Variation of stress with charging rate due to strain-rate sensitivity of silicon electrodes of Li-ion batteries

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1 Variation o stress with charging rate due to strain-rate sensitivity o silicon electrodes o Li-ion batteries The Harvard community has made this article openly available. Please share how this access beneits you. Your story matters Citation Pharr, Matt, Zhigang Suo, and Joost J. Vlassak Variation o Stress with Charging Rate Due to Strain-Rate Sensitivity o Silicon Electrodes o Li-Ion Batteries. Journal o Power Sources 270 (December): doi: /j.jpowsour Published Version doi: /j.jpowsour Citable link Terms o Use This article was downloaded rom Harvard University s DASH repository, and is made available under the terms and conditions applicable to Open Access Policy Articles, as set orth at nrs.harvard.edu/urn-3:hul.instrepos:dash.current.terms-ouse#oap

2 Variation o Stress with Charging Rate due to Strain-Rate Sensitivity o Silicon Electrodes o Li-Ion Batteries Matt Pharr, Zhigang Suo, and Joost J. Vlassak* School o Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA Abstract Silicon is a promising anode material or lithium-ion batteries due to its enormous theoretical energy density. Fracture during electrochemical cycling has limited the practical viability o silicon electrodes, but recent studies indicate that racture can be prevented by taking advantage o lithiation-induced plasticity. In this paper, we provide experimental insight into the nature o plasticity in amorphous Li x Si thin ilms. To do so, we vary the rate o lithiation o amorphous silicon thin ilms and simultaneously measure stresses. An increase in the rate o lithiation results in a corresponding increase in the low stress. These observations indicate that rate-sensitive plasticity occurs in a-li x Si electrodes at room temperature and at charging rates typically used in lithium-ion batteries. Using a simple mechanical model, we extract material parameters rom our experiments, inding a good it to a power law relationship between the plastic strain rate and the stress. These observations provide insight into the unusual ability o a-li x Si to low plastically, but racture in a brittle manner. Moreover, the results have direct ramiications concerning the rate-capabilities o silicon electrodes: aster charging rates (i.e., strain rates) result in larger stresses and hence larger driving orces or racture. Keywords: Lithium-ion batteries, silicon, plasticity, racture, strain rate sensitivity, creep *Corresponding author: vlassak@seas.harvard.edu Tel , Fax

3 1. Introduction Silicon has been identiied as a promising anode material or lithium-ion batteries due to its ability to host a large amount o lithium [1]. Lithiation o silicon, however, causes an enormous volumetric expansion o ~ %, which under constraint can result in large mechanical stresses and racture [2]. Fracture o the electrode leads to loss o active material and creates more surace area or solid-electrolyte interphase (SEI) growth, both o which signiicantly contribute to the ading o the capacity o the system [2-5]. Fortunately, this mechanical damage can be mitigated by nanostructuring the silicon anodes, as has been successully demonstrated in nanowires [6,7], thin ilms [8-12], nanoporous structures [13,14], and hollow nanoparticles [15,16]. Speciically, recent experiments and theories indicate that one can prevent racture by taking advantage o lithiation-induced plasticity [11,17-22]. A number o studies have examined plastic deormation in Li x Si [5,12,18,20-34]. Sethuraman et al. measured stresses during cycling o Li x Si electrodes, inding plastic low, which results in dissipation o energy comparable to that o polarization losses [18]. Zhao et al. suggested that plastic low in a-li x Si reduces the energy release rate (i.e., the crack driving orce), thus preventing racture in nano-sized (~100 nm) electrodes [20]. Hertzberg et al. used nanoindentation to measure the hardness o nanocrystalline Li x Si [23]. They ound a linear variation o the hardness with the volume raction o lithium, decreasing rom 5 GPa or pure nanocrystalline silicon to 1.5 GPa or the ully lithiated phase (Li 15 Si 4 ) [23]. Zhao et al. perormed irst-principles simulations o a-li x Si, attributing plastic low to the continuous breaking and re-orming o Si-Si bonds [24]. Likewise, Fan et al. perormed molecular dynamics simulations, inding lithiationinduced plastic sotening due to the decreasing raction o strong covalent Si-Si bonds and an increase in weak Li-Li bonds that acilitate plastic low [25]. At large concentrations o lithium, they ound that the high mobility o lithium acilitates bond breaking, switching, and re-orming in response to mechanical loading [25]. Beyond these studies, a number o questions remain in regards to the nature o plasticity in Li x Si. For instance, Pharr et al. measured the racture energy o a-li x Si silicon thin ilms, inding that it demonstrates a peculiar ability to both low plastically and racture in a brittle manner [5]. The authors did not, however, speculate as to the physics governing this curious combination o properties. Brassart and Suo have suggested that inelasticity in high-capacity lithium-ion batteries may occur by two processes: low and reaction [30]. The authors deine low as a process driven by deviatoric stress that preserves lithium concentration and volume, similar to plastic low in a metal. By comparison, the authors deine reaction as lithium insertion/removal: a process that changes the concentration and volume o the electrode [30]. One result o their theory is that lithium insertion (or removal) may enable low at a lower stress than that needed or low under pure mechanical loading. The applicability o this reactive 2

4 low theory to a-li x Si remains an open question. A study rom irst-principles calculations ound the lithiation reaction to markedly reduce the low stress o a-li x Si [27], while a molecular dynamics study ound no such eects [25]. There are no experimental studies aimed at investigating these eects. The purpose o this paper is to provide experimental insight into the nature o plasticity in a-li x Si. To do so, we vary the charging rate in amorphous silicon thin-ilm electrodes, while simultaneously measuring stresses. The magnitude o the low stress increases monotonically with the charging rate, indicating that rate-sensitive plasticity occurs at room temperature and at charging rates typical o lithium-ion batteries. These data it well to a power law relationship between the plastic strain rate and the stress. Additionally, our results indicate no evidence o the reactive-low eect in a-li x Si, as has been suggested in literature [27,30]. They do, however, provide insight into the unusual ability o a-li x Si to low plastically, while racturing in a brittle manner. 2. Experimental results Using the substrate curvature technique, we measure stresses in thin-ilm electrodes o amorphous silicon during electrochemical testing. Details can be ound in the experimental procedure section. As a brie summary, 100 nm ilms o amorphous silicon on a glass substrate are lithiated and delithiated galvanostatically at a C/8 rate during the irst cycle (8 hours to ully lithiate and 8 more to ully delithiate). During the second cycle, the lithiation rate is varied systematically to investigate the eect o the charging rate on the stresses that develop in the electrode. Figure 1 shows a typical sequence o the applied charging rate and the response in the measured voltage. During the second lithiation, an increase in the charging rate results in a decrease in the voltage, as is normally observed in Li x Si electrodes. We should note that the horizontal axis in the igure is constructed by integrating the current during the experiment to get the total charge. The amount o charge does not necessarily represent the concentration o lithium in silicon, as SEI ormation may consume lithium during the irst cycle. However, the main ocus o this paper is related to the stress measurements as a unction o charging rate (as shown in Figure 2). These data are obtained during the second cycle, which minimizes the inluence o the SEI on our measurements, as the majority o the SEI is ormed during the initial lithiation [11]. Figure 2 shows the stress measured in the ilm subject to the electrochemical cycling shown in Figure 1. Figure 2b is a zoomed-in view that ocuses on the second lithiation. The legend shows the charging rate during each segment as expressed in the C-rate convention. In this convention, the denominator indicates the number o hours to 3

5 theoretically ully lithiate the electrode. Figure 2b thus demonstrates that increasing the rate o lithiation, e.g., rom C/128 to C/8, results in a quick and sustained increase in the magnitude o the stress (the stress becomes more compressive). Likewise, when the charging rate increases, but by a smaller amount, e.g., rom C/16 to C/8, the stress increases in magnitude, but not as much as compared to, e.g., C/128 to C/8. In other words, the change in stress increases monotonically with charging rate aster charging results in larger stress. We believe that these observations are indicative o a material rate-eect, i.e., plastic deormation o a-li x Si is rate sensitive, even at room temperature. 3. A model o concurrent lithiation and rate-sensitive plasticity We will now outline a simple mechanical model, with the goal o extracting material parameters rom our experimental results. This model extends our previously developed models to account or rate-sensitive plasticity [5,20,21], and is similar to that developed by Bucci et al. [35]. Following these models [5,20,21,35], we take the deormation to consist o contributions rom elasticity, plasticity, and lithiation-induced swelling. The total true strain, ε ij, can then be written as where ε = ε + ε + ε, (1) L E P ij ij ij ij L ε ij represents the true lithiation-induced strain, E ε represents the true elastic ij P strain, and ε represents the true plastic strain. For a thin ilm on a thick substrate, the ij state o stress is equi-biaxial, σ11 = σ22 = σ, with all other components o the stress vanishing. Due to the constraint o the substrate, the total in-plane components o the strain vanish, ε11 = ε22 = ε = 0. Equation (1) becomes L E P ε + ε + ε = 0, (2) where the ε represent the in-plane components o the true strain. Following our previous work [5] and validated by experiments [5,36-38], we take the volume o the ilm, V, to be linear in the state o charge where 0 1 ( β ) V = V + s, (3) 0 V is the initial volume o the ilm, β is related to the atomic volumes ( Ω ) by ( ) β = Ω Ω / Ω Li15Si4 Si Si, and s is the state o charge, with a value o 0 representing pure 4

6 silicon and a value o 1 representing the ully lithiated state (assumed to be a-li 15 Si 4 with a capacity o 3579 mah/g) [36]. Obrovac et al. ound a 280% increase in the volume o silicon upon reaching the ully lithiated state o Li 15 Si 4 [36], i.e., β = 2.8. Using atomic orce microscopy, He et al. and Beaulieu et al. measured similar values in patterned amorphous silicon ilms [37,38]. Both groups ound that the volume increased linearly with lithium concentration [37,38]. Due to the amorphous nature o Li x Si, the lithiation-induced deormation is L L L L assumed to be isotropic, such that the stretch ratios are equal, λ1 = λ2 = λ3 = λ. Since the lithiation-induced volumetric swelling is large compared to volume changes rom 0 L L L elasticity (or plasticity), we set V / V λλλ. Combining with Equation (3) gives ( 1 ) 1/ 3 L λ βs = +. By deinition, the in-plane component o the true lithiation-induced strain L L is ε ln( λ ) =, such that The elastic strains are given by Hooke s law, L 1 ε = ln ( 1 + βs). (4) 3 E ε 1 ν = σ, (5) E where ν is Poisson s ratio o the ilm, E is the elastic modulus o the ilm, and σ is the in-plane component o the true (Cauchy) stress in the ilm. The in-plane component o the true plastic strain will be let in a generalized orm, P P ε = ln( λ ). (6) Combining Equations (2) and (4-6), and taking a time-derivative, leads to P 1 dλ d 1 ν β ds = σ + P λ dt dt E 31 ( + βs) dt. (7) In our experiments, the charging rate, ds/dt, is prescribed, and the stress is measured as a unction o time. Thus, with knowledge o the material properties o the ilm ( ν, E P 1 dλ, and β ), the plastic strain rate in our experiments,, can be calculated as a P λ dt unction o time. Figure 3 shows the experimental values o the terms rom Equation (7) subject to the electrochemical cycling conditions o Figure 1. The elastic component o 5

7 the strain rate is calculated using a representative value o the biaxial modulus o ( ν ) E /1 = 45 GPa, as reported in by Sethuraman et al. [39]. In their study, they ound this quantity to vary only slightly with lithium concentration over the range o concentrations examined in our studies; thus, in constructing Figure 3, we take the biaxial modulus as a constant. The lithiation-induced strain rate is calculated using the prescribed charging rate, ( ds / dt ), and a value o β = 2.8, as previously discussed. The plastic component o the strain rate is then computed using Equation (7). As can be seen rom Figure 3, during the majority o our experiments, the elastic component o the strain rate is small compared to the lithiation-induced strain rate, thereby eliminating the need to know the value o the biaxial modulus to calculate the plastic strain rate. In this limit, Equation (7) becomes P 1 dλ β ds = P λ dt 31 s dt. (8) ( + β ) Equation (8) has a straightorward physical interpretation: the plastic strain rate in the experiments is directly prescribed by the charging rate. By increasing the charging rate, we correspondingly impose an increased rate o plastic strain. Combining our stress measurements with Equation (8), we can obtain a relationship between the rate o plastic strain and the stress in the a-li x Si, thereby extracting material parameters. In particular, we it to our data to a typical viscoplastic power-law [22,40]: where A, m, and P 1 dλ σ σ Y = A P λ dt σy σ are itting parameters. In this equation, Y m, (9) σ can be interpreted as Y the yield stress o the material at a vanishing plastic strain rate. Equation (9) is applicable only when the stress exceeds σ Y. In the equation, σ represents the magnitude o the measured stress (a positive quantity), and the leading negative sign is a result o the compressive stress state during lithiation. One complicating actor in our experiments is that the yield stress varies with the state o charge, as can be seen in the irst cycle in Figure 2. To mitigate this complication, we have perormed the strain-rate experiment over a concentration range in which the yield stress changes slowly with lithium concentration (Figure 2). Upon changing the charging rate rom the nominal rate o C/8 to a dierent rate (e.g., C/16), the stress quickly progresses toward a new value (Figure 2b). To use these data, we rewrite Equation (9) in a convenient orm: 6

8 ( σ σc/8 ) + ( σc/8 σy) ( ) P 1 dλ = A P λ dt σc/8 σc/8 σy m, (10) where σ represents the stress at the nominal charging rate o C/8, and all o the C /8 stresses represent the magnitude o the measured stresses (positive quantities). We assume that ( σ σ ) C/8 is constant, independent o the lithium concentration over the Y range o concentrations considered in the experiments. Ater each change in the. We also measure the quantity σ C /8 charging rate, we measure the quantity ( ) σ σ C /8 just beore (or ater) each change in the charging rate. During the experiment, the charging rate, and hence the plastic strain rate (Equation 8) is prescribed as a unction o time. Thus, we obtain a set o data to which we can it Equation (10) using the quantities A, m, and ( σ σ ) C/8 as itting parameters. Y Figure 4 shows the best it o Equation (10) to the data rom the experiment corresponding to Figure 2. As is evident rom Figure 4, the data it well to the power-law orm. The results o the best its or three such experiments are shown in Table 1. The exponent m, has values in the range o For large values o m, the stress level is insensitive to the applied strain rate and is instead limited by the yield strength, σ Y. In our experiments, the measured values o m demonstrate a moderate level o strain-rate sensitivity. As a inal note, this experimental technique could be used to measure ratesensitive material properties as a unction o lithium concentration by repeating the outlined procedure at various states o charge. In our experiments, we subject the ilm to the same set o (nominal) charging rates twice (Figure 2b). There is little variation in the extracted material parameters rom each set o charging rates. It is o course possible that these properties could vary with the state o charge i we were to study a larger range o lithium concentrations; such work was not pursued in this paper. 4. Discussion Our results indicate that a-li x Si electrodes are rate sensitive at room temperature and that the rate eect is important at charging rates typically used in lithium-ion batteries. These observations have signiicant ramiications or the rate-capabilities o silicon anodes. In particular, due to the strain-rate sensitivity, aster charging will result in larger stresses, which may result in racture o the electrode. 7

9 We should irst mention that a recent work by Boles et al. has investigated creep deormation in lithiated silicon [41]. In their work, constant-orce creep tests were conducted on ully lithiated c-li 15 Si 4 nanowires. Their investigation was primarily perormed at stress levels below the apparent yield stress o the material. Interestingly, at these low stress levels, the authors ound a linear relationship between the strain rate and the stress, suggesting Newtonian viscous low o c-li 15 Si 4 [41]. The nanowires tested by Boles et al. are polycrystalline; thus, the grain boundaries may allow diusional transport as well as creep deormation [41]. Moreover, their testing o the crystalline phase allows or the possibility o creep deormation via dislocation motion. In the present work, we ocus on quantiying the relationship between the charging rate and the resulting stresses during electrochemical cycling o a-li x Si. We use charging rates typical o lithium-ion batteries and examine a range o lithium concentrations. We also limit the depth o discharge in this study to investigate amorphous electrodes, which are typically used in practice due to their improved cycling perormance [42]. Since the material is amorphous, there are no dislocations. Even so, we observe a signiicant rate eect. Due to the large strains associated with lithiation/delithiation, stresses in Li x Si electrodes readily reach the yield stress during cycling. At these large stresses, dierent rom Boles et al., we observe a non-linear relationship between the stress and the strain rate. Our results can answer a number o open questions in literature, as will now be discussed. Our observations o rate-sensitive plasticity in a-li x Si shed light on recent theories and experiments. For instance, Soni et al. ound racture only at large charging rates ( C /2) in 150 nm silicon thin ilms [33]. The authors attributed this observation to diusion limitations through the thickness o the ilm [33]. It is also possible that their observations are due to eects o rate-sensitive plasticity: or the range o charging rates used in our experiments (C/128 to C/2), the stresses varied by over 100 MPa. These variations in stress will signiicantly alter the driving orce or racture, as the energy release rate scales with the square o the stress. Thus, substantially larger crack-driving orces develop at larger charging rates, potentially leading to racture. As another example, Brassart and Suo have suggested that inelasticity in batteries may occur by two processes: low and reaction [30]. Flow changes the shape o the electrode while preserving volume and lithium concentration, and is driven by the deviatoric stress a process similar to plastic low in a metal. By contrast, reaction (i.e., lithium insertion/removal) changes both volume and lithium concentration. Reaction is driven by a combination o the mean stress and the chemical potential o lithium in the environment [30]. These two processes, low and reaction, are intimately coupled as they both involve the same physical processes: breaking and orming atomic bonds. As a result o this chemo-mechanical coupling, larger overpotentials (i.e., larger chemical driving orces) can result in low at a smaller stress [30]. In our experiments, 8

10 larger charging rates result in larger overpotentials. Thus, in the absence o any other rate eects, Brassart and Suo would predict a decrease in the magnitude o the low stress with increasing charging rate. In contrast, we see the exact opposite trend in our experiments larger charging rates result in an increase in the magnitude o the low stress. Thereore, as previously discussed, we believe that our results are indicative o a material rate eect: a-li x Si is strain-rate sensitive. Larger charging rates result in larger strain rates in the material, which in turn generate larger stresses. We should note, however, that our experiments do not entirely preclude the existence o the so-called reactive low eects discussed by Brassart and Suo. It is possible that these eects do indeed exist, but that they are too small relative to the strain-rate dependence o the material to be observed in our experiments. Additionally, Pharr et al. have suggested that a-li x Si demonstrates a peculiar ability to both low plastically and racture in a brittle manner [5]. The authors did not speculate as to the physics governing this curious combination o properties. However, when a crack propagates in a-li x Si, it moves relatively ast, resulting in large strain-rates near the crack tip. In this paper, we provide evidence or rate-sensitivity o a-li x Si: plastic low at larger strain-rates requires larger stresses. Consequently the strains associated with racture are mostly elastic, and the material ails in a brittle ashion, in agreement with ractographic observations and justiying the use o linear elastic racture mechanics as employed in Reerence 5. The discussion herein underscores the importance o determining the exact microscopic picture governing plastic low in a-li x Si. Some potential mechanisms have been investigated through computational atomistic simulations. For instance, Zhao, et al. ound that continuous bond breaking and re-orming assisted by Li insertion can accommodate large plastic deormation [24]. In another study, Zhao, et al. ound bond switching to occur at loosely packed ree-volume regions, leading to localized plastic deormation [34]. Such deormation is reminiscent o that ound in metallic glasses. It has been established that an applied stress can cause local atomic rearrangement in metallic glasses, resulting in macroscopic deormation. In particular, a low equation is usually implemented rom transition-state theory, in which the strain-rate scales with the stress as ε sinh( Ωσ / 2kT ), where Ω is the activation volume, k is the Boltzmann constant, and T is the temperature [43-45]. Over a particular range o stress, this scaling is quite similar to that o Equation 9, which was ound to describe our data well (Table 1). Although this similarity does not deinitively point to a particular microscopic mechanism, it suggests that deormation o a-li x Si shares some o the same eatures as metallic glasses. In another atomistic study, Huang and Zhu attributed plastic deormation during lithiation o silicon to a decrease in strong covalent Si-Si bonds and an increase in weak Li-Li bonds [26]. The high mobility o Li atoms acilitates eective bond switching to accommodate mechanical deormation [26]. Thus, it urther appears 9

11 that bond breaking and re-orming, as well as the mobility o lithium are important or plastic deormation o a-li x Si. These processes take time and thus engender ratedependent plasticity. Unortunately, due to computational limitations o atomistic simulations, the time scales associated with these processes are not well understood. Our current work emphasizes the importance o gaining a better understanding the dynamics o plastic low in a-li x Si, and thus warrants uture theoretical and computational studies. 5. Conclusions We have measured stresses in silicon thin ilms as a unction o the charging rate. Increasing the rate o lithiation resulted in a corresponding increase in the low stress. Our results indicate that rate-sensitive plasticity occurs in a-li x Si at room temperature and at charging rates typically used in lithium-ion batteries. To extract material parameters, we have developed a model o concurrent lithiation and ratesensitive plasticity. The data are ound to it well to an power-law relationship between the strain-rate and the stress, with a value o the stress exponent, m, in the range o , indicating a moderate level o strain-rate sensitivity. These results provide insight into the unusual ability o a-li x Si to low plastically but racture in a brittle manner. Moreover, the results have direct ramiications concerning the rate-capabilities o silicon anodes. In particular, aster charging rates result in larger stresses, which can lead to racture o the electrode. We hope that this work will provide guidance or the design o uture theoretical models that account or material rate eects. Likewise, we hope that this work will inspire uture computational studies aimed at understanding the dynamics o plastic low in a-li x Si. Acknowledgments We thank Frans Spaepen or a number o useul discussions pertaining to this work. This work was supported by the National Science Foundation through a grant on Lithium-ion Batteries (CMMI ). It was perormed in part at the Center or Nanoscale Systems, a member o the National Nanotechnology Inrastructure Network, which is supported by the National Science Foundation under NSF Award No. ECS , and at the Harvard University Materials Research Science and Engineering Center, which is supported by the National Science Foundation under Award No. DMR M.P. acknowledges support by the National Science Foundation Graduate Research Fellowship Program. 10

12 Experimental procedure Cover glass substrates with a thickness o 175 µm were cleaned with acetone and isopropanol and placed into a sputter deposition system (AJA Int. ATC 1800) with a base pressure o <10-8 Torr. All sputtering targets have a 50.8 mm diameter, and depositions were perormed at room temperature (22 C). The samples were plasmacleaned in Ar at 20 mtorr and an RF power o 24 W or 5 minutes. Next, 15 nm o Ti was sputtered onto the substrates using a pressure o 3 mtorr o Ar and a DC power o 100 W or 5 minutes. A 300 nm layer o Cu was then deposited on the Ti underlayer using a pressure o 5 mtorr o Ar and a DC power o 100 W or 15 minutes. The Cu ilm serves as current collector, while the Ti underlayer is used to improve the adhesion between the Cu ilm and the glass substrate. Finally, a 100 nm Si ilm was deposited on the Cu current collector using a pressure o 5 mtorr o Ar and a DC power o 100 W or 27 minutes. The thickness o the ilm was veriied using proilometry, and was ound to be 100 ± 10 nm. The working area o each silicon electrode is 8 mm by 30 mm. We have previously perormed x-ray diraction experiments to conirm that the silicon ilms are amorphous under these sputtering conditions [27]. Electrochemical experiments were conducted in a custom-abricated hermetic Telon electrochemical cell with a glass window. The cell was assembled in a threeelectrode coniguration in an argon-illed glovebox, which was maintained at < 1 ppm moisture. The sputtered silicon ilm was used as the working electrode, and Li oil was used as the reerence electrode and the counter electrode. A 1M solution o LiPF 6 in 4:3:3 (vol %) ethylene carbonate : dimethyl carbonate : diethyl carbonate with a vinylene carbonate additive was used as the electrolyte (MTI Corporation). Electrochemical measurements were perormed with a VersaSTAT 3 galvanostat rom Princeton Applied Research. During the irst cycle, the cell was tested galvanostatically at a current density o 23.6 µa/cm 2 (a C/8 rate assuming a capacity o 3579 mah/g [36]) between 0.8 and 0.05 V vs. Li/Li +. The lower cuto potential o 50 mv vs Li/Li + was employed to prevent crystallization o the a-li x Si electrodes [42]. Relatively thin ilms (100 nm) and an upper cuto potential o 0.8 V vs Li/Li + were employed to prevent racture during delithiation. During the second cycle, the cell was lithiated at a C/8 rate or one hour, ollowed by a number o segments with dierent charging rates. The duration o these segments was ixed such that the total capacity during each segment was 50 mah/g (1.4% o the total capacity o 3579 mah/g). The relatively slow charging rates used in these experiments were selected to allow enough time or diusive equilibrium through the ilms [11,46,47]. We use the data measured during the second cycle to minimize eects o SEI growth [11]. The stress in the ilm was measured by monitoring the curvature o the substrate in-situ during electrochemical testing. The average stress in the ilm was deduced rom the curvature via Stoney s equation [48,49]: 11

13 where σ is the average stress in the ilm, 2 Eh s s σ = σr + ΔK, (11) 6h 1 ( ν ) s E is the elastic modulus o the substrate, h s s is the thickness o the substrate, h is the thickness o the ilm, ν s is Poisson s ratio o the substrate, and Δ K is the change in the curvature o the substrate that results rom the stress in the ilm. σ is the residual stress in the ilm that develops during sputter r deposition and is determined rom substrate curvature measurements beore and ater deposition o the ilm. In our calculations, we have used values o E = 77 GPa and ν = 0.22 or the glass substrates. From Equation (11), it is clear that knowledge o s properties o the ilm other than ilm thickness is not required to evaluate the stress. As in our previous work [5], we take the thickness o the ilm, h, to depend linearly on the state o charge: where ( β ) h = h + s, (12) h is the initial ilm thickness, β is related to the atomic volumes ( Ω ) by ( ) β = Ω Ω / Ω Li15Si4 Si Si, and s is the state o charge, with a value o 0 representing pure silicon and a value o 1 representing the ully lithiated state (assumed to be a-li 15 Si 4 with a capacity o 3579 mah/g) [36]. Obrovac et al. ound a 280% increase in the volume o silicon upon reaching the ully lithiated state o Li 15 Si 4 [36], i.e., β = 2.8. The curvature o the substrate was monitored with a multi-beam optical sensor (MOS) rom k-space Associates. The MOS employs an array o parallel laser beams to measure the curvature o the substrate. The array o laser beams allows simultaneous illumination and detection, which in turn greatly reduces noise in the measurements caused by luid motion in the electrochemical cell or by ambient vibrations. The cell was also placed on an anti-vibration table during testing. The change in the curvature o the substrate was calculated rom the geometric relation d d cosα n d L n 0 Δ K = 0 2 a e s, (13) where d is the distance between two adjacent laser spots measured on the CCD camera, d 0 is the initial distance between the laser spots, α is the angle o relection o the laser beams, L is the distance between the electrochemical cell and the CCD camera, and n a and n e are the indices o reraction o air and the electrolyte, respectively. Since the laser beams pass through air, the electrolyte, and an optical 12

14 window, reraction o the beams at these corresponding interaces must be taken into account, which is the source o the quantity n / n. In the calculation o the stress, we have taken n e = 1.42 or the electrolyte [50] and n a = 1 or air. A combination o Equations (11) - (13) allows or calculation o the average stress in the a-li x Si ilms during electrochemical testing. A more detailed explanation o this technique is provided in Reerence 5. a e 13

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18 (a) 1000 Potential vs Li/Li+ (mv) (b) Potential vs Li/Li+ (mv) x in Li Si x 50 C/8 C/16 C/4 C/32 C/2 C/64 C/ x in Li x Si Fig. 1. Results o electrochemical cycling o a 100 nm Si thin-ilm electrode. (a) Measured potential as a unction o lithium concentration. (b) A zoomed-in view that ocuses on the second lithiation. The vertical dashed line indicates the point at which the set o charging rates is repeated. 17

19 (a) Stress (MPa) (b) x in Li x Si Stress (MPa) C/8 C/16 C/4 C/32 C/2 C/64 C/ x in Li x Si Fig. 2. (a) Stress measured in the 100 nm thin-ilm Si electrode subject to the electrochemical cycling conditions shown in Figure 1. (b) A zoomed-in view that ocuses on the second cycle. The vertical dashed line indicates the point at which the set o charging rates is repeated. A change in the charging rate results in a signiicant change in the stress. 18

20 Strain Rate (1/s) x x in Li Si x Fig. 3. Components o the strain rate in the 100 nm thin-ilm Si electrode subject to the electrochemical cycling conditions shown in Figure 1. The blue line indicates the prescribed lithiation-induced strain rate, ( ) the elastic component o the strain rate, ( ν ) β/ βs ds/ dt. The purple line indicates 1 / E dσ / dt. The orange line represents P P the plastic component o the strain rate, 1/ λ dλ / dt, as calculated rom Equation (7). 19

21 (d λp / dt) / λp σ - σ C/8 Fig. 4. Plastic strain rate (in units o 1/s) versus the change in measured stress relative to the C/8 rate (in units o MPa). The solid red circles represent the experimental data and the open black circles represent the phenomenological model in Equation (10). 20

22 A (1/s) m σ C/8 - σ Y (MPa) R Table 1: Results o the it o the experimental data to Equation (10) or three separate experiments. 21