Kejie Zhao, Matt Pharr, Joost J. Vlassak, and Zhigang School of Engineering and Applied Sciences and Kavli Institute, Harvard University, Cambridge,Massachusetts 02138, USA ͑Received 24 August 2010; accepted 13 November 2010; published online 10 January 2011͒ Silicon can host a large amount of lithium, making it a promising electrode for high-capacitylithium-ion batteries. Upon absorbing lithium, silicon swells several times its volume; thedeformation often induces large stresses and pulverizes silicon. Recent experiments, however,indicate that under certain conditions lithiation causes inelastic deformation. This paper models suchan inelastic host of lithium by considering diffusion, elastic-plastic deformation, and fracture. Themodel shows that fracture is averted for a small and soft host—an inelastic host of a small featuresize and low yield strength. 2011 American Institute of Physics. ͓ Lithium-ion batteries are the batteries of choice for di- in commercial lithium-ion batteries for both cathodes ͑e.g., verse applications, especially for those sensitive to size and 2 and anodes ͑e.g., graphite͒. By contrast, an inelastic weight, such as portable electronics and electric In- host does not fully recover its structure after cycles ͓Fig.
tense efforts are being made to develop lithium-ion batteries . For example, when an electrode is an amorphous solid, of high capacity, fast charging, and long When the such as amorphous silicon, the host atoms may change battery is charged and discharged, the electrode absorbs or neighbors after a cycle of charge and discharge.
desorbs lithium, inducing cyclic deformation and possibly Whether lithiation-induced strain will cause an electrode fracture—a mechanism known to cause the capacity of the to fracture depends on the feature size of the The energy release rate G for a crack in a body takes the Lithiation-induced deformation and fracture is a bottle- form G = Z␴2h / E, where h is the feature size, E Young’s neck in developing lithium-ion batteries of high capacity. For modulus, ␴ a representative stress and Z a dimensionless example, of all known materials for anodes, silicon offers the number of order unity.Fracture is averted if G is below the highest theoretical specific capacityStill, silicon is not used fracture energy of the material, ⌫. Consequently, fracture is in anodes in commercial lithium-ion batteries, mainly be- averted if the feature size is below the critical value cause after a small number of cycles the capacity of siliconfades, often attributed to lithiation-induced fracture.
Recent experiments, however, have shown that the ca- pacity can be maintained over many cycles for silicon anodesof small feature sizes, such as nanowires,thin films,and Representative values for silicon are ⌫ = 10 J / m2 and E porous For such anodes lithiation-induced strain can be accommodated by inelastic deformation. For instance,cyclic lithiation causes silicon thin films and silicon nano- wires to develop Furthermore, the stress in a silicon thin film bonded on a wafer has been measured show-ing that the film deforms plastically upon reaching a yield Existing models of lithiation-induced deformation and fracture have assumed that the electrodes are elastic.Here, we model inelastic electrodes by considering diffusion, elastic-plastic deformation, and fracture. The model showsthat fracture is averted for a small and soft host of lithium—an inelastic host of a small feature size and low We classify hosts of lithium into two types: elastic and inelastic. For an elastic host, the host atoms recover their configurations after cycles of charge and discharge ͓Fig.
. For example, for an electrode of a layered structure, lithium diffuses in the plane between the layers, leaving thestrong bonds within each layer intact. Elastic hosts are used FIG. 1. ͑Color online͒ ͑a͒ For an elastic host of lithium, the host atomsrecover their configurations after a cycle of lithiation. ͑b͒ For an inelastichost of lithium, the host atoms may change neighbors after a cycle of lithia- a͒Electronic mail:
tion. Squares represent host atoms and circles represent lithium atoms.
109, 016110-1
Author complimentary copy. Redistribution subject to AIP license or copyright, see
J. Appl. Phys. 109, 016110 ͑2011͒
If silicon were an elastic host, the lithiation-induced strain ␧ = 100% would cause stress on the order ␴ ϳ E␧.
Equation would predict a subatomic critical This prediction disagrees with the experimental observations—silicon anodes of feature sizes around 100 nm do survivemany cycles of charge and discharge without By contrast, for an inelastic host, lithiation-induced strain can be accommodated by inelastic deformation, For a thin film of silicon bonded on a thick substrate, the measured yield strength is ␴Y = 1.75 For a channel crack in thefilm, Z = 2 is a typical Using these values, Eq. predicts a critical thickness of 130 nm. This prediction agreeswell with available experimental observations: a 250 nm sili- FIG. 2. Evolution of stress in a thin film of an inelastic host during cycliclithiation and delithiation.
con thin film fractured after a few cycles,while a 50 nmsilicon film survived without fractures after 1000 cycles.Ingeneral, for an inelastic electrode of a large capacity, fracture discharge. We consider the limit h Ӷ ͱD␶, where the film is is averted if the feature size is small and the yield strength is so thin that the concentration of lithium is homogeneous low. One extreme is a liquid electrode, which accommodates throughout the thickness of the film. The stresses in the film the absorption-induced strain by flow, and can potentially are given by ␴xx= ␴yy = ␴, ␴zz= 0, where x and y represent the in-plane directions, and z represents the out-of-plane direc- During charge and discharge, the stress in an electrode is tion. The in-plane deformation of the thin film is constrained a time-dependent field. Furthermore, the magnitude of the by the substrate, namely, dxx= dyy = 0. In the elastic stage, stress may exceed the yield strength at places under triaxial constraint. We describe an inelastic host of lithium by adapt- ing an elastic and perfectly plastic model. The increment of the strain is taken to be the sum of three contributions Figure plots the stress evolution as a function of lithium concentration c. Initially, the film deforms elastically and develops a compressive stress, with the slope given by where ␧e is the elastic strain, ␧p the plastic strain, and ␧l the Eq. When the magnitude of compressive stress reaches lithiation-induced strain. Hooke’s law gives the yield strength ␴Y, the film deforms plastically. Upon delithiation, the film unloads elastically, develops a tensile stress, and then deforms plastically in tension. The fully lithi- ated state causes a volume expansion about so that ␤=3. For v=Equation gives the slope d␴/dc= ij = 1 when i = j and ␦ij = 0 other- −103 GPa, which may be compared to the measured value In the thin film, the stress can be induced by the con- straint imposed by the substrate. By contrast, a particle, a nanowire, or a porous structure is almost unconstrained by ␴e = ␴Y, de = dY other materials, and the stress is mainly induced by the in- ij = ␴ij − ␴kk ij 3 homogeneous distribution of lithium.Consequently, the ijsij 2 the equivalent stress. Within the perfectly plastic stress is small when the feature size and charge rate are model, ␭ at each increment is a positive scalar to be deter- mined by the boundary-value problem. The lithiation- To explore the effect of inelastic deformation, we study induced strain is proportional to the concentration of lithium the evolution of the stress field in a spherical particle of silicon. The particle is initiated as pure silicon, and is charged and discharged at a constant current. The dimension- less charge and discharge rate is set to be ina / DCmax where ␤ is a constant analogous to the coefficient of thermal = 0.206, where a is the radius of the particle, in is the current expansion and c denotes the normalized lithium concentra- density for charge and discharge, and Cmax is the theoretical capacity of fully lithiated silicon. This dimensionless rate The concentration of lithium in an electrode is a time- corresponds to in= 0.176 A / m2 for representative values a dependent field, taken to be governed by the diffusion equa- tion, ץc / ץt = Dٌ2c. For simplicity, here, we assume that the diffusivity D is a constant and that diffusion is driven by the Figure and show the distribution radial and hoop stresses during charge. As more lithium is inserted, the As an illustration of the model, consider a thin film of particle expands more near the surface than near the center, amorphous silicon bonded on a substrate. Let h be the thick- resulting in tensile radial stresses. The hoop stress is com- ness of the film, and ␶ the time used to complete charge or pressive near the surface and tensile near the center. For the Author complimentary copy. Redistribution subject to AIP license or copyright, see
J. Appl. Phys. 109, 016110 ͑2011͒
FIG. 3. ͑Color online͒ The evolution of ͑a͒ radial stress and ͑b͒ hoop stress during charge. The evolution of ͑c͒ radial stress and ͑d͒ hoop stress duringdischarge.
spherical particle, the yield condition takes the form ͉␴␪␪ gins, and Y. Cui, 3, 31 ͑2008͒.
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knowledges the support of the Department of Defense 19J. W. Hutchinson and Z. Suo, 29, 63 ͑1991͒.
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2Department of Energy, Basic Research Needs for Electrical Energy Stor- 23D. C. Holzman, 206, 19 ͑2010͒.
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