An Enhanced Ageing Model for Solid-State Batteries

Abstract

The emphasis in the automotive industry towards sustainable mobility has led to a significant interest in hybrid-electric drive-trains with high energy density batteries. Addressing the needs of this strategy, the battery market is exploring new technologies to improve the safety and lifespan of electric vehicles. To this end, there is a focus on the all-solid-state battery (ASSB) technology for its cycle capabilities. Filling the current void in the literature pertaining to accurate ageing models for ASSBs, in the present work, we propose an enhanced version of the numerical ageing model, originally developed for liquid electrolyte based batteries, to forecast the development of the solid electrolyte interface layer that is the major cause of battery capacity fading. The model has been tested on prototype batteries and reveals an accuracy of 99%. The capacity fade in ASSBs has been investigated under different conditions and the enhanced ageing model has been validated using experimental data from these batteries. The findings suggest that there is potential for solid-state batteries to be commercialized, although significant work is needed to match the manufacturing level of lithium-ion batteries embedded with liquid electrolytes.

1. Introduction

Improved energy efficiency and advances in renewable energy are essential to achieve zero emissions. With advances in these areas, a significant reduction in CO₂ emissions can be accomplished in the transport sector. This sector, as indicated by the 2021 IEA study, is currently contributing nearly 24% of the total CO₂ emissions around the globe, and 72% of this comes from passenger cars. This demands a consistent effort towards the decarbonization of the transport sector and energy generation via renewable energy. In response to this, electric vehicles (EVs) are rapidly permeating the market, proposing themselves as an alternative to conventional internal combustion engine (ICE) vehicles. Despite their raising sales numbers, EVs in their current state are not a robust solution for a definite move towards sustainable mobility. That is due to the fact that in terms of life cycle assessment (LCA), commencing from the raw materials and the production to the end of life and recycling, EVs are not a definitive answer compared to ICE-driven vehicles. This assertion is primarily related to the CO₂ level adduced by the battery manufacturing, which represents nearly 40% of the emissions emanating from the manufacturing of the EVs [1]. Further, current Li-ion batteries (LIBs) require high production costs and suffer from a limited lifespan, posing a serious environmental problem. Needless to say, extending the battery’s lifespan is of crucial importance for the electrification of the transport sector. To this end, one of the most promising battery designs is the all solid-state lithium-ion battery (ASSB). The replacement of the liquid electrolyte with a solid electrolyte provides a significant improvement in the safety, energy density, and cycle performance of the battery [2,3,4,5]. With this immense potential, from a computational modelling perspective, to design new and novel ASSBs, there is an urgent need to develop advanced ageing models that can be used to analyze the next-generation ASSBs. However, as in the case of LIBs, the precise ageing mechanism in ASSBs is still largely unknown. Since its introduction in 1991, the graphite anode, with a specific charge capacity of 372 mAh g⁻¹ and a low operating potential of about 0.1 V versus Li+/Li, has become a staple in lithium-ion batteries (LIBs) when used with ethylene carbonate (EC)-based liquid carbonate electrolytes and LiPF6 salt [6]. Despite the inherent instability of these electrolytes at low potentials, their decomposition products form a protective layer known as the solid electrolyte interphase (SEI) on the anode surface, preventing further degradation and graphite exfoliation, thereby ensuring the long-term stability of LIBs [7]. Improvements in SEI through advanced electrolyte compositions, additives, and specific formation cycles have been essential in enhancing the performance of modern LIBs [8]. Current commercial LIBs achieve around 270 Wh kg⁻¹ at the cell level [9]. To exceed this and reach specific energies above 400 Wh kg⁻¹, researchers are exploring alternative anode materials [10]. One approach involves replacing graphite with silicon or lithium metal electrodes. Silicon, with a theoretical specific charge capacity of about 3600 mAh g⁻¹, operates at a similarly low potential but suffers from significant volume expansion during lithiation (up to 400%), which damages the SEI and depletes the cell’s lithium inventory [11,12]. Lithium metal, offering a specific charge capacity of 3860 mAh g⁻¹ and an electrode potential of 0 V (−3.04 V vs. the standard hydrogen electrode), presents another promising option, especially in “anode-free” cells where lithium metal forms in situ. However, challenges such as substantial SEI growth and dendritic lithium formation pose safety risks for lithium metal batteries (LMBs) [13]. To mitigate these issues, research is focused on developing new electrolytes and additives, protecting the current collector and lithium metal surface with appropriate coatings, and using structured host materials for lithium deposition. Evaluating interfacial side reactions, including SEI growth, is crucial for assessing the effectiveness of these strategies [14]. All-solid-state batteries (ASSBs) offer a promising avenue for safer lithium metal anode implementation, as they eliminate flammable liquid electrolytes [14]. The high mechanical strength and high lithium transference number (t(Li+) ≈ 1) of solid electrolytes (SEs) help reduce dendrite growth, making them suitable for lithium metal batteries [15]. However, SEs often have a narrow stability window and react with lithium metal at low potentials, forming unstable or stable SEIs. At high current densities, dendrite growth can occur along cracks and grain boundaries in inorganic SEs, exacerbating interfacial reactions and potentially causing short circuits and cell failure [16].

The SEI layer was originally identified and published by Peled in 1979 [17]. This layer is a consequence of specific reduction and oxidation reactions on the electrode surface. More precisely, when the cathode’s potential is lower and/or the anode’s potential is higher than the electrolyte’s highest occupied molecular orbital (HOMO) and/or lowest unoccupied molecular orbital (LUMO), the electrolyte molecules are oxidized and/or reduced on the electrode surface, respectively. The products of these oxidation and reduction reactions will contribute to the formation of the SEI layer [18,19,20]. It must be noted that the operating environment of a battery, such as the loaded voltage, current, and temperature, affect these HOMO/LUMO windows. The SEI production processes deconstruct the electrolyte molecules and deplete active Li-ions, leading to capacity fading [21]. An SEI layer that is appropriately designed inhibits electron tunneling and electrolyte diffusion at the electrode–electrolyte contact. This reduces the rate of SEI generation and reduction processes, and thus the rate of capacity fading [20]. A typical SEI layer consists of the following sections: (I) an inorganic inner layer that is closer to the electrode contact and is only permeable for Li-ions and (II) a highly permeable organic outer layer that is closer to the electrolyte interface and enables the transit of electrolyte molecules as well as Li-ions. In ASSBs, the first layer has similar thermodynamic and kinetic properties as in the presence of a liquid electrolyte and covers a passivating function towards the anode [15,22]. On the other hand, the second layer can be thermodynamically unstable [15], especially when a lithium anode is implemented, and will facilitate the continuous growth of the SEI layer during the operation of the battery. Consequently, characterization and comprehension of the SEI layer’s formation mechanisms and transport properties are crucial to improving ASSBs’ longevity.

Given that some of the SEI formation reactions take place at picosecond (ps) timescales, they cannot be investigated using the traditional experimental methods. Hence, multi-scale multi-physics modelling, which includes a combination of quantum mechanics (QM) calculations, molecular dynamics (MD) simulations, and macro-scale mathematical modelling, is implemented by researchers to study these mechanisms. In solid-state batteries, various aspects pertaining to the growth of the SEI layer are largely unknown because of its complex structure and unstable formation process [23]. Specifically, the main constraint stems from the absence of knowledge on the diffusion coefficient equation in the SEI layer, which is an issue that is also found in LIBs [24,25,26]. In the absence of an accurate mathematical formulation, Deng et al. [27] hypothesized a constant diffusion coefficient in their reduced order model. A more advanced pseudo-two-dimensional model has been proposed for ASSBs that continues to implement a fixed value for the diffusion coefficient over the anode active materials [28].

Clearly, there is a void in the current literature with regards to an accurate description of the diffusion mechanism in the SEI layer. Addressing this gap, this work aims to develop a model for solid-state batteries where a detailed diffusion equation is incorporated into the macro-scale model. The mathematical formulation presented in this work for ASSBs is inspired by our recent proposition for the liquid electrolyte-based LIBs [29]. More precisely, the model proposed in this work is an adaptation of the one developed by Ekström and Lindbergh [30], i.e., a macro-scale continuum mathematical model that quantifies the influence of SEI layer on the ageing of LIBs with a graphite anode material. Their model is a blend of kinetic and transport control systems and employs a constant diffusion coefficient. Additionally, their model requires three lumped fitting parameters, which are substituted in the equations rather than variables such as a diffusion coefficient. These parameters enhance the accuracy of the model with respect to the experimental data over a variety of temperature and concentration values. However, a major drawback of their model is that it needs to be tuned using battery-specific experimental data for investigating a battery design. Further, the fitting parameters will vary with the material of the LIBs, forbidding us from a computational study of new and unique materials in LIBs. Improving upon their model, in our earlier work [21], we combined quantum mechanics (QM) calculations and molecular dynamics (MD) simulations to propose an equation for the diffusion coefficient, dependent on temperature and Li-ion concentration, for each crystal structure in the SEI’s inner section. To accurately represent the physics within the SEI, this macro-scale mathematical model was refined with a single equation for the diffusion coefficient. Specifically, the diffusion coefficient equation from our previous work [21] was integrated into the macro-scale mathematical model (MSMM) developed by Ekström and Lindbergh [30], which is utilized in commercial engineering software such as Comsol Multiphysics, to analyze the capacity fading of LIBs. In implementing an anologous version in this research for the ASSBs, our formulation is verified in relation to the experimental data of ASSB, predicting the battery’s behaviour for a specific temperature under multiple discharging conditions. Adapting the Ekström and Lindbergh MSMM for ASSB, we present a model with just two simplified fitting parameters, omitting the most complex parameter in the primary model. The updated expression compensates for the influence of temperature and concentration on ageing. As part of model validation, it was employed to study the formation of SEI layers and the consequent capacity fading as a function of time and initial SOC, for an extended range of temperature and concentration conditions.

It must be noted that our earlier model for LIBs was designed for batteries that used graphite anodes [29]. However, in ASSBs, there is an acute shortage of experimental data on batteries with these anodes. This is because these anodes have been proven to not work efficiently with solid electrolytes [31,32,33]. Specifically, it has been shown that there is a loss of contact between the solid electrolyte and this anode. Consequently, a large part of battery research in the past years has been focused on the use of the so-called “soft” electrolytes. The study by Kobayashi [34], the one under our scrutiny, describes the adoption of graphite as an anode in conjunction with two classes of solid polymer electrolytes, a “soft” one and a “hard” one. This work focuses on the “soft” polymer, that will be addressed as “SPE1”. The experimental tests from the literature correspond to a 2032 half coin cell with the architecture [Graphite|SPE1|Li], and a prototype pouch cell [Graphite|SPE1|LiFePO₄]. In the coin cell case, Li is employed as the counter electrode to permit Li intercalation in the graphite anode. The solid-polymer electrolyte is formed by a high-molecular-weight ether-based polymer (P(EO/MEEGE) = 88/12, DAISO). Ethylene oxide (EO) and 2-(2-methoxyethoxy) ethyl glycidyl ether (MEEGE) are its components. Lithium bis(trifluoromethylsulfonyl) imide (LiTFSI, 3 M) was incorporated in a molar ratio of [O]/[Li] = 16/1. Both P(EO/MEEGE) and LiTFSI were dissolved in acetonitrile (AN) [34]. Although specific details regarding the composition of the SEI layer are not provided, the main components that form the permanent layer of the SEI can be identified within the solid polymer electrolyte (SPE). An additional and significant outcome of this study is the validation of the developed model through a real driving cycle, comparing the model’s ageing data with the LIB experimental one.

2. The Theoretical Method and Computational Details

In ordinary LIBs, the inorganic inner layer consists of Li₂CO₃, LiF, and Li₂O [15,21,35]. In contrast, the organic outer layer is made of dilithium ethylene glycol dicarbonate (Li₂EDC) and ROLi (R depends on the solvent). The inner substrate is mainly composed of fixed materials, which do not depend on the electrolyte composition. This is where the diffusion mechanism is deeply investigated and highlighted in order to integrate it with the single-particle model previously described for SEI formation. Experimental research has established that Li₂CO₃ is an outcome of the conversion reaction of CoCO₃ upon Li-ion insertion, when ethylene carbonate is present in the liquid electrolyte. However, Li₂CO₃ is unsteady from a thermodynamic point of view and will reduce to Li₂C₂ and Li₂O [36]. Also, Li₂C₂ will contribute to other processes and create Li⁺, C₂H₂, and C [37]. Hence, Li₂CO₃ cannot be viewed as a permanent element in the inner area of the SEI layer. In the case of an ASSB, the equivalent inner layer composition is confirmed to have LiF and Li₂O [22,23,34]. The diffusion equation of Li ions through the crystal structure of the SEI layer can be achieved by implementing a modified version of the Arrhenius law (Equation (2)), where the Li diffusion coefficient is a function of the concentration (C) and temperature (T). The total diffusion coefficient can be computed with the following relation [21]:

$$D_t={\delta }_{{\mathrm{Li}}_2\mathrm{O}}{D}_{{\mathrm{Li}}_2\mathrm{O}}+{\delta }_{\mathrm{LiF}}{D}_{\mathrm{LiF}},$$

(1)

Furthermore, recent advancements in the characterization of SEI layers have revealed that the interplay between inorganic and organic components significantly impacts the stability and performance of LIBs and ASSBs. In particular, understanding the dynamic evolution of the SEI layer during battery cycling is crucial. The continuous formation and repair of the SEI layer contribute to capacity fading over time. Advanced techniques such as in situ electron microscopy and spectroscopy are being employed to observe these changes at the atomic level, providing deeper insights into the mechanisms driving SEI formation and degradation. Additionally, the integration of computational modelling with experimental data is becoming increasingly important. By simulating the electrochemical environment and interactions within the SEI layer, researchers can predict the behaviour of new electrolyte formulations and SEI compositions. This synergy between theory and experiment accelerates the development of more robust SEI layers, ultimately leading to batteries with longer lifespans and higher efficiency. In summary, the ongoing research into the SEI layers of both LIBs and ASSBs is critical for enhancing battery performance. The integration of experimental studies, advanced characterization methods, and computational modelling provides a thorough approach to understanding and enhancing the SEI layer, setting the stage for future energy storage advancements.

In Equation (1), ${\delta }_i$ and $D_i$ are the fraction of the surface area and the diffusion coefficient of the i^th component, respectively. Additionally, for each species the diffusion coefficient can be computed as a function of temperature and concentration, implementing a modified version of the Arrhenius law:

$$D_i(C,T)=D_{0}exp\left(\frac{-A_{0}EB\left(C\right)}{k_{b}T}\right).$$

(2)

In the above equation, $A_0$ and $D_0$ are component dependent constants. Also, EB stands for the energy barrier, which can be expressed as a function of concentration as:

$$EB\left(C_{Li}\right)=a_2{C}_{L{i}^{+}}^2+a_1{C}_{L{i}^{+}}+a_0,$$

(3)

where the dependent constants, $a_0$, $a_1$, and $a_2$, are summarized in

Table 1. Coefficients of the second-degree polynomial described in Equation (3) for Li₂O and LiF [21].

	${\mathit{a}}_0$ [eV]	${\mathit{a}}_1$ [eV]	${\mathit{a}}_2$ [eV]
Li2O	3.9488	$-8.9294$	12.0460
LiF	1.9886	$-2.5607$	3.5237

.

Since an actual measurement of the fractional surface area of the individual constituents in the SEI layer would be extremely complex [21], the following formulation has been employed in this work:

$${\omega }_i=\frac{m_i}{m_{{\mathrm{Li}}_2\mathrm{O}}+m_{\mathrm{LiF}}},$$

(4)

$$\begin{array}{ccc}\hfill {\delta }_{L{i}_{2}O}& =\frac{{\omega }_{L{i}_{2}O}}{{\omega }_{L{i}_{2}O}+\frac{{\rho }_{L{i}_{2}O}}{{\rho }_{LiF}}{\omega }_{LiF}}\mathrm{and}{\delta }_{LiF}\hfill & \hfill =\frac{{\omega }_{LiF}}{{\omega }_{LiF}+\frac{{\rho }_{LiF}}{{\rho }_{L{i}_{2}O}}{\omega }_{L{i}_{2}O}O},\end{array}$$

(5)

In the above equations, m, L, $\omega$, and $\rho$ are the mass, thickness, mass fraction, and density of the material i (in this case, Li₂O and LiF), respectively.

A Mathematical Model for Ageing

An upgraded version of the Ekström and Lindbergh model, compatible with a solid-state battery is proposed in this research. The model predicts battery ageing by utilizing data on the SEI growth rate, significantly improving the model’s accuracy. In particular, we provide a zero-dimensional model that employs a constant current control in battery cycling, eliminating the need to define a positive electrode or electrolyte. The accumulated charge (Q_SEI (C)), which is lost in ASSBs due to the SEI layer formation processes, is computed as follows:

$$\frac{d{Q}_{SEI}}{dt}=-I_{SEI},$$

(6)

where $I_{SEI}$(A) is the current of the parasitic reactions involved in the SEI layer formation. It can be represented as the aggregate of the currents traversing the surfaces that are completely coated by either an intact or a fractured SEI layer. The mathematical derivation of $I_{SEI}$, presented in the Appendix A, yields the following expression:

$$I_{SEI}=-\left(1+H{K}_{crd}\right)\frac{J{I}_{1C}}{exp\left(\frac{\alpha F{\eta }_{SEI}}{RT}\right)+\frac{fJ{Q}_{SEI}}{I_{1C}}},$$

(7)

where $I_{SEI}$ and $I_{1C}$ denote the currents through the cracked parts and the 1 C-rate charging current, respectively. Furthermore, the original model by Ekström and Lindbergh incorporates the following three lumped fitting parameters:

$$J=\frac{{ϵ}_{cov}{I}_0}{I_{1C}},$$

(8)

$$f=\frac{{\tau }_{cov}V{I}_{1C}^2}{{ϵ}_{cov}\left(1-{ϵ}_{cov}\right)CDF{A}^2},$$

(9)

$$H=\frac{a_{crd}}{{ϵ}_{cov}}.$$

(10)

The parameters in these equations are described in the nomenclature section. Collectively, these parameters compensate for the lack of information on the growth of the SEI layer and the ageing processes. Further, these fitting parameters must be calibrated for each new battery cell and/or operational condition.

As stated previously, the SEI layer comprises various materials and crystal structures, namely, Li₂O and LiF, that allow the diffusion of electrolyte solvent molecules and Li-ions, which are of different sizes and have different charge values. This implies a wide variation in the diffusion coefficients in these constituent structures, which will be governed by the operating temperatures and concentration conditions. Unfortunately, with experimental data for the complex diffusion and reaction processes, Equations (8)–(10) for $J,f$, and H are adopted for better accuracy and agreement with the experimental data. Since the accumulation of Li-ions increases the gradient of the charge distribution in the SEI layer, it increases the electron leakage, increasing $I_0$. Conversely, an increased diffusion coefficient will reduce the gradient of Li-ions within the SEI layer. Specifically, $I_0$ is inversely related to $D_T$ [29]. Thus, in this study, by modifying Equation (8), we utilized the following expression for J, which incorporates a material constant ($J_0$) and $D_T$:

$$J=\frac{J_0}{D_T}$$

(11)

In this equation, D_T is as defined in Equation (1). $J_0$, a material constant, is set to a value of $2.2\times {10}^{-15}$$\left(\frac{m^2}{s}\right)$. Thus, with this definition, the lumped fitted parameter (J), defined in Equation (8) is eliminated from the new model. Similarly, updated expressions for f and H parameters were obtained as follows: The parameters f and H were tuned using the experimental data of Kobayashi et al. [34]. Specifically, the values were tuned to the experimental condition of charge/discharge cycling with 1/8 C-rate and 1/16 C-rate load currents at 60 °C. In doing so, the value of J is as prescribed in Equation (11). The tuned values are reported in

Table 2. Parameters for the modified MSMM ageing model for ASSBs.

ASSB Format	Temperature	f	H
Coin cell	60 °C	1.5 × 105	3.5
Pouch cell	60 °C	3.4 × 104	2.5

.

Although the equation for the diffusion coefficient (Equation (1)) employs the same formulation for all diffusing particles, it appropriately considers the effects of different concentrations, temperatures, and crystal structures on the diffusion coefficient and SEI formation. In the forthcoming section, we establish that our proposed model is accurate with respect to the experimental data, and forecasts the ASSB behaviour under diverse situations, thereby validating the new formulation that requires just two fitting parameters. Furthermore, after considering capacity fading resulting from Li-ion loss during SEI formation, the relative capacity (RC) can be calculated as follows:

$$\mathrm{RC}=\frac{Q_{\mathrm{batt},0}-Q_{SEl}}{Q_{\mathrm{batt},0}},$$

(12)

where Q_batt,0 is the initial battery capacity. Additionally, the thickness of the SEI layer can be expressed as:

$$s=\frac{Q_{SEI}V}{\left(1-{ϵ}_{cov}\right)A}.$$

(13)

The initial charge accumulation (Q_SEI) is set to zero at $t=0$, and the relative capacity RC is accordingly set to 100%. The relative capacity for the previously highlighted ageing scenario is evaluated over a period of 200 days, employing the enhanced Ekström and Lindbergh model, and the results are then compared with the experimental data. Further, a parametric analysis has been conducted, supported by evidence and insights from the literature.

3. Results and Discussion

Model Validation: Due to the lack of sufficient experimental data, we investigated the capacity fade as a function of time for one temperature. The outcome of the investigation is presented in Figure 1. The results demonstrate the ability of the enhanced MSMM in forecasting a capacity fade, consistent with the experimental results for both full-cell and half-cell configurations. This is because of the two fitting parameters considered in this model, which can be adjusted for different cell types and chemistries. Experimental data are from a custom 2032 coin cell whose specific capacity is retrieved from the reference battery.

Furthermore, to demonstrate the ability of the model to adapt to different geometries and cathode compositions, the MSMM is calibrated with an upgraded set of f and H values reported in Table 2, and tested for the ASSB pouch cell made by Kobayashi et al. [34] (see Figure 1). As seen in this figure, there is an excellent agreement with the experimental data with an $R^2$ value of 0.99 in both cases.

Additionally, we also applied our enhanced model to a LIB 63 Ah Nickel/Manganese/Cobalt–Lithium Nickel Manganese Cobalt Oxide(NMC-LMO) battery examined through a Worldwide Harmonized Light Vehicle Test Procedure (WLTP) cycle in Ref. [38]. The testing current profile is retrieved by applying a backward modelling approach to the vehicle data. The predictions from our model along with the experimental data of Micari et al. [39] are shown in Figure 2. As seen in this figure, there is an excellent agreement between our model and the experimental data.

To further establish the accuracy and thereby the validity of our model, we studied the open-circuit storage conditions and a driving cycle test using our model. The results from these evaluations are presented in Figure 3 and Figure 4. To match the experimental data, the values of f and H were retuned for the two operating temperatures of 25 °C and 45 °C (

Table 3. Parameters for the modified MSMM ageing model for LIBs.

Temperature	f	H
25 °C	3.1 × 106	74.8
45 °C	4.9 × 105	42.1

).

As seen in Figure 4, the WLTP results are extremely satisfactory. A certain amount of divergence is registered in the first 2000 cycles because of the oscillatory nature of the experimental data [38]. To summarize, the determination of coefficient is reported in

Table 4. $R^2$ values of the computed relative capacity derived from the modified MSMM models in relation to the experimental results.

Temperature	Operating Condition	${\mathit{R}}^2$ Value Using MSMM
25 °C	Open Circuit-80% SOC	$0.99$
45 °C	Open Circuit-100% SOC	$0.98$
25 °C	WLTP	$0.96$

.

Parametric Analysis: The validated model has been used to evaluate the ageing process of the ASSB coin cell at different operating conditions. In absence of data for additional temperatures, the following temperature-dependent linear profile was assumed for the parameters f and H:

$$f\left(T\right)=f_{0}T+f_1,$$

(14)

$$H\left(T\right)=H_{0}T+H_1,$$

(15)

where $f_0$ and $H_0$ are summarized in

Table 5. The constants in Equations (14) and (15).

${\mathit{f}}_0$$\left[\frac{1}{\mathit{K}}\right]$	${\mathit{f}}_1$	${\mathit{H}}_0$$\left[\frac{1}{\mathit{K}}\right]$	${\mathit{H}}_1$
$-4.5\times {10}^3$	1.7 × 106	$-0.14$	50.1

. The choice of this linear profile is based on our earlier work [29] where we successfully applied an analogous model to study the effect of temperature on LFP commercial battery, assuming a linear temperature-dependent function for the parameters f and H. It must be noted that the values highlighted in our prior work [29] were for liquid electrolyte-based batteries and as such would not be applicable to ASSBs. In fact, with those constants, the produced results were not feasible and in the 40–50 °C range they did not demonstrate sufficient sensitiveness to temperature variation. Hence, with the intent of forecasting a reliable trend for the ASSB capacity fade, the linear profile coefficients, $f_0$, $f_1$, and $H_1$ were re-tuned (see Table 5). The value of $H_0$ was the same as in our earlier model [29].

First, the ASSB cycling capabilities were analyzed for various temperature levels. The battery performance was evaluated within a range limited by the available experimental data. Specifically, the effect of temperature in the range of [40 °C, 60 °C] on the ageing of a battery cycling at $1/8$ C rate has been evaluated, and the results are summarized in Figure 5.

As seen in this figure, lowering the operating temperature significantly retards the ageing of a battery. Though we expect a higher lithiation temperature to not only enhance the ionic conductivity of the SPE but also mitigate the mechanical stress due to the softening effect, there is evidence in the literature that a raise in the temperature has a deleterious effect on solid polymer electrolyte, accelerating the ageing process [40]. Additionally, the electrochemical performance of solid-state lithium batteries significantly declines at extreme temperatures, leading to considerable energy and power losses, reduced cycling lifespan, and increased safety risks [41]. Our trends are in excellent agreement with these propositions. A similar result is obtained when the model is applied to study the effect on the battery cycling at 1 C-rate, the theoretical upper limit value at which the enhanced model has been properly tested and validated (see Figure 6). Comparing the ageing results from the different C-rates shown in Figure 6, it is clear that the raise of the operating c-rate has an adverse effect on the ageing of the solid-state battery. Once again, this is in agreement with the experimental work of Fang et al. [42]. The findings indicate that capacity loss in all-solid-state batteries (ASSBs) is initially driven by a reduction in the contact area between the electrolyte and electrode interfaces. Over time, this degradation is mainly influenced by the mobility of lithium ions within the ageing electrolyte. Furthermore, enhancing the discharge rate can magnify this deterioration [42]. More precisely, their results indicated a 10% increase in capacity loss when they raised the continuous discharging rate of 1 C.

In Figure 7a, the relative capacity is computed for an extended range of initial SOC. The slowest ageing rate was recorded near 50% SOC. However, the initial SOC has a minor influence on the rate of capacity loss with respect to the LIB case. This is due to the higher operating temperature, which mitigates the initial SOC leverage, a tendency consistent with our previous observations [29].

We also investigated the growth of an SEI layer that grows rapidly in the battery. As shown in Figure 7b, in the ASSB coin cell, the maximum layer thickness was 79 nm at the extremes of the SOC window. At 50% SOC, the SEI layer thickness is about 75 nm. This lack of variation in the thickness is consistent with the findings of Kobayashi et al. [34], who stated that unlike the liquid electrolyte systems, the SEI formation occurs over a wider voltage region in the SPE systems. Note that the SEI formation process is more evident in an ASSB because of the high difference in reactivity between the SPE and the graphite anode [34]. In Figure 7a, the relative capacity has been calculated as a function of time and initial state of charge (SOC). Due to the limited experimental data, it is not possible to compare different initial SOCs directly or to confirm the impact of initial SOC variations on battery ageing. However, for an initial SOC of 50%, the trend aligns well with the experimental data, achieving an accuracy of over 98%. Conversely, Figure 7b illustrates the SEI thickness over time. Unfortunately, experimental tests on battery systems do not report SEI thickness, making direct comparisons difficult. In the model, SEI thickness is directly correlated with capacity fade and is a primary factor in capacity retention. Given the high accuracy of capacity retention results relative to the experimental data, the SEI thickness results generated by this model can be considered reliable.

4. Further Investigations

We have reported the efficacy of an enhanced model applied to LIBs evaluated with a fast-varying discharge rate and to solid-state batteries. In the latter instance, implementing only two fitting parameters allows for estimating the ASSB’s electrochemical performance for diverse temperatures and concentrations. Nevertheless, it impels us to acquire new experimental evidence to absolutely validate our model. Furthermore, in the macro-scale model we have assumed the same diffusion profile through the whole SEI layer for all the particles [29], which could explain for both battery technologies the slight divergence from experimental data. On the ASSB’s side, an additional factor to be addressed concerns the composition of the SEI layer. Specifically, the outer SEI layer, whose species are significantly reliant on the solid electrolyte composition, has to be examined as it could experience an uncontrolled expansion. Hence, additional research is needed to gain a better understanding of the SEI layer and to provide a diffusion equation that gathers the effects of particle charge and size, and contact resistance for ASSBs, on the diffusion coefficient and SEI species.

5. Conclusions

We have presented an adaptation to solid-state batteries of the macro-scale model previously developed [29]. The calibration of the two modified lumped fitting parameters in conjunction with the upgraded equation of D_T (Equation (1)) is shown to adequately predict the ASSB capacity decay as a function of time. This model is validated with regard to experimental data at a single temperature under charge/discharge cycling with 1/8 C-rate load current ageing condition. Additionally, acting only on the two lumped parameters allows for adjusting the model to a different ASSB geometry and cathode composition. The model estimates a capacity fade trend in solid agreement with actual data in the instance of an ASSB pouch cell cycled with a 1/16 C-rate charge/discharge load current. Further, the upgraded model can also determine the SEI formation rate as a function of temperature and Li-ion concentration. The LIB analysis gave further insights; in this case, the model is verified at a specific temperature for multiple ageing conditions: (I) open circuit state at 80% and 100% SOC and (II) WLTP cycling with a fast-varying load current. In conclusion, the macro-scale model, whose fitting parameters need to be assessed only for each distinct operating temperature, poses itself as a solid computational tool for accurately predicting the LIBs’ electrochemical performance and conducting a primary study on ASSBs; for the latter, this could be beneficial in the selection of optimal materials.