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C: Energy Conversion and Storage; Energy and Charge Transport

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 J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b04065 • Publication Date (Web): 02 Jul 2018 Downloaded from http://pubs.acs.org on July 6, 2018

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The Journal of Physical Chemistry

Transformation Plasticity provides Insights into Concurrent Phase Transformation and Stress Relaxation Observed during Electrochemical Li-Alloying of Sn Thin Film Aditya Vemulapallya,b, Ravi Kalia, Tanmay K. Bhandakkarc*, Amartya Mukhopadhyaya** a High Temperature and Energy Materials Laboratory, Department of Metallurgical Engineering and Materials Science, b Centre for Research in Nanotechnology & Science, c Department of Mechanical Engineering, Indian Institute of Technology Bombay (IITB), Powai, Mumbai 400076, India 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/de-alloying of electrodes that undergo nucleation-growth induced phase transformations as a function of Li concentration show that such stresses nearly cease to build-up (i.e., hit stress 'plateaus') during the phase transformations, in contrast to the nearly monotonic build-up 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 effect 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 electrochemical alloying process is primarily due to localized plasticity arising from the associated nucleation/growth induced phase transformations.

__________________________________________________________________________________ Corresponding authors: *e-mail id: [email protected], phone (o): (+91)-22-25767537; **e-mail id: [email protected]; phone (o): +912225767612

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1. Introduction The phenomenon of chemical or electrochemical alloying/de-alloying 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/de-alloying process, the internal stress development can be fairly discrete and often localized at the phase transformation font. 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 degradation1–4. From the performance view point, such stress induced degradations result in considerably reduced cycle life (or drastic fade in Li-storage capacity with charge/discharge cycles); thus becoming the major bottleneck towards replacement of the presently used graphitic carbon based anode materials with the higher capacity and safer metallic anode materials for Liion batteries1–4. While experimentally monitoring the average stress developments in Sn film electrode in real time (i.e., in-operando) during electrochemical Li alloying/de-alloying, a very interesting phenomena of no notable increment in the average electrode stress during 1st order phase transformations, as opposed to fairly monotonous build-ups of compressive stresses (during Li alloying) and reversal of the same (during Li de-alloying) was observed by us; as also reported in our previously letter5. 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

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overall dimension changes during Li alloying and de-alloying in the elastic regimes, the very different stress response during the phase transformation regimes were only tentatively attributed to localized stresses associated with nucleation/growth of the new phase leading to localized plasticity5. 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 LiCoO28. Amorphous Si based electrode materials, which do not involve any possibility of phase transformations during Li alloying/de-alloying, 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 modeling9-16. In the realm of modeling, studies by Bower and co-workers14,16 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 references13,17,19,20. In the context of phase-transformation, Song et al.18 have used 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 phase boundary, in contrast to Li insertion sans 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 plateau5; the latter being a direct manifestation of the occurrence of 1st order 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-up5. Nevertheless, other possible causes that include modifications to the Li concentration profiles[21] 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 inoperando 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 electrode 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 non-existent) 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.


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2. Experimental details, Results and Discussions 2.1. In-operando stress measurement during electrochemical Li alloying/de-alloying 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 substrate curvature method. To facilitate the same, quartz disc of 500 µm thickness and 1 inch 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 current collector during electrochemical cycling. β-Sn film (as confirmed from x-ray diffraction studies; see Fig. 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 e-beam technique. Schematic representation of such simple continuous film based electrode architecture is shown in Fig. 1. 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 upwards inside the cell) and thereby facilitate the real-time monitoring of stresses (via monitoring of the substrate curvature) using multi-beam optical stress sensor (MOSS). Similar set-up has been used in our previously published works related to real-time monitoring of stress developments in Li-ion battery electrode materials, including that with Sn electrode1,5,7,8,29-34. For the electrochemical lithiation/delithiation experiments, the cell, having the Sn-quartz film-substrate system as the working electrode, Li foil as counter-cum-reference 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 5 ACS Paragon Plus Environment


the electrolyte, was assembled inside Ar-filled glove-box (Jacomex GP-campus) and electrochemically cycled using Autolab 101 potentiostat/galvanostat. Galvanostatic cycling was performed at a fairly low current density equivalent to C/20. The as-recorded potential profile and corresponding (in-operando) stress profile in the form of a non-dimensional parameter (i.e., normalized by the bi-axial modulus of the substrate, viz., 86 GPa) have been presented in Fig. 2. The potential profile (and associated stress profile) corresponds to the 2nd lithiation half cycle; with the 1st cycle not been used here to avoid additional complications due to irreversible surface reactions, including SEI layer formation, which happens predominantly in the first electrochemical cycle5,30,31,35. The terminology ‘nominal stress’ implies that the stress-thickness (as directly obtained from the substrate curvature) has been normalized by the initial film thickness (and not the instantaneous film thickness). More details on this terminology is available in our previous publications1,5,31-34. It must be noted here that it is the behavior and not the exact stress value(s) that is important for the presently reported work. Prior to lithiation, the active film corresponds to that of single phase β-Sn. During the electrochemical Li-alloying process (i.e., discharge against Li counter/reference electrode), an initial solid solution (i.e., single phase) regime corresponding to lithiation of β-Sn is associated with monotonous decrease in the cell voltage (as shown by the black curve in Fig. 2), following the initial steeper voltage drop due to resistance. The monotonous change in voltage during the solid solution regime is followed by three distinct voltage plateaus, which correspond to the first order phase transformations that take place during Li-alloying, viz., from lithiated Sn → Li2Sn5 (at ~0.69 V; against Li/Li+), Li2Sn5 → LiSn (at ~0.56 V) and LiSn → Li7Sn3 (at ~0.45 V); as indicated in Fig. 2.


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With respect to the stress development, as monitored in-operando (shown by the brown curve in Fig. 2), even though monotonous compressive stress build-up is seen during the single phase lithiation regimes (in either Sn or the subsequently formed Sn-Li intermetallic phases), the stress profile becomes relatively flatter during the two-phase co-existence regimes. Such behavior was reported for the first time by us in the case of Sn film electrodes in our previously published letter5 and was subsequently observed also by Chen et al.6. Furthermore, flattening of stress response during two-phase co-existence while undergoing lithiation/delithitaion has also been observed by us for other electrode materials, such as Al7 and LiCoO28. In all those works, nucleation/growth mediated 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 detailed understanding of the same has been lacking, which form the motivation for the comprehensive mathematical modelling, as detailed in the following sections (i.e., sections 3 and 4). 2.2. Hardness of Sn thin film, as measured via ‘pico-indentation’ As mentioned in the previous sub-section, 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 Fig. S2† in ESI), depth controlled ‘pico-indentation’ (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., < 1/3rd of the film thickness), with maximum normal load reaching ~100 µN. It is to be noted that the confined dimension and fine grain size are expected to render such thin films considerably harder than the



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bulk counterparts. However, comprehensive information in this regard is not yet available for the case of pure Sn. In our case, the average hardness value obtained via the pico-indentation measurements is ~2.3 (± 0.8) GPa, which is greater than that for bulk Sn by an order of magnitude36,37. Assuming that the yield strength is approximately 1/3rd of the hardness38, the yield strength of the un-lithiated (and accordingly, unstrained) Sn film is believed to lie somewhere in between ~0.7-0.8 GPa. It must be mentioned here that, despite thorough cleaning via ultrasonication in acetone and controlled etching using HF prior to the pico-indentation experiments, partial influence of the presence of surface (oxide) layer(s) or impurities on the measured hardness values may not be totally ruled out. Nevertheless, in the context of the necessary parameter here (viz., the yield strength during lithiation), previously reported39 in-situ transmission electron microscopy observations during lithiation of SnO2 nanowires (first getting converted to Sn upon lithiation) indicated a very high dislocation density (viz., ρ of ~1017 m-2) at the lithiation front, which would correspond to increment of flow stress in such strained condition by ~1 GPa (as per αGbρ1/2, where ‘G’ or shear modulus of Sn is ~20 GPa, ‘b’ or Burgers vector is ~0.4 nm and α is ~0.5; as also mentioned in our previously published letter).

In

analogy with the above

observation/inference and in light of the experimentally measured hardness value of the unlithiated/unstrained Sn film in our case (which correspond to yield strength of ~0.7 GPa), the yield strength of partly lithiated Sn (i.e., strained, as well as solution hardened) can be assumed to be close to ~1 GPa (or at least of similar order). Accordingly, this value (of 1 GPa) has been used as the yield strength of lithiated Sn for the computation studies (see YSn in Table 1).


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3. The Mathematical Model and Parameters The schematic representation of the model adopted for capturing the experimentally measured stress evolution during electrochemical lithiation (see Fig. 2) has been presented in Fig. 3. The model comprises of an active thin film electrode (made of β-Sn) of thickness ℓ, perfectly bonded to the substrate (made of quartz) of thickness s. The origin of the co-ordinate system is placed at the film-substrate interface with the x-axis lying along the thickness such that the surface of the film subjected to flux f corresponds to x = ℓ. The adhesive layer (thickness ~20 nm) and the current collector layer (thickness ~100 nm) present in between the Sn film electrode and quartz substrate in the experiments have not been considered in the model (Fig. 1) due to their relatively smaller thickness compared to the thickness of the quartz substrate (thickness ~500 µm). Due to the very large aspect ratio of the thin film [viz., diameter (~2.54 cm) to thickness (~150 nm)], used in the experiment, Li-Flux coming from the side walls of the thin film has been neglected here, with the stress and concentration fields being assumed to be one-dimensional and varying only in the thickness direction (i.e., along x only). The substrate blocks the Li ions and hence do not take part in the process of lithiation. The film and the substrate have been assumed to undergo small quasi-static deformation during lithiation. The film material has been assumed to be linear elastic, perfectly plastic, with the substrate being assumed to be linear elastic solid. Due to influx of Li ions during lithiation, the film expands freely in the thickness (x) direction but is constrained in y-z direction due to perfect bonding with substrate, leading to a equi-biaxial state of stress (i.e., σy = σz ≠ 0); henceforth referred to as diffusion induced stress (DIS). In the present work, the evolution of DIS during Li-alloying has been modeled till the completion of the second phase transformation, i.e., till the formation of LiSn at ~0.56 V (against Li/Li+) (refer to Fig. 1 and section 2.1); with the modelling of the



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second plateau also serving as a confirmation regarding the theories/procedures adopted and understanding developed. In the following paragraphs, the formulation for computing the DIS during each of the stages mentioned above is given. 3a. Calculation of DIS in a single phase (β) film during lithiation The initial phase of the film has been denoted as "β" phase. The film-substrate system shown in Figs. 1 and 3 is a multilayer system, and due to its thinness it is assumed to extend and bend during lithiation40. The total normal strain (ε) in the direction perpendicular to the film thickness has been hence taken as40, ,

(1)

where ε1 and κ are the uniform strain component and uniform curvature of the film-substrate system, respectively. The total normal strain in the film comprises of strain due to elastic and/or plastic deformation and strain induced during lithiation40. The latter has been taken as ΩβCβ/3 by invoking analogy with thermal strain, where thermal expansion coefficient gets substituted by one-third of the atomic volume of β phase Ωβ and temperature gets replaced by concentration Cβ of the β phase22,41–45. The 1/3 in the lithiation strain is owing to the assumption of isotropic volume change due to insertion of Li in β phase. Thus in case the film deforms elastically during lithiation, the total strain (ε) in the film is given as, ε ε

, 0 x l ,

(2)

where is the elastic strain in the film and combination of equations (1, 2) and Hooke’s law

leads to the normal stress

in the film in the x or z direction being,

!

" , 0 # ,

(3)

where Efβ is the bi-axial elastic modulus of the β-phase. The film will deform plastically if the yield criterion is satisfied, i.e., Von-Mises stress σVM = {3(σij-σkkδij/3) (σij-σkkδij/3)/2}1/2 = |σfβ| 10 ACS Paragon Plus Environment

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equals the yield strength Y. The plastic deformation, if at all it occurs will start at the film surface (x = ℓ) and depending upon the applied flux f and the lithiation duration would proceed towards the film-substrate interface (x = 0). The stress distribution in the film in case of elastic plastic deformation is, $ &

!

"

0 ℎ ℎ #

',

(4)

where the elastic-plastic interface in the film at x = h is obtained based on the satisfaction of the yield criterion through equation (3) equaling the yield stress Y. In case the film becomes fully plastic, the stress in the film is equal to Y. It must be mentioned here that, similar to the case of materials like Si1,4 and Graphite17,22, the elastic modulus of Sn and its lithiated phase may also vary with Li-ion concentration23,24. However, for the sake of simplicity, the present work assumes a constant value for Young’s modulus Ef in a given phase of Sn during lithiation. The assumed value is chosen in such a way that it eventually captures the effect of change in elastic modulus of Sn. Since quartz substrate is assumed to undergo elastic deformation throughout the charging process, normal stress in the substrate (σs) in the x or z direction is given as, ( ( ) *,

(5)

where Es is the bi-axial modulus of the quartz substrate. In the absence of any external force acting on the film-substrate system in Fig. 1, the force and moment balance in z and y directions, respectively, implies40; /

(6)

(7)

+, - ( . - . 0(

/

/

1, - ( . - . 0(

/

Given the variation in Li concentration Cβ(x,t) at time t and depending upon the nature of deformation in the film (i.e., elastic or elastic-plastic), equation (3) or (4), along with (5), is 11 ACS Paragon Plus Environment


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substituted in equations (6) and (7) to provide two simultaneous equations for the unknowns ε1 and κ. They are then solved using the symbolic software MAPLEc 46. The solution for ε1 and κ in terms of cβ is too long and hence not presented here. The movement of Li ions in Sn electrode (as depicted in Figs. 1 and 3) under the action of applied flux f at x = ℓ has been modeled as a transport process governed by the chemical potential as given by43, μ μ/ 34#5678 9 : ;

(8)

where R = 8.3144598 J/(mol K) is the universal gas constant, T is the temperature in K and

; 7< 9⁄3 2 ⁄3 is the hydrostatic stress. Following Nernst-Einstein equation,

flux J is given as42–44, @

A 8 Bμ , 34 B

(9)

where Dβ is the diffusivity of Li ion in Sn. Mass conservation of Li ions gives16,44, B8 B@ 0, BC B

)10*

Combining equations (8-10) leads to the diffusion equation16,42; B8 B B8 : 8 B; A E F BC B B 34 B

(11)

Initially the film is assumed to be free of Li ions and the surface x = ℓ is subjected to flux f while the film-substrate interface blocks passage of Li ions. Thus the initial and boundary conditions for equation (11) are; 8 ), 0* 0,

A E

B8 : 8 B; FG 0, B 34 B 1, the concentration and stress solution are coupled

and solved simultaneously. The solution of equation (13) with θβ = 0 for the boundary conditions from Eq (12), ignoring the effect of hydrostatic stress, in the flux is as follows 47; `

[ X \ ]\ ^ K \

)1*X 0J +# A C 3 T # T 2 8 W Z S T A # 6# T VT YT XH

YV _5P a #

(14)

This solution is used as an initial guess for Cβ while iteratively solving equations (6-7) and (1113). The values of ε1, κ and h are calculated based on equations (6) and (7) with the constitutive

response chosen from equation (3) or (4) depending upon elastic or elastic - perfectly plastic deformation and equation (5) for the substrate. The estimated hydrostatic stress σh is substituted

back in to equations (11) to (13) to again determine Cβ numerically using "fsolve" command in the symbolic software MAPLEc

46

. Again, the calculation of ε1, κ, h and σh followed by Cβ is

repeated till the values of ε1, κ, h converge at a given time t. The criterion for convergence is that

the difference in the values of κℓ in the successive iterations is less than or equal to 10-7. The 13 ACS Paragon Plus Environment


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next sub-section deals with the formulation when the film undergoes phase change while being subjected to lithiation. 3b. Stress during phase transformation (β > α) The film undergoes phase transformation from β to α phase when the potential of the electrode reaches the reduction potential of α phase. Since the electrode potential depends upon the concentration at the surface, in the present work it is assumed that phase transformation (βα) occurs when the concentration Cβ at the film surface (x=ℓ) at time t* reaches the maximum

solubility of lithium in the β-phase denoted as Cβe (as per formulation, also used in Refs.15,48. With continued lithiation, the phase-front which originated at the film surface at time t = t*,

progresses in the direction of film-substrate interface. Fig. 5 shows 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 same as equations (3, 4) from the previous sub-section except that the domain at time t is now modified to 0 ≤ x ≤ xi (t). The concentration evolution in the β phase, although governed by equation (11), is approximated by Fick’s law during phase transformation stage (i.e., β → α) as, B8 A B T 8 , BC B T

0 n )C*,

C > C∗

(15)

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


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8 ), t* 8 )C ∗ *,

A

B8 G 0, B C ∗

(17)

The initial condition (t = t*) and boundary condition at the film surface (x = ℓ) and phase front (x = xi (t)) for the α phase are given by; 8r ), t* 8r ,

Ar

B8r +, t B

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