Microstructure-Dependent Macroscopic Electro-Chemo- Mechanical Behaviors of Li-Ion Battery Composite Electrodes

Abstract

The rapid development of the electric vehicle industry has created an urgent need for high-performance Li-ion batteries. Such demand not only requires the development of novel active materials but also requires optimized microstructure of composite electrodes. However, due to complicated heterogeneous electrode microstructure, there still lacks a relationship between the electrode microstructure and the macroscopic electro-chemo-mechanical performance of the battery. In this study, electrochemical and mechanical multi-scale models are developed in order to account for the influence of the heterogeneous microstructure on the macroscopic mechanical and electrochemical behavior of the battery. It is found that porosity and particle size are two important parameters to characterize the microstructure that can affect the macroscopic mechanical and electrochemical behavior. The models developed in this study can be served as designing guidelines for the optimization for the Li-ion battery composite electrodes.

1. Introduction

The development of the Li-ion battery industry has created unprecedented opportunities and challenges at the same time. Therefore, to develop high-performance Li-ion batteries with high energy and power density is of great importance [1]. It is well recognized that the performance of the battery is not only influenced by the active material but also by the electrode microstructure [2,3,4,5,6]. The success of new electrode materials lies in the optimized combination of factors such as energy density, power density, electrochemical stability, safety, material cost, and preparation process, which inevitably demands an extremely long time and great experimental efforts. On the other hand, the existing electrode materials (such as NMC) have a high theoretical capacity and power density but cannot perform their best rate capability due to the randomly designed microstructure of the composite electrode. Therefore, it is practical and effective to design optimized electrode microstructure based on the existing material system in order to improve the performance of Li-ion batteries.

It is widely studied that structural change such as cracks at the particle level can cause severe capacity fade and battery failure [7,8,9,10]. However, the structure characteristics at the electrode level have not been understood in depth due to complicated heterogeneous electrode microstructure. As a matter of fact, it has been experimentally proved that battery performances, including capacity, Coulombic efficiency, cyclic stability, and rate capability, are all greatly affected by, among other factors, particle size distribution, shape, electrode thickness, and porosity, which are characteristic at the electrode scale [11]. These characteristics can be categorized into two groups: those that represent homogeneous characteristics such as porosity and average particle size and those that represent heterogeneous characteristics such as particle size polydispersity and tortuosity.

As one of the important parameters in the microstructure of porous electrodes, porosity has been widely investigated in the literature. Introduction of more pores and increasing porosity can improve electrode transport behavior, decrease ionic resistance, reduce polarization, and elevate the average battery potential. However, in this way, the electrode capacity will be reduced, and the mechanical stiffness and strength will be lowered. This dilemma urgently requires the delicate design of porosity in order to improve battery performance. Apart from porosity, particle size also contributes greatly to the mechanical and electrochemical performance of the batteries. It has been shown that particle size is to a great extent related to aging and degradation behaviors [12,13].

In addition to these homogeneous characteristics, the effect of microstructure heterogeneity on battery performance can also not be underestimated. The spatial distribution of each component in the electrode microstructure is proved to have a significant impact on battery performance [14,15,16]. Structures with smaller pore size or unevenly distributed pores usually come with high tortuosity, which can potentially give rise to longer ionic diffusion paths and higher resistance. This influence is more obvious in thick electrodes with high C-rates. Therefore, the heterogeneous electrochemical mass transfer along the electrode thickness direction should be fully considered in order to achieve accurate estimation of battery performance.

Therefore, it is important to build models that can capture both homogeneous and heterogeneous characteristics of the composite electrodes. As a straightforward approach, the heterogeneous models reconstructed based on X-ray computed tomography images are widely employed by researchers [17,18,19,20,21]. However, due to the experimental cost, the number of these models are limited; thus, they cannot reflect statistical characteristics of the composite electrode. Thus, computed models are developed such as coarse-grained molecular dynamics model, multi-phase smoothed particle model and discrete element method [22,23,24,25,26,27]. In order to capture the macroscopic mechanical behavior, it is common practice to employ homogenized models such as the Mori–Tanaka method [28,29,30].

However, there lacks a thorough investigation into the influence of these parameters because most of the models are either based on heterogeneous models that lose the statistical characteristics or on homogeneous models that lose local information. In this study, electrochemical and mechanical multi-scale models are developed in order to account for the influence of the heterogeneous and homogeneous characteristic of electrode microstructure on the macroscopic behavior of the battery. The remainder of the article is organized as follows: In Section 2, mathematical models are developed based on our previous work. For improved understanding, both heterogeneous and homogeneous models are developed and compared against each other. Numerical simulations are carried out in Section 3, where it is found that porosity and particle size are two important parameters to characterize the microstructure that can affect the macroscopic mechanical and electrochemical behavior. The models developed in this study can be served as design guidelines for the optimization of Li-ion battery composite electrodes.

2. Mathematical Models

For simplicity, we focus our study on a half-cell with lithium metal as the anode, porous polymer as the separator, and a composite cathode. The composite cathode has a porous structure mainly composed of active particles (NMC), a carbon-binder domain (CBD), and pores, which are filled with electrolyte materials. As a homogenized model, the active layer of composite cathode is simplified as a particle-reinforced composite material with active particles as the reinforcement material and the smeared-out domain of the combination of the CBD and pores as the matrix.

2.1. Heterogeneous Model

We first investigate a heterogeneous model that resolves the microstructure of the composite cathode, as shown in Figure 1. It is true that 3D heterogeneous models are more suitable for the correct prediction of electrochemical performance of composite electrodes. However, due to limited computational resources, 3D models are only able to contain a limited number of particles, losing a more general study on the overall performance of the electrode. Therefore, in this study, only 2D structures are modeled. As for the mechanical response, since the structure is assumed sufficiently thick perpendicular to the plane concerned in the model, the plane strain assumption is employed, i.e., the out-of-plane strain vanishes. As an illustrative example, we use NMC111 as the target active material, but the knowledge gained from this study can be easily transferred to other material systems. Also, in the model, since the liquid can have a negligible influence on the mechanical performance of the electrode, we only distinguish liquid from solid electrolyte with the elastic modulus. For this particular example, a liquid electrolyte is employed. Note that the matrix has a porous structure; therefore, the quantities of the matrix are homogenized effective parameters. It is important to consider the heterogeneous structure of CBD in cases such as local fracture in the CBD or electronic conductivity of the CBD. However, in the case discussed in this work, neither electronic conductivity nor the fracture of CBD is the limiting factor of the application of the battery. Therefore, we disregarded the heterogeneous structure and employed a homogeneous CBD model for the sake of simplicity. Then, we build the electro-chemo-mechanical model of the particle and the matrix from our previous work [1,4,31] and briefly summarize here as follows.

We denote the particle and matrix with the superscript p and m, respectively. In both the particle domain $B^p$ and matrix domain $B^m$, there are three sets of coupled fields: the displacement field ${\mathbf{u}}^i$, the concentration field $c^i$, and the electric potential ${\varphi }^i$, where $i=p,m$. The interface between the particle and the matrix is denoted by ${\Gamma }^{*}=B^p\cap B^m$.

Mechanically, force balances are enforced in both domains:

$$\begin{array}{c}\hfill \mathsf{\nabla }·{\mathbf{\sigma }}^i=0\mathrm{in}{B}^i\end{array}$$

(1)

where linear elastic constitutive and linear kinematic relations for both the particle and matrix are assumed.

$${\mathbf{\sigma }}^i={\mathbb{C}}^i\left({\mathbf{\epsilon }}^i-{\displaystyle \frac{1}{3}}{\Omega }^i{c}^i\mathbf{I}\right)$$

(2)

$${\mathbf{\epsilon }}^i={\displaystyle \frac{1}{2}}\left(\mathsf{\nabla }\mathbf{u}+\mathsf{\nabla }{\mathbf{u}}^{\mathrm{T}}\right).$$

(3)

In the constitutive equations, we assume that both the particle and matrix are isotropic, and the expansion upon insertion of Li-ion is also isotropic; thus, the partial molar volumes ${\Omega }^i$ are scalars. If we use the shear modulus $G^i$ and bulk modulus $K^i$ to describe the material stiffness, the fourth-order stiffness tensor becomes

$$\begin{array}{c}\hfill {\mathbb{C}}^i=2G^i\mathbb{I}+\left(K^i-{\displaystyle \frac{2}{3}}{G}^i\right)\mathbf{I}\otimes \mathbf{I}\end{array}$$

(4)

where $\mathbb{I}$ and $\mathbf{I}$ are fourth-order and second-order unit tensors, respectively. At the interface ${\Gamma }^{*}$, the compatibility conditions are enforced, viz.,

$${\mathbf{u}}^p-{\mathbf{u}}^m=0$$

(5)

$${\mathbf{\sigma }}^p{\mathit{n}}^{*}-{\mathbf{\sigma }}^m{\mathit{n}}^{*}=0$$

(6)

with ${\mathit{n}}^{*}$ being the normal vector of ${\Gamma }^{*}$ pointing towards the matrix. Note that in the simulation we assume the volumetric change of the matrix upon insertion/extraction of Li-ion is negligible, which yields ${\Omega }^m=0$.

Electrically, we assume that the particle is electronic conductive with the particle is electronic conductive with the equilibrium potential ${\varphi }^p={\varphi }_{\mathrm{eq}}^p$, and the electric potential in the matrix follows Ohm’s law

$$\begin{array}{c}\hfill \mathsf{\nabla }·{\mathit{i}}^m=0\mathrm{in}{B}^m\end{array}$$

(7)

where ${\mathit{i}}^m$ is the Li⁺ ionic current density in the matrix, defined by

$$\begin{array}{c}\hfill {\mathit{i}}^m=-{\kappa }^m\mathsf{\nabla }{\varphi }^m+\left({\displaystyle \frac{2{\kappa }^{m}RT}{F}}\right)\left(1-t_{+}\right)\mathsf{\nabla }ln{c}^m.\end{array}$$

(8)

Here, ${\kappa }^m$, $t_{+}$, F, R, T denote conductivity, Li⁺ transference number, the Faraday constant, the gas constant, and absolute temperature, respectively. Equation (8) implies that the current density has two driving forces: the applied electric field (first term) and the electric field induced by space charges (second term). We assume that the potential is not influenced by the reaction of the particles locally, meaning there is no interface conditions for the electric field. Instead, boundary conditions at the interfaces at the separator ${\Gamma }_s$ and the current collector ${\Gamma }_{cc}$ sides are imposed for the electric field. For a galvanostatic charge condition of current density $\widehat{i}$, the boundary conditions are

$${\mathit{i}}^m·{\mathit{n}}_s=\widehat{i}\mathrm{on}{\Gamma }_s,$$

(9)

$${\mathit{i}}^m·{\mathit{n}}_{cc}=\widehat{i}\mathrm{on}{\Gamma }_{cc}.$$

(10)

Chemically, the mass conservation law governs the evolution of Li-ion concentration:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial c^i}{\partial t}}=-\mathsf{\nabla }·{\mathbf{j}}^i\mathrm{in}{B}^i\end{array}$$

(11)

where ${\mathbf{j}}^p$ and ${\mathbf{j}}^m$ are Li-ion fluxes in the two respective domains. In the particle, we assume that the particle is electronic conductive with the equilibrium potential ${\varphi }^p={\varphi }_{\mathrm{eq}}^p$ and the stress-assisted diffusion is negligible. Thus, the Li-ions are purely driven by the concentration gradient, and the flux is given by

$$\begin{array}{c}\hfill {\mathbf{j}}^p=-D^p\mathsf{\nabla }{c}^p\end{array}$$

(12)

with $D^p$ being the diffusivity of Li-ions in the active particles. In the matrix, again, we disregard the influence from the stresses on the diffusion; however, the migration due to electric potential is non-negligible. Therefore, the flux is given by

$$\begin{array}{c}\hfill {\mathbf{j}}^m=-D^m\mathsf{\nabla }{c}^m+{\displaystyle \frac{t_{+}}{F}}{\mathit{i}}^m\end{array}$$

(13)

where $D^m$ is the diffusivity and the current density ${\mathit{i}}^m$ is given by Equation (8). This equation indicates that there are two contributions to the flux: the diffusion due to concentration gradient (first term), and the migration due to electric field (second term). At the interface, the current density across the interface is dictated by the Butler–Volmer equation

$${\mathbf{j}}^p·{\mathit{n}}^{*}=j_{\mathrm{BV}}$$

(14)

$${\mathbf{j}}^m·{\mathit{n}}^{*}=j_{\mathrm{BV}}$$

(15)

where

$$\begin{array}{c}\hfill j_{\mathrm{BV}}=i_0\left[exp\left({\displaystyle \frac{{\alpha }_{a}F\eta }{RT}}\right)-exp\left({\displaystyle \frac{-{\alpha }_{c}F\eta }{RT}}\right)\right].\end{array}$$

(16)

Here, $i_0$ is the exchange current density,

$$\begin{array}{c}\hfill i_0=i_{0,ref}{\left({\displaystyle \frac{c^p}{c_{\max }^p}}\right)}^{{\alpha }_c}{\left(1-{\displaystyle \frac{c^p}{c_{\max }^p}}\right)}^{{\alpha }_a}\left({\displaystyle \frac{c^m}{c_{\mathrm{ref}}^m}}\right)\end{array}$$

(17)

where $i_{0,\mathrm{ref}}$, $c_{\max }^p$, $c_{\mathrm{ref}}^m$, ${\alpha }_a$ and ${\alpha }_c$ are the reference exchange current density, maximum Li⁺ in the particle, reference Li⁺ concentration in the matrix, and the anodic and cathodic reaction factors, respectively. $\eta$ is the overpotential, defined as

$$\begin{array}{c}\hfill \eta ={\varphi }^p-{\varphi }^e-E_{\mathrm{eq}}\end{array}$$

(18)

where $E_{\mathrm{eq}}$ is the equilibrium potential.

2.2. Homogeneous Model

In this section, we further smear out the particles in the matrix and consider the composite cathode as a homogeneous material with homogenized parameters to characterize statistically the microstructure, as shown in Figure 2.

Mechanically, we consider the electrode as a particle reinforced composite, whose reinforcement is the active material NMC, and the matrix is the homogenized material of electrolyte and CBD. For the homogenized electrode, the mechanical governing equation in the cathode domain $B^H$ is given as

$$\begin{array}{c}\hfill \mathsf{\nabla }·\mathbf{\sigma }=0\mathrm{in}{B}^H.\end{array}$$

(19)

As for the constitutive relation, the key is to find out effective stiffness tensor ${\mathbb{C}}_{\mathrm{eff}}$ and effective chemical strain ${\mathbf{\epsilon }}^{*}$ such that

$$\begin{array}{c}\hfill \mathbf{\sigma }={\mathbb{C}}_{\mathrm{eff}}\left(\mathbf{\epsilon }-{\mathbf{\epsilon }}^{*}\right).\end{array}$$

(20)

In order to obtain the effective stiffness, we employ the Mori–Tanaka approach [32,33]. We assume that the composite electrode as a dilute dispersion of active particles which are distributed randomly inside the electrode, where interactions between the particles are considered through mean-field theory, as shown in Figure 3. The volume fraction of the active particles is denoted by $\phi$. Under uniform far-field uniform stress and strain boundary conditions, we can obtain the effective stiffness tensor ${\mathbb{C}}_{\mathrm{eff}}$

$$\begin{array}{c}\hfill {\mathbb{C}}_{\mathrm{eff}}=2G_{\mathrm{eff}}\mathbb{I}+\left(K_{\mathrm{eff}}-{\displaystyle \frac{2}{3}}{G}_{\mathrm{eff}}\right)\mathbf{I}\otimes \mathbf{I}\end{array}$$

(21)

where

$$G_{\mathrm{eff}}=G^m+{\displaystyle \frac{\phi \left(G^p-G^m\right)}{1+4\left(1-\phi \right)G^{me}\left(G^p-G^m\right)}}$$

(22)

$$K_{\mathrm{eff}}=K^m+{\displaystyle \frac{\phi \left(K^p-K^m\right)}{1+9\left(1-\phi \right)K^{me}\left(K^p-K^m\right)}}$$

(23)

with

$$\begin{array}{cc}\hfill & K^{me}={\displaystyle \frac{1}{3\left(4G^m+3K^m\right)}},G^{me}={\displaystyle \frac{3\left(2G^m+K^m\right)}{10G^m\left(4G^m+3K^m\right)}}.\hfill \end{array}$$

(24)

As for the macroscopic equivalent chemical strain ${\mathbf{\epsilon }}^{*}$, we also perform mean-field theory and obtain

$$\begin{array}{c}\hfill {\mathbf{\epsilon }}^{*}=\phi {\mathbf{\epsilon }}_c^p+\phi \left[{\left({\mathbb{C}}^p\right)}^{-1}-{\left({\mathbb{C}}^m\right)}^{-1}\right]{\mathbf{b}}^p\end{array}$$

(25)

where

$${\mathbf{b}}^p=\left(\mathbb{I}-{\mathbb{B}}^p\right){\left[{\left({\mathbb{C}}^m\right)}^{-1}-{\left({\mathbb{C}}^p\right)}^{-1}\right]}^{-1}{\mathbf{\epsilon }}_c^p$$

(26)

$${\mathbb{B}}^p={\mathbb{B}}^D{\left[\left(1-\phi \right)\mathbb{I}+\phi {\mathbb{B}}^D\right]}^{-1}$$

(27)

$${\mathbb{B}}^D={\mathbb{C}}^p{\left[\mathbb{I}+\mathbb{S}{\left({\mathbb{C}}^m\right)}^{-1}\left({\mathbb{C}}^p-{\mathbb{C}}^m\right)\right]}^{-1}{\left({\mathbb{C}}^m\right)}^{-1}.$$

(28)

Here, ${\mathbf{\epsilon }}_c^p$ is the chemical strain of the particle, and we disregard the chemical strain of the matrix. $\mathbb{S}$ is the fourth-order Eshelby tensor, which is a function of particle shape and matrix Poisson’s ratio.

Electrochemically, we further disregard the variation along the direction parallel to the separator and only consider the variation along the thickness direction, which is denoted by X. In this way, we come to the modified P2D formulation, where the key parameter is the effective diffusivity $D_{\mathrm{eff}}$ and effective conductivity ${\kappa }_{\mathrm{eff}}$ and are obtained from the microstructure [34]. Based on our previous work [15], we use the volume fraction of active particle $\phi$ and the tortuosity $\tau$ to estimate the effective transport coefficients in the matrix

$$D_{\mathrm{eff}}={\displaystyle \frac{1-\phi }{\tau }}{D}^m$$

(29)

$${\kappa }_{\mathrm{eff}}={\displaystyle \frac{1-\phi }{\tau }}{\kappa }^m.$$

(30)

With the effective transport coefficient, we can write our electrochemical model as following, the more detailed model can be found in our previous work [35,36]. In the absence of the mechanical interactions between the particle and the matrix, there are two sets of coupled fields concerned in this model: the Li-ion concentration $c^i$ and the electric potential ${\phi }^i$ where $i=p,m$. The Li⁺ can only flow in the matrix, contributing to the electric field in the matrix; the electrons are supposed to flow within the particle networks, contributing to the electric field inside the particle phase. Thus, given the thickness of the cathode as L, the governing equations for the electrochemical coupling of the two phases

$$\left(1-\phi \right){\displaystyle \frac{\partial c^m}{\partial t}}=D_{\mathrm{eff}}{\displaystyle \frac{{\partial }^2{c}^m}{\partial X^2}}-{\displaystyle \frac{t_{+}}{F}}{\displaystyle \frac{\partial i^m}{\partial X}}+\left(1-t_{+}^0\right)J_{\mathrm{eff}}(0<X<L)$$

(31)

$${\displaystyle \frac{\partial i^p}{\partial X}}+F{J}_{\mathrm{eff}}=0(0<X<L)$$

(32)

$${\displaystyle \frac{\partial i^m}{\partial X}}-F{J}_{\mathrm{eff}}=0(0<X<L)$$

(33)

where $J_{\mathrm{eff}}$ is the wall flux, representing the reaction at the particle/matrix interface

$$\begin{array}{c}\hfill J_{\mathrm{eff}}=a{j}_{\mathrm{BV}}.\end{array}$$

(34)

Here, a is the specific surface area per unit volume, which is related to the particle size, shape, and volume fraction. $j_{\mathrm{BV}}$ is the flux due to the Butler–Volmer reaction at the interface, given by Equation (16). $t_{+}$ is the transference number, as explained in the last section. The current densities at the particle and matrix phases are given by

$$i^p=-{\kappa }^p{\displaystyle \frac{\partial {\phi }^p}{\partial X}}$$

(35)

$$i^m=-{\kappa }_{\mathrm{eff}}{\displaystyle \frac{\partial {\phi }^m}{\partial X}}+\left({\displaystyle \frac{2{\kappa }_{\mathrm{eff}}RT}{F}}\right)\left(1-t_{+}\right){\displaystyle \frac{\partial ln{c}^m}{\partial X}}.$$

(36)

As for the boundary conditions at the electrode/separator interface, flux boundary conditions can be specified for galvanostatic boundary conditions; at the electrode/current collector interface, no flux of Li⁺ is allowed since it is not permeable to ions. For a more specific discussion, please refer to our previous work [35].

3. Results and Discussion

In this section, composite electrodes with different particle sizes and volume fractions are constructed, both with heterogeneous and homogeneous models, and the mechanical and electrochemical behavior variations between different microstructures are discussed. The parameters used in this study are given in

Table 1. Parameters of the particle packing electrode model used in the simulation.

Definition of Parameters	Symbol (Unit)	Value
Electrode thickness	L ($\mathsf{\mu }\mathrm{m}$)	237
Maximum concentration of Li+ of active material	$c_{\max }^p$ ($\mathrm{mol}\hspace{0.17em}{\mathrm{m}}^{-3}$)	49,500
Li+ diffusion coefficient in matrix	$D^m$ (${\mathrm{m}}^2\hspace{0.17em}{\mathrm{s}}^{-1}$)	Equation (37)
Li+ conductivity in matrix	${\kappa }^m$ ($\mathrm{S}\hspace{0.17em}{\mathrm{m}}^{-1}$)	Equation (38)
Li+ diffusion coefficient in NMC111 particles	$D^p$ (${\mathrm{m}}^2\hspace{0.17em}{\mathrm{s}}^{-1}$)	3 × 10−14
Initial concentration of Li+ in electrolyte	$C_0$ ($\mathrm{mol}\hspace{0.17em}{\mathrm{m}}^{-3}$)	1000
Transference number of Li	$t_{+}$	0.363
Transfer coefficients	${\alpha }_a,{\alpha }_c$	0.5
Temperature	T ($\mathrm{K}$)	298
Young’s modulus of NMC particles	$E^p$ ($\mathrm{G}\mathrm{Pa}$)	78
Poisson’s ratio of NMC particles	${\nu }^p$	0.25
Young’s modulus of matrix *	$E^m\left(\mathrm{GPa}\right)$	5
Poisson’s ratio of matrix *	${\nu }^m$	0.25

* Mixture of electrolyte (volume fraction 70%) and CBD (volume fraction 30%).

.

In addition, the diffusivity $D^m$ and conductivity ${\kappa }^m$ of the matrix are given as a function of Li⁺ concentration This is example 2 of equation:

$$\begin{array}{cc}\hfill D^m& =7.588\times {10}^{-11}{\left(c^m\right)}^2-3.036\times {10}^{-10}{c}^m+3.654\times {10}^{-10}\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill {\sigma }^m& =1.2544\times {10}^{-4}{c}^m\times \left[-8.2488+5.3248\times {10}^{-2}T-2.987\times {10}^{-5}{T}^2+2.6235\times {10}^{-4}{c}^m\right.\hfill \\ \hfill & -9.3063\times {10}^{-6}{c}^{m}T+8.069\times {10}^{-9}{c}^m{T}^2+{\left.2.2002\times {10}^{-7}{\left(c^m\right)}^2-1.765\times {10}^{-10}{\left(c^m\right)}^{2}T\right]}^2\hfill \end{array}$$

(38)

3.1. Mechanical Behavior

As a first example, we examine the effective stiffness in the composite electrode, which is reinforced with spherical particles of three different radii: 3 $\mathsf{\mu }$$\mathrm{m}$, 6 $\mathsf{\mu }$$\mathrm{m}$, and 9 $\mathsf{\mu }$$\mathrm{m}$. For the heterogeneous model, we impose a uniaxial tension test on the electrode as shown in Figure 4 and calculate the effective Young’s modulus $E_{\mathrm{eff}}$.

As for the homogeneous model, we can immediately obtain effective bulk and shear moduli from Equations (22) and (23) and calculate the effective Young’s modulus from the relation

$$\begin{array}{c}\hfill E_{\mathrm{eff}}={\displaystyle \frac{9K_{\mathrm{eff}}}{1+\frac{3K_{\mathrm{eff}}}{G_{\mathrm{eff}}}}}.\end{array}$$

(39)

The results are shown in Figure 5, Figure 6 and Figure 7. We can see that results from heterogeneous model agree very well with the homogeneous model, and for all three cases, the increase in the particle volume fraction gives rise to enhanced stiffness. Furthermore, both heterogeneous and homogeneous models show that particle size play negligible roles in influencing effective macroscopic stiffness of the composite cathode. This can be explained by the fact that volume fraction, rather than local arrangements of the microstructure, plays a dominant role in the macroscopic properties. However, as shown in the contour plots of the heterogeneous models in Figure 5, Figure 6 and Figure 7, particle size plays a key role in local stress state in the composite cathodes. It is obvious that electrodes with smaller particles generally have lower average stresses with smaller fluctuations; electrodes with larger particles suffer from more severe stress concentration and are thus more vulnerable to fracture. This phenomenon shows that smaller particles are generally distributed more uniformly inside the electrode, each bearing similar loads with each other. However, larger particles tend to distribute less uniformly, which leads to higher fluctuations in the loads each particle takes.

We also plotted the macroscopic chemical strain with different volume fractions as shown in Figure 8. It is shown that, in general, more loading of active particles—a higher particle volume fraction—gives rise to increased macroscopic chemical strain. Particle size again plays a negligible role in influencing this strain. The deviation of case for small particles at low volume fraction can be explained by the fact that, in such particular case, the size of particles are comparable with that of the gaps, which may allow rigid-body movements of particles to accommodate volume expansion of particles, resulting a slightly smaller macroscopic volume expansion. As for the stresses, as shown in Figure 8b,c, higher loading of particles will give rise to a more uniformly distributed stress field, which is beneficial in resisting chemical strain-induced cracks.

3.2. Electrochemical Behavior

As a last example, we would like to know how electrode microstructure can influence the electrochemical performance of the composite electrode. We integrate the cathode into a half-cell and run galvanostatic discharge until a cut-off voltage of $2.5$ $\mathrm{V}$. Figure 9a shows the discharge curve estimated by both heterogeneous and homogeneous (P2D) models, both of which agree well with experimental results. Furthermore, in order to evaluate the rate performance, we test the three different cells under different C-rates and measure the depth of discharge (DOD) curves against the particle volume fraction. The results are shown in Figure 9b–d. It is shown that the DOD decreases drastically with increased loading of the active materials. This is an indication that increasing the amount of active material will not necessarily bring about increased capacity. On the contrary, excessive active materials will do harm to the accessible capacity, especially at high discharge rates. Furthermore, the results also show that electrodes with smaller particles generally show much a better rate capability, even with high loading of active materials. The reason for this is that in the battery with liquid electrolyte and NMC particles, the limiting factor of the charge/discharge rate is mostly solid diffusion inside the particles. Therefore, enhancement of particle diffusion efficiency will greatly enhance the rate capability of the complete battery. It should, however, also be noted that in this model we disregard side reactions such as solid-electrolyte-interphase (SEI) formation, which probably favors less exposed surfaces per unit volume, i.e., larger particles, in order to minimize the consumption of lithium and to elongate the lifespan. For these models, the readers are referred to degradation models such as Zhu et al. [37].

The reason is shown in Figure 10, which shows that electrodes with smaller particles in general have more uniform Li⁺ concentration distribution in both the matrix and in the particle, leading to more efficient usage of active material capacity. This conclusion encourages the employment of smaller particles in the composite cathode.

4. Conclusions

In this study, electrochemical and mechanical multi-scale models are developed in order to account for the influence of the heterogeneous and homogeneous characteristics of electrode microstructure on the macroscopic behavior of the battery. To obtain an improved understanding, both heterogeneous and homogeneous models are developed and compared against each other. It is found that porosity and particle size are two important parameters to characterize the microstructure that can affect the macroscopic mechanical and electrochemical behavior. Both from the mechanical and electrochemical point of view, reducing particle size is an effective way to enhance the battery performance, especially at high C-rates. The models developed in this study can be served as designing guidelines for the optimization for the Li-ion battery composite electrodes. Nonetheless, both models cannot capture the failure mechanisms such as fracture at different levels and side reactions, which may lead to discrepancies with experimental results. Therefore, it is highly important to consider failure mechanisms in the model in the future.