Li-ion batteries are ineluctably subjected to external mechanical loading or stress gradient. Such stress can be induced in battery electrode during fabrication and under normal operation . In this paper , we develop a model for stresses generated during lithiation in the thin plate electrode considering the effects of external mechanical loading. It is found that diffusion -indu ced stresses are asymmetrically distributed through the thickness of plate due to the coupling effect s of asymmetrically distributed external mechanical stress. At the very early stage during Li -ions insertion, the effects of the external mechanical loadin g is quite limited and unobvious. With the diffusion time increasing, the external mechanical loading exert s a significant influence on the evolution of stresses generated in the electrode . External compressed electrode is inclined to increase the value of stresses generated during lithiation , while external tensed electrode tends to decrease the value of stresses, and as the diffusion time increases , the effect s of the external mechanical loading on the stresses generated during lithiation become more obvious. Keywords: Lithium -ion battery; Diffusion -induced stress; External mechanical loading; Thin plate electrode *Corresponding author. Email address: w.zhou15@imperial. ac.uk, wbzhou@pku.edu.cn (W. Zhou) 2 1. Introduction Li-ion batter ies have been widely used in mini -type electric instruments such as cell phones and other small portable elec tronic devices due to their high specific capacity, light weight, and no memory effects . However, their further applications such as in the fields of electromotive vehicle, large -scale energy storage and aerospace power supply are limited by the poor cycle of life and the capacity loss of themselves . The main reason is due to the deformation and fracture of the electrode caused by the stress generated in electrodes during cycling [1-4], which can result in electrical disconnects that render electrode active material s incapable of storing Li-ions. One of the critical challenges in Li-ion battery studi es is the internal stress field distribution produced in the process of charging and discharging. Much work has been devoted to studying the stresses resulting f rom cycling of Li -ion batteries due to the intercalation and deintercalation of Li -ions [5-11], however, studies for effects of external mechanical loading on stress generation during lithiation in Li -ion battery electrodes are less . Actually, Li -ion batte ries are ineluctably subjected to external mechanical loading or stress gradient during standard usage or storage . Such stress can arise during fabrication of battery materials , which usually involves compression of the electrode to control its porosity [12-14]. Also, such stress can be induced in electrode materials under normal operation, such as batteries operating in high pressure environments, electric/hybrid vehicles or flexible applications [15]. In many cases, the influence of external stresses on t he diffusion process and the stress generation during lithiation in Li -ion battery electrodes is so significant that it cannot be neglected. Some research ers have stud ied the effect s of electrode compression on capacity and efficiency of various electrode materials . Novak et al. [16] and Gnanaraj et al. [17] found both the capacity and first cycle efficiency of batteries ha d a decrease when subjected to high levels of compression. Th ey explain ed that it was caused by electrode particle fracture for the larg est pressures as well as transport limitations within the liquid path . By studying low-loading electrode s prepared with a natural graphite from Superior Graphite and compressed at a range of moderate pressures, Shim et al. found both the reversible capacit y and irreversible capa city loss (ICL) had a decrease with increases in pressure [18]. They conclude d that the decreased reversible capacity was due to increased stress es generated within the graphite electrode , which also slowed down in Li -ions diffusion process [19]. However, much of these work were conducted using electrochemical characterization. Theoretical studies for the effects of external mechanical loading on stress generation during lithiation in electrodes which are quite necessary for Li-ion batteries have been rare . 3 In this work , we develop a model for stress generation during lithiation in Li -ion battery electrodes with planar geometries coupled with the effects of external mechanical loading . To begin with, two basic approximations for mater ial properties are introduced as follows [9,10]. Isotropic elasticity analysis of infinitesimal deformation is carried out , and large deformation related to plasticity is out of the scope of this research [20]. Mechanical properties and the diffusion coeff icient are independent of Li -ion concentration, which means that Young’s modulus, Poisson’s ratio and the diffusion coefficient are constants. Yang et al. [21] studied the effects of composition -dependent modulus, finite concentration and boundary constrai nt on Li -ion diffusion and stresses in a bilayer Cu -coated Si nano -anode. This is an important work to show that mechanical properties of electrode materials will change and have an effect on diffusion -induced stresses during the lithiation/delithiation pr ocess . Since we focus on studying the effects of external mechanical loading, changes in material properties during the lithiation/delithiation process still need to be further considered by establishing more sophisticated model. 2. Basic Theory 2.1. Diffusion equation We model the insertion and extraction of Li-ions as a diffusion process. The species flux can be defined as [22]  J Mc (1) where M is the mobility of Li-ions, c is the Li-ion concentration and  is the chemical potential, which is given by 0 ln     h RT c (2) where 0 is the initial chemical potential and is assumed to be a constant. R is a gas constant, T is absolute temperature,  is the partial molar volume of the Li-ion, h is the hydrostatic stress. Since atomic diffusion in solids is much slower than elastic deformation, mechanical equilibrium is established much faster than that of diffusion. Mechanical equilibrium is, therefore, treated as a static equilibrium problem , thus the linear elasticity theory is applicable for the coupling of diffusion -induced stresses and external stresses. The hydrostatic stress h can be expressed as ,,1()3    h di ei i x y z (3) 4 where di is the diffusion -induced stress due to the concentration gradient and ei denotes the external mechanical loading induced stress, respectively. Substituting Eq. (2) into Eq. (1), the species flux can be expressed as (1 )    h effcJ MRT c D cRT c (4) where effD is the effective diffusion coefficien t in a stressed isotropic solid. Substituting Eq. (3) into Eq. (4), effD can be given by ,,() (1 ) 13            di ei i x y zh effccD D DRT c RT c (5) where D MRT is the diffusion coefficient in a stress -free isotropic solid . Combining Eq. (4) with the mass conservation equation 0cJt  , we obtain the diffusion equation as follows 22 , , , ,( ) ( )33                            di ei di ei i x y z i x y zc c cDct RT RT (6) 2.2. Diffusion -induced stresses Consider a n electrode p late of thickness l subject ed to a constant uniform charging current density on both of its side faces [2, 2 3], as shown in Fig. 1 . The electrode plate is considered to be an isotropic linear elastic solid and is mainly composed of the active particles . The effect of the electrolyte is neglected though it can be included using a more sophisticated model such as the work conducted by Zhang et al. [24] and Renganathan et al. [25], and the aim of our work is to study the effects of external mechanical loading. Analogous to thermal stresses [26], for a given concentration profile , the nonzero diffusion -induced stress components due to the insertion of solute atoms into host are only dy and dz, which are two equal transverse stresses and can be given by 3001 2(2 )( ) ( )3(1 ) 2 (1 )        ll dy dzE x l lcdx c E c x dxl l (7) where E and  are Young’s modulus and Poiss on’s ratio, respectively. 5 In general , the thin electrode plate is subjected to a unidirectional gradient stress field due to the mechanical loading combination of bending zM and tension yF. As illustrated in Fig. 1 , the unidirectional gradient stress field can be expressed as 00 0,ex ez ey p x a       (8) where 0p denotes the stress gradient due to mechanical bending and 0a is the tensile stress. It should b e noted that Fig.1 only shows the case of 00p , in fact, 0p can be negative and zero and are also studied. Substituting the diffusion -induced stress Eq. (7) and external loading induced stress Eq. (8) into Eq. (6) yields 22 0 3012(1 ) ( ) ( )23          l p clD c c c c x cdx ct RT l (9) where 22 9 (1 )E RT . Hence , the effective diffusion coefficient effD in Eq. (5) becomes 0 30121 ( ) (1 )23         l effcp clD D c x cdx D cRT c lc (10) where 0 30121 ( )23     l p lx cdxRT c lc . It is a positive c onstant and reflects the degree of the effect s of both the concentration gradient and external stress gradient on diffusion flux. Two operations reflect the insertion and extraction of Li-ions: surface Li-ion concentration keep s the maximum concentration a nd surface Li-ion flux remain s a constant flux [7]. In this work , we assume that the surface Li-ion flux is uniform, namely galvanostatic operation, and the initial and boundary conditions are given by 0 ( ,0) , 0 ,0 ,      eff b eff tc x c t D c J x D c J x l (11) where ,bJ tJ are the Li-ion flux at the bottom and top surface , respectively. To perform numerical calculations conveniently, the following dimensionless variables for thickness ˆx , concentration ˆc , time ˆt , parameter ˆ , current density ˆj , stress ˆ ,and external mechanical stress gradient P , are also introduced: max 0 2 max max max3(1 )ˆ ˆ ˆ ˆ ˆ ˆ , , , , , ,3x c tD Jl lx c t c j P pl c Dc E c RT l          (12) Finite difference method is adopt ed here to solve the non -linear Eq. (9) combin ed with definite conditions in Eq . 6 (11), where the time derivative and space derivative of the Li -ion concentration both need to be d iscretized. The time derivative and space derivative are calculated by the forward difference and central difference, respectively. The Simpson’s rule is used to calculate the integral . The dimensionless time increment is set as ˆ=0.001t . The dimensionless thickness of the plate is divided into 200 equal parts, i.e. ˆ=0.005x . Once the diffusion equation is solved, Li-ion concentration is substituted into Eq. (7) for stresses generated in the electrode. 3. Results and Discussion We take the Mn 2O4 system as an example, whose material properties are listed in Table 1 [10]. For simplicity , the surface flux of Li-ions at the bottom surface bJ is assumed to be identical with that of the top surface tJ . As an example, we take the dimensionless current density at the bottom and top surface as max/ 0.1bJ l Dc , and max 0.4 c . We state that different fluxes could influence the stress profile s because o f different Li-ion concentrations at the two surfaces. Eq . (11) can be applied to the Li-ion diffusion during charging and discharging: 00c , b bsJJ , and t tsJJ for the process of Li-ions insertion, while 0 maxcc , b bsJJ , and t tsJJ for Li-ions deinsertion. Fig. 2 shows the results of insertion in the absence of external mechanical stress gradient : (a) normalized concentration profil e through the thickness of the plate , (b) normalized stress profile at different locations and (c) normalized stress over the surface at different charging times. At the early period during Li-ions insertion in Fig. 2(a) , Li-ions transfer across the two su rfaces first and gradually move towards the centre, show ing that Li- ion concentration is much higher around the surface compared with the region close to the centre. There fore, surface region experiences larger volume changes, which causes tensile stress in the centre and compressive stress in the two surface regions as shown in Fig. 2(b) . Also, both the concentration and stress are symmetric ally distributed through the thickness of the plate in the absence of external mechanical stress gradient , and the maximum stress is always located at the entry surface for a given time. Fig. 2(c) illustrate s the evolution of normalized stress at the entry surface versus time, we can see that the magnitude of this stress increases to its peak with increasing time and the n gradually decreases as the charging process proceeds . This is because stress is relevant to the concentration profile of Li-ions inserted in to the plate, at the initial charging stage, the concentration profile is nonuniform, as more and more Li-ions ins ert into the plate, the concentration profile becomes more and more uniform, result ing in gradually decreasing stress. 7 To study the influence of external mechanical stress , we set five different P : 0, -0.5, -1, 0.5 and 1. 03lPpRT is the dimensionless form of 0p , whic h is given by Eq. ( 8) and denotes the stress gradient. Here the negative stress gradient P means applying external compressive loading to the elect rode, while positive P means applying external tensile loading. Fig. 3 shows the results of insertion with different negative external mechanical stress gradient s: (a) normalized concentration profile at the stage of dimensionless time 2/ 0.16 Dt l , (b) normalized stress profile at the early stage of dimensionless time 2/ 0.032 Dt l during Li- ions insertion and (c) normalized stress profile at the later stage of dimensionless time 2/ 0.07 Dt l during Li- ions insertion. From Figs. 3(b) and 3(c), we can see that stress profile through the thickness of the plate is asymmetric due to the coupling effect s of asymmetrically distributed external mechanical stress. At the early stage during Li-ions insertion ( 2/ 0.05Dt l ), the effects of the external mechanical loading is quite limited and unobvious . With the charging time increasing, the external stress gradient exert s a significant influence on the evolution of stresses . Such an i nfluence will increase with increasing external stress gradient , and larger external negative stress gradient tends to increase the value of stress es generated in the electrode compared with that of lower external negative stress gradient as time increases . This result is in accordance with the previous experiments [16-18] that compressed electrode has a decrease in capacity. It also implies that when the external negative stress gradient is applied and becomes larger, the charging at the entry surface will be slower. The external negative stress gradient will impede the diffusion of solute atoms in this case. Consequently , the value of the concentration decrease s and thus leads to more unevenly distributed concentration, as shown in Fig. 3(a) . The results o f insertion with different positive stress gradient s are illustrated in Fig. 4 : (a) normalized concentration profile at the period of dimensionless time 2/ 0.12 Dt l , (b) normalized stress profile at the early period of dimensionless time 2/ 0.032 Dt l during Li-ions insertion and (c) normalized stress profile at the later period of dimensionless time 2/ 0.07 Dt l during Li-ions insertion. In contrast to Fig. 3 , here external tensile loading is applied to the elec trode. As expected, the external positive stress gradient will accelerate the solute penetration and thus leads to more uniform concentration profile , seen in Fig. 4(a) . Therefore, the external tensile stress can be employ ed to modify diffusion barriers and help to decreas e strains /stresses originating from unevenly distributed concentration . Yen et al. found tensile mechanical stress will enlarge atom spacing of silicon and thus enhance the oxidation rate [27]. Moreover, Sanchez et al. show ed that an exter nal tensile stress of 2GPa decreased diffusion barrier s by about 9% , improv ing diffusion rates by about 30% at room temperature [28]. As shown in Figs. 4(b) and 4(c), the external positive stress gradient tends to decrease the value of stresses generated i n the electrode , and as the charging time increas es, the effects of external stress gradient become more obvious. The deformation and fracture of the electrode caused by the excessive internal stress during cycling can result in electrical disconnects, whi ch renders electrode active material incapable of 8 storing lithium -ion and lows the utilization of the active material. From Figs. 3(c) and 4(c), at the same diffusion time 2/ 0.07 Dt l during Li -ions insertion, the maximum tensile stress ( ˆ =0.03) of electrode subjected to positive external stress gradient P=1 are reduced by as much as 8 6% compared to electrode subjected to negative external stress gradient P=-1 ( ˆ=0.22). Therefore, tensed electr ode is superior in fracture resistance, and thus the utilization of the active material should be signiﬁcantly increased because of the decrease of the maximum tensile stresses generated during cycling. By the results mentioned above, diffusion -induced str esses can be tailored by the external stress gradient in order to be kept below material strengths and avoid mechanical fracture. To quantify the effects of the non -uniform distribution of the stresses on the utilization of the active material, more data s uch as the critical fracture strength of electrode material s and fracture energy are needed from a range of further experiments. 4. Conclusions In summary, we develop a model for diffusion -induced stresses of the thin plate electrode and consider the effects of external mechanical loading . The results show that stress profile through the thickness of the plate is asymmetric due to the coupling effects of asymmetrically distributed external mechanical stress. At the early stage during Li-ions insertion, no significant change is observed for different external stress gradient s. With the increas e of the diffusion time , the external stress gradient exert s significant effects on the evolution of stresses . Such effects become more obvious with increasing the external stress gradient, larger external negative stress gradient leads to greater value of stresses as time increases , while larger external positive stress gradient tends to decrease the value of stresses generated in the electrode, and the effects becom e more obvious as time increases.