Hydrostatic Stress Gradient Research Articles

Analytical Investigation of Binder’s Role on the Diffusion Induced Stresses in Lithium Ion Battery through a Representative System of Spherical Isolated Electrode Particle Enclosed by Binder

Binder which serves to maintain mechanical and electrical integrity of Lithium ion batteries and occupies as much as 50% of battery weight is rather relatively less explored in terms of its effect on electrode and binder stresses during charging-discharging operation and hence motivates the present work. Closed form solution of diffusion induced stresses in an isolated spherical linear elastic, perfectly plastic electrode particle encapsulated by a linear viscoelastic binder during galvanostatic charging-discharging cycle under two sets of boundary conditions for the binder surface viz. traction free and fully constrained are developed. It is proved that the stresses in binder are unchanged even if electrode deforms elastically or plastically and whether diffusion in electrode is driven by concentration gradient or combination of concentration gradient and hydrostatic stress. The analysis identifies that current density not only directly controls the diffusion induced stress in electrode but affects it indirectly through change in stiffness of the binder depending upon the cycle time. Furthermore analysis reveals lower viscosity, lower stiffness and faster relaxation behavior as the recommended binder properties for lower diffusion induced stresses in electrode and binder.

(Invited) Thermodynamics, Stress, and Stefan-Maxwell Diffusion in Solids: Application to Small-Strain Materials Used in Commercial Lithium-Ion Batteries

Two different approaches have appeared in the literature in order to derive transport equations for coupled diffusion and stress in lithium host materials. The first approach is taken from solid mechanics and is described in detail, for example, in the work of Larché and Cahn [1] or in the reference text by Gurtin et al. [2]. Other examples of this approach can be found in Li [3] and Zhang et al. [4]. The second approach has been used by electrochemical researchers including Christensen and Newman [5] and Rengenathan et al. [6] to derive transport equations for lithium host materials, based on the Stefan-Maxwell equations as described in Hirschfelder et al. [7] and discussed in Bird and Klingenberg [8] for the treatment of gases and liquids. One of the motivations for this paper was the need to reconcile the different flux relations that result from these two approaches. Gases and liquids do not sustain a gradient in pressure or stress under equilibrium conditions in the absence of other body forces, such as gravity. In contrast, solids can maintain a stress gradient under equilibrium conditions. Hence, the most basic thermodynamic underpinnings in the treatment of a solid must differ from that of a liquid or gas, and this fact led us to employ the methods reviewed in [1,2] (the first approach). The second approach results in a version of the Stefan-Maxwell equations in which driving forces for diffusion are given by gradients of species concentrations, pressure, and temperature. The rigorous basis for these equations has led to their wide application to transport problems involving liquids and gases, but their application to solids is made difficult by the fact that the driving forces do not explicitly consider stress as it appears in solids. In [5] and [6] the pressure gradient is replaced by the hydrostatic-stress gradient in a solid. It is at this point in the derivation of flux relations where differences in the two approaches appear. The derivation of transport relations starts with the second law of thermodynamics, which leads to the formulation of a positive definite transport matrix relating fluxes and driving forces. The choice of which transport matrix to use must then be made based on the specific problem in question. Our approach is to modify the driving forces in the Stefan-Maxwell equations so as to accommodate stress gradients in the solid state. This is done using a generalized Gibbs-Duhem equation that includes gradients of stress within a solid. The transport matrix resulting from the Stefan-Maxwell equations is shown to be positive definite if all of the binary diffusion coefficients have positive values. These two steps demonstrate that the resulting Stefan-Maxwell equations are consistent with the thermodynamics as described in Gurtin et al. [2]. We consider substitutional and interstitial alloys. One of the purposes of this work is to present a rigorous derivation of a simple form of such models, thereby providing a basis on which further complications can be added. In particular, the model considered here ignores creation or annihilation of lattice sites, so that the total amount of all species (which is the same as the number of lattice sites) is conserved. Phase changes are also not considered. Plasticity effects will not be considered and we restrict our attention here only to the case of small strain theory, which is viewed as a preliminary step to the more general finite strain theory. The models considered will allow for the host species to be either mobile or immobile, and some comparisons of these two theories are given by numerical simulation. The model is also assumed to be isothermal. Complete details are given in [9]. F.C. Larché and J.W. Cahn, Acta metall., 30, 1835 (1982).M.E. Gurtin, E. Fried and L. Anand, The Mechanics and Thermodynamics of Continua, Cambridge University Press, New York, 2010.J. C-M. Li, Metallurgical Transactions A, 9A, 1353 (1978).X. Zhang, Wei Shyy, and A. M. Sastry, J. Electrochem. Soc., 154, A910 (2007).J. Christensen and J. Newman, J. Solid State Electrochem., 10, 293 (2006).S. Renganathan, G. Sikha, S. Santhanagopalan, and R. E. White, J. Electrochem. Soc., 157, A155 (2010).J. 0. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1954.R. B. Bird and D. J. Klingenberg, Advances in Water Resources, 62, 238 (2013).D. R. Baker, M. W. Verbrugge and A.F. Bower, J. Solid State Electrochem ., DOI 10.1007/s10008-015-3012-7 (2015).

A Crystal-Plasticity-Based Damage Model Incorporating Material Length-Scale

A crystal-plasticity-based damage model that incorporates material length-scale through use of the slip-plane lattice incompatibility is developed with attention to the physical basis for the evolution of damage in a “bulk” shear deformation and without resort to ad hoc measures of shear deformation. To incorporate the physics of the shear damage process recently found by Kweon et al. (2010, “Experimental Characterization of Damage Processes in Aluminum AA2024-O,” ASME J. Eng. Mater. Technol., 132(3), p. 031008), the development of tensile hydrostatic stress in grains due to grain-to-grain interaction, two existing theories, crystal plasticity, and the void growth equation by Cocks and Ashby (1982, “On Creep Fracture by Void Growth,” Prog. Mater. Sci., 27(3–4), pp. 189–244) is combined to make the model in this study. The effect of the void volume increase onto the constitutive behavior is incorporated by adding the deformation gradient due to the void volume growth into a multiplicatively decomposed kinematics map. Simulations with the proposed model reveal the physics of shear and reproduce the accelerated damage in the shear deformation in lab experiments and industrial processes: the gradient of hydrostatic stress along with the development of macroscopic normal stress (hydrostatic stress) components amplifies the development of the local hydrostatic stress in grains under tensile hydrostatic stress.

Defect-Transport-Induced Stress in Mixed Ionic-Electronic Conducting (MIEC) Ceramic Membranes

An extension of the Nernst–Planck–Poisson (NPP) formulation is developed to include the effects of stress in modeling mixed ionic-electronic conducting (MIEC) ceramics. In addition to diffusion and migration contributions, the Nernst–Planck flux includes the effects of hydrostatic stress gradients on the transport of charged defects. The model uses a thin-plate formulation to determine hydrostatic stress profiles, which depend on the defect-concentration profiles. With oxygen-separation membranes in mind, the model is illustrated using a combination of measured and estimated material properties for La0.6Sr0.4Co0.8Fe0.2O3 − δ (LSCF6482). Under transient conditions, where the oxygen partial pressure on one side of a membrane is suddenly changed, the model predicts concentration and stress profiles. The predicted maximum tensile stress can be very high, suggesting the possibility for membrane damage or failure.

Effect of specimen thickness on DHC velocity for Zr-2.5Nb alloy pressure tube material

Zr-2.5Nb alloy pressure tubes used in pressurized heavy water reactors (PHWR) are susceptible to failure by Delayed Hydride Cracking (DHC), which is a form of localized hydride-embrittlement phenomenon manifested in the presence of a hydrostatic stress gradient and hydrogen concentration above a threshold limit as a sub-critical crack growth process in hydride forming metals. The DHC parameters used for safety assessment are DHC velocity (VDHC) and a threshold stress intensity factor (KIH). In this work DHC velocity was determined for Zr-2.5Nb pressure tube material at 250 and 280 °C using specimens of thickness between 1 mm and 4.5 mm. The DHC velocity was found to increase with increase in specimen thickness at 250 °C and 280 °C. Significant amount of tunneling of the crack was observed for 1-mm and 2-mm thick specimens at 250 °C and 280 °C, with the degree of tunneling increasing with decrease in thickness and increase in test temperature.

Can mechanical stresses noticeably influence the diffusion of hydrogen in zircaloy?

Abstract The diffusion of hydrogen in zircaloy under the influence of mechanical stresses is investigated. The governing equations are derived from the chemical potential of hydrogen including the contribution due to hydrostatic stresses. The obtained differential equation is converted to a system of algebraic equations by applying the finite element method and the weighted residual procedure. The equations are used to study the hydrogen diffusion in a fuel rod cladding tube with an axial crack and in a strip plate. It is demonstrated that the hydrogen tends to flow towards the areas of positive hydrostatic stress gradients where it precipitates in hydrides when the solubility limit is exceeded. The precipitation is itself a mechanism enhancing the transport to these areas by weakening the effect of the diffusion driven by concentration differences. The presented method can readily be applied to the diffusion of other species in metals.

Simulation of hydrogen diffusion affected by stress-strain heterogeneity in polycrystalline stainless steel

The effect of microstructure-induced strain-stress heterogeneity on the evolution of hydrogen distribution was studied on 316 L stainless steel. A crystal plasticity-transient hydrogen diffusion finite element analysis was developed. To reach the stress-strain heterogeneity, synthetic microstructures were computed under uniaxial loading. The results show that hydrostatic stress gradients is the main cause of the hydrogen redistribution and the strain rate has a crucial effect on the redistribution degree. Statistical analysis of H-segregation reveals clustering of H-enriched zones at the mesoscopic scale.

3D Isogeometric Analysis of intercalation-induced stresses in Li-ion battery electrode particles

This paper is concerned with a three-dimensional coupled chemo-mechanical model for intercalation-induced stresses. Thereby, a diffusion model is enhanced by a drifting term involving the gradient of the hydrostatic stress state. A straightforward Finite Element discretization of the coupled diffusion would require C1-continuous shape functions. The implementation is thus based on the concept of Isogeometric Analysis, permitting a discretization purely in terms of the displacements and the concentration field. Furthermore, it allows to set up an operator matrix for the computation of the hydrostatic stress gradient in terms of the primal variables.The model is verified, in a simplified form, using analytical results from literature. It is subsequently employed for the study of the mechanical behavior of spherical and ellipsoidal particles under galvanostatic boundary conditions. The simulation data show a stress relaxation effect that becomes significant both in stiff electrode materials and for high charge rates. During charging, a characteristic tensile core-compressive shell structure develops. Once the stresses have reached a material-dependent threshold, they enhance the diffusion and drive ions away from regions of high compressive hydrostatic stress, that is, towards a particle’s core. Over time, this leads to reduced concentration gradients which, in turn, reduces the stresses in the particle already during the charging process. Models that neglect the stress effect consequently predict stress levels that are not only of higher magnitude, but that also do not decay.A comprehensive study on the influence of geometry, material constants, and charge rate has been performed. It sheds light on the distributions of stresses in ellipsoidal electrode particles. These have been reported to exhibit lower stress levels than spherical particles. Using a transformation of Cartesian stresses into a prolate spheroidal coordinate system, it can be shown that these particles equilibrate stresses by deformations along the semi-major axis. At the same time, a region of high stresses develops as a “belt” around the particles’ equator. Their intensity depends on the shape of the particle but is below that observed in spherical and oblate spheroidal particles. This offers an explanation for the longevity of these particles under cyclic charging processes.

From uni- to multi-axial fretting-fatigue crack nucleation: Development of a stress-gradient-dependent critical distance approach

Fretting fatigue is characterized by combined high stress gradients induced by contact loading and more homogeneous stress gradients induced by bulk fatigue stressing. The stress gradients computed at the “hot-spot” located on the surface at the trailing contact border are very high, usually above 10GPa/mm. For such uncommon stressing conditions, prediction of cracking risk becomes very complex and non-local fatigue approaches must be adopted. The purpose of the present study was to investigate how non-local strategies, such as “critical distance”, developed for medium stress gradient conditions such as “notch” configurations, were transposed to predict fretting cracking risk. Elastic crack nucleation conditions of a 35 Ni Cr Mo 16 low alloyed steel at 10E6 cycles have been identified for various cylinder pad radius, contact pressure and fatigue stress conditions. The experimental crack nucleation conditions were then compared to predictions from analytical simulations coupling uni-axial and Crossland’s multiaxial fatigue descriptions. The local “hot-spot” analysis systematically overestimated cracking risk and induced more than 30% error with respect to the experimental values. The non-local “critical distance method” based on a constant length scale value still displayed more than 10% dispersion suggesting that a non-constant “critical distance” approach must be considered. By expressing the critical distance evolution as a function of the hydrostatic stress gradient operating next to the stress hot-spot, dispersion was reduced below 5%. Established for the Crossland’s stress invariant formulation, this tendency is confirmed by comparing McDiarmid and MWCM critical plane fatigue approaches.

A computational study of systemic hydration in vocal fold collision

Mechanical stresses develop within vocal fold (VF) soft tissues due to phonation-associated vibration and collision. These stresses in turn affect the hydration of VF tissue and thus influence voice health. In this paper, high-fidelity numerical computations are described, taking into account fully 3D geometry, realistic tissue and air properties, and high-amplitude vibration and collision. A segregated solver approach is employed, using sophisticated commercial solvers for both the VF tissue and glottal airflow domains. The tissue viscoelastic properties were derived from a biphasic formulation. Two cases were considered, whereby the tissue viscoelastic properties corresponded to two different volume fractions of the fluid phase of the VF tissue. For each case, hydrostatic stresses occurring as a result of vibration and collision were investigated. Assuming the VF tissue to be poroelastic, interstitial fluid movement within VF tissue was estimated from the hydrostatic stress gradient. Computed measures of overall VF dynamics (peak airflow velocity, magnitude of VF deformation, frequency of vibration and contact pressure) were well within the range of experimentally observed values. The VF motion leading to mechanical stresses within the VFs and their effect on the interstitial fluid flux is detailed. It is found that average deformation and vibration of VFs tend to increase the state of hydration of the VF tissue, whereas VF collision works to reduce hydration.

On the mechanism and the length scales involved in the ductile fracture of a bulk metallic glass

Some bulk metallic glasses (BMGs) exhibit high crack initiation toughness due to shear band mediated plastic flow at the crack tip and yet do not display additional resistance to crack growth due to the lack of a microstructure. Thus, at crack initiation, the fracture behavior of BMGs transits from that of ductile alloys to that of brittle ceramics. In this paper, we attempt to understand the physics behind the characteristic length from the notch root at which this transition occurs, through testing of four-point bend specimens made of a nominally ductile Zr-based BMG in three different structural states. In the as-cast state, both symmetric (mode I) and asymmetric (mixed mode) bend specimens are tested. The process of shear band mediated plastic flow followed by crack initiation at the notch root was monitored through in situ imaging. Results show that stable crack growth occurs inside a dominant shear band through a distance of ∼60μm, irrespective of the structural state and mode mixity, before attaining criticality. Detailed finite element simulations show that this length corresponds to the distance from the notch root over which a positive hydrostatic stress gradient prevails. The mean ridge heights on fractured surfaces are found to correlate with the toughness of the BMG. The Argon and Salama model, which is based on the meniscus instability phenomenon at the notch root, is modified to explain the experimentally observed physics of fracture in ductile BMGs.

Stress relaxation through interdiffusion in amorphous lithium alloy electrodes

At high guest (lithium) atom concentrations, the diffusion of host (e.g., silicon) atoms may become significant in amorphous Li-alloy-based solid electrodes. The effect of this diffusion mechanism on stress development is in addition to guest atom diffusion, stress-induced enhancement of guest atom diffusion and plasticity. The effect of the diffusive migration of host atoms in amorphous Li-alloy-based electrodes is investigated using a continuum model. A mixed-form finite element framework is developed to simulate the full coupling between stress development and interdiffusion. This framework overcomes the challenges associated with the numerical evaluation of the hydrostatic stress gradient. The analysis focuses on the relative importance of the mechanical driving force and chemical driving force for host migration. Calculations show that host migration can cause stress reductions of up to ∼20% in Li–Si electrodes at stress levels below the yield threshold of the material. Analyses also show that the long-term steady state of stress distribution is independent of the host diffusivity and the thermodynamic factor of diffusion which quantifies the tendency of the two species of atoms to chemically mix, even though the transient behavior (in particular, the peak stresses during charging being important quantities) does depend on the thermodynamic factor and the host diffusivity. The diffusion of Si (host) introduces a time scale which, along with the time scale for Li (guest) diffusion, controls the diffusional response of electrodes.

Applying x-ray microscopy and finite element modeling to identify the mechanism of stress-assisted void growth in through-silicon vias

Fabricating through-silicon vias (TSVs) is challenging, especially for conformally filled TSVs, often hampered by the seam line and void inside the TSVs. Stress-assisted void growth in TSVs has been studied by finite element stress modeling and x-ray computed tomography (XCT). Because x-ray imaging does not require TSVs to be physically cross-sectioned, the same TSV can be imaged before and after annealing. Using 8 keV laboratory-based XCT, voids formed during copper electroplating are observed in as-deposited samples and void growth is observed at the void location after annealing. We hypothesize that the mechanism generating voids is hydrostatic stress-assisted void growth. Stresses in a copper-filled TSV with a pre-existing void were simulated by finite element methods. The peaks of the hydrostatic stress and its gradient are shown to be around the edge of the void. Comparing simulated results and experimental data shows that void growth in TSVs is stress-assisted: vacancies diffuse and coalesce at the void as a result of the hydrostatic stress gradient.

Modeling of Oxygen Diffusion Along Grain Boundaries in a Nickel-Based Superalloy

Finite element analyses of oxygen diffusion at the grain level have been carried out for a polycrystalline nickel-based superalloy, aiming to quantify the oxidation damage under surface oxidation conditions at high temperature. Grain microstructures were considered explicitly in the finite element model where the grain boundary was taken as the primary path for oxygen diffusion. The model has been used to simulate natural diffusion of oxygen at temperatures between 650∘C and 800∘C, which are controlled by the parabolic oxidation rate and oxygen diffusivity. To study the effects of mechanical stress on oxygen diffusion, a sequentially coupled deformation-diffusion analysis was carried out for a generic specimen geometry under creep loading condition using a submodeling technique. The material constitutive behavior was described by a crystal plasticity model at the grain level and a unified viscoplasticity model at the global level, respectively. The stress-assisted oxygen diffusion was driven by the gradient of hydrostatic stress in terms of pressure factor. Heterogeneous deformation presented at the grain level imposes a great influence on oxygen diffusion at 750∘C and above, leading to further penetration of oxygen into the bulk material. Increased load level and temperature enhance oxygen concentration and penetration within the material. At 700∘C and below, mechanical loading seems to have negligible influence on the oxygen penetration because of the extremely low values of oxygen diffusivity and pressure factor. In the case of an existing surface microcrack, oxygen tends to accumulate around the crack tip due to the high stress level presented near the crack tip, leading to localized material embrittlement and promotion of rapid crack propagation.

Hydrogen diffusion analyses of a cracked steel pipe under internal pressure

In hydrogen diffusion in metals, we have to consider not only the hydrogen flux driven by the gradient of hydrogen concentration but also the hydrogen flux driven by the gradient of hydrostatic stress. Such hydrogen diffusion is a multi-physics problem coupled hydrogen diffusion with a stress field. We performed such hydrogen diffusion analyses for a steel pipe with a crack on an inner surface under internal pressure. On the inner surface of the pipe, the hydrogen concentration calculated using the fugacity of the hydrogen molecules was used as a boundary condition for a hydrogen diffusion equation. The finite element method was employed both for a hydrogen diffusion analysis and for a stress analysis. We obtained the hydrogen concentration around a crack tip by varying the magnitude of internal pressure and the crack length, and examined the relationship between the maximum hydrogen concentration and a nonlinear fracture mechanics parameter CTOD ( Crack Tip Opening Displacement). Good correlation is found between the maximum hydrogen concentration and CTOD.

Numerical Analysis of Vacancy Transport by Residual Stress in Electromigration on LSI Interconnects

The Li multiplication method for the driving force induced particle migration equation was proposed to solve numerically the stress and electric fields induced vacancy migration equation. On the basis of this method, vacancy migration behaviors were found to be predicted under the competitive relationship between stress and electric field. When a residual stress is dominant, vacancies concentrate around the maximum hydrostatic stress region, such as elastic–plastic boundary. On the other hand, when the electric field is dominant, vacancies do not concentrate around the maximum hydrostatic stress region, but move from the cathode end to the anode end. An in situ observation of electromigration on AlCuSi interconnect was conducted to verify the simulation results. A void nucleated and grew at the tip of a notch on an AlCuSi line without passivation film, while many voids appeared and grew in cathode side in a line with passivation film. Greater hydrostatic stress gradient is considered to occur in the line without passivation film, owing to a small scale of yielding, and this leads to concentrate void formation at the tip of the notch. These results indicate that numerical analysis of electromigration is valid to represent the experimental result. Based on results mentioned above, various failure modes in interconnects cased by electromigration are predictable by the proposed Li multiplication method.

Temperature-dependent stress-induced voiding in dual-damascene Cu interconnects

Temperature-dependent stress-induced voiding (SIV) of Cu dual-damascene interconnects in via-line structures has been studied by stress modeling and focused-ion-beam (FIB) cross-section analysis. The via-line structures have been studied at temperatures ranging from 100 to 250 °C, with highest voiding rate being detected at 200 °C. The hydrostatic stress and stress gradient reach each peak values and lead to void nucleation symmetrically underneath the edge of the via. Stress gradient instead of stress shows crucial effect on the SIV process. Voids tend to grow horizontally along the Cu/SiN interface under larger stress gradient in the line length direction as compared to the line height direction. A voiding model based on Nabarro–Herring equations has been proposed and the relation between the voiding rate and stress gradient has been built. The voiding model shows that the stress-induced voiding is a process by which vacancies diffuse and concentrate to form voids under the force of stress gradient. The stress and the diffusional factors grow oppositely with temperature and the maximum voiding rate is reached at a medium temperature.

Fabrication of Cu Nanowires at Predetermined Positions by Utilising Stress Migration

Abstract: Stress migration (SM) is a phenomenon whereby atoms diffuse from a region of lower stress towards a region of higher stress because of the hydrostatic stress gradient. In this study, the fabrication of Cu nanowires at predetermined positions by controlling the direction of the atomic flux caused by SM is analysed. Cu nanowires are fabricated by rearranging accumulated atoms into wire‐like crystals. First, the sites at which Cu atoms accumulate are found from finite element (FE) analysis and from the results obtained from experiments. Next, a technique is proposed for accumulating atoms by SM at predetermined sites. Finally, the successful fabrication of Cu nanowires at these sites is demonstrated.

Resistance characterization of Cu stress-induced void migration at narrow metal finger connected with wide lead

In this work, copper stress-induced voiding (SIV) at narrow metal finger connected with wide lead is investigated. Three failure modes are explored and discussed with different failure sites. A driving force for void migration resulted from hydrostatic stress gradient is also studied. Meanwhile, in order to assess the impact of copper void on multi-level interconnects, a model based on finite element analysis (FEA) is developed to simulate the resistance change with regard to voiding location, void morphology and interconnect scenario. Finally, a correlation between SIV and resistance change is obtained, which can serve as references for reliability evaluation and risk assessment.

Copper stress migration at narrow metal finger with wide lead

Copper stress migration (SM) at narrow metal finger connected with wide lead is investigated in this work. Voiding phenomenon is explored with respect to specific failure site. Metal geometric factors including lead width, lead length and finger length are also discussed in terms of SM failure rates. Meanwhile, in order to study the failure mechanism, a finite element analysis (FEA) is conducted to evaluate the influences from vacancy migration path, hydrostatic stress gradient and different metal geometries. As a result, a characterization regarding SM at narrow metal finger with wide lead is obtained for the reference of reliability assessment and process improvement.