- Email: [email protected]
Diffusion-induced stresses in graphene-based composite bilayer electrode of lithium-ion battery
Accepted Manuscript Diffusion-induced stresses in graphene-based composite bilayer electrode of lithium-ion battery Dongying Liu, Weiqiu Chen, Xudong Shen PII: DOI: Reference:
S0263-8223(16)32886-0 http://dx.doi.org/10.1016/j.compstruct.2017.01.011 COST 8144
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
16 December 2016 4 January 2017 5 January 2017
Please cite this article as: Liu, D., Chen, W., Shen, X., Diffusion-induced stresses in graphene-based composite bilayer electrode of lithium-ion battery, Composite Structures (2017), doi: http://dx.doi.org/10.1016/j.compstruct. 2017.01.011
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Diffusion-induced stresses in graphene-based composite bilayer electrode of lithium-ion battery Dongying Liua, b, Weiqiu Chena,c,d,e, *, Xudong Shena a
Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China
b
Department of Engineering Mechanics, China University of Petroleum, Qingdao 266580, China
c
State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou 310027, P.R.
China d
Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Zhejiang University, Hangzhou
310027, P.R. China e
Soft Matter Research Center, Zhejiang University, Hangzhou 310027, P.R. China
Abstract The diffusion-induced stresses in a bilayer electrode composed of a current collector and a composite active plate of lithium-ion battery are evaluated analytically. The active plate of the bilayer electrode is reinforced by graphene platelets (GPLs), with the corresponding material properties predicted by the combination of the well-established Halpin-Tsai model and the rule of mixture. The diffusion-induced stresses and curvature of the bilayer electrode under either galvanostatic or potentiostatic charging operation are derived. In addition, modified Stoney formulas for the bilayer electrode are developed for the two charging operations. A parametric study is conducted through numerical examples, with a particular focus on the reinforcing effect of GPLs on the stresses and deformation in the bilayer electrode. The size effects of GPLs, including width, length and thickness, on the stresses and deformation are also discussed. Keywords: Diffusion-induced stresses; graphene-based composite electrode; graphene platelets; Stoney formula.
*
Author for correspondence. Tel.: +86 571 87951866; fax: +86 571 87951866. E-mail address: [email protected] (W.Q. Chen). 1
1. Introduction Lithium-ion batteries (LIBs) have been widely used as power sources in portable electronics, mobile communication devices, as well as in implantable biomedical devices, and have aroused a great deal of interest in scientific and industrial fields due to their high energy density, tiny memory effect, and environmental friendliness [1, 2]. However, the low power capacity of the current LIBs cannot satisfy the ever-growing demand for large-scale applications such as electric vehicles and scalable electricity storage. To overcome this problem, significant attention has been paid to the development of advanced electrode materials with higher energy density and power density [3]. Among the potential materials for electrodes of LIBs, graphene-based composites exhibit many alluring and advantageous characteristics including high charge mobility and electron transport capability, mostly because of the outstanding physical properties of graphene, such as large specific surface area, excellent chemical stability, and electrochemical activity. In such composite electrodes, graphene is highly compatible with other dissimilar active components (such as transitional metal oxides and conductive polymer), and coexists with them in various forms of microstructures such as anchored, wrapped, encapsulated, layered and sandwich-like [4, 5]. Graphene-based composites have been considered as one of the most promising candidates for next-generation electrodes of LIBs, including composite anodes and cathodes [6, 7]. Paek et al. [8] fabricated a composite nanoporous anode by distributing graphene nanosheets in rutile SnO2, and found the graphene-based SnO2 composite electrode exhibits a much higher capacity than that of the bare SnO2 anode. Wang et al. [9] reported a first principle study on the electrochemical performance of the Sn-graphene nanocomposite anode by constructing 3D nanocomposite architecture with the combination of nanosized Sn particles and graphene nanosheets. The Fe3O4-graphene composite anode [10] and CuO-graphene composite anode [11] also show a significant improvement in electrochemical performances of nanostructured transition metal oxides as anode materials for LIBs. Sn-based anode composed of active material encapsulated by graphene was developed, with excellent electrochemical properties including high rate capability and stable cyclic performance [12]. Meanwhile, improvements on cathodes of their fairly low electronic conductivities, rate capability and cyclability could 2
be achieved by using graphene as additives dispersed into the corresponding active material matrix, such as LiFePO4 [13, 14], Li3V2(PO4)3 [15], Li2FeTiO4 [16], and sulfur [17]. For the recent research progress in graphene-based composite electrode for electrochemical capacitors and LIBs, the reader is referred to the overview article by Xu et al. [18]. One bottleneck in developing batteries with high energy density is the damage induced by the insertion/extraction of lithium-ion into/from the active material of electrodes. Taking the Si anode as an example, the volumetric swelling induced by charging will be nearly 400%, which usually leads to fracture [19]. The addition of graphene nanosheets into the Si matrix can alleviate the significant volume changes effectively [20, 21]. Graphene-based composites have been demonstrated to be remarkably effective in enhancing the mechanical properties, including fracture toughness, stiffness, strength, and fatigue resistance, in comparison to the pristine matrix materials [22]. For example, the graphene-nanoribbon/epoxy composites exhibit significant increase in the elastic modulus and tensile strength compared to the pure epoxy [23], and the elastic modulus of epoxy reinforced by a low content of graphene platelets (GPLs) is 31% greater than the pristine epoxy [24]. Therefore, graphene in composite electrodes could not only enhance their electrochemical performances but also improve their mechanical properties. Consequently, graphene-based composite electrodes appear to be superb candidates for improving the performances of anode/cathode to meet the demands of higher energy and power density, larger cyclability and rate capability, and more safety of LIBs. As is well-known, diffusion-induced stress (DIS) arising from insertion/extraction of lithium ions into/from the lattice of active materials is one of the main reasons for mechanical failure of electrodes which may lead to loss of capacity and degradation of cyclic performance of LIBs [25]. Several attempts have been devoted to evaluating DISs in electrodes during charging and discharging. Christensen [26] developed a mathematical model for DISs in a porous electrode made of spherical active materials. Deshpande et al. [27] studied the surface effects on DISs in nanowire electrode structures. Bower and Guduru [28] presented a finite element model to describe the coupled processes of diffusion, deformation and fracture in electrode microstructures. Bhandakkar and Johnson [29] simulated and analyzed 3
morphological changes and DISs in Si honeycomb structured electrode. The work mentioned above all focused on the DISs at particle level and neglected the influence of the current collector. Zhang and his co-authors [30-33] initiated the study of multilayered electrodes composed of active plates and current collector from the viewpoint of mechanics, and systematically developed analytical solutions that are very helpful in designing multilayered electrodes with high performance. Hao and Fang [34] proposed a strategy to reduce the built-in stresses due to lithium-ion diffusion by inducing pre-strain in a bilayer electrode. Zhang et al. studied DISs in transversely isotropic cylindrical electrodes [35] as well as in layered electrode plates with a composition-gradient [36]. Recently, Li et al. [37] carried out an elastoplastic analysis of the lithiation-induced deformation in a bilayer electrode. Although there have been a huge number of research reports either on the material behavior of composite electrodes or on DISs in layered electrodes, as mentioned above, the study on DISs in graphene-based composite electrodes is still unavailable, to the authors’ knowledge. This work attempts to fill the gap by considering the diffusion-induced stresses in a bilayer electrode composed of current collector and active plate. The active plate of the bilayer electrode is a composite containing both active particles and pore-filling electrolyte and simultaneously reinforced by graphene platelets (GPLs). The GPLs are assumed to be dispersed uniformly and randomly in the matrix of the active plate. The corresponding material properties of the composite active plate will be predicted by the well-established Halpin-Tsai model and the rule of mixture. Two charging conditions, i.e. galvanostatic and potentiostatic, will be considered, and the corresponding diffusion-induced stresses and curvature of the bilayer electrode will be derived analytically. Modified Stoney formulas for the bilayer electrode will also be developed. In Zhang et al.’s work [30], the effects of material and geometric parameters of the bilayer electrode on DISs and deformation have been discussed in detail. Hence, we will focus, in numerical calculations, on the effects of GPLs on the behavior of the bilayer composite electrode. In particular, the influences of size and content of GPLs will be studied numerically.
4
2. Basic equations
Fig. 1. Configuration of a bilayer electrode.
A bilayer electrode composed of a current collector with thickness hc and a composite active plate with thickness h p is schematically depicted in Fig. 1. Referred to the Cartesian coordinates
( x, y, z ) ,
the center of the interface between the active plate and the current
collector is taken as the origin of the coordinates, and the z-axis is aligned along the thickness direction of the plate pointing from the bottom surface to the top surface of the active plate. Lithium ions are assumed to be inserted/extracted into/from the active plate along the thickness direction only. Both the current collector and the composite active plate are treated as isotropic and elastic materials at the macroscale, and only linear elastic deformation is considered in this study. The classical plate theory and the assumption of equal biaxial stress for a large electrode plate yield the following in-plane normal strain components
ε x = ε y = ε 0 + zκ
(1)
where ε 0 represents the in-plane normal strain at the interface, and κ is the curvature of the bilayer electrode. The constitutive equations for the current collector and active plate can be written as [30, 38]
1 3
σ p = E ′p ( ε 0 + κ z ) − E ′p Ωc σ c = Ec′ ( ε 0 + κ z ) 5
(2)
where σ is the in-plane equal biaxial stress, c is the molar concentration of lithium ions, Ω is the partial molar volume of solute, and E′ = E (1 −ν ) is the biaxial modulus with E and
ν being the elastic modulus and Poisson’s ratio, respectively. The subscripts p and c indicate the corresponding physical quantities of the active plate and current collector, respectively. The diffusion process of lithium ions in the active plate would change the lattice structure of the active plate, which means the material properties of the active plate depend on the concentration of lithium ions. However, this work will focus on the enhancing effects of GPLs on the mechanical behaviors of the electrode, and hence the diffusion coefficient, elastic modulus and Poisson’s ratio of the active plate are all assumed to be constant during both the charging and discharging processes [30, 31, 39]. In general, the characteristic time for elastic deformation of the electrode is much smaller than that for lithium-ion diffusion. Thus, the elastic deformation induced by diffusion of lithium ions can be reasonably approximated as a static process, and the mechanical equilibrium is expressed by
∫
0
∫
0
− hc
− hc
hp
σ c dz + ∫ σ p dz = 0 0
hp
σ c zdz + ∫ σ p zdz = 0
(3)
0
The substitution of Eq. (2) into Eq. (3) yields
Aε 0 + Bκ − N p = 0 Bε 0 + Iκ − M p = 0
(4)
where hp
0
0
− hc
A = ∫ E ′p dz + ∫
Ec′dz = E ′p hp ( mh + 1)
1 E ′p hp2 (1 − mh2 ) 2 hp 0 1 I = ∫ E′p z 2dz + ∫ Ec′ z 2dz = E ′p hp3 ( mh3 + 1) 0 − hc 3 hp
0
0
− hc
B = ∫ E ′p zdz + ∫
Ec′ zdz =
(5)
in which m = Ec′ E ′p and h = hc h p are the ratios of the biaxial modulus and thickness between the current collector and the active plate, and hp hp 1 1 N p = E ′p Ω∫ cdz, M p = E ′p Ω ∫ czdz 0 0 3 3
(6)
which can be considered as the resultant axial force and moment in the active plate induced 6
by the diffusion of lithium ions. Then solving for ε 0 and κ from Eq. (4) gives
κ =− ε0 =
6 (1 − mh 2 ) N p hp − 2 ( mh + 1) M p E ′p h3p β
(7)
2 2 ( mh3 + 1) N p hp − 3 (1 − mh 2 ) M p 2 E ′p hp β
where β = m2 h4 + 4mh3 + 6mh 2 + 4mh + 1 . It is notable that if the molar concentration c is known, N p and M p , and then the curvature κ and in-plane strain ε 0 can be determined. Two different ways of charging operation are considered here, i.e. the galvanostatic charging with a constant surface lithium-ion flux ( i0 ) and the potentiostatic charging with a constant surface lithium-ion concentration ( c0 ). The lithiation/delithiation process of lithium ions is governed by the Fick’s law, and has been well discussed by Crank [40]. For completeness, a brief introduction to the diffusion analysis is outlined in the appendix. The substitutions of the lithium-ion concentration under the galvanostatic/potentiostatic charging operation in Eqs. (A4) and (A6) of the Appendix, into Eq. (6) and then Eqs. (7) and (2) yield the curvature κ of the bilayer electrode and in-plane equal biaxial stresses σ p and
σ c in the active plate and current collector, respectively. They are given below as 1 2i Ω 2 κ = 0 mh (1 + h ) t + 2 ( mh + 1) + 4 β FD 24 π
∞
∑
( −1)
n
−1
n4
n =1
exp ( −n2π 2 t )
mh 6 (1 + h ) z − ( mh3 + 3h + 4 ) t E ′ Ωi h β σp = p 0 p n 3 FD 3 z 2 − 1 2 ∞ ( −1) cos ( nπ z ) exp ( −n 2π 2 t 2 − 6 + π 2 ∑ n n =1 +
2 E ′p Ωi0 hp
β FD
1 2 2 ( mh + 1) z + mh − 1 + 4 24 π 2
7
∞
∑ n =1
( −1)
n
n4
−1
)
exp ( −n 2π 2 t )
(8)
(9)
4mh3 + 1 + 3mh2 + 6mh (1 + h ) z t Ec′Ωi0 hp σc = 3β FD +6 2 ( mh + 1) z + mh2 − 1 1 + 2 24 π 4
n ∞ ( −1) − 1 exp ( −n2π 2 t ) ∑ 4 n n =1
(10)
for the galvanostatic charging, and 2Ωc0 κ= hp β
8 mh (1 + h ) − 2 π
n 2 ( 2n − 1) 2 π 2 t 4 ( mh + 1)( −1) 1 exp − mh + 2mh + 1 − ∑ 2 2 n − 1 π 4 ( ) 2 n − 1 n =1 ( ) ∞
(11) ∞ 16 ( 2n − 1) 2 π 2 t 1 3 2 2 2 ( 2mh + 3mh − 1) + 3 z (1 + 2mh + mh ) ∑ n=1 2n − 1 2 exp − 4 π ( ) E ′p Ωc0 3 σp =− + mh ( mh + 3h + 4 ) − 6mh (1 + h ) z 3β n 2 ∞ 96 ( −1) exp − ( 2n − 1) π 2 t 2 + mh − 1 + 2 z mh + 1 ( ) ∑ 3 3 4 π n =1 ( 2n − 1)
+
4 E ′p Ωc0 3
( 2n − 1)2 π 2 t ( 2n − 1) π z cos ∑ exp − 2 4 n =1 ( 2n − 1) π ∞
( −1)
n −1
(12)
∞ 16 ( 2n − 1) 2 π 2 t 1 3 2 2 2 ( 2mh + 3mh − 1) + 3 z (1 + 2 mh + mh ) ∑ n =1 2n − 1 2 exp − 4 π ( ) E ′Ωc σ c = − c 0 +mh ( mh3 + 3h + 4 ) − 6mh (1 + h ) z 3β n ∞ ( 2n − 1)2 π 2 t 96 −1) ( 2 exp − + 3 mh − 1 + 2 z ( mh + 1) ∑ 3 4 n =1 ( 2n − 1) π
(13) 2 for the potentiostatic charging, where t = Dt hp and z = z h p are the dimensionless time
and z-coordinate, respectively, F is the Faraday constant, and D is the effective diffusivity of lithium ions in the active plate. It can be found that the curvature for the galvanostatic charging operation in Eq. (8), is identical to that obtained by Zhang et al. [30]. However, the stresses in the current collector and the active plate are different. After a careful check, we found that it is due to the incorrect expressions of α1 and α 2 in Eqs. (23) and (24) in Zhang et al. The correct ones should be
α1 = ( 4mh 3 + 3mh 2 + 1) α 3 and α 2 = 6 ( mh 2 − 1) α 3 using the same notions as in Zhang et 8
al. [30]. The Stoney formula [41] has been widely used in film/substrate structures, which relates the curvature to the stress in the film. Plentiful work has been devoted to extending the limitation of the Stoney formula [42]. Sethuraman et al. [19] performed real-time measurements of stress evolution in a silicon thin film electrode during electrochemical lithiation and delithiation using the multi-beam optical sensor technique, and developed a complex modified Stoney formula based on the principle of minimum potential energy to evaluate the stresses in LIBs [43]. However, the process of lithium diffusion was not considered by Sethuraman et al. [19, 43], and consequently, their modified Stoney formula is independent of time t , which is not rigorous for determining the DISs through measuring the curvature of the electrode. To develop the relation between the stress in the active plate and the curvature of the bilayer electrode, an average biaxial stress through the thickness of the active plate is now defined as
σ p =
1 hp
∫
hp
0
σ p dz
(14)
The substitution of Eqs. (9) and (12) into the above equation yields
1 mhE ′p Ωi0 hp 2 3 σ p = − ( mh + 1) t − 6 (1 + h ) + 4 3 FD β 24 π
∞
∑
( −1)
n =1
n
n4
−1
exp ( −n 2π 2 t )
(15)
for the galvanostatic charging, and ( mh3 − 3h − 2 ) ∞ ( 2n − 1)2 π 2 t 1 exp − ∑ 2 π2 4 n =1 ( 2n − 1) 8mhE ′p Ωc0 σ p = n 3β ( 2 n − 1) 2 π 2 t 12 (1 + h ) ∞ ( −1) 3 exp − − 1 − mh ∑ − 3 3 π 4 n =1 ( 2n − 1)
(16)
for the potentiostatic charging, respectively. From Eqs. (8) and (15), the modified Stoney formula for the bilayer electrode under galvanostatic charging can be expressed as
σ p =
mhE ′p hp 2Ωi0 3 (1 + h ) κ − 6 (1 + mh ) FD 9
t
(17)
Similarly, the modified Stoney formula under potentiostatic charging is mh (1 + h )2 mhE ′p (1 + h ) hp 8 σ p = κ − Ωc0 − 2 β 3π ( mh + 1) 2
( 2n − 1) 2 π 2 t exp − ∑ 2 4 n =1 ( 2n − 1) ∞
1
(18)
These two formulae provide the basis for evaluating the stresses in LIBs induced by diffusion of lithium ions if the curvature of the bilayer electrode has been measured experimentally. 3. Effective elastic properties of GPLs-reinforced composite
In the composite active plate of the bilayer electrode, the GPLs, acting as an effective rectangular solid fiber with width ( wGPL ), length ( lGPL ), and thickness ( hGPL ), are assumed to be dispersed randomly, but uniformly, in the matrix [24]. To evaluate the effective elastic properties of the GPLs-reinforced composite, the following well-established Halpin-Tsai equations [44, 45] are adopted
3 5 E p = E + E⊥ 8 8 E =
1 + ξlηlVGPL 1 + ξ wη wVGPL × EM , E⊥ = × EM 1 − ηlVGPL 1 −η wVGPL
(19) (20)
where E p is the effective Young’s modulus of the composite active plate, and E and E⊥ are the longitudinal and transverse moduli, respectively; EM is the Young’s modulus of the matrix of the active plate, VGPL is the volume fraction of GPLs, and ηl and η w are taken to be
ηl =
( EGPL EM ) − 1 , η = ( EGPL EM ) − 1 ( EGPL EM ) + ξl w ( EGPL EM ) + ξ w
(21)
in which EGPL is the Young’s modulus of GPLs, and ξl and ξw depend on the geometry of GPLs, which can be expressed as lGPL hGPL
ξl = 2
wGPL , ξw = 2 hGPL
(22)
The effective Poisson’s ratio of the composite active plate can be evaluated by using the general rule of mixture [24], which gives 10
ν p = ν GPLVGPL +ν M (1 − VGPL )
(23)
where ν GPL and ν M are the Poisson’s ratios of GPLs and matrix, respectively. Once the effective Young’s modulus and Poisson’s ratio of the active plate are evaluated, the corresponding shear modulus can be obtained as Gp =
Ep 2 (1 + ν p )
(24)
The volume fraction VGPL of GPLs can be calculated using the weight fraction WGPL of GPLs as VGPL =
WGPL WGPL + ( ρ GPL ρ M )(1 − WGPL )
(25)
with ρGPL and ρM being the densities of GPLs and matrix, respectively. The volume and weight fractions of the matrix material in the composite active plate can be obtained by
VM = 1 − VGPL and WM = 1 − WGPL , respectively. The Halpin-Tsai equations are a set of empirical relations for the prediction of the elastic properties of a composite material in terms of the properties of the filler and matrix together with their proportions and geometry, and have been confirmed by experimental measurements. The mechanical properties of the epoxy nanocomposite with GPLs ( WGPL = 0.1% ,
lGPL = 2.5 µm , wGPL = 1.5 µm , and hGPL = 1.5 nm ) were measured by Rafiee et al [24], and the Young’s modulus obtained through uniaxial tensile testing is ~3.7 GPa. The prediction of the Young’s modulus by the Halpin-Tsai equations, i.e. Eq. (19)1, is 3.8 GPa , which overpredicts the experiment data by up to 2.7% only. It should be noted that the evaluation of Young’s modulus by the modified Halpin-Tsai equations presented in [24] (3.23 GPa) is much lower than the experimental result because of the neglect of the width effects of GPLs. It should be stressed that the elastic modulus of the composite active plate evaluated by the Halpin-Tsai equations is an “averaged” property, which is suitable to describe the global characteristics of the composite (such as vibration frequency and buckling load). However, if local phenomena are considered (e.g., local buckling, delamination), some other refined 11
models (e.g. high-order multi-scale models) should be preferred. 4. Numerical results and discussion
Among various electrode materials, Tin (Sn)-based active materials have been considered as one type of the most promising anode materials for high-capacity LIBs due to its high theoretical capacity, widespread availability, and environmentally benign nature [46]. Thus a bilayer anode composed of Tin (Sn)-based composite active plate reinforced by GPLs and Cu current collector under galvanostatic charging operation is taken as the numerical example, whose material properties are EM = 24.7 GPa , ν M = 0.24 , and ρ M = 7.31 g cm 3 [47]; EGPL = 1.01 TPa , ν GPL = 0.186 , and ρ GPL = 1.06 g cm 3 [24]; Ec = 115 GPa and
ν c = 0.35 , respectively. The geometric parameters of GPLs are taken to be lGPL = 2.5 µm , wGPL = 1.5 µm , and tGPL = 1.5 nm [24] in numerical calculations, unless otherwise stated.
Fig. 2. Effective elastic modulus of the composite active plate.
The effective elastic moduli of the composite active plate reinforced by different contents of GPLs are evaluated by the Halpin-Tsai equations, which are shown in Fig. 2. The elastic modulus of the pristine Tin ( WGPL = 0 ) is also given for comparison. It is obvious that the blending of GPLs nanofillers into the Tin-based active material would promote the elastic 12
modulus dramatically, and the more the content of GPLs, the larger the effective modulus. The elastic modulus of the composite active material with WGPL = 1.5% ( E p = 116.9 GPa ) is nearly five times larger than that of the pristine Tin (24.7 GPa).
Fig. 3. Variations of the dimensionless stress σ in the active plate ( 0 ≤ z ≤ 1 ) and the current collector ( −0.1 ≤ z ≤ 0 ).
Fig. 3 presents the dimensionless stresses σ = σ FD ( E ′p Ωin h p ) in the active plate and the current collector for different contents of GPLs. The lithium ions insert into the active plate from the top surface ( z = 1), and make the lithiated region expand. However, the unlithiated region and the current collector will resist the deformation of the electrode. As a consequence, bending deformation is generated. The overall stress in the active plate is produced by the combination of the diffusion-induced stress and bending stress. At the beginning of the charging operation, the overall stress is mainly determined by the diffusion-induced stress. As more and more lithium ions migrate into the active plate, the bending stress becomes dominant. It can be seen that the stress near the bottom surface is always compressive due to the restriction of the current collector, and the stress near the top surface may become tensile due to the increasing of the bending stress. At the earlier stage of the charging operation, the compressive stress at the top surface of the GPLs-reinforced composite active plate is smaller than that of the pristine Tin active plate. However, the 13
compressive stress at the bottom surface behaves oppositely. Thus, an even distribution of the compressive stress is achieved by using the GPLs-reinforced composite active plate. Furthermore, with time increasing, the stress at the top surface becomes tensile, and both the compressive and tensile stresses (at the bottom and top surfaces, respectively) in the GPLs-reinforced composite active plate are smaller than those of the pristine Tin active plate. It also can be found that the overall stress in the GPLs-reinforced composite active plate generally could be lessened in comparison with that in the active plate composed of pristine Tin, except the compressive stress near the bottom surface ( z = 0 ) at the beginning of the charging operation. Results also show that, the maximum compressive stress of the active plate arises at the bottom surface of the active plate, and a large stress drop, i.e. the difference between the tensile stress at the top surface of the current collector and the compressive stress at the bottom surface of the GPLs-reinforced composite active plate, σ c ( z = 0 ) − σ p ( z = 0 ) , also occurs there [34]. The stress drop enlarges as time increases, which however can be reduced by using the GPLs-reinforced composite as the active plate. The maximum compressive stress can be reduced as well by this strategy. Thus, the reinforcement of GPLs is greatly beneficial to preventing the electrode from debonding and delamination.
(a) Bottom surface
(b) Top surface
Fig. 4. Variations of the dimensionless stress σ p at the surfaces of the composite active plate against the thickness ratio.
Fig. 4 gives the variations of the biaxial stress σ p at the bottom ( z = 0 ) and top ( z = 1) surfaces of the active plate against the thickness ratio between the current collector and 14
composite active plate. As pointed out by Zhang et al. [30], the increasing of the thickness ratio hc h p would promote the peak stress due to the stronger restriction of the current collector, which is similar to that in the analysis of thermal stress [42]. However, it can be seen in Fig. 4 that the peak stress has been reduced in the GPLs-reinforced composite electrode, no matter compressive or tensile. Therefore, it can be a practical approach to relaxing the stress by using the GPLs-reinforced composite as an active plate in lithium ion batteries.
(a) Bottom surface, t = 1 .
(b) Top surface, t = 1 .
Fig. 5. Size effects of GPLs on the dimensionless stress σ p in the composite active plate.
As can be observed above, higher lithium-ion concentration corresponds to larger stress level [30]. Thus a relatively typical and larger lithiation time, i.e. t = 1 , is chosen as the dimensionless reference time in the following disscusion. Fig. 5 reveals the size effects of GPLs (i.e. length lGPL , width wGPL , and thickness tGPL ) in terms of the length-to-thickness ratio lGPL tGPL and the length-to-width ratio lGPL wGPL on the dimensionless stress σ p in the composite active plate. The length of GPLs keeps unchanged ( lGPL = 2.5 µm ), and lGPL wGPL = 1
and
lGPL wGPL ≠ 1
correspond
to
a
square-shaped
platelet
and
a
rectangle-shaped platelet, respectively. It is seen that the square-shaped GPLs ( lGPL wGPL = 1 ) as nanofillers can reduce the compressive and/or tensile stresses in the composite active plate more effectively. In other words, GPLs with larger surface area are better-reinforcing nanofillers than their counterparts with smaller surface area due to the larger contact area 15
between the matrix and GPLs. A significant decrease in the dimensionless stress σ p is observed against the length-to-thickness ratio lGPL tGPL , which means GPLs with less graphene layers could be more effective in lessening the stress in the active plate. Thus, a monolayer graphene may be the most outstanding candidate to reinforce the composite electrode.
Fig. 6. Variations of the dimensionless curvature κ of the active plate against the dimensionless time.
The variations of the dimensionless curvature κ = κ FD in Ω of the composite electrode with the dimensionless time t are plotted in Fig. 6. The bending deformation of the composite electrode is due to the restriction of the unlithiated region of the active plate and the current collector. At the beginning of the charging operation, the deformation of the composite electrode is mainly caused by the intercalation of lithium ions. As the lithium ions insert into the active plate continually, the current collector will inhibit the expansion of the active plate more dominantly. It can be observed from Fig. 6 that the blending of GPLs into the Tin active plate would firstly promote, and then reduce the curvature of the bilayer electrode with respect to time. This is the outcome of the competition between the diffusion-induced bending and the bending deformation due to the stiffness mismatch between the active plate and the thick current collector. 16
Fig. 7. Dimensionless curvature κ of the bilayer composite electrode with different weight fractions of GPLs against the thickness ratio h.
The variations of the dimensionless curvature κ = κ FD in Ω of the bilayer electrode at dimensionless times t = 1 against the thickness ratio h = hc h p are plotted in Fig. 7. The curvature first increases, and then decreases with the thickness ratio. This is the result of competition between the diffusion-induced deformation and the bending deformation. The thicker the copper current collector is, the stronger the restriction it exerts on the deformation of the electrode. For a bilayer electrode with a thin current collector (small h), the blending of GPLs nanofillers could reduce the curvature evidently except at the initial period of charging. However, for the case with a thicker current collector, the GPLs-reinforced composite active plate would hold a larger deformation. This is due to the reinforcing GPLs would increase the stiffness mismatch between the active plate and the thick current collector.
17
Fig. 8. Size effects of GPLs on the dimensionless curvature κ of the bilayer electrode.
The size effects of GPLs on the dimensionless curvature κ of the bilayer electrode are shown in Fig. 8. It is found that the square-shaped GPLs would increase the curvature of the electrode at the beginning of charging, and decrease the curvature as more and more lithium ions move into the active plate. It is noted that the curvature of the electrode at the initial period of charging is inconspicuous compared with that at the subsequent charging stage. As can be seen in Fig. 8, the GPLs with larger area could reduce the curvature more effectively. Meanwhile, thinner GPLs would also reduce the curvature. Thus it may be concluded that a large-area monolayer graphene would be preferable to lessen the crucial deformation of the electrode. 5. Conclusions
In this paper, the diffusion-induced stresses in a bilayer electrode composed of a current collector and a composite active plate have been evaluated analytically. The composite active plate is reinforced by graphene platelets (GPLs), which was treated as an isotropic elastic material macroscopically. The corresponding material properties were predicted by adopting the Halpin-Tsai model along with the rule of mixture. The analytical solutions of the diffusion-induced stresses and curvature of the bilayer electrode under galvanostatic and 18
potentiostatic charging conditions were obtained based on the classical thin plate theory. Modified Stoney formulas for the bilayer electrode were also developed, which can be used to evaluate the stresses in the active plate from the deformation of the bilayer electrode in real time. Effects of GPLs on the DISs and deformations of the bilayer electrode were studied extensively through numerical examples. The blending of GPLs nanofillers into the matrix material of the active plate can improve the mechanical properties of the bilayer electrode effectively. From the numerical results, it can be concluded that: (1) The elastic modulus of the active plate can be enhanced observably when reinforced by GPLs. Even low content of GPLs could increase the elastic modulus several times. (2) Generally, both the compressive and tensile stresses in the GPLs-reinforced composite active plate, as well as the stress drop at the interface, could be lessened prominently in comparison with those in the pristine active plate. (3) The blending of GPLs could reduce the deformation of a bilayer electrode with a thin current collector. However, it would increase the deformation of an electrode with a thick current collector. (4) A monolayer graphene would be the most outstanding candidate to reinforce the mechanical behaviors of the bilayer electrode.
Acknowledgements
The work was supported by the National Natural Science Foundation of China (Nos. 11621062, 11532001, and 11402310), the Fundamental Research Funds for the Central Universities (Nos. 2016XZZX001-05 and 15CX08004A), and partly supported by the Foundation for Outstanding Young Scientist in Shandong Province (No. BS2014CL022).
19
References [1] Scrosati B, Hassoun J, Sun YK. Lithium-ion batteries. A look into the future. Energy and Environmental Science. 2011;4:3287-95. [2] Scrosati B, Garche J. Lithium batteries: Status, prospects and future. Journal of Power Sources. 2010;195:2419-30. [3] Armand M, Tarascon JM. Building better batteries. Nature. 2008;451:652-7. [4] Raccichini R, Varzi A, Passerini S, Scrosati B. The role of graphene for electrochemical energy storage. Nature Materials. 2015;14:271-9. [5] Brownson DA, Kampouris DK, Banks CE. An overview of graphene in energy production and storage applications. Journal of Power Sources. 2011;196:4873-85. [6] Huang X, Zeng Z, Fan Z, Liu J, Zhang H. Graphene-based electrodes. Advanced Materials. 2012;24:5979-6004. [7] Wu ZS, Zhou G, Yin LC, Ren W, Li F, Cheng H-M. Graphene/metal oxide composite electrode materials for energy storage. Nano Energy. 2012;1:107-31. [8] Paek SM, Yoo E, Honma I. Enhanced cyclic performance and lithium storage capacity of SnO2/graphene nanoporous electrodes with three-dimensionally delaminated flexible structure. Nano Letter. 2008;9:72-5. [9] Wang G, Wang B, Wang X, Park J, Dou S, Ahn H, Kim K. Sn/graphene nanocomposite with 3D architecture for enhanced reversible lithium storage in lithium ion batteries. Journal of Materials Chemistry. 2009;19:8378-84. [10] Lian P, Zhu X, Xiang H, Li Z, Yang W, Wang H. Enhanced cycling performance of Fe3O4-graphene nanocomposite as an anode material for lithium-ion batteries. Electrochimica Acta. 2010;56:834-40. [11] Mai Y, Wang X, Xiang J, Qiao Y, Zhang D, Gu C, Tu J. CuO/graphene composite as anode materials for lithium-ion batteries. Electrochimica Acta. 2011;56:2306-11. [12] Wu C, Maier J, Yu Y. Sn-based nanoparticles encapsulated in a porous 3D graphene network: Advanced anodes for high-rate and long life Li-ion batteries. Advanced Functional Materials. 2015;25:3488-96. [13] Zhou X, Wang F, Zhu Y, Liu Z. Graphene modified LiFePO4 cathode materials for high power lithium ion batteries. Journal of Materials Chemistry. 2011;21:3353-8. [14] Gong C, Xue Z, Wen S, Ye Y, Xie X. Advanced carbon materials/olivine LiFePO4 composites cathode for lithium ion batteries. Journal of Power Sources. 2016;318:93-112. [15] Jiang Y, Xu W, Chen D, Jiao Z, Zhang H, Ma Q, Cai X, Zhao B, ChuY. Graphene modified Li3V2(PO4)3 as a high-performance cathode material for lithium ion batteries. Electrochimica Acta. 2012;85:377-83. [16] Yang M, Zhao X, Yao C, Kong Y, Ma L, Shen X. Nanostructured cation disordered 20
Li2 FeTiO4/graphene composite as high capacity cathode for lithium-ion batteries. Materials Technology. 2016;31:537-43. [17] Wang JZ, Lu L, Choucair M, Stride JA, Xu X, Liu HK. Sulfur-graphene composite for rechargeable lithium batteries. Journal of Power Sources. 2011;196:7030-4. [18] Xu C, Xu B, Gu Y, Xiong Z, Sun J, Zhao X. Graphene-based electrodes for electrochemical energy storage. Energy & Environmental Science. 2013;6:1388-414. [19] Sethuraman VA, Chon MJ, Shimshak M, Srinivasan V, Guduru PR. In situ measurements of stress evolution in silicon thin films during electrochemical lithiation and delithiation. Journal of Power Sources. 2010;195:5062-6. [20] Xiang H, Zhang K, Ji G, Lee JY, Zou C, Chen X, Wu J. Graphene/nanosized silicon composites for lithium battery anodes with improved cycling stability. Carbon. 2011;49:1787-96. [21] Liang B, Liu YP, Xu YH. Silicon-based materials as high capacity anodes for next generation lithium ion batteries. Journal of Power Sources. 2014;267:469-90. [22] Huang X, Qi X, Boey F, Zhang H. Graphene-based composites. Chemical Society Reviews. 2012;41:666-86. [23] Rafiee MA, Lu W, Thomas AV, Zandiatashbar A, Rafiee J, Tour JM, Koratkar NA. Graphene nanoribbon composites. ACS Nano. 2010;4:7415-20. [24] Rafiee MA, Rafiee J, Wang Z, Song H, Yu ZZ, Koratkar N. Enhanced mechanical properties of nanocomposites at low graphene content. ACS Nano. 2009;3:3884-90. [25] Mukhopadhyay A, Sheldon BW. Deformation and stress in electrode materials for Li-ion batteries. Progress in Materials Science. 2014;63:58-116. [26] Christensen J. Modeling diffusion-induced stress in Li-ion cells with porous electrodes. Journal of The Electrochemical Society. 2010;157:A366-A80. [27] Deshpande R, Cheng YT, Verbrugge MW. Modeling diffusion-induced stress in nanowire electrode structures. Journal of Power Sources. 2010;195:5081-8. [28] Bower AF, Guduru PR. A simple finite element model of diffusion, finite deformation, plasticity and fracture in lithium ion insertion electrode materials. Modelling and Simulation in Materials Science and Engineering. 2012;20:045004. [29] Bhandakkar TK, Johnson HT. Diffusion induced stresses in buckling battery electrodes. Journal of the Mechanics and Physics of Solids. 2012;60:1103-21. [30] Zhang J, Lu B, Song Y, Ji X. Diffusion induced stress in layered Li-ion battery electrode plates. Journal of Power Sources. 2012;209:220-7. [31] Song Y, Lu B, Ji X, Zhang J. Diffusion induced stresses in cylindrical lithium-ion batteries: Analytical solutions and design insights. Journal of The Electrochemical Society. 2012;159:A2060-A8. [32] Lu B, Song Y, Guo Z, Zhang J. Modeling of progressive delamination in a thin film driven by diffusion-induced stresses. International Journal of Solids and Structures. 2013;50:2495-507. 21
[33] He YL, Hu HJ, Song YC, Guo ZS, Liu C, Zhang JQ. Effects of concentration-dependent elastic modulus on the diffusion of lithium ions and diffusion induced stress in layered battery electrodes. Journal of Power Sources. 2014;248:517-23. [34] Hao F, Fang D. Reducing diffusion-induced stresses of electrode-collector bilayer in lithium-ion battery by pre-strain. Journal of Power Sources. 2013;242:415-20. [35] Zhang XY, Hao F, Chen HS, Fang DN. Diffusion-induced stresses in transversely isotropic cylindrical electrodes of lithium-lon batteries. Journal of The Electrochemical Society. 2014;161:A2243-A9. [36] Zhang XY, Hao F, Chen HS, Fang DN. Diffusion-induced stress and delamination of layered electrode plates with composition-gradient. Mechanics of Materials. 2015;91, Part 2:351-62. [37] Li D, Li Z, Song Y, Zhang J. Analysis of diffusion induced elastoplastic bending of bilayer lithium-ion battery electrodes. Applied Mathematics and Mechanics. 2016;37:659-70. [38] Yang F, Li JCM. Diffusion-induced beam bending in hydrogen sensors. Journal of Applied Physics. 2003;93:9304-9. [39] Hao F, Gao X, Fang D. Diffusion-induced stresses of electrode nanomaterials in lithium-ion battery: The effects of surface stress. Journal of Applied Physics. 2012;112:103507. [40] Crank J. The mathematics of diffusion. Oxford, UK: Oxford University Press, 1979. [41] Stoney GG. The tension of metallic films deposited by electrolysis. Proceedings of the Royal Society of London Series A. 1909;82:172-5. [42] Freund LB, Suresh S. Thin film materials: Stress, defect formation and surface evolution. Cambridge, UK: Cambridge University Press, 2004. [43] Sethuraman VA, Van Winkle N, Abraham DP, Bower AF, Guduru PR. Real-time stress measurements in lithium-ion battery negative-electrodes. Journal of Power Sources. 2012;206:334-42. [44] Hull D. An introduction to composite materials. Cambridge, UK: Cambridge University Press, 1981. [45] Halpin JC, Kardos JL. The Halpin-Tsai equations: A review. Polymer Engineering & Science. 1976;16:344-52. [46] Wang B, Luo B, Li X, Zhi L. The dimensionality of Sn anodes in Li-ion batteries. Materials Today. 2012;15:544-52. [47] Qi Y, Hector LG, James C, Kim KJ. Lithium concentration dependent elastic properties of battery electrode materials from first principles calculations. Journal of The Electrochemical Society. 2014;161:F3010-F3018.
22
Appendix The lithium ions are assumed to immigrate along the thickness direction of the electrode only, and governed by the following one-dimensional Fick’s law
D
(A1)
∂ 2c ∂c − =0 ∂z 2 ∂t
where D is the effective diffusivity of lithium ions in the active plate, which is taken to be a constant in this study, and c is the concentration of lithium ions, which depends on the thickness variable z and time t. The initial concentration of lithium ions in the active plate is assumed to be zero, i.e., c = 0, for t = 0
(A2)
The boundary conditions for the active plate under the charging with a constant surface lithium-ion flux i0, i.e. for the galvanostatic operation, are ∂c = 0 at z = 0 ∂z ∂c i D = 0 at z = hp ∂z F
(A3)
where F is the Faraday constant. Thus, the lithium-ion concentration in the active plate can be obtained as follows [40] i0 h p Dt 3 z 2 − h p2 2 c ( z, t ) = − 2 + FD h p2 6 hp2 π
∞
∑
( −1)
n =1
n2
n
nπ z n 2π 2 Dt cos exp − hp h 2p
(A4)
The boundary conditions for the active plate under charging with a constant surface lithium-ion concentration c0 , i.e. for the potentiostatic operation, are given as ∂c = 0 at z = 0 ∂z c = c0 at z = hp
(A5)
The corresponding solution is [40] ( 2 m − 1) 2 π 2 Dt ( 2m − 1) π z c ( z, t ) = c0 − c0 ∑ cos exp − 2hp 4hp2 m =1 ( 2m − 1) π ∞
4 ( −1)
m −1
23
(A6)
S0263-8223(16)32886-0 http://dx.doi.org/10.1016/j.compstruct.2017.01.011 COST 8144
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
16 December 2016 4 January 2017 5 January 2017
Please cite this article as: Liu, D., Chen, W., Shen, X., Diffusion-induced stresses in graphene-based composite bilayer electrode of lithium-ion battery, Composite Structures (2017), doi: http://dx.doi.org/10.1016/j.compstruct. 2017.01.011
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Diffusion-induced stresses in graphene-based composite bilayer electrode of lithium-ion battery Dongying Liua, b, Weiqiu Chena,c,d,e, *, Xudong Shena a
Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China
b
Department of Engineering Mechanics, China University of Petroleum, Qingdao 266580, China
c
State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou 310027, P.R.
China d
Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province, Zhejiang University, Hangzhou
310027, P.R. China e
Soft Matter Research Center, Zhejiang University, Hangzhou 310027, P.R. China
Abstract The diffusion-induced stresses in a bilayer electrode composed of a current collector and a composite active plate of lithium-ion battery are evaluated analytically. The active plate of the bilayer electrode is reinforced by graphene platelets (GPLs), with the corresponding material properties predicted by the combination of the well-established Halpin-Tsai model and the rule of mixture. The diffusion-induced stresses and curvature of the bilayer electrode under either galvanostatic or potentiostatic charging operation are derived. In addition, modified Stoney formulas for the bilayer electrode are developed for the two charging operations. A parametric study is conducted through numerical examples, with a particular focus on the reinforcing effect of GPLs on the stresses and deformation in the bilayer electrode. The size effects of GPLs, including width, length and thickness, on the stresses and deformation are also discussed. Keywords: Diffusion-induced stresses; graphene-based composite electrode; graphene platelets; Stoney formula.
*
Author for correspondence. Tel.: +86 571 87951866; fax: +86 571 87951866. E-mail address: [email protected] (W.Q. Chen). 1
1. Introduction Lithium-ion batteries (LIBs) have been widely used as power sources in portable electronics, mobile communication devices, as well as in implantable biomedical devices, and have aroused a great deal of interest in scientific and industrial fields due to their high energy density, tiny memory effect, and environmental friendliness [1, 2]. However, the low power capacity of the current LIBs cannot satisfy the ever-growing demand for large-scale applications such as electric vehicles and scalable electricity storage. To overcome this problem, significant attention has been paid to the development of advanced electrode materials with higher energy density and power density [3]. Among the potential materials for electrodes of LIBs, graphene-based composites exhibit many alluring and advantageous characteristics including high charge mobility and electron transport capability, mostly because of the outstanding physical properties of graphene, such as large specific surface area, excellent chemical stability, and electrochemical activity. In such composite electrodes, graphene is highly compatible with other dissimilar active components (such as transitional metal oxides and conductive polymer), and coexists with them in various forms of microstructures such as anchored, wrapped, encapsulated, layered and sandwich-like [4, 5]. Graphene-based composites have been considered as one of the most promising candidates for next-generation electrodes of LIBs, including composite anodes and cathodes [6, 7]. Paek et al. [8] fabricated a composite nanoporous anode by distributing graphene nanosheets in rutile SnO2, and found the graphene-based SnO2 composite electrode exhibits a much higher capacity than that of the bare SnO2 anode. Wang et al. [9] reported a first principle study on the electrochemical performance of the Sn-graphene nanocomposite anode by constructing 3D nanocomposite architecture with the combination of nanosized Sn particles and graphene nanosheets. The Fe3O4-graphene composite anode [10] and CuO-graphene composite anode [11] also show a significant improvement in electrochemical performances of nanostructured transition metal oxides as anode materials for LIBs. Sn-based anode composed of active material encapsulated by graphene was developed, with excellent electrochemical properties including high rate capability and stable cyclic performance [12]. Meanwhile, improvements on cathodes of their fairly low electronic conductivities, rate capability and cyclability could 2
be achieved by using graphene as additives dispersed into the corresponding active material matrix, such as LiFePO4 [13, 14], Li3V2(PO4)3 [15], Li2FeTiO4 [16], and sulfur [17]. For the recent research progress in graphene-based composite electrode for electrochemical capacitors and LIBs, the reader is referred to the overview article by Xu et al. [18]. One bottleneck in developing batteries with high energy density is the damage induced by the insertion/extraction of lithium-ion into/from the active material of electrodes. Taking the Si anode as an example, the volumetric swelling induced by charging will be nearly 400%, which usually leads to fracture [19]. The addition of graphene nanosheets into the Si matrix can alleviate the significant volume changes effectively [20, 21]. Graphene-based composites have been demonstrated to be remarkably effective in enhancing the mechanical properties, including fracture toughness, stiffness, strength, and fatigue resistance, in comparison to the pristine matrix materials [22]. For example, the graphene-nanoribbon/epoxy composites exhibit significant increase in the elastic modulus and tensile strength compared to the pure epoxy [23], and the elastic modulus of epoxy reinforced by a low content of graphene platelets (GPLs) is 31% greater than the pristine epoxy [24]. Therefore, graphene in composite electrodes could not only enhance their electrochemical performances but also improve their mechanical properties. Consequently, graphene-based composite electrodes appear to be superb candidates for improving the performances of anode/cathode to meet the demands of higher energy and power density, larger cyclability and rate capability, and more safety of LIBs. As is well-known, diffusion-induced stress (DIS) arising from insertion/extraction of lithium ions into/from the lattice of active materials is one of the main reasons for mechanical failure of electrodes which may lead to loss of capacity and degradation of cyclic performance of LIBs [25]. Several attempts have been devoted to evaluating DISs in electrodes during charging and discharging. Christensen [26] developed a mathematical model for DISs in a porous electrode made of spherical active materials. Deshpande et al. [27] studied the surface effects on DISs in nanowire electrode structures. Bower and Guduru [28] presented a finite element model to describe the coupled processes of diffusion, deformation and fracture in electrode microstructures. Bhandakkar and Johnson [29] simulated and analyzed 3
morphological changes and DISs in Si honeycomb structured electrode. The work mentioned above all focused on the DISs at particle level and neglected the influence of the current collector. Zhang and his co-authors [30-33] initiated the study of multilayered electrodes composed of active plates and current collector from the viewpoint of mechanics, and systematically developed analytical solutions that are very helpful in designing multilayered electrodes with high performance. Hao and Fang [34] proposed a strategy to reduce the built-in stresses due to lithium-ion diffusion by inducing pre-strain in a bilayer electrode. Zhang et al. studied DISs in transversely isotropic cylindrical electrodes [35] as well as in layered electrode plates with a composition-gradient [36]. Recently, Li et al. [37] carried out an elastoplastic analysis of the lithiation-induced deformation in a bilayer electrode. Although there have been a huge number of research reports either on the material behavior of composite electrodes or on DISs in layered electrodes, as mentioned above, the study on DISs in graphene-based composite electrodes is still unavailable, to the authors’ knowledge. This work attempts to fill the gap by considering the diffusion-induced stresses in a bilayer electrode composed of current collector and active plate. The active plate of the bilayer electrode is a composite containing both active particles and pore-filling electrolyte and simultaneously reinforced by graphene platelets (GPLs). The GPLs are assumed to be dispersed uniformly and randomly in the matrix of the active plate. The corresponding material properties of the composite active plate will be predicted by the well-established Halpin-Tsai model and the rule of mixture. Two charging conditions, i.e. galvanostatic and potentiostatic, will be considered, and the corresponding diffusion-induced stresses and curvature of the bilayer electrode will be derived analytically. Modified Stoney formulas for the bilayer electrode will also be developed. In Zhang et al.’s work [30], the effects of material and geometric parameters of the bilayer electrode on DISs and deformation have been discussed in detail. Hence, we will focus, in numerical calculations, on the effects of GPLs on the behavior of the bilayer composite electrode. In particular, the influences of size and content of GPLs will be studied numerically.
4
2. Basic equations
Fig. 1. Configuration of a bilayer electrode.
A bilayer electrode composed of a current collector with thickness hc and a composite active plate with thickness h p is schematically depicted in Fig. 1. Referred to the Cartesian coordinates
( x, y, z ) ,
the center of the interface between the active plate and the current
collector is taken as the origin of the coordinates, and the z-axis is aligned along the thickness direction of the plate pointing from the bottom surface to the top surface of the active plate. Lithium ions are assumed to be inserted/extracted into/from the active plate along the thickness direction only. Both the current collector and the composite active plate are treated as isotropic and elastic materials at the macroscale, and only linear elastic deformation is considered in this study. The classical plate theory and the assumption of equal biaxial stress for a large electrode plate yield the following in-plane normal strain components
ε x = ε y = ε 0 + zκ
(1)
where ε 0 represents the in-plane normal strain at the interface, and κ is the curvature of the bilayer electrode. The constitutive equations for the current collector and active plate can be written as [30, 38]
1 3
σ p = E ′p ( ε 0 + κ z ) − E ′p Ωc σ c = Ec′ ( ε 0 + κ z ) 5
(2)
where σ is the in-plane equal biaxial stress, c is the molar concentration of lithium ions, Ω is the partial molar volume of solute, and E′ = E (1 −ν ) is the biaxial modulus with E and
ν being the elastic modulus and Poisson’s ratio, respectively. The subscripts p and c indicate the corresponding physical quantities of the active plate and current collector, respectively. The diffusion process of lithium ions in the active plate would change the lattice structure of the active plate, which means the material properties of the active plate depend on the concentration of lithium ions. However, this work will focus on the enhancing effects of GPLs on the mechanical behaviors of the electrode, and hence the diffusion coefficient, elastic modulus and Poisson’s ratio of the active plate are all assumed to be constant during both the charging and discharging processes [30, 31, 39]. In general, the characteristic time for elastic deformation of the electrode is much smaller than that for lithium-ion diffusion. Thus, the elastic deformation induced by diffusion of lithium ions can be reasonably approximated as a static process, and the mechanical equilibrium is expressed by
∫
0
∫
0
− hc
− hc
hp
σ c dz + ∫ σ p dz = 0 0
hp
σ c zdz + ∫ σ p zdz = 0
(3)
0
The substitution of Eq. (2) into Eq. (3) yields
Aε 0 + Bκ − N p = 0 Bε 0 + Iκ − M p = 0
(4)
where hp
0
0
− hc
A = ∫ E ′p dz + ∫
Ec′dz = E ′p hp ( mh + 1)
1 E ′p hp2 (1 − mh2 ) 2 hp 0 1 I = ∫ E′p z 2dz + ∫ Ec′ z 2dz = E ′p hp3 ( mh3 + 1) 0 − hc 3 hp
0
0
− hc
B = ∫ E ′p zdz + ∫
Ec′ zdz =
(5)
in which m = Ec′ E ′p and h = hc h p are the ratios of the biaxial modulus and thickness between the current collector and the active plate, and hp hp 1 1 N p = E ′p Ω∫ cdz, M p = E ′p Ω ∫ czdz 0 0 3 3
(6)
which can be considered as the resultant axial force and moment in the active plate induced 6
by the diffusion of lithium ions. Then solving for ε 0 and κ from Eq. (4) gives
κ =− ε0 =
6 (1 − mh 2 ) N p hp − 2 ( mh + 1) M p E ′p h3p β
(7)
2 2 ( mh3 + 1) N p hp − 3 (1 − mh 2 ) M p 2 E ′p hp β
where β = m2 h4 + 4mh3 + 6mh 2 + 4mh + 1 . It is notable that if the molar concentration c is known, N p and M p , and then the curvature κ and in-plane strain ε 0 can be determined. Two different ways of charging operation are considered here, i.e. the galvanostatic charging with a constant surface lithium-ion flux ( i0 ) and the potentiostatic charging with a constant surface lithium-ion concentration ( c0 ). The lithiation/delithiation process of lithium ions is governed by the Fick’s law, and has been well discussed by Crank [40]. For completeness, a brief introduction to the diffusion analysis is outlined in the appendix. The substitutions of the lithium-ion concentration under the galvanostatic/potentiostatic charging operation in Eqs. (A4) and (A6) of the Appendix, into Eq. (6) and then Eqs. (7) and (2) yield the curvature κ of the bilayer electrode and in-plane equal biaxial stresses σ p and
σ c in the active plate and current collector, respectively. They are given below as 1 2i Ω 2 κ = 0 mh (1 + h ) t + 2 ( mh + 1) + 4 β FD 24 π
∞
∑
( −1)
n
−1
n4
n =1
exp ( −n2π 2 t )
mh 6 (1 + h ) z − ( mh3 + 3h + 4 ) t E ′ Ωi h β σp = p 0 p n 3 FD 3 z 2 − 1 2 ∞ ( −1) cos ( nπ z ) exp ( −n 2π 2 t 2 − 6 + π 2 ∑ n n =1 +
2 E ′p Ωi0 hp
β FD
1 2 2 ( mh + 1) z + mh − 1 + 4 24 π 2
7
∞
∑ n =1
( −1)
n
n4
−1
)
exp ( −n 2π 2 t )
(8)
(9)
4mh3 + 1 + 3mh2 + 6mh (1 + h ) z t Ec′Ωi0 hp σc = 3β FD +6 2 ( mh + 1) z + mh2 − 1 1 + 2 24 π 4
n ∞ ( −1) − 1 exp ( −n2π 2 t ) ∑ 4 n n =1
(10)
for the galvanostatic charging, and 2Ωc0 κ= hp β
8 mh (1 + h ) − 2 π
n 2 ( 2n − 1) 2 π 2 t 4 ( mh + 1)( −1) 1 exp − mh + 2mh + 1 − ∑ 2 2 n − 1 π 4 ( ) 2 n − 1 n =1 ( ) ∞
(11) ∞ 16 ( 2n − 1) 2 π 2 t 1 3 2 2 2 ( 2mh + 3mh − 1) + 3 z (1 + 2mh + mh ) ∑ n=1 2n − 1 2 exp − 4 π ( ) E ′p Ωc0 3 σp =− + mh ( mh + 3h + 4 ) − 6mh (1 + h ) z 3β n 2 ∞ 96 ( −1) exp − ( 2n − 1) π 2 t 2 + mh − 1 + 2 z mh + 1 ( ) ∑ 3 3 4 π n =1 ( 2n − 1)
+
4 E ′p Ωc0 3
( 2n − 1)2 π 2 t ( 2n − 1) π z cos ∑ exp − 2 4 n =1 ( 2n − 1) π ∞
( −1)
n −1
(12)
∞ 16 ( 2n − 1) 2 π 2 t 1 3 2 2 2 ( 2mh + 3mh − 1) + 3 z (1 + 2 mh + mh ) ∑ n =1 2n − 1 2 exp − 4 π ( ) E ′Ωc σ c = − c 0 +mh ( mh3 + 3h + 4 ) − 6mh (1 + h ) z 3β n ∞ ( 2n − 1)2 π 2 t 96 −1) ( 2 exp − + 3 mh − 1 + 2 z ( mh + 1) ∑ 3 4 n =1 ( 2n − 1) π
(13) 2 for the potentiostatic charging, where t = Dt hp and z = z h p are the dimensionless time
and z-coordinate, respectively, F is the Faraday constant, and D is the effective diffusivity of lithium ions in the active plate. It can be found that the curvature for the galvanostatic charging operation in Eq. (8), is identical to that obtained by Zhang et al. [30]. However, the stresses in the current collector and the active plate are different. After a careful check, we found that it is due to the incorrect expressions of α1 and α 2 in Eqs. (23) and (24) in Zhang et al. The correct ones should be
α1 = ( 4mh 3 + 3mh 2 + 1) α 3 and α 2 = 6 ( mh 2 − 1) α 3 using the same notions as in Zhang et 8
al. [30]. The Stoney formula [41] has been widely used in film/substrate structures, which relates the curvature to the stress in the film. Plentiful work has been devoted to extending the limitation of the Stoney formula [42]. Sethuraman et al. [19] performed real-time measurements of stress evolution in a silicon thin film electrode during electrochemical lithiation and delithiation using the multi-beam optical sensor technique, and developed a complex modified Stoney formula based on the principle of minimum potential energy to evaluate the stresses in LIBs [43]. However, the process of lithium diffusion was not considered by Sethuraman et al. [19, 43], and consequently, their modified Stoney formula is independent of time t , which is not rigorous for determining the DISs through measuring the curvature of the electrode. To develop the relation between the stress in the active plate and the curvature of the bilayer electrode, an average biaxial stress through the thickness of the active plate is now defined as
σ p =
1 hp
∫
hp
0
σ p dz
(14)
The substitution of Eqs. (9) and (12) into the above equation yields
1 mhE ′p Ωi0 hp 2 3 σ p = − ( mh + 1) t − 6 (1 + h ) + 4 3 FD β 24 π
∞
∑
( −1)
n =1
n
n4
−1
exp ( −n 2π 2 t )
(15)
for the galvanostatic charging, and ( mh3 − 3h − 2 ) ∞ ( 2n − 1)2 π 2 t 1 exp − ∑ 2 π2 4 n =1 ( 2n − 1) 8mhE ′p Ωc0 σ p = n 3β ( 2 n − 1) 2 π 2 t 12 (1 + h ) ∞ ( −1) 3 exp − − 1 − mh ∑ − 3 3 π 4 n =1 ( 2n − 1)
(16)
for the potentiostatic charging, respectively. From Eqs. (8) and (15), the modified Stoney formula for the bilayer electrode under galvanostatic charging can be expressed as
σ p =
mhE ′p hp 2Ωi0 3 (1 + h ) κ − 6 (1 + mh ) FD 9
t
(17)
Similarly, the modified Stoney formula under potentiostatic charging is mh (1 + h )2 mhE ′p (1 + h ) hp 8 σ p = κ − Ωc0 − 2 β 3π ( mh + 1) 2
( 2n − 1) 2 π 2 t exp − ∑ 2 4 n =1 ( 2n − 1) ∞
1
(18)
These two formulae provide the basis for evaluating the stresses in LIBs induced by diffusion of lithium ions if the curvature of the bilayer electrode has been measured experimentally. 3. Effective elastic properties of GPLs-reinforced composite
In the composite active plate of the bilayer electrode, the GPLs, acting as an effective rectangular solid fiber with width ( wGPL ), length ( lGPL ), and thickness ( hGPL ), are assumed to be dispersed randomly, but uniformly, in the matrix [24]. To evaluate the effective elastic properties of the GPLs-reinforced composite, the following well-established Halpin-Tsai equations [44, 45] are adopted
3 5 E p = E + E⊥ 8 8 E =
1 + ξlηlVGPL 1 + ξ wη wVGPL × EM , E⊥ = × EM 1 − ηlVGPL 1 −η wVGPL
(19) (20)
where E p is the effective Young’s modulus of the composite active plate, and E and E⊥ are the longitudinal and transverse moduli, respectively; EM is the Young’s modulus of the matrix of the active plate, VGPL is the volume fraction of GPLs, and ηl and η w are taken to be
ηl =
( EGPL EM ) − 1 , η = ( EGPL EM ) − 1 ( EGPL EM ) + ξl w ( EGPL EM ) + ξ w
(21)
in which EGPL is the Young’s modulus of GPLs, and ξl and ξw depend on the geometry of GPLs, which can be expressed as lGPL hGPL
ξl = 2
wGPL , ξw = 2 hGPL
(22)
The effective Poisson’s ratio of the composite active plate can be evaluated by using the general rule of mixture [24], which gives 10
ν p = ν GPLVGPL +ν M (1 − VGPL )
(23)
where ν GPL and ν M are the Poisson’s ratios of GPLs and matrix, respectively. Once the effective Young’s modulus and Poisson’s ratio of the active plate are evaluated, the corresponding shear modulus can be obtained as Gp =
Ep 2 (1 + ν p )
(24)
The volume fraction VGPL of GPLs can be calculated using the weight fraction WGPL of GPLs as VGPL =
WGPL WGPL + ( ρ GPL ρ M )(1 − WGPL )
(25)
with ρGPL and ρM being the densities of GPLs and matrix, respectively. The volume and weight fractions of the matrix material in the composite active plate can be obtained by
VM = 1 − VGPL and WM = 1 − WGPL , respectively. The Halpin-Tsai equations are a set of empirical relations for the prediction of the elastic properties of a composite material in terms of the properties of the filler and matrix together with their proportions and geometry, and have been confirmed by experimental measurements. The mechanical properties of the epoxy nanocomposite with GPLs ( WGPL = 0.1% ,
lGPL = 2.5 µm , wGPL = 1.5 µm , and hGPL = 1.5 nm ) were measured by Rafiee et al [24], and the Young’s modulus obtained through uniaxial tensile testing is ~3.7 GPa. The prediction of the Young’s modulus by the Halpin-Tsai equations, i.e. Eq. (19)1, is 3.8 GPa , which overpredicts the experiment data by up to 2.7% only. It should be noted that the evaluation of Young’s modulus by the modified Halpin-Tsai equations presented in [24] (3.23 GPa) is much lower than the experimental result because of the neglect of the width effects of GPLs. It should be stressed that the elastic modulus of the composite active plate evaluated by the Halpin-Tsai equations is an “averaged” property, which is suitable to describe the global characteristics of the composite (such as vibration frequency and buckling load). However, if local phenomena are considered (e.g., local buckling, delamination), some other refined 11
models (e.g. high-order multi-scale models) should be preferred. 4. Numerical results and discussion
Among various electrode materials, Tin (Sn)-based active materials have been considered as one type of the most promising anode materials for high-capacity LIBs due to its high theoretical capacity, widespread availability, and environmentally benign nature [46]. Thus a bilayer anode composed of Tin (Sn)-based composite active plate reinforced by GPLs and Cu current collector under galvanostatic charging operation is taken as the numerical example, whose material properties are EM = 24.7 GPa , ν M = 0.24 , and ρ M = 7.31 g cm 3 [47]; EGPL = 1.01 TPa , ν GPL = 0.186 , and ρ GPL = 1.06 g cm 3 [24]; Ec = 115 GPa and
ν c = 0.35 , respectively. The geometric parameters of GPLs are taken to be lGPL = 2.5 µm , wGPL = 1.5 µm , and tGPL = 1.5 nm [24] in numerical calculations, unless otherwise stated.
Fig. 2. Effective elastic modulus of the composite active plate.
The effective elastic moduli of the composite active plate reinforced by different contents of GPLs are evaluated by the Halpin-Tsai equations, which are shown in Fig. 2. The elastic modulus of the pristine Tin ( WGPL = 0 ) is also given for comparison. It is obvious that the blending of GPLs nanofillers into the Tin-based active material would promote the elastic 12
modulus dramatically, and the more the content of GPLs, the larger the effective modulus. The elastic modulus of the composite active material with WGPL = 1.5% ( E p = 116.9 GPa ) is nearly five times larger than that of the pristine Tin (24.7 GPa).
Fig. 3. Variations of the dimensionless stress σ in the active plate ( 0 ≤ z ≤ 1 ) and the current collector ( −0.1 ≤ z ≤ 0 ).
Fig. 3 presents the dimensionless stresses σ = σ FD ( E ′p Ωin h p ) in the active plate and the current collector for different contents of GPLs. The lithium ions insert into the active plate from the top surface ( z = 1), and make the lithiated region expand. However, the unlithiated region and the current collector will resist the deformation of the electrode. As a consequence, bending deformation is generated. The overall stress in the active plate is produced by the combination of the diffusion-induced stress and bending stress. At the beginning of the charging operation, the overall stress is mainly determined by the diffusion-induced stress. As more and more lithium ions migrate into the active plate, the bending stress becomes dominant. It can be seen that the stress near the bottom surface is always compressive due to the restriction of the current collector, and the stress near the top surface may become tensile due to the increasing of the bending stress. At the earlier stage of the charging operation, the compressive stress at the top surface of the GPLs-reinforced composite active plate is smaller than that of the pristine Tin active plate. However, the 13
compressive stress at the bottom surface behaves oppositely. Thus, an even distribution of the compressive stress is achieved by using the GPLs-reinforced composite active plate. Furthermore, with time increasing, the stress at the top surface becomes tensile, and both the compressive and tensile stresses (at the bottom and top surfaces, respectively) in the GPLs-reinforced composite active plate are smaller than those of the pristine Tin active plate. It also can be found that the overall stress in the GPLs-reinforced composite active plate generally could be lessened in comparison with that in the active plate composed of pristine Tin, except the compressive stress near the bottom surface ( z = 0 ) at the beginning of the charging operation. Results also show that, the maximum compressive stress of the active plate arises at the bottom surface of the active plate, and a large stress drop, i.e. the difference between the tensile stress at the top surface of the current collector and the compressive stress at the bottom surface of the GPLs-reinforced composite active plate, σ c ( z = 0 ) − σ p ( z = 0 ) , also occurs there [34]. The stress drop enlarges as time increases, which however can be reduced by using the GPLs-reinforced composite as the active plate. The maximum compressive stress can be reduced as well by this strategy. Thus, the reinforcement of GPLs is greatly beneficial to preventing the electrode from debonding and delamination.
(a) Bottom surface
(b) Top surface
Fig. 4. Variations of the dimensionless stress σ p at the surfaces of the composite active plate against the thickness ratio.
Fig. 4 gives the variations of the biaxial stress σ p at the bottom ( z = 0 ) and top ( z = 1) surfaces of the active plate against the thickness ratio between the current collector and 14
composite active plate. As pointed out by Zhang et al. [30], the increasing of the thickness ratio hc h p would promote the peak stress due to the stronger restriction of the current collector, which is similar to that in the analysis of thermal stress [42]. However, it can be seen in Fig. 4 that the peak stress has been reduced in the GPLs-reinforced composite electrode, no matter compressive or tensile. Therefore, it can be a practical approach to relaxing the stress by using the GPLs-reinforced composite as an active plate in lithium ion batteries.
(a) Bottom surface, t = 1 .
(b) Top surface, t = 1 .
Fig. 5. Size effects of GPLs on the dimensionless stress σ p in the composite active plate.
As can be observed above, higher lithium-ion concentration corresponds to larger stress level [30]. Thus a relatively typical and larger lithiation time, i.e. t = 1 , is chosen as the dimensionless reference time in the following disscusion. Fig. 5 reveals the size effects of GPLs (i.e. length lGPL , width wGPL , and thickness tGPL ) in terms of the length-to-thickness ratio lGPL tGPL and the length-to-width ratio lGPL wGPL on the dimensionless stress σ p in the composite active plate. The length of GPLs keeps unchanged ( lGPL = 2.5 µm ), and lGPL wGPL = 1
and
lGPL wGPL ≠ 1
correspond
to
a
square-shaped
platelet
and
a
rectangle-shaped platelet, respectively. It is seen that the square-shaped GPLs ( lGPL wGPL = 1 ) as nanofillers can reduce the compressive and/or tensile stresses in the composite active plate more effectively. In other words, GPLs with larger surface area are better-reinforcing nanofillers than their counterparts with smaller surface area due to the larger contact area 15
between the matrix and GPLs. A significant decrease in the dimensionless stress σ p is observed against the length-to-thickness ratio lGPL tGPL , which means GPLs with less graphene layers could be more effective in lessening the stress in the active plate. Thus, a monolayer graphene may be the most outstanding candidate to reinforce the composite electrode.
Fig. 6. Variations of the dimensionless curvature κ of the active plate against the dimensionless time.
The variations of the dimensionless curvature κ = κ FD in Ω of the composite electrode with the dimensionless time t are plotted in Fig. 6. The bending deformation of the composite electrode is due to the restriction of the unlithiated region of the active plate and the current collector. At the beginning of the charging operation, the deformation of the composite electrode is mainly caused by the intercalation of lithium ions. As the lithium ions insert into the active plate continually, the current collector will inhibit the expansion of the active plate more dominantly. It can be observed from Fig. 6 that the blending of GPLs into the Tin active plate would firstly promote, and then reduce the curvature of the bilayer electrode with respect to time. This is the outcome of the competition between the diffusion-induced bending and the bending deformation due to the stiffness mismatch between the active plate and the thick current collector. 16
Fig. 7. Dimensionless curvature κ of the bilayer composite electrode with different weight fractions of GPLs against the thickness ratio h.
The variations of the dimensionless curvature κ = κ FD in Ω of the bilayer electrode at dimensionless times t = 1 against the thickness ratio h = hc h p are plotted in Fig. 7. The curvature first increases, and then decreases with the thickness ratio. This is the result of competition between the diffusion-induced deformation and the bending deformation. The thicker the copper current collector is, the stronger the restriction it exerts on the deformation of the electrode. For a bilayer electrode with a thin current collector (small h), the blending of GPLs nanofillers could reduce the curvature evidently except at the initial period of charging. However, for the case with a thicker current collector, the GPLs-reinforced composite active plate would hold a larger deformation. This is due to the reinforcing GPLs would increase the stiffness mismatch between the active plate and the thick current collector.
17
Fig. 8. Size effects of GPLs on the dimensionless curvature κ of the bilayer electrode.
The size effects of GPLs on the dimensionless curvature κ of the bilayer electrode are shown in Fig. 8. It is found that the square-shaped GPLs would increase the curvature of the electrode at the beginning of charging, and decrease the curvature as more and more lithium ions move into the active plate. It is noted that the curvature of the electrode at the initial period of charging is inconspicuous compared with that at the subsequent charging stage. As can be seen in Fig. 8, the GPLs with larger area could reduce the curvature more effectively. Meanwhile, thinner GPLs would also reduce the curvature. Thus it may be concluded that a large-area monolayer graphene would be preferable to lessen the crucial deformation of the electrode. 5. Conclusions
In this paper, the diffusion-induced stresses in a bilayer electrode composed of a current collector and a composite active plate have been evaluated analytically. The composite active plate is reinforced by graphene platelets (GPLs), which was treated as an isotropic elastic material macroscopically. The corresponding material properties were predicted by adopting the Halpin-Tsai model along with the rule of mixture. The analytical solutions of the diffusion-induced stresses and curvature of the bilayer electrode under galvanostatic and 18
potentiostatic charging conditions were obtained based on the classical thin plate theory. Modified Stoney formulas for the bilayer electrode were also developed, which can be used to evaluate the stresses in the active plate from the deformation of the bilayer electrode in real time. Effects of GPLs on the DISs and deformations of the bilayer electrode were studied extensively through numerical examples. The blending of GPLs nanofillers into the matrix material of the active plate can improve the mechanical properties of the bilayer electrode effectively. From the numerical results, it can be concluded that: (1) The elastic modulus of the active plate can be enhanced observably when reinforced by GPLs. Even low content of GPLs could increase the elastic modulus several times. (2) Generally, both the compressive and tensile stresses in the GPLs-reinforced composite active plate, as well as the stress drop at the interface, could be lessened prominently in comparison with those in the pristine active plate. (3) The blending of GPLs could reduce the deformation of a bilayer electrode with a thin current collector. However, it would increase the deformation of an electrode with a thick current collector. (4) A monolayer graphene would be the most outstanding candidate to reinforce the mechanical behaviors of the bilayer electrode.
Acknowledgements
The work was supported by the National Natural Science Foundation of China (Nos. 11621062, 11532001, and 11402310), the Fundamental Research Funds for the Central Universities (Nos. 2016XZZX001-05 and 15CX08004A), and partly supported by the Foundation for Outstanding Young Scientist in Shandong Province (No. BS2014CL022).
19
References [1] Scrosati B, Hassoun J, Sun YK. Lithium-ion batteries. A look into the future. Energy and Environmental Science. 2011;4:3287-95. [2] Scrosati B, Garche J. Lithium batteries: Status, prospects and future. Journal of Power Sources. 2010;195:2419-30. [3] Armand M, Tarascon JM. Building better batteries. Nature. 2008;451:652-7. [4] Raccichini R, Varzi A, Passerini S, Scrosati B. The role of graphene for electrochemical energy storage. Nature Materials. 2015;14:271-9. [5] Brownson DA, Kampouris DK, Banks CE. An overview of graphene in energy production and storage applications. Journal of Power Sources. 2011;196:4873-85. [6] Huang X, Zeng Z, Fan Z, Liu J, Zhang H. Graphene-based electrodes. Advanced Materials. 2012;24:5979-6004. [7] Wu ZS, Zhou G, Yin LC, Ren W, Li F, Cheng H-M. Graphene/metal oxide composite electrode materials for energy storage. Nano Energy. 2012;1:107-31. [8] Paek SM, Yoo E, Honma I. Enhanced cyclic performance and lithium storage capacity of SnO2/graphene nanoporous electrodes with three-dimensionally delaminated flexible structure. Nano Letter. 2008;9:72-5. [9] Wang G, Wang B, Wang X, Park J, Dou S, Ahn H, Kim K. Sn/graphene nanocomposite with 3D architecture for enhanced reversible lithium storage in lithium ion batteries. Journal of Materials Chemistry. 2009;19:8378-84. [10] Lian P, Zhu X, Xiang H, Li Z, Yang W, Wang H. Enhanced cycling performance of Fe3O4-graphene nanocomposite as an anode material for lithium-ion batteries. Electrochimica Acta. 2010;56:834-40. [11] Mai Y, Wang X, Xiang J, Qiao Y, Zhang D, Gu C, Tu J. CuO/graphene composite as anode materials for lithium-ion batteries. Electrochimica Acta. 2011;56:2306-11. [12] Wu C, Maier J, Yu Y. Sn-based nanoparticles encapsulated in a porous 3D graphene network: Advanced anodes for high-rate and long life Li-ion batteries. Advanced Functional Materials. 2015;25:3488-96. [13] Zhou X, Wang F, Zhu Y, Liu Z. Graphene modified LiFePO4 cathode materials for high power lithium ion batteries. Journal of Materials Chemistry. 2011;21:3353-8. [14] Gong C, Xue Z, Wen S, Ye Y, Xie X. Advanced carbon materials/olivine LiFePO4 composites cathode for lithium ion batteries. Journal of Power Sources. 2016;318:93-112. [15] Jiang Y, Xu W, Chen D, Jiao Z, Zhang H, Ma Q, Cai X, Zhao B, ChuY. Graphene modified Li3V2(PO4)3 as a high-performance cathode material for lithium ion batteries. Electrochimica Acta. 2012;85:377-83. [16] Yang M, Zhao X, Yao C, Kong Y, Ma L, Shen X. Nanostructured cation disordered 20
Li2 FeTiO4/graphene composite as high capacity cathode for lithium-ion batteries. Materials Technology. 2016;31:537-43. [17] Wang JZ, Lu L, Choucair M, Stride JA, Xu X, Liu HK. Sulfur-graphene composite for rechargeable lithium batteries. Journal of Power Sources. 2011;196:7030-4. [18] Xu C, Xu B, Gu Y, Xiong Z, Sun J, Zhao X. Graphene-based electrodes for electrochemical energy storage. Energy & Environmental Science. 2013;6:1388-414. [19] Sethuraman VA, Chon MJ, Shimshak M, Srinivasan V, Guduru PR. In situ measurements of stress evolution in silicon thin films during electrochemical lithiation and delithiation. Journal of Power Sources. 2010;195:5062-6. [20] Xiang H, Zhang K, Ji G, Lee JY, Zou C, Chen X, Wu J. Graphene/nanosized silicon composites for lithium battery anodes with improved cycling stability. Carbon. 2011;49:1787-96. [21] Liang B, Liu YP, Xu YH. Silicon-based materials as high capacity anodes for next generation lithium ion batteries. Journal of Power Sources. 2014;267:469-90. [22] Huang X, Qi X, Boey F, Zhang H. Graphene-based composites. Chemical Society Reviews. 2012;41:666-86. [23] Rafiee MA, Lu W, Thomas AV, Zandiatashbar A, Rafiee J, Tour JM, Koratkar NA. Graphene nanoribbon composites. ACS Nano. 2010;4:7415-20. [24] Rafiee MA, Rafiee J, Wang Z, Song H, Yu ZZ, Koratkar N. Enhanced mechanical properties of nanocomposites at low graphene content. ACS Nano. 2009;3:3884-90. [25] Mukhopadhyay A, Sheldon BW. Deformation and stress in electrode materials for Li-ion batteries. Progress in Materials Science. 2014;63:58-116. [26] Christensen J. Modeling diffusion-induced stress in Li-ion cells with porous electrodes. Journal of The Electrochemical Society. 2010;157:A366-A80. [27] Deshpande R, Cheng YT, Verbrugge MW. Modeling diffusion-induced stress in nanowire electrode structures. Journal of Power Sources. 2010;195:5081-8. [28] Bower AF, Guduru PR. A simple finite element model of diffusion, finite deformation, plasticity and fracture in lithium ion insertion electrode materials. Modelling and Simulation in Materials Science and Engineering. 2012;20:045004. [29] Bhandakkar TK, Johnson HT. Diffusion induced stresses in buckling battery electrodes. Journal of the Mechanics and Physics of Solids. 2012;60:1103-21. [30] Zhang J, Lu B, Song Y, Ji X. Diffusion induced stress in layered Li-ion battery electrode plates. Journal of Power Sources. 2012;209:220-7. [31] Song Y, Lu B, Ji X, Zhang J. Diffusion induced stresses in cylindrical lithium-ion batteries: Analytical solutions and design insights. Journal of The Electrochemical Society. 2012;159:A2060-A8. [32] Lu B, Song Y, Guo Z, Zhang J. Modeling of progressive delamination in a thin film driven by diffusion-induced stresses. International Journal of Solids and Structures. 2013;50:2495-507. 21
[33] He YL, Hu HJ, Song YC, Guo ZS, Liu C, Zhang JQ. Effects of concentration-dependent elastic modulus on the diffusion of lithium ions and diffusion induced stress in layered battery electrodes. Journal of Power Sources. 2014;248:517-23. [34] Hao F, Fang D. Reducing diffusion-induced stresses of electrode-collector bilayer in lithium-ion battery by pre-strain. Journal of Power Sources. 2013;242:415-20. [35] Zhang XY, Hao F, Chen HS, Fang DN. Diffusion-induced stresses in transversely isotropic cylindrical electrodes of lithium-lon batteries. Journal of The Electrochemical Society. 2014;161:A2243-A9. [36] Zhang XY, Hao F, Chen HS, Fang DN. Diffusion-induced stress and delamination of layered electrode plates with composition-gradient. Mechanics of Materials. 2015;91, Part 2:351-62. [37] Li D, Li Z, Song Y, Zhang J. Analysis of diffusion induced elastoplastic bending of bilayer lithium-ion battery electrodes. Applied Mathematics and Mechanics. 2016;37:659-70. [38] Yang F, Li JCM. Diffusion-induced beam bending in hydrogen sensors. Journal of Applied Physics. 2003;93:9304-9. [39] Hao F, Gao X, Fang D. Diffusion-induced stresses of electrode nanomaterials in lithium-ion battery: The effects of surface stress. Journal of Applied Physics. 2012;112:103507. [40] Crank J. The mathematics of diffusion. Oxford, UK: Oxford University Press, 1979. [41] Stoney GG. The tension of metallic films deposited by electrolysis. Proceedings of the Royal Society of London Series A. 1909;82:172-5. [42] Freund LB, Suresh S. Thin film materials: Stress, defect formation and surface evolution. Cambridge, UK: Cambridge University Press, 2004. [43] Sethuraman VA, Van Winkle N, Abraham DP, Bower AF, Guduru PR. Real-time stress measurements in lithium-ion battery negative-electrodes. Journal of Power Sources. 2012;206:334-42. [44] Hull D. An introduction to composite materials. Cambridge, UK: Cambridge University Press, 1981. [45] Halpin JC, Kardos JL. The Halpin-Tsai equations: A review. Polymer Engineering & Science. 1976;16:344-52. [46] Wang B, Luo B, Li X, Zhi L. The dimensionality of Sn anodes in Li-ion batteries. Materials Today. 2012;15:544-52. [47] Qi Y, Hector LG, James C, Kim KJ. Lithium concentration dependent elastic properties of battery electrode materials from first principles calculations. Journal of The Electrochemical Society. 2014;161:F3010-F3018.
22
Appendix The lithium ions are assumed to immigrate along the thickness direction of the electrode only, and governed by the following one-dimensional Fick’s law
D
(A1)
∂ 2c ∂c − =0 ∂z 2 ∂t
where D is the effective diffusivity of lithium ions in the active plate, which is taken to be a constant in this study, and c is the concentration of lithium ions, which depends on the thickness variable z and time t. The initial concentration of lithium ions in the active plate is assumed to be zero, i.e., c = 0, for t = 0
(A2)
The boundary conditions for the active plate under the charging with a constant surface lithium-ion flux i0, i.e. for the galvanostatic operation, are ∂c = 0 at z = 0 ∂z ∂c i D = 0 at z = hp ∂z F
(A3)
where F is the Faraday constant. Thus, the lithium-ion concentration in the active plate can be obtained as follows [40] i0 h p Dt 3 z 2 − h p2 2 c ( z, t ) = − 2 + FD h p2 6 hp2 π
∞
∑
( −1)
n =1
n2
n
nπ z n 2π 2 Dt cos exp − hp h 2p
(A4)
The boundary conditions for the active plate under charging with a constant surface lithium-ion concentration c0 , i.e. for the potentiostatic operation, are given as ∂c = 0 at z = 0 ∂z c = c0 at z = hp
(A5)
The corresponding solution is [40] ( 2 m − 1) 2 π 2 Dt ( 2m − 1) π z c ( z, t ) = c0 − c0 ∑ cos exp − 2hp 4hp2 m =1 ( 2m − 1) π ∞
4 ( −1)
m −1
23
(A6)