## 1. Introduction

With the rapid development of portable electronic devices and electric vehicles, as one of the main choices for energy storage and energy supplied systems, lithium-ion batteries are deemed to be in urgent need of achieving long lifetimes [

1,

2]. In order to fulfill this, industrial and the scientific sectors have oriented efforts towards understanding their mechanical behavior and the underlying damage mechanisms of battery components from active particles [

3,

4] to cell structures [

5,

6]. One has recognized that the primary cause for capacity attenuation of rechargeable lithium batteries is that the Vegard stress induced by the electrochemical reaction cannot only damage the active substances [

7], but also polymer bonding materials in the composite electrode [

8,

9,

10]. Therefore, it is extremely significant to study the stress distribution and mechanical stability of the electrode particle–binder system.

Conventionally, polymer materials play two pivotal roles in lithium-ion batteries. They are used as separators to separate positive and negative electrodes, and as adhesives for binding isolated active particles to the current collector [

11]. For the former, microporous membranes based on semi-crystalline polyolefin materials such as polyethylene (PE), polypropylene (PP) and their blends are widely used in liquid electrolyte lithium-ion batteries [

12,

13]. As for all-solid-state batteries, with great efforts made by the scientific community, various poly(ethylene oxide) (PEO)-based solid polymer electrolytes obtained by crosslinking [

14,

15,

16], blending [

17,

18] or grafting [

19] have been applied. For the latter, poly(vinylidene fluoride) (PVdF) are the most commonly used and investigated binding materials mainly because of good electrochemical stability. In the literatures, focusing on the mechanical stability of particle–binder system, Rahani and Shenoy first studied the mechanical degradation in the graphite anode bonded by PVdF using finite element methods. They found that the yield stress level of PVdF determined the average stress of the composite electrode [

20]. Takahashi also analyzed the stress evolution of an isolated graphite sphere enclosed by conductive additive filled PVdF in the lithiation process [

21]. It was indicated that the polymeric and conductive composite (BCC) had certain constraints on the lithiation deformation of active material, leading to a decrease in the tensile stress of graphite particle. Moreover, the PVdF-based BCC is more likely to take place mechanical degradation in the circumferential direction compared to the active materials. In addition, Singh and Bhandakkar studied the stress evolution of a spherical electrode particle/PVdF system during galvanostatic electrochemical cycling [

22]. The viscoelasticity of polymer binders was proven to affect the diffusion induced stress (DIS) in active material and binder and decreasing viscosity and characteristic relaxation time could weaken DIS of the composite electrode. In contrast with above one-dimensional particle models, Higa and Srinivasan calculated the stress of axisymmetric silicon particle sandwiched between two cylinder of PVdF in the course of charging [

23]. The simulation disclosed that most of the strain energy of electrode system was stored by PVdF and the energy per interfacial area decreased with particle size and binder stiffness. For this reason, the debonding between active material and BCC may be one of the causes for the degradation of silicon electrode capacity. Lee et al., further investigated the interface failure of a spherical graphite particle and a cylindrical PVdF binder [

24]. It demonstrated that the delamination of binders and active particles in the lithiation process was diametrically opposite to the damage mechanism of that inside the active particle. The high lithium-concentration gradient, caused by the large particle size and high charging rate, resulted in the increase of maximum principal stress in the active particles. However, it could help to decrease the interface stress between the binder and electrode material. In consideration of the complex mesostructures of electrode particles and binders, the coupled electrochemical–mechanical simulation was carried out using the experimentally reconstructed microstructure, which was captured by the scanning electron microscope [

25] or nanocomputed tomography (X-ray nano-CT) techniques [

26]. These works well revealed the roles of electrode geometry of active materials, binder loading and boundary conditions on its surface on the stresses in electrode and PVdF binder under lithiation–delithiation cycling. It is noted that the aforementioned investigations neglected the influence of solid electrolyte interface (SEI) film and conductive agent content in BCC. However, the published reports indicated that SEI formed on the surface of active materials during the lithiation process was strongly associated with the mechanical stability of the electrode material [

27] and the conductive additives acted upon a complicated mechanical role when added into the polymer binders for composite electrodes [

28,

29,

30,

31,

32].

Due to the popular PVdF binder contains toxic organic solvent, sodium alginate (SA) [

33,

34,

35], polyacrylic acid (PAA) [

36,

37], sodium carboxymethyl cellulose (CMC) [

38], styrene–butadiene rubber (SBR) [

39], which use water as solution, may be regarded as the potential constitutes in composite electrode for advanced secondary battery. Considerable efforts have been devoted to clarifying the influence of binder nature on the cycle stability and rate performance of cell. Owing to high bonding strength of aqueous polymer binders, high-capacity electrodes also exhibited mechanical stability, good capacity retentions and rate capabilities [

36,

37]. It is noticed that the cohesion properties of PAA binder acted a pivotal role on the mechanical integrity and electrochemical stability during charging–discharging process. Recently, Li et al., compared the influence of binder stiffness on bending deformation and DIS in Si anodes with SA, Nafion and PVdF with the results suggesting that the binder plays an important role in lithiation-induced deformation and the cracking of composite electrodes [

40]. Wang et al., further observed that the elastic modulus and hardness of Si composite electrodes were mainly related to the mechanical properties of water-soluble binders, instead of the adhesion between binders and active particles. These findings may help to understand how the aqueous polymer adhesive system impacts the mechanical stability of electrode materials and vigorously promote the development of high performance and durable composite electrodes. Nevertheless, due to the lack of mechanical properties of conductive agent filled water soluble polymer composite under liquid electrolyte, the fracture mechanism is still not clear for the kind of binder and conductive materials in the composite electrodes. To date, there has been little exploration into the structural integrity of electrode particles system with SEI film, which is enclosed by the binder and conductive composite (BCC).

In this study, we establish an electrochemical-mechanical model for the multilayer spherical particles that consist of an active material, SEI and BCC. The lithium concentration and diffusion induced stress distribution in the electrode system have been emphatically discussed by coupling the effects of SEI and the viscoelasticity of polymer binder. In order to clarify the mechanical failure mechanism of aqueous BCC under realistic condition, the evolution of peak stress in BCC is investigated systematically under different water–based polymer binders, loading of conductive carbon black, elastic modulus and thickness of SEI, as well as charging rates. In contrast to the single-particle or particle-BCC coating structures, we found—possibly for the first time—that the circumferential cracks induced by lithiation may primarily initiate in BCC rather than in other electrode components.

## 2. Model Description

The multilayer electrode particle system composed of active material, SEI film and BCC as shown in

Figure 1 are considered. It is assumed that the particles are spherical in structure, SEI and BCC are uniformly deposited and coated on them, and the corresponding radii are

$a$,

$b$ and

$c$, respectively. Due to high porosity in the graphite electrode composite, the mechanical interaction among the active particles may be negligible. This is to say, there are no external forces on the exposed particle surface. Similar boundary condition at outer surface were adopted for determining the stress evolution of electrode particle system in many published investigations [

21,

22,

23,

24]. Therefore, the spherically symmetric particle with free traction at the outer surface of the BCC is applied to depict the composite graphite electrode. Based on the hypothesis, the thickness of SEI is

${h}_{SEI}=b-a$, while the thickness of BCC is

${h}_{BCC}=c-b$.

Under the spherical coordinates, the diffusion of lithium in active particles is determined by the following equation:

Here,

$c\text{}\left(\mathrm{mol}/{\mathrm{m}}^{3}\right)$ is the molar concentration of lithium and

$J=Dc\nabla \mu /\left({R}_{g}T\right)$ is the related lithium flux, where

$\text{}D\text{}\left({\mathrm{m}}^{2}/\mathrm{s}\right)$,

$\mu \text{}\left(\mathrm{J}/\mathrm{mol}\right)$,

${R}_{g}\text{}\left(\mathrm{J}/\mathrm{K}/\mathrm{mol}\right)$ and

$T\text{}\left(\mathrm{K}\right)$ represent the diffusion coefficient of lithium, chemical potential, universal gas constant and temperature, respectively. Taking the influence of mechanical energy caused by stress on chemical potential into consideration, the chemo-mechanical potential

$\mu $ can be further expressed as:

where

${\mu}_{0}$ is the an invariant reference potential,

$\mathsf{\Omega}\text{}({m}^{3}/mol)$ is the partial molar volume of lithium and

${\sigma}_{h}$ is the hydrostatic stress, which can be calculated by radial stress

${\sigma}_{r}$ and circumferential stress

${\sigma}_{\theta}$ under the spherical coordinates, i.e.,

${\sigma}_{h}=\left({\sigma}_{r}+2{\sigma}_{\theta}\right)/3$.

The following governing equation of lithium diffusion can be obtained through substituting Equation (2) into Equation (1).

Driven by the gradient of chemical potential, the lithiation and delithiation on the surface of the particles are assumed to take place at galvanostatic or potentiostatic conditions and the corresponding initial and boundary conditions are expressed as [

41,

42,

43]:

where

${c}_{0}$ is initial molar concentration of lithium,

$n$ is the surface normal vector,

$F\text{}$= 96,485.3 C/mol represents the Faraday’s constant,

${i}_{n}\text{}\left(\mathrm{A}/{\mathrm{m}}^{2}\right)$ is the surface current density of active particles and

${c}_{b}$ is the boundary molar concentration of lithium under constant voltage operation.

For convenience, the state of charge (SOC) is introduced to intuitively reflect the lithiation state of active material, which can be acquired by the following equation:

When the active material is lithiated, the lithiation deformation will generate the Vegard stress, which then triggers the strain in surrounded SEI and BCC. The structural stress may endanger the mechanical integrity of the electrode particle system and ultimately results in the degradation of battery performance. Regarding the graphite particles that are considered here, it is assumed that both the active materials and SEI are linear elastic materials and thus the corresponding relationship between stress and strain can be expressed as:

where

$E\text{}\left(GPa\right)$ and

$v$ are the elastic modulus and Poisson’s ratio of the lithium compounds, respectively.

$\theta ={\epsilon}_{r}+2{\epsilon}_{\theta}$ is the volumetric strain.

${\epsilon}_{r}$ and

${\epsilon}_{\theta}$ are the radial and hoop strains. The first term on the right side of Equation (6) is the mechanical elastic stress, while the latter term is related to the lithium concentration and represents the stress induced by atomic diffusion. It is noticeable that only active materials are lithiated during lithium solid-phase diffusion, and thus the above equation is only valid for active particles. For SEI, the term of diffusion induced stress in Equation (6) must be omitted.

Previously, we carried out the tensile stress relaxation experiments on SA and CMC/SBR doped by Super-S carbon black at the weight ratio of 0%, 20%, 35%, 50% and 60% in 1.1-M LiPF6-EC/DMC, respectively. In terms of the evolution of normalized stress versus time, it was found all curves exhibited typical linear viscoelastic behavior (time-dependent stress reduction), but with different degrees of relaxation [

32]. To this end, the deformation response of BCC is thereby characterized in term of a rheological model composed of two Maxwell elements and a spring in parallel (see

Figure 2). The corresponding constitute equations are given by

where

$\sigma $ and

$\epsilon $ represent stress and strain, respectively.

${K}^{BCC}$ is the volume modulus and

$\mathsf{\Gamma}\left(t\right)$ is the function of relaxation modulus. The subscript ‘vol’ and ‘dev’ indicate the spherical tensors and deviator tensors, respectively. The superscript BCC means that this parameter corresponds to the binder and conductive composite. The function of the relaxation modulus can be written in terms of the Pony series:

where

${G}_{0}$,

${G}_{1}$ and

${G}_{2}$ represent the shear modulus of springs in the Maxwell model, respectively.

${\tau}_{1}$ and

${\tau}_{2}$ are the relaxation time values of corresponding dashpot.

According to the tensile stress relaxation curves of Super-S carbon black (SS) filled SA and CMC/SBR films [

32] and Equation (8), the relaxation modulus and characteristic time are obtained by 1stOpt

^{®} nonlinear regression software (7D-soft high technology incorporation, Beijing, China), and its evolution against SS content are separately listed in

Table 1 and

Table 2.

Neglecting the body force, the equilibrium equation in spherical coordinates can be presented as:

On condition that the lithiation deformation is in the infinitesimal range, the radial strain

${\epsilon}_{r}$ and hoop strain

${\epsilon}_{\theta}$ can be expressed as functions of radial displacement

$\text{}u$:

As to the spherical symmetric structure under consideration, the radial displacement at the center of the sphere is inevitably zero in the deformation process, as the Equation (11). In the following equations, superscripts A, SEI and BCC represent the active materials, solid electrolyte interface and carbon black-filled aqueous polymer binder materials, respectively.

The active material–SEI interface and SEI–BCC interface meet the corresponding displacement and stress continuity conditions as presented by Equation (12):

Finally, the boundary condition at the BCC surface is expressed as:

To numerically drive the aforementioned model in the following simulations, one can perform coupled analysis in commercial numerical software COMSOL. Here, noting that the stress within the isotropic active particle is obtained via the analogy between thermal stresses and diffusion induced stresses, the partial derivative of hydrostatic stress versus lithium concentration depends only on the material constant, i.e.,

$\partial {\sigma}_{h}/\partial c=-2E\mathsf{\Omega}/\left[9\left(1-v\right)\right]$. The lithium concentration can thus be solved beforehand by substituting it into the governing Equation (3) and applying the corresponding initial and boundary conditions shown in Equation (4). Through the above processing, the two-way coupling between lithium diffusion and mechanical stress can be decoupled. Hence, the problem degenerates into a traditional viscoelastic problem, which is defined by Equations (6)–(13), and it can be easily solved by numerical methods everywhere [

44].