Diffusion-induced Stress Research Articles

Examination of Graphite Particle Cracking as a Failure Mode in Lithium-Ion Batteries: A Model-Experimental Study

Capacity fade in lithium-ion batteries remains an area of active research, with failure of the graphite anode thought to be an important contributor. While the formation of the solid electrolyte interphase and the subsequent loss of cyclable lithium have been well studied, mechanical degradation remains an area where ambiguity remains. While there appears to be little experimental evidence that suggest that macroscopic particle cracking occurs, mathematical models have suggested that this phenomenon is likely. The goal of this paper is to clarify this ambiguity by combining experimental cycling, mathematical stress modeling, and post-mortem microscopy. We experimentally determine an average diffusion coefficient of lithium in graphite using a thin-layer electrode and use this information in a diffusion-induced stress model. Our results suggest that cracking is not likely during lithiation due to the proximity of equilibrium potential to the cutoff potential. On delithiation, at 25°C, even at 30 C rate cracking is unlikely while at −10°C, a rate of 10 C can lead to particle cracking. By extrapolating the results of the thin-layer electrode to a thick porous electrode, we found that graphite cracking is unlikely to occur during typical vehicle operations.

Multiscale analysis of prelithiated silicon nanowire for Li-ion battery

A diffusion induced stress (DIS) model based on the finite element method was used to analyze mechanical stress within a multiscale framework for silicon nanowire anodes designed for use in Li-ion batteries. With a prelithiated nanowire, the mechanical moduli of lithium silicide and the migration energy barriers of lithium were calculated by density functional theory method, while the diffusion constants for lithium silicide were obtained using kinetic Monte Carlo simulations. Unlike previous DIS analyses, this multiscale approach reveals a decreasing hoop stress with increasing lithium concentration. Furthermore, an increase in the diffusion coefficient with Li concentration was also found to be a much more significant factor than the reduction in the mechanical moduli, causing a prelithiated silicon anode to experience lower stress levels than pristine silicon. On the basis of this finding, it is concluded that prelithiation reduces the loss in cycling performance that typically occurs due largely to an excess of induced stress.

Finite element analysis of lithium insertion-induced expansion of a silicon thin film on a rigid substrate under potentiostatic operation

Diffusion-induced stress and volumetric expansion under potentiostatic operation are investigated with an axisymmetric finite element model taking account of plastic yielding, coupling effects between diffusion and stress, diffusion from the edge surface, and concentration dependence of material properties. Significant differences on stresses, displacements, and fracture energies between purely elastic and elastic–plastic materials are found. Plasticity based on von-Mises criterion has no effect on concentration variation. The critical regions for fracture are the edge surface, and the regions near the edges on both the top surface and the interface.

Variations of boundary reaction rate and particle size on the diffusion-induced stress in a phase separating electrode

In contrast to the case of single-phase delithiation wherein faster discharging leads to higher diffusion-induced stress (DIS), this paper reports nonmonotonous dependency of the boundary reaction rate on the DIS in nanosized spherical electrode accompanying phase separation. It is attributed to a transition from two-phase to single-phase delithiation driven by increase of the boundary reaction rate leading to narrowing and vanishing of the miscibility gap in a range of the particle size. The profiles of lithium concentration and the DIS are identified during the transition based on a continuum model. The resultant maximum DIS first decreases in the region of two-phase delithiation and later returns to increase in the region of single-phase delithiation with the increase of the boundary reaction rate. A map for the failure behavior in the spherical electrode particle is constructed based on the Tresca failure criterion. These results indicate that the failure caused by the DIS can be avoided by appropriate selection of the said parameters in such electrodes.

Comment on "Effects of dislocation mechanics on diffusion-induced stresses within a spherical insertion particle electrode" by P. Wei, J. Zhou, X. Pang, H. Liu, K. Deng, G. Wang, Y. Wu and B. Chen, J. Mater. Chem. A., 2014, 2, 1128

In this work, it is shown that the analysis by Wei et al. [Wei et al., J. Mater. Chem. A, 2014, 2, 1128–1136] is inappropriate for studying the effect of dislocation mechanics on diffusion-induced stresses. Using the theories of dislocation mechanics and plasticity, a set of equations for analyzing the effect of dislocations on diffusion-induced stresses are proposed.

Reply to the ‘Comment on “Effects of dislocation mechanics on diffusion-induced stress within a spherical insertion particle electrode”’ by F. Yang, J. Mater. Chem. A., 2014, 2, DOI: 10.1039/c4ta01983k

We reply to the comments raised by Yang’s discussion of our original article studying the effects of dislocation mechanics on diffusion-induced stresses within a spherical insertion particle electrode (Wei et al., J. Mater. Chem. A, 2014, 2, 1128–1136).

Graphite Particle Cracking as a Failure Mode during Li-Ion Battery Cycling

In recent years, there has been aggressive research on lithium ion batteries for vehicle applications. For better understanding of the battery lifetime, the degradation mechanisms have attracted much interest because vehicle batteries are generally operated under more severe conditions than consumer electronics applications, i.e. wider temperature ranges and higher rates. One likely degradation mechanism is the mechanical breakdown of the particles. To investigate this potential degradation mechanism, numerical predictions, which are based on the diffusion-induced stress, are proposed 1-5.For example, in graphite anodes, the crack generation at the particle surface may accelerate self-discharge due to the exposure of the new surface to electrolyte solvents, leading to further formation of the solid electrolyte interphase (SEI). In previous papers1,3,4, numerical predictions have been used to estimate tendencies of graphite particles to crack, suggesting that this phenomenon is likely at reasonable rates. However, these papers use a value of lithium diffusion coefficient in graphite that is approximately two to three orders of magnitude lower than that reported experimentally by Dokko et al.6, where the diffusion coefficient was measured with a single graphite particle. In order to reconcile this difference, a closed loop system combining the determination of diffusion coefficients from experimental data and model predictions based on the determined value is required to consistently understand the particle crack generation criteria. In this study, we focus on stresses in a graphite particle and identify propensities of the particle crack generation criteria at high cycling rates, which is particularly important for vehicle batteries. To understand the criteria, we will propose the following approach that combines experimental tests and mathematical model predictions:Determination of transport property (diffusion coefficient) in the graphite particle from experimental dataMathematical predictions of the stresses in the particle using the above transport propertyDurability tests of the graphite electrodeIdentification of the propensities for the crack generation by model-experimental comparisons The experiments were performed using 2325-typed coin cells. To conduct durability tests at high cycling rates, a thin graphite electrode (ca. 20 um) was prepared. The graphite electrode, polypropylene (Celgard 2400), and lithium foil were used as a working electrode, a separator, and a counter electrode, respectively. A few drops of 1 M LiPF6in EC:DEC (1:2) were added as an electrolyte solvent. In the durability tests, the graphite-lithium half-cells were cycled at different C-rates and temperatures. Then, the particle surface was observed with a scanning electron microscope.The mathematical model to predict the stresses in the particle was constructed by the coupling of the diffusion equation, graphite particle volume change, and mechanical equations. All the equations were numerically solved using finite-element package COMSOL Multiphysics.The presentation will discuss the experimental results, the diffusion coefficient that should be used, the predictions of the model, and the agreement of the model with experimental observations on durability of graphite. References J. Christensen and J. Newman, J. Solid State Electrochem., 10, 293 (2006)X. Zhang, W. Shyy, and A. Sastry, J. Electrochem. Soc., 154, A910 (2007)S. Renganathan, G. Sikha, S. Santhanagopalan, and R. White, J. Electrochem. Soc., 157, A155 (2010)P. Barai and P. Mukherjee, J. Electrochem. Soc., 160, A955 (2013)Y. Dai, L. Cai, and R. White, J. Power. Sources, 247, 365 (2014)K. Dokko, N. Nakata, Y. Suzuki, and K. Kanamura, J. Phys. Chem. C, 114, 8646 (2010)

Degradation of lithium ion batteries employing graphite negatives and nickel–cobalt–manganese oxide + spinel manganese oxide positives: Part 2, chemical–mechanical degradation model

Capacity fade is reported for 1.5 Ah Li-ion batteries containing a mixture of Li–Ni–Co–Mn oxide (NCM) + Li–Mn oxide spinel (LMO) as positive electrode material and a graphite negative electrode. The batteries were cycled at a wide range of temperatures (10 °C–46 °C) and discharge currents (0.5C–6.5C). The measured capacity losses were fit to a simple physics-based model which calculates lithium inventory loss from two related mechanisms: (1) mechanical degradation at the graphite anode particle surface caused by diffusion-induced stresses (DIS) and (2) chemical degradation caused by lithium loss to continued growth of the solid-electrolyte interphase (SEI). These two mechanisms are coupled because lithium is consumed through SEI formation on newly exposed crack surfaces. The growth of crack surface area is modeled as a fatigue phenomenon due to the cyclic stresses generated by repeated lithium insertion and de-insertion of graphite particles. This coupled chemical–mechanical degradation model is consistent with the observed capacity loss features for the NCM + LMO/graphite cells.

Interface diffusion-induced creep and stress relaxation in unidirectional metal matrix composites under biaxial loading

Analytical solutions are developed for interface diffusion-induced creep and stress relaxation in unidirectional metal matrix composites under biaxial transverse loading. The driving force for the interface diffusion is the normal stress acting on the interface, which is obtained from rigorous Eshelby inclusion theory. The solutions are a function of the applied stress, volume fraction and radius of the reinforced-fiber, the modulus ratio between the fiber and the matrix, specially, exhibit a strong dependence of creep rate and stress relaxation behavior on the biaxial stress ratio. Moreover, the solution for the interface stress presented in this study also gives some insight into the relationship between the interface diffusion and interface slip. For the application of the solutions in the realistic composites, the scale effect is taken into account by detailed finite element analysis based on a unit cell model.

MEASUREMENT AND CALCULATION OF CARBON CONCENTRATION AND DIFFUSION-INDUCED STRESS IN STAINLESS STEEL AFTER LOW TEMPERATURE GAS CARBURIZING

Low temperature gas carburizing has been established as a surface hardening process to improve the wear and fatigue resistance of austenitic stainless steel.In the gas carburizing treatment of stainless steel at low temperature,carbon diffuses into the steel to a depth up to dozens of or even hundreds of micrometers.In industrial practice,it would be useful to establish a model that can predict the carburizing depth from carburizing condition.However,to date a satisfactory model does not exist,and the classic Fick's law has been proved to be inaccurate to describe the carbon diffusion during carburation.It has been observed that the insertion of carbon can lead to the evolution of very high compressive stresses in the surface layer of stainless steel.Since it has well established that the stresses induced by diffusion of atoms can in turn affect the diffusion behavior,the high compressive stresses due to carbon diffusion during carburation are supposed to play a role in the further diffusion of carbon.In this work,316L stainless steel block specimens were gas carburized at low temperature,and the carbon concentration,diffusion-induced stress along the depth direction of specimens were measured.Based on the coupled stress-diffusion theory,a model was built to calculate the carbon concentration and diffusion-induced stress in the specimens after carburizing.Then the calculated carbon concentration and stress distribution were compared to that obtained by above measurement.The results show as follow:(1) After carburizing,a diffusion layer containing high amounts of carbon and compressive stress was formed near the surface of specimens.The concentration and compressive stress were the maximum at the surface,and decreased with increasing depth.There was a linear relationship between the stress and carbon concentration.With increasing carburizing time,the depth and concentration level of the carbon diffusion layer increased.(2) To describe the carbon diffusion during carburation of stainless steel at low temperature,the model based on coupled stress-diffusion theory is more appropriate than the classical Fick's second law.The diffusion-induced compressive stress played an important role in diffusion of carbon.(3) The compressive stress highly increased the apparent diffusion rate of carbon.This explained the phenomenon that measured diffusion depth of carbon is much higher than that excepted from Fick's law.The interaction between diffusion and diffusion-induced stress should be considered when studying diffusion mechanism of carbon or nitride in carburizing or nitriding similar to the gas carburizing at low temperature.

Computational Study of Effect of Si Morphology on Mechanical Integrity of Si/CNT Hetetostructure Anode for Li Ion Battery

Over the past decade, lithium ion batteries have emerged as the most important and flag ship energy storage devices among other battery technologies such as Ni-Cd, Ni-MH, Lead acid, etc. Further improvement in battery technology is constantly being sought, since the demand for better energy storage devices is rising continuously. Current state of the art Li ion batteries have graphitic anode, which provides capacity of 372 mAh/g. Silicon has emerged as the most promising candidate to replace graphite due to its high theoretical gravimetric capacity of 4200 mAh/g. However, commercialization of Si based anodes is hindered by mechanical degradation of the Si based anodes resulting from Li intercalation induced stresses. Different approaches have been suggested to improve the mechanical integrity of the Si based anodes and the retention of electrode capacity. Experimental as well as modeling studies reveal that the use of nano-sized silicon (nanoparticles, nanowires, etc.) can significantly improve the anode performance. However, loss of electronic conductivity due to mechanical failure is still a challenge that prevents commercialization of Si based anodes.Recently, nanostructured heterostructures of Si with carbon nanotube (CNTs) are being explored in literature. CNTs are known to have very good mechanical strength along with excellent electrical and thermal properties. Li intercalation is known to result in reasonable capacity of ~300 mAh/g. Thus, heterostructures comprised of CNTs and nanostructured Si can be expected to give excellent electrochemical performance. Different Si/CNT heterostructures ranging from core-shell CNT/Si structure to Si nano-globules adhered to the CNT with sufficient separation between the adjacent globules can be synthesized. However, Si morphology in the heterostructure is known to significantly alter the electrode performance as shown by Epur et al. [1]. Understanding the role of morphologies and the resulting diffusion-induced stresses (see Figure 1) on the mechanical degradation of Si/CNT heterostructure needs to be explored to understand the plausible mechanisms contributing to capacity fade.We utilize a thermodynamically consistent theoretical framework employed in a finite element setting [2] to model the Li intercalation induced deformation processes as well as failure occurring in different Si/CNT heterostructure anodes configurations (see Figure 1). CNT/Si interface is modeled using a novel cohesive law [3]. Two-phase lithiation of the amorphous Si is considered. Different Si/CNT heterostructure morphologies are compared in terms of their mechanical durability during electrochemical cycling to identify the most stable Si/CNT electrode configuration. Qualitative comparison of performed simulation studies with experimental results is made. Results from this study are expected to aid in the fabrication of improved Si/CNT heterostructure anodes. Results of the modeling studies and the comparison to experimental reports with resultant validation of the experimental findings will be presented and discussed.

Invited Presentation: Mathematical Modeling of Diffusion-Induced Stress and Other Coupled Physics in Li-Ion Batteries

The pursuit of high-energy-density, low-cost electrochemical energy storage systems for electric vehicles inevitably requires careful examination of the coupling between mechanics, electrochemistry, thermodynamics, and mass transport in batteries. High-capacity Li battery materials such as Li alloys and Li metal undergo large volume changes during battery cycling, and the stress induced by these volume changes must be accommodated at the material, composite, or device level in order to reduce the rate of battery degradation. Here we review some early work in the modeling of diffusion-induced stress in Li-ion insertion materials,[1,2] showcase other examples of multiscale, multidomain battery models,[3,4] and discuss future research directions in the field of battery modeling.1. J. Christensen and J. Newman, "Stress generation and fracture in lithium insertion materials," Journal of Solid State Electrochemistry, 2006. 10(5): p. 293-319.2. J. Christensen and J. Newman, "A Mathematical Model of Stress Generation and Fracture in Lithium Manganese Oxide," Journal of the Electrochemical Society, 2006. 153(6): p. A1019-A1030.3. J. Christensen, D. Cook, and P. Albertus, "An Efficient Parallelizable 3D Thermoelectrochemical Model of a Li-Ion Cell," Journal of The Electrochemical Society, 2013. 160(11): p. A2258-A2267.4. S. Patnaik, An Electrical Network Model for Computing Current Distribution in a Spirally Wound Lithium Ion Cell, MS Thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology: Cambridge, MA (2012).

Mechanical behavior of a fluid-sensitive material during liquid diffusion

This paper described the analytical study that we performed in an attempt to understand the combined effect of liquid diffusion and temperature on the mechanical response of viscoelastic liquid-sensitive material. A constitutive equation for linear viscoelasticity, which includes the effect of liquid diffusion, is used to model the mechanical response of a fluid-sensitive polymer such as PLA-based bioplastic. The viscoelastic characteristics which represent material degradation due to liquid diffusion were expressed using a creep-based formulation represented by Burger’s model. Creep experiment data were fitted to the Burgers model to provide a liquid content-dependent set of input data for subsequent time-dependent analysis. Further, analytical solutions for stresses and deformations were obtained from the corresponding elastic solution by applying the Correspondence Principle, using previously defined material characteristics. Spatial and time variations of stress and deformation were evaluated to give a precise description of the material behavior under hygroscopic conditions. We propose a stress concentration factor to take into account the liquid diffusion-induced stress that may result in a failure of an application. The results emphasize the importance of considering liquid diffusion and viscoelastic properties in the design of components using liquid-absorbable material.

Effects of electrode properties and fabricated pressure on Li ion diffusion and diffusion-induced stresses in cylindrical Li-ion batteries

The effects of electrode properties and fabricated pressure on Li ion diffusion and diffusion-induced stress in a cylindrical Li-ion battery are studied. It is found that hydrostatic pressure or elastic modulus variation in the active layer have little effect on the distribution of Li ions for a higher diffusivity coefficient, but both can facilitate Li ion diffusion for a lower diffusivity coefficient. The elastic modulus variation has a significant effect on the distribution of stress and hydrostatic pressure can reduce the surface stress for the lower diffusivity coefficient. A higher charging rate causes a more transient response in the stress history, but a linear charging history is observed for slow charging rates. A higher charging rate would not inflict extra damage on the electrode for the higher diffusivity coefficient and the stress history becomes highly transient and charging rate dependent for the lower diffusivity coefficient. The effect of fabricated pressure can be neglected.

Effects of dislocation mechanics on diffusion-induced stresses within a spherical insertion particle electrode

A sphere dislocation model is developed to describe the effect of dislocation on diffusion-induced stress in battery electrodes. We follow the conventional assumption that the diffusion-induced stresses are symmetrical during intercalation and de-intercalation. The results show that dislocation-induced stress can lower the tensile stress, convert the state of stress from tensile to compressive, and the tendency to fracture can be reduced with decreasing the particle radius down to the nanometer range. We conclude that dislocation mechanics are beneficial to avoid the formation of cracks during cycling. Here, we suggest a possibility of developing a new strategy to optimize the diffusion-induced stresses and prolong battery life.

Diffusion-Induced Stresses in Transversely Isotropic Cylindrical Electrodes of Lithium-Ion Batteries

As novel electrodes of lithium-ion batteries, anisotropic nanocrystal materials with improved electrochemical performance can experience anisotropic swelling and sustain stresses during charging and discharging. However, previous diffusion-induced stress models have rarely considered the effects of anisotropic properties. In this study, an analytical model of diffusion-induced stresses is developed for transversely isotropic cylindrical electrodes during lithiation under galvanostatic operation. The results show that the transversely isotropic stress solutions for cylindrical electrodes can degrade into isotropic stress solutions, and the radial and tangential stresses depend slightly on the Young's modulus ratio and lithiation expansion coefficient ratio of axial direction to isotropic plane, while the axial stress depends strongly on the two ratios. In addition, we introduce two correction factors which establish a bridge between the stress solutions for transversely isotropic electrodes and isotropic electrodes, and further investigate them from two special cases. It is indicated that transversely isotropic electrodes can be designed more flexibly compared with isotropic ones due to tunable anisotropic properties.

Fracture damage of nanowire lithium-ion battery electrode affected by diffusion-induced stress and bending during lithiation

Lithium-ion battery electrode materials generally experience significant volume changes during lithium diffusion during charging and discharging.

A simple model for diffusion-induced dislocations during the lithiation of crystalline materials

Assuming that the lithiation reaction occurs randomly in individual small particles in the vicinity of the reaction front, a simple model of diffusion-induced dislocations was developed. The diffusion-induced dislocations are controlled by the misfit strain created by the diffusion of solute atoms or the phase transformation in the vicinity of the reaction front. The dislocation density is proportional to the total surface area of the “lithiated particle” and inversely proportional to the particle volume. The diffusion-induced dislocations relieve the diffusion-induced stresses.

The Influence of Crystallographic Orientation on the Chemo-Elastic Response of Reconstructed LixCoO2 Cathode Particles

This work investigates the inhomogeneous and anisotropic, chemo-elastic response of a particular, but representative, reconstructed 3.6-μm LixCoO2 single-crystal cathode particle from a Li-ion battery. The results are based on a three-dimensional computational model that represents coupled electrochemical and mechanical response throughout the lithiation process (0.50 ⩽ x ⩽ 0.94) associated with a 1C galvanostatic discharge. The study focuses on the roles that phase transformation and crystallographic orientation play in the Li-concentration and stress-strain fields. Results show very high local peaks of diffusion-induced stresses, indicating the significant potential for particle fracture. Anisotropic chemo-elastic fields give rise to local Li-concentration bands, causing chemically induced misfit strains and high stresses, and such occurrences are determined to be irrespective of the particle morphology. The locally high stresses are associated primarily with a crystallographic phase transformation between two hexagonal phases that is prominent at x ≈ 0.75, being closely related to bands of chemical-misfit strains developed in the Li0.75CoO2 material. The chemo-mechanical band structure depends significantly on crystallographic orientation within the electrode particle.

Probing the Thermal Implications in Mechanical Degradation of Lithium-Ion Battery Electrodes

In a lithium-ion cell, heat generation and temperature evolution during operation pose a significant bearing on the mechanical degradation and cell performance. The thermal implications on the electrode mechano-electrochemical behavior have been elucidated. Crack formation due to diffusion-induced stress in the active particles has been analyzed. Temperature dependence of the mechano-physicochemical parameters has been taken into account. Total amount of diffusion-induced damage has been estimated for different current density, ambient temperature and particle size. For subzero temperatures, adiabatic operation can boost the cell performance significantly. Increased mechanical degradation has been observed for high C-rate and larger particle sizes. Decreasing ambient temperature results in aggravated crack formation resulting in severe capacity loss. However, at subzero temperatures and under high C-rate conditions, significant concentration gradient exists near the active particle peripheral region resulting in reduced damage penetration. The cell performance analysis reveals that the impact of mechanical degradation on the capacity loss is most prominent at subzero temperatures. The effect of cycling shows accelerated damage in the first few cycles followed by a plateau in the damage evolution. Existence of a critical particle size for maximum damage has been suggested which depends significantly on the cell temperature.