Effects of Diffusion-Induced Nonlinear Local Volume Change on the Structural Stability of NMC Cathode Materials of Lithium-Ion Batteries

Abstract

Electrochemical stress induced by the charging/discharging of electrode materials strongly affects the lifetime of lithium-ion batteries (LIBs) by regulating mechanical failures. Electrochemical stress is caused by a change in the local volume of the active materials associated with the lithium-ion concentration. The local volume change of certain active materials, such as nickel-rich LiNi_xMn_yCo_zO₂ (NMC), varies nonlinearly with the lithium content, which has not been considered in the stress calculations in previous studies. In this paper, the influence of nonlinear local volume change on the mechanical response of NMC-active materials is investigated numerically. The goal is achieved by using a concentration-dependent partial molar volume calculated from the previously obtained local volume change experimental results. A two-dimensional axisymmetric model was developed to perform finite element simulations by fully coupling lithium diffusion and stress generation at a single particle level. The numerical results demonstrate that (1) the global volume change of the particle evolves nonlinearly, (2) the stress response correlates with the rate of change of the active particle’s volume, and (3) stress–concentration coupling strongly affects the concentration levels inside the particle. We believe this is the first simulation study that highlights the effect of a concentration-dependent partial molar volume on diffusion-induced stresses in NMC materials. The proposed model provides insight into the design of next-generation NMC electrode materials to achieve better structural stability by reducing mechanical cracking issues.

1. Introduction

With the growing demand for energy storage devices, lithium-ion batteries (LIBs) are gaining more interest due to their higher capacity and longer cycle life [1]. LIB applications range from small medical and portable electronic devices to electric vehicles (EVs); however, their capacity decreases over time [2]. One of the main goals of the development of the next-generation batteries is to increase their efficiency by limiting the capacity-fading mechanisms [3]. Layered cathode materials, such as lithium nickel manganese cobalt oxide LiNi_xMn_yCo_zO₂ (NMC), are the most promising active materials with lower costs and higher energy density [4,5]. However, the battery capacity quickly decreases due to various mechanical failures caused by diffusion-induced stress (DIS) when lithium is introduced into (lithiation) or extracted from active materials (delithiation) [6,7].

A lot of effort has gone into the development of charging/discharging models to understand the mechanisms underlying DIS generation. One of the pioneering works are the studies of Christensen and Newman [8]. They investigated the lithiation-induced stresses and predicted mechanical failure in a single spherical graphite anode. A thermal analogy was then employed to calculate the DIS using finite element method (FEM) simulations [9]. Later, the DIS model was further updated by including the phase transition effects [10,11], grain boundaries [12,13,14], charging/discharging rates [15], material properties [16,17], active material morphology [18], surface stresses [19,20], yielding and plastic deformations of the active material [21,22,23,24], and solid–electrolyte interface [25,26,27]. Furthermore, several researchers have developed various stress-regulated charging/discharging strategies to reduce stresses using such DIS models [28,29,30]. Going a step further, the dynamic growth of failures in LIB model systems under the influence of DIS has also been simulated [31,32,33]. However, in order to reduce the simulation complexity, the mentioned studies have mainly used constant material properties, whereas most of the active material properties depend on the lithium concentration. Therefore, to accurately predict the charging/discharging behavior of the material, precise property values should be used when performing simulations.

In the last few years, many researchers have devoted themselves to determining the effect of concentration-dependent material properties on the charging and discharging processes of lithium-ion batteries. The researchers included the lithium concentration-dependent elastic properties especially Young’s modulus [34,35,36,37,38,39,40,41], yield stress [24,42,43], Poisson’s ratio [44], coefficient of chemical expansion or partial molar volume [45,46], toughness [47], and lithium diffusion coefficient [48,49] to DIS simulation models. For example, Deshpande et al. [50] used a simple cylindrical electrode particle to find that Young’s modulus variation has a significant effect on the evolution of DIS. Zhang et al. [51] examined the effect of the concentration-dependent elastic modulus on lithium diffusion and DIS generation using composition-gradient LCO cathode particles. They found that lithiation-induced stiffening regulates DIS in composition-gradient electrodes. Hong et al. [44] investigated the effects of concentration-dependent diffusivity, Young’s modulus, and Poisson’s ratio on stress evolution during lithiation of the Sn particle and concluded that the change in material properties with lithium content significantly alters mechanical failure modes. Cai and Guo [48] examined the effect of changing the diffusion coefficient and elastic modulus hardening with the lithium concentration on DIS in an anisotropic graphite anode particle. They found that diffusivity as a function of concentration increased concentration gradients and thus enhanced DIS because the change in the volume of the active material induced by the lithium concentration change is the main source of DIS. However, the effects of the concentration-dependent chemical expansion coefficient have rarely been investigated in the past. The study of nonlinear volume change and its effects on DIS is essential to determine strategies for avoiding mechanical failures in high-energy-density cathode materials such as NMC.

During charging, both the lattice parameters $a$ and $c$ of the NMC unit cell change strongly, which ultimately leads to a large change in the unit cell volume [52]. This volume change evolves nonlinearly with the lithium content. Distinct changes in the lattice structure are the main sources of stress generation that lead to various modes of mechanical cracks [53]. The disintegration and cracks induced by lithium diffusion in NMC materials disrupt the ionic and electronic conduction pathways [54]. Consequently, structural degradation accelerates the capacity decrease of NMC cathodes. However, the effects of the nonlinear volume change of NMC materials in stress diffusion problems have not yet received much attention.

In this paper, we focus on the effects of diffusion-induced nonlinear volume change on the evolution of lithium concentration distributions and the related stress development. To achieve the goal, we first calculated the partial molar volume (chemical expansion coefficient) based on experimentally obtained volume change values given in reference [52]. Next, we simulated the charging (delithiation) phenomenon in a single NMC particle using the concentration-dependent partial molar volume under galvanostatic charging conditions. We then compared the results of the constant and variable (concentration-dependent) partial molar volumes to evaluate the effects of the nonlinear volume change on the chemomechanical response of the NMC-active material. The main objectives of this paper are as follows:

1.

To construct a fully coupled finite element chemomechanical model in order to investigate the effects of a concentration-dependent partial molar volume on the mechanical response of NMC particle.

2.

To study and compare the simulation results for the constant and concentration-dependent partial molar volume of the NMC particle during the charging (delithiation) process.

3.

To investigate the simultaneous effects of the particle size or the charging rate and the concentration-dependent partial molar volume on the mechanical response of the NMC-active material.

The rest of the paper is organized as follows: we first developed a fully coupled chemomechanical model that considered the local effects of a nonlinear volume change on the distribution of stress and lithium concentration. We then validated the application of the concentration-dependent partial molar volume by comparing the finite element simulation results with the experimentally obtained volume changes. Afterwards, we compared the stress and lithium concentration results for the constant and variable (concentration-dependent) partial molar volumes for the delithiation (charging) process. We then evaluated the effects of the particle size and charge rate. Finally, the article is concluded in the last section.

2. Methodology

For numerical simulations, an isolated active particle with a radius ${\mathrm{R}}_{\mathrm{s}}$ was considered, as shown in Figure 1. The axisymmetric model system and its spatial discretization for a representative case of lithiation are illustrated in Figure 1a,b, respectively. We employed full coupling between lithiation/delithiation kinetics and mechanics of diffusion-induced deformations. The computational modeling of this fully coupled chemomechanical model is presented in the following subsections.

2.1. Fully Coupled Chemomechanical Model

2.1.1. Modeling of Lithium Ion Diffusion in an Active Particle

The diffusion of lithium ions inside the active particle was assumed to follow Fick’s law [55]:

$$\frac{\partial c}{\partial t}+\nabla \cdot J=0$$

(1)

where $c$ is the lithium ion molar concentration; $t$ is the lithiation/delithiation time, and $J$ is the lithium ion flux inside the active material. The flux $J$ is defined as a function of chemomechanical potential as [9]:

$$J=-M c \nabla \mu$$

(2)

where $M$ is the mobility factor given by $M=D/RT$, and $\mu$ is the stress-dependent chemical potential, which can be defined by hydrostatic stress (${\sigma }_h$) as [9]:

$$\mu ={\mu }^o+RT\ln X-\Omega {\sigma }_h$$

(3)

where ${\mu }^o$ is the reference state potential; $R$ is the universal gas constant; $T$ is the absolute temperature; $X$ is the molar fraction, and $\mathsf{\Omega }$ is the partial molar volume. Combining Equations (2) and (3) leads to:

$$J=-D\left(\nabla c-\frac{c}{RT}\nabla \left(\Omega {\sigma }_h\right)\right)$$

(4)

Finally, putting Equation (4) to Equation (1) gives the final form of the partial differential equation for solving lithium concentrations in a coupled stress–concentration manner [56]:

$$\frac{\partial c}{\partial t}-D{\nabla }^{2}c+\frac{D}{RT}\nabla c\cdot \nabla \left(\Omega {\sigma }_h\right)+\frac{Dc}{RT}{\nabla }^2\left(\Omega {\sigma }_h\right)=0$$

(5)

The third and fourth terms on the left-hand side of Equation (5) result from the stress–concentration coupling. Thus, without stress–concentration coupling, the partial differential equation above reduces to:

$$\frac{\partial c}{\partial t}-D{\nabla }^{2}c=0$$

(6)

2.1.2. Modeling of Diffusion-Induced Stress

The DIS in the active particle was solved by the following partial differential equation for mechanical equilibrium [20]:

$${}_{ }\nabla \cdot \mathsf{\sigma }+F_b=0$$

(7)

where $\mathsf{\sigma }$ is the Cauchy stress tensor, and $F_{\mathit{b}}$ is the body fore. In this work, no body force was assumed, so $F_{\mathit{b}}=\mathbf{0}$. For elastic deformations during the charging/discharging process, the stress–strain relationship is governed by Hook’s law as [33]:

$$\mathsf{\sigma }=C:{\mathsf{\epsilon }}_e$$

(8)

where $C$ is the fourth-order stiffness matrix, and ${\mathsf{\epsilon }}_e$ is the elastic strain. The total strain caused by the elastic and diffusion-induced deformations is given as [33]:

$${\mathsf{\epsilon }}_t={\mathsf{\epsilon }}_e+{\mathsf{\epsilon }}_d=\frac{1}{2}\left({\left(\nabla u\right)}^{\mathrm{T}}+\nabla u\right)$$

(9)

where ${\mathsf{\epsilon }}_t$ and ${\mathsf{\epsilon }}_d$ are the total and diffusion-induced strains, and $u$ denotes the displacement field. Thus, the elastic strain is calculated by subtracting the diffusion-induced strain from the total strain as [9,14]:

$${\mathsf{\epsilon }}_e={\mathsf{\epsilon }}_t-{\mathsf{\epsilon }}_d$$

(10)

For the DIS calculations, the diffusion strain was calculated using the thermal analogy [8,9,57] as:

$${\mathsf{\epsilon }}_d=\frac{1}{3}\tilde{\mathrm{c}}\Omega I$$

(11)

where $\tilde{\mathrm{c}}$ is the concentration difference between the current and the initial state, and $I$ is the identity matrix. In another method, using directly the volume change ($\mathsf{\Delta }\mathrm{V}/{\mathrm{V}}_{\mathrm{o}}$) obtained from the experiments, the diffusion-induced strain can be represented by the following diffusion-induced deformation gradient $F_d$:

$$F_d={\left(1+\frac{\mathsf{\Delta }\mathrm{V}}{{\mathrm{V}}_{\mathrm{o}}}\right)}^{1/3}I$$

(12)

2.2. Material Properties and Numerical Simulations

To perform the finite element fully coupled chemomechanical simulations, the structural mechanics and transport of the dilute species modules of COMSOL Multiphysics (version 6.0) was employed in this work. The partial differential equation of the mechanical equilibrium was solved in the structural mechanics’ module, while the partial differential equations of the mass balance of lithium ions inside the active material were solved using the transport of the dilute species module. The built-in parallel sparse direct solver (PARDISO) transient solver was used to solve the weak forms of partial differential equations in a fully coupled manner. The delithiation phenomenon was modeled by applying a constant negative flux to the exposed surfaces of the active particle. The partial differential equations were solved in time increments until the local minimum state of charge reached the lower limit of the respective NMC-active material. To explore the effects of the concentration-dependent volume change of the active material, a representative case of 2 µm particle size of LiNi_1/3Mn_1/3Co_1/3O₂ (NMC-111) at 1C charge/discharge rate was considered, unless otherwise specified. Furthermore, the following initial and boundary conditions were considered to solve the partial differential equations given in Equations (5) and (7).

Initial conditions: At the start of the simulation, it was assumed that the maximum lithium concentrations ($c_{max}$) were homogeneously distributed throughout the NMC active particles as:

$${}_{ }c=c_{total} \mathrm{at} t=0$$

(13)

where $c_{total}$ is the stoichiometric lithium concentration in the NMC material. To solve the partial differential equation of mechanical equilibrium Equation (7), initially no stress was considered in the particle as:

$${\left(\nabla \cdot \mathsf{\sigma }\right)}_{t=0}=0$$

(14)

Boundary conditions: To solve the issue of lithium diffusion inside the particle, a constant lithium flux was applied to the particle surface, such as:

$$J\cdot n=-\frac{i_{\mathrm{n}}}{\mathrm{F}}=-\frac{{\mathrm{C}}_{\mathrm{rate}}\alpha \rho }{\mathrm{F}}\times \frac{R_s}{3}$$

(15)

where $n$, $i_n$, $\mathrm{F}$, and ${\mathrm{C}}_{\mathrm{rate}}$ denote the outward normal unit vector on the external surface of the particle, the current density, the Faraday’s constant, and the charging rate, respectively. The $\alpha$, $\rho$, and $R_s$ are the specific capacity, density, and radius of the active particle, respectively.

To quantify the effects of the nonlinear volume change, during the simulation the total volume change of the active material was calculated as:

$$\frac{\mathsf{\Delta }\mathrm{V}}{{\mathrm{V}}_{\mathrm{o}}}=\frac{{\left({\displaystyle \int dv}\right)}_{t=t}-{\mathrm{V}}_{t=0}}{{\mathrm{V}}_{t=0}}$$

(16)

where $\mathsf{\Delta }\mathrm{V}$ is the change in particle volume between the current (at $t=t$) and initial (at $t=0$) delithiation state. To explore the mechanical response of active materials, the maximum first principal stress (${\sigma }_{max}$) was calculated and compared for various cases. Further, to investigate the change in lithium concentration behavior, the difference between the maximum and minimum states of charge ($\mathsf{\Delta }\mathrm{SOC}$) was evaluated as:

$$\mathsf{\Delta }\mathrm{SOC}=\frac{c_{\max }-c_{\min }}{c_{total}}\times 100\%$$

(17)

where $c_{\max }$ and $c_{\min }$ are local maximum and minimum lithium concentrations. Finally, to assess the change in charge storage, the normalized capacity ($\mathsf{\Pi }$) of the active particle was calculated as follows:

$$\mathsf{\Pi }=\frac{\mathrm{F}{\displaystyle \int cdv}}{\alpha \rho {\mathrm{V}}_{\mathrm{o}}}$$

(18)

where $\mathsf{\Pi }$ will change between 1 and 0 during delithiation.

The convergence of both the maximum stress (${\sigma }_{max}$) and change in state of charge ($\mathsf{\Delta }\mathrm{SOC}$) parameters was confirmed by increasing the degrees of freedom by refining the mesh size. According to the mesh independence test, the simulations were carried out with 5673 mesh elements and 46,356 degrees of freedom.

Concentration-Dependent Partial Molar Volume

To calculate the concentration-dependent partial molar volume, the volume change data during the delithiation process of NMC materials were obtained from reference [52]. The concentration-dependent partial molar volume of each NMC structure (i.e., NMC-111, LiNi_0.5Mn_0.2Co_0.3O₂ (NMC-523), LiNi_0.6Mn_0.2Co_0.2O₂ (NMC-622), and LiNi_0.8Mn_0.1Co_0.1O₂ (NMC-811)) was calculated separately. The volumetric strain ($\lambda$) due to the volume change caused by the delithiation was calculated as:

$$\lambda ={\left(\frac{\mathsf{\Delta }\mathrm{V}}{{\mathrm{V}}_{\mathrm{o}}}+1\right)}^{1/3}-1={\left(\frac{{\mathrm{V}}_{\mathrm{x}=\mathrm{x}}-{\mathrm{V}}_{\mathrm{x}={\mathrm{x}}_{\max }}}{{\mathrm{V}}_{\mathrm{x}={\mathrm{x}}_{\max }}}+1\right)}^{1/3}-1$$

(19)

where $\mathsf{\Delta }\mathrm{V}$ is the volume change that is calculated as the delithiation states increase. The ${\mathrm{V}}_{\mathrm{x}=\mathrm{x}}$ is current, and ${\mathrm{V}}_{\mathrm{x}={\mathrm{x}}_{\max }}$ is the initial (with maximum lithium content) unit cell volume. Based on this strain value, the partial molar volume is given by:

$$\Omega =\frac{3\times \lambda }{\left(\mathrm{x}-{\mathrm{x}}_{\max }\right)\times c_{total}}$$

(20)

Using Equation (20), the concentration-dependent partial molar volume of each NMC was calculated and plotted in Figure 2. For simplicity, we proposed that other material properties, such as Young’s modulus and diffusion coefficient of the active material, were independent of the lithium-ion concentrations. Other material properties and simulation parameters used in this work are listed in

Table 1. Summary of simulation parameters and material properties for NMC materials used for numerical simulations.

Parameters	Symbols	Units	Values
			NMC-111	NMC-523	NMC-622	NMC-811
Young’s modulus 1	$E$	GPa	202.98	191.79	181.52	194.4
Diffusion coefficient 2	$D$	m2 s−1	3.39 × 10−15	3.89 × 10−15	7.5 × 10−15	4.0 × 10−14
Specific capacity 3	$\alpha$	mAh g−1	188.75	194.89	203.18	213.42
Stoichiometric lithium concentration 4	$c_{total}$	mol m−3	33,452	34,542	36,009	37,825
Minimum SOC 5	${\mathrm{SOC}}_{\min }$	%	21	19	18	11
Maximum SOC 6	${\mathrm{SOC}}_{\max }$	%	94	93	93	90
Poisson’s ratio 7	$\nu$	-	0.25
Density 8	$\rho$	kg m−3	4750
Faraday’s constant	$F$	C mol−1	96,487
Absolute temperature	$T$	K	300
Universal gas constant	$R$	J mol−1 K−1	8.314

¹ Sun and Zhao [58]; ² Wei et al. [59] and Huang et al. [60]; ^3,5,6 de Biasi et al. [52]; ⁴ Calculated by $c_{total}=\alpha \rho /F$; ⁷ Cheng et al. [61]; ⁸ Mistry et al. [62].

.

3. Results and Discussions

3.1. Validations of Numerical Results

In this section, we validate the calculations of the concentration-dependent partial molar volume of NMC with different nickel contents; NMC-111, NMC-523, NMC-622, and NMC-811. If the distributions of lithium concentration inside the active particle are sufficiently homogenized, then the local volume change will equal the percentage of the total volume change of the active particle. Therefore, in Figure 3, we compare the results by plotting the global volume change obtained by simulating a small particle under a lower charge rate (i.e., $R_s=2 \mathsf{\mu }\mathrm{m}$ and $C_{rate}=1C$) against the normalized charge capacity and the local volume change of the respective NMC material against the lithium content obtained from [52]. The close volume change values obtained from the simulations and the experiments suggest that this concentration-dependent partial molar volume can be used for stress calculations of NMC-active materials.

Then, the simulation results obtained by directly using the local volume change as the intercalation strain (Equation (12)) and using the calculated concentration-dependent partial molar volume as the thermal analogy (Equation (11)) are compared in Figure 4. To avoid changes caused by stress–concentration coupling, only uncoupled simulations were performed in this section using Equation (6). All simulation results are similar either by using the intercalation strains or thermal analogy. This, therefore, proves the validity of the thermal analogy method and the use of concentration-dependent partial molar volume instead of the direct use of the local volume change as the volumetric strain in finite element simulations.

3.2. Effects of Variable Partial Molar Volume

Figure 5 compares the simulation results obtained for a representative case of a $2 \mu \mathrm{m}$ NMC-111 particle at a discharge rate of 1C with and without considering the nonlinear local volume change during the delithiation process. Figure 5a shows that the total volume of the active particle ($\mathsf{\Delta }\mathrm{V}/{\mathrm{V}}_{\mathrm{o}}$) changes nonlinearly when the variable (concentration-dependent) partial molar volume (${\mathsf{\Omega }}_{var}$) is considered. On the other hand, using a constant partial volume (${\mathsf{\Omega }}_{const}$), the change of $\mathsf{\Delta }\mathrm{V}/{\mathrm{V}}_{\mathrm{o}}$ is linear. This indicates that the local volume change significantly affects the global volume change, suggesting that the local volume change will also increase the stress due to the surrounding constraints provided by the binder, other active particles, current collector, and separator.

The evaluation of the stress generation caused by lithium diffusion is essential for the analysis of mechanical failure. For this purpose, the local maximum of the first principal stress (${\sigma }_{max}$) was analyzed here. DIS developed because of the local mismatch of lithium concentration levels between the inner and outer regions of the active particle. This mismatch caused the regions to expand/contract based on the magnitude of the partial molar volume. Consequently, using ${\mathsf{\Omega }}_{var},$ the rate of volume change induced by lithium diffusion is different at different concentration levels, and the associated stress increase is different along the radial direction. As a result, ${\sigma }_{max}$ evolves differently for ${\mathsf{\Omega }}_{var}$, as illustrated in Figure 5b. The evolution of ${\sigma }_{max}$ reveals several smaller peaks before a larger peak toward the end of the delithiation process, while these peaks are not visible when ${\mathsf{\Omega }}_{const}$ is used. Instead, ${\sigma }_{max}$ increases gradually after achieving a distinct peak in the early stages of the delithiation process. Figure 5b also shows that the peak magnitude of ${\sigma }_{max}$ is almost three times higher for the ${\mathsf{\Omega }}_{var}$ case compared to the ${\mathsf{\Omega }}_{const}$ case indicating a higher probability of mechanical failures. In summary, the evolution and levels of diffusion-induced stress in the active particle are significantly affected by the use of concentration-dependent partial molar volume. The reason for the appearance of such smaller peaks in the evolution of ${\sigma }_{max}$ is given in the following paragraphs.

The maximum concentration difference in the active particle measures concentration gradients to some extent. Therefore, the maximum SOC difference ($\mathsf{\Delta }\mathrm{SOC}$) was evaluated to express the change in the evolution of the concentration gradients. Figure 5c compares the evolution of $\mathsf{\Delta }\mathrm{SOC}$ using ${\mathsf{\Omega }}_{const}$ and ${\mathsf{\Omega }}_{var}$. For ${\mathsf{\Omega }}_{const}$, $\mathsf{\Delta }\mathrm{SOC}$ initially increases sharply, and after reaching a distinct peak, the increase becomes more gradual. In the initial stages of a particle’s delithiation, only the outermost regions undergo the deintercalation process, so, over time, the SOC difference between the surface and the core increases rapidly, but when the deintercalation front reaches the particle’s core, the core starts to delithiate. The change in $\mathsf{\Delta }\mathrm{SOC}$ drops, which is the second stage of a gradual increase in $\mathsf{\Delta }\mathrm{SOC}$. In contrast, when ${\mathsf{\Omega }}_{var}$ is included, the change of $\mathsf{\Delta }\mathrm{SOC}$ becomes irregular. It is affected by either a local volume change or a stress change through stress–concentration coupling.

To explore the effects of the concentration-dependent partial molar volume on the capacity stored inside the active particle, the normalized capacity ($\mathsf{\Pi }$) was compared for both the ${\mathsf{\Omega }}_{const}$ and ${\mathsf{\Omega }}_{var}$ cases, as shown in Figure 5c. The same capacity value was observed for both partial volume cases. This indicates that although the nonlinear local volume affects the particle’s response to stress, the total stored charge remains the same.

Uncoupled numerical simulations were performed to isolate the effects of stress–concentration coupling and concentration-dependent partial molar volume on the SOC differences. In Figure 6a, the $\mathsf{\Delta }\mathrm{SOC}$ evolution is again plotted for ${\mathsf{\Omega }}_{const}$ and ${\mathsf{\Omega }}_{var}$, without considering the coupling between the stress and concentration. The $\mathsf{\Delta }\mathrm{SOC}$ is the same for both partial molar volumes, although changes are still present in the evolution of ${\sigma }_{max}$, as shown in Figure 6b when ${\mathsf{\Omega }}_{var}$ is considered. Thus, this proves that (1) changes in $\mathsf{\Delta }\mathrm{SOC}$ are caused by stress–concentration coupling, meaning that inhomogeneous stress levels due to nonlinear volume changes affect the lithium concentration levels, and (2) ${\sigma }_{max}$ is not directly affected by the change in $\mathsf{\Delta }\mathrm{SOC}$.

Further, to figure out the mechanism underlying the stress response caused by the concentration-dependent partial molar volume, we investigated the rate of change of $\mathsf{\Delta }\mathrm{V}/{\mathrm{V}}_{\mathrm{o}}$ over time. The evolution of the 1st temporal derivative of $\mathsf{\Delta }\mathrm{V}/{\mathrm{V}}_{\mathrm{o}}$ was plotted with the evolution of ${\sigma }_{max}$ in Figure 7. The trend of ${\sigma }_{max}$ perfectly matches the trend of the 1st derivative of $\mathsf{\Delta }\mathrm{V}/{\mathrm{V}}_{\mathrm{o}}.$ This proves that the change in maximum stress behavior is directly affected by the rate of change in $\mathsf{\Delta }\mathrm{V}/{\mathrm{V}}_{\mathrm{o}}$.

We then investigated the effects of the concentration-dependent partial molar volume on the distribution patterns of the lithium concentration and the associated stress generation. It is clear from the comparison that the distribution patterns of the local lithium concentration and the first principal stress are consistent as the partial molar volume changes from constant to a variable. As shown in Figure 8a,b, the maximum local concentration of lithium occurs at the center of the particle, while the minimum occurs at the surface of the particle. Similarly, the local maximum first principal stress occurs at the surface of the particle, and the minimum stress occurs at the center of the particle, as shown in Figure 8c,d. This indicates that although the use of concentration-dependent partial volume enhances stress levels, it does not change the location of maximum stress.

3.3. Effects of Particle Size

Usually, the particle size (${\mathrm{R}}_{\mathrm{s}}$) inside the electrode varies. Particle size plays an essential role in the stress increase caused by lithium diffusion. Therefore, in this section, the particle size was changed from ${\mathrm{R}}_{\mathrm{s}}=1 \mu \mathrm{m}$ to ${\mathrm{R}}_{\mathrm{s}}=30 \mu \mathrm{m}$ in order to study the effects of concentration-dependent partial molar volume on the stress increase. Figure 9 compares the simulation results for various particle sizes with and without the use of a diffusion-induced nonlinear local volume change. The red dots to the right of each 3D plot represent the peak value during the delithiation time evolution of the corresponding parameter. The absolute (positive) volume change values are plotted in Figure 9 for better visibility and comparison of trends.

In both cases, active particles with different ${\mathrm{R}}_{\mathrm{s}}$ shrink in the same manner. However, for ${\mathsf{\Omega }}_{const}$, the volume change is linear, and for ${\mathsf{\Omega }}_{var,}$ the volume change is nonlinear, as illustrated in Figure 9a,d, respectively. Moreover, as ${\mathrm{R}}_{\mathrm{s}}$ increases, the peak values of volume change decrease, which is consistent for both cases of partial molar volume. The peak values of $\mathsf{\Delta }\mathrm{V}/{\mathrm{V}}_{\mathrm{o}}$ decrease because the stop condition for larger particles is achieved earlier, and the maximum volume achieved decreases. This indicates that the nonlinear volume change will have minimal effects for larger particles.

The $\mathsf{\Delta }\mathrm{SOC}$ curves in Figure 9b,e show that as ${\mathrm{R}}_{\mathrm{s}}$ increases, the magnitude of $\mathsf{\Delta }\mathrm{SOC}$ increases significantly. Larger particles take longer to start delithiating the core, resulting in enhanced concentration gradients within the particle that remain for a longer period of time. As a consequence, $\mathsf{\Delta }\mathrm{SOC}$ increases. However, the trends and magnitude for both ${\mathsf{\Omega }}_{const}$ and ${\mathsf{\Omega }}_{var}$ remain almost the same, suggesting that the concentration-dependent partial molar volume has a minimal effect on $\mathsf{\Delta }\mathrm{SOC}$ with increasing particle size.

Meanwhile, as the concentration gradients increase, ${\sigma }_{max}$ also increases, as shown in Figure 9c,f. Although the evolution in time and magnitude of ${\sigma }_{max}$ for both the constants ${\mathsf{\Omega }}_{const}$ and ${\mathsf{\Omega }}_{var}$ are different, the trend in the rise of the peak value of ${\sigma }_{max}$ with increasing ${\mathrm{R}}_{\mathrm{s}}$ is the same. For larger particles, the difference between the peak values of ${\sigma }_{max}$ is due to a very small volume change. In summary, the trends in the evolution of $\mathsf{\Delta }\mathrm{V}/{\mathrm{V}}_{\mathrm{o}}$, $\mathsf{\Delta }\mathrm{SOC}$, and ${\sigma }_{max}$ remain consistent, while ${\mathrm{R}}_{\mathrm{s}}$ increases for both the ${\mathsf{\Omega }}_{const}$ and ${\mathsf{\Omega }}_{var}$ cases. However, the propensity for mechanical fracture increases with increasing particle size. Meanwhile the effects of concentration-dependent partial molar volume on the stress rise decrease for larger particles.

3.4. Effects of Charge Rate

Fast charging is the most desirable aspect of EVs. However, fast charging significantly increases the capacity degradation of lithium-ion batteries. Therefore, to explore the effects of fast charging in combination with the use of a concentration-dependent partial molar volume, the ${\mathrm{C}}_{\mathrm{rate}}$ was varied from 0.5 C to 10 C with a difference of 0.5 C. Simulations were performed for ${\mathrm{R}}_{\mathrm{s}}=2 \mu \mathrm{m}$ of the NMC-111-active particle.

Figure 10 shows that the evolution trends of $\mathsf{\Delta }\mathrm{V}/{\mathrm{V}}_{\mathrm{o}}$, $\mathsf{\Delta }\mathrm{SOC}$, and ${\sigma }_{max}$ are consistent for different ${\mathrm{C}}_{\mathrm{rates}}$, even when ${\mathsf{\Omega }}_{var}$ was used. As the stopping condition of the numerical simulations was achieved earlier with an increasing ${\mathrm{C}}_{\mathrm{rate}}$, the peak magnitude of $\mathsf{\Delta }\mathrm{V}/{\mathrm{V}}_{\mathrm{o}}$ decreases, as presented in Figure 10a,d for ${\mathsf{\Omega }}_{const}$ and ${\mathsf{\Omega }}_{var}$, respectively. Although the trends are consistent, the evolution of $\mathsf{\Delta }\mathrm{V}/{\mathrm{V}}_{\mathrm{o}}$ becomes nonlinear once the concentration-dependent partial molar volume is used.

Furthermore, increasing the ${\mathrm{C}}_{\mathrm{rate}}$ causes an increase in $\mathsf{\Delta }\mathrm{SOC}$ due to enhanced concentration gradients. Figure 10b,e show a slight increase in the $\mathsf{\Delta }\mathrm{SOC}$ values for ${\mathsf{\Omega }}_{var}$ compared to ${\mathsf{\Omega }}_{const}$. This is caused by this stress–concentration coupling effect.

Moreover, small SOC differences for a lower ${\mathrm{C}}_{\mathrm{rate}}$ indirectly indicate a more uniform distribution of lithium inside the particles and thus exhibit reduced concentration gradients. Since the lithium diffusion-induced strain mismatch is the main cause of stress generation, with smaller SOC differences, stress increase is also reduced for the lower ${\mathrm{C}}_{\mathrm{rate}}$. As the ${\mathrm{C}}_{\mathrm{rate}}$ increases, ${\sigma }_{max}$ in a given delithiation time increases. The increase in the stress raises the probability of fracture. Although trends in the evolution of ${\sigma }_{max}$ remain consistent for ${\mathsf{\Omega }}_{const}$ and ${\mathsf{\Omega }}_{var}$, using ${\mathsf{\Omega }}_{var}$, the stress rise is significantly higher, as shown in Figure 10c,f for ${\mathsf{\Omega }}_{const}$ and ${\mathsf{\Omega }}_{var}$, respectively. Thus, the propensity for mechanical failure increases significantly when using a concentration-dependent partial molar volume under fast charging conditions.

4. Conclusions

In this paper, the influence of a concentration-dependently local volume change of active materials on the lithium concentration evolution and diffusion-induced stress generation was studied by performing finite element simulations using a fully coupled chemomechanical model at the particle level. The concentration-dependent partial molar volume was calculated based on the previously obtained local nonlinear volume change of the NMC-active materials. We first validated the calculations of the partial molar volume by comparing the results of the global volume change obtained by finite element simulations with the local volume change values that were previously received in experiments described in the literature. We then compared the chemical and mechanical response of the active material with and without considering the concentration-dependent partial molar volume.

The main findings of this work are given below:

1.

The local volume change induced by the concentration-dependent chemical expansion of the active material significantly alters the global volume change of the active particles, which suggests that the stress increase due to the surrounding materials in electrodes will be affected by the concentration-dependent partial molar volume of the active materials.

2.

The concentration-dependent partial molar volume significantly changes the stress evolution trends and SOC differences. The peak stress due to diffusion is almost three times greater for a variable partial molar volume. However, the accumulated capacity within the particle remains independent of the change in partial molar volume.

3.

The trends of the maximum diffusion-induced stress in the particle correlate with variations in the time rate of $\mathsf{\Delta }\mathrm{V}/{\mathrm{V}}_{\mathrm{o}}$ change (total volume change of particle). Although changes in the partial molar volume affect the stress response of the active particles, the lithium concentration patterns and stress distributions remain the same.

4.

As the particle size increases, the propensity for mechanical failure increases with the use of the concentration-dependent partial molar volume. However, the effect of changing the partial molar volume on the stress rise decreases for larger particles.

5.

Faster charging with a concentration-dependent partial molar volume increases diffusion-induced stress levels compared to using a constant partial molar volume.