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Article Cite This: J. Phys. Chem. C 2018, 122, 16561−16573

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Transformation Plasticity Provides Insights into Concurrent Phase Transformation and Stress Relaxation Observed during Electrochemical Li Alloying of Sn Thin Film Aditya Vemulapally,†,‡ Ravi Kali,† Tanmay K. Bhandakkar,*,§ and Amartya Mukhopadhyay*,† High Temperature and Energy Materials Laboratory, Department of Metallurgical Engineering and Materials Science, ‡Centre for Research in Nanotechnology & Science, and §Department of Mechanical Engineering, Indian Institute of Technology Bombay (IITB), Powai, Mumbai 400076, India

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ABSTRACT: Stress development during electrochemical alloying/insertion of “guest species” into electrode materials is known to considerably affect the performance and integrity of the concerned electrode. Monitoring of the average in-plane stress developments in-operando during electrochemical Li alloying/dealloying of electrodes that undergo nucleation/growth-induced phase transformations as a function of Li concentration shows that such stresses nearly cease to build-up (i.e., hit stress “plateaus”) during the phase transformations, in contrast to the nearly monotonic buildup of stresses during the single-phase (i.e., solid solution forming) regimes. To understand such observation and establish correlations between the composition, phase transformation, and stress development (in the case of lithiated Sn and Sn−Li intermetallic phases), comprehensive sets of mathematical modeling have been performed by considering lithiation as a diffusion process and the film to be elastic, perfectly plastic material. The observation of stress “plateaus” in the in-operando experiments has been satisfactorily explained and matched with the computed stress profiles based on the combined effects of Eigen strain during phase transformation and transformation-induced plasticity. Overall, it can now be concluded that the “stress flattening” observed in electrode materials during the electrochemical alloying process is primarily due to localized plasticity arising from the associated nucleation/growth-induced phase transformations.

1. INTRODUCTION The phenomenon of chemical or electrochemical alloying/ dealloying of materials often leads to considerable changes in dimensions with respect to the pristine state, which, in turn, leads to the development of stresses. Such stresses can be either extraneous due to constraining of the overall change in dimension externally or internal due to concentration gradient and/or phase transformation, even in the absence of any external constraining effect. For materials undergoing phase transformation(s) during the alloying/dealloying process, the internal stress development can be fairly discrete and often localized at the phase transformation front. One of the more direct implications of the above phenomena is on the performance of some metallic materials (such as Sn), which, despite forming a potential class of anode materials for Li-ion batteries due to the ability to reversibly alloy with Li, suffer from stress-induced degradation.1−4 From the performance viewpoint, such stress-induced degradations result in considerably reduced cycle life (or drastic fade in Listorage capacity with charge/discharge cycles), thus becoming the major bottleneck toward replacement of the presently used graphitic carbon based anode materials with the higher capacity and safer metallic anode materials for Li-ion batteries.1−4 © 2018 American Chemical Society

While experimentally monitoring the average stress developments in Sn film electrode in real time (i.e., in-operando) during electrochemical Li alloying/dealloying, a very interesting phenomenon of no notable increment in the average electrode stress during first-order phase transformations, as opposed to fairly monotonous build-ups of compressive stresses (during Li alloying) and reversal of the same (during Li dealloying), was observed by us, as also reported in our previous letter.5 While the monotonous stress build-ups and reversals in the single-phase regimes were easy to explain based on the (external) constraints provided by the substrate to the overall dimension changes during Li alloying and dealloying in the elastic regimes, the very different stress response during the phase transformation regimes was only tentatively attributed to localized stresses associated with nucleation/growth of the new phase leading to localized plasticity.5 Similar observations made during real-time monitoring of stress developments in thicker Sn film electrodes were also reported more recently by Chen et al.6 Additionally, observations of similar nature were reported by us also for other electrode materials, viz., Al7 and even LiCoO2.8 Amorphous Si-based electrode materials, which Received: April 29, 2018 Revised: June 19, 2018 Published: July 2, 2018 16561

DOI: 10.1021/acs.jpcc.8b04065 J. Phys. Chem. C 2018, 122, 16561−16573

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The Journal of Physical Chemistry C

2. EXPERIMENTAL DETAILS, RESULTS, AND DISCUSSIONS 2.1. In-Operando Stress Measurement During Electrochemical Li Alloying/Dealloying of Sn Film Electrode. The in-plane stress developments in SN films were monitored in real time (i.e., in-operando) during electrochemical Li alloying via the substrate curvature method. To facilitate the same, a quartz disc of 500 μm thickness and 1 in. diameter was used as the (stiff) substrate. Ti film of 20 nm thickness was used as the adhesive layer between the quartz substrate and Cu film of 100 nm thickness, with the Cu film acting as a current collector during electrochemical cycling. β-Sn film (as confirmed from X-ray diffraction studies; see Figure S1 in ESI) of 150 nm thickness, viz., the active material for electrochemical Li alloying, was further deposited on the Cu film. All the above-mentioned metallic films in this work were vapor deposited using the e-beam technique. A schematic representation of such simple continuous film-based electrode architecture is shown in Figure 1.

do not involve any possibility of phase transformations during Li alloying/dealloying, also show signatures of suppressed stress build-up beyond a certain Li content in similar in situ stress experiments. However, in that case viscous flow has been established as the cause via a set of experimental observations and associated mathematical modeling.9−16 In the realm of modeling, studies by Bower and coworkers14,16 have used finite inelastic deformation, coupling between stress, voltage, and thermal activation based free lattice sites to model stress relaxation, asymmetry in tensile, and compressive yield stress observed in thin films undergoing charging−discharging. Other notable modeling efforts involving film geometry can be found in the references.13,17,19,20 In the context of phase transformation, Song et al.18 have used the phase-field method to model concentration and stress evolution in phase-separating layered electrodes and showed that Li diffusion depends significantly on the phase separation profile and location of the phase boundary, in contrast to Li insertion without phase transformation, and is insensitive to the type of charging operation. As mentioned earlier, the recent in-operando experiments on thin-film electrodes have revealed that there exists a somewhat one to one correlation between the variation of the electrode potential and stress, i.e., the stress in the film also flattens out at almost the same time as the potential hits a plateau,5 the latter being a direct manifestation of the occurrence of firstorder phase transformation. The formation of the “stress plateau” in those works has been tentatively ascribed to localized plastic deformation at the phase transformation font due to “huge” Eigen strain build-up.5 Nevertheless, other possible causes that include modifications to the Li concentration profiles21 and variable changes in the elastic modulus1,17,22−24 during the occurrence of phase transformations may not be ruled out. However, as of now, evaluation of the impacts/influences of the aforementioned possible causes and understanding on these aspects, or in other words, establishment of correlations between the dynamic processes involving continuous changes in composition, Li concentration profiles, (intermittent) occurrences of phase transformations, localized stresses, and average (overall) stress responses (as observed in the in-operando experiments), is still lacking. In this context, a mathematical model is developed in the present work, which accounts for and develops correlation between elastic−plastic deformation, phase transformation, coupled stress, and electrochemical potential to calculate stresses in thin-film electrodes during electrochemical Li alloying. In addition to using concepts related to Eigen stress, for the first time, the present work introduces the concept of transformation-induced plasticity (i.e., TRIP)25−28 to the electrochemical alloying process. It must be mentioned here that TRIP has traditionally been used for explaining yielding in metallic materials undergoing phase transformation during heat treatment subjected to fairly low (or even nonexistent) average external stresses. In the present manuscript, the important experimental results (as under consideration here) have been mentioned and briefly discussed (along with the associated experimental details) in section 2. The mathematical model developed here to explain the experimental observations and the results/insights obtained based on the computation studies have been detailed in the subsequent sections 3 and 4.

Figure 1. Simple schematic representation of the electrode architecture (upside down; as assembled inside the cell) showing the quartz substrate (of ∼500 μm thickness), on which Cu current collector film (of thickness ∼100 nm) gets deposited with a thin adhesive interlayer of Ti (of thickness ∼20 nm). The active film for electrochemical Li alloying (viz., Sn of thickness ∼150 nm) gets deposited as a continuous layer on the Cu film.

The film electrode was then assembled in a custom-made electrochemical cell, which facilitates the in-operando monitoring of the in-plane stress development in the Sn film during electrochemical cycling. The custom-made electrochemical cell has a transparent quartz window on the top to allow optical access to the back side of the quartz substrate (facing upward inside the cell) and thereby facilitate the real-time monitoring of stresses (via monitoring of the substrate curvature) using a multibeam optical stress sensor (MOSS). A similar setup has been used in our previously published works related to realtime monitoring of stress developments in Li-ion battery electrode materials, including that with the Sn electrode.1,5,7,8,29−34 For the electrochemical lithiation/delithiation experiments, the cell, having the Sn-quartz film−substrate system as the working electrode, Li foil as counter-cumreference electrode, polymer membrane (from Celgard) as separator, and LP-30 (i.e., 1 M LiPF6 in 1:1 (v/v) EC:DMC) (from Merck) as the electrolyte, was assembled inside an Arfilled glovebox (Jacomex GP-campus) and electrochemically cycled using an Autolab 101 potentiostat/galvanostat. Galvanostatic cycling was performed at a fairly low current density equivalent to C/20. 16562


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Figure 2. Coupled cell voltage and nominal stress plot showing the development of in-plane stresses in Sn film (as monitored in situ) during electrochemical Li alloying for the 2nd discharge half cycle against Li.

phase coexistence while undergoing lithiation/delithitaion has also been observed by us for other electrode materials, such as Al7 and LiCoO2.8 In all those works, nucleation/growthmediated first-order phase transformation and the associated internal stresses leading to localized plastic deformation and/or stress relaxation have been tentatively proposed as the possible cause(s) behind such behavior. However, identifying the phenomena/mechanism primarily responsible for the observed behavior and developing a detailed understanding of the same have been lacking, which form the motivation for the comprehensive mathematical modeling, as detailed in the following sections (i.e., sections 3 and 4). 2.2. Hardness of Sn Thin Film, As Measured via “PicoIndentation”. As mentioned in the previous subsection, yielding is likely to be a very relevant and important phenomenon in the context of the observed stress response. Hence, for developing some insight into the yield strength (or flow stress) of the 150 nm thick Sn films (having grain size of ∼150 nm; see Figure S2 in ESI), depth-controlled “picoindentation” (using Pi 85L, Hystiron; inside Fei Quanta 3D FEG-SEM) was performed on the as-deposited films. In order to obtain reliable values with such thin films, the indentation depth was limited to 40 nm (i.e., t *

(15)

since it was found that eqs 11 and 15 yield results which are indistinguishable due to almost negligible concentration gradient also in the β phase (see Figure 4a) and the shrinking of the domain with time (xi > 0 as t increases). The initial and boundary conditions are Cβ(x , t ) = Cβ(t *), − Dβ

∂Cβ ∂x

= 0, Cβ(xi , t ) = Cβ e x=0

(16)

l o o fl o 3x 2 − l 2 2 o Dβ t Cβ = + − 2 m 2 2 o o Dβ o 6l π o l n ∞

=

where Cβ(t) is the concentration in the film at the onset of phase transformation at the film surface (x = S ) at the instant of start of phase transformation (i.e., at time t*). Equation 16b, i.e., the second of the three parts of eq 16, implies no transfer of Li ions across the film−substrate interface (i.e., at x = 0). Similar to the β phase, the evolution of concentration in the α phase during phase transformation has also been modeled as a diffusion process [see eqs 8−11]. In the context of Sn, the coupling parameter θα is usually smaller than θβ. Due to the fact that the concentration gradient δCα/δx is almost zero (see Figure 4a), the stress coupling can be ignored, and the transport process in the α phase can be described as a concentration-driven process through Fick’s law, viz.

(14)

This solution is used as an initial guess for Cβ while iteratively solving eqs 6 and 7 and (11−13. The values of ε1, κ, and h are calculated based on eqs 6 and 7 with the constitutive response chosen from eq 3 or 4 depending upon elastic or elastic− perfectly plastic deformation and eq 5 for the substrate. The estimated hydrostatic stress σh is substituted back into eq 11−13 to again determine Cβ numerically using “fsolve” command in the symbolic computation software MAPLEc.46 Again, the calculation of ε1, κ, h, and σh followed by Cβ is repeated until the values of ε1, κ, and h converge at a given time t. The criterion for convergence is that the difference in the values of κS in the successive iterations is less than or equal to 10−7. The next subsection deals with the formulation when the film undergoes phase change while being subjected to lithiation. 3b. Stress during Phase Transformation (βàα). The film undergoes phase transformation from the β to α phase when the potential of the electrode reaches the reduction potential of the α phase. Since the electrode potential depends upon the concentration at the surface, in the present work it is assumed that phase transformation (β → α) occurs when the concentration Cβ at the film surface (x = S ) at time t* reaches the maximum solubility of lithium in the β-phase denoted as Cβe (as per formulation, also used in refs 15 and 48). With continued lithiation, the phase-front which originated at the film surface at time t = t* progresses in the direction of the film−substrate interface. Figure 5 shows a snapshot of the film−substrate system with phase-front located at x = xi at a time t > t*. The concentration variation on either side of the phase front is different and is one of the factors governing the propagation of the phase front. In such a case, the stress in the film depends not only upon the concentration gradient in each phase but also on the thickness of the region corresponding to each phase. In the β phase, the expression for stress is the same

∂Cα D ∂ 2C = α 2 α , xi(t ) ≤ x ≤ S ; ∂t ∂x

t > t*

(17)

The initial condition (t = t*) and boundary condition at the film surface (x = S ) and phase front (x = xi(t)) for the α phase are given by Cα(x , t ) = Cαe , − Dα

∂Cα ∂x

= f , Cα(xi , t ) = Cαe x=S

(18)

Equation 18, analogous to eq 12, corresponds to the flux f applied at the film surface while at the phase front is the condition adopted in the present work during phase transformation (β → α).43,48,49 The concentration Cαe is the minimum solubility of Li in the α-phase that is in equilibrium with the maximum solubility of Li in β-phase Cβe.48 Estimating Cβe and Cαe independently would require a dedicated separate study, which is beyond the scope of the present work. Thus, recourse is taken to the experiments to establish Cαe and Cβe, a strategy adopted previously also.48 The values of Cαe and Cβe are selected such that the time taken for completion of the phase transformation (β →α) in the model matches with the respective time taken in Sn experiments (as in Figure 2 here and ref 5). It is worthwhile to mention that an excellent comparison between the present solution for the α-phase and the detailed coupled calculation not presented here also supports neglecting the coupling term. Mass balance across the interface xi(t) leads to 16566


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The Journal of Physical Chemistry C Dβ

∂Cβ ∂x

+ Dα x = xi−

∂Cα ∂x

= [Cβ e − Cαe] x = xi+

dxi , t > t* dt (19)

and offers an equation for the velocity of the interface dxi/dt (as also formulated in refs 43, 48, and 49). Conventionally the velocity of the planar interface is assumed to be based only on the flux emanating from one side, i.e., from the β phase, since the other side (i.e., α phase) does not have any diffusing species.41,50−53 On the contrary, in the present case, there exists incoming flux f in the α phase which causes the lithium to also diffuse from the α-phase, thus accounting for the second term in the left-hand side of eq 19. We would like to point out that Cβe and Cαe used in this work are different from the stress-free stoichiometric equilibrium values as has been emphasized by Bower et al.16 Equation 19 can be integrated to yield the interface position xi(t) subject to the initial condition xi(t*) = S . Equations 15−19 have been solved numerically to calculate the evolution of concentration Cα, Cβ, and xi with time. The time has been discretized and at every instant, and eq 19 has been used to estimate the position of interface based on the concentration in the previous time step. The new interface front position has been used to solve for the concentration variations in α and β phases based on eqs 15−18. The concentration solution has been used again to calculate the interface position based on eq 19. These steps have been repeated iteratively until the interface position and concentration in each phase converged at a given time step. Similar to a previous subsection, the criterion for convergence has been that the differences in the values of xi/S in the successive iterations become less than or equal to 10−7. The solution thus progresses in time until the phase front coincides with the film−substrate interface (xi = S ), thereby signaling completion of transformation from the β to α phase. In line with the β phase, the α phase has also been modeled as an elastic−perfectly plastic, isotropic, and homogeneous solid undergoing small deformations. As mentioned in section 2.2, yield strength of the lithiated Sn is assumed to be greater compared to that of pure Sn, and so the transformed phase α is likely to deform elastically throughout the lithiation process. The correctness of the elastic deformation assumption in the α phase will be confirmed in the Results section. Applying Hooke’s law, the normal component of stress in the α phase σfα in the x- or z-direction is given by Ω C i y σfα = Efαjjjε1 + κx − α α − εt zzz 3 k {

Figure 4. (a) Snapshot of the concentration profile C(x,t) along the film thickness until the completion of the first phase transformation, i.e., Sn → Li2Sn5 (0 ≤ t ≤ tcLi2Sn5). (b) Comparison of the temporal evolution of the in-plane average normal stress σavg(t) as predicted by the model [i.e., as per eq 25] with the experimental results (as presented in Figure 2). Here, Cb is the molar concentration of Li in Li2Sn5; DSn is the diffusion coefficient of Li in Sn; S is the thickness of the thin film; and Es is the elastic modulus of the substrate. Note that the number next to the concentration profile, as mentioned in (a), is the normalized time Dsnt/l2. The model prediction is based on three mechanisms, viz., purely elastic deformation, elastic−plastic deformation, and transformation plasticity invoked progressively with time and shown through dot, dash−dot, and solid lines, respectively. In particular, normalized times 0.0419, 0.203, and 0.474 shown in (a) correspond to the end of the elastic regime, end of the single-phase region comprised of elastic−plastic deformation and beginning of the first phase transformation (Sn → Li2Sn5), and completion of the first phase transformation, respectively.

(20)

where Efα, Ωα, and εt are the elastic modulus, partial molar volume of lithium in the α-phase, and transformation strain, respectively. The transformation strain εt induced in the α phase due to phase transformation has been quantified mainly with respect to two contributions, viz., Eigen strain41,43 and transformation plasticity.25−28,54,55 Eigen strain, here also referred to as volumetric mismatch strain, arises due to change in lattice volume in the two phases under consideration, viz., β and α. Thus, the Eigen strain is given as εt = εs =

Vα − Vβ * 3Vβ *

(21)

where Vα is the unit cell volume of the α phase and Vβ* corresponds to the unit cell volume of the β phase, having lithium ions inserted at the instant of phase change. In contrast 16567


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In the present work, the film−substrate system (Figure 1) is not subjected to any external force other than the constraint provided by the curvature [as already considered in eq 25], and so the stress in the system considered here is only due to the internal stress in the film. The internal stress in the film is the sum of the residual stress due to deposition and diffusioninduced stress developed in the thin film. The magnitude of the former is much smaller than the latter, and hence diffusioninduced stress is the internal stress used for the calculation of transformation plasticity-based strain. Thus the average stress in the film σavg from eq 25 at the instant of beginning of phase transformation (t = t*) is considered equivalent to the stress applicable in the model of transformation plasticity. Thus, the inelastic strain εtp induced due to the transformation plasticity, as originally proposed by Leblond et al.,25 and modified for the present work is ε εtp = −3 s Sz(ln z − 1) Yβ (26)

to the literature,26 where the expression for εt is based on the unit cell volume of the β phase (Vβ), the present work uses Vβ*, which is given by Vβ * = Vβ + C(xi , t )Ωβ

(22)

The additional term in the right-hand side of eq 22 corresponds to the volume of the Li ions diffused into the active material until the beginning of phase transformation at time t = t*. The choice of Vβ* instead of Vβ is mainly due to the following two reasons: 1. First, in true sense, it is the β phase in the presence of diffused Li which undergoes phase change as opposed to the pure β phase. 2. Second, the transformation plasticity strain,25−28 as presented a little later, assumes a constant volume for the transforming phase; while in the present work, the volume of the transforming β phase undergoes significant volume change due to Li diffusion. In the situation where part of the β phase gets yielded before the phase transformation gets initiated and the newer phase α has greater yield strength as compared to the parent phase, the plastic strain εpfβ(x,t) = (ε1(t*) + κ(t*)x − ΩβCβ(x,t*)/3 + Yβ/ Efβ) (xi(t) ≤ h(t*) ≤ x ≤ S ) incurred in the β phase region at the instant of phase transformation (t = t*) needs to be accounted for in eq 20 while evaluating the stress σfα(x,t) at the same point, but as part of the transformed material α. Enforcing mechanical equilibrium through force balance along the z direction and moment balance along the y direction leads to 0

fb =

∫−s σsdx + ∫0 0

mb =

xi

∫−s σsxdx + ∫0

σfβ dx +

∫x

l

σfα dx

(23)

i

xi

σfβx dx +

where S = 2σavg/3 is the deviatoric part of the average diffusion-induced stress in the film and z(t) = (1 − xi/S ) is the proportion of the β-phase in the film, respectively. This model is valid for the case when the applied stress is much smaller compared to the yield stress of the phase having the lower yield strength,25 i.e., when S/Yβ < 0.5. Zwigl and Dunand54,55 have extended the model of Leblond and co-workers25 for situations where the stresses are comparable to the yield strength of any one of the phases, and based on their numerical model the value for transformation plasticity-based strain εtp is found through the solution of the transcendental equation given below

∫x

δ=

l

σfαx dx

i

(24)

Using eqs 3 or 4 depending upon whether the β phase does yield or not, along with eqs 5 and 20 and the concentration profiles Cα and Cβ, eqs 23 and 24 can be solved for the curvature κ and uniform strain ε1. The average value of the in-plane normal component of stress in the thin film is given by σavg

(∫ =

0

h

l

xi

l

(27)

where δ = S/Yβ and γ = εtp/2εt. For a given value of δ, eq 27 is solved using the in-built function “Newtons Method” in MAPLEc46 to find γ and consequently εtp. The initial guess for the function “Newtons Method” is obtained using eq 26. Depending upon the value of S/Yβ, the value of εtp calculated from either eq 26 or 27 is then appended to the right-hand side of eq 20 to find out stress in the α phase.

)

σfβ dx + ∫ σyβ dx + ∫ σfα dx x h i

1 1 1 ijj 3γ 1 1 yzz + + − − z jj 4 6γ 2 2γ k 4 6 9γ z{ ÄÅ ÉÑ ÅÅÅ (3γ + 3 2γ + 2)2 ÑÑÑ ÑÑ lnÅÅÅ Ñ 2 ÅÅÅÇ 9γ − 6γ + 4 ÑÑÑÖ

(25)

4. RESULTS AND DISCUSSION CORRELATING THE EXPERIMENTAL OBSERVATIONS AND MODELING STUDIES In this section, the mathematical formulation, as detailed in the previous section, has been applied to the experimental results obtained via in-operando monitoring of the in-plane average stress development during electrochemical Li alloying in Sn film, until the end of the second-phase transformation, viz., lithiated Sn → Li2Sn5 and Li2Sn5 → LiSn (see Figure 2 and section 2.1). The material properties and the requisite parameters, as used in the mathematical model, can be found in Table 1. Initially at time t = 0, the film has been assumed to be free of Li, and both the film and the substrate have been assumed to be stress free. Figure 4a and b shows the best possible comparison of the temporal evolution of the average in-plane normal stress σavg(t) with the experimental result (as in Figure 2) and the concentration profile C(x,t) in the film at

In cases where the newly formed α phase has a higher yield strength (Yα > Yβ) and the stress in either or both the phases is within the elastic limit, the flattening of the average stress during phase transformation of the active material, as observed in the experiments (see section 2.1, Figure 2, and also ref 5), cannot be fully captured through stresses predicted based on eq 20. Accordingly, the present work proposes the mechanism of transformation-induced plasticity (or TRIP)25−28,54−56 to be operational in such situations, as above. Transformation plasticity was originally developed by Leblond and coworkers25 to explain the anomalous plastic behavior during phase transformation in steels undergoing heat treatment when the stress in the system is much lower than the yield stress of the original and the transforming phase. The stress in the system can be due to the applied load and the internal stresses. 16568


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perfectly plastic deformation in film. It can be clearly seen from Figure 4b that plastic deformation lowers the slope of the average stress−time curve compared to the initial elastic part and in fact bounds its value through yield stress YSn. Once the concentration of the film surface becomes equal to Cβe at time t = t*Sn, as justified in detail in the preceding section, the phase transformation from the β(Sn) → α(Li2Sn5) phase is assumed to be initiated. At the instant of first phase transformation t = t*Sn, the film is not completely yielded, and the plastic deformation front has reached until x = 0. The value of Cαe1 is chosen to be equal to Cb, and the value of Cβe1 is chosen to be 0.82Cb such that the time taken for first phase transformation tcLi2Sn5 matches with the experimentally noted value of 6750 s, a procedure analogous to the work of Zhang et al.48 The corresponding concentration and average stress evaluation are based on eqs 15−27 and are shown through solid lines in Figure 4a and b. During the phase transformation of β(Sn) → α(Li2Sn5), since the diffusivity of Li in Li2Sn5 is 6 orders of magnitude greater than that of Li in Sn (see Table 1), the concentration gradient is almost zero, which is confirmed by the flat concentration profiles seen in Figure 4a. Despite the significant value of the coupling parameter θβ = 19.4416 ≫ 1, the very high diffusivity renders the coupling between stress and concentration immaterial. Hence, during the course of the first phase transformation the concentration solution is obtained based on eqs 15−19 independently of stress. Subsequently, eqs 20−27 are used to calculate the stress shown in Figure 4b. It must be mentioned here that the coupled calculation during the first phase transformation, as not shown here, yielded results similar to Figure 4 based on uncoupled formulation. The importance of consideration of plastic deformation, phase-transformation-induced Eigen strain, and transformation-induced plasticity (TRIP) in the model is evident through excellent qualitative match between the prediction of the model and the experimentally obtained variation in average stress in the film with time as shown in Figure 4b. In the first phase transformation regime, had the TRIP not been used, the average stress variation predicted by the model would be closer in terms of magnitude but would fail to capture the slope seen in the experiment (as in Figure 2). In fact, the stress evaluation based solely on phasetransformation-induced Eigen strain leads to a flatter stress profile. Once the first phase transformation gets completed, lithiation continues in Li2Sn5 film as shown in Figure 5a. Owing to the very high diffusivity of Li in Li2Sn5 (see Table 1), the concentration of Li is homogeneous throughout the film. In the absence of any concentration gradient and higher yield strength compared to Sn (Table 1), Li2Sn5 deforms elastically, and the corresponding average stress in the film shown in Figure 5b by dots has been calculated using eqs 1− 7 and 14). The values of ELi2Sn5 and ΩLi2Sn5, as used and mentioned in Table 1, are slightly less than the values reported in the literature for the reasons similar to the ones mentioned for the case of pure Sn. Although the model qualitatively captures the time variation of average stress during lithiation in Li2Sn5, there is considerable difference in the exact value of average stress prediction between the model and the experiments, as seen in Figure 5b. One possible reason for such deviation could be delamination of some portion of the electrode−current collector interface, leading to reduction of the amount of active material actually available for lithiation, and consequently lesser value of average film stress. In the context of

different time instances as predicted by the model during lithiation until t = tcLi2Sn5 (where tcLi2Sn5 is the time signaling the completion of the first phase transformation from β (i.e., Sn) to α (i.e., Li2Sn5) in the film), respectively. The average stress, concentration, time, and position (x) in Figure 4 are normalized by Es, Cb, S )2/Dβ, and S ), respectively. The result, on the basis of the model, as shown in Figure 4b until the onset of the first phase transformation, is based on elastic−perfectly plastic deformation and coupled concentration−stress formulation in the film through eqs 1−13, while the part involving phase transformation uses eqs 15−27. The value of the coupling parameter θβ = 15.577 during lithiation in the β phase until the start of the first phase transformation justifies the coupling of stress and diffusion adopted during the β → α transformation stage. The interdependent stress and concentration evolution eqs 3, 4, and 11−13 have been solved numerically in an iterative manner with a nondimensional time step ΔtDβSn/S2 = 10−3 and nondimensional spatial grid size Δx/S ) = 10−3 to generate Figure 4. The chosen time step and grid size ensured accuracy and convergence of the concentration and stress solution, and it was found that any further reduction in their values did not yield any significant improvement and rather increased only the computational time. In line with the assumption of constant elastic modulus used in the present work and coupled with the observation that Sn may become less stiff with an increase in Li concentration,23 a reduced value of elastic modulus Eβ instead of the value corresponding to that of pure Sn is used and noted in Table 1. Note that the selected value of Eβ is also in accordance with the elastic modulus of the newer phase Li2Sn5 (as per ref 23). Similar to the case of the elastic modulus, the partial molar volume of Li may also change with its concentration57,58 because Li present in the electrode may try to repel the additional Li coming into the electrode.42 So, as compared to the value of 9.0 × 10−6 mol/m3 based on stoichiometry, a reduced value of partial molar volume of Li in the β-phase Ωβ is used for our calculations, and the same has been mentioned in Table 1. The chosen value of Ωβ allows for the best possible fit of average stress, as predicted by the model, with the experimental results (see Figure 4b), and is also consistent with the volume of Li per unit mole of Sn of the transforming phases. The chosen value of Ωβ allows for the best possible fit of average stress, as predicted by the model, with the experimental results (see Figure 4b) and is consistent with the volume of Li per unit mole of Sn of the transforming phases. The distinct change in the slope of the average stress with respect to time in Figure 4b is reflective of the change in the deformation mechanism in Sn film. At the start, with progress in Li-alloying, the concentration builds up in the film, as shown by dotted lines in Figure 4a, and correspondingly a compressive elastic stress develops in the film, as also shown by dotted lines in Figure 4b. Despite the coupled stress− concentration calculation, a significant gradient exists in Li concentration C(x,t) in the film owing to the very low diffusivity of Li in Sn, as noted in Table 1. With progress in time the film undergoes yielding, starting at the film surface (i.e., x = S ) and progressing toward the film−substrate interface (x = 0). The corresponding solution for concentration and stress, as involved in eqs 4−13, is shown by dash−dotted lines in Figures 4a and b, respectively. Note that the coupling between stress and concentration persisted during the elastic− 16569


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and the underlying substrate/layer becomes energetically favorable when σf = {(2Ef Γi)/h}1/2

(28)

where Ef is the plane strain modulus (∼57 GPa for Sn and ∼53 GPa for Li2Sn523); Γi is the interfacial resistance or resistance toward separation at the film/substrate interface (taken to be ∼1 J m−2 for the Sn/Cu interface as per ref 60); and h is the film thickness (here ∼150 nm). Based on the above eq 28, delamination is expected when the biaxial stress in the film (i.e., σf) is at least ∼0.9 GPa (taking the upper bound value of Ef, viz., 57 GPa). It may be recalled here that the experimentally measured biaxial stress (as measured inoperando during electrochemical lithiation of Sn film) is ∼1.3 GPa (compressive) at the beginning of the second voltage plateau associated with the Li2Sn5 → Sn phase transformation (see Figure 2 here and also ref 5). Hence, even the experimental data support the possibility of initiation of buckling-induced delamination of the active film under consideration here by the beginning of the second voltage plateau. In fact, earlier we reported observations concerning near complete delamination of similar Sn films upon electrochemical lithiation/delithiation for 35 cycles.32 Furthermore, formations of pores and voids during lithiation of Sn have also been reported via in situ X-ray tomography experiments.61,62 Initiation/occurrence of all the above phenomena is expected to lower the actual stress level in the Sn film compared to what can be captured in the present model, for which considerations of the occurrences of delamination and void/pore formation lie beyond the scope. The lithiation continues in Li2Sn5 film until the surface concentration equals the equilibrium concentration Cβe2. Beyond this time, the second phase transformation, i.e., from Li2Sn5 (β2 phase) to LiSn (α2 phase), is initiated. The value of molar concentration of Li in LiSn has been calculated using the molar volume of LiSn and number of moles of Li in LiSn, as shown in Table 1. The value of Cβ2 is unknown and found iteratively. A value for Cβ2 is chosen, and eqs 15−19 are solved for concentration in the alpha and beta phases. The β2/α2 interface position is numerically solved, and the time taken for phase transformation in the model (viz., tcLiSn) is noted. The process gets repeated until the value of time taken (i.e., tcLiSn), as predicted by the model, matches with the experimental value (as in Figure 2 and ref 5). The value in the final step is the equilibrium concentration Cβ2. During the second phase transformation (Li2Sn5 → LiSn), stress in the transformed (α) region is calculated using eqs 20−24 and 27, while the stress in the nontransformed (β) region, i.e., the region between the current collector and interface position, is calculated using eqs 4 and 14 and is shown by a dash-dotted line in Figure 5b. Here also the model is able to capture the slope of the average stress variation with time as in the experiments, but there is difference in the magnitude predicted by the model and that measured in the experiment. Here also the model is able to capture the slope of the average stress variation with time as in the experiments, but there is a difference in the magnitude predicted by the model and that measured in the experiment. The primary reason for the difference in this case is the error accrued during the elastic stage (as shown through a solid line in Figure 5b). The same error is inherited at the start of the Li2Sn5 → LiSn phase transformation stage (as shown through a dash-dotted line in Figure 5b), which further grows slightly during the phase transformation stage. It may be further noted

Figure 5. (a) Snapshot of the concentration profile C(x,t) along the film thickness, up to the completion of the second phase transformation Li2Sn5 → LiSn (t ≤ tcLiSn). (b) Comparison of the temporal evolution of the in-plane average normal stress σavg(t) predicted by the model [as per eq 25] with the experimental results (see Figure 2). Here, DSn is the diffusion coefficient of Li in Sn; S is the thickness of thin film; Es is the elastic modulus of substrate; and Cb is the molar concentration of Li in Li2Sn5. Note that the number next to the concentration profile mentioned in (a) is the normalized time DSnt/S2 . The model prediction post time t > tcLi2Sn5 is based on purely elastic deformation followed by transformation plasticity in Li2Sn5 shown through dashed and dash-dotted lines, respectively. In particular, normalized times 0.475, 0.603, and 0.848 correspond to the start of the lithiation of the second phase Li2Sn5 (tcLi2Sn5), end of the single-phase region for Li2Sn5, beginning of the second phase transformation (Li2Sn5 → LiSn) (t*Li2Sn5), and completion of the second phase transformation (tcLiSn), respectively.

the proposed delamination, basic thin-film mechanics59 predicts that when a film is subjected to a certain biaxial stress (σf) delamination between the film (here Sn or Li2Sn5) 16570


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applicable also to metallic materials undergoing phase transformations while being subjected to electrochemical insertion of alloying elements. Overall, the results obtained in this work are encouraging toward application of the as-developed model to other insertion/alloying-based electrode materials, leading to better understanding of the correlation between composition− concentration profiles−phase assemblage/transformation− stress development; which also being one-dimensional in nature (allowing better flexibility toward the variation of parameters) and computationally efficient, additionally renders itself quite attractive (especially, with respect to the possibility of a direct comparison with the experiments). Hence, in the context of one of the immediate practical implications, this has immense significance toward the selection and usage of electrodes based on materials which undergo dimensional changes, as well as first-order phase transformations, during alkali metal cation insertion/removal. Nevertheless, it is believed here that the quantitative difference between the experimental and model prediction in this case, which increases beyond the first phase transformation (Sn → Li2Sn5), is most likely due to film delamination occurring during the experiment. Accordingly, the model can be made even more realistic through consideration of large deformation, two-dimensional geometry, surface roughness, finite thickness of phase front to account for the interfacial energy, as well as independent estimates for the occurrences of phase nucleation and propagation through appropriate forms of Gibb’s free energy.

that, even though the experiment has been carried out for a complete discharging (or lithiation) half cycle, the modeling of stress evolution in the film is terminated at the end of the second phase transformation because of the increasing difference between experimental and model prediction caused most likely by extraneous factors that are not possible to account for in the model.

5. CONCLUSIONS AND OUTLOOK In-operando monitoring of stress developments during electrochemical Li alloying led to interesting observations concerning flattening of stress response (i.e., no notable change in average in-plane stress) only during nucleation/growth-induced phase transformations between lithiated Sn and (subsequently formed) Sn−Li intermetallic compounds. Such flattening of the stress response during phase transformations was in contrast to the fairly monotonous build-up of compressive stress during the single-phase lithiation regimes (i.e., during formation of solid solution with Li). To better understand these aspects and establish a correlation between the dynamic processes involving changes in composition (i.e., Li content or concentration profiles), occurrence/nonoccurrence of phase transformations, and associated stress developments during Li alloying, we have modeled here the time evolution of the average in-plane stresses during electrochemical lithiation of Sn film electrode during the solid solution (i.e., Sn → lithiated Sn) and the first two phase transformation (i.e., lithiated Sn → Li2Sn5 and Li2Sn5 → LiSn) regimes, while being subjected to constant incoming Li flux. The lithiation has been modeled as a diffusion process, and the film has been considered to be elastic−perfectly plastic material, undergoing small deformation. The resulting diffusion-induced stresses caused due to elastic and elastic−plastic deformation have been accounted for, and whenever required coupling between concentration and stress has been taken into consideration. The notable feature of stress constancy observed during phase transformation in the in-operando experiments has been satisfactorily explained based on the involvement of transformation induced plasticity (TRIP) and Eigen strain. We have shown that it is the combined effect of Eigen strain during phase transformation and TRIP in the model which leads to computed stress profiles that match very well, at least in a qualitative sense, with the experimentally obtained average stress variations during Li-alloying. Overall, the simulations also support our hypothesis that the “stress flattening” observed in metallic electrode materials is primarily due to localized plasticity during phase transformation. In this context, it is important to note here that, as confirmed from the in-operando measurements and detailed analysis, plastic deformation does take place despite the nanoscaled confining dimensions in thin-film form and the ultrafine grain size (∼150 nm) having led to increment in the yield strength of the concerned Sn film to the order of almost ∼1 GPa; as indicated by “pico-indentation” measurements conducted here. This in itself indicates the severity of the stresses that get generated during the nucleation-growthinduced phase transformations that take place during electrochemical alkali metal-ion insertion/removal to/from the electrode material. On a different note, this also indicates that the theory of TRIP, as developed erstwhile primarily for explaining the localized plasticity observed during phase transformations induced by heat treatments of metallic materials at otherwise fairly low average stress values, is

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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.8b04065. X-ray diffraction pattern and scanning electron microscope image obtained with the as-deposited Sn film (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Phone: (+91)-22-25767537. *E-mail: [email protected]. Phone: (+91)22-25767612. ORCID

Amartya Mukhopadhyay: 0000-0002-9368-0935 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS AM would like to thank SERB-DST (vide: EMR/2016/ 000760), GOI, for financial assistance, and CEN, SAIF, ICCF (IIT Bombay), and DST-FIST for enabling the usage of some of the experimental/analytical facilities. The help received from the “Oriental Imaging Microscopy (OIM) & Texture Lab” at the Department of Metallurgical Engineering and Materials Science towards conducting the pico-indentation measurements is duly acknowledged.

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REFERENCES

(1) Mukhopadhyay, A.; Sheldon, B. W. Deformation and stress in electrode materials for Li-ion batteries. Prog. Mater. Sci. 2014, 63, 58− 116. 16571


Article

The Journal of Physical Chemistry C (2) Besenhard, J. O.; Yang, J.; Winter, M. Will advanced lithiumalloy anodes have a chance in lithium-ion batteries? J. Power Sources 1997, 68, 87−90. (3) Winter, M.; Besenhard, J. O. Electrochemical lithiation of tin and tin-based intermetallics and composites. Electrochim. Acta 1999, 45, 31−50. (4) Kasavajjula, U.; Wang, C.; Appleby, A. J. Nano- and bulk-siliconbased insertion anodes for lithium-ion secondary cells. J. Power Sources 2007, 163, 1003−1039. (5) Mukhopadhyay, A.; Kali, R.; Badjate, S.; Tokranov, A.; Sheldon, B. W. Plastic deformation associated with phase transformations during lithiation/delithiation of Sn. Scr. Mater. 2014, 92, 47−50. (6) Chen, C. H.; Chason, E.; Guduru, P. R. Measurements of the Phase and Stress Evolution during Initial Lithiation of Sn Electrodes. J. Electrochem. Soc. 2017, 164, A574−A579. (7) Kali, R.; Badjate, S.; Mukhopadhyay, A. Effects of residual stress on overpotentials and mechanical integrity during electrochemical Lialloying of Al film electrodes. J. Appl. Electrochem. 2017, 47, 479−486. (8) Malav, V.; Jangid, M. K.; Hait, I.; Mukhopadhyay, A. In Situ Monitoring of Stress Developments and Mechanical Integrity during Galvanostatic Cycling of LiCoO2 Thin Films. ECS Electrochem. Lett. 2015, 4, A148−A150. (9) Soni, S. K.; Sheldon, B. W.; Xiao, X.; Tokranov, A. Thickness effects on the lithiation of amorphous silicon thin films. Scr. Mater. 2011, 64, 307−310. (10) Sethuraman, V. A.; Chon, M. J.; Shimshak, M.; Srinivasan, V.; Guduru, P. R. In situ measurements of stress evolution in silicon thin films during electrochemical lithiation and delithiation. J. Power Sources 2010, 195, 5062−5066. (11) Soni, S. K.; Sheldon, B. W.; Xiao, X.; Verbrugge, M. W.; Ahn, D.; Haftbaradaran, H.; Gao, H. Stress Mitigation during the Lithiation of Patterned Amorphous Si Islands. J. Electrochem. Soc. 2012, 159, A38−A43. (12) Soni, S. K.; Sheldon, B. W.; Xiao, X.; Bower, A. F.; Verbrugge, M. W. Diffusion Mediated Lithiation Stresses in Si Thin Film Electrodes. J. Electrochem. Soc. 2012, 159, A1520−A1527. (13) Bucci, G.; Nadimpalli, S. P. V.; Sethuraman, V. A.; Bower, A. F.; Guduru, P. R. Measurement and modeling of the mechanical and electrochemical response of amorphous Si thin film electrodes during cyclic lithiation. J. Mech. Phys. Solids 2014, 62, 276−294. (14) Bower, A. F.; Guduru, P. R.; Sethuraman, V. A. A finite strain model of stress, diffusion, plastic flow, and electrochemical reactions in a lithium-ion half-cell. J. Mech. Phys. Solids 2011, 59, 804−828. (15) Bower, A. F.; Guduru, P. R. A simple finite element model of diffusion, finite deformation, plasticity and fracture in lithium ion insertion electrode materials. Modell. Simul. Mater. Sci. Eng. 2012, 20, 45004. (16) Bower, A. F.; Chason, E.; Guduru, P. R.; Sheldon, B. W. A continuum model of deformation, transport and irreversible changes in atomic structure in amorphous lithium−silicon electrodes. Acta Mater. 2015, 98, 229−241. (17) He, Y. L.; Hu, H. J.; Song, Y. C.; Guo, Z. S.; Liu, C.; Zhang, J. Q. Effects of concentration-dependent elastic modulus on the diffusion of lithium ions and diffusion induced stress in layered battery electrodes. J. Power Sources 2014, 248, 517−523. (18) Song, Y. C.; Li, Z. Z.; Soh, A. K.; Zhang, J. Q. Diffusion of lithium ions and diffusion-induced stresses in a phase separating electrode under galvanostatic and potentiostatic operations: Phase field simulations. Mech. Mater. 2015, 91, 363−371. (19) Ji, L.; Guo, Z.; Du, S.; Chen, L. Stress induced by diffusion curvature and reversible electrochemical reaction in bilayer lithiumion battery electrode plates. Int. J. Mech. Sci. 2017, 134, 599−609. (20) Chen, B.; Zhou, J.; Zhu, J.; Liu, Z. Diffusion induced stress and the distribution of dislocations in a nanostructured thin film electrode during lithiation. RSC Adv. 2014, 4, 64216−64224. (21) Zhang, L.; Song, Y.; He, L.; Ni, Y. Variations of boundary reaction rate and particle size on the diffusion-induced stress in a phase separating electrode. J. Appl. Phys. 2014, 116, 143506.

(22) Qi, Y.; Guo, H.; Hector, L. G., Jr; Timmons, A. Threefold Increase in the Young’s Modulus of Graphite Negative Electrode during Lithium Intercalation. J. Electrochem. Soc. 2010, 157, A558− A566. (23) Stournara, M. E.; Guduru, P. R.; Shenoy, V. B. Elastic behavior of crystalline Li-Sn phases with increasing Li concentration. J. Power Sources 2012, 208, 165−169. (24) Sen, R.; Johari, P. Understanding the Lithiation of the Sn Anode for High-Performance Li-Ion Batteries with Exploration of Novel Li−Sn Compounds at Ambient and Moderately High Pressure. ACS Appl. Mater. Interfaces 2017, 9, 40197−40206. (25) Leblond, J. B.; Devaux, J.; Devaux, J. C. Mathematical modelling of transformation plasticity in steels I: Case of ideal-plastic phases. Int. J. Plast. 1989, 5, 551−572. (26) Taleb, L.; Sidoroff, F. A micromechanical modeling of the Greenwood-Johnson mechanism in transformation induced plasticity. Int. J. Plast. 2003, 19, 1821−1842. (27) Volkov, A. E.; Likhachev, V. A.; Nikolaev, P. I. Theory of transformation plasticity. Sov. Phys. J. 1984, 27, 223−225. (28) Cho, Y. G.; Kim, J. Y.; Cho, H. H.; Cha, P.-R.; Suh, D.-W.; Lee, J. K.; Han, H. N. Analysis of Transformation Plasticity in Steel Using a Finite Element Method Coupled with a Phase Field Model. PLoS One 2012, 7, e35987. (29) Mukhopadhyay, A.; Tokranov, A.; Guo, F.; Xiao, X.; Hurt, R. H.; Sheldon, B. W. Engineering of graphene layer orientation to attain high rate capability and anisotropic properties. Adv. Funct. Mater. 2013, 23, 2397−2404. (30) Mukhopadhyay, A.; Tokranov, A.; Xiao, X.; Sheldon, B. W. Stress development due to surface processes in graphite electrodes for Li-ion batteries: A first report. Electrochim. Acta 2012, 66, 28−37. (31) Sonia, F. J.; Ananthoju, B.; Jangid, M. K.; Kali, R.; Aslam, M.; Mukhopadhyay, A. Insight into the mechanical integrity of few-layers graphene upon lithiation/delithiation via in situ monitoring of stress development. Carbon 2015, 88, 206−214. (32) Kali, R.; Krishnan, Y.; Mukhopadhyay, A. Effects of phase assemblage and microstructure-type for Sn/intermetallic “composite” films on stress developments and cyclic stability upon lithiation/ delithiation. Scr. Mater. 2017, 130, 105−109. (33) Mukhopadhyay, A.; Tokranov, A.; Sena, K.; Xiao, X.; Sheldon, B. W. Thin film graphite electrodes with low stress generation during Li-intercalation. Carbon 2011, 49, 2742−2749. (34) Jangid, M. K.; Sonia, F. J.; Kali, R.; Ananthoju, B.; Mukhopadhyay, A. Insights into the effects of multi-layered graphene as buffer/interlayer for a-Si during lithiation/delithiation. Carbon 2017, 111, 602−616. (35) Seo, D. M.; Nguyen, C. C.; Young, B. T.; Heskett, D. R.; Woicik, J. C.; Lucht, B. L. Characterizing Solid Electrolyte Interphase on Sn Anode in Lithium Ion Battery. J. Electrochem. Soc. 2015, 162, A7091−A7095. (36) Sahin, O.; Uzun, O.; Kolemen, U.; Ucar, N. Vickers microindentation hardness studies of β-Sn single crystals. Mater. Charact. 2007, 58, 197−204. (37) Deng, X.; Chawla, N.; Chawla, K. K.; Koopman, M. Deformation behavior of (Cu, Ag)-Sn intermetallics by nanoindentation. Acta Mater. 2004, 52, 4291−4303. (38) Mukhopadhyay, A.; Todd, R. I. Effect of Yttria Doping on the Microstructure and Mechanical Properties of Al2O3-FeAl2O4 Nanocomposites Developed via Solid State Precipitation. J. Eur. Ceram. Soc. 2010, 30, 2905−2915. (39) Huang, J. Y.; Zhong, L.; Wang, C. M.; Sullivan, J.; Xu, W.; Zhang, L. Q.; Mao, S. X.; Hudak, N. S.; Liu, X. H.; Subrmanian, A.; Fan, H.; Qi, L.; Kushima, A.; Li, J. In Situ Observation of the Electrochemical Lithiation of a Single SnO2 Nanowire Electrode. Science 2010, 330, 1515−1520. (40) Hsueh, C. H. Modeling of elastic deformation of multilayers due to residual stresses and external bending. J. Appl. Phys. 2002, 91, 9652−9656. (41) Deshpande, R.; Cheng, Y.-T.; Verbrugge, M. W.; Timmons, A. Diffusion Induced Stresses and Strain Energy in a Phase-Trans16572


Article

The Journal of Physical Chemistry C forming Spherical Electrode Particle. J. Electrochem. Soc. 2011, 158, A718. (42) Bohn, E.; Eckl, T.; Kamlah, M.; McMeeking, R. A Model for Lithium Diffusion and Stress Generation in an Intercalation Storage Particle with Phase Change. J. Electrochem. Soc. 2013, 160, A1638− A1652. (43) Park, J.; Lu, W.; Sastry, A. M. Numerical Simulation of Stress Evolution in Lithium Manganese Dioxide Particles due to Coupled Phase Transition and Intercalation. J. Electrochem. Soc. 2011, 158, A201−A206. (44) Bower, A. F.; Guduru, P. R.; Chason, E. Analytical solutions for composition and stress in spherical elastic−plastic lithium-ion electrode particles containing a propagating phase boundary. Int. J. Solids Struct. 2015, 69−70, 328−342. (45) Huttin, M.; Kamlah, M. Phase-field modeling of stress generation in electrode particles of lithium ion batteries. Appl. Phys. Lett. 2012, 101, 133902. (46) Maple (18); Maplesoft, a division of Waterloo Maple Inc.: Waterloo, Ontario. (47) Park, G. S. The mathematics of diffusion. Polymer 1975, 16, 414. (48) Zhang, Q.; White, R. E. Moving Boundary Model for the Discharge of a LiCoO2 Electrode. J. Electrochem. Soc. 2007, 154, A587−A596. (49) Renganathan, S.; Sikha, G.; Santhanagopalan, S.; White, R. E. Theoretical Analysis of Stresses in a Lithium Ion Cell. J. Electrochem. Soc. 2010, 157, A155. (50) Subramanian, V. R.; Ploehn, H. J.; White, R. E. Shrinking Core Model for the Discharge of a Metal Hydride Electrode. J. Electrochem. Soc. 2000, 147, 2868. (51) Bard, A. J.; Faulkner, L. R. Electrochemical methods: fundamentals and applications; John Wiley & Sons, Inc.: New York, 2001. (52) Christensen, J.; Newman, J. A Mathematical Model of Stress Generation and Fracture in Lithium Manganese Oxide. J. Electrochem. Soc. 2006, 153, A1019−A1030. (53) Porter, D. A.; Easterling, K. Phase transformations in metals and alloys; Chapman & Hall: London, NY, 1992. (54) Zwigl, P.; Dunand, D. C. A non-linear model for internal stress superplasticity. Acta Mater. 1997, 45, 5285−5294. (55) Zwigl, P.; Dunand, D. C. A numerical model of transformation superplasticity for iron. Mater. Sci. Eng., A 1999, 262, 166−172. (56) Leblond, J. B.; Mottet, G.; Devaux, J. C. A theoretical and numerical approach to the plastic behaviour of steels during phase transformations-II. Study of classical plasticity for ideal-plastic phases. J. Mech. Phys. Solids 1986, 34, 411−432. (57) Malavé, V.; Berger, J. R.; Martin, P. A. ConcentrationDependent Chemical Expansion in Lithium-Ion Battery Cathode Particles. J. Appl. Mech. 2014, 81, 91005. (58) Courtney, I. A.; Dahn, J. R. Electrochemical and In Situ X-Ray Diffraction Studies of the Reaction of Lithium with Tin Oxide Composites. J. Electrochem. Soc. 1997, 144 (6), 2045−2052. (59) Freund, L. B.; Suresh, S. Thin Film Materials; Cambridge University Press: Cambridge, 2003. (60) Choi, W. K.; Lee, H. M. Effect of Soldering and Aging Time on Interfacial Microstructure and Growth of lntermetallic Compounds between Sn-3.5Ag Solder Alloy and Cu Substrate. J. Electron. Mater. 2000, 29 (10), 1207−1213. (61) Gonzalez, J.; Antartis, D. A.; Chasiotis, I.; Dillon, S. J.; Lambros, J. In situ X-ray micro-CT characterization of chemomechanical relaxations during Sn lithiation. J. Power Sources 2018, 381, 181−189. (62) Gonzalez, J.; Sun, K.; Huang, M.; Dillon, S.; Chasiotis, I.; Lambros, J. X-ray microtomography characterization of Sn particle evolution during lithiation/delithiation in lithium ion batteries. J. Power Sources 2015, 285, 205−209.

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