Micromechanisms of Capacity Fade in Silicon Anode for Lithium-Ion Batteries

Micromechanisms of Capacity Fade in Silicon Anode for Lithium-Ion Batteries

S. Pala, S. Damleb, S. Patelc, M. K. Duttaa, P. N. Kumtaa,b,d, and S. Maitia

a Department of Bioengineering, University of Pittsburgh, PA 15261, USA b Department of Chemical Engineering, University of Pittsburgh, PA 15261, USA

c Department of Mechanical Engineering, Michigan Technological University, MI 49931, USA

d Mechanical Engineering and Materials Science , University of Pittsburgh, PA 15261, USA

Large volume change and associated stress generation is known to cause failure of the silicon thin film anode used for Lithium-ion batteries after a few cycles. Experimental observations suggest that plastic deformation of the underlying Cu substrate and degradation of the active/inactive interface are the primary reasons responsible for the capacity fade. The goal of the present study is to examine the interplay between these mechanisms using a computational mechanics approach. In the present study, a novel multi-physics finite element framework has been developed to simulate the lithiation and de-lithiation induced failure of amorphous Si (a-Si) thin film on Cu foil. The numerical framework is based on the finite deformation of the active silicon wherein diffusion of lithium occurs, plastic deformation of the Cu foil, and debonding of the active/inactive interface. The effect of substrate property, interfacial energy and kinetics of interface degradation has been examined.

Introduction Lithium ion batteries are flagship rechargeable systems for a variety of portable and consumer electronics employing graphite as the anode material of choice. In recent years silicon anodes have been researched extensively as potential alternative anode material owing to its very high theoretical capacity (~4200 mAh/g). However, the enormous volume expansion (~300 %) of silicon during lithiation leads to the mechanical failure of anode (1). Consequently, the loss of electrical contact within the active material as well as with the current collector results in poor cyclic performance (2). Capacity retention over a large number of cycles coupled with minimization of the first cycle irreversible loss and improved coulombic efficiency by avoiding the ensuing mechanical failure is the primary concern preventing the large-scale deployment of silicon based electrodes in future rechargeable lithium-ion batteries.

A wide range of design strategies has been adopted for the manufacture silicon based anode materials with increased cycle life. All these anode designs, through various modifications of material composition and/or anode geometry, aim to preserve its mechanical integrity in face of severe volume expansion (3). However, an in depth understanding of the complex interplay between electrochemistry and mechanics that leads to the ultimate failure of anode is still lacking. A number of theoretical and

ECS Transactions, 41 (11) 87-99 (2012)10.1149/1.3687394 The Electrochemical Society

87Downloaded 08 Jun 2012 to 138.67.180.167. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp

numerical reports based on simple geometries have been published over the years to study various aspects of this complicated problem (4, 5, 6, 7). However, though fracture and failure of anode is the primary reason for performance degradation, this aspect is not studied in detail. Few recent studies have incorporated the principal stress based failure law that may predict the nucleation of cracks in a quasi-brittle material, but not the actual propagation and its effect on the capacity retention. While they are useful for qualitative understanding of the mechanisms responsible for failure, these models are not predictive in nature, which can guide the battery designers towards the design of optimal configuration and achieving the desired materials properties.

We present herein a thermodynamically consistent theoretical framework that couples electro-kinetics with the deformation failure response of the anode materials system. We postulate that a damage zone precedes the actual nucleation of a crack. Segregation of Li atoms in this zone reduces the strength of the material in the damage zone progressively thus resulting in crack formation. A multi-physics cohesive zone model has been developed to model the process of crack nucleation and propagation. Transport of Li in Si is coupled with the mechanical behavior in a finite deformation setting. Transport of Lithium from the electrolyte to the anode is modeled by the well-known Butler-Volmer equation. Our model includes mechanical behavior of the current collector, and in turn examines the interaction of its passive response with the active response of a-Si based anode. The input to our model is the C-rate of charging and the output is the consequent resulting voltage capacity curve. Our model is capable of simulating the multiple electrochemical cycles, and thus can predict the interfacial crack propagation leading to capacity fade mimicking the experimental current-voltage response for an amorphous Si anode. We describe the details of the model along with its experimental validation against a-Si thin film anode deposited on copper, the established anode current collector used in experiments in the next section. The article is concluded with a discussion of simulations results combined with suggestions for future improvements of the model.

Model description

Transport of Li atoms from electrolyte to anode During galvanostatic electrochemical cycling of thin film anode, if the anode with maximum theoretical Li alloying capacity C mAh/g is to be completely lithiated in n h, then the Li flux across the electrolyte-anode surface JLi (mole m

-2 s-1) is given by , where A is the surface area per unit volume of the active

material, F as the Faraday constant and m is mass of the active material. The overpotential s at the a-Si anode- electrolyte interface can be estimated through the Butler-Volmer equation as

[1]

with an estimated exchange current density . Exchange current density at the anode surface depends on the maximum concentration of the anode material, cmax , the surface concentration, cs, and the concentration of the electrolyte, cewith reaction rate constant k . Values of k depend on the type of reaction

ECS Transactions, 41 (11) 87-99 (2012)


(lithiation/delithiation) occurring at the anode and are selected to match the experimentally observed voltage profile of the anode half-cell. Delamination of the thin film from the current collector increases the contact resistance leading to additional voltage drop across the interface. The half-cell voltage Vis thus given by

, where, UOCP and Rc denoted open circuit potential and contact

resistance.

To account for the non-ideality in the charge/discharge process, the Uocp determined experimentally as a function of state of charge (SOC) by Galvanostatic Intermittent Titration Technique (GITT) has been used in all the simulations (8). Transport of Li in the anode and the attendant deformation of the anode material Diffusion of Li atoms in a-Si anode induces stress due to volume expansion, and this in turn influences the transport of lithium. Thus, these two phenomena are strongly coupled. We have developed a thermodynamically consistent model for these interacting phenomena. The salient features of this model are discussed in the following. Kinematics of diffusion induced expansion and balance laws To account for the large deformation of the silicon anode due to Li transport, we introduce a deformation map , which maps a material point X in the reference configuration 0 to the spatial point x in spatial configuration t at any given time t as

. Therefore, the displacement field u(X,t) can be obtained from x = u + X . We assume a multiplicative decomposition of the deformation gradient of the form F = FeF with . Deformation gradient solely due to the insertion of lithium in the anode material at zero stress is given as F = (1+c)I , with I being the identity tensor, c(X, t) is the concentration of the lithium atom in the material configuration, and the expansion coefficient assuming isotropic volume expansion. The elastic distortion of the lattice is characterized by Fe component of the total deformation gradient. To find the concentration field of lithium in the anode during alloying, we consider the mass conservation of lithium atoms in the reference configuration given as t c + X >J = 0 [2] where J is the flux of Lithium at a material point X(t). Initial and boundary conditions of alloying of lithium in the anode can be described as

c(X, t) = c0 (X) on 0c , and J >N=J0 (X, t) on0J [3] with c0 (X)as the initial concentration of lithium in the anode, J0 as the applied time varying flux of lithium atom at point X on the boundary 0with normal N . Atomic diffusion occurs at a much slower time scale; hence, mechanical equilibrium is assumed to be already attained. Therefore, the balance of linear momentum can be expressed as

X >P = 0 with P = FS [4] Where, Pand S are the first and second Piola-Kirchhoff stress tensor, respectively. The boundary conditions for mechanical equilibrium are given as PN = t on 0t , and



u = u0 on 0u . Thermodynamical considerations The free energy functional of the anode material can be additively decomposed as (F,c) = 1(Fe ) +2 (c) [5] with 1(Fe )the elastic free energy density and 2 (c) the chemical energy density. The elastic free energy depends on the overall deformation gradient (through F) as well as the concentration of the lithium atom. However, chemical energy density of the anode material depends only on the concentration of lithium. The material time derivative of the free energy can thus be written as

= 2F-1 Ce

F-T :

C2

Me :L +c

c [6] The second Piola-Kirchhoff stress tensor Se and Mandel stress tensor Me at the intermediate configuration is expressed as and Me = CeSe , respectively, with as the right Cauchy-Green tensor. From the second term, we define p as a pressure like quantity at the reference configuration, with In the limit of small deformation, this term reduces to where, is the Cauchy stress tensor. For an irreversible process, the second law of thermodynamic states that:

with s as the entropy density, J as the diffusion flux and as the chemical potential. Incorporating Helmholtz free energy function and mass conservation of lithium in the anode, the Clausius-Duhem inequality for anode with isothermal condition can be written as

D = S :C2

+c J 0 [7] Substituting the expression for free energy functional [6], the above inequality can be expressed as

[8]

From the first part of the above inequality, we derive the thermodynamically consistent definition of second Piola-Kirchhoff stress tensor S in the reference configuration as S = F

-1SeF-T, The second part of the inequality offers the expression of the chemical

potential in the reference configuration as Furthermore, the inequality above can be ensured only when the third component is negative semi-definite, which provides the definition of the lithium flux of anode in the reference configuration as

where the mobility is a symmetric and positive definite tensor. In the present study, we assume isotropic mobility tensor such that . Thus, the atomic flux can be recast as with D = MRT / c as the diffusivity of the lithium atom in the anode material. Therefore, the equation for the transport of lithium in the anode material can be expressed as

[9]



This equation suggests that the change of concentration of lithium in the anode is driven by the concentration gradient as well as the gradient of pressure. Continuum modeling of debonding and interface degradation kinetics To model the crack initiation and propagation, we resorted to a popular method called the cohesive element technique. In its classical form, this method is formulated only for the deformation field. It relates the displacement jump ahead of a crack tip to the traction T the material exerts to resist fracture through a cohesive failure law. However, segregation of Li atoms in the interfacial zone can cause embrittlement and thus enhance failure. Moreover, Maranchi et. al., (9) have reported the presence of lithium atoms in the interfacial region after failure. We extend the classical formulation to account for this effect from a thermodynamic viewpoint (10). In the spirit of Gibbs isotherm, the excess of the energy per unit area can be written as [10] assuming isothermal conditions. In this expression, is the surface energy of each separating surface, and can be related to the work of separation as in the

absence of any segregation. The last term of the above equation stems from the segregation of Li atoms, with is the chemical potential and as the excess of these atoms on the interface. Note that, in the absence of this term, our formulation coincides with the classical displacement based cohesive technique. When segregation is present, is a function of both and . We assume a separable form as follows: [11]

where, is a parameter varying with the fraction of absorbed atoms cint (equivalent to ), and degradation constant Sb . To estimate adsorption of Li atoms from the bulk to the interface, the Langmuir-McLean model is assumed to be valid:

[12]

where Sm is the segregation factor and cb is the surface concentration of Li on the bulk. Finally, the cohesive traction can be found as

[13]

In the thin film anode system, the delamination of the active material from the inactive current collector is one of the primary reasons contributing to capacity fade and increase in the electrical resistance between the active/ inactive interfaces. Therefore, an additional voltage drop occurs at the interface, which reduces charge discharge capacity.



Continuum modeling of substrate The anode is attached to the current collector to provide an electrical pathway. During alloying of the anode, the current collector undergoes severe deformation often in the inelastic regime. Such inelastic deformations may contribute to delamination of the active material as discussed previously. The present analysis accounts for such inelastic deformation considering finite deformation kinematics through multiplicative decomposition of the total deformation gradient as F = FeFp . To determine the plastic part

of the deformation gradient Fp , a flow rule Fp Fp1 = Np is considered with the flow direction Np = f / Me where f is the yield function and is the plastic parameter. In the present formulation, the yield function with isotropic linearly hardening is assumed and is of following type

f Me , p( ) = 32 Med :Med y + Hp = 0 [] with Me

d is the deviatoric part of Mandel stress tensor Me . The yield stress and hardening moduli are represented as y and H , respectively.

Materials and Methods As mentioned earlier, prediction of the solid phase potential requires an accurate estimation of the overpotential and the state of charge dependent open circuit voltage Uocp through experiments. However, the actual experimental results depend on the anode configuration. We have performed GITT experiment for estimating the Uocp for 250 nm thick a-Si thin films deposited on Cu substrate (11). The 250 nm thick film of a-Si was prepared by radio frequency magnetron sputtering. The details of the deposition conditions and the fabrication of the 2016 coin cells for electrochemical testing can be found in the literature (12). Prior to GITT, the half-cell was cycled at C/4 charge rate for 5 charge/discharge cycles to ascertain the formation of the SEI layer. For GITT, galvanostatic lithiation of fully discharged a-Si anode was carried out in repeated segments of 1 h at C/25 current followed by relaxation period of 10 h to ensure equilibration. The cut-off Voltage was set to 0.02 V vs. Li/Li+ electrode. Similar process for discharge of the fully lithiatiated anode was carried out and the cut off voltage was set to 1.2 V vs. Li/Li+ electrode. The Uocp data obtained from GITT experiments was fitted as a function of state of charge (SOC) using cubic splines and is used for all the simulations executed in this study. As we have the open circuit voltage experimental data only for a-Si thin film anode, computational simulations reported in this article were performed on this set-up only. A finite element framework has been developed to study the coupled transport and stress generation problem in a domain. Variational forms of the coupled equilibrium and transport equations were obtained to perform finite element semi-discretization in space. A Newton-Raphson scheme based linearization procedure was used to solve the resulting algebraic equations in an iterative manner. A backward Euler implicit time stepping algorithm was employed to solve the resulting temporal ordinary differential equations numerically which is presented here.

s



Problem description

The numerical model presented in the previous section has been used to simulate the thin film anode combined with the base and the interface between them during the lithiation/delithiation cycles and the ensuing chemo-mechanical response has been studied. After the first cycle of charging and discharging, the a-Si film undergoes vertical fracture and forms islands attached to the current collector (12). During further cycling, the stresses generated in the film are not sufficient to generate additional vertical cracks and the island structure is therefore preserved during further electrochemical cycling (13). Thus, in this study we will simulate the mechanical response of a single a-Si island along with the elasto-plastic deformation of the substrate attached to it. To study the anode structure, we have considered a three-dimensional domain as shown in Figure 1(c), where the bottom substrate represents the current collector and the top island represents the Si thin film having a cohesive layer in between. The separation between the vertical cracks formed in the thin film (9) structure is observed to be in the micron range. The a-Si thin film thickness was considered as 250 nm with 1 m x 1 m sized island to mimic the experimental conditions, and 3 times larger dimensions were considered for the substrate to sustain the volume expansion of the island having a rigid support at the bottom surface. Selection of domain for simulation is explained in Figure 1. As we are considering galvanostatic charging and discharging, a constant Li-ion flux is applied through the top surface of the a-Si island as shown in Figure 1.

Figure 1: (a) Schematics of a-Si thin film anode half cell. (b) a-Si thin film anode after 1st electrochemical cycle. (c) Domain selected for simulation.

In this numerical study, material parameters for the a-Si island are given as Diffusion coefficient (7), expansion co-efficient (8), Youngs



modulus E = 80 GPa and Poisson ratio (9). Taking advantage of symmetry we have considered only 1/4th of the domain presented in Figure 1(c) to minimize the computational time. The finite element discretization of the domain is shown in Figure 4(a). Appropriate symmetric boundary conditions have been applied on the surfaces of the symmetry.

Results Simulation of voltage-capacity curves and experimental validation Figure 2 shows a comparison of the experimental and simulated Voltage-Capacity plots for electrochemical cycling of 250 nm thin film of a-Si anode at a C/2.5 charge/discharge cycling rate. Since the formation of SEI layer and the irreversible capacity loss during the first electrochemical cycle is neglected in the present study, the simulation result shown is only for the 2nd charge/discharge cycle. The electrochemical parameters used for the simulation are given in Table I. It should be noted that the cell potential predicted by the simulation are in excellent agreement with the experimental values except at the onset of lithiation.

Figure 2. Comparison of Simulated and Experimental Voltage-Capacity plot for C/2.5

charge/discharge rate

Table I: Parameters and model properties for the intercalation model

F (Faraday constant) 96485.34 C.mol-1 R (Universal gas constant) 8.314 m2.kg.s-2.K-1.mol-1 T (Room temperature) 398 K a,c (Symmetry factor) 0.5 ce (Li concentration in electrolyte)

1000 mol.m-3

cmax (Max. Li concentration in a-Si)

3.651 105 mol.m-3 (corresponding to 4200 mAh/g)

k (Reaction rate constant) m2.5.s-1.mol-0.5 (Lithiation) m2.5.s-1.mol-0.5 (Delithiation)



Simulation of crack propagation Figure 3 shows the simulation of the interfacial crack propagation at the a-Si island - current collector interface. The simulation was performed employing the mechanical properties of the current collector similar to that of Copper with E =100 GPa, ,

and (14) for 40 consecutive electrochemical cycles. The study was conducted with the interface degrading due to Li segregation at various rates ( corresponding to no interface degradation). Sm was kept constant at 0.05. It can be seen that when there is no effect of lithium segregation on the interface strength, crack propagation stopped after the 4th cycle. However, for all other cases where interfacial segregation of Li has been taken into account, crack propagates slowly in an intermittent manner progressively degrading the electrical contact between a-Si and Cu. This slow delamination of the interface is a major contributing factor towards the gradual capacity fade observed in Si-thin film anodes.

Figure 3: Propagation of interfacial crack with electrochemical cycles. An interfacial strength of 2 GPa and a fracture toughness of 25 J/m2 have been assumed. Crack propagation is presented as the percentage of delamination of the original area. Effect of current collector material property Datta et. al., (15) recently showed that the presence of an amorphous carbon intermediate layer between the a-Si thin film and the current collector can significantly improve the cycling performance of the anode. High Columbic efficiency and low capacity fade during electrochemical cycling indicates that the interfacial crack propagation is suppressed due to presence of the soft intermediate layer. This leads to the conclusion that modifying the mechanical properties of the substrate can significantly alter the anode stability during electrochemical cycling. To study the effect of mechanical properties of the current collector on the stability of a-Si island, we simulate the electrochemical cycling of a-Si island on elastic substrates (current collectors) with different stiffness mismatch (ESubstrate/ESi) between the active material (silicon) and the substrate (Figure 5). For elastic substrates, due to absence of plastic flow in the current collector, crack



initiates and propagates when the stress at the middle of the island exceeds the interfacial strength (see Figure 4(b)). As the maximum tensile stress at the interface occurs at the completion of lithiation, if the interface survives the stress state at complete lithiation (~280% volume expansion), the island structure will survive the consecutive electrochemical cycles (assuming that fatigue and any other mechanism does not degrade the interfacial strength appreciably). Figure 5 shows the normalized interfacial crack initiation time for anode configurations with elastic current collector for different ratios of ESubstrate/ESi. Active material is cycled at C/2.5 charge rate and the crack initiation time is normalized with the theoretical time required for complete lithiation of 250nm a-Si thin film; =0 indicating start of lithiation and =1 indicating completion of lithiation or start of delithiation. The interfacial strength is taken as 1 GPa and values of fracture toughness are varied between 15 to 50 J/m2.

Figure 4: (a) Finite element discretization of an intact a-Si island and current collector. (b) Interfacial crack (at middle) between a-Si island and the elastic substrate ( ). (c) Simulated interfacial crack (at corner) between a-Si island and the Cu substrate after undergoing severe plastic deformation.



Figure 5: Effect of elastic substrate stiffness on normalized crack initiation time ( ). Interfacial strength is taken as 1 GPa. Simulations indicate that having a compliant elastic substrate can improve mechanical stability of the a-Si island during electrochemical cycling. For the cases where Esub / Esi was less than 0.18, there was no interfacial crack propagation. Also, having higher interfacial energy can delay the interfacial crack initiation to some extent.

Discussion and conclusions

In this paper, we report a novel multiphysics model taking into account the coupled effect of lithium transport in the silicon anode and the associated stress generation, combined with the ultimate failure of the anode. We have validated our model against experimental results obtained from half-cell experiments conducted on anodes of similar configuration. The simulated voltage-capacity curve is compared against the experimental results in Figure 2. A sudden drop in voltage at the onset of lithiation in the experimental data indicates that a higher overpotential is encountered. We have experimentally determined the open circuit potential to minimize this error. However, in our theory we have not considered the stress-potential coupling that is present in the physical configuration. We also have not considered the presence of SEI layer on top of the thin film and the side reactions known to occur during Li intercalation. These facts may explain the discrepancy of our simulation results with the experiments at the early stage of lithiation. Apart from this, the model prediction is in close agreement with the experimental results.

While prediction of single cycle performance demonstrates the capability of a theoretical model, batteries are typically subjected to multiple charge/discharge cycles under practical service condition. Cycling performance and capacity fade combined with first cycle irreversible loss and coulombic efficiency are the overall primary design concerns for Li-ion batteries. To show the predictive capability of our model over many



cycles, we have chosen two particular experimental findings. One of them is concerned with the stability of the Si thin film of Cu Substrate. During lithiation, plastic flow of the base occurred at the corner of the island that reduced the tensile stress at the center as shown in Figure 4(c). However, during delithiation or de-alloying, the permanent set due to plasticity of the substrate attempts to pull back on the corner of the island that was trying to relax to its original configuration. This mechanism produced very high tensile stress at the interface and caused the nucleation of primarily mode I crack at the corner. However, after a few cycles the plastic shake down occurred; the stress at the crack tip fell just below the interfacial strength, and the crack was arrested. But subsequent segregation of Li in the interfacial region, as experimentally observed in (9), weakens the interface gradually causing an intermittent motion of the crack (and attendant loss in capacity). Finally the crack propagates all through the interfacial region from the corner towards the center causing loss of contact and drastic fade in capacity. For the second case, the low modulus of carbon enabled very large deformation of the substrate. During lithiation as shown in Figure 4(b), the middle of the island arched upward along with the substrate material thus preventing the build up of high stress. The maximum principal stress remained far below the interfacial strength. During delithiation, the whole system comes back to its original configuration elastically. Interfacial stress was low enough such that strength reduction due to segregation was not enough to result in failure of the interface over many cycles. These simulations thus reveal that the stiffness of the base is an important design parameter that influences long term performance of silicon anode. It is hoped that these computational mechanics based models will help the experimentalists design improved silicon based anode systems.

Acknowledgments

The authors gratefully acknowledge support to this work by the Centre for Complex Engineered Multifunctional Materials (CCEMM), University of Pittsburgh. PNK acknowledges support of the DOE-BATT program and the Edward R. Weidlein Chair Professorship funds for partial support of this research.

References 1. L. Y. Beaulieu, K. W. Eberman, R. L. Turner, L. J. Krause and J. R. Dahn,

Electrochemical and Solid-State Letters, 4, A137 (2001). 2. Z. Chen, L. Christensen and J. R. Dahn, Electrochem. Communications, 5, 919

(2003). 3. U. Kasavajjula, C. Wang and A. J. Appleby, Journal of Power Sources, 163, 1003

(2007). 4. A. F. Bower, P. R. Guduru and V. A. Sethuraman, Journal of the Mechanics and

Physics of Solids 59, 804 (2011). 5. T.K. Bhandakkar and H. Gao, International Journal of Solids and Structures, 47,

1424 (2010). 6. R. Deshpande, Y. -T. Cheng and M.W. Verbrugge, Journal of Power Sources, 195,

5081 (2010). 7. K. Zhao, M. Pharr, J. J. Vlassak and Z. Suo, Journal of Applied Physics, 109, 016110

(2011). 8. R. Chandrasekaran, A. Magasinski, G. Yushin and T. F. Fuller, Journal of The

Electrochemical Society, 157, A1139 (2010).



9. J. P. Maranchi, F. Hepp, G. Evans, N. T. Nuhfer and P. N. Kumta, Journal of The Electrochemical Society, 153, A1246 (2006).

10. R. J. Asaro, Philosophical Transactions of the Royal Society of London. Series A. 295, 151 (1980).

11. L. Baggetto, J. F .M. Oudenhoven, T. van Dongen, J. H. Klootwijk, M. Mulder, R. H. Niessen, M. H. J. M. de Croon and P. H. L. Notten, Journal of Power Sources, 189, 402 (2009).

12. J. P. Maranchi, A. F. Hepp and P. N. Kumta, Electrochemical and Solid-State Letters, 6, A198 (2003).

13. J. Li, A. K. Dozier, Y. Li, F. Yangand and Y. -T. Cheng, Journal of The Electrochemical Society, 158, A689 (2011).

14. Y. W. Y. Denis and S. Frans, Journal of Applied Physics, 95, 2991 (2004) 15. M. K. Datta, J. Maranchi, S. J. Chung, R. Epur, K. Kadakia, P. Jampani and P. N.

Kumta, Electrochimica Acta, 56, 4717 (2011).



Micromechanisms of Capacity Fade in Silicon Anode for Lithium-Ion Batteries

Documents

Transcript of Micromechanisms of Capacity Fade in Silicon Anode for Lithium-Ion Batteries