Reduced Order Modeling of Mechanical Degradation Induced Performance Decay in Lithium-Ion Battery Porous Electrodes

If an active particle of fixed radius is delithiated and lithiated at a constant rate for multiple cycles, it will experience mechanical degradation due to diffusion induced stresses. Several acoustic emission based experiments have been conducted to understand how fracture evolves within solid active particles.^48,49,62 In all the three research articles, it has been reported that mechanical degradation initiates at the first lithiation-delithiation cycle. Maximum amount of microcrack evolves in the first one or two cycles. After three to four cycles, damage evolution almost saturates. Insignificant amount of mechanical degradation occurs in subsequent lithiation-delithiation cycles. A similar behavior in terms of saturation in microcrack formation has been reported by the authors in an earlier article.⁵⁰ How fast the damage grows, and at what magnitude it saturates, depends on the particle size and rate of operation. Since the evolution of microcrack initially increases and eventually saturates, the best numerical approximation of such a behavior will be provided by the increasing form of an exponential decay curve.

A reduced order model has been developed to estimate the amount of microcrack formation under certain C-rate operating conditions and for particular particle sizes (R_s). A simplified expression for microcrack density (or fraction of broken bonds, f_bb) as a function of amp-hour-throughput (Ahtp) can be written as,

Here A_max signifies the maximum amount of damage that can occur in an active particle and m_rate stands for the rate at which damage evolution occurs. Both the A_max and m_rate parameters are functions of C-rate (C_rate) and particle size (R_s). More detailed expressions of the maximum damage and rate of damage will be provided in sub-section Development of a reduced order model of the Results and discussion section. It has been argued in Barai and Mukherjee²¹ that formation of a microcrack increases the tortuosity of the diffusion pathway resulting in reduced diffusivity of the active particle. An expression for effective diffusivity has been developed to correlate the microcrack density with the diffusivity of the active particle, which is given as,

where D_s denotes the solid phase diffusivity in the anode active materials without any mechanical degradation. The solid phase diffusivity (D_s) (also used in Eqs. A1a and A1b) is replaced by the expression of effective diffusivity as obtained from Eq. 2. It has been demonstrated that reducing the local diffusivity of the active particle could capture the effect of mechanical degradation on the diffusion of lithium species.²¹ The simulation of diffusion inside active particles and mechanical degradation has been conducted in a circular 2D domain. For different particle sizes and different C-rates, certain values of surface concentration were obtained from the 2D simulations taking into account the effect of microcracks. For the same particle sizes and C-rates, 1D simulations were conducted with constant values of diffusivity that can generate the same surface concentration as that obtained from the 2D simulations. The diffusivity value for the 1D simulations has been obtained from Eq. 2. Estimation of the exponent γ has been conducted by comparing the surface concentrations from 1D and 2D analyses. More elaborate explanation behind the exact value of the exponent (γ) will be provided towards the end of sub-section Development of a reduced order model of the Results and discussion section.

In the computational analysis, fraction of broken bonds or microcrack density (f_bb) is defined as the ratio of broken elements over total number of elements. Here elements refer to a computational entity and does not have any experimentally observable counterpart. Thus, the term f_bb turns out to be a dimensionless number. Based on dimensional analysis, the term A_max turns out to be dimensionless. Whereas, the term m_rate must maintain the dimension of (Ahtp)^{− 1} to achieve dimensional equality within the exponent. However, the two defining terms (A_max, m_rate) used in the reduced order model does not have any direct experimentally measurable counterparts. Acoustic emission is a methodology to measure the extent of mechanical degradation in solid materials. Cumulative strain energy release measured form acoustic emission experiments is equivalent to the damage profile observed in the present context.^48,62 Hence, an experimental counterpart of the maximum amount of mechanical degradation parameter (A_max) is the maximum cumulative strain energy release observed from acoustic emission experiments. Saturation in the strain energy release has been reported in several experimental articles,^48,49,62 which is equivalent to the saturation in evolution of microcrack density observed by the authors.⁵⁰

Most of the reduced order models are derived from systematic reduction of governing differential equations. But if the phenomenon under consideration cannot be characterized by a governing differential equation, then based on the trend, phenomenological models can be developed to capture the variation.¹³ In the present context, evolution of microcrack density cannot be captured using any differential equation. It is possible to develop reduced order models for stress generation, but fracture formation is rather a stochastic process because material heterogeneity is also involved there.⁶³ Due to the lack of governing equations, it is important to have phenomenological models for predicting damage evolution. Usually empirical models are mathematical expressions that are developed entirely based on data. No physical significance exists behind particular mathematical expressions. However, in the present context, the saturation phenomenon in mechanical degradation can be explained from the strain energy release perspective.^21,47,50 Under externally applied load, evolution of mechanical degradation happens to release the excessive strain energy that the system cannot sustain. During lithiation – delithiation process, the same amount of diffusion induced load acts on the system. The strain energy release required for sustaining the concentration gradient induced load is achieved within the first few discharge – charge cycles. As a result, during subsequent lithiation – delithiation process, extra strain energy release is not required. This leads to saturation in the amount of mechanical degradation. Here, an inherent assumption is that the lithiation – delithiation process occurs at a constant rate. The maximum amount of mechanical degradation depends on the particle size and C-rate of operation, through the magnitude of concentration gradient term. Thus there exists some form of physical significance behind the equations adopted in this study. To calculate the amount of microcrack formation, the governing differential equations are not being solved in details. As a result, the physics based mathematical representation of damage evolution can be described as a reduced order model of a complicated phenomenon.

The entire computational methodology adopted in the present context can be divided into several smaller components. Firstly, diffusion of lithium inside a circular cross section of a spherical particle have been solved using the time dependent Fick's law. The concentration gradient inside the active particles give rise to diffusion induced stress (DIS). Secondly, generation of DIS can lead to formation of microcracks, which can be captured by solving the momentum balance equation. Nucleation and propagation of microcracks produce spanning cracks. Details of the computational methodology used to obtain the spanning cracks have been described by the authors in earlier articles.^21,47 A brief description of the same computational technique will be provided below. Time dependent diffusion equation is solved using the finite volume method on a 2D square grid. Constant flux boundary condition is applied on the surface of the circular active particle. Two dimensional lattice spring methods have been used to capture the microcrack formation within the active particles. The main essence of this methodology is that the entire mass of the continuum can be assumed to be discretized within uniformly distributed nodes. Each of the nodes is connected by spring elements. This spring elements display axial as well as shear resistance. Under externally applied diffusion induced load, these lattice spring elements deform to maintain equilibrium within the structure. This gives rise to evolution of strain energy within each of the springs. If the energy stored in an element exceeds its fracture threshold, it is assumed to be broken and irreversibly removed from the network of lattice springs. Subsequent rupture of adjacent spring elements due to diffusion induced stress can give rise to formation of spanning cracks within the active particles. The variable "fraction of broken bonds" (f_bb) is defined as the ratio of number of broken springs over the total number of springs that exist within the domain.

Formation of a microcrack hinders the diffusion pathway of active particles, and increases the tortuosity within the material.²¹ This is taken into account within the 2D diffusion solve by modifying the diffusivity parameter only at the point where mechanical degradation occurred. Such modification of the local diffusivity resulted in higher values of concentration gradient within the active particles. Surface concentration decreases much faster during the delithiation process due to formation of microcracks. Consequently, reduction in effective capacity is observed due to increased mass transport resistance.

To solve for this capacity fade due to mechanical degradation in the anode active particles, the effective solid phase diffusivity has been extracted from Eqs. 1a and 2. The values of diffusion coefficient in individual active particles under different amounts of microcrack density have been estimated. After that, it is implemented within the 1D computational framework for electrode level analysis (Eq. A1a). Because the evolution of voltage and capacity within an electrode is a transient process, the governing differential equations (Eqs. A1a, B4, B5 and B6) are solved in an incremental fashion by taking small steps over the time variable. At each time steps, the increment in microcrack density for every particle in the negative electrode is estimated according to the following rate equation:

Eq. 3 has been obtained by taking the first derivative of Eq. A1a with respect to the independent variable Ahtp. In a small time increment, the incremental amp-hour-throughput (Ahtp) is also small. Thus, the total amount of microcrack density after a small time increment can be obtained as:

Here, Ahtp is the amp-hour-throughput at the end of the present step, Ahtp₀ is the amp-hour-throughput at the end of the previous step, and ΔAhtp is the incremental amp-hour-throughput. The effective diffusivity under this modified amount of microcrack density has been estimated from Eq. 2, and fed into Eq. A1a of the 1D coupled-electrode level model. Reduction in solid phase diffusivity increases the resistance due to mass transport and results in an effective capacity fade of the lithium ion cell. The capacity fade due to reduction in solid phase diffusivity becomes more prominent under high C-rate operating conditions.

For the single discharge simulations, the initial state of charge (SOC) is assumed to be 0.9 in the negative electrode and 0.35 in the positive electrode. These values of SOC resulted in an initial voltage of approximately 4.3 V. The lower cut-off voltage for lithium ion cells has been assumed to be 2.4 V for hard-carbon based anodes. For analyzing cycling performance, the lithium ion battery is assumed to be in a discharged state. It is charged to an upper voltage limit of 4.2 V at a very slow rate (0.05 C). Discharge-charge cycles are conducted on these electrodes to estimate the cycling performance of an LIB under different design (particle size) and operating conditions (C-rate). The parameters used in the simulations are listed in Table I. Evolution of mechanical degradation within single active particles as well as different particles located at different portions of the electrode are elaborated in the Results and discussion section.

According to the authors, evolution of microcrack density occurs toward the beginning of the delithiation process.^21,47 Eventually, the amount of microcrack formation reaches a state of saturation, and no further increase in mechanical degradation is observed during subsequent discharge-charge cycles. Thus, an exponential increase in damage evolution followed by saturation can be successfully captured by Eq. 1a provided in the Methodology section. The maximum amount of damage (A_max) and the rate of damage evolution (m_rate) depend on the particle size (R_s) and the C-rate at which the simulation is being conducted. The purpose of reduced order modeling is to develop an analytical expression that can approximately predict the microcrack density (f_bb) under certain particle size and C-rate operating condition. Following Eq. 1a and 1 a, the unfinished task is to estimate an analytical representation of A_max and m_rate as a function of particle size and C-rate. Evolution of microcracks has been simulated for six different particle sizes, R_s = [2.5 μm, 5.0 μm, 7.5 μm, 10.0 μm, 12.5 μm, and 15.0 μm], and eight different C-rates [1 C, 2 C, 3 C, 4 C, 5 C, 6 C, 8 C and 10 C], for each of the particles. Damage evolution for each of these cases were estimated by solving the detailed 2D models developed by the authors in an earlier article.²¹ The damage evolution vs. amp-hour-throughput curve for each of the particle sizes at every C-rate has been plotted separately. The optimum values of A_max and m_rate has been estimated using a least-square minimization based fitting technique. Different particle size operating at different rates produce different magnitudes of A_max and m_rate. Two generalized analytical expressions have been developed that can capture the variation in A_max and m_rate for various particle sizes and C-rates, which are also provided below,

and

where, R_s represents particle size and C_rate signifies how fast the active particles are delithiated and lithiated. The two expressions provided in the Eqs. 5a and 5b have been estimated by using a least square fitting method. The general form of this expression has been obtained based on the physics of the problem. Figure 4a demonstrates that Eq. 5a can estimate the values of A_max with R²accuracy equal to 0.9066. Similarly, as depicted in Figure 4b, Eq. 5b can predict the values of m_rate with R²accuracy equal to 0.8051. The different parameters used in Eqs. 5a and 5b are obtained using the "nlinfit" function embedded in MATLAB. These analytical expressions given in Eqs. 5a and 5b along with Eq. 1a constitute the reduced order model for predicting microcrack density inside active particles. The R² value for A_max is 0.9, which is definitely good for prediction purposes. However the R² value of m_rate is only 0.8 that is not sufficiently good for prognosis purpose. Since the maximum value of mechanical degradation depends on A_max, and m_rate just dictates how quickly/slowly the maximum value is reached, not very accurate prediction for m_rate can still be applied for prediction purposes. The inaccuracy introduced by R² value of 0.8 for the m_rate parameter, will have minor impact on the final prognosis. This reduced order model is applicable to active particles of different size and operating at different C-rates but maintained at a fixed room temperature (T = 25°C) condition. Reduced order models of microcrack density applicable to different operating temperatures were not investigated in this study and will be reported as part of a separate article.

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Once an approximate expression for the evolution of microcrack density is established, it is important to characterize how the mechanical degradation affects the solid phase diffusivity of the active particles. In earlier articles, it was argued by the authors that formation of a microcrack hinders the diffusion pathway of lithium.^21,47,50,65 In the presence of a microcrack, the ions take a more tortuous pathway to traverse from one point to another, which eventually results in reduction of the diffusivity of the active particles. To capture this deterioration in the diffusion coefficient due to evolution of microcracks, an analytical expression is suggested in Eq. 2. The only unknown term in the right hand side of that equation is the exponent γ. In the 2D simulations reported here, the impact of local microcrack formation on the diffusion of lithium species is taken into consideration by decreasing the local diffusivity. Thus, the effect of increased tortuosity is incorporated within the 2D simulations. The concentration gradient obtained from the 2D simulation incorporates the effect of microcrack formation within itself. Here concentration gradient refers to the difference between bulk concentration and the surface concentration. Hence, the variable concentration gradient has the units as mol/m³. In Figure 5, the symbols denote the concentration gradient at the end of single delithiation process for different particles operating at various C-rates obtained from the 2D simulations.

One-dimensional simulations are also conducted with different values of effective diffusivity that can predict the concentration gradient obtained from the 2D simulations. For the 1D model, the Fick's law has been solved along the radial direction of a cylindrical particle. The effective diffusivities used in the 1D simulations were evaluated using Eq. 2. The main purpose of this exercise is to estimate a value of the exponent γ that can most accurately predict the values of the concentration gradient obtained from the 2D simulations. The analysis is being conducted for a fixed particle size (R_s) and a particular rate of delithiation, denoted by C_rate. Under these operating conditions, the magnitude of A_max and m_rate are estimated from Eqs. 5a and 5b, respectively. Using the value of A_max and m_rate, the amount of mechanical degradation (f_bb) has been estimated from Eq. 1a). Using this amount of microcrack formation, the magnitude of diffusivity has been obtained from Eq. 2 (the parameter γ is used in this step). This updated diffusivity is used to conduct the 1D simulation. The concentration gradient extracted from these 1D simulations should correlate properly with the concentration gradients obtained from 2D simulations. Whichever value of the exponent γ provides the best comparison for a wide range of operating conditions, that value will be adopted in the remaining simulations of this article.

In Figure 5, the lines denote values of concentration gradients as estimated by the 1D simulation (the symbols correspond to results from 2D analysis). Figures 5a, 5b, and 5c report the comparison between the concentration gradients obtained from 2D and 1D simulations for γ = 5.0, γ = 7.5 and γ = 9.5, respectively. As can be observed in Figure 5a that the 1D simulation with γ = 5.0 significantly under-predicts the concentration gradient for large particles. On the contrary, Figure 5c clearly shows that γ = 9.5 over-predicts the concentration gradient for large particle sizes operating at high C-rate conditions. The best correlation between the concentration gradients from the 1D and 2D simulations can be obtained with γ = 7.5, also depicted in Figure 5b. Usage of least square based fitting methodology would definitely be mathematically more accurate. However, it does not render any physical understanding of how variation of the parameters changes the effective diffusivity value. As a result, relatively more brute force type methodology have been adopted to calculate the exact value of exponent γ. It can be concluded that, to correlate the effect of microcrack density between 2D and 1D simulations, an optimum estimate of the exponent γ is 7.5. However, the active particles observed inside the LIB electrodes are spherical in shape and definitely require 3D consideration. To extend the estimate of exponent γ from 2D to 3D applications, it is raised by a factor of 3/2. The magnitude of this factor 3/2 has been estimated from the experience that the concentration gradients observed in 3D spherical active particles are approximately 3/2 times larger in magnitude than the concentration gradients observed in 2D cylindrical particles. Thus, for 3D applications, the optimum value of the exponent will be γ_3D = 3/2 · γ = 11.25. For all the subsequent applications, the optimum value of γ in 3D will be used (unless otherwise mentioned). Variation in exponent γ with changes in different physical properties, such as, diffusivity, elastic modulus or fracture threshold, has not been investigated yet. It will be considered as a future exercise.

All the simulations and analysis reported until now were conducted on a single active particle. The theory behind the 2D single particle simulations can be found in a previous article by the same authors.²¹ A realistic electrode consists of several spherical particles. The electrolyte concentration and electrolyte potential also change along the thickness of the electrode, which becomes more prominent under high C-rate conditions. As observed by the authors (see Refs. 21 and 50) as well as other researchers (see Refs. 66–68), larger diffusion-induced stress acts on the active particles under higher C-rates. Large diffusion-induced stress has the potential to induce enhanced amounts of mechanical degradation. Different C-rates affect the lithium flux within the active particles. Flux of lithium in or out of the active particle changes along the thickness of the electrode. To capture the variations in applied lithium flux, or in other words the C-rate, it is very important to solve the coupled 1D mass and charge transport equations (Eqs. A1a, A4a, A5a and A6a) provided in the Appendix A.

Performance of a lithium-ion cell depends on the open circuit potential of the active materials used in the electrode. Lithium nickel manganese cobalt oxide (LiNiMnCoO₂), also known as NMC, has been used as the cathode material. The expression of open-circuit-potential (OCP) for NMC has been adopted from Awarke et al. Ref. 69 Because damage evolution inside anode is being analyzed here, two different OCP curves have been taken into consideration, which correspond to two different anode materials: (i) Hard carbon, and (ii) Graphite. The OCP of hard-carbon has been adopted from Gu and Wang (see Ref. 32), whereas the OCP for graphite has been adopted from Srinivasan and Newman (see Ref. 70). Comparative reproduction of the OCP profiles for hard-carbon and graphite has been reported in Figure C1. Graphite shows a much flatter open circuit potential than hard carbon. As a result, the reaction current density at the anode shows a much higher gradient for graphite as compared to hard-carbon. For computational simplicity, the OCP for hard carbon has been adopted for the full charge-discharge simulations. The drive cycle simulations are conducted using both the OCP profiles (graphite and hard carbon). A comparative analysis, of which material leads to reduced mechanical degradation under drive cycle conditions, is presented towards the end of this article.

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The linearized governing differential equations given in Eqs. A1a, B4, B5, and B6 have been discretized using the finite-difference method and solved by implementing it in MATLAB. Coupling between these governing differential equations has been conducted through the nonlinear Butler-Volmer equation provided in Eq. A2. The parameters used to solve these coupled differential equations are provided in Table I. The voltage vs. capacity performance curve during the first constant current discharge process obtained by solving the 1D electrode level model is demonstrated in Figure 6a. Here, hard-carbon has been used as the anode active material and NMC as the cathode active material. Four different C-rates are taken into consideration. Higher values of C-rate resulted in reduced capacity due to enhanced kinetic and mass transport limitations. The maximum amount of lithium flux observed in the anode and cathode during the first CC discharge at different C-rates is reported in Figure 6b. In the present analysis, during the discharge process, the migration current is assumed to be positive. Outflux of lithium is assumed to have a positive sign, and the influx is signified by a negative value of the flux variable. During the discharge process, lithium species move out of the negative graphite electrode and enter the cathode. According to the convention followed in this research, during the discharge process, the anode experiences a positive flux of lithium, and negative flux is observed inside the cathode. As depicted in Figure 6b, the lithium flux in both the anode and cathode increases with the increase in applied C-rate. It can also be concluded from Figure 6b that the magnitude of the maximum lithium flux in both the anode and cathode is highest at the beginning of the discharge process. It eventually reduces and saturates at a particular value. Towards the end of the discharge process, maximum flux at the cathode experiences some fluctuation. The maximum lithium flux traverses along the thickness of the electrode during the discharge process, which is not shown in Figure 6b.

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During the first discharge process, the variation in the electrolyte potential and the electrolyte concentration plays a major role in determining the performance of the LIB. Figure 6c demonstrates the distribution of the electrolyte potential at the end of the first discharge state along the thickness of the entire electrode. Increasing the C-rate at which the cell is being operated results in an increased electrolyte potential at the negative electrode. The electrolyte potential is kept fixed at zero at the positive electrode-current collector interface (see boundary condition Eq. A6b). Similarly, variations in the electrolyte concentration along the thickness of the electrode at the end of the discharge process are displayed in Figure 6d. The portion inside the vertical dashed line signifies the region that lies inside the separator. The initial concentration inside the electrolyte is assumed to be 2,000 mol/m³. During the discharge process, inside the anode the lithium atoms come out of the solid active particles and enter the electrolyte. Within the cathode, the lithium atoms travel from the electrolyte into the solid active particles. During discharge, transport of lithium ions from anode to cathode through the separator happens via the diffusion and migration process. Due to diffusion-induced limitations, at a high C-rate (4 C), a significant amount of lithium ions is depleted from the cathode electrolyte. It is important to note that, during discharge lithium ion concentration in the anode may reach values as high as 3 M. There are chances of salt precipitation within the electrolyte, which can lead to loss of cyclable lithium and subsequently capacity fade. Also variation in lithium ion concentration may impact the conductivity within the electrolyte. However, in the present simulations dependence of electrolyte conductivity on the lithium ion concentration has not been incorporated. During the charge process, depletion of lithium ion concentration would occur in the anode and large concentration may be observed in the cathode. Hence, an appropriate value of the initial concentration of lithium salt within the electrolyte should be considered. Proper care must be taken while charging or discharging an LIB at very high C-rates to prevent situations where lithium ions are completely deleted from the electrolyte. Mechanical degradation within the solid active particles is not taken into consideration in any of the simulation results reported in Figures 6a–6d.

To establish the validity of the developed computational model, voltage vs. capacity performance curves predicted by the simulation is compared with experimentally observed results. Such a comparison between the performance curves obtained at 1 C and 3 C are shown in Figure 7. The experimentally observed voltage vs. capacity plots have been adopted form Figure 2 in Ji et al. (JES 2013).³⁴ Even though the comparison is not one on top of the other, they match quiet well in a qualitative sense. Graphite anode and NMC cathode has been used in both experiment and simulation. But the OCP curves for NMC and graphite used in computational analysis are not the same as reported in the experimental article. Isothermal condition has not been maintained in the experimentally observed result. Whereas, the simulations are conducted under isothermal operating conditions at T = 25°C. The change in temperature for 1 C is not significant, as reported in the experimental results. For operation at 3 C, almost 20°C increase in temperature is observed. The increased capacity for the cell at 3 C can be attributed to this increase in internal temperature. Difference in the overall voltage profile can be due to the mismatch in OCP curves for both the anode and cathode materials. Also in the present computational analysis, electrolyte conductivity (κ) has been assumed to be constant, and not dependent on lithium ion concentration. In a realistic electrolyte, conductivity changes strongly as a function of lithium ion content. The difference between the experimentally observed and simulated voltage curve can also be attributed to this parametric discrepancy.

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For solid active particles where transport of lithium species can be reasonably approximated by the diffusion process, delithiation gives rise to tensile stress and evolution of microcrack density along the peripheral region. During the lithiation process, microcrack evolution takes place close to the center of the active particles. The analytical expressions in Eqs. 1a, 5a, and 5b estimate the amount of microcrack density during delithiation for wide range of C-rates (1 C – 10 C) and particle sizes (2.5 μm – 15.0 μm). This relation is derived based on mechanical damage evolution only within the graphite active particles during the delithiation process. At the time of discharge, the graphite active particles within the anode experience delithiation. Hence, the analytical expression derived for microcrack evolution can be applicable to the anode active particles during the discharge process. For the present study, the cathode particles are assumed to be free of mechanical degradation. According to Eq. 2, diffusivity of the solid active phase decreases due to evolution of mechanical damage. Figure 8 demonstrates the distribution of mechanical degradation along the thickness of the negative electrode at the end of the first discharge process. Capacity fade due to increasing microcrack density has been analyzed in Figure 9. Strictly speaking, there should be a feedback of cell performance and capacity fade on the mechanical degradation. Based on some earlier investigations conducted by the authors, impact of capacity fade on further mechanical degradation is negligible (see Ref. 21). Because of this minor feedback effect, while deriving the reduced order model, only the impact of mechanical degradation on change in solid-state diffusivity has been taken into consideration. Point to be noted here is that, microcrack formation happens due to formation of concentration gradient within the active particles during operation. Effect of further microcrack formation due to performance decay has been neglected here.

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Under uniform distribution of particle sizes, the flux of lithium atoms experienced by the active particles determines the amount of diffusion-induced stress and subsequently the evolution of mechanical degradation. The location where maximum lithium flux is observed, experiences the largest extent of microcrack density. Thus, it is very important to have a prior knowledge of the location of the maximum reaction current density to properly understand the evolution of microcrack density along the thickness of the electrode. Figure 8a demonstrates the variations in lithium flux along the thickness of anode during CC discharge at 3 C containing active particles with a radius of 15 μm. Towards the beginning of the discharge simulation, at time t = 33.33 sec, a significantly large reaction current density and lithium flux are observed close to the separator. The lithium flux observed close to the current collector is significantly smaller than the value observed at the separator. With increasing time, the lithium flux close to the separator decreases and the flux at the current collector increases. At around t = 133.0 sec, the lithium atom flux at the current collector and the separator becomes almost equal. Close to the end of the discharge process, at t = 500.0 sec, the active particles close to the current collector experience slightly higher lithium flux than the particles located close to the separator. Thus, during the discharge process, there is a shift in the maximum reaction front from the separator to the current collector over time.

Because of the variation in the reaction current density over time, mechanical degradation also evolves accordingly along the thickness of the electrode. Figure 8b depicts how the microcrack density increases during CC discharge process at 3 C in an anode containing uniformly distributed active particles with a radius of 15 μm. Initially, at t = 33.33 sec, active particles close to the separator experience the maximum amount of lithium flux, which results in enhanced damage evolution near the separator. A similar pattern of higher microcrack density close to the separator and less damage near the current collector are observed until t = 166.66 sec. Then, due to the shift of the reaction current front towards the current collector, enhanced mechanical degradation is observed inside the active particles close to the current collector. Finally, close to the end of the discharge process, at t = 500.0 sec, almost uniform microcrack density is observed from the separator to the current collector. Distribution of mechanical damage at the end of the first discharge process is reported along the thickness of the negative electrode in Figure 8c for two different particle sizes (10.0 μm and 15.0 μm) and three different C-rates (1 C, 2 C, and 3 C). Less damage evolution is observed for smaller particles operating under low C-rate conditions. Larger particles operating at high C-rates display enhanced microcrack density. However, for all the particle sizes and all the operating conditions, damage evolution is very much uniform along the thickness of the electrode (from separator to current collector). This uniformity in microcrack density appears due to shifting of the maximum reaction current front from the separator to the current collector during the constant current discharge process.

To analyze the impact of microcrack density on the overall performance of the LIB electrode, multiple charge discharge cycles were conducted taking into consideration the effect of mechanical degradation on diffusivity of anode active particles. A correlation between the solid phase diffusivity of the anode active particles and microcrack density can be obtained from Eq. 2. Figure 9a demonstrates five charge-discharge cycles with (red dashed line) and without (black solid line) taking the damage evolution in the anode active particles into consideration. To maintain consistency in the capacity values, the battery is charged first from a very low state-of-charge condition in a CCCV fashion until a maximum voltage of 4.2 V is reached. Incorporation of mechanical degradation of the anode active particles results in reduction of effective solid phase diffusivity, and subsequently the resistance due to ion transport increases. Hence, a reduction in effective capacity is observed due to the evolution of microcracks inside the active particles. Capacity during a discharge process is estimated by subtracting the Ahtp at the beginning of discharge from the Ahtp at the end of the discharge process. Referring to Figure 9a, to estimate the discharge capacity without damage during the third cycle, Ahtp at point B is subtracted from the Ahtp at point A.

Figure 9b demonstrates the discharge capacity while operating at 2 C and 4 C during five subsequent charge-discharge cycles. If the evolution of mechanical degradation in anode active particles is not taken into consideration, the capacity values during all five discharge phenomena are exactly the same. Capacity at 4 C discharge (blue solid line) is less than that observed at 2 C (red solid line) due to the rise in kinetic and transport resistance at higher rates of operation. If mechanical degradation is taken into account, the discharge capacity keeps decreasing as the battery is cycled more and more (dashed line). Capacity fade during operation at 4 C is much greater than that observed at 2 C because at higher rates enhanced mechanical degradation occurs. A higher fraction of microcrack density (f_bb) results in smaller values of effective diffusivity of the anode active particles. Reduced diffusivity increases the transport resistance and subsequently enhances the capacity fade. Hence, the capacity fade due to mechanical degradation observed at higher rates of operation are much larger than that experienced at low C-rates.

Capacity fade due to only the mechanical degradation can be estimated by obtaining the difference between capacity without and with damage. The extent of capacity fade solely due to mechanical degradation is reported in Figure 9c. Maximum amount of damage at four different C-rates are also reported. Operation at lower values of C-rate (1 C or 2 C) gives rise to less damage and subsequently smaller capacity fade. However, a larger extent of damage and enhanced capacity fade are observed for high rate (3 C and 4 C) operations. Irrespective of the rate of operation, capacity fade tends to saturate at a certain limit. For smaller rates of operation, the capacity fade saturates much earlier than the batteries operating at higher C-rates. The maximum capacity fade also increases with an increasing rate of operation. All the simulations reported in Figure 9 used a particle radius of 10.0 μm in the anode. Extremely close values of microcrack density observed at 3 C and 4 C lead to the conclusion that the particle has almost reached the maximum amount of damage it will ever experience during CCCV charge–CC discharge cycles.

Instead of having a constant particle size along the thickness of the negative electrode, implementation of a gradient in particle size from the current collector to the separator (ascending or descending) may impact the evolution of microcrack density and subsequently capacity fade of the LIB. The two different particle size distributions taken into consideration are as follows: i) linearly increasing particle size from 5 μm at the current collector to 15 μm at the separator, and ii) linearly decreasing particle size from 15 μm at the current collector to 5 μm at the separator. While using different particle size distributions, the total volume of the electrode and the volume fraction of solid active material have been kept constant. For changing particle radius, the electroactive surface area changes accordingly, which is taken into account by modifying the specific surface area parameter (a_s). Since the total amount of active material dictates the overall capacity of the electrode, maintaining a fixed volume fraction of the solid phase ensures consistency of capacity. The cathode and anode parameters used in the simulations have been adopted from existing literature and listed in Tables I. Here, "N" refers to the variables/parameters corresponding to the negative electrode, "P" refers to those relevant to the positive electrode. Figure 10a depicts a comparative analysis of voltage vs. capacity performance curve for the two different particle size distributions. The solid line corresponds to the case where particle size decreases from current collector to separator. The dashed line signifies the other particle size distribution of smaller particles close to the current collector and larger particles close to the separator. The performance curves at lower C-rates (1 C and 2 C) show insignificant difference between the two particle size distributions. At higher values of the C-rate, the case with descending particle size from current collector to separator displays slightly larger capacity than its counterpart. For example, at 4 C the particle size distribution with 5.0-μm particles close to the current collector and 15.0 μm particles at separator [case (i)] shows 0.4-Ah lower capacity than the particle size distribution with 15.0 μm particles at the current collector and 5.0-μm particles near the separator [case (ii)].

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When the variation in particle size in the negative electrode is taken into account, evolution of mechanical degradation inside the active particles of anode deserves investigation. Figure 10b demonstrates the damage profile after the first discharge for two different particle size distributions: i) 5.0-μm particles close to the current collector and 15.0-μm particles at the separator (denoted by dashed lines), and ii) 15.0-μm particles at current collector and 5.0-μm particles close to the separator (denoted by solid lines). Because larger particles experience enhanced mechanical degradation for both types of particle size distributions, a greater amount of microcrack density is observed wherever there exist large-sized particles. Thus, for case (i), enhanced mechanical damage occurs close to the separator. Similarly, for case (ii), evolution of higher amounts of microcrack density appears close to the current collector. It is evident from Figure 10b that the extent of damage evolution is independent of the location of the particle. For example, particles with a radius 15.0 μm experience around 9% microcrack densities at the end of the first discharge at 4 C, irrespective of whether it is located near the separator or the current collector. Similar behavior can be observed for other particle sizes operating at other C-rate conditions as well. This saturation in mechanical degradation happens because the maximum reaction front shifts from the separator to the current collector during the discharge process.

LIBs are usually operated in multiple charge-discharge cycles. Thus, it is important to investigate the evolution of microcrack density and capacity fade due to particle size distribution inside the anode. For the cycling analysis smaller magnitude of particle size distributions are taken into consideration: i) linearly increasing particle size from 2.5 μm at the current collector to 12.5 μm at the separator (denoted by black lines), and ii) linearly decreasing particle size from 12.5 μm at the current collector to 2.5 μm at the separator (denoted by red lines). For the cycling analysis, the cell is initially charged in a CCCV fashion from a very low SOC to 4.2 V. Then, the LIB is operated under CC discharge–CCCV charge conditions for five subsequent cycles. The voltage vs. capacity performance curve for operation at 3 C is shown in Figure 11a for both particle size distributions. The two curves almost overlap, indicating a minor difference in capacity fade observed by the two different particle size distributions. A closer look at the discharge curves for the fifth cycle shows that the particle size distribution with 2.5-μm particles close to the separator [case (ii)] leads to a capacity 0.23 Ah larger than case (i), which contains 12.5-μm particles close to the separator.

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When a distribution of particle size is used, evolution of microcrack density inside the anode active particles during multiple cycles deserves investigation. Two different particle size distributions considered in this study are same as that reported in the previous paragraph; the first one involves 2.5 μm–12.5 μm particles with increasing size and the other one consists of 12.5 μm–2.5 μm particles with decreasing size from the current collector to the separator. Figure 11b demonstrates the extent of microcrack density at the end of the first and the fifth discharge process for both particle size distributions. Irrespective of the location of the particles, larger particles experience higher microcrack density. Equivalently, less mechanical degradation is observed in smaller particles. Damage observed in the active particles after the fifth discharge is almost double of what occurred in the first discharge process. The extent of microcrack density after the first discharge process reported in Figure 10b is much greater than that observed in Figure 11b. This difference appears because for the single discharge experiments, the lithium ion cells are discharged from a voltage of 4.75 V to a lower cutoff limit of 3.0 V. In contrast, during the charge-discharge cycles, at the time of the first discharge the lithium ion cells are discharged from 4.2 V to 3.0 V. Because the lithium ion cell operates in a smaller voltage range in the second case, the anode active particles experience the delithiation process for shorter amount of time. Thus, the extent of mechanical degradation is much smaller after the first discharge for multiple charge-discharge cycles.

Until now, all the simulations are conducted under the assumption that during operation the lithium ion cells experience complete discharge and then complete charge at a constant C-rate, and the discharge-charge cycle goes on. However, in a realistic drive cycle, the lithium ion cells rarely experience complete discharge at a constant rate. Under drive-cycle operation, discharge-charge pulses occur depending on the driving conditions. Figures F1a and F1b included in Appendix F, demonstrate the C-rate experienced by LIBs in hybrid electric vehicles (HEV) and plugin-hybrid-electric-vehicles (PHEV), respectively. Positive C-rates correspond to discharging, and negative C-rates signify charging. HEVs operate mostly in a charge sustaining (CS) mode and experience equal amounts of discharge and charge pulses. The discharge-charge pulses are extremely strong and range between −10 C and 20 C. On the other hand, PHEVs can operate in both charge sustaining (CS) and charge depleting (CD) modes. The C-rate profile demonstrated in Figure F1b corresponds to a PHEV operating under CD condition. There exist both discharge and charge pulses in CD operations. PHEVs experience much milder discharge-charge pulses, which range between −3.5 C and 6 C.

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Evolution of mechanical degradation under HEV and PHEV drive cycle conditions for different particle sizes is reported in Figure 12a and 12b, respectively. Two different open circuit potential (OCP) profiles are used inside the anode: i) hard-carbon, and ii) graphite. The OCP of a hard-carbon anode shows a steep profile (see Figure C1). Thus, the reaction current density within the negative electrode remains flatter during the discharge process. In contrast, graphite has an extremely flat OCP profile. This leads to a large gradient in the reaction current density inside the anode. In the graphite anode, the active particles close to the separator experience significantly large reaction current density than those located near the current collector. Thus, mechanical degradation should be much larger near the separator for graphite than for hard-carbon materials. This trend is more prominent in Figure 12b, where the hard-carbon chemistry shows a relatively flat damage profile (solid line) whereas under graphite chemistry, a steeper damage profile is observed (dashed line) while traveling from the current collector to the separator.

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In both Figures 12a and 12b, a lower microcrack density is observed close to the current collector, and larger microcracks appear near the separator for all the different particle sizes considered. Under drive cycle operating conditions, the lithium ion cells experience high C-rates in multiple pulses. As shown in Figure 8a, at the beginning of the discharge process, a maximum reaction current is experienced close to the separator. As the discharge process continues the maximum reaction current front shifts towards the current collector. In drive cycles, because the C-rates act in pulses, the maximum reaction current only acts close to the separator; it never gets the opportunity to shift towards the current collector. As a result, in drive cycle scenarios, the maximum reaction current front confines itself near the separator only. Thus, a significant gradient in mechanical degradation is observed while between the current collector and the separator. This observation is applicable to both the hard-carbon and graphite chemistries. Because the HEV vehicles experience a much larger magnitude of C-rates [-10 C to 20 C, see Figure F1a], the mechanical degradation reported in Figure 12a shows a large gradient along the thickness of the electrode. For example, under HEV drive conditions, 7.5-μm particles experience only 2% microcrack density near the current collector whereas, near the separator, the mechanical degradation can be as high as 6%. The PHEVs operate under less severe C-rate conditions. Figure 12b demonstrates that for the same 7.5-μm particles, under PHEV driving conditions only 1.8% damage evolves close to the current collector, which increases to 3% microcrack density near the separator. Hence, HEV operating conditions are relatively more detrimental for the LIBs from a mechanical degradation perspective.

For 12.5-μm particles, some irregularity in the usual pattern of highest microcrack density close to the separator is observed in Figure 12a. The microcrack density drops below the maximum value for the active particles located extremely close to the separator. It was argued in earlier articles (see Refs. 47 and 50) that for extremely large active particles under very high C-rate operations, the concentration gradient confines itself very much close to the surface of the particles and cannot penetrate into the interior. Thus, the mechanical degradation is also observed close to the peripheral region only, which results in a reduction in microcrack density. Figure 12a reports the evolution of microcrack density under HEV drive cycles, where the C-rates can be as high as 20 C, being applied in multiple pulses of very short duration. Under such large magnitude of C-rates, the reaction current density close to the separator will be extremely high. For the 12.5-μm particles, the lithium concentration will mostly be confined extremely close to the surface of the particle. Due to a lack of penetration of the concentration gradient, reduced evolution of microcrack density is observed in 12.5-μm particles under HEV operating conditions at locations extremely close to the separator.

It has already been argued that the OCP profile for graphite is more flat than hard-carbon. This leads to formation of higher gradient in reaction current density along the thickness of the graphite anode. Also, as observed in Figure 12b, mechanical degradation for graphite is slightly larger than that observed for hard-carbon. According to the OCP profile, due to higher gradient in reaction current density, mechanical degradation in graphite should be significantly larger than hard-carbon. However, the small difference in microcrack formation can be attributed to the fact that the mechanical degradation reaches a peak value with increasing reaction current density. Further increase in reaction current density results in reduction of microcrack density. This maximum value of microcrack density is observed due to the fact that under extremely high reaction current density, the lithium concentration gradient cannot penetrate significantly within the solid active particles. Lithium concentration remains confined close to the particle surface, resulting in reduced mechanical degradation at extremely high rates of operation. Because of the existence of a maximum amount of microcrack formation, the difference in mechanical degradation is minimal for hard-carbon and graphite anode materials.

Evolution of microcrack density inside the active particles during the drive cycles is definitely not sufficient to characterize the impact of mechanical degradation on the performance of LIBs. Analysis of capacity fade due to the evolution of microcrack density deserves elaboration to complete the investigation process. A computational reference performance test (RPT) has been defined to characterize the amount of capacity fade under certain drive cycle conditions for different particle sizes. According to this computational RPT, the lithium ion cells are discharged at 1 C from a high SOC, x_p = 0.2 and x_n = 0.9, which corresponds to a voltage usually greater than 4.5 V. The CC discharge process is continued until the lower cutoff voltage limit of 3.0 V is reached. The Ahtp during the entire discharge process is reported as the capacity of the cell. Distribution of microcrack density that occurred during the drive cycle was kept constant while conducting the computational RPT test. Reduced diffusivity due to the formation of microcracks increases the resistance due to mass transport and eventually results in deterioration of the effective capacity. For different particle sizes, the net discharge capacities are reported in Figure 12c. The small squares signify the magnitude of the discharge capacity without the presence of any mechanical degradation. Five different drive cycles are considered: i) an EV under US06 driving conditions (denoted by cyan lines), ii) an HEV operating on highway (magenta lines), iii) an HEV operating under Urban Dynamometer Driving Schedule (UDDS) driving conditions (blue lines), iv) a PHEV operating in CD mode on US06 (denoted by red lines), and v) a PHEV operating in CS mode (denoted by the thick black line). Both the anode chemistries of hard carbon (denoted by triangles) and graphite (denoted by big circles) have been investigated. Among all the drive cycles, the HEVs operating in UDDS driving conditions experience the maximum amount of capacity fade, solely due to mechanical degradation. In contrast, EVs and PHEVs operate in a less severe fashion from the capacity fade perspective. Particle size also impacts the amount of capacity fade during drive cycle operations. As can be observed in Figure 12c, lithium ion cells with particles less than 10.0 μm do not experience any severe capacity fade even under the HEV driving conditions. However, particles larger than 10.0 μm indeed experience severe capacity fade under HEV operating conditions. For example, HEVs operating under UDDS driving conditions, experience 10% capacity fade for 12.5-μm particles and approximately 16% capacity fade for particles with radius around 15.0 μm. From this analysis, it is clearly evident that usage of particles smaller than 10.0 μm is beneficial for drive cycle applications.

Capacity fade due to mechanical degradation while using hard-carbon anode or graphite anode is approximately the same. But from their OCP curves, graphite is supposed to have significantly higher gradient in reaction current density along the thickness of the active particles, which would give rise to enhanced mechanical degradation. Slightly larger microcrack formation for graphite than hard-carbon anode material has already been demonstrated in Figures 12a and 12b. The capacity-fade reported in Figure 12c is estimated using the computational RPT technique, where the cell is discharged at a rate of 1C. At such low rates of operation, minor spatial variation in mechanical degradation along the thickness of the electrode, does not affect the overall cell resistance. As a result, capacity-fade for both graphite and hard-carbon is almost the same. However, the overall mechanical degradation has a significant impact on the effective capacity of the electrode. It has been demonstrated in Figure 12c that large particles experience more severe mechanical degradation and subsequently higher capacity fade. This conclusion is applicable to both the hard carbon and graphite anode chemistries.

Instead of using a constant particle size throughout the thickness of the negative electrode, the impact of a gradient in particle size from the current collector to the separator on the mechanical degradation and subsequently the capacity of the lithium ion cell are worth investigating. Two different particle size distributions have been considered here: i) linearly increasing particles size from 5.0 μm close to the current collector to 15.0 μm at the separator, and ii) linearly decreasing particle size from 15.0 μm at the current collector to 5.0 μm close to the separator. Damage profiles at the end of drive cycles observed in four different types of vehicles (EV, HEV, PHEV-CD and PHEV-CS) and for two different anode chemistries (hard-carbon and graphite) are shown in Figures 13a and 13b. The extent of microcrack density for increasing (5–15 μm) particle size [case (i)] is shown in Figure 13a. Larger particle sizes and higher reaction current densities close to the separator significantly enhanced mechanical degradation in the active particles that are located near the separator. The magnitude of the microcrack density close to the separator can be as high as 10% under the most severe HEV operating conditions. Smaller active particles located close to the current collector rarely experience high reaction current density, and the damage evolution within them is significantly low.

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The damage profile for decreasing active particle sizes from 15 μm at the current collector to 5 μm at the separator is depicted in Figure 13b at the end of four different drive cycles. Evolution of microcrack density under both hard-carbon (denoted by a solid line) and graphite (denoted by a dashed line) chemistries of the anode was investigated, and the damage profiles for both of them are very close to each other. It was argued earlier that under drive cycle conditions, current is extracted in pulses, and the maximum reaction front is confined to the region very close to the separator. Thus, maximum damage should happen to the particles near the separator only. However, for the decreasing particle size distribution with the smallest particles located close to the separator, less damage should occur near the separator region. Thus, for this particular case of decreasing particle size operating under drive cycle conditions, there exist competitions with regard to where the minimum amount of damage should occur. Figure 13b clearly demonstrates that the amount of damage is not the minimum extremely close to the separator even though the smallest particles reside there. Due to the extremely high reaction current density, microcrack density evolves to a larger extent near the separator. Hence, the minimum amount of damage occurs somewhere at the interior of the negative electrode. The exact location of the minimum microcrack density depends on the drive cycle under which the battery is being operated. For example, HEVs operating under UDDS driving conditions (blue line) will experience the minimum damage somewhere close to the center of the electrode. However, for other PHEV and EV driving conditions, the minimum damage occurs near the separator somewhere at the interior of the electrode (see the red, cyan, and black lines). The active particles adjacent to the separator still experience higher microcrack density due to enhanced reaction current density. The maximum amount of damage is observed wherever the largest particles reside. For the particular case under investigation, the largest particles (15 μm) are located near the current collector. As a result, the maximum amount of damage is observed close to the current collector itself.

Some knowledge about the extent of mechanical degradation is important to estimate the durability of the active particles. However, the real impact of the mechanical degradation is reflected in analyzing the capacity fade after operation under different drive cycle conditions. Computational RPT tests were conducted on both distributions of particle sizes [linearly increasing from 5 μm to 15 μm (denoted as case (i)), and linearly decreasing from 15 μm to 5 μm (represented as case (ii))] after operating under five different drive cycles. The capacities obtained from the computational RPT are shown in Tables IIa and IIb for the two anode chemistries of hard-carbon and graphite, respectively. For case (ii), where 5-μm sized particles reside close to the separator, under no mechanical degradation, this distribution shows a higher magnitude of capacity (0.05 Ah larger) than its counterpart that contains 15-μm sized particles (case (i)) near the separator. Under the most severe HEV operating conditions, the particle size distribution with decreasing particle size (current collector to separator) shows 0.75 Ah larger capacity than its counterpart with increasing particle size. This difference in capacity fade will be even more under high C-rate operations. Utilization of graphite chemistry leads to more capacity fade than that observed for hard-carbon anodes. Thus, particle size distributions with large particles close to the separator should be avoided from the perspective of mechanical degradation. Similarly, capacity fade is less severe for those distributions that contain smaller particles close to the separator.

Figure 1 shows schematic diagram of a cell that is usually adopted in "porous electrode theory". According to this homogenized model, two different phases are considered in each electrode, the active particles and the electrolyte. The separator acts as a medium through which only the electrolyte diffuses. The electrons cannot pass through the polymeric separator. The active particles are considered as spherical particles, and lithium diffusion within them are modeled using Fick's law of diffusion with a flux prescribed boundary condition on the surface.

Here, c_s is the concentration of lithium atoms within the solid phase, D_s signifies the diffusivity of lithium within the solid, and j is the reaction current density that flows at the solid electrolyte interface. The expression for the reaction current density is obtained from the Butler- Volmer equation, which is also written as,

Where j is the reaction current density, j₀ signifies the exchange current density, α_a and α_c are the anodic and cathodic transfer coefficients of electrode reaction, respectively, F signifies the Faraday constant, R represents the universal gas constant, and T corresponds to the temperature at which the reaction occurs. For the simulations conducted in the present article, the temperature has been kept fixed at room temperature or 298 K. The overpotential in the anode and cathode are denoted by η and defined by the expression,

Where ϕ_s signifies the solid phase potential, ϕ_e denotes the electrolyte phase potential, and finally, U signifies the open circuit potential within the electrodes.

Similar to the solid phase, diffusion of lithium ions also occurs within the electrolyte phase. Usually the diffusivity of lithium ions within the electrolyte is much faster than that observed within the solid phase. Accumulation of lithium ions within the electrolyte due to the reaction current acts as a source term.

Here, L = L_a + L_c + δ_sep and ε signifies the porosity. Also, c_e represents the concentration of lithium ions within the electrolyte, D^eff_e( = D_eε^1.5) represents effective diffusivity of lithium ions within the electrolyte in terms of the actual diffusivity and the porosity of the medium, a_s is the active surface area per unit volume, and t₊ represents the transference number corresponding to lithium ions. Transference number signifies the fraction of migration current that is carried by the lithium ions. That fraction of lithium ions within the electrolyte does not accumulate and simply migrate from one electrode to the other. As a result, that fraction is deducted from the total amount of lithium ions that gets released during the flow of reaction current.

Electro-neutrality is a necessary condition to be satisfied at every point within the electrochemical cell. Thus, quasistatic charge conservation must occur within both the solid phase as well as the electrolyte phase of the cell. Electric potential within the solid phase follows Ohm's law,

Here, σ^eff( = σ(1 − ε)^1.5) is the effective conductivity written as a function of free conductivity and porosity, ϕ_s is the solid phase potential, I signifies the applied current, and A represents the area at the electrode current collector interface. Similarly charge conservation within the electrolyte gives,

Here, the electrolyte potential is denoted by ϕ_e. Also, L_a − ε denotes the left side of the anode-separator interface and L_a + ε represents the right side of the anode separator interface. Similarly, L_a + δ_sep − ε and L_a + δ_sep + ε corresponds to the left and right side of the separator-cathode interface, respectively. For reference purpose, the electrolyte potential is set to zero at the cathode/current-collector interface. The effective electrolyte conductivity written as κ^eff( = κε^1.5) in terms of the free electrolyte conductivity (κ) and the porosity of the electrode (ε). The second term in left hand side of Eq. A6a signifies diffusion of the charged particles associated with concentration gradients. Also κ^eff_d = κ_dε^1.5, where κ_d is a constant. While calculating conservation of charge within the electrolyte, the contribution from both the charge transport via Ohm's law and the diffusion of charged particles must be taken into consideration. As a result, there exist two terms in the left hand side (LHS) of the governing equation provided in Eq. A6a. At the active particle-electrolyte interface, lithium ions are consumed as they react with the electron and diffuse inside the active particles as lithium atoms. Thus, the reaction current acts as a sink term at the electrode electrolyte interface. The conductivity of electrolyte (κ) and κ_d are not independent parameters. They depend on the diffusivity of the electrolyte phase (D_e) and the initial electrolyte concentration (c_{e, init}).⁶⁰

The assumption that ionic conductivity (κ) is a function of initial electrolyte concentration (c_{e, init}) is not fully correct. It should rather be a function of local electrolyte concentration (c_e(x, t)). However, that modification would introduce another set of nonlinearity within the governing differential equations. The developed computational methodology is indeed capable of solving this type of nonlinear equations. The major purpose of this article is to analyze the capacity fade due to microcrack formation within the solid phase. As a result, some approximation regarding the actual magnitude of ionic conductivity is acceptable. Hence, estimation of κ has been conducted based on the initial electrolyte concentration (c_{e, init}) is justified to some extent.

Cell voltage is determined by calculating the difference between the solid phase potential at the positive and the negative electrodes. The drop in cell voltage due to the internal resistance (R_cell) should also be incorporated.

The governing differential equations described above have been discretized along the thickness direction of the electrode. All three different regions of the anode, separator, and cathode have been taken into consideration. There exist four unknown variables at each node of the computational domain, which are given as follows, (i) solid phase concentration on the surface of the active particles (c_{s, s} = c_s(r = R)), (ii) electrolyte phase concentration (c_e), (iii) solid phase potential (ϕ_s), and (iv) electrolyte phase potential (ϕ_e). The electrolyte concentration, solid potential, and the electrolyte potential have been solved in a coupled fashion using the governing differential equations, Eqs. A4a, A5a, and A6a. Solution of the solid phase concentration was conducted separately (Eq. A1a). A spherical active particle is assumed to exist at each of the nodes. The solid phase concentration distribution has been obtained by solving the diffusion equation for spherical particles at each of the nodes individually. All the terms in Eqs. A4a, A5a, and A6a except the reaction current density (j) are linear in nature. A Taylor series expansion has been introduced to linearize the nonlinear component of the reaction current density (j), which is a function of solid phase potential (ϕ_s), electrolyte potential (ϕ_e), and solid phase surface concentration (c_{s, s}). Because the solid phase concentration has been computed separately, the reaction current density has not been linearized with respect to the c_{s, s} term. The Taylor series expansion of the reaction current density gives,

Here, ϕ_s and ϕ_e are the values of solid and electrolyte potential, which are unknown and being solved for. ϕ_{s, 0} and ϕ_{e, 0} are the values of solid and electrolyte phase potential at the previous time step, which is known. Derivative of the reaction current density with respect to the solid and electrolyte phase potential can be written as,

Please see Eq. A3 for an expression of η as a function of ϕ_s and ϕ_e. The derivative of reaction current density with respect to the overpotential is provided in Eq. B3:

Substituting Eqs. B1, B2, and B3 into Eqs. A4a, A5a, and A6a, linearized versions of the governing differential equations have been obtained as follows:

Discretizing the linear equations Eqs. B4, B5, and B6 by using a finite difference scheme, the coupled matrix for solid and electrolyte potential and electrolyte concentration are obtained.