Concentration-dependent Material Properties Research Articles

Influence of concentration-dependent material properties on the fracture and debonding of electrode particles with core–shell structure

Core–shell electrode particle designs offer a route to improved lithium-ion battery performance. However, they are susceptible to mechanical damage such as fracture and debonding, which can significantly reduce their lifetime. Using a coupled finite element model, we explore the impacts of diffusion-induced stresses on the failure mechanisms of an exemplar system with an NMC811 core and an NMC111 shell. In particular, we systematically compare the implications of assuming constant material properties against using Li concentration-dependent diffusion coefficient and partial molar volume. With constant material properties, our results show that smaller cores with thinner shells avoid debonding and fracture regimes. When factoring in a concentration-dependent partial molar volume, the maximum values of tensile hoop stress in the shell are found to be significantly lower than those predicted with constant properties, reducing the likelihood of fracture. Furthermore, with a concentration-dependent diffusion coefficient, significant barriers to full electrode utilisation are observed due to reduced lithium mobility at high states of lithiation. This provides a possible explanation for the reduced accessible capacity observed in experiments. Shell thickness is found to be the dominant factor in precluding structural integrity once the concentration dependency is accounted for. These findings shed new light on the performance and effective design of core–shell electrode particles.

A coupled mechanical-electrochemical phase-field formulation for understanding the evolution of lithiated-silicon sponge

The unexpected formation of lithiated-silicon sponge during cyclic charge of lithium-ion batteries with silicon anodes has been observed in recent reports. The severe hollowing of silicon electrodes leads to an irreversible deformation and the generation of new surfaces, which greatly affects the structural integrity and cycling stability of the cells. However, the understanding of lithiated-silicon sponge still remains as a great challenge. In this work, a coupled mechanical-electrochemical model based on the phase-field method is proposed, which can describe the void evolution and the surface instability in Si electrodes. The model combines a description of the lithium diffusion, vacancy nucleation and annihilation, and an elastic-plastic constitutive model with concentration-dependent material properties. The simulation results reveal that the void within/on the silicon electrode tends to grow toward/along the electrode surface and toward other voids, leading to the electrode surface disruption and void merging, as well as the intra-electrode tunnel formation and electrolyte intrusion. Importantly, it is found that mechanical pressure can not only suppress the growth of a single void but also prevent the linkage and merging between multiple voids. The proposed theoretical framework and the findings of this work provide an in-depth understanding of the complicated mechanical-electrochemical coupling behavior of Si electrodes.

Modulus Estimation of Composites with High Porosity, High Particle Volume Fraction, and Particle Eigenstrain: Application to the LIB Active Layer with a Bridged-Particle Mesostructure

Due to the complex mesostructure and components of composite active layers in lithium-ion battery (LIB) electrodes, coupled with the concentration-dependent material properties and eigenstrains, efficiently estimating the effective modulus of the active layers remains a great challenge. In this work, the classic Mori–Tanaka method is found to be unable to estimate the modulus of the active layer. By realizing the importance of the mesostructure feature, a rod-rod model is proposed. The resulting modulus is expressed analytically. It is shown that the rod-rod model can accurately estimate the modulus evolution of the active layer if the material properties of the components and the evolution of volume fractions are known in advance. Moreover, a simplified rod-rod model is also developed to reduce the complexity of the proposed method. By knowing the volume fractions at two arbitrary states of charge and subsequently determining two constants, the simplified model can estimate the modulus efficiently. Considering both its accuracy and its simplicity, the simplified rod-rod model is the most suitable for the estimation. Thus, the methods developed in this work provide a new perspective for analyzing the material properties of composite active layers in LIB electrodes.

An Improved Experiment for Measuring Lithium Concentration-Dependent Material Properties of Graphite Composite Electrodes.

The in situ curvature measurement of bilayer beam electrodes is widely used to measure the lithium concentration-dependent material properties of lithium-ion battery electrodes, and further understand the mechano-electrochemical coupling behaviors during electrochemical cycling. The application of this method relies on the basic assumption that lithium is uniformly distributed along the length and thickness of the curved active composite layer. However, when the electrode undergoes large bending deformation, the distribution of lithium concentration in the electrolyte and active composite layer challenges the reliability of the experimental measurements. In this paper, an improved experiment for simultaneously measuring the partial molar volume and the elastic modulus of the graphite composite electrode is proposed. The distance between the two electrodes in the optical electrochemical cell is designed and graphite composite electrodes with four different thickness ratios are measured. The quantitative experimental data indicate that the improved experiment can better satisfy the basic assumptions. The partial molar volume and the elastic modulus of the graphite composite electrode evolve nonlinearly with the increase of lithium concentration, which are related to the phase transition of graphite and also affected by the other components in the composite active layer. This improved experiment is valuable for the reliable characterization of the Li concentration-dependent material properties in commercial electrodes, and developing next-generation lithium batteries with more stable structures and longer lifetimes.

Chemo-mechanical analysis of ratcheting deformation in silicon particle electrode under cyclic charging and discharging

Evidences have accumulated that high capacity silicon electrode will expand/shrink dramatically during cyclic charging/discharging operation. In this process, the electrode may undergo elastic-inelastic deformation, and cyclic asymmetric behavior of tension and compression can naturally result in the electrode ratcheting deformation. In this paper, a chemo-mechanical semi-analytical model is established to describe the ratcheting behavior of the silicon particle electrode based on concentration-dependent material properties. The results show that inelastic deformation can reduce the rapid increase of the stresses and further reduce the possibility of surface crack. However, the accumulated irreversible inelastic strain may become one of the important factors for mechanical degradation and even electrode failure. The influence of hydrostatic stress on electrode ratcheting behavior is discussed in detail. It can decrease the concentration gradient and stress level, thereby reducing the inelastic region and ratcheting deformation, which will prolong the predicted lifetime of the electrode. The diffusion coefficient, particle size and the charging rate have also been investigated. The results reveal that ratcheting deformation may be decreased by the high diffusivity, the small particle size and slow charging strategy, which will improve the mechanical stability and capacity of the electrode. Meanwhile, interparticle contact may worsen the ratcheting deformation. Increasing the electrode porosity appropriately can provide space for silicon particles to expand, which may reduce the ratcheting deformation. The present work gives a new explanation for ratcheting deformation in silicon particle electrode and provides a theoretical basis for guiding one in designing next-generation high capacity lithium-ion batteries.

Capturing the stress evolution in electrode materials that undergo phase transformations during electrochemical cycling

The present work sheds light on the stresses generated in a spherical particle subjected to phase transformations during ion-insertion. In order to account for the physical process that occurs during electrochemical cycling, the models used are those of small deformation and account for the effects of phase transformation, chemo-mechanical coupling and concentration-dependent material properties. The two-phase lithiation is modeled by the Cahn–Hilliard equation. It is found that the DISs arise from the inhomogeneous volume expansions resulting from Li concentration gradients and the hydrostatic stress facilitates the diffusion of Li-ions under elastic deformation while it hinders diffusion in the plastic case. When the elastic modulus is reduced the magnitude of the diffusion-induced stress decreases but the strain increases under elastic deformation whereas the opposite occurs for the plastic case. Furthermore, if the electrode is assumed to undergo strain softening during plastic deformation, smaller stresses and higher plastic strains are predicted than when strain hardening is assumed. The novelty of this work is that the proposed models highlight the importance of chemo-mechanical coupling effects, concentration-dependent material properties and plastic deformation on diffusion-induced stresses. These findings render prospective insights for designing next-generation mechanically stable phase transforming electrode materials.

Precise residual stress profile in ion-exchanged silicate glass by modified contour method

In this work, the contour method is modified and used to precisely estimate the residual stress profiles in ion-exchanged aluminosilicate glasses, including the effects of non-homogeneous material properties. The method requires the precise measurements of the fracture surface’s topography and exploits fractographic features to mate and accurately determine the out-of-plane deformation after fracture. Residual stresses are computed from the out-of-plane deformation through a weight function approach rather than by numerical methods, as weight functions can easily account for concentration-dependent material properties. The stress profiles in aluminosilicate glass with three different molten salt soaking times were calculated and compared with the approximate profiles measured by optical waveguide. It was found that the optical waveguide’s estimations under-predicts the surface stress by as much as 10–30 %, likely due to the assumption that the stress optic coefficient is unaffected by the K+↔Na+ exchange.

Influence of Li Concentration-Dependent Diffusion Coefficient and Modulus Hardening on Diffusion-Induced Stresses in Anisotropic Anode Particles

Particle cracking caused by diffusion-induced stresses (DISs) is an important reason for lithium-ion battery (LIB) capacity fading. In this study, concentration-dependent material properties are introduced to model the distribution of the concentrations and evolution of DISs in anisotropic active particles. The concentration-dependent diffusion coefficient increases the concentration gradient and thus the DISs, and the concentration-dependent elastic modulus hardening increases the internal DISs and thus the stress-enhanced diffusion of Li ions. Diffusion in the direction of a large diffusion coefficient enhances the diffusion in the direction of a small diffusion coefficient, which leads to an anisotropic concentration, concentration gradient and DISs. The greater the anisotropic difference within the particles is, the more obvious the decrease in the radial stress and hoop stress. The results can be comparable with many published experimental results of graphite and indicate that the role of concentration-dependent material properties and anisotropy in the particles cannot be ignored.

Direct Osmotic Pressure Measurements in Articular Cartilage Demonstrate Nonideal and Concentration-Dependent Phenomena.

The osmotic pressure in articular cartilage serves an important mechanical function in healthy tissue. Its magnitude is thought to play a role in advancing osteoarthritis. The aims of this study were to: (1) isolate and quantify the magnitude of cartilage swelling pressure in situ; and (2) identify the effect of salt concentration on material parameters. Confined compression stress-relaxation testing was performed on 18 immature bovine and six mature human cartilage samples in solutions of varying osmolarities. Direct measurements of osmotic pressure revealed nonideal and concentration-dependent osmotic behavior, with magnitudes approximately 1/3 those predicted by ideal Donnan law. A modified Donnan constitutive behavior was able to capture the aggregate behavior of all samples with a single adjustable parameter. Results of curve-fitting transient stress-relaxation data with triphasic theory in febio demonstrated concentration-dependent material properties. The aggregate modulus HA increased threefold as the external concentration decreased from hypertonic 2 M to hypotonic 0.001 M NaCl (bovine: HA=0.420±0.109 MPa to 1.266±0.438 MPa; human: HA=0.499±0.208 MPa to 1.597±0.455 MPa), within a triphasic theory inclusive of osmotic effects. This study provides a novel and simple analytical model for cartilage osmotic pressure which may be used in computational simulations, validated with direct in situ measurements. A key finding is the simultaneous existence of Donnan osmotic and Poisson-Boltzmann electrostatic interactions within cartilage.

Effect of temperature and moisture on free vibration characteristics of skew laminated hybrid composite and sandwich plates

This paper is concerned with the effect of variation in temperature and moisture concentration on free vibration response of skew laminated hybrid composite and sandwich plates. The coupled thermo-elastic and hygro-elastic finite element model is formulated using the first-order shear deformation theory (FSDT). Uniform temperature and moisture concentration rise is considered for the analysis. Soft-core viscoelastic materials are considered for the sandwich plates and are modeled using the complex modulus approach. Linear strain-displacement relations are used to develop a mechanical stiffness matrix, and the initial stress stiffness matrix is generated using non-linear strain-displacement relations to represent the non-mechanical stiffness matrix. Numerical examples for the generated finite element model are presented and discussed comprehensively to understand the effect of temperature, moisture concentration, skew angle, length to width ratio, length to thickness ratio, and boundary conditions on the vibration response of the laminated hybrid composite and sandwich plates. Further investigation is devoted to studying the influence of temperature and moisture concentration-dependent material properties, stacking sequence, core to face sheet thickness ratio, and fiber orientation on vibration behavioral response of sandwich and hybrid composite plates.

A comparative study on ratcheting deformation between negative Poisson's ratio electrode and thin film electrode in Li-ion battery cyclic operation

In the charging/discharging cycle of battery, electrode may expand/shrink cyclically and be in an asymmetric behavior of tension and compression. Based on this, electrode may undergo elastic deformation and inelastic deformation, which can naturally result in the electrode ratcheting deformation and capacity fading. In this paper, a new analytical model is developed to study the ratcheting deformation of thin film electrode and negative Poisson's ratio electrode (hollow re-entrant honeycomb shape) in battery cyclic operation. The influence of concentration-dependent material properties on electrode ratcheting deformation is discussed in detail. By the comparison between thin film electrode and negative Poisson's ratio electrode, it is found that the stress level in negative Poisson's ratio electrode is relatively low due to the faster appearance of the inelastic deformation. Furthermore, the structure optimization of negative Poisson's ratio electrode can suppress the ratcheting deformation (produce less accumulated deformation) in battery operating cycles and increase the battery cycle life about 8.1% when compared with thin film electrode. Our work shows that the electrode mechanical degradation is in the form of ratcheting deformation when subjected to cyclic loading. The developed analytical model of negative Poisson's ratio electrode may help for the electrode design and provide a theoretical way to prolong the cycle life of the battery.

Closed-form solution to the residual stresses in ion-exchanged silicate glass including concentration-dependent material properties

In this work, the closed-form solution to the residual stresses in ion-exchanged, silicate glasses accounting for composition-dependent material properties, is derived. The solution is applicable to physical systems where the self-diffusion coefficients of the exchanged, ionic species are nearly identical, such as for Na+ ↔ K+ in both soda-lime and alumino-silicate glass. In the proposed solution, the relaxation function is expressed in terms of the Prony series and the relaxation times, Young's modulus, and the dilation coefficient are assumed as analytic functions of the chemical concentration. A significant, practical aspect of this research is that the derived closed-form solution could be used to establish the relationship between the material properties and the local ion concentration through the regression analysis of experimentally-determined, residual stress profiles. Practical illustrations of how the relationship between the ion concentration and both the dilation coefficient and the relaxation times could be established in ion-exchanged, soda-lime glass are finally presented.

Effect of Li concentration-dependent material properties on diffusion induced stresses of a Sn anode.

Sn is one of the promising Li ion battery anode materials with high theoretical capacity and mechanical properties that allow for effective relaxation of Li diffusion-induced stresses. Sn is a low melting point metal with a low modulus and strength and has the ability to relax stresses via plasticity and creep deformations. In this study, concentration-dependent material properties are used in numerical simulations to model the Li diffusion-induced stress evolution in Sn micropillars. Simulation results using concentration-dependent material properties resulted in a completely different failure mode in comparison to that of concentration-independent simulation results. Tensile hoop stress needed for crack propagation was analyzed to be at the core for concentration-independent material properties, and switched to being at the surface for concentration-dependent simulation results. In addition, by incorporating these maximum tensile DIS results, the critical size for the failure of Sn micropillars was determined to be 5.3 μm at C/10 charging rate. This was then correlated with experimental observations, where fracture occurred in Sn micropillars with sizes larger than 6 μm, while 4.4 μm sized Sn micropillars survived the lithiation cycle.

Effects of plastic deformation in current collector on lithium diffusion and stress in bilayer lithium-ion battery electrode

Lithium-ion batteries (LIBs) have already become indispensable energy storage devices, as they can meet urgent requirements for higher energy and power density in the applications ranging from portable electronics to electric vehicles. However, in the process of charging and discharging of LIB, the diffusion-induced stress associated with inhomogeneous Li concentration in the electrode may cause the electrode material to damage, and then further degrade storage capacity and cycling performance of LIB. Therefore, it is important to quantitatively understand the mechanism relating to the stress evolution in electrode during electrochemical cycling, which will be conducive to developing effective methods of relieving the diffusion induced stress. In this work, a bilayer electrode model is proposed by taking into account Li diffusion, built-in stress, concentration-dependent material properties and elastoplastic deformation of current collector. Based on the established model, the influences of the possible plastic deformation in the current collector on the lithium diffusion and stress evolution of bilayer electrode during charging are investigated. The numerical results show that the plastic deformation of current collector can weaken the constraint between current collector and active layer, which leads to a smaller electrode curvature and more homogeneous lithium concentration in the active layer. The relaxation effect of the plastic deformation not only significantly relieves the stresses at the bottom and top surface of active layer, but also promotes the diffusion of lithium into active layer, which can improve the structural reliability of the electrode and increase the effective capacity of the active layer. Furthermore, the influences of the yield strength and plastic modulus of the current collector are discussed. The results indicate that the constraint between the current collector and active layer becomes weaker with reducing yield strength and plastic modulus of current collector, respectively. In other words, the further stress relaxation in the electrode indicates that the capacity can be enhanced upon reducing the yield strength and plastic modulus of current collector, respectively. Considering our results, it is expected that a bilayer electrode composed of the current collector with smaller mechanical strength enjoys simultaneous improvement in battery usable capacity and structural reliability. Consequently, the results of this paper provide a route to improving the cycle performance of bilayer lithium-ion battery electrode.

Effects of electrode particle morphology on stress generation in silicon during lithium insertion

We study the effects of particle morphology and size on stress generation during Li insertion into Si particles using a fully coupled diffusion-elasticity model implemented in a finite element formulation. The model includes electrochemical reaction kinetics through a Butler–Volmer equation, concentration-dependent material properties, and surface elasticity. Focusing on two idealized geometries (hollow spheres and cylinders), we simulate stresses during Li insertion in Si. These systems describe a wide variety of morphologies that have been fabricated and studied experimentally, including particles, nanowires, nanotubes, and porous solids. We find that stresses generated in solid particles during Li insertion decrease as particle radii decrease from μm-scale, but reach a minimum at about 150 nm. Surface stresses then begin to dominate the stress states as the particle size continues to decrease. The minimum occurs at larger radii for hollow particles. We also find that hollow particles experience lower stresses than solid ones, but our results suggest that there is not a significant difference in maximum stress magnitudes for spherical and cylindrical particles. Studying the influence of concentration-dependent elastic moduli we find that while they can significantly influence stress generation for potentiostatic insertion, their role is minimal when surface reaction kinetics are considered.

Ultrafiltration of macromolecular solutions and cross-flow microfiltration of colloidal suspensions. A contribution to permeate flux calculations

Ultrafiltration and cross-flow microfiltration are pressure-driven membrane processes which exhibit concentration polarisation in front of the membrane or gel layer, respectively. In ultrafiltration the concentration polarization is essentially determined by diffusion in the boundary layer. With the usual assumption of constant material properties, the film theory predicts fluxes quite accurately in the case of ultrafiltration of low molecular weight solutions. The calculated fluxes are, however, significantly lower than the observed figures in case of ultrafiltration of macromolecular solutions. It will be shown that with concentration-dependent material properties, especially a variable density (!) and a variable diffusion coefficient, the film theory can predict fluxes of ultrafiltration of macromolecular solutions very accurately. Such a theory fails, however, in case of cross-flow filtration of colloidal suspensions. Even with the assumption of a variable density in the boundary layer, the calculated fluxes are too low by a factor of about 10. For cross-flow microfiltration of colloidal suspensions, a model is presented considering basically the membrane-parallel drag forces acting on a particle in the vicinity of the membrane and a “friction” force as a consequence of the lateral drag forces. The model has been tested with different module systems and with different colloidal suspensions.

Swelling of articular cartilage and other connective tissues: electromechanochemical forces.

We have measured the relationship between tissue swelling stress and consolidation for bovine articular cartilage and corneal stroma in uniaxial confined compression as a function of bath ionic strength. Our experimental protocol and results clearly demonstrate that two concentration-dependent material properties are necessary to describe the chemical dependence of tissue swelling stress in uniaxial compression over the range of deformations and concentrations explored. A general electromechanochemical model for the swelling stress of charged connective tissues is developed. The model focuses on the role of charged matrix macromolecules in determining the mechanical behavior of the tissue. A constitutive relation for the swelling stress in uniaxial confined compression is formulated and the concentration dependence of the material properties of articular cartilage and corneal stroma is determined. The associated free swelling behavior of cartilage and cornea specimens is computed from these results and is found to compare favorably with data from the literature.