Phase-field model for the two-phase lithiation of silicon

As an ideal anode material, silicon has the highest lithium-ion capacity in theory, but the broader application is limited by the huge volumetric strain caused by lithium insertion and extraction. To better understand the physical process and to resolve the related reliability issue, enormous efforts have been made. Recent experiments observed sharp reaction fronts in both crystalline and amorphous silicon during the first lithiation half-cycle. Such a concentration profile indicates that the process is likely to be reaction limited. Based on this postulation, a phase-field model is developed and implemented into a finite-element code to simulate the coupled large inelastic deformation and motion of the reaction front in a silicon electrode. In contrast to most existing models, the model treats both volumetric and deviatoric inelastic deformation in silicon as a direct consequence of the lithiation at the reaction front. The amount of deviatoric deformation is determined by using the recently developed kinetic model of stress-induced anisotropic reaction. By considering the role of stress in the lithiation process, this model successfully recovers the self-limiting phenomenon of silicon electrodes, and relates it to the local geometry of electrodes. The model is also used to evaluate the energy-release rate of the surface crack on a spherical electrode, and the result suggests a critical size of silicon nanoparticles to avert fracture. As examples, the morphology evolution of a silicon disk and a Si nanowire during lithiation are also investigated.     

Cracking mechanisms in lithiated silicon thin film electrodes

The massive cracking of silicon thin film electrodes in lithium ion batteries is associated with the colossal volume changes that occur during lithiation and delithiation cycles. However, the underlying cracking mechanism or even whether fracture initiates during lithiation or delithiation is still unknown. Here, we model the stress generation in amorphous silicon thin films during lithium insertion, fully accounting for the effects of finite strains, plastic flow, and pressure-gradients on the diffusion of lithium. Our finite element analyses demonstrate that the fracture of lithiated silicon films occurs by a sequential cracking mechanism which is distinct from fracture induced by residual tension in conventional thin films. During early-stage lithiation, the expansion of the lithium-silicon subsurface layer bends the film near the edges, and generates a high tensile stress zone at a critical distance away within the lithium-free silicon. Fracture initiates at this high tension zone and creates new film edges, which in turn bend and generate high tensile stresses a further critical distance away. Under repeated lithiation and delithiation cycles, this sequential cracking mechanism creates silicon islands of uniform diameter, which scales with the film thickness. The predicted island sizes, as well as the abrupt mitigation of fracture below a critical film thickness due to the diminishing tensile stress zone, is quantitatively in good agreement with experiments.     

Cyclic plasticity and shakedown in high-capacity electrodes of lithium-ion batteries

Of all materials, silicon has the highest capacity to store lithium, and is being developed as an electrode for lithium-ion batteries. Upon absorbing a large amount of lithium, the electrode swells greatly, with a volumetric change up to 300%. The swelling is inevitably constrained in practice, often leading to stress and fracture. Evidence has accumulated that the swelling-induced stress can be partially relieved by plastic flow, and that electrodes of small feature sizes can survive many cycles of lithiation and delithiation without fracture. Here we simulate a particle of an electrode subject to cyclic lithiation and delithiation. A recently developed theory of concurrent large swelling and finite-strain plasticity is used to co-evolve fields of stress, deformation, concentration of lithium, and chemical potential of lithium. We identify three types of behavior. When the yield strength is high and the charging rate is low, the entire particle deforms elastically in all cycles. When the yield strength is low and the charging rate is high, the particle (or part of it) undergoes cyclic plasticity. Under intermediate conditions, the particle exhibits the shakedown behavior: part of the particle flows plastically in a certain number of initial cycles, and then the entire particle remains elastic in subsequent cycles. We discuss the effect of the three types of behavior on the capacity and the electrochemical efficiency.     

Lithiation-enhanced charge transfer and sliding strength at the silicon-graphene interface:A first-principles study

The application of silicon as ultrahigh capacity electrodes in lithium-ion batteries has been limited by a number of mechanical degradation mechanisms including fracture, delamination and plastic ratcheting, as a result of its large volumetric change during lithiation and delithiation. Graphene coating is one feasible technique to mitigate the mechanical degradation of Si anode and improve its conductivity. In this paper, first-principles calculations are performed to study the atomic structure, charge transfer and sliding strength of the interface between lithiated silicon and graphene. Our results show that Li atoms segregate at the(lithiated) Si-graphene interface preferentially, donating electrons to graphene and enhancing the interfacial sliding resistance. Moreover, the interfacial cohesion and sliding strength can be further enhanced by introducing single-vacancy defects into graphene.These findings provide insights that can guide the design of stable and efficient anodes of silicon/graphene hybrids for energy storage applications.     

Analysis of diffusion induced elastoplastic bending of bilayer lithium-ion battery electrodes

Bilayer electrode, composed of a current collector layer and an active material layer, has great potential in applications of in-situ electrochemical experiments due to the bending upon lithiation. This paper establishes an elastoplastic theory for the lithiation induced deformation of bilayer electrode with consideration of the plastic yield of current collector. It is found that the plastic yield of current collector reduces the restriction of current collector to an active layer, and therefore, enhances in-plane stretching while lowers down the rate of electrode bending. Key parameters that influence the elastoplastic deformation are identified. It is found that the smaller thickness ratio and lower elastic modulus ratio of current collector to an active layer would lead to an earlier plastic yield of the current collector, the larger in-plane strain, and the smaller bending curvature, due to balance between the current collector and the active layer. The smaller yield stress and plastic modulus of current collector have similar impacts on the electrode deformation.     

Size-dependent fracture of Si nanowire battery anodes

We use a unique transmission electron microscope (TEM) technique to show that Si nanowires (NWs) with diameters in the range of a few hundred nanometers can be fully lithiated and delithiated without fracture, in spite of the large volume changes that occur in this process. By analyzing the stresses associated with lithiation and delithiation we conclude that the process does not occur by the growth of discrete crystalline phases; rather it occurs by amorphization of the Si NWs followed by diffusion of Li into the structure. By accounting for the large deformation associated with this process and by including the effects of pressure gradients on the diffusion of Li, we show that Si NWs with diameters less than about 300 nm could not fracture even if pre-existing cracks were present in the NW. These predictions appear to be in good agreement with the experiment.     

Bauschinger and size effects in thin-film plasticity due to defect-energy of geometrical necessary dislocations

The Bauschinger and size effects in the thinfilm plasticity theory arising from the defect-energy of geometrically necessary dislocations (GNDs) are analytically investigated in this paper. Firstly, this defect-energy is deduced based on the elastic interactions of coupling dislocations (or pile-ups) moving on the closed neighboring slip plane. This energy is a quadratic function of the GNDs density, and includes an elastic interaction coefficient and an energetic length scale L. By incorporating it into the work- conjugate strain gradient plasticity theory of Gurtin, an energetic stress associated with this defect energy is obtained, which just plays the role of back stress in the kinematic hardening model. Then this back-stress hardening model is used to investigate the Bauschinger and size effects in the tension problem of single crystal Al films with passivation layers. The tension stress in the film shows a reverse dependence on the film thickness h. By comparing it with discrete-dislocation simulation results, the length scale L is determined, which is just several slip plane spacing, and accords well with our physical interpretation for the defect- energy. The Bauschinger effect after unloading is analyzed by combining this back-stress hardening model with a friction model. The effects of film thickness and pre-strain on the reversed plastic strain after unloading are quantified and qualitatively compared with experiment results.     

Nonlinearity analysis of piezoelectric micromachined ultrasonic transducers based on couple stress theory

This paper studies the static deformation behavior of a piezoelectric micromachined ultrasonic transducer (PMUT) actuated by a strong external electric field. The transducer membrane consists of a piezoelectric layer, a passive layer and two electrode layers. The nonlinearities of the piezoelectric layer caused by electrostriction under a strong electric field are analyzed. Because the thickness of the transducer membrane is on the microscale, the size dependence of the deformation behavior is evaluated using the couple stress theory. The results show that the optimal ratio of the top electrode diameter and the membrane diameter is around 0.674. It is also found that this optimal value does not depend on any other parameters if the thicknesses of the two electrodes are negligible compared with those of the piezoelectric and passive layers. In addition, the nonlinearities of the piezoelectric layer will become stronger along with the increase of the electric field, which means that softening of the membrane stiffness occurs when a strong external electric field is applied. Meanwhile, the optimal thickness ratio for the passive layer and the piezoelectric layer is not equal to 1.0 which is usually adopted by previous researchers. Because there exists size dependence of membrane deformation, the optimal value of this thickness ratio needs to be greater than 1.0 on the microscale.     

Multiaxial yield surface of transversely isotropic foams: Part II—Experimental

A robust understanding and modeling of the yield behavior in solid foams under complex stress states is essential to design and analysis of optimal structures using these lightweight materials. In pursuit of this objective a new custom-built Multi-Axial Testing Apparatus (MATA) is developed to probe the yield surface of transversely isotropic Divinycell H-100 PVC foam under a multitude of uniaxial, biaxial and triaxial strain paths. Experimental yield data produced constitutes the most comprehensive data set ever produced for any foam as it covers the entire spectrum of stress paths from hydrostatic compression to hydrostatic tension. Experimental results reveal that yielding in foams exhibits not only a quadratic pressure dependence, which is widely recognized in literature, but also a significant linear pressure dependence, which has been largely overlooked in previous studies. A new energy-based yield criterion developed for transversely isotropic foams is also validated using the experimental yield data.     

Real-Time Stress Measurement in SiO<Subscript>2</Subscript> Thin Films during Electrochemical Lithiation/Delithiation Cycling

Oxide coatings have been shown to improve the cyclic performance of high-energy density electrode materials such as Si. However, no study exists on the mechanical characterization of these oxide coatings. Here, thin film SiO₂ electrodes are cycled under galvanostatic conditions (at C/9 rate) in a half-cell configuration with lithium metal foil as counter/reference electrode, with 1 M LiPF₆ in ethylene carbonate, diethyl carbonate, dimethyl carbonate solution (1:1:1, wt%) as electrolyte. Stress evolution in the SiO₂ thin film electrodes during electrochemical lithiation/delithiation is measured in situ by monitoring the substrate curvature using a multi-beam optical sensing method. Upon lithiation SiO₂ undergoes extensive inelastic deformation, with a peak compressive stress of 3.1 GPa, and upon delithiation the stress became tensile with a peak stress of 0.7 GPa. A simple plane strain finite element model of Si nanotube coated with SiO₂ shell was developed to understand the mechanical response of the core-shell type microstructures under electrochemical cycling; measured stress response was used in the model to represent SiO₂ constitutive behavior while Si was treated as an elastic-plastic material with concentration dependent mechanical properties obtained from the literature. The results reported here provide insights and quantitative understanding as to why the highly brittle SiO₂ coatings are able to sustain significant volume expansion (300%) of Si core without fracture and enhance cyclic performance of Si reported in the literature. Also, the basic mechanical properties presented here are necessary first step for future design and development of durable Si/SiO₂ core shell structures or SiO₂-based electrodes.     

Assessment of composite failure and ultimate strength without experiment on composite

This paper attempts to estimate the ultimate strength of a laminated composite only based on its con- stituent properties measured independently. Three important issues involved have been systematically addressed, i.e., stress calculation for the constituent fiber and matrix materials, failure detection for the lamina and laminate upon the internal stresses in their constituents, and input data determination of the constituents from monolithic measurements. There are three important factors to influence the accuracy of the strength prediction. One is the stress concentration factor （SCF） in the matrix. Another is matrix plasticity. The third is thermal residual stresses in the constituents. It is these three factors, however, that have not been sufficiently well realized in the composite community. One can easily find out the elastic and strength parameters of a great many laminae and laminates in the current literature. Unfortunately, necessary information to determine the SCF, the matrix plasticity, and the thermal residual stresses of the composites is rare or incomplete. A useful design methodology is demonstrated in the paper.     

Ductile versus brittle behavior of amorphous metals

Ductile behavior of amorphous metals, their ability to sustain localized flow at high nominal stresses, is attributed to a mechanism which alleviates the severe stress conditions prevailing near potential cleavage flaws. The model problem of the plane strain deformation of an infinite block of non-linear visco-elastic material containing an elliptical hole is studied numerically and analytically. A strong dependence of the viscosity on the hydrostatic tension, a result of the increase in the number of viscous flow defects with dilatation, is the principal source of non-linearity. The analysis reveals that, under a constant remote strainrate, the initial elastic stress distribution ahead of the hole gives way, with time, to a more uniform stress distribution. Altering the stress distribution permits the remote (nominal) stress to achieve higher values before critical stress conditions are reached locally at the concentrator. At ordinary temperatures, amorphous metals are not in thermodynamic equilibrium; this motivates a modification of the constitutive law that reflects the kinetic difficulty of maintaining thermodynamic equilibrium under conditions of varying hydrostatic tension. Re-solving the elliptical hole problem with the modified constitutive law reveals a delay in the stress redistribution in front of the concentrator which may explain brittle fracture.     

Plane stress state problems for notched bodies whose plastic properties depend on the form of the stress state

We consider one possible approach to the problem of describing the dependence of material plastic strain characteristics on the stress hydrostatic component arising in many porous, fractured, and other inhomogeneous materials. The plastic strain of the media under study is investigated under the plasticity assumption in the corresponding generalized form with the use of the form parameter of the stress state. The plasticity constitutive relations are stated on the basis of the plastic flow law associated with the accepted plasticity condition. For the conditions of plane stress state in the framework of the material rigid-plastic model, a system of partial differential equations is obtained and conditions for its hyperbolicity are determined. The relations for determining the stress fields and velocity fields in plastic domains are obtained, and their properties are investigated. The problem of tension of a strip with symmetric angular notches is solved, where the stress fields are determined and the continuous displacement rate field is constructed. The problem of uniform symmetric tension of a plane with a circular hole is considered. The stress fields in a strip with symmetric circular notches are examined. A comparison with solutions for plastically incompressible media whose properties are invariant with respect to the form of the stress state is performed.     

Generalizing J 2 flow theory: Fundamental issues in strain gradient plasticity

It has not been a simple matter to obtain a sound extension of the classical J₂ flow theory of plasticity that in- corporates a dependence on plastic strain gradients and that is capable of capturing size-dependent behaviour of metals at the micron scale. Two classes of basic extensions of clas- sical J₂ theory have been proposed: one with increments in higher order stresses related to increments of strain gradi- ents and the other characterized by the higher order stresses themselves expressed in terms of increments of strain gra- dients. The theories proposed by Muhlhaus and Aifantis in 1991 and Fleck and Hutchinson in 2001 are in the first class, and, as formulated, these do not always satisfy ther- modynamic requirements on plastic dissipation. On the other hand, theories of the second class proposed by Gudmundson in 2004 and Gurtin and Anand in 2009 have the physical deficiency that the higher order stress quantities can change discontinuously for bodies subject to arbitrarily small load changes. The present paper lays out this background to the quest for a sound phenomenological extension of the rate- independent J₂ flow theory of plasticity to include a de- pendence on gradients of plastic strain. A modification of the Fleck-Hutchinson formulation that ensures its thermo- dynamic integrity is presented and contrasted with a compa- rable formulation of the second class where in the higher or- der stresses are expressed in terms of the plastic strain rate. Both versions are constructed to reduce to the classical J₂ flow theory of plasticity when the gradients can be neglected and to coincide with the simpler and more readily formulated J₂ deformation theory of gradient plasticity for deformation histories characterized by proportional straining.     

A plasticity model for cellular materials with open-celled structure

Based on a rigid-plastic material model that obeys the von Mises yield criterion, the plastic behavior of foams with an open-celled structure is studied in this paper using a single unit cell. An approximate continuum plasticity model is developed within the framework of the upper bound theorem of plasticity to describe the yield behavior of foams. The microscopic velocity fields are derived for the unit cell, which satisfy the incompressibility and the kinematic boundary conditions, and expressed in macroscopic rate of deformation. From the microscopic velocity fields, a macroscopic yield function is developed for foams under multi-axial stresses and includes the effects of the hydrostatic stress due to the void presence and growth. The dependency of the derived yield surfaces of foams on their relative densities is studied. The plastic behavior of foams is also studied numerically using the finite element method. The newly developed plasticity model is compared with the finite element analysis results and other available foam models and then correlated with the finite element results.     

单晶硅滑动磨损性能及其相变研究

考察了单晶硅在室温和低接触应力条件下的摩擦磨损行为随速度的变化情况．结果表明，单晶硅的摩擦系数和磨损率随滑动速度的提高而呈现降低趋势；单品硅在低速、短滑动时间下的磨损表面形貌特征以微断裂为主，并伴有一定程度的塑性变形;随着滑动时间的延长，塑性变形特征逐渐减弱，塑性变形同具有金属延性特征的p—Sn(简称si—II)相密切相关，Si—II在滑动过程中可转变为体心立方结构(简称Si-III)、斜方六面体结构(简称Si—XII)和非晶硅相;在高速条件下，单晶硅磨损表面呈现微断裂和较弱的塑性变形特征；尽管通过Raman分析证实磨损表面存在si—III相，但其对磨损机制的影响有待于进一步研究．     

Three-Dimensional Study of Graphite-Composite Electrode Chemo-Mechanical Response using Digital Volume Correlation

A custom built reusable cell for in situ lithiation and mechanical deformation studies while in an X-ray tomograph was demonstrated, and the strain and volume changes of a composite graphite anode were computed from 3D X-ray microcomputed tomography data via Digital Volume Correlation (DVC). The test anode was a composite electrode comprised of a porous compliant matrix, graphite as the Li⁺ host material, 5-μm ZrO₂ marker particles for use with DVC, and active carbon black to enhance electrical conductivity. The composite electrodes were hot-pressed to control their porosity, and in turn the mechanical integrity of the resulting material. This composite anode was included in a half-cell and lithiated in situ while in a tomograph, and intermittent 3D data were collected at different lithiation levels up to full gravimetric capacity. Strain measurements by DVC demonstrated relatively uniform expansion of the freestanding electrode with average normal strains in the three directions varying by 20%, while the internal shear strains were found to be negligible. The average experimental strains were about 75% of the theoretical value, as estimated by the rule of mixtures, which implies that ~25% of the normal strains in graphite, due to lithiation, are accommodated by the surrounding matrix.     

In Situ Measurements of Strains in Composite Battery Electrodes during Electrochemical Cycling

The cyclic stress in lithium-ion battery electrodes induced by repeated charge and discharge cycles causes electrode degradation and fracture, resulting in reduced battery performance and lifetime. To investigate electrode mechanics as a function of electrochemical cycling, we utilize digital image correlation (DIC) to measure the strains that develop in lithium-ion battery electrodes during lithiation and delithiation processes. A composite graphite electrode is cycled galvanostatically (with constant current) in a custom battery cell while optical images of the electrode surface are captured in situ. The strain in the electrode is computed using an in-house DIC code. On average, an unconstrained composite graphite electrode expands 1.41 % during lithiation and contracts 1.33 % during delithiation. These strain values compare favorably with predictions based on the elastic properties of the composite electrode and the expansion of graphite-lithium intercalation compounds (G-LICs). The establishment of this experimental protocol will enable future studies of the relationship between electrode mechanics and battery performance.     

柱状电极屈曲或裂纹扩展的临界力学参数研究

柱形结构电极是近年来使用最为广泛的锂电池电极结构之一．本文以硅材料细长柱形电极为例，研究了充电电流大小、电极长径比、初始裂纹长度以及断裂韧性对于电极的屈曲现象和裂纹扩展现象发生时间的影响．计算结果表明，屈曲与裂纹扩展现象出现的先后顺序与充电电流大小无关；具有小的长径比，大的初始裂纹长度以及较小断裂韧性的电极，裂纹扩展比屈曲现象更早发生．对于硅材料，不同长径比的电极具有不同临界断裂韧性值，当材料的断裂韧性小于该临界值，在锂化过程中裂纹扩展会先于屈曲现象发生；该临界断裂韧性值随初始裂纹长度的增加而增加．本文的结论对于电极的结构设计以及材料选取具有一定指导意义．