Vulnerable Window of Yield Strength forSwelling-Driven Fracture of Phase-Transforming

Battery Materials

Atoallah Mesgarnejad1 and Alain Karma∗1

1Center for Inter-disciplinary Research on Complex Systems,Department of Physics, Northeastern University, Boston, MA.

02115, U.S.A.

August 19, 2019

Abstract

Despite numerous experimental and theoretical investigations of the me-chanical behavior of high-capacity Si and Ge Li-ion battery anodes, our basicunderstanding of swelling-driven fracture in these materials remains limited.Existing theoretical studies have provided insights into elasto-plastic defor-mations caused by large volume change phase transformations, but havenot modeled fracture explicitly beyond Griffith’s criterion. Here we usea multi-physics phase-field approach to model self-consistently anisotropicphase transformation, elasto-plastic deformation, and crack initiation andpropagation during lithiation of Si nanopillars. Our computational resultsreveal that fracture occurs within a “vulnerable window” inside the two-dimensional parameter space of yield strength and fracture energy and high-light the importance of taking into account the surface localization of plasticdeformation to accurately predict the magnitude of tensile stresses at theonset of fracture. They further demonstrate how the increased robustnessof hollow nanopillars can be understood as a direct effect of anode geometryon the size of this vulnerable window. Those insights provide an improvedtheoretical basis for designing next-generation mechanically stable phase-transforming battery materials undergoing large volume changes.

Introduction

Increasing demand for portable energy storage has motivated a large researchactivity focused on high-capacity Li-ion battery anodes. Current carbon-based

∗a.karma@northeastern.edu

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anodes have limited theoretical capacity (372 mA h g−1 for Li6C [1]). Silicon andgermanium have an order of magnitude larger theoretical capacity (3579 mA h g−1

for Li15Si4, 4200 mA h g−1 for Li22Si5 [1], 1384 mA h g−1 for Li15Si4 [2]) but areprone to fracture due to the high, approximately 300%, volume expansion duringlithiation [3, 4], which limits their use. Different designs have been explored toovercome this limitation including silicon nanopillars [5, 6], thin films [7, 8, 9], opennano-porous crystalline Si structures with ultra-high interfacial area produced bydealloying of Si-based alloys [10, 11], combinations of these [12, 13], as well ascomposite designs that embed silicon particles inside a more mechanically stablematrix [14, 15, 16, 17, 18].

Basic studies of the lithiation process have shown that crystalline silicon (c-Si) transforms to an amorphous lithiated alloy (a-LixSi) [19, 20, 2]. The kinet-ics of this large volume change phase transformation is understood to be bothinterface-reaction limited [20] and highly anisotropic, reflecting the two key ob-servations that the velocity of the c-Si/a-LixSi interface remains approximatelyconstant during lithiation, and that this velocity depends strongly on crystallo-graphic orientation [19]. Numerical studies of elasto-plastic deformations of c-Siparticles (nanowires, nano-/micro-pillars, etc) have demonstrated that the result-ing anisotropic swelling can produce both large shape changes of the particles and,as a non-trivial effect of compressive yielding, tensile stresses on their outer sur-face that can potentially drive fracture [3, 21, 22, 23]. Those insights have alreadyproven useful to test new anode designs to mitigate fracture [23]. However, ourability to predict when and how fracture occurs in different large volume changematerials (e.g. Si versus Ge) and different anode geometries (e.g. solid versushollow nanopillars [13]) is still limited. To date, the onset of fracture has beenpredicted using analytical solutions for stresses obtained in idealized geometries,assuming purely plastic deformation and isotropic swelling, and by applying aGriffith criterion to predict fracture onset for a flaw size comparable to the par-ticle dimension [24, 25]. However, crack initiation and propagation in the settingof large elasto-plastic deformation and anisotropic swelling in phase-transformingmaterials remain largely unexplored.

Here, we use a multi-physics phase-field approach to simulate both anisotropicswelling and fracture of solid and hollow c-Si nanopillars within a unified theoreti-cal framework and derive from our simulations an understanding of when and howfracture occurs as a function of key materials parameters, including yield strengthand fracture energy, and geometric parameters such as nanopillar radius and slen-derness. Phase transformation is modeled using a phase-field ψ that distinguishesthe c-Si and a-LixSi phases and is evolved dynamically to describe the interface-reaction-limited anisotropic motion of the c-Si/a-LixSi interface. Fracture, in turn,is modeled using the well-established variational approach that couples elasticityto a phase-field φ, which distinguishes pristine and broken regions of the mate-rial [26, 27]. To realistically model large volume changes, this variational approachis implemented using a large deformation formulation of elasto-plasticity combin-ing neo-Hookean nonlinear elasticity and J2 plasticity to quasi-statically evolve φtogether with the material displacement field and the plastic deformation gradi-

2

ent tensor. The phase-field approach offers several advantages in the present con-text. It provides a self-consistent formulation to model simultaneously anisotropicswelling, large elasto-plastic deformation, and fracture. Furthermore, it can de-scribe the evolution of phase boundaries and cracks of arbitrarily complex shapes,as demonstrated in applications to other phase transformations [28] and fractureproblems such as thermal shock fracture [29], mixed mode fracture [30], ductilefracture [31, 32], and the simpler chemo-mechanical fracture of single-phase bat-tery cathode particles [33, 34, 35, 36], also driven by volume expansion due toLi intercalation but only involving small elastic stresses and no phase change. Inaddition, in contrast to Griffith theory, the phase-field approach is able to describecrack initiation without pre-existing flaws. This property stems from the fact thatφ varies smoothly in space on a length scale ξ, thereby enabling crack formationon the scale of the “process zone” where elastic energy is transformed into newfracture surfaces. Hence, directly relevant to the present study, the phase-fieldapproach can quantitatively describe crack initiation from surface imperfectionssuch as U- or V-shape notches by treating ξ treated as a material-dependent pa-rameter [37]. V-shape notches, in particular, bear close similarity to surface shapedeformations of lithiated Si particles undergoing elasto-plastic deformation duringanisotropic swelling [3, 21, 22, 23].

To keep computations tractable, we perform 2D plane-strain simulations (∂z ≡0) on a cross-section of an unconstrained nanopillar (τzz = 0) lithiated from itssurface (i.e. outer boundary for a solid nanopillar and both outer and inner bound-aries for hollow nanopillars). Furthermore, to dissect the contributions of multipleinteracting physical effects (including compressive and tensile yielding, anisotropicswelling, localization of plastic deformation, and crack initiation and propagation),we carry out three different types of computations of increasing complexity. In afirst step, we model stress evolution without fracture by assuming that swelling isisotropic and that the stress fields and plastic hardening parameter α only varyradially and are independent of the azimuthal angle θ as depicted schematically inFigure 1a. This axisymmetric approximation reduces the 2D problem to a 1D ra-dial problem. Stress evolution in a similar idealized geometry has been previouslystudied analytically by taking into account only plastic deformation [24, 25]. Bytaking into account here both elastic and plastic deformations, we demonstratethat tensile stresses generated on the particle surface by volume expansion reacha maximum value as a function of yield strength σy. Even though the criticalσy value corresponding to this maximum is outside the experimentally [8, 38, 39]or theoretically estimated [40] range σy ∼ 0.5–2 GPa for Si, the existence of thiscritical yield strength provides a valuable theoretical framework to understandfracture behavior inside this lower estimated range of 0.5–2 GPa. For this rea-son, we investigate stress evolution over a wide range of σy that encompasses theentire vulnerable window for fracture. In a second step, we carry out a simi-lar computation, still without fracture, but for the full 2D problem without theaxisymmetric approximation in which the stress fields and α can vary both radi-ally and azimuthally inside the pillar cross section. This enables us to asses howanisotropic swelling modifies tensile stresses on the pillar surface. We find that

3

tensile stresses become amplified by localization of plastic deformation, but stillexhibit a maximum as a function of increasing σy. Those 1D and 2D computationsdemonstrate the existence of a vulnerable window of yield strength inside whichpillars are prone to fracture. More crucially, for the relevant experimentally re-ported yield strengths (σy = 0.5–2 GPa) of Si, the difference between the materialand the most critical yield strength controls the magnification of generated tensilestresses. In a last step, we validate the existence of this window by repeating our2D computations with fracture, showing that pillars fracture only over an inter-mediate range of yield strength. We compare the results of full 2D simulationswith estimates based on our numerically calculated stresses in the previous stepsusing the Griffith theory framework. We then use experimental estimates of safepillar radius (i.e. largest pillar radius without fracture) to quantitatively validateour findings by calculating our estimate of yield strength.

Results and discussion

Model

We model the swelling-driven deformation of the material using the finite J2 elasto-plasticity framework [41, 42, 43] and account for the fracture of the materialby coupling it to a phase-field fracture model [26, 27]. Furthermore, we modelthe anisotropic motion of the c-Si/a-LixSi interface during lithiation using a non-conserved phase field ψ where ψ = 0 in the crystalline phase and ψ = 1 in theamorphous phase. The material properties are approximated using a linear role ofmixture between the crystalline and amorphous phases. We define the deformationgradient tensor as Fij = ∂xi/∂Xj = 1 + ∂ui/∂Xj where Xi are the undeformedcoordinates and xi = Xi + ui are the deformed coordinates of material pointsand u is the displacement field. We use a multiplicative decomposition of thedeformation gradient tensor such that

Fij =√JψF

eikF

pkj (1)

where Jψ = (1 + βψ)2 is the phase dependent volumetric expansion due to phasechange with linear Vegard expansion coefficient β, F p is the plastic deformation,and F e is the elastic deformation. We use the framework of the phase-field methodfor fracture [27, 26] by introducing the fracture phase field φ along with the processzone size ξ. To forbid interpenetration of the fracture faces similar to [32], we writethe free energy with using an isochoric-volumetric split in undeformed coordinatesas:

F (F e, ψ, φ) =

ˆΩ0

((φ2 + ηξ)W

+(F e, ψ) +W−(F e, ψ))dx

+3Gc

8

ˆΩ0

(1− φξ

+ ξ|∇0φ|2)dx

+e0

ˆΩ0

(fdw(ψ) + w2|∇0ψ|2

)dx (2)

4

where ∇0• = (∂ • /∂Xi), Gc is the fracture energy and e0 is the energy cost phasechange for unit undeformed volume, and w is the characteristic phase changethickness. Furthermore, we assume a neo-Hookean hyper-elastic material

W+(F e, ψ)

µ(ψ)

2

(bekk − 2

)+κ(ψ)

4

(Je2 − 2 ln(Je)− 1

)if Je ≥ 1

µ(ψ)

2

(bekk − 2

)otherwise

(3)

W−(F e, ψ) =

0 if Je ≥ 1κ(ψ)

4

(Je2 − 2 ln(Je)− 1

)otherwise

(4)

where for Je = det(F e), we define isochoric left Cauchy-Green deformation tensorbeij = F e

ikFejk/J

e. In this model, the shear modulus µ(ψ) = ψµa + (1 − ψ)µc andbulk modulus κ(ψ) = ψκa + (1 − ψ)κc are extrapolated between the shear andbulk moduli of amorphous (µa, κa) and crystalline phase (µc, κc). Following classicJ2 plasticity we assume that the von Mises equivalent stress τeq =

√3sijsij/2 ≤

(σy +Kα) where sij = µ(ψ)(beij − bekkδij/N) is the deviatoric part of the Kirchhoffstress tensor τ (τ = Jσ where σ is the Cauchy stress tensor), σy is the yieldstrength and α is the isotropic hardening parameter (at the infinitesimal strainlimit the isotropic hardening parameter reduces to the equivalent plastic strain).We write the governing equations for the displacement and fracture phase field asthe minimizers of (2):

δF

δui= 0 s.t.

√3sijsij/2 ≤ (σy +Kα) (5)

δF

δφ= 0 (6)

where δF/δ• is the Frechet derivative of the free energy F with respect to field•. Moreover, we model the curvature-independent interface-reaction-controlledanisotropic motion of the c-Si/a-Si interface as [44]:

∂ψ

∂t= −M

( ∇0ψ

|∇0ψ|

)(1

e0

δF

δψ− w2|∇0ψ|∇0 ·

( ∇0ψ

|∇0ψ|

))(7)

where M(∇0ψ/|∇0ψ|) is the anisotropic mobility of phase change where we usethe same form of the anisotropic mobility M(∇0ψ/|∇0ψ|) as An et al. [23].

Vulnerable window of yield strength

To understand the basic mechanism of stress generation in these components, asthe first step, we performed an exhaustive series of axisymmetric simulations with-out fracture (i.e simulations in which stress fields and the plastic flow hardeningparameter are assumed to only vary radially). Results of the axisymmetric com-putations are shown in Figure 1. Figure 1(c-d) shows the evolution of the hoop

5

a b

c d

a-Si

c-Si

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Figure 1: Results of axisymmetric simulations of lithiation of hollow cylindricalcrystalline nanopillars of outer radius R = 85 nm and variable thickness t. (a)Schematic representation of the hollow pillar geometry. (b) Plots of maximumhoop stress reached during complete lithiation on the nanopillar outer (r = R,solid lines) and inner (r = R− t, dashed lines) boundaries versus yield strength σyfor different t/R ratios. Plots predict the existence of a “vulnerable window” of σyinside which the maximum hoop stress can exceed the threshold for fracture. (c)Radial profiles at different times of the phase transformation ψ field (right verticalaxis, gray lines with the c-Si/a-LixSi interface located at ψ = 0.5), the Kirchhoffhoop stress

√3τθθ/2 (left vertical axis, red lines with earlier stages shown using

lighter red) and Von Mises stress τeq (left vertical axis, blue lines) for σy = 3 GPa.Plots show compressive yielding followed by subsequent reversal of the sign ofthe hoop stress and yielding under tension. (d) Same as (c) but for higher yieldstrength σy = 7 GPa where tensile yielding does not occur.

stresses and equivalent von Mises stress during lithiation of a solid nanopillarwhere we can identify three regimes in these figures. As the phase transforma-tion boundary (ψ = 0.5) invades inside the particle from the outer boundary(r = R), it creates compressive stresses due to the large volumetric expansionof the a-LixSi phase. The resulting compressive stresses generate plastic flowthat caps the von Mises stress at σy. As the crystalline core shrinks further, thecompressive stresses on the outer boundary subside and change sign due to theinitial compressive yield. Consequently, the hoop stress on the outer boundary

6

changes sign and becomes tensile (Figure 1(c-d)), thereby confirming the knock-on effect of compressive yielding on the creation of tensile stresses that has beenhypothesized to cause cracking [3, 21, 22, 23]. Importantly, for σy smaller thanapproximately 4 GPa for the present parameters, the tensile hoop stress reachesthe yield strength before the c-Si core has vanished, which results in secondaryplastic yielding under tension (Figure 1c). In this range (σy ≤ 4 GPa), the maxi-mum hoop stress reached during complete lithiation, max(τθθ), increases linearlywith σy as shown in Figure 1b for t/R = 1 corresponding to a solid pillar; sincethe outer boundary is traction free (τrr ≡ 0) and the von Mises stress is cappedby σy, max(τθθ) = 2σy/

√3 on that boundary for plane-strain. In contrast, for

larger σy, compressive yielding requires a larger lithiated fraction, which reducesthe amount of volumetric expansion available to create tensile stresses duringshrinkage of the remaining c-Si core. Therefore, max(τθθ) remains below the yieldstrength and decreases with increasing σy as shown in Figure 1b. We should alsohighlight that although all simulations presented in this article were performedusing β = 0.7 that corresponds to ∼ 280% volume change at full lithiation, ourresults show that there exists a universal relationship between the dimensionlessmaximum hoop stress max(τθθ)/µaβ and the dimensionelss yield strength σy/µaβ(see Figure ??), such that our findings can be extended to other materials whosephase-transformation result in smaller volume changes. This universality can bereadily understood noticing that at smaller expansion coefficients, smaller stressesare generated; therefore, the knock-on effect of the compressive yielding only takesplace at smaller yield strength. Crucially, these results show that the vulnerablewindow of yield strength is shifted to smaller values of yield strength for smallerexpansion coefficient. The resulting hat shape of the max(τθθ) versus σy plot inFigure 1b suggests the existence of a vulnerable window for fracture correspond-ing to the range of σy where the maximum hoop stress becomes large enough toinitiate fracture. Specifically, existence of a maximum generated hoop stress at acritical yield strength increases the available energetic driving force for fracture atlower yield strength as confirmed below by our full 2D simulations (Figures 3–5).Figure 1b also shows plots of max(τθθ) versus σy on the inner (r = R − t) andouter (r = R) boundaries of hollow nanopillars. The maximum hoop stress on theouter boundary still exhibits a maximum. However, since the compliance of theannulus is inversely related to its slenderness t/R, the maximum stresses on bothboundaries decrease with increasing slenderness.

Next, we performed 2D simulations of anisotropic swelling without fracture forpillars oriented in two crystallographic directions [001] and [112]. In these simula-tions, the crystalline core is no longer circular and the anisotropic mobility of theamorphization front creates a crystalline silicon core with sharp corners. Duringlithiation, those crystalline corners concentrate stresses and localize plastic flowin their vicinity. When the stresses change sign and become tensile on the pillarouter boundary, shear localization produces V-shaped notches at orientations cor-responding to these corners for lower yield strength, which allows tensile yieldingto occur on the periphery subsequent to compressive yielding, but not larger yieldstrength where tensile yielding does not occur. This difference can be seen in the

7

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0 2 4 6 8 10σy [Gpa]

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θθ)[Gpa]

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Figure 2: Comparison of 2D and 1D axisymmetric simulations of lithiation ofsolid nanopillars of radius R = 85 nm without fracture. Plots of maximum hoopstress τθθ vs. yield stress σy for [001] (a) and [112] (b) oriented nanopillars (r isthe radial coordinate in the undeformed frame). In 2D simulations, localization ofplastic flow at orientations corresponding to sharp corners of the crystalline Si coreand concomitant creation of V-shaped notches (Figure 3) magnifies the magnitudeof stresses, thereby enlarging the size of the vulnerable window for fracture. Thelargest stresses are created along the crystalline corners at a small distance fromthe surface (red circles).

pillar morphologies in Figure 3 for σy = 1 GPa and σy = 10 GPa (σy = 7 GPafor [112] oriented pillar) that did not fracture. Those notches further concen-trate stresses, thereby augmenting the magnitude of hoop stresses several fold atthose orientations. This magnification is shown in Figure 2 where we comparemax(τθθ), defined as before as the maximum hoop stress reached in time duringcomplete lithiation, from 1D axisymmetric computations of isotropic swelling andthe present 2D computations of anisotropic swelling. For the latter case, we reportmax(τθθ) both on the outer surface (blue and yellow diamonds) and at a positioninside the particle close to the outer surface (red circles) where max(τθθ) reachesits maximum value along a vertical axis that contains the corners of the c-Si core.The maximum hoop stress is seen to be magnified both by localization of plasticdeformation during compressive yielding, which occurs for all σy reported, andV-shape notches that form for lower σy due to tensile yielding.

The 2D results in Figure 2 confirm the existence of a critical yield strengththat generates maximal tensile stresses but were obtained from simulations with-out fracture. To investigate the effect of stress augmentation on fracture, werepeated a series of simulations with fracture for different σy and the [001] and[112] crystallographic orientations and for fixed process zone size to radius ratioξ/R = 0.02. The time evolutions of particle morphologies in Figure 3 show thatnanopillar fracture during lithiation over an intermediate range of σy inside the vul-nerable window centered at the critical yield strength. For σy = 1 GPa, V-shapednotches are created along the crystalline corner directions but the magnitude of

8

surface tensile stresses are insufficient for crack initiation. For σy = 3 GPa, cracksinitiate due to stress concentration at V-shaped notches and propagate unstablytowards the crystalline core. For σy = 5 GPa, tensile yielding and hence V-shapednotches are absent but tensile stresses grow sufficiently large (at a later stage ofcharging compared to σy = 3 GPa) to create two pairs of cracks that propagateunstably after initiation. The horizontal pair subsequently arrests and the verti-cal pair propagates further due to spontaneous symmetry breaking. Finally, forσy = 10 GPa (σy = 7 GPa for [112] oriented pillar), reduced compressive plasticflow in the system suppresses the subsequent increase of tensile stresses, therebypreventing crack initiation. In [001] oriented pillars, crack initiate at the four cor-ners corresponding to [100] and [010] directions where stresses are largest. In [112]oriented pillars, crack initiate at the two corners along [111] directions.

9

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[001]<latexit sha1_base64="67LEu7Io2yipq636iljPxFG3l/o=">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</latexit>

[112]<latexit sha1_base64="5T4hd7zMUDKGrTkzcAW6sGekOXI=">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</latexit>

σy = 3Gpa<latexit sha1_base64="Z0S3Oa8nkzt7cHv8koTd54ReDJk=">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</latexit><latexit sha1_base64="Z0S3Oa8nkzt7cHv8koTd54ReDJk=">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</latexit><latexit sha1_base64="Z0S3Oa8nkzt7cHv8koTd54ReDJk=">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</latexit><latexit sha1_base64="Z0S3Oa8nkzt7cHv8koTd54ReDJk=">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</latexit>

σy = 5Gpa<latexit sha1_base64="ZVI2dISJA8+z7hm84lfGqbzR3gs=">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</latexit><latexit sha1_base64="ZVI2dISJA8+z7hm84lfGqbzR3gs=">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</latexit><latexit sha1_base64="ZVI2dISJA8+z7hm84lfGqbzR3gs=">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</latexit><latexit sha1_base64="ZVI2dISJA8+z7hm84lfGqbzR3gs=">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</latexit>

σy = 7Gpa<latexit sha1_base64="L6Mq4UoEKhj3UwQLBWTmnTp1O1U=">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</latexit><latexit sha1_base64="L6Mq4UoEKhj3UwQLBWTmnTp1O1U=">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</latexit><latexit sha1_base64="L6Mq4UoEKhj3UwQLBWTmnTp1O1U=">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</latexit><latexit sha1_base64="L6Mq4UoEKhj3UwQLBWTmnTp1O1U=">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</latexit>

σy = 10Gpa<latexit sha1_base64="yDjpcL1lX7Zei0kNFyWrgoG2/ro=">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</latexit><latexit sha1_base64="yDjpcL1lX7Zei0kNFyWrgoG2/ro=">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</latexit><latexit sha1_base64="yDjpcL1lX7Zei0kNFyWrgoG2/ro=">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</latexit><latexit sha1_base64="yDjpcL1lX7Zei0kNFyWrgoG2/ro=">AAADW3icdVJdaxNBFJ1m/aix2lTxQXxZDIKUsOyKYF8KxaTVF7FC0xa6IdydvUmGzscyM6sNw+Kv8VV/jw/+Fyfpqs0GLwycOefcmXsvNys4MzaOf260glu379zdvNe+v/Xg4XZn59GpUaWmOKSKK32egUHOJA4tsxzPC40gMo5n2WV/oZ99Rm2Ykid2XuBIwFSyCaNgPTXuPE0NmwoYz/eTOO2lAuxMC/eugGrc6cZRvIxwHSQ16JI6jsc7rd00V7QUKC3lYMxFEhd25EBbRjlW7bQ0WAC9hCleeChBoBm5ZQ9V+MIzeThR2h9pwyV7M8MtSzMTs/KOA2HsTKxxC/MaaeYia1RhJ3sjx2RRWpT0uohJyUOrwsWwwpxppJbPPQCqme8jpDPQQK0faTsdoO9T4wf/2ccCNVild10KeiqYrNyfYdb3tFf9L+NE/3N77H0Sv1AlBMjcpYNapMDdoCke3hAPm+JRLRqq3VFTLDOvZorni7Eo7sqm4aphuGoa+qv63zr6VdPJ0FtRFDPHIox6YbXYrqS5S+vg9FWUxFHy6XX34G29Z5vkGXlOXpKEvCEH5D05JkNCyVfyjXwnP1q/giBoB1vX1tZGnfOYrETw5DcAGR9H</latexit>

σy = 5Gpa<latexit sha1_base64="ZVI2dISJA8+z7hm84lfGqbzR3gs=">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</latexit><latexit sha1_base64="ZVI2dISJA8+z7hm84lfGqbzR3gs=">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</latexit><latexit sha1_base64="ZVI2dISJA8+z7hm84lfGqbzR3gs=">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</latexit><latexit sha1_base64="ZVI2dISJA8+z7hm84lfGqbzR3gs=">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</latexit>

σy = 3Gpa<latexit sha1_base64="Z0S3Oa8nkzt7cHv8koTd54ReDJk=">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</latexit><latexit sha1_base64="Z0S3Oa8nkzt7cHv8koTd54ReDJk=">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</latexit><latexit sha1_base64="Z0S3Oa8nkzt7cHv8koTd54ReDJk=">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</latexit><latexit sha1_base64="Z0S3Oa8nkzt7cHv8koTd54ReDJk=">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</latexit>

σy = 1Gpa<latexit sha1_base64="VfycIy4HJIq0osScvj7P/7e8vwA=">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</latexit>

σy = 1Gpa<latexit sha1_base64="VfycIy4HJIq0osScvj7P/7e8vwA=">AAAFE3icdZTPbtNAEMa3bYDiAk3hgsTFIlRCKLLicoALUkrSglQhSpW0leooWm8myape29pdl0Qr8xa8AVc4cuEGXHkAzjwDnFnnD3HWZSVL4/l9s5r5PLIfB1TIWu3nyupa6crVa+vXrY0bN29tlrduH4so4QTaJAoifupjAQENoS2pDOA05oCZH8CJf97I+MkFcEGjsCXHMXQYHoS0TwmWOtUt3/UEHTDcHT9zvarHsBxypl7EOO2WKzWnNjl2MXBnQaX+4Pfnbxcbfw67W2u/vF5EEgahJAEW4sytxbKjMJeUBJBa214iIMbkHA/gTIchZiA6ajJDam/rTM/uR1w/obQnWStXoTATYsx8rcy6FCbLkpeyyUyiL1LL6ED2n3YUDeNEQkimDfSTwJaRnRll9ygHIoOxDjDhVA9hkyHmmEhtp+WF8JZEjOGwp7xWqibWCcJVK02XISEz6vuqUYQ4mGEdFfnuonbXZAcLdmAyAA39KOhllkWBAlNAMwGweKioA07VLlwwmHNwBpdxSf4JJHFMvDefSm/TXgFGedhVNVMw5AuBjg3KEpbhZK5gpuAo5+hRwXFdTLp6K0YZye3JaLqIluU1Qe8vh1f6itcxcCwj/kh5mA8YDReNzd69avq/ilZujFZhjGauy2bBohws+Lefg/smTPzlL5+YgpEhGJmCxjLPL2dq6d+Ca/4EisHxjuM+dnbeuJX6czQ96+geuo8eIhc9QXX0Eh2iNiLoHfqAPqJPpfelL6Wvpe9T6erKrOYOWjqlH38BpJPV/g==</latexit>

Figure 3: Phase-field simulation of fracture during the lithiation of [001] (left)and [112] (right) oriented R = 85 nm nanopillars for different σy and for fixedprocess zone size to radius ratio ξ/R = 0.02. Color map depicts of the hardeningparameter α and the thick black line shows the amorphous-crystalline boundary(i.e., ψ = 0.5 contour line). The red dashed boxes show snapshots just beforecrack initiation and after unstable penetration towards the crystalline core. Theresults confirm the prediction of a window of σy for fracture and distinguishesmodes of fracture with (second row) and without (third row) localization of plasticdeformation creating a surface V-shaped notch prior to fracture (see also SI movies1a,b to 6a,b).

Size effects

Experimental observations of Si anodic components show a clear size dependencywhere, for example, nanospheres with radii smaller than ∼ 75 nm [5] and nanopil-lars with radii smaller than 120 nm [6] do not break during lithiation. Such sizedependencies, prevalent in brittle and quasi-brittle materials [45, 46, 47, 48], aretypically characterized by a power law relationship between the stress to fractureand component size, τc ∼ 1/

√R, for R much larger than the process zone size.

Within the theoretical framework of Linear Elastic Fracture Mechanics (LEFM),which treats crack surfaces as sharp boundaries, this power law is readily ob-tained from the expression for the energy release rate at the tip of a crack of

10

length a under a spatially homogeneous critical stress τc, which can be written asG = Caσ2(1− ν)/µ for plane-strain where C is a dimensionless constant that gen-erally depends on particle geometry and load configuration. Equating the energyrelease rate with the fracture energy (G = Gc), we obtain the expression

τc =

√Gcµ

Ca(1− ν)(8)

and thus the scaling τc ∼ 1/√R by further assuming that the maximum flaw size a

increases proportionally to the particle size (a ∼ R and a R). In the phase-fieldmodel used in this article, which describes the state of the material with a spatiallyvarying scalar field φ, crack nucleation is an inherent property of the model andoccurs via an instability that causes φ to develop a local dip (φ → 0) when thelocal stress exceeds a critical value τc ∼

√Gcµ/ξ [49] (up to a numerical prefactor

that also depends on particle geometry and load configuration). Consequently, bycomparing the above scaling expression for τc to Eq. (8), we can physically interpretξ as playing an analogous role to the dominant flaw size in the LEFM framework.Furthermore, by using the result of a stability analysis of a 1D stretched strip inthe phase-field model, which yields the prediction τc =

√3Gcµ/4ξ(1− ν) [49], we

obtain the constant C = 4/3 by comparison with Eq. 8 with a = ξ. This value isclose to the standard LEFM value C = π/2 for a crack of length 2a in a uniformstress. One main qualitative difference, however, is that crack nucleation in thephase-field model occurs through an instability of the pristine state in which φis spatially uniform and hence does not require the introduction of a flaw in theform of a finite length seed crack as in LEFM.

We take advantage of this property to investigate the particle size dependenceof fracture onset by performing simulations at fixed process zone size to parti-cle size ratio ξ = ξ/R. From the dimensional analysis, it is natural to definethe dimensionless fracture energy as Gc/(µaR) where µa is the shear modulus ofthe amorphous a-LixSi phase. The dimensionless fracture energy can be readilyinterpreted as the ratio of the Griffith length scale Gc/µa and the nanopillar ra-dius. Figure 4a reports the results of an extensive series of 2D phase-field fracturesimulations for [001] oriented nanopillars for ξ = 0.02, which identifies regions ofthe two-dimensional parameters space Gc/(µaR) and σy where fracture does (redcircles) or does not (blue crosses) occur. These results confirm the existence of avulnerable window of σy where cracks initiate and propagate in nanopillars dur-ing lithiation. This window is centered around the critical value of σy generatingmaximal tensile stresses and shrinks in size with increasing Gc/(µaR).

We now assess if the onset of fracture in 2D phase-field simulations can be pre-dicted within Griffith theory by using Eq. (8) together with values of the maximumhoop stress during lithiation obtained in 1D or 2D phase-field simulations withoutfracture (Figure 2). For this, we assume stress-free surfaces (τrr(R) = τrθ(R) = 0)and small elastic strains, which allows to use a quadratic approximation for thestored elastic energy W+ ' τ 2

θθ(1 − ν2a)/2Ea where Ea = 9κaµa/(3κa + µa),

νa = (3κa − 2µa)/(6κa + 2µa) are the elastic modulus and Poisson ratio of the

11

0 2 4 6 8σy [GPa]

0.00

0.01

0.02

0.03

0.04

0.05

0.06G

c/µaR

Broken

Not broken

Eq. (1) max(τθθ) from axisymmetric simulations

Eq. (1) max(τθθ) = 2σy/√3

Eq. (1) max(τθθ) from 2D simulations

Safe particle limit

0.5 1.0 1.5 2.0σy [Gpa]

10−2

10−1

ξ/R

Broken

Not broken

Eq. (1) max(τθθ) = 2σy/√3

Eq. (1) max(τθθ) from 2D simulations

a b

Figure 4: (a) Vulnerable window of fracture plotted for yield strength σy vs. di-mensionless fracture energy (particle size) Gc/(µaR) for [001] oriented nanopillarsfor fixed process zone size to radius ratio ξ/R = 0.02. Circles depict cases with, andcrosses show cases without fracture. The results confirm the existence of a vulner-able window of fracture energy (size) and yield strength for fracture. Lines showthe comparison with the closed-form approximation (Eq. (9)) based on max(τθθ)using full 2D simulation results (red ciricles in Figure 2a, black line), 1D axisym-metric results shown in Figure 1b (dashed gray line), and max(τθθ) = 2σy/

√3

(dashed orange line). The green shaded region shows the approximate value ofthe dimensionless fracture energy calculated for the safe particle size 120 nm [6]and fracture energies in the range 5–7 J m−2 [39]. (b) Phase diagram of fracture inthe plane of dimensionless process zone size 0.01 ≤ ξ/R ≤ 0.2 and yield strengthσy obtained from full 2D simulation using Gc/(µaR) = 0.01 (corresponding toR = 120 nm and fracture energy Gc = 6 J m−2). The results are consistent withthe experimentally estimated range σy = 0.5− 2 GPa for fracture.

a-Si. We found that this small strain quadratic form provides an accurate de-scription of the elastic energy in phase-field simulations calculated using nonlinearneo-Hookean elasticity (Eq. (3)). We can then rewrite Eq. (8) as a function of themaximum hoop stress as

Gc

µaR= Cξ(1− ν)

(max(τθθ)

µa

)2

(9)

where C = 4/3. Substituting in the above expression the values of max(τθθ)obtained from 1D axisymmetric simulations, we obtain the gray dashed line inFigure 4a that falls significantly below the boundary, comprised between red cir-cles and blue crosses, corresponding to the onset of fracture in 2D simulationswith fracture. Consequently, Griffith theory with axisymmetric tensile stressesunderestimates the critical value of Gc/(µaR) for fracture at fixed σy and, hence,overestimates the safe pillar radius. This discrepancy can be attributed to the factthat the 1D axisymmetric simulations lack the stress amplification due to plasticlocalization and instability. We therefore conclude that localization of plasticitycaused by anisotropic volumetric expansion plays a significant role in fracture of Si

12

nanopillars. This conclusion is further supported by the finding that the predictionof Eq. (9) is significantly improved when we use values of maximum hoop stressesobtained from 2D simulations without fracture, which exhibit stress concentrationat V-shaped notches. Unlike in 1D axisymmetric simulations, where the tensilehoop stress is always maximum at the pillar surface, hoop stresses in 2D simula-tions with localization of plastic deformation reach their maximal values inside theparticle at a short distance away from the V-shaped notch (Figure 2). Therefore,we can reasonably use Eq. (9) together with values of max(τθθ) both inside theparticle and at the tip of the V-shaped notch (corresponding to the red circles andblue squares in Figure 2a, respectively) to obtain lower and upper bounds for thefracture boundary in the plane of Gc/(µaR) and σy. The results are depicted bythe gray shaded region in Figure 4a that is comprised between the lower and up-per bounds computed in this fashion. The fracture boundary between red circlesand blue crosses in 2D phase-field simulations falls for the most part inside thisgray shaded region (in particular over the range σy ∼ 0.5–2 GPa of experimentalrelevance), thereby confirming that stress concentration near V-shaped notches isan important mechanism promoting fracture.

We can now relate our numerical findings to experimental observations of safenanopillar sizes. Figure 4a shows that the safe nanopillar radius (where pillarswith radii less than the safe value do not break) decreases with increasing σyover the estimated range σy = 0.5–2 GPa for a-LixSi. The experimental rangeof safe nanopillar radius is highlighted by the green shaded region in Figure 4a.This region was computed using the experimentally observed safe nanopillar ra-dius (120 nm) [6] and the estimated range of fracture energy (5–7 J m−2) fromexperimental measurements [39]. 2D phase-field simulations predict that, insidethis green shaded region of Figure 4a, fracture occurs for σy between 1.5 GPa(blue crosses) and 2 GPa (red circles), which falls in the upper part of the rangeσy = 0.5–2 GPa estimated from experimental measurements [8, 38, 39]. Phase-fieldmodeling prediction also depend generally on flaw size through the ratio ξ/R. Totest this dependence, we repeated a series of simulations by varying ξ/R over therange 0.01 to 0.2, which encompasses the value 0.02 used in all simulations pre-sented so far. These simulations were carried out at fixed dimensionless fractureenergy Gc/(µaR) = 0.01 calculated using the average reported fracture energy fora-LixSi Gc = 6 J m−2 [39] and the observed safe nanopillar radius R ' 120 nm [6].The results reported in Figure 4b show that nanopillars become more vulnerableto fracture with increasing ξ/R as theoretically expected. For the largest valueξ/R = 0.2 studied here, fracture occurs for σy between 0.5 GPa (blue cross) and1 GPa (red circle), which falls in the lower part of the range σy = 0.5–2 GPaestimated from experimental measurements [8, 38, 39]. While the precise valueof ξ/R is not known, its value is presumably much smaller than unity given thatnanopillars do not typically exhibit large visible flaw sizes prior to lithiation [20, 6].

We can further confirm the above estimate of σy by redoing the calculationsfor isotropic lithiation of amorphous Si. Experimental observations demonstratethat a-Si lithiates isotropically [50, 51, 52]. It is therefore reasonable to use our1D axisymmetric simulations to estimate the magnitude of hoop stresses gen-

13

erated during lithiation of a-Si. We can simplify Eq. (9) further using the ex-pression max(τθθ) ' 2σy/

√3 valid for small σy (see Figure 1b), which yields the

prediction Gc/µaR = (4C/3)ξ(1− ν)(σy/µa)2 for isotropic lithiation of a-LixSi.

Zhao et al. [24] obtained a similar expression of the form of Gc/µaR ∼ (σy/µa)2

previously by an analysis of lithiation that only considers plastic deformationand computes the dimensionless prefactor numerically assuming an initial flawsize comparable to R. In contrast, here, the prefactor is obtained analyticallyfrom the aforementioned 1D stability analysis of crack initiation in the phase-fieldmodel [49]. We can use this isotropic estimate along with the safe experimentallyobserved safe nanopillar radius 1 µm [52] to calculate an upper bound for its yieldstrength 0.4–1.2 GPa using the process zone size 0.02 ≤ ξ/R ≤ 0.2. We can usea similar analysis for Ge. Since lithiation of Ge is observed to be isotropic, usingthe estimates of its shear modulus µa ' 19 GPa we can estimate σy in the range1.5–4.6 GPa. We calculated the above range similarly using the process zone size0.02 ≤ ξ/R ≤ 0.2 and based on observed safe nanopillar radius of 250 nm [53].

Geometrical effects

Finally, to highlight the non-trivial role of geometry beyond size effects, we inves-tigate the fracture of hollow nanopillars that have been shown experimentally tobe more resistant to fracture [13]. Our axisymmetric computations predict thatthis geometrical protective effect is present for large enough yield strength dueto a decrease of the maximum hoop stress reached during complete lithiation ofcrystalline Si as a function of σy (i.e., the decrease of the peak value of plots in Fig-ure 1b with increasing annulus slenderness). However, for low yield strength, themaximum hoop stress remains bounded by σy even for large slenderness. This im-plies that the hollow nanopillar design can mitigate fracture only for materials withmoderately high yield strength. To test these predictions, we modeled the lithia-tion and fracture of hollow nanopillars with a constant cross-sectional area equalto solid nanopillars with radii 85–170 nm and different slenderness 0.08 ≤ t/R ≤ 1for σy = 3 GPa. The results illustrated in Figure 5 show that increased slender-ness has a protective effect for this σy value. For example, nanopillars break fort/R = 0.4 but only exhibit minor cracking near the interior surface of t/R ' 0.29.Even more slender nanopillars (t/R ' 0.08) do not fracture. This protective geo-metrical effect, however, does not persist at lower yield strength for ξ/R = 0.02.Our axisymmetric computations predict that the maximum hoop stress generatedin even the thinnest annulus is equal to that of a solid nanopillar for σy = 1 GPa(Figure 1b). Our 2D simulations confirm this prediction by showing that the hol-low nanopillar for t/R ' 0.29, which is protected for σy = 3 GPa, fractures forσy = 1 GPa.

In summary, we have used a multi-physics phase-field approach to model simul-taneously anisotropic phase transformation, elasto-plastic deformation, and crackinitiation and propagation during lithiation of Si nanopillars. Our results identifya vulnerable window of yield strength inside which pillars fracture during lithia-tion and distinguish two different modes of fracture inside that window with and

14

t/R

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Figure 5: Final cross section and fracture pattern as a function of dimension-less fracture energy Gc/(µaR) and slenderness t/R for hollow nanopillars withequivalent cross-section area as solid nanopillars of radius R = 85, 121 and 170(Gc = 6 J m−2, σy = 3 GPa, and ξ/R = 0.02). Color map depicts the harden-ing parameter α. The parameter range where the nanopillar fractured is shownwith the red background. One can see that at this yield strength the more slen-der nanopillar mitigates the failure. Also increasing particle size (decreasing thefracture energy) promotes initiation of cracks in-line with the experimental obser-vations (see SI movies 7a,b and 8a,b for results of simulations for Gc/(µaR) = 0.01and t/R ' 0.29 and t/R = 0.4).

without surface localization of plastic deformation prior to fracture for lower andhigher yield strength, respectively. Those two modes follow from the existence ofa critical yield strength that generates maximal tensile stresses during lithiation.Combined with experimental measurements of fracture energy [39] and observa-tions of size dependent fracture [6, 5, 52], our results yield an estimate of yieldstrength within a range σy ' 0.5–2 GPa consistent with experimental [8, 38, 39]and theoretical [40] estimates. This range is smaller than the critical yield strengthconsistent with the observation of localized plastic deformation during lithiation

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of Si nanopillars [3, 6, 5]. Over this range, plastic deformation mitigates fractureby energy dissipation but, at the same time, promotes it by the creation of stress-concentrating V-shaped notches that precede quasi-brittle fracture. Our resultsalso suggest that the observed increased robustness of hollow Si nanopillars [13]is due to a reduction of the critical yield strength generating maximal tensilestresses with increasing slenderness. This interpretation, however, warrants fur-ther investigation since this protective effect is only significant in simulations withlarge enough σy values. The present study highlights the importance of compu-tationally informed geometric design that takes into account the subtle interplaybetween material properties and geometry to generate reliable predictions of me-chanical stability of high-capacity battery materials, paving the way for designsthat exploit more complex geometries such as open nanoporous structures withultra-high interfacial area [11].

Methods

All the equations are solved using Galerkin Finite Element Method. Furthermore,to ensure the robustness of solution only 1/4 of each geometry was simulated andappropriate boundary conditions are applied on the symmetry axises. Our imple-mentation is based on PETSc [54] as the linear algebra backbone and libMesh [55]for finite elements bookkeeping. Eq. (5) is solved using a Newton method wherewe calculate the consistent tangent moduli explicitly at each iteration. Further-more, Eq. (7) is integrated explicitly using 50 substeps during each time step toensure the accuracy and stability of integration. Table 1 summarizes the valuesof different parameters used in our simulations.

Table 1: Material properties of a-Si and c-Si used in the simulations.

Material property Value Units

κc [23] 108 GPaκa [23] 10.8 GPaµc [23] 50 GPaµa [23] 5 GPaK 50 MPaGc [39] 6 J m−2

w 0.85 nmβ 0.7 -

Author contributions

A.M. and A.K. conceived the theoretical study and jointly interpreted the numer-ical results and wrote the paper. A.M. carried out the numerical study.

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Acknowledgements

A.M. and A.K. acknowledge the support of Grant No. DE-FG02-07ER46400 fromthe U.S. Department of Energy, Office of Basic Energy Sciences. The majorityof the numerical simulations were performed on the Northeastern University Dis-covery cluster at the Massachusetts Green High Performance Computing Center(MGHPCC).

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