Mechanical contact in composite electrodes of lithium-ion batteries

Journal of Power Sources 440 (2019) 227115

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Mechanical contact in composite electrodes of lithium-ion batteries Bo Lu a, b, c, *, Yanfei Zhao d, **, Jiemin Feng a, c, Yicheng Song a, c, Junqian Zhang a, c a

Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai, 200444, China Engineering Research Center of Nano-Geo Materials of Ministry of Education, China University of Geosciences, Wuhan, 430074, China c Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai, 200072, China d Department of Civil Engineering, Shanghai University, Shanghai, 200444, China b

H I G H L I G H T S

� Analytical solutions of PP and PC contact are established. � The contact stress can be higher than the diffusion-induced stress. � The stress within the active particle is mainly induced by mechanical contacts. A R T I C L E I N F O

A B S T R A C T

Keywords: Mechanical contact Active particle Current collector Contact stress Lithium ion battery

An analytical model of mechanical contact problems in composite electrodes of lithium-ion batteries is developed in this article. Two typical types of mechanical contact, namely contact between particles and contact between particle and current collector, are investigated. Key parameters that affect the contact problem are identified from the analytical solution. High uniformity of the particle size is found to be critical to the electrode. Furthermore, a soft current collector could significantly reduce the contact stress and hence is also suggested. It is figured out that contact stress is comparable to or even higher than the diffusion-induced stress under freeexpansion state, even if the mechanical constraint in electrodes is weak. To highlight the significance of con­ tact stress, an electrochemical cycling verification experiment which involves a charging pause is conducted. Both analytical and experimental results indicate that the mechanical contact plays a crucial role in the evalu­ ation of mechanical stability of lithium-ion battery electrodes.

1. Introduction Stresses and subsequent mechanical failures exert considerable ef­ fects on electrochemical performances of lithium-ion batteries (LIBs). The mechanical-electrochemical coupling behaviors in LIBs, such as stress effect on electron and mass transfer [1–4], on voltage profile [5], on capacity [6] and on life-time [7–12], have been widely investigated. However, most of these analyses propose a single ideal active particle [1, 8,13] or an ideally homogenized layered electrode [14–16]. The complicated interaction between different components within the composite electrode has not been fully revealed yet. Active particles composed of active materials are main basic com­ ponents of commercial composite electrodes which are commonly calendered to improve the energy density. The scanning electron

microscope (SEM) images reported in Refs. [17–19] showed the com­ posite electrodes present compact porous structures, in which contact between active particles is unavoidable. The active particles will swell upon lithiation, leading to contact stress between each other. Ex/in situ TEM studies made by Lee et al. also revealed the extra contact stress could significantly change the reaction kinetics of lithiated silicon [20]. Furthermore, Luo et al. experimentally proved the existence of lithium flux between contacted neighboring particles [21]. These experimental results indicate the mechanical-electrochemical coupling interaction between active particles should receive intense attention. Obviously, such interaction cannot be simply described by a single particle model [1,8,13] or a homogenized electrode model [14–16] which are adopted in most coupling models of composite electrodes for the sake of simplification [22,23].

* Corresponding author. Shanghai Institute of Applied Mathematics and Mechanics, School of Mechanics and Engineering Science, Shanghai University, Shanghai, 200444, China. ** Corresponding author. E-mail addresses: [email protected] (B. Lu), [email protected] (Y. Zhao). https://doi.org/10.1016/j.jpowsour.2019.227115 Received 15 April 2019; Received in revised form 27 August 2019; Accepted 4 September 2019 Available online 13 September 2019 0378-7753/© 2019 Elsevier B.V. All rights reserved.

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Journal of Power Sources 440 (2019) 227115

Numerical analyses of the mechanical contact between active par­ ticles have been reported in some studies, but the contact problem is still not comprehensively understood. Rahani et al. investigated the stress profile in porous electrodes for two different microstructures: active particles connected with binder bridges and active particles encased in binder shells [24]. Wang et al. demonstrated inelastic shape changes of Si particles caused by mechanical contact and traced stress evolution at binder/particle interface [25]. It is noted that direct contact between active particles was not involved in both studies. However, Kumar et al. suggested that particle-to-particle contact would occur when the mass ratio of the active material in the active layer is higher than 60% which is already quite low for most battery systems [26]. Therefore, occurrence of direct contact between active particles should be expected during charge-discharge cycling of LIBs. The simulation of 3D reconstructed electrodes further confirmed the existence of particle-to-particle con­ tacts and found that the extremely high stress could be induced by the particle-to-particle contact [27]. Besides contact between active particles, contact between active and inactive components such as that between particles and the current collector is also crucial. On one hand, contact areas provide electronic path which relates closely to the electrical resistance in LIBs [3]. For instance, by applying extra pressure on the electrode, mechanical con­ tact is enhanced and the electrical conductivity is improved. On the other hand, mechanical contact between the active particle and the current collector, which is similar to indentation, is possibly detrimental to the integrity of the current collector and may further lead to the in­ ternal short circuit within the LIBs [28,29]. A similar contact problem in the all-solid-state LIBs, namely the contact between the active material and the solid electrolyte, was addressed by Tian et al. recently [30]. Nevertheless, contact between the active and inactive components is still lack of sufficient investigation. In the present study, an analytical model will be provided to discuss mechanical contact problems in composite electrodes for LIBs. Particleto-particle (PP contact) and particle-to-current-collector contact (PC contact) which are two typical types of contact in the electrodes will be investigated. Key parameters will be indentified and their effects on mechanical performances will be evaluated. A charge-discharge exper­ iment which includes a charging pause will be conducted to highlight the importance of mechanical contact in electrodes.

electrochemical cycling, it is assumed that the two particles touch each other without contact stress, as shown in Fig. 1a. In other words, the initial state for these two particles is stress-free. Here, the expansion of active particle upon lithiation is focused on. The two active particles start to swell upon lithiation and at a certain time their radii are sup­ 0 0 posed to increase to R 1 and R 2 , respectively. However, the two active particles cannot expand freely due to the existing mechanical constraint in the battery and consequently they squeeze each other. To demon­ strate the role of contact stress, normal contact is focused here, indi­ cating the center of the contact area and the two particle centers invariably lie on a same straight line. The size of the contact area is assumed much smaller than the sizes of two particles. The contact area can be approximately described by a disc with radius a. A constraint factor λ (0 � λ � 1) is introduced here to describe the mechanical constraint. Higher λ represents stronger constraint in the battery. If the mechanical constraint is ignored (λ ¼ 0), the active par­ ticle would expand freely, as demonstrated in Fig. 1b. Displacements of the two particle centers in the free-expansion condition are denoted by δ1 and δ2, respectively. If the electrode is well calendered and the cell is tightly packed, a rigid-constraint condition (λ ¼ 1) is made: the distance between the two particle centers remains the same after deformation, as shown in Fig. 1c. λ is a measurable parameter and can be estimated by measuring the electrode thickness [31–33]. For instance, if the thickness change of the graphite electrode under the free expansion condition (denoted by εf) is approximately 10% after the first lithiation [31] and the thickness change of that in a pouch cell (denoted by εc) is about 5% [33], then the constraint factor of the graphite electrode in the pouch cell can be estimated as λ ¼ 1-εc/εf ¼ 0.5. In coin and cylindrical cells which generally have stronger constraints, λ is certainly higher than 0.5 and possibly close to 1. On one hand, the composite of polymeric binders and conductive additives only covers partial surfaces of the active particle and is commonly extremely soft [17,26,34,35]. On the other hand, direct contact between active particles was revealed in Refs. [26,27]. There­ fore, the influences of binders/conductive additives are negligible and direct PP/PC contact is considered in this study. However, for the case that the deformation constraint from the composite of binders and conductive additives cannot be neglected, the present analysis can also serve as a conservative estimation for mechanical contact in composite electrodes. It is assumed that the particles undergo elastic deformation. There­ fore, the problem of the contact between the two active particles shown in Fig. 1 can be solved by the classic contact solution between two elastic bodies [36,37]. The radius a of the contact area can be read as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R0 1 R0 2 a¼ δ; (1) R0 1 þ R0 2

2. Analysis The spherical geometry of the active particle is widely accepted for modeling the actual active particle in composite electrodes [6,8,13,22]. Here we firstly consider two spherical active particles with initial radii denoted by R1 and R2, respectively. Their Young’s moduli and Poisson’s ratios are respectively denoted by E1, E2, υ1 and υ2. Before

Fig. 1. Illustrations of mechanical contact of two spherical active particles upon lithiation: (a) initial stress-free state, (b) free-expansion condition (λ ¼ 0) and (c) rigid-constraint condition (λ ¼ 1). 2

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Journal of Power Sources 440 (2019) 227115

where csurf is the surface concentration. Interaction of diffusion between the two particles is not considered so that the concentration profile in each particle can be solved individually. By solving equations (5)–(8), we have the evolution of concentration as follows [13,39]: # " ∞ 2 sinðλ r Þ 2 3 1 3 2 X n 1 ; (9a) c1 ¼ IR 2 t þ r1 2 e λn t=R 2 2 10 r1 n¼1 λn sinðλn Þ R

where δ is the relative displacement of two particles. According to the definition of λ, the relationship δ ¼ λðδ1 þ δ2 Þ is established. Large contact areas could improve electronic conductivity in the composite electrode by providing convenient electronic pathway between active particles. The total contact force P applied on particles is given by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 R0 1 R0 2 3 (2) P¼ δ; 3ðk1 þ k2 Þ R0 1 þ R0 2

" 1 c2 ¼ I 3t þ r2 2 2

where k1 ¼ ð1 υ1 2 Þ=E1 , k2 ¼ ð1 υ2 2 Þ=E2 are elastic material con­ stants of the two spheres, respectively. The force P, which describes the resultant of contact stress, can be considered as the total constraint force applied on particles. In battery systems, the constraint force P reflects mechanical loading within composite electrodes. The highest contact stress, namely the characteristic contact stress appears at the center of the contact area and can be expressed by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R0 1 þ R0 2 p0 ¼ δ: (3) πðk1 þ k2 Þ R0 1 R0 2

Ωcmax Ri Q i 3

∞ X

c1 ¼ csurf þ 2csurf

1 ∂2 ½ri ci ðri ; tÞ� ∂ci ðri ; tÞ ¼ ; ri ∂ri 2 ∂t

∞ X

c2 ¼ csurf þ 2csurf

∂ci in ¼ with ri ¼ Ri ∂ri F

Q1 ¼

(9b)

e

ðmπ Þ2 t ð

2

ð

� 1Þm � sin mπr1 ; mπr1

(10a)

� 1Þm � sin mπ r2 mπr2

3 It; R

(10b)

(11a) (11b)

Q2 ¼ 3It for the galvanostatic cases and

(4)

Q1 ¼ csurf

csurf

Q2 ¼ csurf

csurf

∞ 6 X 1 e 2 π m¼1 m2 ∞ 6 X 1 e m2 m¼1

π2

ðmπ Þ2 t=R

ðmπ Þ2 t

2

;

(12a)

(12b)

for the potentiostatic cases. It is noted the interaction of diffusion between particles is not involved for the above two cases. However, for reason of comparison, we consider here an additional ideal case where the interaction of diffusion between the two particles is extremely rapid, which means their normalized capacities should be identical all the time, namely Q1 ¼ Q2 ¼ Q. Besides, in this ideal case, concentrations within the two particles are uniformly distributed. To make this uniform distribution case comparable with the galvanostatic case, the average capacities of these two cases are set identical at any given time t. Then the relation­ ship between the formal current density Iu and capacity Q can be ob­ tained for the uniform cases as: � 2 3 R þ 1 Iu t Q¼ (13) 3 R þ1

(5)

(6)

where Iu is identical to I in the galvanostatic cases, i.e. Iu ¼ I. In gal­ vanostatic cases, charging processes between particles are not syn­ chronous, i.e. Q1 6¼ Q2 , which was reported in some studies [19,21]. Therefore, by comparing the uniform distribution and galvanostatic cases, the effect of charging synchronization between active particles can be revealed. The lithiation process terminates once the capacity of particle 2 reaches the limit, i.e. Q2 ¼ 1. The stress-assisted diffusion is not considered here due to its insig­ nificant effect in graphite and most common cathode active materials [2]. Even if the stress-assisted diffusion is considered, equations (11) and (13) are still available, since capacities in galvanostatic and uniform

(7)

where F ¼ 96485.3C/mol is the Faraday constant and in is the surface current density; the boundary condition for potentiostatic case is ci ¼ csurf with ​ ri ¼ Ri

# sinðλn r2 Þ λn 2 sinðλn Þ

λn 2 t

for potentiostatic cases where ci ¼ ci =cmax , csurf ¼ csurf =cmax , t ¼ Dt=R2 2 , ri ¼ r=Ri , λn are the positive solutions of λn cotλn ¼ 1, I ¼ in R2 =ðFDcmax Þ is the dimensionless current density and R ¼ R1 =R2 is the particle size ratio. In below, only the cases of R � 1 is discussed. The normalized R1 capacity Qi ¼ 3 0 ci r2 dr reads as follows:

is assumed. Galvanostatic and potentiostatic operations will be dis­ cussed in this study. The boundary condition for galvanostatic case is D

ðmπ Þ2 t=R

m¼1

where c is the concentration of lithium, D is the diffusion coefficient. The initial condition of the concentration profile ci ðri ; 0Þ ¼ 0

e m¼1

where Ω is the partial molar volume of the active material, cmax is the RR saturation concentration of lithium and Qi ¼ ð3=cmax Ri 3 Þ 0 i ci r2 dr is the normalized amount of lithium or the normalized capacity of the active particle. Interestingly, equation (4) indicates that contact of the two active particles only depends on the amount of lithium rather than its detailed distribution. Evolution of the normalized capacity can be determined by solving the diffusion problem. The spherically symmetric diffusion within the active particle is assumed to be governed by Fick’s laws [40]: D

∞ 2 X e r2 n¼1

for galvanostatic cases and

High contact stress would certainly risk the integrity of active particles. Contact size a, contact force P and characteristic contact stress p0 are three key factors which characterize the contact problem and are commonly focused on [36,37]. It is noted that only p0 and a are not enough to fully characterize a contact problem. Comparing to p0 which only stands for the stress at the center of the contact area, P demon­ strates a resultant mechanical response at the particle level and could be useful for multi-scale analyses [38]. According to Fig. 1, the geometric 0 relationship R i ¼ Ri þ δi exists where i(¼1,2) represents different par­ ticles. In this case, the contact problem described by equations (1)–(3) can be solved once δ1 and δ2 are known. The impact of contact on spherically symmetric diffusion of lithium within spherical active particles is neglected, owing to the very small contact area. Thus, the displacements δi which are related to the con­ centration profile has the expression [39]: δi ¼

3 10

(8)

3

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Journal of Power Sources 440 (2019) 227115

distribution cases only depend on boundary fluxes. By taking the above equations, the PP contact problem can be solved analytically, which will be demonstrated in Section 3. Furthermore, by letting R1 →∞ and δ1 ¼ 0, the presented analysis can be extended to analyze the PC contact which will be discussed in Section 4. In addition, the calendering force and the corresponding mechanical behavior of active particle [41,42] can also be estimated, by considering constant radii and an applied displacement. Since the calendering is a pure me­ chanical problem without the volume change of active particle, it can be directly solved by equations (1)–(3), and hence a detailed discussion will not be included in this work.

force and contact stress depend upon four dimensionless parameters, namely, λ, Ωcmax , R and I. λ describes the mechanical constraint, Ωcmax represents the maximum volume change of the active material, R gives the uniformity of active particle sizes, the loading factor I stands for the charging speed of the battery. According to the expression of τ ¼ Ωcmax It, it can be seen that Ωcmax and I have identical effects on the evolution of a, P and p0 in the galvanostatic and the uniform cases. For the potentiostatic case, there are only three key parameters, i.e. λ, Ωcmax and R. The evolutions of the capacity Q, the contact size a, the contact force P and the characteristic contact stress p0 for the galvanostatic, the potentiostatic and the uniform cases are demonstrated in Fig. 2. Regardless of specific cases, a, P and p0 always increase monotonically as lithiation proceeds. The increase in a provides a possible explanation why the resistance of battery decreases directly during the charging process, since the contact area a is related to the electronic conductivity as previously mentioned [43]. The increases in P and p0 indicate the raising risk of mechanical failures of electrodes as well as active particles. The average capacities of the uniform case and the galvanostatic case are identical at any given time, but it can be seen from Fig. 2a that the two particles’ capacities increase inconsistently in the galvanostatic case. The inconsistent capacity growing leads consequently to slightly larger contact sizes, slightly higher contact stresses as well as evidently higher contact forces, as shown in Fig. 2b–d. This indicates the effect exerted by charging synchronization of active particles needs to be considered in the mechanical analysis of LIBs. Low synchronization risks the mechanical stability of electrodes. Furthermore, as shown in Fig. 2a, a non-synchronous lithiation causes a premature quit of charging and a consequent loss of capacity utilization in the particle of a bigger size. It is noted that by making R ¼ 1, equation (17) is identical exactly to equation (14), which indicates a relatively high uniformity of particle sizes can avoid problems caused by the non-synchronous lithiation. The mechanical contact process under potentiostatic operation

3. Particle-to-particle contact (PP contact) Firstly, the PP contact in which two active particles share the same material properties will be discussed. It is noted that υ ¼ υ1 ¼ υ2 and E ¼ E1 ¼ E2 lead to k ¼ k1 ¼ k2 . Substituting the expression of the normalized capacity in equations (11)–(13) into equations (1)–(3), the contact size a, the contact force P and the characteristic contact stress p0 can be obtained as follows. For the galvanostatic cases, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2λðR þ τÞð1 þ τÞ (14a) τ; a¼ 1 þ R þ 2τ 4 P¼ 3

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2λ3 ðR þ τÞð1 þ τÞ 3 τ; 1 þ R þ 2τ

(14b)

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2λð1 þ R þ 2τÞ p0 ¼ τ π ðR þ τÞð1 þ τÞ 1

(14c)

where τ ¼ Ωcmax It, a ¼ a=R2 , P ¼ Pð1 υ2 Þ=E. For the potentiostatic cases, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ η0 Þð1 þ ηÞðRη0 þ ηÞ ; a ¼ λR 1 þ R þ Rη 0 þ η

υ2 Þ=ðER2 2 Þ and p0 ¼ p0 ð1

(15a)

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ η0 Þð1 þ ηÞðRη0 þ ηÞ3 λ3 R ; 1 þ R þ Rη 0 þ η

(15b)

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ R þ Rη0 þ ηÞðRη0 þ ηÞ λ p0 ¼ π Rð1 þ η0 Þð1 þ ηÞ

(15c)

2 P¼ 3

1

where

ηðtÞ ¼

" Ωcmax 1 3

∞ 6 X 1 e 2 π m¼1 m2

# ðmπ Þ2 t

� � t ; η’ ¼ η 2 R

and here csurf ¼ 1 is assumed. For the uniform cases, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2� 2 � 1þR 1þR a ¼ λR τ τ; 1þ 3 3 1þR 1þR 2 P¼ 3

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � 2 �� 2 �3 1þR 1þR λ3 Rð1 þ RÞ2 1 þ τ τ ; 3 3 1þR 1þR

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2� λð1 þ RÞ2 1 þ R τ 1u p0 ¼ t � � �: π R 1 þ R3 þ 1 þ R2 τ

(16ab)

(17a)

(17b)

(17c)

It is noted that equations (14a,b,c) are identical, respectively, to equa­ tions (17a,b,c) when R ¼ 1. From the analytical solutions presented in equations (14)–(17), key influence parameters which dominate the PP contact can be extracted. For the galvanostatic and the uniform cases, the contact size, contact

Fig. 2. Evolution of (a) capacity Q, (b) contact size a, (c) contact force P and (d) characteristic contact stress p0 with respect to lithiation time t in PP contact. The dot lines represent the maximum values, i.e. amax , Pmax and p0;max , respectively. 4

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Journal of Power Sources 440 (2019) 227115

evolves rapidly at first and then becomes steady. The dot lines in Fig. 2b–d represent the steady values of a, P and p0 , which are denoted by amax , Pmax and p0;max , respectively. The peak values of a, P and p0 in

σ d;max ¼

the uniform case are found to be identical to amax , Pmax and p0;max , respectively. This is because as t→∞, capacities of both particles approach to 1, which becomes exactly the same condition as the contact process in the uniform case ends. Actually, amax , Pmax and p0;max are the theoretical maximum values and can be given according to equations (15) and (16) by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 λRΩcmax ð3 þ Ωcmax Þ; amax ¼ (18a) 3 Pmax ¼

p0;max

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð1 þ RÞ λ3 RðΩcmax Þ3 ð3 þ Ωcmax Þ; 27

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λΩcmax ¼ ð1 þ RÞ : π Rð3 þ Ωcmax Þ 1

EΩcmax : 3ð1 υÞ

(19)

Its dimensionless version pd;max ¼ σd;max ð1 υ2 Þ=E ¼ ð1 þ υÞΩcmax =3 is plotted in Fig. 3a where υ ¼ 0:3 is chosen. The maximum contact stress p0;max is found to be higher than the highest diffusion-induced stress pd;max . For instance, the ratio p0;max =pd;max is as high as 8.65 when Ωcmax ¼ 0:01. Even for Ωcmax ¼ 0:1 which is typical for graphite, p0;max is still more than two times the value of pd;max . In other words, the contact stress is at least comparable to and could be even much higher than the diffusion-induced stress, which emphasizes the contact stress plays an even more important role than the diffusion induced stress does in mechanical analyses of composite electrodes. This will be further demonstrated by experiments in Section 5. Fig. 3b illustrates the effects of particle size ratio R on amax , Pmax and p0;max . Small R implies the active particles are similar in size. On the

(18b) (18c)

contrary, large R means low uniformity of particle sizes. Both Pmax and p0;max increase monotonically with an increase in R, indicating that the large difference of particle sizes will be detrimental to mechanical sta­ bility of battery. It can be seen that the maximum contact force Pmax varies dramatically. When R increases from 1 to 10, Pmax rises by more than sixteen times. This indicates that the slightly undermining unifor­ mity of particle sizes will lead to a significant rise of mechanical loading within electrodes and consequently a great risk of battery safety. Hence, a high uniformity of particle size is suggested for material design. However, the maximum contact stress p0;max is nearly insensitive to the

It is noted that amax , Pmax and p0;max are determined by λ, Ωcmax and R. Larger maximum volume change will lead to larger deformation and severer contact. This is demonstrated by Fig. 3a in which the maximum contact size amax , the maximum contact force Pmax and the maximum contact stress p0;max all increase monotonically with increasing Ωcmax . For the sake of comparison, the free expansion state (Fig. 1b) is recalled and the diffusion-induced stress under this state will be compared with the contact stress. The highest diffusion-induced stress during a lithiation process under the free-expansion state can be esti­ mated by Refs. [13,39].

variation of R. When R increases from 1 to 10, p0;max only increases by less than 74%. The highest possible diffusion-induced stress pd;max is also presented

Fig. 3. Impacts of (a) maximum volume change Ωcmax , (b) particle size ratio R and (c) constraint factor λ on maximum contact size amax , maximum contact force Pmax and maximum contact stress p0;max in PP contact. The dash lines represent the highest possible diffusion induced stress pd;max in free expansion state. 5

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Journal of Power Sources 440 (2019) 227115

by the dash line in Fig. 3b. According to equation (19), pd;max is inde­

softer current collector or a harder active material. Since there is only one particle involved in this section and equations (20)–(22) only depend on Q ¼ Q2 rather than the detailed distribution of concentration, there is no necessity to discuss impacts of different operations. The evolution of contact size a, contact force P and characteristic contact stress p0 with respect to the normalized capacity Q is demon­ strated in Fig. 4. a, P and p0 all increase monotonically with increasing Q and reach their maximum values respectively when the particle is fully lithiated (Q ¼ 1). Similarly, we denote amax , Pmax and p0;max as the

pendent of R. Being consistent with the result of Fig. 3a, p0;max is much

higher than pd;max and the difference between them increases as R in­ creases. This indicates the contact stress should be particularly consid­ ered when the difference of particle sizes is large. The effects of the constraint factor λ on PP contact are demonstrated in Fig. 3c. The contact stress p0;max is higher than the diffusion-induced stress pd;max even for a very weak mechanical constraint, e.g. λ ¼ 0:1. For the case of graphite electrode in the pouch cell, namely λ ¼ 0:5 [31, 33], the value of p0;max is about one time higher than that of pd;max . Therefore, the contact stress should not be neglected even for electrodes with weak mechanical constraints. In addition, the weak constraint certainly reduces the contact force and the contact stress, but at the same time the contact size shrunk significantly, potentially leading to con­ ductivity loss [26,30]. It is noted that the constraint factor here is treated as an independent parameter, though λ may be related to the state of charge in actual LIBs. Spherical particles are mainly focused on in this work. Stress con­ centration occurred in non-spherical contact could lead to even much higher contact stress. Hence, in analysis of reconstructed composite electrodes [27], contact stress should be given enough attention for evaluation of active particle integrity and electrode stability.

maximum values of a, P and p0 , respectively. By letting Q ¼ 1, equations (20)–(22) reduce to

P¼

4 1 27 k þ 1

(21)

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λΩcmax Q ; 3 þ Ωcmax Q

(22)

2 p0 ¼ πðk þ 1Þ

4 1 27 k þ 1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ3 ð3 þ Ωcmax ÞðΩcmax Þ3 ;

2 πðk þ 1Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λΩcmax : 3 þ Ωcmax

(24) (25)

expansion Ωcmax and the ratio of elastic constants k. If p0;max exceeds the yield strength of a current collector, a local plastic deformation starting from the contact part would become inevitable and subsequent damages such as fracture would follow. Thus, equation (25) provides a design principle in terms of the above three parameters for avoid mechanical failures of the current collector. The impacts of Ωcmax on amax , Pmax and p0;max are illustrated in Fig. 5a. Similar to the PP contact, a larger maximum volume change Ωcmax of active materials leads to a higher contact force Pmax and a higher contact stress p0;max . This indicates that local mechanical dam­ ages of current collectors are likely to occur when the volume change of active materials is large. The highest possible diffusion-induced stress pd;max in the free expansion state is also plotted in Fig. 5a. In the case of PC contact, p0;max is still higher than pd;max . Therefore, again, the significance of mechan­ ical contacts in composite electrodes is emphasized. Moreover, it is also suggested that stress localization may contribute greatly to interfacial damages in composite electrodes. But this issue has not been considered yet in recent related studies [44,45].

(20)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ3 ð3 þ Ωcmax QÞðΩcmax QÞ3 ;

Pmax ¼

(23)

It is clear that amax , Pmax and p0;max for PC contact only depend on three parameters, namely the constraint factor λ, the maximum volume

As mentioned in Section 2, equations (1)–(3) are also capable to analyze the PC contact by letting R1 →∞ and δ1 ¼ 0. Here, R1 → ∞ in­ dicates the particle 1 becomes to a substrate which represents the cur­ rent collector [44,45]. Because active particles and current collectors are made of different materials, k1 does not equal to k2 in this section. Equations (1)–(3) are rewritten as: 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λð3 þ Ωcmax QÞΩcmax Q; 3

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λð3 þ Ωcmax ÞΩcmax ; 3

p0;max ¼

4. Particle-to-current-collector contact (PC contact)

a¼

amax ¼

where k ¼ k1 =k2 reveals the difference of elastic constants between the

active particle and the current collector. A larger k represents either a

Fig. 5b demonstrates the impacts of k on amax , Pmax and p0;max . If the

Fig. 4. Evolution of contact size a, contact force P and characteristic contact stress p0 with respect to normalized capacity Q in PC contact. 6

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Journal of Power Sources 440 (2019) 227115

Fig. 5. Impacts of (a) maximum volume change Ωcmax , (b) elastic constants ratio k and (c) constraint factor λ on maximum contact size amax , maximum contact force Pmax and maximum contact stress p0;max in PC contact. The dash lines represent the highest possible diffusion induced stress pd;max in the free expansion state.

active material is given, a softer current collector would lead to a larger

stability and safety in LIBs. The effects of the constraint factor λ on the PC contact are demon­ strated in Fig. 5c. The conclusion is similar to that of Fig. 3c, i.e. the contact stress p0;max becomes higher than the diffusion-induced stress pd;max for λ � 0:1. This implies the contact stress is also crucial for PC contact even for a very weak mechanical constraint. In fact, equations (1)–(3) are also capable to analyze other types of contact between active and inactive components, for instance, the contact between active particle and conductive additive (by letting δ1 ¼ 0). However, the analysis would be quite similar to the presented PP or PC contact. Thus, no more details will be described repeatedly.

k. The maximum contact size amax is found to be barely affected by k. On

the contrary, Pmax and p0;max highly depend on k. They both decrease

monotonically with increasing k, as shown in Fig. 5b. For k ¼ 0 which corresponds to a rigid current collector (E1 →∞), Pmax and p0;max reach

their highest values. When k is small, a small increase in k would cause a significant reduction in Pmax and p0;max . p0;max becomes even lower than the highest possible diffusion-induced stress pd;max when the current

collector is soft enough (say k > 2:17). This indicates that a soft current collector is beneficial to the mechanical stability of electrodes. However,

when k > 10, Pmax and p0;max becomes trivial and the effect of k becomes insignificant. This tells that too soft current collector is not necessary for maintaining the mechanical stability. Currently, Young’s moduli of most active materials are close to or even lower than the moduli of copper and aluminum which are typical materials for the current collector [2]. This means for most common

5. Charging pause To illustrate the above-mentioned conclusion that the contact stress could be more critical than the diffusion induced stress, we design special charging schemes by introducing a charging pause (Fig. 6a). Analytical modeling of the diffusion-induced stress is given in Appendix A and the evolution of diffusion-induced stress at the surface of a particle for the designed charging schemes is presented in Fig. 6d. Whichever charging scheme is adopted, Fig. 6d illustrates the diffusion-induced stress pd is much lower than its highest possible value pd;max (about 0.03). More importantly, the interruption of the charging process (Fig. 6a) leads to the concentration relaxation (Fig. 6c) which results in a drop of the diffusion-induced stress (Fig. 6d). Once the charging process is recovered, the diffusion-induced stress goes back up rapidly (Fig. 6d). In other words, a pause in the charging operations induces an undulation of the diffusion-induced stress. It is noted that cyclic stresses during the charge and discharge operations would commonly lead to fatigue of the electrodes, which consequently results in the degradation of batteries [8, 47–50]. Definitely, such additional undulations of the diffusion-induced

electrode structures, k < 1. In this case, besides the mechanical failure in active particles, the local mechanical damage of current collectors is also

likely to occur. For instance, k is approximately 0.1 for an electrode composed of graphite particles and a copper current collector whose moduli are about 10Gpa and 100Gpa, respectively. Ωcmax for graphite is about 0.09 [2]. According to equation (25), the calculated maximum contact stress p0;max in this case can be as high as 1Gpa which is much higher than the yield strength of the copper current collector (commonly no more than 300Mpa for copper [46]). This will definitely cause me­ chanical failures of the current collector and such mechanical failures bring about unexpected severe conditions such as internal short circuit and thermal runaway [28,29]. Thus, the presented result suggests stress localization caused by mechanical contacts between particles and the current collector should play a crucial role as well in the evaluation of 7

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Journal of Power Sources 440 (2019) 227115

evolution of voltage when the charging process is paused. After a 5-min pause, the battery is almost fully relaxed and a considerable decrease of diffusion-induced stress can be expected. Hence, the 6-min and 12-min (approximately 20% and 40% of the whole charging time) pause are chosen in the electrochemical cycling tests. The cyclic performance in terms of the discharge capacity during the charge-discharge processes with/without charging pauses is demonstrated in Fig. 8. Batteries were tested for each strategy (with/without 6/12-min charging pause). As shown in Fig. 8, all lines almost coincide with each other, and no accelerating degradation is observed in the interrupted charging scheme. The discharge capacity retention rates at the 500th cycle of all batteries are approximately 82%, and the difference between them is less than 0.5%. Therefore, consistent with our analytical predictions, the experimental results indicate as well the stress within the active particle could be induced mainly by mechanical contacts. 6. Concluding remarks In this paper, analytical solutions of mechanical contact problems in composite electrodes for LIBs have been provided. Two typical types of mechanical contacts, namely PP contact and PC contact, have been investigated. The dimensionless key parameters which characterize the contact problem have been identified according to the analytical solu­ tions. The evolution of the contact size a, the contact force P and the characteristic contact stress p0 has been discussed. Regardless of the specific contact type (PP or PC) and the specific operation type (galva­ nostatic, potentiostatic or uniform), a, P and p0 increase monotonically as the lithiation proceeds and will reach their corresponding peak values, namely amax , Pmax and p0;max , once the active particle is fully

Fig. 6. Illustration of charging pause: (a) current, (b) capacity, (c) surface concentration and (d) surface stress evolve with charging time.

stress as shown in Fig. 6d would accelerate fatigue process and causes extra capacity fading. However, contact stress would not undulate as the diffusion-induced stress does when the charging process is interrupted. Since as revealed by Fig. 6b, the normalized capacity Q keeps constant during the pause stage and contact stress only depends on this Q. This means if contact stress contributes the most part of mechanical stresses, no evidently extra capacity loss would be observed by introducing charging pauses. Therefore, by checking the electrochemical cyclic performances especially at the charging pause in the designed cycling tests, it will be illustrated that contact stress may play a crucial role in mechanical analysis of electrodes for a conventional LIB system. According to Fig. 6d, the strongest stress undulation generates when the charging process is interrupted at the point of galvanostaticpotentiostatic transition. Therefore, in the electrochemical cycling test, the pause is designed at the transition point to maximize the pause effect, as shown in Fig. 7a. Details of the experimental method and procedure can be found in Appendix B. Fig. 7b demonstrates the

lithiated. The analytical expressions of amax , Pmax and p0;max have also been obtained. In the PP contact, the constraint factor λ, the maximum volume change Ωcmax and the radius ratio R determine the maximum contact size amax , the maximum contact force Pmax and the maximum contact stress p0;max . amax , Pmax and p0;max all increase monotonically with increasing λ, increasing Ωcmax or increasing R. By comparing galvano­ static and uniform cases, charging synchronization between different active particles has been evaluated. Low synchronization causes pre­ mature ending of the charging process, loss of the capacity utilization as well as increase in the contact stress. These drawbacks can be overcome by providing a high degree of uniformity in the active particle size. Additionally, it also reveals that great uniformity of particle size is

Fig. 7. Illustration of charge-pause experiment: (a) evolution of current in electrochemical cycling test (first three cycles), black line stands for the normal chargedischarge process, and red line stands for the process which includes charging pauses at galvanostatic-potentiostatic transition; (b) evolution of voltage when the charging process is paused. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) 8

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Journal of Power Sources 440 (2019) 227115

Fig. 8. Cyclic performances of batteries with/without charging pauses: (a) discharge capacity vs. cycle number; (b) the enlarged version. Numbers following "#" represent different batteries in one group of experiments.

responsible for preventing increase in the maximum contact stress. Therefore, from both electrochemical and mechanical points of view, a high uniformity of particle size is suggested. In the PC contact, Pmax and p0;max depend on λ, Ωcmax and the ratio of

and safety of LIBs. In consideration of distinct reactions of the contact stress and the diffusion-induced stress to a charging pause, the interrupted chargedischarge cycling tests have been conducted to verify behaviors of these two stresses within the active particle. No obvious accelerating degradation caused by the pause indicates the contact stress can account for a great part of the mechanical stress and plays a significant role in mechanical analysis of electrodes in LIBs.

elastic constants k while amax is only related to λ and Ωcmax . Larger Ωcmax causes bigger amax , higher Pmax and higher p0;max . Pmax and p0;max

decrease monotonically with increasing k. Therefore, to suppress the growing of the contact stress, soft current collectors are suggested. A copper current collector is not soft enough for the graphite active par­ ticle so that local mechanical damages due to large contact stress are very likely to occur to the copper current collector. In addition, the maximum contact stress p0;max has been compared with the highest possible diffusion-induced stress pd;max in both PP and PC cases. The contact stress is comparable to and could even be much higher than the diffusion-induced stress, especially when the current collector is stiffer than the active material. The contact stress is crucially high even for very weak mechanical constraints. Hence, mechanical contact plays an important role in evaluations of mechanical stability

Acknowledgements We would like to acknowledge the support of the National Key Research and Development Program of China (No. 2017YFB0701604), the National Natural Science Foundation of China (Nos. 11702166, 11702164, 11332005 and 11672170), the Shanghai Sailing Program (No. 17YF1606000) and the Engineering Research Center of Nano-Geo Materials of Ministry of Education, China (No. NGM2019KF010).

Appendix A. Diffusion-induced stress under charging pause A spherical active particle of radius R under free expansion state is considered. The diffusion of Li-ions within this particle is assumed to be governed by Fick’s law: 0

D

0

1 ∂2 ½rcðr; t Þ� ∂cðr; t Þ ¼ : r ∂r2 ∂t0

(A.1)

The duration time of a single elementary operation, which could be galvanostatic, potentiostatic or pause, is denoted by t ; while t stands for the total time of the whole charging process. The boundary conditions for galvanostatic and potentiostatic operation are given by equations (7) and (8), respectively. The boundary condition for a pause operation can be treated as a special case of galvanostatic operations with in ¼ 0. The initial condition of concentration profile is in a general form of 0

(A.2)

cðr; 0Þ ¼ c0 ðrÞ: If the present operation is galvanostatic, we have the evolution of concentration as follows [39]: # " ∞ ’ 1 2 3 2X ’ λ2n t sinðλn rÞ c ¼ I 3t þ r e 2 10 r n¼1 λn 2 sinðλn Þ ∞ 2X þQ0 þ e r n¼1

’ λ2n t

sinðλn rÞ sin2 ðλn Þ

Z

1

sinðλn ζÞζc0 ðζÞdζ: 0

If the present operation is potentiostatic, the concentration at the present time can be obtained as [39]: 9

(A.3)

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Journal of Power Sources 440 (2019) 227115

"

∞ X

c ¼ csurf 1 þ 2

ðmπ Þ2 t

e

’

ð

m¼1

2 þ r

1Þm sinðmπ rÞ mπ r Z

∞ X

e

ðmπ Þ2 t

’

# (A.4)

1

sinðmπrÞ

c0 ðςÞςsinðmπςÞdς: 0

m¼1

If the present operation is pause, equation (A.3) is reduced to Z 1 ∞ 2X 2 ’ sinðλn rÞ c ¼ Q0 þ e λn t 2 sinðλn ζÞζc0 ðζÞdζ: r n¼1 sin ðλn Þ 0

(A.5)

The initial concentration c ¼ 0 for the entire charging process is assumed. The diffusion-induced stresses in the free expansion state can be expressed as [39]:

σr ¼

2EΩcmax ½Q 9ð1 υÞ

σθ ¼

EΩcmax ½2Q þ cave ðrÞ 9ð1 υÞ

where cave ðrÞ ¼ ð3=r3 Þ pd ¼

ðυ2

(A.6a)

cave ðrÞ�;

Rr 0

(A.6b)

3cðrÞ�;

2

ζ cðζÞdζ. The maximum stress commonly appears at the surface of a particle [13], which is given by Ref. [39]:

1Þσθ ð1Þ ð1 þ υÞΩcmax ¼ ½cð1Þ E 3

(A.7)

Q�:

If cð1Þ ¼ 1 and Q ¼ 0 which only occurs when the initial charging operation is potentiostatic, pd would reach its highest possible value pd;max . Combining equations (A.3)-(A.5) and (A.7), the stress evolution of charging pause can be calculated. It is noted that the diffusion-induced stress upon delithiation simply has an opposite sign to the stress upon lithiation. Thus, the impact of pause on diffusion-induced stress during discharge step is expected to be similar to that during charge step. Appendix B. Experimental methods and procedures A group of commercial 18650 cylindrical LIBs manufactured by Sony Co. were tested. This type of batteries, US18650VTC5A, is presented by the manufacturer as a high power, versatile and long calendar life battery, suitable for portable high power devices and commercial electric vehicles. Its main characteristics are summarized in Table B1. Table B1 Battery characteristics Characteristics

Content

negative electrode positive electrode Nominal capacity nominal voltage Internal resistance Recommended standard charge method Recommended charge and cutoff voltage Cell weight

graphite NMC 2.5Ah 4.25 V 7–15 mΩ 2.5 A to 4.25 V CC/CV 4.25 V–2 V 40.3 g

The cycling tests were carried out by using a Neware BTS-5V10A battery test system. The batteries were tested in a temperature chamber to maintain a constant ambient temperature of 23 � C. The temperatures both in the climate chamber and on the battery surface were measured with thermocouple temperature sensor. Two battery charging strategies were selected: constant current followed by constant voltage (CC-CV) and CCPause-CV strategies. For the CC-CV strategy, the constant current process was charging at 2C to the cut-off voltage of 4.25 V and the constant voltage process was charging at 4.25 V to the cut-off current of 0.25 A. For the CC-Pause-CV strategy, the CC and CV states were the same to the CC-CV strategy and the charging process added a 6-min or 12-min pause between the CC and CV stage. Details of the duration of charging pause can be found in Section 5. The discharging rate for all the strategies was 1C to the cut-off voltage of 2 V. Charge-discharge cycling tests with pause during discharge step were also conducted. Since there is commonly no CV stage during discharge step, the pause was designed to occur after 30-min discharge. The cyclic performance in terms of the discharge capacity during the charge-discharge processes with/without discharging pauses is demonstrated in Fig. B1. As expected, all lines still coincide with each other, and also no acceler­ ating degradation is observed in the interrupted discharging scheme.

10

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Journal of Power Sources 440 (2019) 227115

Fig. B1. Discharge capacity of batteries with/without discharging pauses.

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