Time needed for the movement of phase interface.
This chapter reviews several theoretical models that are used to compute the stress fields inside the electrode particles of lithium-ion batteries during discharging/charging process and provides a guideline for researchers to choose the appropriate models. Due to the limitation of the existing models, a general electrochemo-mechanical framework is presented to model the concentration and stress fields of the electrode during the phase transformation. The interaction between stresses fields and phase transformation is addressed, which is a novel discovery in the research of lithium-ion batteries. The electrodes with different sizes and geometries are compared. The structural and electrochemical advantages of hollow core-shell structure particles are highlighted. The present work could help to accurate predict stress profile in electrode particles with different sizes, geometries, and charging operations and contributes to finding the optimal electrode. Therefore, this chapter is helpful for the material and structure design of electrodes of lithium-ion batteries.
- lithium-ion batteries
- surface/interface stress
- hydrostatic stress
- phase transformation
- core-shell electrode
Lithium-ion batteries are the choices of diverse applications, such as electronics and electric cars because of their high capacity, high voltage, and long lifetime, and attract wide research interest in the community of chemistry, electro-chemistry, and mechanics [1–6]. During the process of charging/discharging, lithium ions insert into/extract from electrodes and induce high stresses . The stress could cause the fracture of electrode when it exceeds the ultimate strength of the material [8–10]. This intercalation-induced fracture is indeed a key mechanism for lithium-ion battery's capacity fade.
To prevent this stress-induced electrode failure, accurate prediction of the stress fields is the first step. Therefore, multiple models are developed to address the effects of different factors on the stress field, e.g., hydrostatic stress , surface stress , charging operation [12, 13], material imperfection , plasticity [15, 16], heat generation [17, 18], particles' dissolution [19, 20]. Inserting the stress field into the fracture mechanics model, some novel models are proposed to investigate the electrode fracture and battery failure. For instance, Woodford et al.  calculates the stress intensity factor of the initial crack inside the electrode and the stability of crack growth. Several novel models are developed to predict the dynamic crack propagation [21–23]. The effects of some factors cannot be characterized by continuum models, which motivates some atomic scale research. Gao et al. developed the atomistic models to study the strong coupling of diffusion, stress, and solute concentration, and the surface locking instability during atomic intercalation into electrode [24, 25]. Suo et al. employed the first principle calculation to investigate the microscopic deformation and lithiation induced plasticity of silicon electrode [26, 27].
Phase transformation during the discharging/charging process has been widely reported in different electrode active materials [28, 29]. During the phase transformation, the electrode is divided into two phases by a phase interface. Because the equilibrium concentration of the two phases at the interface is different, an interface concentration discontinuity is observed. This phase interface is moving during the phase transformation, whose movement is characterized by an interface mass balance condition. Though this phase transformation can be tracked by the moving boundary models [30–33], the stress fields during the phase transformation are not well studied. Moreover, the effect of stress fields on phase transformation remains unknown. This chapter will systematically investigate the interactions between phase transformation and stress fields.
The second step to avoid the stress-induced electrode failure is to find an optimal electrode to lower the stress. The main research interests focus on the size and shape of the electrodes. Considerable efforts are put into studying the size of electrodes [25, 34]. The major conclusions are that nanoelectrode particles are not as easy to fail as micro ones, and the batteries with nanoelectrode particles have better cyclability. However, researchers obtain this conclusion by using the same models to analyze the behaviors of different-sized particles, which is incorrect. For example, the linear diffusion model  is only accurate for nanoelectrode particles while loses accuracy for large electrode particles. A comprehensive model that could provide accurate stress prediction for different sized electrode particles is needed. In terms of electrode shape, though researchers have tested the electrode particles of different shapes, e.g., spherical , core-shell [35, 36], nanowire , thin film [38, 39], layered plates , thin strip , and cylinder , it is still an open question what an optimal electrode is.
Hollow structure particle is a good candidate for the electrode of lithium-ion batteries, due to its unique structural properties, e.g., doubled surface area, core-shell structure, and large internal void . Researchers have analyzed the mechanical properties of core-shell structure [44, 45], and emphasized its significant influence on the stress fields. Furthermore, the advantages of hollow structure particles are highlighted by numerous experiments, when they are used as electrode particles. For example, hollow Si anode shows high initial discharging capacity and low-capacity degradation . High capacity, good cyclability, and high rate capability are reported for hollow core-shell mesoporous TiO2 spheres . High coulombic efficiency, great rate performance, and excellent stability are observed in hollow Fe2O3 particles . However, these works reveal few reasons for the optimal behaviors of hollow structure particles. Therefore, a comprehensive analysis is needed to investigate the properties of hollow structure electrode. In this chapter, a theoretical model on the hollow particles is introduced which considers the effects of hydrostatic stress, surface/interface stress, and phase transformation simultaneously.
In what follows, Section 2 briefly reviews the recently developed models on the stress fields of electrode particles. A guideline is proposed for researchers to choose appropriate models. Section 3 presents an electrochemo-mechanical framework to model the concentration and stress profile in the electrode of Lithium-ion batteries. The effects of hydrostatic stress, surface/interface stress, and phase transformation are fully coupled. Section 4 applies this framework to the hollow spherical electrode particle and calculates its concentrations and stress fields. In Section 5, the size and shape effects of electrode are analyzed to find the optimal electrode. Size effects induced by hydrostatic stress and surface/interface stresses are investigated through a cross-scale analysis, and shape effect is studied by varying the shell thickness. The structural and electrochemical advantages of hollow structure electrodes are investigated. Finally, some remarkable conclusions and discussions are provided in Section 6.
2. Review of existing models
To provide accurate predictions of the stress fields in the electrodes, multiple models are developed to address the effects of different factors on the stress field. This section briefly introduces some recently developed models and provides a guideline for researchers to choose appropriate models.
Zhang et al.  considers the effect of hydrostatic stress on the lithium flux, which provides accurate predictions on the stress fields. However, this effect is only important for micro electrode particles and can be ignored for nano ones. When the particle size is in nanoscale, Cheng et al.  reports the effect of surface stress, which is inversely proportional to the particle size. This surface effect is essential for nanoelectrode particles because it changes the stress state from traction free into compression and therefore prevents the growth of manufacture induced cracks. Therefore, the model with hydrostatic stress suits the research on micro electrode particles while the surface stress is a good choice for nano ones. Cheng et al.  and Lu et al.  investigate the influence of charging operations on stress fields, i.e., potentiostatic and galvanostatic operation. The galvanostatic process is the first stage of the charging process and occupies over 90% of the entire charging time and is commonly used in different models. However, there is some unused lithium ions at the end of galvanostatic process. To avoid the waste of the unused lithium, potentiostatic operation is needed. Therefore, potentiostatic operation is appropriate for the research on improving the battery efficiency.
The strain approximation is slightly different in modeling the cathode and anode. When modeling the cathode, since the deformation is small, the infinitesimal strain assumption is well accepted [14, 18, 35, 36]. However, during the charging/discharging process, the deformation of anode is large, especially when the anode material is silicon. Therefore, finite strain approximation is more reasonable in modeling the deformation of anode. The constitutive law is most important for the stress modeling, since it directly relates the stress state and deformation. Linear elastic assumption is usually the choice, since it is the simplest and the deformation of electrode particles usually stays in the elastic range. However, when the deformation is large enough, the stress can exceed the yielding criterion and the deformation may reach plastic scope. Therefore, linear elastic models are used for small deformation modeling, while plastic models are usually employed in modeling large deformation of electrode [15, 16].
Manufacture-induced initial flaws or cracks are important reasons for the failure of electrodes and therefore need to be carefully addressed. Stress concentration appears at these imperfections, and the growth of the imperfections could lead to the fracture of the electrodes. Woodford et al. proposes a fracture mechanics model to predict the stress fields at the initial crack . However, this model can only determine whether the initial crack would grow. To model the dynamic crack propagation progress, multiple fracture models are developed [21–23]. In addition to cracks, one can investigate other failure mechanisms of the electrodes, e.g., delamination [13, 50], by inserting the stress fields into the corresponding models.
Phase transformation is experimentally observed in the electrode active material during discharging/charging process and could significantly affect the stress fields [30–32]. Multiple models have been developed to address the phase transformation-induced discontinuities in concentration and stress fields. However, the inverse effect of the stress fields on the phase transformation has not been well studied, until the recent work of current authors . Liu et al.  proposes a fully coupled system which investigates the interactions between phase transformation and stress fields, which will be discussed in this chapter.
In the following, this chapter proposes an electrochemo-mechanical framework to model the concentration and stress fields of electrode particles. The interactions between stress fields and phase transformations are characterized. The effect of hydrostatic stress, surface stress, and interface stress are fully addressed.
3. Electrochemo-mechanical framework
There are three different stages in the whole discharging process, which are schematically illustrated in Figure 1. In the first stage, there is
3.1. Mechanical equations
As illustrated in Figure 1b, a phase interface
Classical elasticity theory yields the equilibrium equation,
and the infinitesimal strain geometric equation
Combination of Eqs. (1)–(3) yields the governing equation of each phase. In addition to the equations of two bulk phases, the deformation of interface needs to be characterized. The equilibrium equation of the phase interface S
The interface constitutive law yields the relation between interface stress tensor
where τ0 is the strain-independent interface stress,
The two phases are fully bonded at the interface, which yields the no jump condition
3.2. Electrochemical equations
Lithium ions extract from/insert into electrode particles during the charging/discharging process. This process is usually modeled as the diffusion of lithium ions, driven by the gradient of chemical potential. The velocity (
where the electrochemical potential ϕp in an ideal solid solution is ,
where is a constant, R is gas constant, T is absolute temperature, X is molar fraction of lithium ion, and is hydrostatic stress,
where Dp = MpRT is diffusivity. Please note that the effect of hydrostatic stress on the species flux was neglected by previous work [11, 12, 41, 42], and is first proposed in . The equation of substance conservation is written as
The boundary conditions are determined by the charging operation. Under galvanostatic operation,
The movement of the phase interface is assumed to be under the control of diffusion process in the adjacent phases, and the interface position is tracked by a jump material balance ,
The electrode particle is initially at the stress free state, which implies that the initial concentration is uniformly distributed, i.e.,
A general electrochemo-mechanical framework is developed for an electrode particle of arbitrary geometry in lithium‐ion batteries. The discharging/charging process can be regarded as a quasi-static process. In each time step, by inserting the concentration of Li ions into Eqs. (1)–(6), one can compute the stress field. Substituting the stress field into Eqs. (13)–(17), one can obtain the concentration field.
4. Stress field in hollow spherical electrode particle
Section 3 provides a electrochemo-mechanical framework to compute the concentration and stress profile for an electrode particle of arbitrary geometry in lithium ion batteries. This section will perform analysis on the concentration and stress fields of a specific electrode particle. Hollow structures have great potential as electrode of lithium-ion batteries, due to their better cyclability, higher capability, and lower capacity degradation [46, 48, 49]. Therefore, in the following, we will analyze the concentration and stress fields of the hollow electrode particles.
Figures 3 illustrate the structure of hollow spherical particles, r1 and r2 are the inner and outer radii of the hollow particles, respectively. ζ = r1/r2 denotes the ratio of inner and outer radii. As emphasized in last section,
4.1. Electrochemo-mechanical framework of hollow particle
As an axisymmetric problem, the displacement vector only has radial component (u) and the stress and strain tensors only contain two independent components σr , σθ and εr , εθ (note σϕ = σθ, εϕ = εθ). Under the spherical coordinate, Eqs. (1)–(3) become
where Please note that and , and ξ is the position of interface. Ap, Bp are constants that need to be determined via the boundary conditions.
For the hollow spherical electrode particle, the interface equilibrium Eq. (4) is given as
For the isotropic interface, the interface constitutive law Eq. (5) is
where Ks is interface modulus (). The displacement continuity Eq. (6) reduces to,
The surface stress is effected via the boundary condition ,
It is noted that this boundary condition can be regarded as the special case of Eq. (4), when the interface is located at the inner and outer surfaces.
The constants A
The stresses inside the electrode particles are induced by the non-uniform distribution of concentration, which is studied by an electrochemical model. In the first and third stages of the discharging process, there is only
The gradient of the hydrostatic stress can be rewritten as
where θp = 2Ep(Ωp)2/[9(1 − νp)RT]. Rewrite the dimensionless parameter λp = θpcp = θpcmax ⋅ cp/cmax. Considering cp/cmax ∈ (0, 1), one can expect that the effect of hydrostatic stress depends on the value of λp. Define the effective diffusivity , and rewrite Eq. (29) as
One can observe that the governing equation could reduce to a similar form to the classical diffusion equation. The effects of hydrostatic stress are effected via a concentration dependent diffusivity. The lithium ion flux is inserted into and extracted from the electrode through the outer surface. So the outer surface is active, while inner surface is inactive. Therefore, the specific form of boundary condition Eq. (14) is
Eq. (31) implies that the concentration gradient at the outer surface is inversely proportional to the factor 1 +
When the concentration at the outer surface reaches the equilibrium value of the
For the hollow spherical particles, the governing equation of the moving interface Eq. (17) reduces to
where dξ/dt is the moving velocity of the phase interface. Note that the contributions of θ
The concentrations at two sides of the interface are equal to their own equilibrium values, i.e.,
At this stage, the
During the phase transformation process, the
Initially, the lithium ions are uniformly distributed inside the electrode particle, i.e.,
The beauty of the formulation for the hollow spherical particle is that the governing equations Eqs. (29), (32), and (33) are decoupled from the stress field explicitly. This implies that one can directly compute the concentration field by using Eqs. (29)–(36) instead of solving a fully coupled system, and then calculate the stress field using Eqs. (22)–(26) at each step.
4.2. Concentration and stress fields
Section 4.1 provides the model to compute concentration and stress fields. The effect of hydrostatic stress is coupled into governing equation Eq. (32), and the surface/interface stress are effected via Eq. (23) and (26). To reveal the interactions between stress profile and phase transformation, a hollow spherical electrode particle is analyzed, for example. The inner and outer radii of electrode particle are 10 and 20 μm, and the particle is discharged under 10 A/m2 currency density. Here and in the following, LiCoO2 is used as the electrode material unless otherwise specified because it is the mostly reported electrode material with phase transformation phenomenon and most widely used in industry. The material properties of LiCoO2 are provide in .
4.2.1. Concentration field
The concentration field of the electrode particle at the three stages are calculated (c.f. Figure 4). At the first stage, there is
When the phase transformation happens, a phase interface appears between the
To address the interactions between phase transformation and stress field, one can first investigate the effect of hydrostatic stress on the phase transformation. Please recall that the governing equation of the phase interface is modified to include the gradient of hydrostatic stress. Table 1 lists the time it needs for the phase interface to arrive the positions illustrated in Figure 4b. Compared with the conventional equations without considering the effect of hydrostatic stress, the moving speed of phase interface is much faster, which implies that the time during the phase transformation stage is overestimated by the previous models . This is because the factor (1 +
|Position||Without hydrostatic stress||With hydrostatic stress|
|0.9 r2||23,500 s||171 s|
|0.8 r2||44,950 s||328 s|
|0.7r2||61,210 s||452 s|
|0.6r2||73,700 s||550 s|
When the phase interface arrives at the inner surface, the phase transformation ends and the third stage begins. There is
4.2.2. Stress profile
Lithium ion is non-uniformly distributed in the electrode and induces stresses. The stress field of a single phase electrode particle has been investigated by many researchers [18, 21, 35, 36], while the stress field of the biphase one is still unknown. Eq. (22) implies that the stress fields are explicitly related to the concentration field. One can expect the interface concentration discontinuity to have a significant influence on the stress field, which is illustrated in Figure 5. The constant surface stress τ0 = 1J/m2 and surface modulus Ks = 5N/m, unless otherwise specified . The radial stress of the single phase electrode particle monotonically increases/decreases in the discharging/charging process [14, 18]. But the radial stress of this biphase particle increases from the inner surface to the phase interface and decreases from the interface to the outer surface. The stresses predicted from the present model are compared with those from the one without considering surface/interface effect. Without considering the surface effect, radial stress is zero at the surfaces because Eq. (26) reduces to the traction free boundary condition. Under the effect of surface stress, the radial stress is positive at the inner surface, and negative at the outer one. This implies that the electrode particle boundaries can still under tension or compression though no external load is applied. Without considering the interface effect, the radial stress is continuous at the interface since Eq. (23) reduces to the continuity condition and is automatically satisfied. Under the interface effect, a positive stress jump is observed at the interface. Please note the surface/interface effects are highlighted for this hollow biphase particle, due to its double surfaces and sharp interface.
Hoop stress is the driven force for the propagation of manufacture-induced imperfection. Therefore, hoop stress is more important in the stress analysis of electrode particles. One can observe that the hoop stress decreases from the inner surface to the outer one in the
The interactions between stress fields and phase transformation is systematically investigated in the above. One can conclude that the hydrostatic stress could accelerate the phase transformation and the time during phase transformation is overestimated by the conventional model without considering the effect of hydrostatic stress . Moreover, the during the phase transformation, there is a sharp phase interface. The phase interface induced discontinuity in both radial and hoop stresses, and made the stress field in the electrode significantly higher. In summary, the phase transformation increases the stress field and therefore threatens the safety of electrode particle. But the hydrostatic stress could help the electrode to finish the phase transformation faster, and is important to avoid the failure of electrode particle.
5. Electrode geometry analysis
A key concept to avoid the stress-induced electrode failure is to find an optimal electrode, in which the stress is lower. In order to optimize the electrode, size and shape of the electrodes are two major topics to investigate. Different sized electrodes are studied during the last decades and considerable efforts are put into searching electrodes with optimal shapes, and researchers have tested multiple shapes [13, 34–36, 40–42]. However, those work mainly focus on the investigation of single phase solid electrodes. Hollow particle, with a unique doubled surface area, core-shell structure and large internal void, has great potentials in applications of lithium-ion batteries . In Section 4, the concentration and stress profile of hollow electrode particles are analyzed. This section will study the size and shape effects of this hollow spherical particles.
5.1. Size effect
Size effect implies the dependence of the stress fields on the size of electrode particles, which has been investigated by researchers [11, 14, 18]. They conclude that larger particles lead to higher concentration gradient and induce higher stress. However, the model used in the above work is fairly simple. In the present model, the size effect can interact with the effects of hydrostatic stress and interface/surface stress, and have more complicated behaviors. To investigate this effect, a cross scale analysis is presented here, with particle size ranging from 10 nm to 20 μm and being discharged under 10 A/m2 are simulated. The inner radii are half of the outer ones for these electrode particles.
5.1.1. Hydrostatic stress-induced size effect
Section 4 highlights the effect of hydrostatic stress on concentration and stress. In what follows, we will discuss the dependence of its effect on electrode size. Figures 6a illustrates the maximum hoop stresses of different sized electrode particles, when the concentration at the outer surface reaches . The stress fields with/without considering the effect of hydrostatic stress are compared. The stress fields predicted by two models are almost the same for the nanoelectrode particles, but are significantly different for the micro electrode particles. Without the effect of σh, large concentration gradient exists in large electrode particles, which induces the stress as high as 160 MPa. However, the stress field is not that high in reality. This is because σh can lower the concentration gradient and, therefore lower the stress fields. One can also explain the effect of σh as follows: the “effective” inward flux at outer surface D∂c/∂r = in/F(1 + θc) decreases with c, which is equivalent to the case that the battery is discharged in a lower currency density. And it is well known that the stress of the electrode particle is lower when discharged under a lower currency density. Therefore, one can conclude that the effect of σh is necessary in stress predictions for the micro particles, but can be neglected for the nanoparticles.
5.1.2. Surface stress-induced size effect
In the community of solid mechanics, the surface stress-induced size effect is widely observed [45, 51, 52]. In 2008, Cheng et al.  introduces the surface stress effect to stress analysis of nano LiMn2O4 solid electrode particles in lithium-ion batteries. Here the size effect with surface stress is presented for the hollow spherical electrodes. We analyze the surface stress-induced size effect in the first stage of discharging process, since there is only
5.1.3. Interface stress-induced size effect
Although the surface effect of electrode particles has been reported for several years , the interface stress is not considered until the recent work of the authors . In fact, for hollow electrode particles, the surface stress at the inner and outer surfaces are just special cases of interface stress Eq. (3). Since the size effect of surface stresses is reported above, the size effect due to interface stresses is expected.
Figures 7 illustrates the stress fields of different sized biphase electrode particles, that are discharged under 10 A/m2 currency density. For the micro electrode particle, the interface stress is not significant and the discontinuity of
In summary, hydrostatic stress, surface stress, and interface stress can all induce size effect. The effect of hydrostatic stress increases with particle size while the effects of surface and interface stresses decrease with particle size. This is because the hydrostatic stress is inserted into bulk equations while the surface/interface stress is considered in the surface/interface equations. The ratio of surface/interface over volume decreases with the particle size, so do the effects of surface/interface stresses.
5.2. Shape effect
Because of its better cyclability, higher rate capability, and less capacity degradation compared to the solid sphere electrodes [46, 48, 49], hollow electrode particle attracts wide research attention. The structural feature of hollow particles, i.e., shape effect, can be characterize by a parameter ζ = r1/r2. The hollow sphere could reduce to solid ones when ζ = 0 (r1 = 0), and become a thin shell when ζ ≈ 1 (r1 ≈ r2). The shape effect on the mechanical and electrochemical properties of hollow electrode particles are systematically investigated in the following.
5.2.1. Shape effect on stress
The maximum and minimum hoop stresses of hollow particles with different
In conclusion, the hollow structure electrodes can significantly reduce the stress field of electrode. The hoop stress decreases with the thickness of the shell. Furthermore, because the tensional hoop stress is the driven force of crack growth in the electrode, hollow structure can help to lower the failure possibility of lithium ion batteries.
5.2.2. Shape effect in efficiency
At the end of galvanostatic process, there is some residual capacity. The residual lithium will be in waste without the potentiostatic operation. To characterize the effective capacity of the battery, a variable
where Capt and Capre are total and residual capacities. Please note that η ≤ 1 since Capre ≥ 0. By performing the potentiostatic charging, the battery gets fully charged/discharged, i.e., Capre can reduce to 0 and η will reach 1.
For this special electrode, the total capacity is
where Q is the theoretical capacity of the electrode material. The residual capacity of the electrode particle is
LixMn2O4 is taken as an example to analyze the efficiency of the electrodes, whose material properties are provided in . Variation of η with ζ is illustrated (Figure 9). One can observe that η increases with ζ, and approaches 1 when ζ is close to 1. This implies that the effective capacity can be high enough without potentiostatic charging as long as the shell is thin enough.
The particle sizes and charging densities have significant influence on the stress profile of electrode particles. In what follows, their effects on the efficiency are investigated. Figure 9a illustrates the dependence of efficiency on charging currency density when the electrode particles have the same size r2 = 20 μm. Observations show that η becomes higher when the electrode is charged under lower currency density. The efficiency η is above 96% regardless of ζ , when in = 5 A/m2. The efficiencies of different-sized electrode particles are compared in Figure 9b. The particles are all charged under the same currency density in = 20 A/m2 . One can observe that smaller electrode particles have higher efficiency. When the electrode particle is small (i.e. r2 = 5 μm), the efficiency is above 96%.
In summary, the efficiency of electrode particle depends on particle size, shell thickness, and charging currency density. High effective capacity can be obtained by using thin shell hollow particle, small electrode particle, and low currency density.
The size and shape effect of hollow spherical particles are systematically investigated above. The size effect is observed for the hydrostatic stress, surface stress, and interface stress, which implies that hydrostatic stress is important in predicting the stress of micro particles while surface and interface stresses are essential in computing the stress fields of nanoparticles. The advantages of hollow electrode particles are highlight in significantly reducing the stress and increasing the efficiency. In summary, hollow structure particle can significantly improve the mechanical and electrochemical properties and is a good candidate for the optimal electrode particle.
This chapter reviews the models on stress analysis of electrode particles in lithium-ion batteries, and then provides an electrochemo-mechanical framework to model the concentration and stress of an electrode with arbitrary geometry. The hydrostatic stress and surface/interface stress are considered. The interactions between phase transformation and stress profile are investigated. The equations are then reformulated for the specific hollow spherical electrode particles. Conclusions are summarized as follows:
The interaction between stress fields and phase transformation is fully addressed. Under the effect of hydrostatic stress, the phase transformation process is much faster, which implies the time of phase transformation is overestimated in previous publications. Due to the existence of phase interface, the stress becomes discontinuous at the interface. The stresses filed of the whole electrode is much larger in the biphase stage.
The size and shape effects for hollow spherical particle are investigated. Through a cross-scale analysis, we conclude that hydrostatic stress, surface stress and interface stress can all induce size effect. Hydrostatic stress is important in predicting the stress of micro particles while surface and interface stresses are essential in computing the stress fields of nanoparticles.
The shape effect of the hollow electrode particle is analyzed, which highlights its structural advantages. One can find that the stress field highly depends on ζ (=r1/r2). The maximum hoop stress decreases with ζ. When ζ is close to 1, the stress is approaching zero. A variable η, to characterize the efficiency (ratio of effective capacity over total capacity) of battery is defined. η also decreases with ζ. When the thickness of hollow electrode particle is small enough, η is close to 1. Therefore, the hollow particle is a good candidate for the optimal electrode particle.
This chapter presents an electrochemo-mechanical framework to accurate predict the stress profile in the electrode particles with different sizes and geometries. The proposed framework considers different effects simultaneously, and quantitatively compares their contributions. This chapter also proposes an optimal electrode particle, i.e., the hollow spherical particle, and systematically analyzes its size and shape effects. Therefore, the present chapter is helpful for the material and structure design of electrode.
This work is sponsored by the Alexander von Humboldt (AvH) foundation through project ‘Mechanics theory of materials with complex surfaces and its applications’; Huiling Duan appreciates the support of following agencies: Major State Basic Research Development Programme of China, National Natural Science Foundation of China.
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