Numerical Modeling of Fracture-Resistant Sn Micropillars as Anode for

Mar 15, 2016 - Numerical Modeling of Fracture-Resistant Sn Micropillars as Anode for Lithium Ion Batteries. Nadeem Qaiser†, Yong Jae Kim†‡, Chun...
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Numerical Modeling of Fracture-Resistant Sn Micropillars as Anode for Lithium Ion Batteries Nadeem Qaiser,† Yong Jae Kim,†,‡ Chung Su Hong,† and Seung Min Han*,† †

Graduate School of Energy Environment Water Sustainability (EEWS), Korea Advanced Institute of Science and Technology (KAIST), Daejeon 305-701, Korea ‡ Frontier in Extreme Physics, Korea Research Institute of Standards and Science, Daejeon 305-340, Korea S Supporting Information *

ABSTRACT: Sn possesses three times higher capacity in comparison to graphite anode (372 mAhg−1) that makes it a promising candidate for enhanced performance Li ion batteries. Contrary to Si, Sn is compliant and ductile in nature and thus is expected to readily relax the Li diffusion-induced stresses. The low melting point of Sn additionally allows for stress relaxations from time-dependent or creep deformations even at room temperature. In this study, numerical modeling is used to reveal the significance of plasticity and creep-based stress relaxations in the Sn working electrode. The maximum elastic tensile hoop stresses for 1 μm micropillar size with 1C charging rate conditions reduces down from ∼1 GPa to ∼200 MPa when Sn is allowed to plastically deform at a yield strength of ∼150 MPa. After experimentally determining the creep response of Sn micropillars, creep deformations are incorporated in numerical modeling to show that the maximum tensile hoop stress is further reduced to ∼0.45 MPa under the same conditions. Lastly, the Li-induced stresses are analyzed for different micropillar sizes to evaluate the critical size to prevent fracture, which is determined to be ∼5.3 μm for C/10 charging rate, which is significantly larger than that in Si. density.10 Different nanoscale morphologies were experimentally studied including nanoparticles11 and nanowires12,13 that have the benefits of stress relaxation, enhanced contact with the electrolyte, and a direct conduction pathway to the current collector for the case of nanowires. Although a clear size dependency in the Li-induced stresses are expected, experimental verification of size dependency to ensure the mechanical stability has been a challenge. In the study by Lee et al. pillars were etched from doped Si wafer to determine the critical size for failure, which was approximately 240−360 nm depending on charging rate.14 However, gathering sufficient results for statistical analysis was difficult since the experiments are time consuming and requires detailed characterization. In order to address the limitations in experimental determination of length scale for mechanical stability, analytical and numerical modeling techniques have been used previously to determine the stresses for different sized nanostructures of varying morphologies. For example, the stress fields built up in the shell were analytically calculated, and conditions for fracture and debonding were discussed for core−shell structures of spherical and cylindrical geometries of Si anodes.15 The charging conditions of anode material influence the value and distribution of Li diffusion-induced stresses (DIS) in the anode. Cheng et al. studied stress evolutions in Si spherical particle under galvanostatic or potentiostatic charging conditions through analytical expressions.16,17 A numerical model for

1. INTRODUCTION A high energy density lithium ion battery (LIB) is considered as a promising candidate for future energy storage systems ranging from portable consumer applications to high power applications such as electric vehicles. Current commercialized LIBs are based on the graphite anode, which has a specific capacity of 372 mAhg−1, but there exists a demand for an increase in the capacity of up to ∼600 mAh g−1.1 Among different anode materials being proposed for LIBs, silicon (Si) and tin (Sn) have received extensive interest as anode materials due to higher theoretical capacity and environmental benignity.2−5 Si has a high theoretical charging capacity of 4200 mAhg−1 and is considered as one of the most promising anode materials for LIBs. However, Si experiences large volume expansion by ∼400% during lithiation, i.e., the process of alloying with Li, resulting in extreme stress build up and crack formations.6 During delithiation, the removal of lithium from the LixSi phases causes a large tensile stress, and thus further crack propagation can occur. As multiple cycles progress, structural degradations and morphological changes caused by repeated volume changes can cause loss of electrical contact and thus result in poor cyclic performances in Si-based LIBs despite their high theoretical capacity.7−9 One method of relaxing the Li-induced stresses is to make use of nanostructures that can have benefits in relaxing the stresses due to the short diffusion lengths for Li transport resulting in lower stress build up. In addition, the high surface to volume ratio increases the presence of active sites for Li storage, thus improving the specific capacity, while higher electrode/electrolyte contact area leads to higher power © XXXX American Chemical Society

Received: January 1, 2016 Revised: March 15, 2016

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Figure 1. (a) Schematic diagram showing the imposed boundary conditions used to calculate the lithium concentration gradients and DIS. Normalized concentration profiles as a function of (b) SOC for 1 μm Sn micropillar at constant c rate of 1C and (c) comparison to Si micropillar at SOC = 100%. (d) Effect of varying c rates for 1 μm Sn micropillar at SOC = 100%, showing more inhomogeneous concentration profiles for faster c rates.

be ideal if the core is a compliant and soft material, but Si, unfortunately being a hard and stiff material as well as having anisotropic volume expansion, is difficult to suppress the volume expansions using other hard shells thereby leading to poor cyclic performance.27,28 Sn is also a promising anode material but not explored as much as Si due to a lower theoretical capacity of ∼994 mAh g−1; however, it is known to be abundant and inexpensive and still possess three times higher theoretical reversible capacity than graphite.29−31 The lithiated Sn, Li4.4Sn, also is known to have smaller and isotropic volume expansion up to 260% compared to that of Si.32 The significant advantage of Sn in comparison to Si, however, is due to the intrinsic mechanical properties of Sn; Sn is compliant and ductile in nature (i.e., the low stiffness and yield strength), leading to smaller DIS. Sn also is known to have time-dependent creep deformation due to its low melting point (Tm = 505 K) that can relax the stresses further. Wang et al.33 conducted an in situ TEM study of Sn nanoparticles and found Sn expands isotropically upon lithiation, and nanoparticles with size up to ∼526 nm showed no fracture. In addition, volume expansion is much smaller, providing lower hoop stress in the surface layer. Although Wang et al. referred to Sn as having “self-healing” capability, no further analysis was provided. Therefore, merit of the experimentally observed mechanical stability of Sn demands theoretical understanding of DIS in Sn during lithiation. In this study, the DIS in Sn during lithiation is investigated for elastic stresses first followed by plastic relaxations as plasticity occurs in Sn when the stress exceeds its low yield strength. Most importantly, the creep behavior of Sn micropillars was determined experimentally and incorporated in the numerical model to determine the role of creep in reduction of DIS in Sn micropillars. The results from our numerical analysis setup then allowed for prediction of critical size below which Sn becomes fracture resistant. This modeling

lithiation was performed by taking a thermal-strain approach for corresponding DIS. An earlier approach in numerical modeling was to analyze the stress generation along the radius of spherical electrode18 but was later expanded to DIS for various three-dimensional (3D) shapes and sizes of electrodes.19,20 In the work of Ryu et al., numerical modeling has been used to analyze Si nanopillars to determine the critical size, below which stress-induced fracture does not occur. Ryu et al. concluded that the critical size for fracture is in the range of 100−300 nm,21 and Si nanopillars with small diameters below ∼100 nm, therefore, are expected to be fracture resistant. Although there exists a critical size for fracture resistance in Si, realizing commercial fabrication of nanowires with small diameters, however, remains a challenge. One of the existing challenges in ensuring cyclic stability is in managing formations of solid−electrolyte interphase (SEI) layers that eventually cause loss in electrical contact of the active material. Nanostructures, which can be engineered to have fracture resistance below a critical dimension, are still vulnerable to thick SEI formations due to large surface area to volume ratios. In order to stabilize the outer surface of the nanostructures against SEI formations, coating Si nanoparticles and nanowires with other materials such as Cu or graphitized C has been implemented in an attempt to suppress the large volume expansion.22−24 Although enhanced cyclic life was reported in comparison to bare Si counterpart, preventing volume expansion/contraction of solid Si nanostructures is not possible. Stiffer and harder oxide shells on nanotubes were then introduced, and the radial volume expansions can be minimized since there is space for Si to expand inward toward the center of the tube25 as the outer stiff, hard shell prevents outward expansion. As the outer surface is more stabilized, the SEI formations will also be minimized, leading to enhanced cyclability. Stiff carbon coating on Si nanotubes was also reported to have similar effects.26 This idea of clamping would B

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An axisymmetric analysis using COMSOL Multiphysics was used to model the Li concentration gradient, and the evolution of radial and hoop stresses was estimated by using the Structural Mechanics module in COMSOL. The height to diameter ratio was taken to be 5:1 with the bottom end fixed while the top end free to move axially, which is equivalent to a generalized plane-strain condition along the z axis for such a long aspect ratio pillar or essentially a nanowire.42,43 Additionally, Sn lithiation occurs by a diffusion-controlled mechanism unlike the reaction-controlled lithiation in Si, and a single-phase diffusion model was utilized. To the best of our knowledge, no study has reported that Sn deforms via anisotropic volume expansion during lithiation; thus, isotropic volume expansion is assumed for these simulations. First, Li insertion typically dictates the reliability; thus, the first charging process was evaluated in this study. For elastic analysis, the elastic modulus and Poisson’s ratio were taken to be ∼50 GPa and ∼0.36, respectively, and the elastic−plastic simulations assumed elastic to perfectly plastic governing behavior with yield strength that was determined based on our nanocompression test of FIBmilled Sn micropillars to1 μm diameter. In these models, in order to find the effect of pillar size on DIS, the underlying effect of plasticity and creep deformation was studied on Sn micropillars with sizes of 0.8 μm and 1 μm, whereas to estimate the critical size to prevent the fracture; a series of simulations was run with different sizes of anode. To show the effect of charging rate (c rate) on DIS, various c rates from C/5 to 1C were adopted in our modeling (1C = the surface will reach to Cmax in 1 h). To extract the radial and hoop stresses within Sn micropillars, a dimensionless radial position is defined as r/R, where r = position along the radial direction at the halfway point of the pillar’s height and R = total radius of Sn micropillar. The simulation domain is carefully meshed to guarantee numerical stability. To incorporate the creep effect, the creep deformations are allowed in Sn anode using the Structural Mechanics module in COMSOL. To ensure the convergence the relative and absolute tolerances are set to 10−12 and 10−2, respectively. For the DIS modeling with and without creep relaxations and calculation of the corresponding critical size for fracture resistance, the material properties are assumed to be constant during lithiation. The diffusion coefficient (D), elastic modulus (E), yield strength (σy), partial molar volume (Ω), and Poisson’s ratio (υ) are taken as constants (no Li concentration and stress dependency is considered) as follows: D = 10−15 m2 s−1,44 E = 50 GPa, υ = 0.36, σy = 150 MPa (by our experiments), Ω = 1.63 × 10−5 m3 mol−1, and Cmax = 1.84 × 105 mol m−3. 2.2. Nanocompression Creep Testing for Calculating the Parameters. Bulk polycrystalline Sn sample with a grain size of ∼1−2 mm was mechanically polished down to grit size of #4000 and then annealing at 378 k (∼0.75Tm) for 5 h in vacuum followed by aging at room temperature for more than 1 week. Then Sn micropillars were fabricated in a single grain of bulk sample using focused ion beam (FIB) milling with a Quanta 3D FEG (FEI Co., Hillsboro, OR, USA). These milled micropillars have a diameter of ∼1 μm with an aspect ratio of ∼3:1 and taper angle of ∼5°. Compression creep tests were performed using a TI-750 Ubi Nanoindenter (Hysitron Inc. Minneapolis, MN) with a flat punch indenter tip. Before creep tests, Sn micropillars were compressed under the displacement control at an engineering strain rate of 0.001 s−1. By calculating the engineering stress (σ) vs engineering strain (ε) curves from the load (P) vs

provides insight into the intrinsic material properties for practical designing of the Sn-based lithium ion battery with higher cyclic and rate performance.

2. EXPERIMENTAL SECTION 2.1. Numerical Modeling. This study utilized a numerical analysis to model the Li diffusion into Sn micropillars and corresponding DIS. The transport of lithium in host is a timedependent concentration-driven diffusion process and is governed by Fick’s second law (eq 1),34,35 where the concentration gradient from the surface to the core causes Li to diffuse in along the radial direction of Sn micropillar ⎡ ∂ 2c ∂c 1 ∂c ⎤ = D∇2 c = D⎢ 2 + ⎥ ∂t r ∂r ⎦ ⎣ ∂r

(1)

The Li flux is controlled by specifying the constant current density on the surface of a micropillar, where normalized lithium concentration c is given by C/Cmax, i.e. the actual lithium concentration (C) divided by the concentration of lithium in the fully lithiated state (Cmax). Charging is completed once the concentration at the surface of micropillar attains the maximum theoretical capacity (Cmax) of Sn. Current definition for fully lithiated state does not imply a homogeneous concentration profile, as seen in Figure 1. During charging, Li ions diffuse into the host that changes the lattice constants of the host material. These lattice constants may be assumed to vary linearly with volume of the inserted Li ions.36,37 DIS is calculated by using the thermal-strain approach, i.e., by taking the partial molar volume equivalent to an arbitrary thermal expansion coefficient to result in the desired volumetric strains. In the work by Yang et al. the anisotropic morphologies in Si NWs were modeled using the small strain and finite-strain theories and found that both theories converged to similar DIS trends and morphology changes for all Si NWs except the concentration of hoop tension at particular angular sites.38 However, given no angular sites or anisotropic expansion in the Sn anode, using the small strain model in this study seems reasonable.39,40 The governing equations used for this study were described in detail in the work by Zhao et al.41 and so will be reviewed briefly here. The total strain (dεtij) induced in the host material during lithiation is taken as dεtij = dεeij + dεpij + dεIij, where dεeij is elastic strain, dεpij is plastic strain, and dεIij is lithiation-induced strain that is proportional to the normalized concentration of Li into the Sn host. Hooke’s law gives dεije = d

{ E1 [(1 + v)σ − vσ δ ]} ij

kk ij

(2)

where v is Poisson’s ratio. For i = j, δij = 1; otherwise, δij = 0. The material starts to yield under the von Mises condition. For modeling the plastic deformations, the J2-flow rule is assumed that gives ⎧ 0, σe < σy ⎪ ⎪ dεijp = ⎨ 0, σe = σy , dσe < dσy ⎪ ⎪ Sij , σe = σy , dσe = dσy ⎩

where Sij = σij − σkk δij/3 is the deviatoric stress, σe = the equivalent stress, and σy the yield strength.

(3)

3SijSij/2 C

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Figure 2. DIS evolutions for elastic deformations. (a) Radial and (b) hoop stresses as a function of SOC for Sn micropillar with a diameter of 1 μm at constant c rate of 1C. At SOC = 100%; effect of c rates on radial stresses of Sn micropillar size for (c) 1 and (e) 0.8 μm and hoop stresses for (d) 1 and (f) 0.8 μm. Larger pillar size and faster c rate results in higher DIS due to formation of more inhomogeneous concentration profiles.

displacement (h) curves as σ ≈ P/A and ε ≈ h/l, respectively, where A is the initial cross sectional area of micropillars, the elastic limit or yield strength of Sn micropillars was defined as ∼150 MPa (Figure S2, Supporting Information). A series of constant-load creep experiments was performed on 1 μm Sn micropillars within the elastic regime under the load control. During the creep tests, the loading/unloading rate was fixed as 10 μN s−1 and load was held at various elastic stress levels (30, 70, and 100 MPa) for 200 s. To be careful for thermal drift issues in creep tests with nanoindentation equipment, the thermal drift rate was measured again in the unloading sequences by holding the load at 2 μN for 20s. The thermal drift rate was checked before and after each test to confirm that the drift can be neglected in further creep analysis. Steady-state creep rate (ε•) is defined as the creep rate at holding = 200 s, which is obtained by fitting engineering creep strain (ε creep) vs holding time curves with Garofalo’s mathematical fitting45,46 and differentiating the fitting data with respect to “t”. The stress−creep strain is governed by the well-known power law formulation given by eq 4 ε ̇ = Aσ n

3. RESULTS AND DISCUSSION 3.1. Concentration Profiles and Li Diffusion-Induced Stress Evolution. During lithiation, the outer surface of the anode has a higher concentration of Li that results in a concentration gradient, which acts as a driving force for Li diffusion. The galvanostatic charging with a constant Li flux applied on the surface is used to model the Li insertion in Sn micropillars. The Li ions are inserted into the Sn micropillar from the outer surface as shown in the schematic diagram in Figure 1a. Unlike in the case for Si, where reaction-controlled lithiation is required for two-phase sharp boundary modeling, Sn has fast isotropic diffusion that allows for diffusioncontrolled single-phase modeling. The Li concentration is normalized with a maximum theoretical capacity (Cmax) of Sn, i.e., at Li4.4Sn. For the pure Sn case (no Li insertion), c = 0; c = 1 denotes when the Sn anode surface reaches its full capacity of Cmax. The normalized concentration profiles of Li at the c rate of 1C as a function of lithiation percentage or state of charge (SOC) is shown in Figure 1b. As expected, the Li concentration gradient in host material depends on not only the diffusivity of anode material but also the c rate and size of the anode. It should be noted that the c rates are defined as a dimensionless parameter, i.e., = Dt/R2, where D, t, and R are the diffusion coefficient, elapsed time at the end of lithiation, and radius of Sn micropillars, respectively. As the concentration gradient is formed, the mismatch in the volume of the material results in DIS. Therefore, a material with lower diffusivity will result in higher DIS since a steeper

(4)

The values of the stress exponent (n) and creep constant (A) were calculated and found to be ∼1.5 and 2.3 × 10−8, respectively (Figure S3, Supporting Information) and were incorporated in the numerical modeling to determine the DIS relaxation arising from creep deformations. D

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Figure 3. DIS evolutions upon allowing for plastic relaxation via elastic-purely plastic deformations. (a) Radial and (b) hoop stresses as a function of SOC showing that the steady-state stress reduces to ∼200 MPa for 1 μm Sn micropillar at constant c rate of 1C. At SOC = 100%; effect of c rates on radial stresses of Sn micropillar size for (c) 1 and (e) 0.8 μm and hoop stresses for (d) 1 and (f) 0.8 μm.

region) cause tensile hoop and radial stress to develop throughout the micropillar. Since the free surface is traction free, the radial stresses with the SOC at any given time vanish at the surface and increase monotonically to the center as shown in Figure 2a. The expanded region near the free surface in the hoop direction is in compression with respect to the center region as seen in Figure 2b. At early stages of lithiation, the radial and hoop DIS starts to elevate rapidly. The maximum tensile hoop DIS at the center of the micropillar, for example, increases from ∼340 to ∼960 MPa in a slight change of lithiation percent, i.e., SOC from 0.3% to 1.7%. The DIS reach the steady state at ∼SOC = 4.0% and then attains the steady states for the remaining SOC as shown in Figure 2a and 2b. A larger Sn micropillar exhibits longer Li diffusion paths, and this causes higher DIS, while a smaller micropillar has a more homogeneous concentration gradient leading to smaller DIS. In addition to inhomogeneous concentration gradients, Sn micropillar with a large size is also stiffer in resisting the radial volume expansion and results in higher DIS. DIS for two different micropillar sizes, i.e., 1.0 and 0.8 μm, were examined, and the elastic radial and hoop DIS as a function of c rates at SOC = 100% are shown in Figure 2c−f. Faster c rate corresponds to higher Li flux at the surface and leads to more inhomogeneous concentration gradients and thus higher DIS. The maximum tensile DIS was typically found at the center, and the maximum tensile stress of Sn micropillar with a diameter of 1.0 μm, for example, reaches ∼1000 MPa at a c rate of 1C; this is about ∼5 times higher as compared to the

concentration gradient will be formed. On the other hand, a material with higher diffusivity will result in a less steep concentration gradient or more homogeneous concentration profile and thus lead to lower DIS. Sn is known to have a lower activation barrier for Li diffusion and thus higher diffusivity (10−15 m2 s−1) in comparison to that of Si (10−16 m2 s−1), and thus, the concentration gradient is not as steep in Sn; comparison of the two materials at SOC = 100% for the constant micropillar size of 1 μm and c rate of 1C is shown in Figure 1c. For different c rates, the normalized lithium concentration profiles at the end of charge, i.e., SOC = 100%, are shown in Figure 1d. Higher c rate results in a more inhomogeneous concentration gradient in the Sn that is expected to result in higher DIS due to the severe volumetric mismatch. It is known that the estimated concentration gradients and the corresponding induced stresses increase with time and eventually reach a steady-state concentration profile when modeled using galvanostatic charging.21,41 This happens because the degree of inhomogeneous concentration gradients depends on the competition between the Li diffusivity and the c rate and is not bounded by the finite solubility of Li in Sn. The elastic stress distributions were first analyzed as shown in Figure 2a−f. Since the Li concentration is not homogeneous in the micropillar that leads to lattice mismatch, the higher Li contents near the surface lead to more expansion near the surface than at the center. As the outer shell expands outward, pulling forces acting on the inner region (relatively Li poor E

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Figure 4. DIS relaxations through creep deformations. (a) Radial and (b) hoop stress relaxations as a function of SOC, revealing the DIS increases first and then starts to relax with the passage of time for 1 μm Sn micropillar at constant c rate of 1C. At SOC = 4%; effect of c rates on radial stresses of Sn micropillar size for (c) 1 and (e) 0.8 μm and hoop stresses for (d) 1 and (f) 0.8 μm, indicating higher DIS for higher c rates due to an inhomogeneous concentration profile.

By using the experimentally yield strength, the numerical model was used to determine the radial and hoop stress evolutions as depicted in Figure 3a−f. In order to compare directly with elastic deformations and reveal the plasticity relaxations in Sn micropillar, identical conditions were chosen to show the DIS as a function of SOC (Figure 3a and 3b), micropillar sizes, and c rates (Figure 3c−f). The radial stresses are tensile at the center similar to the elastic case, but maximum tensile stress at SOC = 100% has been reduced from ∼1000 to ∼200 MPa for a constant diameter of 1 μm and c rate of 1C. The hoop stresses are also analogous to its distribution in the elastic case, i.e., tensile at the center and compressive at the surface, but a significantly lower magnitude of DIS was observed due to plastic relaxations. Similar to the elastic deformation case, the maximum tensile DIS (∼200 MPa in this case) at the center of the micropillar reaches steady state at ∼SOC = 4.0%. The radial and hoop stress evolutions for micropillar sizes of 1 and 0.8 μm for different c rates at the end of lithiation (SOC = 100%) are shown in Figure 3c−f. The slower lithiation rate of C/5 has more homogeneous gradients and thus results in lower radial and hoop DIS. Note that all of these cases have resulted in compressive hoop stress at the surface and tensile at the center, and the overall trend in stresses is well matched with other work.21,41 As noted earlier, Sn is a low melting point metal (Tm = 505 K), and room temperature (∼0.6Tm) is already high enough to have significant creep or time-dependent plastic deformations

maximum tensile stress of the same sized micropillar at a slower c rate of C/5 (∼200 MPa). Likewise, the maximum tensile stress of Sn micropillar with a diameter of 0.8 μm reaches ∼650 MPa at a c rate of 1C, as compared to a slower c rate of C/5 (∼130 MPa). For constant c rate, for example, at C/5, the maximum tensile DIS at the center for a 1.0 μm micropillar is higher (∼200 MPa) as compared to a micropillar size of 0.8 μm (∼130 MPa). It should be noted that Sn being a relatively low modulus metal is also part of the reason why the DIS in general are lower in Sn in comparison to Si. The impact of the arbitrary Young’s modulus (from ∼50 to ∼80 GPa) of Sn on radial and hoop stresses at SOC = 100% for elastic deformations at constant Sn micropillar size of 1.0 μm is revealed in Supplementary Figure S1(a and b). 3.2. Plastic and Creep Deformations-Based Stress Relaxations. Consideration of only elastic response results in unrealistically high DIS, and thus, the effect of allowing for plasticity was analyzed next. Elastic−perfectly plastic governing equation was used for plasticity, where the plastic flow occurs at a yield strength that was experimentally determined from compression of Sn micropillars with a diameter of 1 μm, as shown in Supplementary Figure S2(a). The SEM images of Sn micropillar before and after testing are shown in Supplementary Figure S2(b and c). The loading slope in σ−ε curves for Sn micropillars indicated an elastic modulus of ∼45 GPa, which is close to that of bulk Sn (∼50 GPa), and the yield strength was calculated as ∼150 MPa on average from 5 compression tests. F

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Figure 5. Comparison of elastic, plasticity, and creep relaxations in Sn micropillar with a size of 1 μm at constant c rate of 1C. Radial stress at SOC of (a) 4% and (c) 100% and hoop stress at SOC of (b) 4% and (d) 100%. Snapshots of axisymmetric modeling for Sn micropillar size of 1 μm for (e) radial stress distribution with maximum tensile stress shown at the center of the pillar and (f) hoop stress distribution with maximum tensile stress shown at the center that switches to compressive at the surface of Sn micropillar.

is governed by the underlying power law formulation, and the stress exponent (n) and material constant (A) are calculated as 1.5 and 2.3 × 10−8, respectively. By using these measured creep parameters, the creep-based stress relaxation is numerically modeled in COMSOL Multiphysics. All material properties, c rates, and micropillar sizes are identical with the abovementioned simulations. Radial and hoop DIS is expected to be readily relaxed from plasticity with additional relaxations from creep deformations. The radial and hoop stresses as a function of SOC are shown in Figure 4a and 4b. At the early stages of lithiation, the maximum tensile hoop DIS at the center starts to increase from ∼130 (SOC = 0.3%) to ∼200 MPa (SOC = 1.7%). In contrast to steady-state stresses for elastic-purely plastic deformations case (at SOC = 4.0%), the DIS starts to relax due to creep deformations in Sn micropillar. The tensile hoop stress at the

that can cause further relaxations of DIS in Sn micropillars. To investigate the coupled DIS-creep effect within a single analysis, the power law creep (ε̇ = Aσ″) parameters, i.e., stress exponent, n, and material constant, A, are required. However, due to the high surface-to-volume ratio of micropillars and the enhanced atomic diffusion along the surface, creep behavior of Sn micropillars is significantly different with their bulk counterparts.45 For this reason, the creep parameters of Sn micropillars were obtained through the micropillar creep tests rather than simply adopting the results of bulk Sn creep from the literature. It is worth noting that although the nanoscale creep behaviors are frequently explored through nanoindentation creep tests, uniaxial creep tests are used to avoid complications arising from the complex stress state and fixed plastic strains in self-similar indentations.45,46 The results shown in Figure S3 (Supporting Information) verify that creep behavior of 1 μm Sn micropillar G

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existing flaw was positioned at the center of the micropillar to determine the critical size for fracture. Stress intensity factor (KI) was calculated by inputting the maximum tensile hoop stress determined previously from COMSOL and then using the J-integral method of ABAQUS FEM simulations with the lithiated Sn elastic material properties from the work of Wolfenstine et al.47 Criterion for failure was determined when the KI exceeded KIc (KI > KIc), where KIc is the fracture toughness of Sn micropillar. More details of numerical modeling for fracture analysis in ABAQUS can be found in the Supporting Information. It should be noted that the results of this fracture analysis are heavily dependent on the literature value of KIc, but there still is a lack of literature to sufficiently support the fracture toughness (KIc) for lithiated Sn. In the work by Wolfenstine et al.47 KIc for Li4.4Sn alloy was experimentally measured using nanoindentation to be ∼0.8 MPa m1/2; however, the drawback of this work is that during transfer from the glovebox to the indenter the samples were exposed to ambient air that can cause surface oxidation to result in variance in the measured KIc. In addition, the needed fracture-related material properties for LixSn alloy at various SOC are not available in the literature yet. Nevertheless, we take the reported KIc from the work by Wolfenstine et al. to provide an estimate for the critical size that will serve as a preliminarily understanding of the fracture resistance of Snbased anode materials. As noted earlier, for faster c rates the maximum tensile hoop DIS at the center is higher; therefore, the critical size is expected to be smaller. To determine the critical size at a particular c rate, i.e., C/10, the KI values with different Sn micropillars sizes were calculated and the micropillar size at which the KI reaches KIc, i.e., the critical size is shown in Figure 6a. The critical diameter of Sn micropillar for a c rate of C/10

center has decreased from ∼200 (SOC = 1.7%) to ∼140 MPa (SOC = 3.0%). Also, these creep relaxations continue for every lithiation percent increment until DIS becomes negligible at SOC = 100%. For instance, the maximum tensile hoop stress at the center has lowered to ∼100 MPa at SOC = 4.0% and further to ∼0.45 MPa at SOC = 100%. The effects of micropillar size and c rates now with creep deformation were also analyzed. The radial and hoop DIS for micropillar sizes of 1 and 0.8 μm for different c rates at SOC = 4% are shown in Figure 4c−f. The creep deformations have resulted in lower DIS as compared to elastic and plastic deformations; however, at a particular lithiation state, i.e., SOC = 4%, and with combined elastic-plastic-creep deformations the larger micropillar size and faster c rate cause higher DIS relative to each other. For example, with a constant micropillar size of 1 μm, the maximum tensile hoop DIS at the center is higher (∼110 MPa) for a c rate of 1C as compared to a c rate of C/5 (∼60 MPa). Similarly, for a constant c rate of C/5, the maximum tensile hoop DIS at the center for a micropillar with a larger size of 1 μm is higher (∼62 MPa) as compared to a micropillar size of 0.8 μm (∼42 MPa). At two different lithiation states (SOC = 4% and 100%), the overall comparison of radial and hoop DIS with and without relaxations in a Sn micropillar of a constant size of 1 μm and c rate of 1C are shown in Figure 5a−d. The distribution of radial and hoop DIS has a similar trend to that in the elastic and elastic−plastic case, but the DIS was further reduced by allowing for creep relaxations. As aforementioned, the DIS depends on the Li concentration gradient, which in turn is related to the time scale for diffusion as well as the c rate. Li diffusion time scale can be represented as ∼R2/D, and c rate can be described in terms of a dimensionless parameter given by Dt/R2. In addition to DIS from the above time scales, Sn pillars have another scale associated with creep rate that will determine the stress relaxation over time. A plot of the stress evolution over time at the center of pillar is shown in Supplementary Figure S4 for the case of 1 μm pillar size, c rate of C/5. In the early stages of lithiation, the stress increases rapidly beyond the von Misses stress that initiates plastic relaxation. Supplementary Figure S4 shows the comparison of what happens with and without creep in which the stress remains the same for the elastic−perfectly plastic case, but the stress starts to decrease once the creep relaxation is initiated. Therefore, the resulting DIS in Sn pillars are dictated by not only the time scale associated with diffusion and c rate but also creep relaxations. The distributions of radial and hoop DIS in the 3D axisymmetric model of Sn micropillars used in this study are shown in Figure 5e and 5f. The color legend of the DIS distribution shows the tensile hoop stress at the center and compressive at the surface of the micropillar. Interestingly, radial and hoop DIS at lithium-rich phases (Li4.4Sn) are significantly lower, which prevent further material degradation, and Sn anode-based LIBs are expected to have high cyclic stability. These creep-based relaxed stresses divulge the selfhealing ability of Sn micropillars during lithium progression that would lead to exhibit enhanced structural stability and performance for higher energy density devices. 3.3. Critical Size To Prevent the Fracture. The failure of LixM (M = Si, Sn) alloy is dependent on the tensile DIS that can drive the crack propagation. The numerical modeling results presented above indicate that the maximum tensile hoop stress is located at the center of the Sn micropillar, so a pre-

Figure 6. Conditions of fracture for Sn micropillars. (a) Calculations of KI with constant c rate of C/10, and (b) critical sizes for different c rates, ranges from C/10 to 1C. The critical sizes for 1C and C/10 are ∼1 and ∼5.3 μm, respectively. H

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The Journal of Physical Chemistry C was found to be ∼5.3 μm, which is many orders larger than that of c-Si (∼300 nm) from the work of Ryu et al. under the same c rate of C/10.21 The critical size of Sn micropillar to prevent fracture at various c rates is shown in Figure 6b. Faster c rate leads to higher DIS, and thus, at a c rate of 1C the critical size of the Sn micropillar reduces up to ∼1 μm. The size-dependent fracture is as expected for our case of Sn due to the different degree of DIS as a function of micropillar size, and this is in agreement with the work by Cheng et al.48 In their work, Cheng et al. reported that fracture is expected to be size dependent for the cases of small Biot number (i.e., B ≪ 1), a dimensionless parameter that defines the ratio between Li diffusion resistance to that of charge transfer, but no size dependency is expected for the case of a large Biot number (i.e., B ≫ 1). Large B corresponds to higher diffusion resistance to result in highly inhomogeneous concentration gradients. For potentiostatic operation, this is analogous to constant Li concentration on the surface of a micropillar and does not cause size dependency in fracture.48 Contrary to potentiostatic operation, our Sn lithiation was under galvanostatic operation, where the c rate is controlled by specifying the constant Li flux at the surface and corresponds to the case with small “B” value (B ≪ 1) to result in significant size dependency in fracture of Sn anode. As noted earlier, the critical size determined is heavily dependent on the properties of lithiated Sn such as yield strength, work hardening rate, and KIc that is inputted into the numerical simulations. Therefore, the critical size of ∼5.3 μm is expected to change as the underlying material parameter is changed. However, the provided analysis and mechanisms for why Sn micropillars would be fracture resistant are of merit and can explain the recent experimental observations of “selfhealing” Sn nanostructured anode. For example, the in situ TEM study by Wang et al. concluded that the Sn nanoparticles with ∼526 nm prevented fracture during lithiation,33 which is significantly larger than the size experimentally determined for Si. The fracture resistance of Sn via plastic and creep relaxations as well as an estimate for the critical size for failure provided in this study is expected to serve as a stepping stone in more future experimental studies to confirm the simulation results to potentially lead to commercially realizable micrometer-sized Sn anode that is more practical than nanoscale Si anode for LIBs with enhanced reliability.

melting point metal Sn was shown to be excellent at relaxing DIS, which is in agreement with experimental reports of “selfhealing” Sn nanoparticles for LIBs. Lastly, the critical size of Sn to prevent the fracture was estimated based on the literature value of KIc, and fracture is expected to occur when the micropillar size is larger than ∼5.3 and ∼1 μm for c rates of C/ 10 and 1C, respectively. Naturally, due to its stress relaxation mechanisms discussed above, the critical size for failure is many orders of magnitude larger than that for Si. Therefore, the results of this study suggest that Sn is a mechanically robust and highly stable material that can more easily be commercialized owing to the larger fabrication window of sizes for mechanical stability in comparison to Si.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b00002. Effect of Young’s modulus on DIS, results of nanocompression and creep testing, comparison of diffusion, c rate and creep relaxation time scales, and numerical modeling procedure for critical size calculations for Sn pillars (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +82-42-350-1716. Fax: +82-42-350-1710. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work at KAIST was graciously supported by the R&D Convergence Program of MSIP (Ministry of Science, ICT and Future Planning) NST (National Research Council of Science & Technology) of Korea (Grant # CAP-13-1-KITECH), and the National Research Foundation of Korea (NRF) under the Ministry of Science (NRF-2014R1A4A1003712).

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REFERENCES

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4. CONCLUSIONS In summary, the numerical modeling has been utilized to provide insight to diffusion-induced stresses, and the effect of stress relaxations arising from plasticity and creep in Sn micropillar during lithiation was studied. The DIS distributions were analyzed as a function of SOC, rate of charging, as well as micropillar size. As lithiation progressed, maximum tensile hoop and radial stresses were shown to develop at the center of the Sn micropillar. Faster charging rate and larger anode size was shown to cause higher stresses due to more inhomogeneous Li concentration gradients. The experimentally determined yield strength of 150 MPa and creep constants (n = 1.5, A = 2.3 × 10−8) of Sn micropillars were used to determine the stress relaxations from plasticity and creep. A Sn micropillar with a constant diameter of 1 μm and c rate of 1C at 100% SOC showed maximum elastic tensile hoop DIS at the center of ∼1 GPa that was lowered to ∼200 MPa upon allowing for plasticity and then relaxed further to ∼0.45 MPa with creep relaxations. Therefore, the inherent deformation behavior of the low I

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