium chemical potentials. 1 Moreover, the lattice structure thus also dictates various possible diffusion paths in the host matrix. Consequently, a phase-separating LiFePO 4 (LFP) cathode material features fast Li transport along 1D crystal diffusion paths, 2 while NCM features isotropic transport of Li in 2D crystal planes. 3,4 The size distribution of individual primary particles and secondary particles, i.e. agglomerates, is another parameter influencing (de)intercalation dynamics, and therefore, also influences overpotentials at different rates of (de)lithiation. [5][6][7] In battery-grade LFP materials the primary particles are usually packed into secondary particles (agglomerates) of various geometries, e.g. platelet-like agglomerates (Fig. 1a), characterising this type of LFP as a suitable material for high power applications. On the other hand, NCM materials in general are prepared in form of much larger micro-meter sized secondary particles, reminiscent of a sphere as discernible in the SEM image (Fig. 1b) of a pristine material.
One of the promising approaches to tackling the large diversity of requirements and materials relies on virtual prototyping with models featuring a high level of prediction capability in the early development phase. Virtual prototyping enables high fidelity exploration of the design space and pinpointing the most promising designs in a virtual environment before the first real prototypes are available. This enables a significant reduction in development time and costs. However, virtual prototyping in the early development phase also imposes a significant challenge on the applied modelling framework, which should feature a sufficient level of detail and a sound physical basis to ensure both sufficiently high prediction capabilities and generality.
A promising tool for supporting the early stage design of batteries are continuum porous electrode battery models, based on or inspired by the pioneering work in the field of porous electrode theory published by Newman et al. in 1975. 8 Even though Newman theory forms the basis of multiple research models, e.g., [9][10][11][12][13][14] and commercial battery models, e.g., 15,16 it possesses certain deficiencies in the context of the present requirements on advanced battery models. The deficiencies of Newman based continuum models on the level of an electrode sandwich can be summarized as: • models do not adequately represent electrode topology including ion and electron wiring by (a) assuming ideally immersed particles in the electrolyte and neglecting the connectivity between particles, this being of particular importance in the LPF material (Figs. 1 and 3 as well as Fig. 2a) and (b) commonly assuming a constant size of particles; • models rely on Fickʼs diffusion equation, 5,14,15,17 which oversimplifies intra-and inter-particle Li transport; • models rely on multiple empirical correlations, where in particular the empirical correlations of potentials as a function of lithiation frequently hinder modelling specific intra-electrode phenomena 5,12,14,18 ; Due to the before mentioned deficiencies, Newman based continuum models fail to predict some recent experimental observations, e.g. particle-by-particle (de)lithiation forming a so-called mosaic pattern at low (dis)charge rates, 19-21 preferential diffusion of Li along 1D channels in an orthorhombic crystal structure of LFP, 2 preferential initial lithiation of smaller particles at low C-rates and nearly uniform particle lithiation at higher C-rates, 22 existence of both Li-rich and Li-poor phases in the surface layer of an active particle, 23,24 incoherent nanoscale domains hindering Li transport, 22 memory effect, 25 etc.
To tackle these challenges, the literature offers several models on lower scales or phenomenological modelling representations aimed at modelling specific phenomena in the batteries. On the particle level, a paradigm shift in the understanding of Li insertion mechanisms in LFP was initiated by Delmas et al. 20 with the "domino-cascade" mechanism in which the LiFePO 4 /FePO 4 interface propagates as a wave through the crystal as a consequence of the minimisation of elastic energy, and Dreyer et al., 19 where the particle-by-particle intercalation mechanism in a phase separating LFP material at low currents is explained through an insightful thermodynamically based theory. Bai et al. 26 demonstrated, using an electrochemical phase-field model, that phase separation is suppressed and particles are filled homogeneously when applying currents above a certain critical value, while recently Abdellahi et al. 27 provided a first principles reasoning on the preferential formation of a diffuse ac interface with a large intermediate solid solution region after spinodal decomposition, as opposed to the commonly assumed bc interface.
On a particle ensemble level, Orvananos et al. published a series of modelling papers [28][29][30] studying interactions between active particles in the LFP cathode. In 28 they investigated intra-particle phase separation. Reference 29 demonstrates a continuum model of Li transport kinetics between active particles, which proceeds via the electrolyte only or additionally via direct contact between particles, whereas their modelling domain was limited only to a fraction of the electrode with direct modelling of a very limited number of particles. Kondo et al. 31 used phenomenological modelling representations to explain the voltage response during the memory effect and the voltage response during relaxation induced polarization (RIP) while assuming constant concentration and potential fields in the cathode and by simplifying the influence of particle size on the equilibrium chemical potential of the particle. Recently, Zelič et al. 32 revealed a Figure 1. SEM image of (a) pristine commercial LFP cathode material and (b) pristine commercial NCM811 cathode material. consistent phenomenological background for the onset or absence of the memory effect using an advanced continuum Li-ion battery modelling framework featuring sequential multi-scale model linking.
On the electrode level, Farkhondeh et al. 33,34 published a simple mesoscopic model, which accounts for the inhomogeneity of physical properties in phase-changing insertion materials. With a model relying on a simple linear relation between the overpotential and the particle current, they have succeeded at representing a voltage vs capacity trend of LFP at different rates. 33,34 This was made possible by the introduction of a kind of ad hoc bin specific ohmic hindrance that impedes the insertion-de-insertion reaction. Although the authors state that the model does not include any geometric detail, 33 it will be reasoned later in the paper that within a more rigorous modelling framework proposed in this paper, the approach proposed in 33,34 mimics some of inherent geometrical properties of the LFP. Literature also offers a limited number of Newman based models that consider particle size variation, e.g. [5][6][7] Guhlke et al. 7 followed a similar approach as 33,34 when modelling LFP material. They introduced a multi-particle size distribution and considered stochastic distributions of volumes and surfaces with a possibility of adjusting the ratio between the active surface and particle surface, without considering its direct relation to the topology of a real electrode and thus particle connectivity. Focusing only on different particle sizes, the model of Mao et al. 5 accounts for multiple particle sizes of the active materials in terms of three distributions: one for LMO particles, one for NCM primary and one for NCM secondary particles. Thereby, they have accounted for a material and electrode specific design featuring two distinct size distributions of NCM particles. Among other findings, Mao et al. 5 indicate that the assumption of non-porous secondary particles is reasonable in the operating condition of interest in their study, i.e. at least up to 2C applied current, and for the applied material. This, in specific cases, significantly simplifies the model and relaxes the need for characterising the actual secondary porous particles. Despite introducing particle size variations, 5,6 neglect particle connectivity and assume Fickian diffusion.
To present significant progress in the area of continuum porous electrode Li-ion battery modelling, an advanced modelling framework inspired by Newmanʼs approach is proposed in this paper. To adequately address the aforementioned recent experimental observations and resulting modelling challenges, the presented advanced porous electrode modelling framework, addressing the entire single cell or half-cell, focuses on including mesoscopic effects through: • a more consistent representation of the electrode topology including particle connectivity and particle size distribution capable of plausible virtual representations of different materials, • redistribution of Li between particles via electrolyte or via direct contact by considering the electrode topology and material properties, • the application of a thermodynamically derived active particle equilibrium potential from regular solution theory for phase separating materials.
The proposed modelling framework, therefore, establishes the missing link between the mesoscopic scale with a detailed 3D representation of the electrode topology and the continuum scale, where interrelation to the real electrode topology is missing. The proposed modelling framework namely enables the consideration of basic electrode topology characteristics at the continuum scale, thus opening possibilities for more profound electrode topology analyses on the computationally efficient continuum scale.
The capabilities of the proposed modelling framework are primarily validated on the LFP cathode material, which features a non-monotonous particle equilibrium chemical potential as a function of particle lithiation, small primary particles and fast diffusion in 1D, thus being a representative of a very challenging cathode material for Li-ion battery model validation. To demonstrate the generality of the proposed modelling framework, the results of the NCM cathode material, calculated by the same modelling framework and adapting only intrinsic material properties, are also presented. Therefore, the application of the proposed generic porous electrode modelling framework on the level of the entire single cell or half-cell Li-ion battery, for the first time: • represents a unified approach for modelling materials and with significantly different electrode designs by establishing a link to their topological characteristics, • allows for the modelling of the following phenomena in LFP batteries: • topologically driven low Li utilization at high current densities for materials featuring small particles and highly non-uniform diffusion; • topologically driven voltage response during the memory effect. 31

Battery Model
This section comprises the governing equations of the advanced continuum modelling framework that considers mesoscopic effects related to electrode topology. The basic governing equations are given for an arbitrary electrode material, whereas material specific approaches are clearly exposed separately. On a more detailed level, this section focuses on the LFP and NCM materials, both being model materials with very distinctive characteristics on both the material and electrode topology level. The focus on these two materials is further justified by the fact that Lithium titanate Li Ti O 4 5 12 (LTO), although being an anode material, features certain similarities with the LFP material, 35,36 and by the fact that NCM is in a broader sense representative for a wide spectrum of layered materials, while advanced graphite models are already available in the literature, e.g. 37,38 Although porous modelling theory represents a relatively sophisticated modelling approach, the existing porous electrode theory including all the required sub-models does not provide a fully credible virtual representation of a real battery. If the deficiencies of the porous electrode modelling framework are inspected more closely, they can be divided into: (a) modelling framework specific deficiencies which cover: an assumption in the Bruggeman equation, used in homogenized models, that does not fully consistently predict tortuosity in the porous medium, 39 oversimplified electron transport, an oversimplified electrochemical description of the process, neglecting or oversimplifying the treatment of capacitive effects, and the isothermal assumption that is applied in this model but is not inherent to all porous electrode models; (b) sub-model specific deficiencies, which cover the application of the Butler-Volmer equation for modelling interfacial currents, featuring limited accuracy 40,41 ; and (c) material specific deficiencies, which cover all material parameters, including their temperature and concentration dependencies, parameters related to irreversible and reversible heat generation (if it is modelled), and active particle potentials as a function of the lithiation level. The innovative contributions proposed in this paper that further enhance the consistency of a virtual representation of the electrode topology on the continuum level should thus be assessed considering these facts.
More consistent representation of the electrode topology.-The modelling framework was designed in such a way as to enable a seamless extension or exchange of multiple sub-models and thus enable the tailoring of modelling depth and dimensionality to comply with the requirements on a sufficient level of detail and generality to give an adequate virtual representation of multiple real materials. In line with the focus on the electrode (cathode) side, Fig. 2a shows a schematic representation of the battery cell building blocks available in the proposed framework ranging from a complete half-cell with metal Li anode to a full cell with both electrodes discretised. The complete half-cell system allows for a focused study of the underlying phenomena specific to the cathode material from the modelling as well as from the experimental aspect.
A crucial step toward a more consistent representation of the electrode topology was to more adequately represent the topology of active material. The proposed modelling framework is sufficiently generic to additionally support modelling spatial location of carbon/ binder as reported in Refs. 42-44, which is of importance at determining the overall response of the electrodes in particular upon high C-rate operation, however such an extension is beyond the scope of this paper. Therefore, to efficiently support adequate representation of the topology of active material while achieving short computational times that allow for long them virtual analyses of the entire electrode, an innovative virtual representation of the electrode topology, including active particle connectivity and particle size distribution, is proposed in this paper. The main aim of this enhanced electrode topology representation within the porous electrode theory based model is to plausibly represent the electrode topology and material properties. Figures 1a, 1b and 3a as well as 3b clearly indicate two very different material and material processing specific features: a) NCM is characterised by much larger secondary particles compared to the LFP and b) the topology of the secondary particles of both materials is very different, which profoundly influences interactions of primary particles with the electrolyte and with other primary particles as well as their ionic transport.
Due to the fact that computational times should be as decoupled from the modelled material (and thus virtual representation of active material) as possible, the proposed modelling framework decouples particle size from the size of the computational cell as schematically represented in (Fig. 3c). This assumption imposes a constant electrolyte and solid potential as well as a constant electrolyte concentration in a particular computational cell, which is a valid assumption for moderately dense computational meshes as variation in the listed parameters is, in general, not abrupt. An important merit of this innovative size decoupling approach arises from the fact that it allows for heterogeneous particle activities and thus heterogeneous particle lithiation levels and equilibrium potentials within a particular computational cell, which enables the unprecedented combination of short computational times and the resolving of specific particle phenomena on the level of the entire single cell or half-cell.
Another important aspect in properly representing the topological characteristics of the electrodes includes the consideration of a realistic particles size distribution. To preserve as high a level of (c) a schematic representation of the decoupling of the particle size and the size of the computational cell and a representation of particle connectivity; (d) a representation of a NCM agglomerate composed of primary particles, where surface exposed to the electrolyte is indicated with cyan and surface in direct contact with neighbouring particles is indicated with red; (e) a simplified modelling approach of topology represented in (d) with a large homogeneous secondary particle with internal discretisation; (f) a detailed particle connectivity scheme of the LFP with clearly indicated division of the total surface of the primary particle to the surface exposed to the electrolyte (cyan) and to the surface in direct contact with other primary particles (red).
consistency with the real electrode as possible, measured multiparticles size distributions (Fig. 6) can be utilised in this study.
A second innovative contribution, on the level of porous electrode models and cell simulations, arises from the more adequate topological representation of particle surfaces, which in addition to the particle size distribution also comprises an innovative particle connectivity scheme (Fig. 3c). Figures 3a and 3b clearly reveal that the primary particles in both materials feature contact with other primary particles. In addition, for the LFP material most of the primary particles are also, at least partially, in contact with the electrolyte, whereas this is not necessarily the case for the NCM material. These facts clearly limit applicability of the general assumption of the porous electrode theory assuming ideally immersed particles in the electrolyte 12,15 at least for the LFP material. Furthermore, assumption of ideally immersed particles also neglects the inherent connectivity between particles (Figs. 3a and 3b). This is again more pronounced in the LFP material, where it significantly influences the electrode dynamics. 19,[28][29][30] Therefore, if relevant, the proposed modelling framework can consider this inherent particleparticle connectivity and its impact on particle-electrolyte interaction (Fig. 2c), which will later in the paper turn out as crucial for plausible modelling of the LFP material.
Proposed modelling framework, therefore, establishes the missing link between realistic topological characteristics of electrodes and their plausible virtual representation on the level of the continuum model. This link is established by elaborating an unified approach for modelling materials with significantly different electrode topologies through virtual creation of agglomerates from primary particles, as presented in Fig. 3. Furthermore, proposed generic modelling framework also allows for seamless replication of previously published porous electrode modelling approaches including Newman based modelling approach 9-15 as shown in Fig. 3e and as elaborated later in this section.
More consistent representation of the electrode topology ( Fig. 3c), therefore, complies with the advances of the proposed porous electrode modelling framework announced in the Section Introduction. It ensures a more consistent virtual representations of electrode topologies of different materials, the modelling of the redistribution of Li between particles via electrolyte or via direct contact and the capability of implementing a thermodynamically derived active particle equilibrium potential that is dependent on particle size. Specifically, the multi particle size distribution is implemented in such a way that each computational cell can host an arbitrary number of representative (denoted with letter R) particles N R with sizes  R arbitrarily distributed by the distribution function  . Representative particles are introduced, as despite modelling large number of particles in individual agglomerates, it is obviously not possible to model all particles and thus each particle can represent multiple real particles in the electrode, which further supports the scalability of the modelling framework. The size of the k-th representative particle in j-th computational cell is therefore denoted as The relation between a particleʼs size  j k , R , its surface A j k , R and volume V j k , R can be arbitrarily defined. In this paper, a sphere topology is used to represent particles and therefore the size  j k , R is equivalent to the diameter of the sphere. Representative particles are then further grouped into individual agglomerate structures. A volume of representative agglomerate, usually one per computational cell, is calculated with the sum of volumes of the representative particles To ensure a credible electrode representation, the number of actual agglomerates (denoted with letter A) per computational cell is calculated where V j cc represents the volume of the computational cell,  j and j V,b cf j + represent porosity and the electrode volume fraction of binder and conductive fillers in the computational cell, respectively. The number of actual particles N A that correspond to each representative particle is therefore defined to be equal to the number of actual agglomerates in each computational cell Equation 4 allows for the calculation of the total volume of active particles V cell tot in the electrode, which can be determined by a summation across all computational cells and particles within each computational cell The value of V cell tot is used in the calculation of global cell depth of discharge (DoD) that is calculated using the volume weighted DoD of the representative particle with the following equation An important step toward a more consistent representation of the electrode topology is based on an innovative particle connectivity scheme that is compliant with SEM images (Fig. 1), which clearly indicate that primary particles form complex topological structures. The modelling framework thus enables connectivity of particles within a computational cell and connectivity of selected particles between neighbouring computational cells (Figs. 3c and 3e). The modelling framework again supports the implementation of arbitrary routines for particle connectivity, which allows for the replication of multiple electrode topologies and material properties. Consequently, the total reaction surface of an individual primary particle is divided into the surface exposed to the electrolyte A j k , ely and the sum of surfaces in direct contacts (denoted with DC) with neighbouring particles A l j k l , , DC å , written as where l denotes the index of each direct particle contact. The total molar flux j at the boundary of each individual representative particle is therefore the contribution of molar flux between the particle and the electrolyte, and the molar flux between particles in direct contact as shown in Fig. 3e weighted by the corresponding surfaces The definitions of j ely and j DC specific to this paper are presented later in Section Governing equations by Eqs. 15 and 18, respectively. With the introduction of a multi-particle model and connectivity scheme, it is crucial to properly perform an upscaling of molar fluxes on the solid/electrolyte interphase from the particle level to the electrode level. The contribution of the molar flux j j k , of an individual representative particle in the ensemble within each computational cell, is being governed by the surface, which is exposed to the electrolyte with respect to the total surface of all particles where the surface ratio of the k-th representative particle can be written as: The upscaled molar flux j j cc is used as the source term in Eqs. 11, 12 and 13 in Section Governing equations.
An important feature of the proposed generic modelling framework arises from the fact that it allows for seamless replication of previously published porous electrode modelling approaches including previous tailored modelling approaches for different materials, material processing methods and electrode topologies within a single codebase. By neglecting transport via direct contact with other particles Eq. 18 and by not considering the real particle size distribution of the electrode but rather a stochastic distribution, which reduces the consistency of the virtual representation of the electrode topology, the approach proposed by 7 can be replicated. Similarly, by neglecting transport via direct contact with other particles Eq. 18 and reducing multi particle size distribution to two or three different sizes, a modelling depth comparable to the one presented in 5,6 can be replicated. With a further reduction to a single particle size distribution an original Newman based modelling approach 15 is obtained.
In addition, Fig. 3b also provides reasoning on the suitability of the original Newman based modelling approach 15 for modelling NCM cells. Due to the isotropic transport of Li in 2D crystal planes, 3,4 due to the fact that primary particles are densely packed within secondary particles with a very low intra-secondary particle electrolyte share and due to minor inter-secondary particle connectivity, secondary particles could be assumed as plausible representation of ideally immersed particles in the electrolyte (Fig. 2b) that approach Fickian diffusion, as modelled by the Newman based modelling approach. 15 To demonstrate generality and applicability of the proposed modelling framework, this approach, being consistent with assumptions elaborated by Mao et al., 5 will be applied when modelling the NCM material.
As a Newman based representation is not valid for the LFP material, unlike for the NCM, (Fig. 3b), a bin specific ohmic hindrance that impedes the insertion-de-insertion reaction was proposed in. 33,34 This approach can also be covered by the proposed modelling framework, as the approach in 33,34 can be interpreted as a kind of ad hoc representation of a reduced surface exposed to the electrolyte (Figs. 3a, 3e and Eq. 7) and thus features increased hindrance that impedes the insertion-de-insertion reaction. However, the approach proposed in 33,34 neglects inter-particle transport via direct contact with other particles Eq. 7.
Governing equations.-The main governing equations that govern the evolution of the concentration and potential fields are based on Newmanʼs porous theory based model with boundary conditions presented in Appendix A, e.g. 8,9,12 The equation for material balance (or mass conservation) of ionic species in a battery cell is defined as: Although the modelling framework support arbitrary formulations for modelling current density at the solid/electrolyte interphase, similarly as in most porous electrode models, 9,12,15,16 the Butler-Volmer formulation j ely is applied in this study where i 0 is the exchange current density and η is overpotential defined as represents the equilibrium solid potential and can be further defined as where U 0 is the standard equilibrium potential and μ represents the equilibrium chemical potential of an active material/individual particle.
The current density at the interface between two particles in direct contact was derived from Eq. 19 and is similar to equation used by Orvananos It is driven by a difference in the chemical potential between particles k 1 and k 2 . c s represents the surface concentration of Li in active particles. P represents the permeability of the direct contact surface and can be varied to account for the misalignment of 1D fast diffusion channel between particles. Mass conservation in active particle can in its most general form be written as where c s represents concentration of Li in solid and D s represents diffusion constant in solid.
Specifics of the NCM material.-NCM primary particles form spherically shaped secondary particles as seen in Figs. 1b and 3b. As elaborated in previous section, considering 2D preferential diffusion paths within the crystal lattice, a dense arrangement of primary particles leaving almost no voids for the electrolyte to penetrate through, and a topology close to a sphere with diameter in a range of 10 μm as well as a monotonous chemical potential as a function of lithiation, the modelling of intra-particle Li transport with the commonly used Fickʼs diffusion equation, in spherical coordinates, is a reasonable approximation (Fig. 3e) at lower to moderate C rates. 5 However, such pristine spherically shaped material can rarely be observed in commercial batteries, where, due to the manufacturing process, particles tend to crack (Fig. B·1). Agglomerate cracks yield increased active surface, which has larger impact in the results at high C rates thus further supporting statement on suitability of Fickʼs diffusion equation, in spherical coordinates, at lower to moderate C rates.
NCM compositions used nowadays, e.g. 8-1-1, 6-2-2, 5-2-3, ..., are significantly more diverse compared to the single LFP composition. Each NCM composition has, in general, different properties on the atomistic level and there is lack of mechanistically or ab-initio based functional dependencies of open circuit potential as a function of lithiation. The general Eq. 17 is therefore substituted with an empirical correlation of open circuit potential as a function of lithiation, e.g. 5,12 U which is usually fitted to the experimental results.
Calculation of the current density at the particleʼs surface is governed by the Butler-Volmer Eq. 15 with a widely used expression for exchange current density i 0 , e.g. 5 and is also used in models featuring other non-phase separating electrode chemistries, such as LCO. 12,16 The relation between equilibrium chemical potential μ and lithium concentration c s is given by the Nernst relation 46 With an assumption of spherical particles and a constant isotropic diffusion coefficient, the following form of the diffusion equation written in spherical coordinates, which is widely applied, e.g., 5,14-17 is obtained Specifics of the graphite material.-Graphite is one of the most commonly used anode materials and hosts Li atoms within its layered structure. It is a phase-separating material 37 which poses additional challenges in modelling its chemical potential. There have already been some insightful attempts to model graphiteʼs chemical potential mechanistically by applying a phase-field model to each individual layer and considering their interactions in a periodic 2-layer arrangement, e.g., 37,38 however these models are still in the early development phases with some deficiencies in modelling open-circuit voltage potential in the whole range of Li concentration. 37 Consequently, most electrochemical models still use empiric correlations for anodeʼs open-circuit potential as a function of lithiation, e.g., 12,16 similarly as in the case of the NCM cathode material (Section Specifics of the NCM material).
Specifics of the LFP material.-The modelling of intra-particle transport of Li in phase separating materials, e.g. LFP, LTO, graphite, with Fickʼs law of diffusion, introduces fundamental thermodynamic inaccuracies to the model. 47 The Cahn-Hilliard Eq. 26 inherently allows for the modelling of the phenomena characteristic for phase separating materials, e.g. spinodal decomposition, movement of the phase boundary and uphill diffusion within the active particles. 26,49,50 Further transformation of Eq. 26 can be made by considering two facts: (i) characteristic diffusion times in LFP nano-particles are significantly shorter than (dis)charge times of the battery, 26,32 (ii) active particles of a battery are subjected to higher overpotentials than 40 mV, which preferences solid solution intercalation against a phase separated domino cascade, under normal operating condition. [51][52][53] The first listed fact (short diffusion times) enables the transformation of Eq. 26 to 0D without losing much physical relevance and accuracy. This step is crucial for achieving computational efficiency of the model, since solving a fourth order Eq. 26 with non-linear terms is already computationally demanding for a single particle 50 and it would thus impose a significant computational burden on a level of a full discretised electrode with many representative particles. The procedure of transforming Eq. 26 to 0D relies on volume integration that is described in detail in reference 53 representing a multi-scale approach based on sequential linking. 54,55 As indicated in item ii), the volume integration procedure can be done analytically in the limit of high overpotentials, where a solid solution pathway of intercalation is realized. The obtained analytic expression for the chemical potential (Fig. 4) can be written as 53 where c s is the average filling fraction of the particle. After combining the equation for mass conservation Eq. 19, the equation for the boundary condition at the particleʼs surface Eq. A·22, the Butler-Volmer Eqs. 15, and 27 for a 0D simplified equilibrium chemical potential, the variation of particle concentration is evaluated as represents the filling fraction. The proposed modelling framework also features the size dependent regular solution parameter  ( ) W which was initially determined by Ferguson et al. 51 from the voltage gap (DF) between spinodal points from regular solution theory. For platelet-like particles they calculated that particles smaller than 20 nm do not tend to phase separate due to the fact that energy required to establish a two-phase regime inside the particle exceeds the energy for homogeneous lithiation. Recently, Zelič et al. 53

Results
This section presents various types of results covering: constant current discharging of different cathode materials, i.e. NCM and LFP, at different rates, constant current discharging of the LFP material for different cathode loadings, and memory effect, as a specific phenomenon of phase separating materials. The results for constant current discharging and the memory effect are also validated by experimental results proving the capabilities of the proposed modelling framework to model a wide range of materials, cell designs and operating conditions.
Non-phase separating materials-NCM results.-As reasoned in Section Specifics of the NCM material, the model of the NCM material considers homogeneous spherical secondary particles which obey Fickian diffusion and which are ideally immersed in the electrolyte. This modelling depth is consistent with Newmanʼs original modelling approach, as discussed at the end of Section More consistent representation of the electrode topology. Modelling topology features full cell sandwich, i.e. discretised cathode NCM, separator and graphite anode. Details of the model parameters are given in Tables C·I and C·II. Experimental results were obtained from EiG ePLB C020 cellʼs specifications. Cell DoD values were calculated relative to the nominal capacity (20 Ah) of the cell.
The results in Fig. 5a are expectedly similar to the ones in the existing literature, e.g., 16,56 and show good agreement between simulated and measured results. As announced in Section More consistent representation of the electrode topology, these results indicate the capability of Newmanʼs approach to model NCM materials at different lower to moderate C rates. Figure 5b provides more insight into the sub-electrode level phenomena at around x = 0.5 of a low and higher discharge rate. This figure clearly exposes that the low cell voltage at higher rates, which is characteristic for this rate and state of lithiation, originates from the higher polarisations associated with inter-facial current and transport losses in the electrolyte as well as in particular with large transport losses secondary particles. This indicates that in NCM electrodes, which are characterised with large secondary particles with very low intra-secondary particle electrolyte share, low Li utilization at higher rates is indeed mainly limited by the bulk particle diffusion, which is consistent with findings in Ref. 57.
Although the simulated and measured results (Fig. 5a) are in good agreement, which is crucial for confirming the credibility of the modelling framework, they feature certain minor discrepancies. These discrepancies could certainly be further minimized by an excessive tuning of model parameters, however, approaching an agreement of simulated and measured results within line thickness does not necessarily prove a more credible modelling basis, as model calibration could convert to fault compensation, as discussed in 2. It is thus much more important that good agreement was obtained with an unified set of model parameters applied in all operating conditions. Figure 5b reveals the Li + concentration distribution across the entire cell sandwich width with more pronounced gradients at higher C-rates. Figure 5c with the electrolyte phase potential is closely linked to the concentration distribution in Fig. 5b featuring increased potential gradients at higher C-rates. Differences in internal gradients in both Figs. 5b and 5c arise from different porosities in the cathode, separator and anode. Figure 5d shows the solid phase potentials in both electrodes. Due to the relatively large conductivity of the NCM cathode material and thin electrode, the solid phase potentials s F yield approximately 2.5 mV and 0.2 mV of potential difference across the cathode for 3C and C/5 rate, respectively. Solid phase potential gradients in the graphite anode are even lower and range from approximately 0.09 mV to 0.01 mV for 3C and C/5 rate, respectively. Insight into particle surface stoichiometries during discharge provides Fig. 5e where particles close to separator (de)lithiate first, then particles closer to the current collectors follow.
Normalized concentration profiles within the particles are shown in Fig. 5f at different C-rates. The surface of the particle is located at a normalized particle radius equal to 1. The results follow a parabolic profile with gradients corresponding to the magnitude of the solid/electrolyte interface current boundary condition at various C-rates. With higher C-rates, the surface concentration saturates faster, yielding higher polarization and consequently lower electrode utilization (Fig. 5a) compared to the low C-rate where the concentration profile is nearly flat.
Phase separating materials-LiFePO 4 results.-It is important to note that unlike some tailored modelling approaches for the LFP material, e.g., 33,34 the proposed modelling framework allows for the credible modelling of very different materials, as for example the NCM and the LFP material, while adapting intrinsic material properties and considering the proper topological representation of the electrode.
As reasoned in Sections More consistent representation of the electrode topology and Specifics of the LFP material, the model of the LFP material considers both varying particle size distribution (7) and particle connectivity. 58 The size of primary and secondary particles was measured experimentally with TEM imaging on a statistically representative number of particles, shown in Fig. 6. The virtual representation of size distributed primary particles was generated with a modified Box-Muller algorithm resulting in a log-normal size distribution following the experimentally determined size distribution. These primary particles were then virtually arranged into secondary particles, i.e. agglomerates, as depicted in Fig. 3. Virtual arrangement of primary particles into these secondary particles again follows experimentally determined topologies of secondary particles. Secondary particles of different topologies were generated in different computational cells to more credibly resemble measured electrode topology and number of these agglomerates was determined as outlined in Eq. 4. Obviously, outer primary particles in these agglomerates feature larger surface in contact with the electrolyte while inner primary particles feature mainly of entirely direct contacts with other primary particles.
The modelling framework was applied on a tailor built 25 μm thick electrode and a 250 μm thick separator considering an experimentally determined primary particle size distribution with a mean value of approximately 275 nm and porosity of 49% for normal electrode, estimated value of 90% for a dilute electrode and 92% for separator. 50 primary particles were assumed to form a representative agglomerate in each computational cell. Li metal was used as a negative electrode and represented with the interface current boundary condition. Electrode DoD values were calculated relative to the measured capacity (164 mAh/g) obtained during C/100 discharge.
In addition, the LFP material was modelled considering a thermodynamically derived active particle equilibrium potential from regular solution theory Eq. 27 and Fig. 4, which on one hand increases the prediction capability of the model and on the other hand allows for the consideration of an equilibrium chemical potential as a function of particle size, which is inherent to the LFP material 51 and which decisively influences Li transport.
Constant current discharging.-As for the NCM case, similar results of constant current discharging will be analysed first. To ensure a more profound insight into the phenomenology of the processes and to allow for more consistent model calibration, the constant current discharging of a dilute electrode containing only 35 μg of active material (Fig. 7a) is analysed in addition to an electrode with normal active material loading (Fig. 7b). On one hand, this allows for better model calibration, but on the other hand, it confirms the capability of the proposed modelling framework to Figure 5. (a) A comparison between experimentally measured and simulated discharge curves; (b-e) spatially resolved results within the battery cell sandwich at a SoC = 50% after being discharged from SoC = 100% with four different C-rates, separator region is depicted with vertical green lines, with (b) spatially resolved Li + concentration, (c) spatially resolved e F potential, (d) spatially resolved s F potential and (e) spatially resolved surface particle stoichiometry; (f) normalized particle concentration profile of Li in the representative cathode particle closest to the separator. model very different electrode topologies at different discharge rates by only adapting the porosity of the electrode.
It is discernible from Fig. 7a that the total cell polarization is significantly lower for the dilute electrode, which is more exposed for high rates, i.e. 5C. As LFP particles are characterized with very short diffusional time constants (Section Specifics of the LFP material) and as for very dilute electrodes the electrolyte polarization is very low, this cell topology provides a very good basis for calibrating the Butler-Volmer model Eq. 29 in the flat part of the discharge curve. Figure 7a features certain discrepancies at the beginning of discharge. This can mainly be attributed to the simplicity of the regular solution model which does not capture the non-idealities of the Li-poor and Li-rich solid solution end-members. 34 However, besides this initial discrepancy, very good agreement of the simulated and measured voltage in the flat part of the discharge curve, especially at low C-rates, confirms the adequacy of the particle potential model to replicate experimental results while considering the calibrated Butler-Volmer model.
After the calibration of the Butler-Volmer model and by only adapting the porosity of the electrode, the model was used to model an electrode with normal loading. Good agreement between simulated and measured voltage traces at low C-rates (Fig. 7b) further confirms the adequacy of the particle potential model and the capability of the model to adequately simulate a slight slope of the voltage curve. This is to a large extent driven by inter-particle phase separation (Fig. 7d) of the phase separating material, which is mostly driven by the size dependency of the particle's chemical potentials Eq. 27. Figure 7c elucidates an important aspect of phase separating materials by presenting the parameters of two particles in direct contact with each other for a C/100 discharge rate. Figure 7c clearly shows that at low C rates, particle currents reach high peak values despite a very low cell current, which is the consequence of a very small active particle size population, being in-line with several experimental observations, e.g. 59 The DoD curves (Fig. 7d) show a particle transition from the Li-poor to the Li-rich phase occurring at x 0.1 » for a smaller, and at x 0.7 » for a larger particle. Pronounced partial cycling, which originates from interactions with other particles, is visible before the transition from low to high Li concentration. The redistribution of Li between particles occurs during phase separation when a lithiating particle extracts Li from neighbouring particles that are in direct contact with this particle, consequently lowering their DoD, and/or from the electrolyte. Between x 0.7 » and x 0.8 » Fig. 7d additionally exposes an extreme case of complete extraction of Li from a smaller particle (Par#2), when the larger particle (Par#1), featuring approximately 8 times larger volume, undergoes a phase change and extracts Li from the electrolyte as well as from this neighbouring particle in direct contact. A second re-lithiation of the smaller particle follows immediately after this phase change of the larger particle. These phenomena are clearly seen in the traces of molar fluxes on the solidelectrolyte interface denoted by j ely and direct contact fluxes denoted by j DC , of both particles in Fig. 7c.
A comparison of results in Figs. 7c-7f clearly shows that the qualitative trends of particle currents and consequent particle DoDs are significantly different for low and high rates, which is also in-line with in situ X-ray diffraction observations in Ref. 59 and synchrotron-based liquid scanning transmission X-ray microscopy (STXM). 24 This can be explained by high local overpotentials that are associated with high rates. These high overpotentials namely result in a homogeneous solid solution lithiation of particles upon discharging as opposed to a kind of particle-by particle lithiation 19 of phase separated particles at lower local overpotentials. 26 To further support better visualisation and thus provide another insight into the phenomenology of the processes, the supplementary material comprises a movie of a particle lithiation, i.e. DoD, as a function of time for low and high rates that correspond to the ones presented in Figs. 7d and 7f. To enhance the visual comparison, the movies are synchronized based on the current throughput rather than in time to have a direct visual comparison of the electrode state at the same DoD of the electrode (see supplementary material is available online at stacks.iop.org/JES/167/060531/mmedia).
Good agreement between the measured and modelled curves at high C-rates (Fig. 7b) arises from an adequate description of active particle potential and the corresponding particle currents. This inherently confirms the adequacy of the modelling representation of the topology of the cathode within the proposed modelling framework (Figs. 2 and 3). Besides the direct Li transport between particles, this modelling representation reduces the surface of a particle that is available for Li transfer across the solid/electrolyte interface Eq. 7. This modelling representation also increases the losses of direct Li transport between particles which do not have well aligned channels for fast diffusion 60 Eq. 18. Both effects, therefore, increase the overpotentials required to transport Li into particles featuring small or negligible solid/electrolyte surface and which are connected to other particles by not well-aligned channels for fast diffusion. This is the main mechanism for low Li utilization at high rates in the LFP cathodes composed of small primary particles.
Although Li transport in secondary particles limits Li utilization in both, NCM and LFP, materials, plausible, material specific, topological representation of the electrode combined with adequate dependencies of the chemical potential as a function of particle lithiation are required for proper virtual representation of materials. Layered crystal structure with 2D preferential diffusion paths, nearly homogeneous spherical secondary particles and a monotonous chemical potential as a function of lithiation enables application of Fickʼs diffusion equation, in spherical coordinates, as a reasonable approach for modelling NCM material. However, Figs. 7d and 7f clearly indicate that proper modelling of the LFP material calls for more profound topological representation of the electrode, being one of the main contributions of this paper, and application of adequate chemical potential dependencies as a function of particle lithiation.
To further support this statement, Fig. 8a presents another insightful result, which reveals the need for plausible topological representation of the electrode when modelling LFP material. It namely exposes a significant electrode topology driven difference in Li utilization at high rates, which clearly differentiates the proposed approach to the Newmanʼs based approach. Unlike the blue curve "Adequate el. topology: 0D", which is identical to the one presented in Fig. 7b, red curve 'Ideally immersed par.: 0D", which depicts results of equally sized particles that were ideally immersed in the electrolyte, thus neglecting the reduced surface due to particle agglomeration Eq. 7 and particle to particle transport, fully fails in predicting Li utilization at high rates. These results indicate that the Figure 6. The frequency histogram of LFP Targray primary particle sizes experimentally obtained from TEM measurements with last bin representing the rest of the larger particles (>700 nm).
application of the Newman based approach on the level of primary particles results in the wrong prediction of a nearly full Li utilization at a high 5C rate (Fig. 8a). Additionally, Fig. 8b clearly presents that these results are also not influenced by the selection of the modelling approach for simulating intra-particle diffusion, as results are nearly identical for the Fickian (diffusion equation in radial coordinate system) approach and the 0D diffusion approach Eq. 28. This further confirms that intra-particle diffusion is not a rate-determining step for small primary particles, as reasoned in Section Specifics of the LFP material. This finding also enlightens, as indicated in Section Specifics of the LFP material, the need of introducing empirical approaches to limit insertion-de-insertion reaction between active particles and the electrolyte when modelling LFP material, 7,33,34 which deviates from the basic Newmanʼs approach. 9,10 Unlike approaches of Guhlke et al. 7 and Farkhondeh et al., 33,34 introducing a kind of ad hoc bin specific ohmic hindrance and claiming that the model does not include any geometric detail, the proposed approach clearly interrelates different levels of Li utilization at different rates to topological characteristics of the electrode. Inspired by real topologies of electrodes (SEM image in Fig. 1), a unified approach for modelling materials by virtually creating agglomerates, representing secondary particles, from primary particles establishes a link between realistic topological characteristics of electrodes and their virtual representation. Therefore, complying with the SEM image in Fig. 1 and previous studies, e.g., 60,61 analysis in Fig. 8b clearly indicates that low Li utilization at high rates is to a large extent related to the topological characteristics of the electrodes, which are also schematically depicted in the Fig. 8a.
Based on presented results, it can be concluded that the proposed innovative particle connectivity scheme (Fig. 3e) offers a good basis for a more consistent representation of the electrode topology. These results show, for the first time, the significance of the electrode topology and thus the significance of the particle size distribution and connectivity for the adequate prediction of low Li utilization in small particles at high rates at the level of continuum models of the entire electrode. The proposed innovative modelling framework, therefore, represents a foremost continuum modelling approach, enabling a more consistent reproduction of real multi-particle size distributions and particle connectivity, thus enabling more consistent modelling of transport phenomena.
Memory effect.-Memory effect refers to an unusual polarization bump during charge/discharge sequence, near the SoC where a partial charge/discharge had been terminated in a previous step. 31 This phenomenon presented by Sasaki et al. 25 was later explained by Kondo et al. 31 using phenomenological modelling representations while assuming constant concentration and potential fields in the cathode and by simplifying the influence of particle size o the chemical potential of the particle. By upgrading the study presented in, 32 this section presents the first results of electrode topology driven variations in the magnitude of the memory effect related polarization bump, i.e. different magnitudes of the polarization bump (Fig. 10a) due to different topologies of two LFP materials (Fig. 9). This achievement further confirms the capability and generality of the proposed modelling framework to model complex electrode phenomena using a single modelling basis.
To support this analysis, experiments were performed on two laboratory-built half-cells with different LFP materials. First one was the aforementioned cell prepared by using commercial Targray LFP cathode material. The second cell was assembled by using cathode based on another type of LFP material (LFP-pcrm) that was synthesised in-house by a novel pulse combustion reactor method in a slightly reductive environment, as described in detail in reference. 62 The SEM image comparison (Fig. 9) of the two LFP cathode materials reveals that commercial Targray LFP material (Fig. 9a) exhibits preferred 2D agglomeration forming platelet-like carbon-coated secondary particles (agglomerates). On the contrary, the LFP-pcrm material shows evident 3D agglomeration forming porous carbon-coated sphere-like agglomerates with diameter ranging from about 2 to 10 μm, two of such (larger) agglomerates being shown in Fig. 9b. Clearly, the LFP-pcrm material can be regarded as being notably strongly agglomerated compared to the Targray LFP. Consequently, a higher percentage of particles have limited access to the main conductive pathways (along electrolyte or carbon black). As presented in Fig. 10a, electrode topology has an impact on the magnitude of the polarization bump. The presented experimental results thus reveal that magnitude of the polarization bump can serve as an assessment of the level of particle agglomeration and direct inter-particle Li distribution.
To comply with the outlined objectives on model generality, the same modelling framework, presented in Section Constant current discharging, was applied to model the memory effect with the following charge/discharge sequence: (I) partial charge from SoC = 0% to SoC = 50%; (II) one-hour rest with no current applied; (III) partial discharge from SoC = 50% to SoC = 10%; (IV) one minute rest with no current applied; (V) memory release charge from SoC = 10% to SoC = 100%. The applied currents were consistent with the ones used in the, 25,31 i.e. ±C/2. Models of the two cells differ in the connectivity of primary particles in agglomerates. Inspired by the SEM image (Fig. 9), connectivity of primary particles in the LFP-pcr electrode, therefore, features larger agglomerate structures. The resulting voltage response comparison between the experimental and simulated results is shown in Fig. 10 as a function of SoC in order to preserve the consistency of the representation with the original references. Figure 10 indicates that the proposed model is capable of replicating the measured memory effect related phenomena of different materials. LFP-pcr electrode, which is characterized by larger agglomerate structures, features larger magnitude of the polarization bump and comparison of Figs. 10a and 10b reveals that modelling framework is capable of simulating this trend by considering proper virtual topological representation of the electrode. Furthermore, the results of active particle population (Fig. 11) are in agreement with the results proposed by Kondo et al., 31 supporting the finding that the memory effect in the LFP material is associated with a sudden drop of the active particle population after re-entering the SoC range where a partial charge had been terminated previously.
Application of a half-cell model to simulate the memory effect has another benefit, as it allows for a deeper insight into the underlying phenomena. Figures 12a-12d provide a graphical representation of particles' DoD as a function of their individual crystallite size at different time instances during the memory writing/releasing sequence. In-line with the schematic representation provided in Ref. 31, the initial partial charge (Fig. 12a) starting from lithiated particles, smaller particles (characterized by higher activity) delithiate preferentially, while larger particles remain in the lithiated state. During the following rest period, the characteristic non-monotonous potential of LFP particles (Fig. 4 and Eq. 27) within the spinodal region results in phase separation thus minimising the free energy of the electrode yielding the DoD distribution presented in 12b. It can be seen from the comparison of Figs. 12a and 12b that most of particles did phase separate during the rest period of one hour, whereby some (small amount) of the particles still remain within the spinodal region. During a subsequent partial discharge (Fig. 12c), after which the electrode is nearly fully lithiated, the so called "Shim-shaped distribution" is obtained (Fig. 12c). This Shim-shaped distribution is a precursor of the memory effect. Namely, during the following memory releasing charge, this Shim-shaped distribution provokes a sudden drop of active particle population as shown in Figs. 12d and 11.
Unlike the schematic representation provided by Kondo et al., 31 Figs. 12a-12d that are modelled with a complete half-cell model indicate that the division of lithiated and delithiated particles at the end of the one hour rest does not feature an abrupt change at a certain particle size. This difference arises from more consistent topological representation of the electrode, inherent coupling between particle size and its chemical potential 51,53 and varying potential and concentration fields in the electrolyte that characterize the proposed modelling framework. As demonstrated, the application of more realistic simulation conditions still produces the anticipated particle size-dependent phase separation outcome, however the division does not feature an abrupt change at a certain particle size. To further support better visualisation and thus provide another insight into phenomenology of the processes, the supplementary material comprises a movie of the representation of the particles' DoD as a function of their size and a corresponding movie of the representation of particles' DoD as a function of electrode

Conclusions
An advanced continuum level modelling framework characterized by a more consistent virtual representation of the electrode topology was presented in the paper. Proposed modelling framework establishes the missing link between realistic topological characteristics of the active material and their plausible virtual representation on the level of the continuum model. This link is established by elaborating an unified approach for modelling materials with significantly different topologies of active material through virtual creation of agglomerates from primary particles. Proposed innovative approach comprises multi-particle size distribution of primary particles and particle-to-particle connectivity scheme.
Generality and applicability of the proposed modelling framework was demonstrated on two different cathode materials, being LFP and NCM, and two different anode materials, being graphite and Li metal anode. All analysed cases were simulated with the same modelling framework by adapting only virtual representation of electrode's topologies and intrinsic material properties. Credibility and applicability of the proposed modelling framework is confirmed through: • good agreement with experimental results for various discharge curves of both cathode materials, • revealing several insightful topologically driven inter-and intra-particle phenomena in the LFP material, which comprise Figure 11. Cell potential and active particle population during a charge/discharge series of sequences that result in the memory effect. Sequences are labelled as: (I) partial charge; (II) first rest; (III) partial discharge; (IV+V) second rest and memory release. enlightening background of the topologically driven low Li utilization at high current densities and voltage response difference during the memory effect of the two different LFP materials.
In addition to the enhanced modelling fidelity of electrode topologies, proposed generic modelling framework allows also for seamless replication of previously published porous electrode modelling approaches. Due to enhanced prediction capability and generality, proposed computationally efficient continuum level battery modelling framework enables more consistent virtual prototyping on the level of the single cell and thus tailoring battery design to a specific application. A.3. Solid phase potential s F .-Solving Eq. 13 requires the following boundary conditions. Applied (dis)charge current density I app is imposed at the current collector/cathode boundary condition. Zero-flux boundary conditions are imposed at the cathode/separator boundary (Figs. A·1a and A·1b) and at the separator/anode boundary in the case of the full cell sandwich (Fig. A·1a). Finally, a reference voltage value V ref is set at the anode/current collector boundary (Fig. A·1a).   (Figs. B·1a-B·1d). Namely, practically all of the top-most agglomerates in commercial cathode are mechanically deformed (flattened) with partial or total separation into the individual primary particles. Additionally, Figure B·1. Comparison of pristine commercial NCM811 (Targray) material (a)-(d) and NCM811 cathode material obtained from a non-cycled commercial 18 650 Li-ion battery (e)-(h). SEM images of particles in the non-treated pristine Targray material (a)-(c) clearly shows that the non-affected primary particles are compactly integrated into the corresponding agglomerates that do not show some observable delamination of the primary particles (d). Material obtained from a commercial 18 650 Li-ion battery (e)-(h) reveals extensive damage of (especially) top-most agglomerates exhibiting separation of primary particles that additionally show significant crack formation. considerable damage of primary particles is observed where significant crack formation is initiated. All of this damage is ascribed to the consequences of the calendering procedure during battery (cathode) manufacturing. In return, cracked particles provide a larger solid/electrolyte interface surface and shorter diffusion paths.