Numerical computation of central crack growth in an active particle of electrodes influenced by multiple factors

Yuwei Zhang·Zhansheng Guo,2

Abstract Mechanical degradation,especially fractures in active particles in an electrode,is a major reason why the capacity of lithiumion batteries fades.This paper proposes a model that couples Li-ion diffusion,stress evolution,and damage mechanics to simulate the growth of central cracks in cathode particles(LiMn2O4)by an extended finite element method by considering the influence of multiple factors.The simulation shows that particles are likely to crack at a high discharge rate,when the particle radius is large,or when the initial central crack is longer.It also shows that the maximum principal tensile stress decreases and cracking becomes more difficult when the influence of crack surface diffusion is considered.The fracturing process occurs according to the following stages:no crack grow th,stable crack grow th,and unstable crack grow th.Changing the charge/discharge strategy before unstable crack growth sets in is beneficial to prevent further capacity fading during electrochemical cycling.

Keywords Li-ion battery ·Active particle of electrodes·Central crack and growth ·Extended finite element method ·Crack surface diffusion

1 Introduction

Li-ion batteries(LIBs)play an essential role in the battery market of portable electronics and electric/hybrid electric vehicles due to their high energy densities,small size,and long service life[1–4].However,one of the main problems of current commercial LIBs is capacity fading along with an increase in the cycle time.The degradation mechanisms of LIBs can be either electrochemical[1–3,5]or mechanical[1–3,6],which are the most important and popular mechanisms,and the mechanism manifests itself by capacity loss,power fading,or both[7].Dendrite grow th,which is not discussed in this paper,represents another important degradation mechanism in LIBs[1,2,8,9].Mechanical degradation mechanisms are mainly related to the volume changes and subsequent stress generated in the active material particles of the anode or cathode within lithium de/-intercalation[7].As a consequence of tensile stress,active particles may develop cracks,loss of contact between each other or from the current collector,and isolation[10].Fractures among active particles increase the resistance to electron transport and therefore reduce lithium intercalation or deintercalation[11].Additionally,fractures also generate new free surface areas,which,when they come into contact with the electrolyte,cause new side reactions[12]and a new solid electrolyte interface(SEI)layer is generated[13].These phenomena cause further electrochemical degradation.

Early researchers discussed the evolution of mechanical stress and the Li-ion concentration.For example,Lee et al.[14]investigated the stress evolution within thin plates,hollow cylinders,and composites.Then,Christensen and Newman[15,16]developed a detailed mathematical model for diffusion-induced stress in intercalated active particles and applied a tensile stress failure criterion to predict particle cracking.Furthermore,Zhang et.al[17]studied the interplay between mechanical stress and Li-ion diffusion in active particles.The work of Cheng and Verbrugge[18]studied the evolution of strain energy and stress within host sphere particles of an electrode under potentiostatic and galvanostatic operation.Also,Korsunsky et al.[19]devel-oped explicit formulae for internal stress and concentration within active sphere particles of the cathode during charging and discharging.The study by Zhang et al.[20]developed analytical diffusion-induced stresses for transverse isotropic cylindrical electrodes during intercalation process. The effect of chemo-mechanical coupling,surface reaction and particle size on stress evolution and phase interface mitigation in an elasto-plastic silicon particle with complex geometry was investigated by Lu and Ni[21]by using a phase- filed model.The Li-ion distribution and diffusion-induced stress in porous electrodes was analyzed by Ji and Guo[22].Many researcher shave been investigating the fracture phenomenon and diffusion-induced damage in recent years.For example,Deshpande et al.[13]explained the number of battery cycles with coupled chemical degradation based on Paris’s low formulation of mechanical fatigue.The effect of the aspect ratio and the position of the initial defect on crack propagation within LiMn2O4elliptical particles during charging and discharging was explored by Zhu et al.[23]by using the extended finite element method(X-FEM).The effect of pressure-gradient dependent diffusion on the distribution of the Li-ion concentration and crack propagation was investigated Grantab and Shenoy[24].Crack growth within active particles in the cathode during the intercalation and de-intercalation processes was simulated by Klinsmann et al.[25,26]by using a phase field model.The effect of the material properties,particle size,and discharge rate on fractures of active particles were studied by Zhao et al.[27]by calculating the energy release rate.Then,Woodford et al.[28,29]proposed an “electrochemical shock”model to calculate the critical cracking conditions from the initial surface semicircular defect and proposed a design criterion for electrochemical shock in Li-ion battery electrodes.The mechano-diffusional driving force behind fractures was analyzed by Gao and Zhou[30]by using a finite element method to calculate the J-integral.Hierarchical crack distribution patterns in their thermal-shock experiment by analogizing the thermal shock and electro chemistry shock was found by Lei et al.[31].

Considerable amounts of research involving experimental observations of particle fractures for LiCoO2[19,32],LiMn2O4[33,34],LiFePO4[35,36],silicon[37],and graphite[38–40]after certain electrochemical cycles,have been reported.To this end,Ebner et al.[41]used X-ray tomography to find many cracked particles of lithium nickel manganese cobalt oxide before electrochemical cycles.Therefore,it can be concluded that the fracturing of active particles,which is a type of mechanical degradation,should receive more attention.Both central[19,38]and surface[37,42,43]cracks are found in micron-sized active particles.However,Harris et al.[44]emphasized that electrode active particles tend to experience internal cracking rather than surface cracking.Thus,in this paper,central cracking rather than surface cracking in active particles is considered.

The present work entailed an extensive study of crack propagation during Li-ion intercalation in active particles.We consider the combined effect of Li-ion diffusion,mechanical stress,and damage mechanics to solve the fracture problem induced by diffusion.The effect of the particle size,C-rate,and initial central crack length on fracture behavior is studied by X-FEM combined with ABAQUS 6.16 software.Our study also took into account the diffusion of Li-ions from the crack surface,a phenomenon that was disregarded in all published papers.Thus,the influence of crack surface diffusion(CSD)on the Li-ion concentration,stress,and fracture behavior of particles is also discussed in this paper.LiMn2O4particles were chosen because they are widely used in commercial LIBs and because the properties of this material are known[15,17,25,45].The particle size,C-rate,initial central crack length,and CSD have a significant influence on the fracture behavior of LiMn2O4.Three crack growth stages are identified and discussed.Our results showed that it is necessary to change the charge/discharge strategy to reduce the degradation of battery performance by avoiding unstable crack growth in the particle.

2 Theory

2.1 Concentration and diffusion induced stress

Figure 1 depicts a LiMn2O4particle with a central crack of length 2a.Figure 1a,b illustrates situations in which CSD is either disregarded or taken into account,respectively.The crack direction is along the initial crack direction because the geometry and diffusion boundary are symmetrical along the central horizontal axis,so the lithium ion concentration field and stress field is also symmetrical along the central horizontal axis.During the intercalation process,lithium ions diffuse only from the surface of the electrode particles to the center along the radial direction(Fig.1a).However,if CSD is considered,Li-ions diffuse from both the surface and from the crack surface as shown in Fig.1b.In fact,when the crack propagates,Li-ions intercalate from the surface of the initial central crack,as well as from the newly generated crack surface.Li-ion concentration distribution will influence crack growth and crack growth will influence Liion concentration redistribution.To focus on the influence of r,C-rate,ā,and CSD on fracture behavior,we ignored the influence of a newly generated crack on the Li-ion concentration,stress,and fracture behavior.For simplification,the impact of the Li-ion diffusion process of the lithium concentration on the material properties is ignored.Therefore,the elastic modulus E,Poisson’s ratio ν,and diffusion coefficient D of LiMn2O4all remain constant throughout the electro-chemical cycle.Furthermore,Mn2O4is also assumed to be a homogeneous linear elastomeric material[46].

Fig.1 Geometry for Li-ion intercalation with a central crack:a excluding CSD,b including CSD

The diffusion of Li-ions in the electrode particles is expressed as follows[24]:

where c represents the concentration and J represents the diffusion flux.As stated by Fick’s law,the diffusion flux is proportional to the concentration gradient as follows[24]:

Combining Eqs.(1)and(2),we have:

The initial boundary condition is c=c0,and the constant current boundary condition is:

where inis the current density and F is Faraday’s constant.The relationship between flux J and the C-rate is[25]:

where V is the volume of the particle(are a under two dimensional(2D)condition),and S is the surface area of the particle(perimeter under 2D condition).Further,C denotes the C-rate,i.e.,the theoretically required time in hours until the battery is either fully charged or discharged and cmaxis the saturated concentration of Li-ions in the host particle.

The stress–strain relationship induced by the concentration gradient is given by:

2.2 Cohesive damage model in X-FEM

Since the governing equations of the Li-ion diffusion are similar to that of heat transfer,2D plane strain problem is solved by the coupled thermal stress modules of ABAQUS in this work.Thus,the transient heat transfer process is used to analogize the process of Li-ion diffusion.The fracture behavior is analyzed by X-FEM,which describes the crack behavior by adding an additional function to the shape function.In X-FEM,the crack surface element uses the Heaviside function as the extended shape function to describe the discontinuous behavior.Figure 2 describes the linear softening cohesive damage traction-separation model,which is used to describe the damage behavior of crack tipelements.The maximum principal tensile stress criterion is used to determine the moment at which damage starts.When the maximum principal tensile stress of an element satisfies the following Eq.(7),the element begins to enter the damage softening stage[47]:

Fig.2 Linear softening cohesive damage traction-separation model

where d is the damage variable,which ranges from 0(for the undamaged stage,the maximum principal tensile stress σ reaches the maximum allowable principal tensile stress σi)to 1(for a completely damaged stage,the separation δ reaches the maximum value δf).It should be pointed out that cracks do not propagate from the tip along the initial crack direction until damage variable d increases to 1.

3 Results and discussion

Different particle radii(r=5,10,and 15μm),initial central crack lengths(a/r= 0.02,0.05,0.10,and 0.20),C-rates(0.5C,1C,3C,and 5C),and CSD were studied to explore their effects on the fracture behavior of particles.In the following analysis,the crack length,intercalation time,and horizontal coordinate along the crack direction are normalized as,and=x/R.The material properties of Mn2O4are summarized in Table 1.

Table1 Material properties of Mn2O4

3.1 Concentration and stress analysis

Figure 3a,b shows the results of the Li-ion concentration under=0.127 with or without the influence of CSD,and Fig.3cand d show the results when the normalized intercalation time equals 0.849.It can be clearly seen that the surface region of the particle exhibited high Li-ion concentrations,whereas the central region exhibited lower Li-ion concentrations.A comparison of Fig.3a,c or b,d shows that the Li-ion concentration increases whenincreases from 0.127 to 0.849.A comparison of Fig.3a,b or c,d reveals that the minimum Li-ion concentration no longer occurs at the center point of the particle with CSD,because Li-ions not only diffuse from the surface to the center of a particle but also from the surface of the central crack to the surface of the particle.The Li-ion concentration is redistributed based on the CSD introduced result when the Li-ion concentration gradient decreases.

Figure 4 shows the Li-ion concentration profile at=0.127 with or without CSD considered.The black solid and dashed lines represent the Li-ion concentration profiles when CSD is not considered and for different values ofwhen CSD is considered,respectively.A comparison of the solid and dashed lines indicates that CSD significantly decreases the Li-ion concentration gradient near the center of the particle,but the effect of CSD is gradually alleviated along the radius direction.A comparison of the results with different values ofshows that the effect of a decreasing Li-ion concentration gradient is more significant whenis larger.The increase of Li-ion concentration near the center of the particle under largeris also found in Fig.4.Whenis larger,the increase of lithium ion concentration is more obvious.A largermeans larger free crack surface;hence,more Li-ions can diffuse from the central crack surface and CSD is more prominent.

Fig.3 Concentration of Li-ion within a particle at C-rate=1 and r=5μm.CSD is included in a and c,excluded from b and d.a and b Results when=0.127,c and d results when=0.849

Fig.4 Li-ion concentration profile of four different values of and no CSD considered(black solid line)at=0.127.Here=0 represents the center of the electrode particle,and=1 represents the surface of the electrode particle along the crack direction.“W”represents with,“O”represents without

Figure 5a,b shows the results of the maximum principal stress distribution under=0.127 with or without the influence of CSD,whereas Fig.5c,d shows the results when the normalized discharging time equals 0.849.In order to explore the influence of CSD on stress,the no crack growth stage is chosen for the stress analysis.The maximum principal tensile stress appears at the front end of the central crack tip and can be expected to decrease the chemical potential and enhance Li-ion diffusion[24].A comparison of Fig.5a,corb,d shows that the maximum principal tensile stress continues increasing during the intercalation process,but it still does not exceed σi(100 MPa).This means that the element in front of the tip of the central crack tip does not enter the softening stage,and the crack does not propagate.If the maximum principal tensile stress in the crack tip element fulfills Eq.(7),the element will enter the damage softening stage,and d begins to increase from zero.When d increases to 1,the crack begins to propagate.A comparison of Fig.5a,b or c,d indicates that the maximum principal tensile stress in front of the crack tip decreases significantly when CSD is considered.Diffusion from the central crack surface can cause stress redistribution,decrease the maximum principal tensile stress of the crack tip,and even change the fracture behavior.Harris et al.[44]also found that the presence of a new surface,as a result of the presence of even a small internal crack or pores,can decease the stress value near the new surface.

Fig.5 Maximum principal stress distribution in an active particle with C-rate=1 and r=5μm.CSD is included in a and c,excluded from b and d.a and b Results when=0.127,c and d results when=0.849

3.2 Fracture behavior analysis

In order to understand fracture behavior during intercalation more clearly,we analyzed the crack development of a particle as a function of time during Li-ion intercalation process when r equals10μm,equals0.1,and the C-rateequals1(shown in Fig.6).The crack behavior during intercalation from the initial to the final state was found to occur in three stages:no crack grow th(stagestable crack grow th(stage),and unstable crack grow th(stage).Sun et al.[48]who studied the cracks induced by the lithiation/de-lithiation cycles of a graphite electrode by carrying out experiments also identified stable crack growth and unstable crack growth stages.The three types of stage is divided by red dash line in Fig.6,and the corresponding transition time are 0.044(stageto stage)and 0.116(stageto stage.It should be emphasized that the transition time from a different stage can be determined for every certain case(C-rate,r,andIn stage,the maximum tensile principal stress σmaxin front of the crack tip is lower than the maximum allowable tensile principal stress σi,the material does not enter the damage stage or enter the damage stage,the damage variable is less than 1,and the crack does not experience grow th.Electrodes can be safely used during this stage.

Fig.6 Crack growth profile during Li-ion intercalation process with r=10μm,=0.1,and C-rate=1 without CSD considered

3.3 Multi-factor influence on fracture behavior

Table 2 presents the results of the different types of fracture behavior of particles with or without CSD considered for r=5,10,and15μm.Region I represents that the crack does not propagate(i.e.,only stageoccurs)during the entire intercalation process.Region II represents that the crack proceeded through stageand is just entering stageat the end of intercalation process.And region III represents that the crack experienced stagesandduring the intercalation process.The optimal combination of the C-rate,particle radius,and initial crack length with or without CSD can be obtained from Table 2.LIBs can be cycled safely in region I(surrounded by the solid line)and can also be cycled around the dashed line between I and II.Region III represents dangerous conditions(surrounded by the dashed line),which is unacceptable in the cycling life of LIBs.Therefore,the condi-tionsin region III or around the dashed line between II and III should be avoided during our daily use of LIBs.This optimal window enables us to determine the appropriate long-term use C-rate without reducing the capacity of LIBs.

Table 2 Fracture behavior distribution of particle under different conditions

3.3.1 Influence of C-rate

As the region in which r=5μm in Table 2 indicates,when=0.05(the fourth column),the crack remains in region I when the C-rate=0.5 and 1.As the C-rate increases to 3,the crack begins to propagate,enters fracture behavior,and remains in region II.When the C-rate continues increasing to 5,crack growth enters the unstable crack growth stage and the fracture behavior of the particle changes from II to III.A high C-rate,which results in a large concentration gradient and produces a large amount of stress within the particle,facilitates cracking and unstable crack grow th.This conclusion is also valid for the other values ofshown in Table 2.

3.3.2 Influence of normalized length of the initial central crack

As the results for r=10μm in Table 2 shown,when the C-rate remains at 0.5,the fracture behavior of the particle is I with=0.02.Asincreases to 0.05 and 0.1,the crack begins to propagate and stays at II.Whencontinues increasing to 0.2,the fracture behavior changes from II to III.A large value ofcauses increasing maximum principle tensile stress in front of the crack tip,facilitating cracking,and unstable crack grow th.

3.3.3 Influence of particle radius

A comparison of r=5,10,and 15μm in Table 2 shows that,for the sameand C-rate,a large particle radius causes cracking and unstable crack growth to occur easily,i.e.,when r=5μm and C-rate=3,the fracture behavior of the particle is II with=0.05,0.1,and 0.2 and I with=0.02.In comparison,when r=10,15μm and C-rate=3,the crack behavior remains in III regardless of the value of.A larger particle radius results in an increase in both the Li-ion concentration gradient and the maximum principal tensile stress such that cracking and the phenomenon of unstable crack growth easily occur.

3.3.4 Influence of CSD

The influence of CSD differs from that of the other factors we discussed above.As the area in which r=5μm in Table 2 shows,the fracture behavior changes from II to I with C-rate=1 and=0.2 and 0.1.It also changes from III to II with C-rate=5 and=0.2.The CSD can reduce the maximum principle tensile stress near the crack tip(shown in Fig.5)by decreasing the Li-ion concentration gradient(shown in Fig.4)within the particle to prevent the crack from easily propagating or entering the unstable growth stage.Comparing the results with and without CSD in Table 2,the results of the fracture behavior are not changed when the value ofis small(such as 0.02 and 0.05),whereas part of the results changed when the value ofis large(such as0.1and0.2).The effects of CSD are more significant asincreases,because largermeans more Li-ions can diffuse from the central crack surface to decrease the concentration gradient(shown in Fig.4),and then influence the fracture behavior results.

In Table 2,there is a very interesting phenomenon when the crack length changes from 0.02 to 0.1 for r=10μm and C-rate=0.5 with CSD considered.The fracture behavior is I→II→I and it is not monotonous with crack length increasing.As initial crack length increases,the maximum principle stress near the crack tip increases according to the relationship between stress and crack length of fracture mechanics.A larger crack length causes a larger stress increase(we call this factor A).Once CSD is introduced,a larger crack length will decrease the Li-ion concentration gradient obviously(see Fig.4)which causes stress to decrease obviously(see Fig.5)(we call this factor B).The role of factors A and B is reverse with crack length change.When I firstly appears=0.02),crack length is small and both factors affect stress only a little.When II firstly appears(=0.05),the effect of factor A plays a leading role.When the crack length continues increasing(=0.1),the effect of factor B will play a key role,which causes I to appear again.When r=10μm and C-rate=0.5 with CSD considered,III changes I when=0.2;the effect of factor B plays a completely dominant role.

The sensitivity of different particle radii to the CSD is different.The difference between the results of r=5μm with and without CSD is dramatic,but the difference of r=10μm is less than that of r=5μm.The results of r=15μm with and without CSD are completely the same.The Li-ion concentration and maximum principal tensile stress in an electrode particle increases sharply as the particle radius increases.As the particle radius is sufficiently large(e.g.,from 10 to 15μm),the particle is more likely to experience cracking or to enter the unstable crack growth stage at a low C-rate and when the value ofā is small.The radius has a larger influence on the fracture behavior than the CSD when the particle radius is larger.In this case,the radius becomes the new dominant factor affecting the fracture behavior of a particle rather than the CSD.

Fig.7 The changes of normalized crack length over normalized time during Li-ion intercalation process in an active particle with r=5μm and C-rate=5

Figure7 shows the changes in the normalized crack length over time with different initial crack lengths.To focus on the influence of CSD and initial crack length on fracture behaviors,the stage transition time is not discussed in that figure,but is shown in Fig.6.This enables the effect of the CSD on the time required for stable or unstable crack growth to be explored.The dashed and solid lines represent the results with and without considering CSD, respectively. A comparison of the dashed and solid lines in all four colors indicates that both types of normalized transition time(stageto stageand stageto stageare delayed to a different degree.A larger value ofincreases the delay time,but does not change the three-stage fracture behavior.Particularly,stagedoes not appear during the entire intercalation process when considering CSD atwhereas it appears atand 0.1.This is becausecan increase to approximate 1 in a very short time in stage,which is unacceptable for LIBs.Hence,CSD is the major factor affecting the time at which stageis entered.The results can guide the charge and discharge strategy in a battery management system to prevent the generation of long cracks inside the particle.Reducing the charging or discharging rate should be applied before the time of transition from stageto stage

4 Conclusions

A model that couples Li-ion diffusion,mechanical stress,and damage mechanics is presented to simulate the central crack growth of active particles by using X-FEM and by taking into account the influence of multiple factors.CSD was shown to significantly reduce the Li-ion concentration gradient near the center of the particle,but to alleviate gradually along the radius direction.The effect of decreasing the Li-ion concentration gradient is more significant when the initial crack is longer.CSD can also reduce the maximum principle tensile stress near the crack tip.Crack growth was found to occur in a three-stage process(no crack grow th,stable crack grow th,and unstable crack grow th)during the Li-ion intercalation process.The unstable crack growth stage is unacceptable when a long cycling lifetime is required for LIBs.The optimal combination of the C-rate,particle radius,and initial crack length with or without CSD was obtained.Without considering CSD,a smaller radius,lower C-rate,and shorter initial central crack length can prevent cracking or the unstable crack growth stage from being entered.With CSD considered,the result of fracture behavior is safer than that without CSD.In addition,it can delay the transition time within the three-stage process.In this respect,larger values ofā increase the delay time,but do not change the three-stage fracture behavior.

Our results can be used in a battery management system to avoid unstable crack growth leading to battery capacity degradation.Reducing the charging/discharging rate or changing to potentiostatic charging before the unstable crack growth stage is entered is suggested as a good way to realize charge/discharge strategy optimization.CSD plays an important role in particle fracture analysis.Thus,guidelines need to be accompanied by the statistical analysis of the initial flaw sizes in the active material.

AcknowledgementsAuthors gratefully acknowledge the financial support of the National Natural Science Foundation of China(11472165 and 11332005).