#  Film strains enhance the reversible cycling of intercalation electrodes

Delin Zhang  Aerospace and Mechanical Engineering, University of Southern
California, Los Angeles, CA 90089 Jay Sheth  School of Engineering, Brown
University, Providence, RI 02912  Brian W. Sheldon  School of Engineering,
Brown University, Providence, RI 02912  Ananya Renuka Balakrishna

###### Abstract

A key cause of chemo-mechanical degradation in battery electrodes is that they
undergo abrupt phase transformation during the charging/discharging cycle.
This phase transformation is accompanied by lattice misfit strains that
nucleate microcracks, induce fracture and, in extreme cases, amorphize the
intercalation electrode. In this work, we propose a strategy to prevent the
chemo-mechanical degradation of intercalation electrodes: we show that by
engineering suitable film strains we can regulate the phase transformations in
thin-film intercalation electrodes and circumvent the large volume changes. We
test this strategy using a combination of theory and experiment: we first
analytically derive the effect of film strain on the electrochemical response
of a thin-film intercalation electrode and next apply our analytical model to
a representative example ($\mathrm{Li_{x}V_{2}O_{5}}$ with multiple phase
transformations). We then test our theoretical predictions experimentally.
Specifically, we electrochemically cycle thin-film $\mathrm{V_{2}O_{5}}$
electrodes with different film strains and measure their structure, voltage,
and stress responses. Our findings show that tensile film strains lower the
voltage for phase transformations in thin-film $\mathrm{V_{2}O_{5}}$
electrodes and facilitate their reversible cycling across a wider voltage
window without chemo-mechanical degradation. These results suggest that film
strain engineering is an alternative approach to preventing chemo-mechanical
degradation in intercalation electrodes. Beyond thin-film electrodes, our
findings from this study are applicable to the study of stress-induced phase
transformations in particle-based electrodes and the thin surface layers
forming on cathode particles.
Keywords: Intercalation electrodes; Thin films; Phase transformations; Chemo-
mechanical degradation; Stress evolution;

##  1 Introduction

Lithium-ion batteries (LIBs) are promising candidates for sustainable energy
storage, although their use in high-energy-density applications, such as
powering electric vehicles and aircrafts, is still a challenge. One of the key
limitations for battery performance is the structural degradation of
electrodes with repeated cycling, ultimately resulting in capacity loss (Radin
et al., 2017; Lewis et al., 2019; Christensen et al., 2018; Mukhopadhyay and
Sheldon, 2014). When charging/discharging a battery, most intercalation
cathode materials undergo volume changes of $\sim 10-15\%$ (Xiang et al.,
2017). These are particularly problematic when they are associated with abrupt
lattice transformations that induce misfit strains (Poluektov et al., 2019).
Upon repeated cycling, the corresponding stresses can nucleate microcracks and
in some cases amorphize the materials (Chen et al., 2006; Shpigel et al.,
2019; Zhang et al., 2018). These mechanisms severely limit the reversible
cycling of a variety of electrodes and contribute to the structural
degradation of batteries (De Jesus et al., 2018; Zhang et al., 2019).

In the past, researchers have developed several approaches to limit the
structural degradation of intercalation materials. For example, operating a
vanadium pentaoxide ($\mathrm{V_{2}O_{5}}$) electrode in a smaller voltage
window 2.5–4V can prevent the large volume changes at $\gamma-\omega$
transformation (Shimizu et al., 1993; Baddour-Hadjean et al., 2012;
Whittingham, 1976). In other intercalation electrodes like
$\mathrm{FePO_{4}}$, phase transformation is controlled by regulating the Li-
diffusion kinetics (Woodford et al., 2012; Kao et al., 2010; Tang et al.,
2009; Bai et al., 2011) and/or by fabricating electrodes as nanoparticles (Liu
et al., 2011; Yue and Liang, 2017; Wang et al., 2006). These experiments have
resulted in a solid-solution type of microstructure that does not contain
phase boundaries and thus avoids abrupt changes in elastic strains. While
these strategies avoid large transformation strains in intercalation
electrodes and have improved their lifespans, they do so at the cost of the
material’s energy storage capacity and performance (Radin et al., 2017;
Mukhopadhyay and Sheldon, 2014).

Alternatively, we may look into other materials systems, such as
semiconductors, ferroelectrics, and dichalcogenides, in which phase
transformations are regulated by film strains (Sun et al., 2017; Roytburd et
al., 2015; Ota et al., 2016; Graz et al., 2009). In thin-film ferroelectrics
$\mathrm{SrTiO_{3}}$, the misfit strains are precisely controlled to increase
the phase transformation temperature from cryogenic values
($\approx\mathrm{0K}$) to room temperature (Haeni et al., 2004; Chen, 2008).
The phase transformation microstructures such as periodic arrangement of
metal-insulator phases in $\mathrm{VO_{2}}$ are modulated using film strains
(Wu et al., 2006; Shao et al., 2018). Additionally, in thin-film Co electrodes
tensile stresses are used to delay the crack onset during phase
transformations (Marx et al., 2016). However, for intercalation electrodes
such as $\mathrm{V_{2}O_{5}}$, little is known about the extent to which
engineering film strains can alter the phase transformation voltages. Doing so
could expand the operational voltage window for $\mathrm{V_{2}O_{5}}$
electrodes while retaining the structural integrity of electrodes.

Today, most practical batteries use a particle-based electrode with the
intercalation material as its active particle (Mohanty et al., 2016; Crabtree
et al., 2015). In these electrode configurations, it is challenging to
investigate the effect of mechanical stresses on phase changes and voltage
curves in the intercalation material. This is largely because of the
difficulty in controlling and assessing the mechanical constraints and the
corresponding stress states that occur in particle-based electrodes. These
issues are generally easier to evaluate with thin-film electrode
configurations because of their large surface area and relatively easy-to-
control morphology. Thin-film electrode configurations are used in
microbatteries (Fehse et al., 2017; Nathan et al., 2005), and resemble the
thin surface layers that form on many cathode particles (e.g., in
$\mathrm{LiNi_{1-y-z}Co_{y}Mn_{z}O_{2}}$ (NMC) cathodes, where phase
transformations create surface damage layers that lead to capacity loss
(Sallis et al., 2016)). Previously, Bucci et al. (2014) have investigated the
electrochemical-mechanical response of thin-film amorphous Si electrodes, but
the links between film strains and structural degradation of crystalline
electrodes is not well understood. With this in mind, for this study, we
propose to investigate the role of film strains on phase transformations in
thin-film intercalation electrode geometry. To be specific, we will evaluate
thin film configurations where the volume changes of the electrode are
constrained by an underlying stiff substrate and investigate the role of film
strains on phase transformation voltages in intercalation electrodes.

We hypothesize that the film strains can regulate the voltages for phase
transformations in electrode materials, and thus can improve the reversible
cycling over a wider voltage range. We test this hypothesis using a
combination of theory and experiments: First, we analytically derive the
effect of film strain on the electrochemical response of cathodes. Our
analytical framework is general and in principle can be adapted to any
intercalation material. As an example for the present study, we choose the
$\mathrm{V_{2}O_{5}}$ compound. $\mathrm{V_{2}O_{5}}$ is a relevant
intercalation electrode for Li-batteries and can store up to three Li-ions per
formula unit via a series of phase transformations.111The phases of interest
include the $\alpha,\epsilon,\delta,\gamma$ and $\omega$ phases, which
respectively correspond to Li-contents, $x<0.01$, $0.35<x<0.7$, $x=1$,
$1<x<3$, $x=3$. These phase transformations occur over a wide range of voltage
windows (Wang and Cao, 2006; Christensen et al., 2018) and are accompanied by
substantial volume changes in $\mathrm{V_{2}O_{5}}$.222For example, the
$\gamma-\omega$ $(x\approx 3)$ transformation is particularly problematic
because it leads to irreversible structural damage and amorphization of the
material (Christensen et al., 2018; Huang et al., 2019). Consequently,
although vanadium pentaoxide ($\mathrm{V_{2}O_{5}}$) promises high power
densities and high capacities (Jeon et al., 2001; Wang and Cao, 2006), its
structural irreversibility limits its widespread usage. We theoretically
investigate whether thin-film straining of $\mathrm{Li_{x}V_{2}O_{5}}$
electrode can regulate its multiple phase transformation stages and thus
enable the structural reversibility of the intercalation material. Second, we
test our theoretical predictions using experiments. Here, we electrochemically
cycle two types of thin-film $\mathrm{V_{2}O_{5}}$ electrodes with different
film strains and investigate their corresponding mechanical degradation. Doing
so enables us to measure the in-situ stress evolution in electrodes during
electrochemical cycling. Our experimental findings validate our theoretical
predictions that film strains modify the voltage for phase transformation and
help improve the structural reversibility of intercalation cathodes. Overall,
our work presents a theoretical and experimental framework to engineer film
strains as design criteria to enhance the lifespans of intercalation
materials.

##  2 Theory

In this section we derive the effect of film strain on phase transformations
in thin-film intercalation electrodes, and show how film strains can be
engineered to enhance the structural reversibility and lifespans of
electrodes. First, we describe the free energy function for an intercalation
electrode and analytically derive the effect of film strain on the free energy
landscape. Next, we investigate how film strain affects the chemical potential
and stress evolution, in-situ, in thin-film electrodes.

In principle, our theoretical framework can be adapted to any phase separating
electrode material, but for the present work, we choose
$\mathrm{Li_{x}V_{2}O_{5}}$ as the reference electrode. Crystalline
$\mathrm{Li_{x}V_{2}O_{5}}$ undergoes series of phase transformations as a
function of Li-content and each phase transformation stage is characterized by
a constant voltage plateau describing a two-phase coexistence (Christensen et
al., 2018). In our analytical calculations we model such two-phase coexistence
and investigate the effect of elastic energy on individual voltage plateaus.
For further details on the phase transformations in
$\mathrm{Li_{x}V_{2}O_{5}}$ and its corresponding material parameters see
Section 7.1 of the Appendix.

###  2.1 Free energy

The free energy $\Psi$ of an electrode $\mathcal{E}$ as a function of Li-
content $c$ and strain $\mathbf{E}$ is given by:

$\displaystyle\Psi(c,\nabla c,\mathbf{E})$ $\displaystyle=\psi_{grad}(\nabla
c)+\psi_{bulk}(c)+\psi_{elas}(\mathbf{E},c)$
$\displaystyle=\int_{\mathcal{E}}\frac{1}{2}\kappa|\nabla
c|^{2}+\frac{\Omega}{V_{m}}c(1-c)+\frac{RT}{V_{m}}[c\mathrm{ln}(c)+(1-c)\mathrm{ln}(1-c)]$
$\displaystyle+\frac{1}{2}[\mathbf{E-E}_{0}(c)]{\color[rgb]{0,0,0}\ :\
}\mathbb{C}[\mathbf{E-E}_{0}(c)]\thinspace\mathrm{d}\mathbf{x}$ (1)

The total free energy of a heterogeneous system given by Eq. 1 is the volume
integral of the gradient energy, bulk energy and the elastic energy of the
system. The Li-content field $c(\mathbf{x})$ represents the fraction of
interstitial sites in the electrode occupied by Li per unit volume. The
gradient energy $\int\kappa|\nabla c|^{2}\mathrm{d\mathbf{x}}$ penalizes
changes in Li-content and its coefficient $\kappa$ is determined in
numerics/computations to define the interface (e.g., phase boundaries)
properties. This coefficient is typically small and its contribution to the
total energy at larger length scales is often neglected. The bulk energy (a
regular solution model) describes the free energy landscape with minima
corresponding to the lithiated $(c=1)$ and unlithiated $(c=0)$ phases of the
electrode. The regular solution coefficients are calibrated to fit the
thermodynamic properties to voltage measurements of
$\mathrm{Li_{x}V_{2}O_{5}}$. The elastic energy of the system accounts for
elastic strains arising from lattice misfit between the lithiated/unlithiated
phases and the film strains arising from the mismatch between the film and
substrate. These strains deviate from the preferred strain tensor
$\mathbf{E}_{0}(c)$ that describes the transformation of lattice geometry
between the lithiated and unlithiated phases. In this paper, we assume
$\mathbf{E}_{0}(c)$ to vary as a function of Li-content $c$ and is given by:

$\displaystyle\mathbf{E}_{0}(c)$
$\displaystyle=\left[\begin{array}[]{ccc}\epsilon_{\alpha}(c)&0&0\\\
0&\epsilon_{\beta}(c)&0\\\ 0&0&\epsilon_{\gamma}(c)\end{array}\right].$ (5)

Here, $\epsilon_{\alpha}(c),\epsilon_{\beta}(c),\epsilon_{\gamma}(c)$ are
polynomials of third-order in $c$ that are fitted to the lattice parameter
measurements of $\mathrm{Li_{x}V_{2}O_{5}}$, and are listed in Table. 1 of
Appendix. These polynomials account for the anisotropic lattice strains in the
intercalation material, which differs from the elastic strains defined for
alloying-type of electrodes (Sheldon et al., 2011). From here on we note that
the preferred strain tensor is a function of Li-content, and we drop $c$ from
the notation. Similarly the polynomials of the preferred strains will be
represented by $\epsilon_{\alpha},\epsilon_{\beta}$ and $\epsilon_{\gamma}$.

###  2.2 Film strain modifies the free energy landscape

Next, we introduce the boundary conditions for a thin-film electrode in Eq. 1.
A thin-film of a single-crystal electrode constrained to a rigid substrate is
shown in Fig. 1, where the mismatch between the substrate and the electrode
lattices imposes a film strain $\mathrm{E_{s}}$ along the
$\mathrm{e_{1}-e_{2}}$ directions. Such mismatch could arise from differences
in coefficient of thermal expansions (CTE) between the film and substrate. For
the thin-film shown in Fig. 1, we model inplane axial strains
$\mathrm{E_{11}=E_{22}=E_{s}}$ and $\mathrm{E_{12}=E_{21}=0}$. We calculate
the film strain $\mathrm{E_{s}}$ as a CTE mismatch between the unlithiated
film and the substrate material.

Figure 1: (a) Schematic illustration of a thin-film electrode (e.g.,
$\mathrm{V_{2}O_{5}}$) constrained to a much thicker substrate. The substrate
typically consists of two layers, namely the current collectors (e.g.,
aluminum (Al) or gold (Au)) and a thick quartz substrate. An additional
titanium layer (Ti) is used with the gold collector for better adhesivity. (b)
The film strains arise from the CTE mismatch between the thin film and the
substrate. For example, after annealing the thin-film shrinks more than the
quartz substrate. (c) When the film is bound to the substrate the CTE mismatch
induces a film strain in the electrode.

In thin-film electrodes, the film thickness is far smaller than its in-plane
dimensions and is acted only upon by loads (e.g., CTE misfit) that are
parallel to the film. With this electrode geometry and boundary conditions, we
assume that thin-films are under the plane stress condition, and thus satisfy
the stress-free boundary conditions along the $\mathrm{e_{3}}-$axis. That is,
the stresses involving the out-of-plane components are zero,
$\sigma_{13}=\sigma_{23}=\sigma_{31}=\sigma_{32}=0.$ Consequently, the strains
$\mathrm{E_{13},E_{23},E_{31},E_{32}}$ corresponding to their respective
stress components are zero. The strain tensor for a thin film reduces to:

$\displaystyle\mathbf{E}$
$\displaystyle=\left[\begin{array}[]{ccc}\mathrm{E_{s}}&0&0\\\
0&\mathrm{E_{s}}&0\\\ 0&0&\mathrm{E_{33}}\end{array}\right].$ (9)

During lithiation and delithiation, the thin-film deforms in the out-of-plane
direction to minimize the total elastic energy of the system. In other words,
the out-of-plane strain $\mathrm{E_{33}}$ satisfies the plane stress condition
$\sigma_{33}=0$ and is obtained by minimizing the free energy with respect to
$\mathrm{E_{33}}.$

$\displaystyle\frac{\delta\Psi}{\delta\mathrm{E_{33}}}$
$\displaystyle=\mathrm{C_{11}}[\mathrm{E_{33}}-\epsilon_{\gamma}]+\mathrm{C_{12}}[2\mathrm{E_{s}}-\epsilon_{\beta}-\epsilon_{\alpha}]$
(10)

Next, in order to satisfy $\sigma_{33}=0,$ we have:

$\displaystyle\frac{\delta\Psi}{\delta\mathrm{E_{33}}}$ $\displaystyle=0$
$\displaystyle\mathrm{E_{33}}$
$\displaystyle=\mathrm{\frac{C_{12}}{C_{11}}}[\epsilon_{\alpha}+\epsilon_{\beta}-2\mathrm{E_{s}}]+\epsilon_{\gamma}$
(11)

The out-of-plane strain $\mathrm{E_{33}}$ is a function of the film strain and
Li-content (i.e., $\epsilon_{\alpha},\epsilon_{\beta},\epsilon_{\gamma}$ are
polynomials in $c$). Substituting Eq. 11 in Eq. 1, and assuming a linear
elastic cubic crystal333The indicial form of the elastic energy for a linear
cubic crystal is derived in Section 7.1 of Appendix. we can express the free
energy of the electrodes as a function of Li-content $c$ and film strain
$\mathrm{E_{s}}$ as follows:

$\displaystyle\Psi(c,E_{\mathit{s}})=\int_{\mathcal{E}}\bigg{\\{}$
$\displaystyle\frac{1}{2}\kappa|\nabla
c|^{2}+\frac{\Omega}{V_{m}}c(1-c)+\frac{RT}{V_{m}}[c\mathrm{ln}(c)+(1-c)\mathrm{ln}(1-c)]$
$\displaystyle+\mathrm{\frac{C_{11}-C_{12}}{2}}\bigg{[}(\mathrm{E_{s}}-\epsilon_{\alpha})^{2}+(\mathrm{E_{s}}-\epsilon_{\beta}])^{2}+\bigg{(}\mathrm{\frac{C_{12}}{C_{11}}}[\epsilon_{\alpha}+\epsilon_{\beta}-2\mathrm{E_{s}}]\bigg{)}^{2}\bigg{]}$
$\displaystyle+\frac{\mathrm{C_{12}}}{2}[\mathrm{E_{s}}-\epsilon_{\alpha}+\mathrm{E_{s}}-\epsilon_{\beta}+\mathrm{\frac{C_{12}}{C_{11}}}(\epsilon_{\alpha}+\epsilon_{\beta}-2\mathrm{E_{s}})]^{2}\bigg{\\}}\mathrm{d}\mathbf{x}$
(12)

Figure 2: A 3D plot of the free energy expression in Eq. 12 as a function of
Li-content and film strain. The coefficients correspond to the
$\alpha-\epsilon$ phase transformation in $\mathrm{Li_{x}V_{2}O_{5}}$
electrode. Film strains affect the height of wells and alter the energy
barrier between two phases. As can be seen, increasing tensile strains shifts
the height of minimum energy wells and lowers the energy barrier between the
$\alpha$ and $\epsilon$ phases.

The coefficients in Eq. 12 correspond to the $\alpha-\epsilon$ phase
transformation stage in $\mathrm{V_{2}O_{5}}$ and are listed in Appendix. We
next plot Eq. 12 as a function of $c$ and $\mathrm{E_{s}}$ (see Fig. 2). At
$\mathrm{E_{s}}=0$, a double-well landscape with minima located at the
unlithiated $c/c\mathrm{{}_{0}}=0$ and the lithiated phases
$c/c\mathrm{{}_{0}}=1$ is shown. Here $c_{\mathrm{0}}$ denotes the Li-mole
fraction of the $\epsilon$ phase. This double-well landscape changes in two
ways under non-zero film strains: First, increasing tensile (or compressive)
strains shift the minimum energy wells. For example, the height of the energy
wells changes with increasing tensile strain, and at $\mathrm{E_{s}=0.008}$
the global minima is located at $c=0$. Second, increasing tensile (or
compressive) strains affects the height of the energy barrier between the two
wells. That is, under tensile strains the energy barrier between $c=0$ and
$c=c_{\mathrm{0}}$ wells is reduced and thus would require a smaller driving
force for phase transformation. In the next subsection, we will show how these
changes to the free energy landscape can be engineered to facilitate
reversible phase transformations in battery electrodes.

###  2.3 Chemical potential

We define the chemical potential as the variational derivative of the free
energy functional in Eq. 1:

$\displaystyle\mu$ $\displaystyle=\frac{\delta\Psi}{\delta c}$
$\displaystyle=\frac{\partial\psi_{bulk}}{\partial
c}+\frac{\partial\psi_{elas}}{\partial c}-\kappa\nabla^{2}c$ (13)

Fig. 3(a) depicts the chemical potential as a function of Li-content in the
$\mathrm{V_{2}O_{5}}$ intercalation cathode. The flat voltage plateau at
$-\mu/\mathrm{F}=3.4\mathrm{V}$ represents phase separation during the
$\alpha-\epsilon$ phase transformation for a free-standing
$\mathrm{V_{2}O_{5}}$ electrode (Note $\mathrm{F=96485\thinspace C/mol}$ is
the Faraday constant). This flat voltage plateau gains an upward (or downward)
slope with increasing compressive (or tensile) film strains. Additionally, the
plateau shifts to lower voltages with increasing tensile strains.444Note that
the effect of tensile and compressive strains on the voltage plateau in Fig.
3(a) is specific to the $\alpha-\epsilon$ phase transformation of
$\mathrm{V_{2}O_{5}}$ and its corresponding anisotropic lattice strains. For
example, the lattice strains $\epsilon_{\alpha}$ decreases by 0.5% on
intercalation while $\epsilon_{\beta}$ and $\epsilon_{\gamma}$ increase by
0.1% and 3.2% respectively on intercalation. This anisotropic expansion of
$\mathrm{Li_{x}V_{2}O_{5}}$ affects the observed shift of the voltage plateau.

Figure 3: (a) The film strain affects the voltage plateaus corresponding to
the phase transformation stage. Specifically, increasing tensile strains
introduce a downward slope to the chemical potential curve and shift the phase
transformation voltage to lower values. (b) Using Eq. 13 we predict the
chemical potential landscape for each phase transformation stage in
$\mathrm{Li_{x}V_{2}O_{5}}$ electrode. We note that in addition to voltage
plateaus lowering, tensile strains also introduce an additional slope to the
chemical potential curves. (c) Inset figure showing the $\alpha-\epsilon$ and
$\epsilon-\delta$ phase transformations in $\mathrm{Li_{x}V_{2}O_{5}}$
electrode. The experimental data corresponding to the electrochemical
measurements for a free-standing $\mathrm{V_{2}O_{5}}$ electrode is provided
by Luo et al. (2018).

Fig. 3(b-c) shows four distinct voltage plateaus corresponding to four stages
of phase transformation of $\mathrm{Li_{x}V_{2}O_{5}}$. Although all four
voltage plateaus are lowered with increasing film strains, significant shifts
in voltage curves are observed for the $\delta-\gamma$ and the $\gamma-\omega$
phase transformation stages. We attribute these relatively large shifts of
voltage plateaus to the abrupt lattice transformation at the $\delta-\gamma$
and the $\gamma-\omega$ phase transformations, which generate lattice strains
of up to $\sim 10\%$ (see Fig. 10 in Appendix). This abrupt change in lattice
strains, in particular the $\epsilon_{\alpha}(c)$, affects the chemical
potential via the elastic energy term.

Figure 4: The effect of $\psi_{elas}1$ and $\psi_{elas}2$ to the percentage
voltage shift. Here relative percentage voltage shift is derived by comparing
with reference voltage values of a free-standing $\mathrm{Li_{x}V_{2}O_{5}}$
in each phase transformation stage.

We next quantify the effect of individual elastic energy terms in Eq. 12 on
the voltage plateaus shifts. For a free standing $\mathrm{Li_{x}V_{2}O_{5}}$
intercalation electrode the strain tensor will closely satisfy the preferred
lattice strains, i.e., $\mathbf{E}=\mathbf{E}_{0}(c)$ to minimize the elastic
energy of the system. However, the thin-film geometry of the electrode imposes
a mechanical constraint with $\mathrm{E_{11}}=\mathrm{E_{22}}=\mathrm{E_{s}}$,
which gives rise to a finite elastic energy in the system. Fig. 4 shows how
the elastic energy terms, namely the terms on lines two ($\psi_{elas}1$) and
three ($\psi_{elas}2$) of Eq. 12, affect the voltage plateau
shifts.555$\psi_{elas}1=\frac{c_{11}-c_{12}}{2}[(\mathrm{E_{s}}-\epsilon_{\alpha})^{2}+(\mathrm{E_{s}}-\epsilon_{\beta}])^{2}+(\frac{c_{12}}{c_{11}}[\epsilon_{\alpha}+\epsilon_{\beta}-2\mathrm{E_{s}}])^{2}$,
$\psi_{elas}2=\frac{1}{2}c_{12}[\mathrm{E_{s}}-\epsilon_{\alpha}+\mathrm{E_{s}}-\epsilon_{\beta}+\frac{c_{12}}{c_{11}}(\epsilon_{\alpha}+\epsilon_{\beta}-2\mathrm{E_{s}})]^{2}$
On the Al substrate, the voltage plateaus shift lower by $\sim 10\%$ and $\sim
4\%$, respectively, for the $\delta-\gamma$ and $\gamma-\omega$ stages, when
compared to the reference voltage plateaus for a free standing
$\mathrm{Li_{x}V_{2}O_{5}}$. Similarly, on the Au substrate, large voltage
plateau shifts of $8\%$ and $3\%$ are observed, respectively, for the
$\delta-\gamma$ and $\gamma-\omega$ stages. In these voltage shifts, we note
that $\psi_{elas}1$ plays an important role and accounts for up to $\sim 65\%$
of the total voltage plateau shifts. We attribute this effect to the increase
in misfit strains between the imposed film strain and the preferred lattice
strains (e.g., ($\mathrm{E_{s}}-\epsilon_{\alpha}$),
($\mathrm{E_{s}}-\epsilon_{\beta}$)) at the $\delta-\gamma-\omega$ phase
transformations.

It should also be noted that the final $\gamma-\omega$ phase transformation is
accompanied by severe puckering and large structural changes that are known to
amorphize the electrode (Christensen et al., 2018; Cocciantelli et al., 1992).
This irreversible structural damage to the electrode can be prevented by
modeling $\mathrm{V_{2}O_{5}}$ electrodes as thin-films with suitable film
strains. Specifically, film stresses of $\sim 2\mathrm{GPa}$ (i.e.,
$\mathrm{E_{s}\approx 1\%}$) lowers the $\gamma-\omega$ voltage plateau to
below $2$V and can thus suppress phase transformation when operated in the
2–4V voltage window. This strain engineering of thin-film
$\mathrm{V_{2}O_{5}}$ electrodes facilitates its structural reversibility.

In short, these changes of voltage plateau caused by film strains can
facilitate reversible phase transformation in two ways: First, the downward
sloping voltage plateaus suggest suppressed phase separations. This
suppression of phase separation reduces coherency stresses, which are known to
nucleate microcracks in electrodes. Second, the shifting of voltage plateaus
to lower values and outside of the typical operational window (2.5–4 V)
prevents phase transformation. This shift in voltage plateaus can be
engineered to prevent phase transformations with large volume changes. We will
apply these concepts to engineer reversible phase transformation in
$\mathrm{V_{2}O_{5}}$ electrode in Section 3.

###  2.4 Stress response

We assume a linear elastic cubic relation and describe the stress tensor
$\boldsymbol{\sigma}$ as a function of the stiffness tensor $\mathbb{C}$, the
preferred strain tensor $\mathbf{E}_{\mathrm{0}}$, and the Li-content $c$:

$\displaystyle\boldsymbol{\sigma}(c,\mathbf{E})$
$\displaystyle=\frac{\partial\Psi}{\partial\mathbf{E}}$
$\displaystyle=\mathbb{C}:[\mathbf{E-E}_{0}(c)]$ (14)

The stress components are individually given in Eq. (14) of the Appendix. As
discussed previously we assume a single crystal electrode and neglect the
effect of grain structure and the kinetics of phase transformation on the
stress response. We model the preferred strain tensor to describe the affine
lattice parameter changes in $\mathrm{Li_{x}V_{2}O_{5}}$ during phase
transformation as a function of Li-content $c$. We calibrate the preferred
strains with the experimental lattice parameter measurements for
$\mathrm{Li_{x}V_{2}O_{5}}$ as shown in Fig. 10(a-c). We fit the preferred
strain data using a third-order polynomial that captures the general trend of
preferred strains as a function of Li-content. While a higher-order polynomial
would capture the abrupt changes in lattice strains (shown by the experimental
data in Fig. 8), we use a lower order polynomial to prevent overfitting of the
strain data. Although this polynomial approximation of the stresses does not
capture the abrupt changes in the in-plane stresses as measured
experimentally, it provides a continuum formulation of in-plane stresses as a
function of Li-content and crystal orientation.

Figure 5:  In-plane stresses $\sigma_{11}$ and $\sigma_{22}$ for a
$\mathrm{Li_{\mathit{x}}V_{2}O_{5}}$ thin-film electrode as a function of Li-
content for three crystal orientations: (a) with the $c$–axis perpendicular to
the plane of the film, (b) with the $a$–axis perpendicular to the plane of the
film, and (c) $b$–axis perpendicular to the plane of the film. The inset
schematics illustrate the crystal orientations and the dashed line corresponds
to the film stress $\sigma_{\mathrm{s}}$.

Fig. 5(a-c) depicts the in-plane stress components $\sigma_{11},\sigma_{22}$
as a function of Li-content for three different crystallographic orientations
of $\mathrm{Li_{x}V_{2}O_{5}}$. To be specific, the crystal in Fig. 5(a) is
oriented such that its $c-$axis is perpendicular to the thin-film plane. In
this case, the in-plane stress $\sigma_{11}$ goes up with increasing Li-
content. By contrast, for a different crystal orientation as in Fig. 5(c),
$\sigma_{11}$ decreases initially and then shows a upward tendency with
increasing Li-content. A similar variation of $\sigma_{22}$ is also observed.
These contrasting stress responses for different in-plane stress components
and for different crystal orientations will affect the net stress evolution in
a polycrystalline material. In such cases, a detailed measurement of grain
orientations in the electrode is necessary to estimate the net stress
response.

##  3 Experiment

In this section, we present results from an experiment investigating the
effect of film strains on the reversible cycling of $\mathrm{V_{2}O_{5}}$
thin-film electrodes. As such this study serves as an experimental test of the
theoretical results presented in Section 2.

###  3.1 Methods

We designed four custom-made electrochemical cells, each comprising of a thin-
film $\mathrm{V_{2}O_{5}}$ cathode, Li-metal anode, and a liquid
electrolyte.666equimolar mixture of Ethylene Carbonate, Dimethyl Carbonate,
and 1M $\mathrm{LiPF_{6}}$. In two of the cells, we deposited
$\mathrm{V_{2}O_{5}}$ on Aluminum-coated-quartz (Al), and in the remaining two
cells we deposited $\mathrm{V_{2}O_{5}}$ on Gold-coated-quartz (Au) substrate,
see Fig. 1(a). We next annealed the thin-film cathodes at 550∘C in the air for
three hours. On cooling the cathodes the $\mathrm{V_{2}O_{5}}$ thin-film
shrinks more than the quartz glass substrate, see Fig. 1(b-c). This difference
in lattice geometries between the thin-film and the substrate gives rise to a
lattice misfit and induces tensile strains in the $\mathrm{V_{2}O_{5}}$
electrodes. Using the MOSS technique, we measure the in-plane stress values of
the $\mathrm{V_{2}O_{5}}$ film on Al and Au substrates to be 2GPa and 1GPa
respectively. In this work, we do not study the mechanistic cause for the
difference in in-plane stresses for the Al and Au substrates, however, we use
this difference to investigate the effect of film stresses on reversible
cycling of $\mathrm{V_{2}O_{5}}$ thin films. The $\mathrm{V_{2}O_{5}}$ films
on both Al and Au substrates are crystalline in the initial state and were
characterized using transmission electron microscopy (TEM), scanning electron
microscopy (SEM) and X-ray diffractometer (XRD), see Fig. 6(a). For further
details on the thin-film preparation, characterization, and MOSS technique,
please see Appendix 7.2.

Next, we divided the cells into two groups (each group containing two cells
with a $\mathrm{V_{2}O_{5}}$ cathode deposited on Al and Au substrates), and
tested our theory that film strain enhances the structural reversibility of
thin-film cathodes during electrochemical cycling. In the first group, we
potentiostatically cycled the cells at constant voltages 4V, 3V, and 2V. Each
of these holds were maintained until the films were equilibrated. After three
charge/discharge cycles we conducted XRD measurements and SEM imaging for the
$\mathrm{V_{2}O_{5}}$ films on Al and Au substrates, respectively. In the
second group, we galvanostatically cycled the cells at a C-rate of C/20 and
typically in the voltage window of 2-4V. During galvanostatic cycling, we
measured the voltage and stress values for the $\mathrm{V_{2}O_{5}}$ films on
Al and Au substrates, respectively. Further details on the electrochemical
cycling and the in-situ voltage-stress measurement techniques are described in
Appendix 7.2.

###  3.2 Potentiostatic cycling

Fig. 6(a-c) shows the SEM images and XRD measurements of the
$\mathrm{V_{2}O_{5}}$ cathode on Au and Al substrates respectively. The SEM
images in Fig. 6(a-b) show the $\mathrm{V_{2}O_{5}}$ electrode surface before
and after three potentionstatic cycles. The $\mathrm{V_{2}O_{5}}$ electrode on
Au ($\sim 1\mathrm{GPa}$ film stress) amorphizes after three cycles while the
$\mathrm{V_{2}O_{5}}$ electrode on Al ($\sim 2\mathrm{GPa}$ film stress)
remains crystalline after three cycles. The XRD measurements in Fig. 6(c) show
the structural evolution of crystallographic peaks corresponding to
$\mathrm{V_{2}O_{5}}$ cathode on Au and Al substrates respectively.
Specifically, the $\mathrm{V_{2}O_{5}}$ cathode on Au substrate loses
crystallinity after three cycles. This is confirmed by the loss of the (001)
peak in its XRD measurement, see Fig. 6(c). By contrast, for the
$\mathrm{V_{2}O_{5}}$ cathode on Al substrate, the (001) peak gradually
broadens and shifts towards lower theta values with repeated cycling. But this
cathode still remains crystalline even after five cycles and maintains its
orthorhombic structure. Overall, these potentiostatic experiments show that
the increased film straining enhances the structural reversibility of
$\mathrm{V_{2}O_{5}}$ electrodes.

Figure 6: (a) SEM images of the $\mathrm{V_{2}O_{5}}$ electrode surface
deposited on a Au substrate (top) and an Al substrate (below). These images
show the microstructure on $\mathrm{V_{2}O_{5}}$ surface before
electrochemical cycling. (b) SEM images of $\mathrm{V_{2}O_{5}}$ surface after
potentiostatic cycling (3 cycles). Our experiments show that the
$\mathrm{V_{2}O_{5}}$ film deposited on Al retains crystallinity on cycling
while the $\mathrm{V_{2}O_{5}}$ film deposited on Au loses crystallinity or
amorphizes on potentiostatic cycling. (c) The X-ray diffraction measurements
of the $\mathrm{V_{2}O_{5}}$ film on Al and Au substrates cycled
potentiostatically. In the Al case, the X-ray peak corresponding to
$\mathrm{V_{2}O_{5}}$ (001) broadens upon repeated cycling. But in the Au
case, the X-ray peak disappears after 3 cycles. These measurements further
confirm the loss of crystallinity in $\mathrm{V_{2}O_{5}}$ film deposited on
Au.

###  3.3 Galvanostatic cycling

Fig. 7(a-b) shows the in-situ voltage measurements of $\mathrm{V_{2}O_{5}}$ on
Au and Al substrates respectively. Both the electrochemical cells were cycled
galvanostatically, under a constant current density of $\mathrm{0.2\mu
A/cm^{2}}$ and at a rate of C/20, in a fixed voltage window of 2–4V. The
voltage traces in this range demonstrate that altering the initial stress
state leads to substantial differences in the electrochemical response. This
comparison also provides an explanation for the amorphization observed during
potentiostatic cycling with an Au current collector in Fig. 6. Here the
voltage responses show plateaus corresponding to the phase transformations at
different Li-intercalation stages. In Fig. 7(a), the $\mathrm{V_{2}O_{5}}$
cathode on Au substrate undergoes four distinct phase transformation stages
that are characterized by one voltage slope (at 3.4V) and three voltage
plateaus (at 3.20V, 2.31V, and 2.04V). The final $\gamma-\omega$ phase
transformation stage is accompanied by a large volume change that amorphizes
the $\mathrm{V_{2}O_{5}}$ cathode. This amorphization contributes to the
structural irreversibility of the cathode and thus no distinct voltage
plateaus are observed in the subsequent discharge cycles. These features of
structural irreversibility and loss of voltage plateaus on the $\gamma-\omega$
phase transformation in $\mathrm{V_{2}O_{5}}$ are consistent with results on
$\mathrm{V_{2}O_{5}}$ amorphization reported by other researchers
(Whittingham, 1976; Christensen et al., 2018).

Figure 7: Voltage-capacity plots of $\mathrm{V_{2}O_{5}}$ cathode deposited on
Au and Al substrates and cycled galvanostatictically in a fixed 2–4V voltage
window. Both the thin-film cathodes were cycled with a constant current
density of $\mathrm{0.2\mu A/cm^{2}}$ and at a rate of C/20. (a) The voltage
curve for $\mathrm{V_{2}O_{5}}$ on Au substrate with
$\mathrm{\sigma_{s}\approx 1GPa}$ shows four voltage plateaus in the first
cycle. These voltage plateaus are not observed in subsequent cycles showing
structural irreversibility. (b) The voltage curve for $\mathrm{V_{2}O_{5}}$ on
Al substrate with $\mathrm{\sigma_{s}\approx 2GPa}$ shows only three voltage
plateaus in the first cycle. These voltage plateaus are observed in the
subsequent cycles showing structural reversibility.

By comparison, with a larger tensile film stress ($\mathrm{\sigma_{s}\approx
2GPa}$), the $\mathrm{V_{2}O_{5}}$ cathode on the Al substrate undergoes only
three phase transformation stages that are characterized by voltage
slopes/plateaus at 3.33V, 3.18V, and 2.12V respectively. These phase
transformations occur at lower voltage values when compared to their
corresponding transformations on the Au substrate. The final $\gamma-\omega$
phase transformation stage is not observed in $\mathrm{V_{2}O_{5}}$ on Al
substrates in the 2–4V operational window. Consequently, the large volume
changes associated with the $\gamma-\omega$ phase transformation does not
occur in the 2–4V voltage window, and the structural reversibility of the
cathode is preserved. Thus, in the subsequent discharge cycles on Al
substrate, the $\mathrm{V_{2}O_{5}}$ electrode retains its structural
reversibility as shown by the recurrence of the voltage plateaus in Fig. 7(b).

###  3.4 Voltage plateaus and in-plane stresses

We next do a deep discharge of the $\mathrm{V_{2}O_{5}}$ cathodes on both the
Au and Al substrate until all four phase transformations have occurred. That
is, unlike in Section 3.3, we do not fix the operational voltage window to
2–4V, but do a deep discharge until all four phase transformations occur in
$\mathrm{V_{2}O_{5}}$ on both the Au and Al substrates. During these discharge
cycles we measure both the voltage and stress responses as a function of Li-
content, see Fig. 8(a-b).

Figure 8: In-situ voltage and stress measurements for the crystalline
$\mathrm{V_{2}O_{5}}$ film on (a) Al and (b) Au substrates respectively. Both
$\mathrm{V_{2}O_{5}}$ films are cycled galvanostatically, with a constant
current density of $\mathrm{0.2\mu A/cm^{2}}$ and at a rate of C/20. The
voltage plateaus corresponding to multiple phase transformations in
$\mathrm{V_{2}O_{5}}$, and the inplane stresses measured using the MOSS
technique are shown in subfigures (a) and (b).

Voltage Plateaus: Fig. 8(a-b) shows four distinct voltage plateaus
corresponding to the $\alpha-\epsilon-\delta-\gamma-\omega$ phase
transformations in $\mathrm{V_{2}O_{5}}$ cathodes on both the Au and Al
substrates.777Note that the $\mathrm{Li_{x}V_{2}O_{5}}$ cathode expands in
volume (i.e., positive volume change) during initial intercalation up to
$\mathrm{x}\leq 2$, and then contracts in volume (i.e., negative volume
change) during the final stages of intercalation for $\mathrm{x}>2$, see Fig.
10(a-c) in the Appendix. Here, we note two distinguishing features of the
voltage curves: The first feature is that the voltage plateaus for
$\mathrm{V_{2}O_{5}}$ on the Al substrate are lower than the corresponding
voltage plateaus on the Au substrate in Fig. 8(a). Take the $\gamma-\omega$
stage as an example, the voltage plateau on the Al substrate is at 1.94V which
is $\sim 5\%$ smaller than the voltage plateau at 2.038V on the Au substrate.
Similarly, the voltage plateau at the $\delta-\gamma$ phase transformation
stage is lowered to 2.118V on the Al substrate.888Note, that the
$\delta-\gamma$ voltage plateau of $\mathrm{Li_{x}V_{2}O_{5}}$ on the Au
substrate shifts by only $\sim 3\%$ compared with 2.374V of the free-standing
one. We explain this surprisingly small voltage shift to the combined tensile
and compressive stress responses measured during the $\delta-\gamma$ phase
transformation stage, see Fig. 8(b). These voltage shifts in Fig. 8(a-b) are
consistent with our theoretical predictions in Section 2.2. We explain the
downward (and relatively large) shifts of the voltage plateaus to the larger
misfit stresses on the Al substrate. These large stresses combined with the
negative volume change (e.g., for $\gamma-\omega$ phase transformation)
contribute to the downward shifts of the voltage plateaus (from Eq. 13).

The second feature is that the voltage plateau shifts downward for both the
$\delta-\gamma$ and $\gamma-\omega$ phase transformation stages despite their
opposing stress responses (and/or volume changes) during intercalation. That
is, the $\delta-\gamma$ voltage plateau at 2.31V on Au substrate is lowered to
2.12V on the Al substrate, despite the positive volume change during the
$\delta-\gamma$ phase transformation. We attribute this discrepancy to the
polycrystalline nature of the $\mathrm{V_{2}O_{5}}$ electrode in our
experiment (i.e., electrode contains grains with multiple orientations), and
the large tensile film stresses that contribute to the net lowering of the
voltage plateaus. For example, the film stress in $\mathrm{V_{2}O_{5}}$ on Al
substrate ($\mathrm{\sigma_{s}\approx 2GPa}$) is nearly twice as large as the
film stress in $\mathrm{V_{2}O_{5}}$ on the Au substrate
($\mathrm{\sigma_{s}\approx 1GPa}$). This difference in in-plane stresses
lowers the voltage values for phase transformations despite their positive
volume change as shown theoretically in Section 2.2 and 2.3.

Stress response: During Li-insertion, the stress responses measured using the
MOSS technique exhibit features that are similar to both the Al and Au current
collectors, see Fig. 8(a-b). For example, the cathodes in both cases show
compressive stresses during the $\alpha-\epsilon-\delta-\gamma$ phase
transformations $(\mathrm{x}<2)$, which suggest positive volume change
(expansion) of the cathode; During the $\gamma-\omega$ phase transformation
stage $\mathrm{x>2}$ we observe tensile stresses that suggests a negative
volume change (contraction) of the cathode. Surprisingly, during Li-extraction
the stress response is tensile during the initial stages. For reversible
elastic behavior in the $\gamma-\omega$ phase transformation, one would expect
the Li-extraction to be compressive. With continual delithiation $(x<1.8)$,
the stresses eventually decrease towards their initial states.

We next note that the net stress response, measured using the MOSS technique
in Fig. 8, is an order of magnitude smaller than our theoretical predictions
for an ideal single-crystal thin-film electrode, see Fig. 5(a-c). We attribute
this difference in stress responses between theory and experiment to the
polycrystalline texture of the electrode. As discussed in Section 2.4, single
crystals with different crystallographic orientations generate distinct in-
plane stresses. In our experiment, the polycrystalline electrode contains
grains of different orientations and we measure its net stress response using
the MOSS technique. The average stress response in Fig. 8(a-b) first decreases
during lithiation (for $x\leq 2$) and increases thereafter (for $x>2$). This
stress evolution as a function of Li-content is similar to the stresses
predicted by our analytical model for a single crystal
$\mathrm{Li_{x}V_{2}O_{5}}$ with $c-$axis in the $\mathrm{e_{1}-e_{2}}$ plane,
see Fig. 5(b-c). This similarity in stress responses suggests that the
majority of the grains in the polycrystalline electrode are oriented with
their $c-$axis in the $\mathrm{e_{1}-e_{2}}$ plane. In a future study, we plan
to map the stress evolution as a function of the electrode’s individual grain
orientations.

Overall, we note three key observations in our experiments: First, in the
potentiostatic experiment, the $\mathrm{V_{2}O_{5}}$ electrode on Al substrate
retains crystallinity upon repeated cycling in the 2–4V operational window
while the $\mathrm{V_{2}O_{5}}$ electrode on Au substrate loses crystallinity
(or amorphizes) after three cycles. Second, in the galvanostatic experiment,
the voltage plateaus corresponding to $\mathrm{V_{2}O_{5}}$ phase
transformations are shifted to lower values with an increase in the initial
film strain. This shift of voltage plateaus, specifically the lowering of
$\gamma-\omega$ plateau, expands the operational voltage window for lithium
batteries. Third, the in-plane stresses, in both potentiostatic and
galvanostatic experiments, evolve as a function of Li-content and are
anisotropic in the $\mathrm{Li_{\mathit{x}}V_{2}O_{5}}$ thin-film electrode.

##  4 Discussion

Our theoretical and experimental studies demonstrate that the film strain
enhances the reversible cycling of thin-film $\mathrm{V_{2}O_{5}}$ cathodes in
the 2–4V voltage window. The theoretical calculations predict that the film
strains shift the voltage plateaus to lower values and circumvent the enormous
volume changing phase transformation
($\mathrm{Li_{2}V_{2}O_{5}-Li_{3}V_{2}O_{5}}$). This prediction was confirmed
in our experiments that demonstrate structural reversibility of
$\mathrm{V_{2}O_{5}}$, with film stresses $\sim 2\mathrm{GPa}$ and across a
wider voltage window 2–4V. In the remainder of this discussion, we discuss the
limiting conditions on our theoretical and experimental framework and then
compare our results with prior work on phase transformation electrodes.

Two features of this work limit the conclusion we can draw about the effect of
film strain on reversible cycling of thin-film electrodes. First, we derive
the theoretical model by assuming the thin-film electrode to be a single
crystal, and thus our predictions on voltage curves and stress responses are
subject to the shortcomings associated with the presence of grain boundaries
and other defects commonly found in a polycrystalline material. Second, we
account for the effect of film stress in the variational derivative of the
free energy (i.e., chemical potential) and do not consider the effect of
diffusion kinetics on voltage measurements. Prior research shows that the
internal stress states (e.g., arising from film strains) can alter diffusion
rates (Ganser et al., 2019; Sultanova and Figiel, 2021) and affect the
mechano-chemistry of batteries (Bucci et al., 2017). In order to capture this
effect, the diffusion kinetics would need to be coupled with the evolving
stresses in the electrode. This effect is currently not captured in the
present theory and we plan to address it in a future study.

Our findings differ from prior research on suppressing phase transformation in
intercalation electrodes. First, unlike previous studies (Shimizu et al.,
1993; Baddour-Hadjean et al., 2012), we do not constrain the operational
voltage window between 3–4V to prevent the $\gamma-\omega$ phase
transformation in $\mathrm{V_{2}O_{5}}$ electrodes. The film strains enable us
to electrochemically cycle$\mathrm{V_{2}O_{5}}$ across a wider voltage window
2–4V, which increases the specific capacity of the battery. Prior research on
thin-film $\mathrm{V_{2}O_{5}}$ electrodes similarly report a wider voltage
window (McGraw et al., 1998). Second, we need not engineer electrode geometry
(for example nano-sized particles) or Li-kinetics (increased C-rates) to
suppress phase separation. The film strains can be engineered to shift the
voltage plateaus to lower values, which enables the reversible cycling of
$\mathrm{V_{2}O_{5}}$ despite its thin-film geometry or small C-rates.

Compared to other electrode materials such as Si-Li alloys, our findings show
that, for $\mathrm{V_{2}O_{5}}$, the elastic energies have a more complex
impact on the two-phase equilibria that are associated with voltage plateaus.
In Si-Li alloys deleterious phase transformations are known to limit their
voltage range for reversible cycling. That is, voltages below 0.05 V are
generally avoided to prevent its transformation to the crystalline
$\mathrm{Li_{4}Si_{15}}$ phase (Obrovac and Christensen, 2004). Since this
transformation involves a structural change at a fixed composition, the
voltage plateaus that feature prominently in $\mathrm{V_{2}O_{5}}$ are not
observed. However, elastic energy effects are still potentially relevant for
Si-Li alloys and tensile strains should inhibit the formation of the
$\mathrm{Li_{4}Si_{15}}$ phase which is accompanied by a volume decrease.
Another distinguishing feature between $\mathrm{V_{2}O_{5}}$ and Si-Li alloys
is that the stress-induced voltage shifts in the $\mathrm{V_{2}O_{5}}$
experiments are relatively large when compared to Si-Li alloys. For example,
in a single phase electrode material like Si, a 0.5 GPa change in the in-plane
stress corresponds to a voltage shift of roughly 0.03 V, whereas in our
experiments the comparable shifts for the $\delta-\gamma$ and $\gamma-\omega$
plateaus are much larger than this. These examples show that elastic energies
play an important role in regulating phase transformation voltages in
electrodes beyond $\mathrm{V_{2}O_{5}}$.

More generally, the present work establishes a theoretical and experimental
framework to enhance the lifespans of intercalation electrodes. Specifically,
we show that film strains can modulate the free energy landscape of phase
transformation electrodes. These strains can be engineered precisely (e.g., by
using shape-memory alloy substrates (Muralidharan et al., 2017)) to control
the voltage plateaus for phase transformation, and can be calibrated to offer
wider voltage windows for battery operation. Beyond $\mathrm{V_{2}O_{5}}$, the
study serves as a theoretical guide to tailor film strains on thin-film
electrodes and could be used as a design tool to engineer the mechano-
chemistry of batteries. Applying this to thin film electrodes is directly
relevant in certain microbatteries. Strain engineering strategies are also
potentially relevant in a broader range of practical electrodes, and here
implementation will require approaches that reflect the structures of the
specific materials.

##  5 Conclusion

To conclude, we use a combination of theory and experiments to show that film
strains regulate phase transformations in thin-film intercalation electrodes
and that film strains can be engineered to prevent chemo-mechanical
degradation of electrodes. Specifically, our findings show that tensile
strains lower the voltage for phase transformation in $\mathrm{V_{2}O_{5}}$
thin-film electrodes and facilitate their reversible cycling across a 2–4V
voltage window that is wider than previously achieved. More broadly, our
research shows a novel route of engineering film strains of thin-film
electrodes that make intercalation electrodes with longer lifespans possible.

##  6 Acknowledgement

The authors acknowledge the High-Performance computing center at the
University of Southern California for providing resources that contributed to
the research results reported within this paper. A.R.B acknowledges the
support of the Provost Fellowship and start-up funds from the University of
Southern California. Research at Brown University was supported by the
Ceramics Program at the National Science Foundation (DMR-1832829).

##  7 Appendix

###  7.1 Theoretical calculations

#### Phase transformation in $\mathrm{Li_{x}V_{2}O_{5}}$

Crystalline $\mathrm{V_{2}O_{5}}$ undergoes multiple phase transformations as
a function of Li-content that are characterized by constant voltage plateaus
(Christensen et al., 2018). Each voltage plateau corresponds to the co-
existence of two phases of $\mathrm{Li_{x}V_{2}O_{5}}$ as shown in Fig. 7. For
example, consider the $\alpha-\epsilon$ phase transformation stage (see Fig.
9(a)), at the beginning of this voltage plateau Li is inserted into the host
material that induces phase transformation from the $\alpha$ to the $\epsilon$
phase. On further lithiation the composition of the $\alpha$ or $\epsilon$
phases does not change, however the $\epsilon$ phase grows in size at the
expense of the $\alpha$ phase. This growth is marked by the movement of a
phase boundary and both the $\alpha,\epsilon$ phases coexist on this voltage
plateau. At the end of the voltage plateau all of the $\alpha$ phase is
transformed to the $\epsilon$ phase. This two phase behaviour is widely
reported for the $\mathrm{Li_{x}V_{2}O_{5}}$ compound for all its phase
transformation stages (Wang et al., 2006; Christensen et al., 2018).

In our analytical calculations we assume a double-well potential to describe
the two phase coexistence at each voltage plateau in
$\mathrm{Li_{x}V_{2}O_{5}}$. Other possible configurations, such as a pre-
existing nucleus of a third phase at a grain boundary are possible, however,
in the absence of specific data on these microstructures, the two phase
configuration described above provides a fundamental framework to investigate
the role of elastic energy on the voltage plateaus. Similar two phase
configurations are assumed to model phase transformations in other materials,
such as ferroelectrics, shape memory alloys, and ferromagnets that show series
of solid-solid phase transformation stages as a function of temperature (Chen,
2008; Mamivand et al., 2013). In line with previous research, we model the
two-phase coexistence of $\mathrm{Li_{x}V_{2}O_{5}}$ at each voltage plateau
using a double-well potential.

We next describe how the solubility limits and enthalpy of mixing are derived
for a two phase transformation in the $\mathrm{Li_{x}V_{2}O_{5}}$ system.
These thermodynamic properties of the material are fitted with the voltage
measurements for $\mathrm{Li_{x}V_{2}O_{5}}$ at each transformation stage.
Fig. 9(b) shows a free energy curves for a representative two phase
transformation. In our analysis we describe this free energy curve using the
standard regular solution model, see Eq. (6). The regular solution model has
been previously used to describe phase transformations in intercalation
electrodes, such as LiFePO4, LiCoO2, $\mathrm{Li_{x}V_{2}O_{5}}$ (Tang et al.,
2011; Nadkarni et al., 2019; Bai et al., 2021). We calibrate the coefficients
of the regular solution model to fit with the experimentally measured
thermodynamic and elastic properties of the material.

Figure 9: Schematic showing (a) the representative phase transformation (from
phase $\alpha$ to phase $\epsilon$) upon lithiation in a intercalation
material, and (b) graphical representation of the free energy curves of two
phases, $\alpha$ and $\epsilon$ with (solid lines) and without (dashed lines)
strain energy contributions. $G_{\alpha}^{eqm}$ and $G_{\epsilon}^{eqm}$ are
equilibrium molar Gibbs free energy for phase $\alpha$ and $\epsilon$ in the
case without strain energy contribution. $c_{\alpha}$ and $c_{\epsilon}$ are
the equilibrium Li-content in the $\alpha$ and $\epsilon$ phases respectively.
$\mathrm{F}\Delta\phi^{*}$ and $\mathrm{F}\Delta\phi$ are the chemical driving
forces with and without the strain energy, respectively.

For example, in Fig. 9(b), the $c_{\alpha}$ and $c_{\epsilon}$ are the
equilibrium Li-content in the $\alpha$ and $\epsilon$ phases
respectively.999In our calculation we normalize this Li-content to vary as
$0\leq c\leq 1$. These equilibrium Li-contents are derived from the voltage
curves and serve as the solubility limits for the binary model. In our
calculations we normalize the solubility limits for each two-phase
transformation stage, i.e., $0\leq c\leq 1$. Next, the chemical potential
difference between the active material (i.e., $\mathrm{V_{2}O_{5}}$) and the
counter electrode (i.e., Li-metal in our experiments) serves as the chemical
driving force for Li-ion exchange in the active material. We model this
driving force in terms of the Gibbs free energy change per mole of reaction:
$\Delta\psi=-z\mathrm{F}\Delta\phi$. Here, $z=+1$ is the charge number of
mobile Li-ion species, $\mathrm{F}$ is the Faraday constant, and $\Delta\phi$
is the potential difference between the two electrodes. The balance between
the chemical and electrical forces on lithium exchange imply that the
$\Delta\phi$ corresponds to the two phase equilibrium voltage plateaus in
$\mathrm{Li_{x}V_{2}O_{5}}$ during phase transformation. That is the common
tangent in Fig. 9(b) corresponds to the equilibrium voltage for the two phase
existence and is used to fit the values for the enthalpy of mixing. We use
this strategy to fit the solubility limits and equilibrium voltage values for
the remainder of the phase transformation stages (e.g.,
$\epsilon-\delta,\delta-\gamma,\gamma-\omega$ stages). While the exact shape
of the curves and the energy barrier is approximate, this two-phase
configuration provides the basic analytical framework to investigate the role
of elastic energy on phase transformation voltages. Furthermore, a detailed
model following the multi phase field approach (Steinbach and Apel, 2006)
would be necessary to construct a free energy functional to describe a
functional for a multi-phase transformation, however, this approach is beyond
the current scope of our work.

#### Model parameters

$\mathrm{V_{2}O_{5}}$ undergoes multiple phase transformation stages, and we
assume piecewise functions to describe the energy landscape at each phase
transformation stage. The thermodynamic and elastic constants are fitted to
the electrochemical (Luo et al., 2018) and lattice parameter measurements
(Baddour-Hadjean et al., 2012; Murphy et al., 1979; Cava et al., 1986;
Enjalbert and Galy, 1986; Cocciantelli et al., 1991; Dickens et al., 1979;
Meulenkamp et al., 1999; Satto et al., 1999) accordingly. The table below
lists the constants and polynomial functions used in our calculations.

Thermodynamic constants | Value
---|---
Gradient energy constant | $\kappa=3.084\times 10^{-18}\mathrm{J/m}$
Enthalpy of mixing |
$\alpha-\epsilon$ | $\Omega_{1}={\color[rgb]{0,0,0}5800}\mathrm{J/mol}$
$\epsilon-\delta$ | $\Omega_{2}={\color[rgb]{0,0,0}3571}\mathrm{J/mol}$
$\delta-\gamma$ | $\Omega_{3}={\color[rgb]{0,0,0}2890}\mathrm{J/mol}$
$\gamma-\omega$ | $\Omega_{4}={\color[rgb]{0,0,0}3587}\mathrm{J/mol}$
Temperature | $T=298\mathrm{K}$
Gas constant | $R=8.314\mathrm{J/mol\cdot K}$
|
Elastic constants | Value
Stiffness constants | $c_{11}={\color[rgb]{0,0,0}153.685}\mathrm{GPa}$
| $c_{12}={\color[rgb]{0,0,0}65.865}\mathrm{GPa}$
| $c_{44}={\color[rgb]{0,0,0}43.67}\mathrm{GPa}$
Preferred strain, see Fig. 10 | $\epsilon_{\alpha}={\color[rgb]{0,0,0}0.024c^{3}-0.105c^{2}+0.035c}$
| $\epsilon_{\beta}={\color[rgb]{0,0,0}0.001c^{3}-0.007c^{2}+0.015c}$
| $\epsilon_{\gamma}={\color[rgb]{0,0,0}-0.025c^{3}+0.053c^{2}+0.119c}$

Table 1: Specific values of coefficients used for analytical calculations in
section 2. These values correspond to phase transformations in
$\mathrm{V_{2}O_{5}}$ electrode and are fitted to the experimental data
provided by Luo et al. (2018)

.

Figure 10: Lattice strains in $\mathrm{Li_{\mathit{x}}V_{2}O_{5}}$ as a
function of Li-content. We fit polynomials with the strains provided by
Baddour-Hadjean et al. (2012); Murphy et al. (1979); Cava et al. (1986);
Enjalbert and Galy (1986); Cocciantelli et al. (1991); Dickens et al. (1979);
Meulenkamp et al. (1999); Satto et al. (1999).

### Analytical calculation of elastic energy with thin film conditions

The general form of elastic energy $\psi_{elas}$:

$\displaystyle\psi_{elas}(c,\mathbf{E})$ $\displaystyle=\frac{1}{2}[{\bf
E}-{\bf E}_{\rm 0}(c)]\ :\ \mathbb{C}[{\bf E}-{\bf E}_{\rm 0}(c)]$
$\displaystyle=\mathrm{C_{44}}[{\bf E}-{\bf E}_{\rm 0}({c})]\ :\ [{\bf E}-{\bf
E}_{\rm 0}({c})]+{\rm{\frac{\mathrm{C_{12}}}{2}}}[({\bf E}-{\bf E}_{\rm
0}({c})]{\color[rgb]{0,0,0}\ :\ }{\rm tr}[{\bf E}-{\bf E}_{\rm 0}({c})]{\bf
I}$
$\displaystyle+\big{(}\frac{\mathrm{C_{11}-C_{12}}}{2}-\mathrm{C_{44}}\big{)}\sum_{i=1}^{3}({\bf
e}_{i}\cdot[{\bf E}-{\bf E}_{\rm 0}({c})]{\bf e}_{\it i})^{2}$ (15)

Note, $[{\bf E-\bf E}_{\rm 0}(c)]{\color[rgb]{0,0,0}\ :\ }[{\bf E-\bf E}_{\rm
0}(c)]$ expands to:

$\displaystyle[{\bf E}-{\bf E}_{\rm 0}(c)]{\color[rgb]{0,0,0}\ :\ }[{\bf
E}-{\bf E}_{\rm 0}(c)]=\sum_{{\rm i=1}}^{\rm 3}({{\bf e}}_{\it i}\cdot[{\bf
E}-{\bf E}_{\rm 0}(c)]{{\bf e}}_{\it i})^{\rm 2})+\
2(\mathrm{E^{2}_{12}}+\mathrm{E^{2}_{23}}+\mathrm{E^{2}_{13}})$ (16)

By substituting Eq. 16 in Eq. 15 we get:

$\displaystyle\psi_{elas}(c,\mathbf{E})$ $\displaystyle=\frac{1}{2}[{\bf
E}-{\bf E}_{\rm 0}(c)]{\color[rgb]{0,0,0}\ :\ }\mathbb{C}[{\bf E}-{\bf E}_{\rm
0}(c)]$
$\displaystyle=2\mathrm{C_{44}}(\mathrm{E^{2}_{12}}+\mathrm{E^{2}_{23}}+\mathrm{E^{2}_{13}})+\frac{\mathrm{C_{12}}}{2}[\mathrm{E_{11}}-\epsilon_{\alpha}+\epsilon_{22}-\epsilon_{\beta}+\mathrm{E_{33}}-\epsilon_{\gamma}]^{2}$
$\displaystyle+\frac{\mathrm{C_{11}}-\mathrm{C_{12}}}{2}\bigg{[}(\mathrm{E_{11}}-\epsilon_{\alpha})^{2}+(\mathrm{E_{22}}-\epsilon_{\beta})^{2}+(\mathrm{E_{33}}-\epsilon_{\gamma})^{2}\bigg{]}$
(17)

Next taking plane stress conditions for the thin film, we have
$\mathrm{E_{13}}=\mathrm{E_{23}}=0$, and $\mathrm{E_{33}}$ is derived from Eq.
11. Furthermore, the mechanical constraints for the thin film conditions are
given by $\mathrm{E_{11}}=\mathrm{E_{22}}=\mathrm{E_{s}}$ and
$\mathrm{E_{12}}=0$. We substitute these conditions in the above elastic
energy expression:

$\displaystyle\psi_{elas}[c,\mathrm{E_{s}}]=\int_{{\mathcal{E}}}$
$\displaystyle\frac{\mathrm{C_{11}}-\mathrm{C_{12}}}{2}[(\mathrm{E_{11}}-\epsilon_{\alpha})^{2}+(\mathrm{E_{22}}-\epsilon_{\beta})^{2}+(\mathrm{E_{33}}-\epsilon_{\gamma})^{2}]$
$\displaystyle+\frac{\mathrm{C_{12}}}{2}[\mathrm{E_{11}}-\epsilon_{\alpha}+\mathrm{E_{22}}-\epsilon_{\beta}+\mathrm{E_{33}}-\epsilon_{\gamma}]^{2}\mathrm{d}\mathbf{x}.$
(18)

Substituting for film strains:

$\displaystyle\psi_{elas}[c,\mathrm{E_{s}}]=\int_{\mathcal{E}}$
$\displaystyle\frac{\mathrm{C_{11}}-\mathrm{C_{12}}}{2}\bigg{[}(\mathrm{E_{s}}-\epsilon_{\alpha})^{2}+(\mathrm{E_{s}}-\epsilon_{\beta}])^{2}+\bigg{(}\frac{\mathrm{C_{12}}}{\mathrm{C_{11}}}[\epsilon_{\alpha}+\epsilon_{\beta}-2\mathrm{E_{s}}]\bigg{)}^{2}\bigg{]}$
$\displaystyle+\frac{\mathrm{C_{12}}}{2}[\mathrm{E_{s}}-\epsilon_{\alpha}+\mathrm{E_{s}}-\epsilon_{\beta}+\frac{\mathrm{C_{12}}}{\mathrm{C_{11}}}(\epsilon_{\alpha}+\epsilon_{\beta}-2\mathrm{E_{s}})]^{2}\mathrm{d}\mathbf{x}.$
(19)

### Analytical calculation of in-plane stresses

We define stresses as
$\mathbf{\sigma}=\frac{\partial\Psi}{\partial\mathbf{E}}$, and assuming a
linear elastic cubic relation, we have:

$\displaystyle\sigma_{11}=\mathrm{C_{11}}[\mathrm{E_{11}-\epsilon_{\alpha}}]+\mathrm{C_{12}}[\mathrm{E_{33}-\epsilon_{\gamma}}+\mathrm{E_{22}-\epsilon_{\beta}}]$
$\displaystyle\sigma_{22}=\mathrm{C_{11}}[\mathrm{E_{22}-\epsilon_{\beta}}]+\mathrm{C_{12}}[\mathrm{E_{33}-\epsilon_{\gamma}}+\mathrm{E_{11}-\epsilon_{\alpha}}]$
$\displaystyle\sigma_{33}=\mathrm{C_{11}}[\mathrm{E_{33}-\epsilon_{\gamma}}]+\mathrm{C_{12}}[\mathrm{E_{11}-\epsilon_{\alpha}}+\mathrm{E_{22}-\epsilon_{\beta}}]$
(20)

Next, we substitute $\mathrm{E_{11}=E_{s}}$, $\mathrm{E_{22}=E_{s}}$, and from
Eq. 11
$\mathrm{E_{33}=\frac{\mathrm{C_{12}}}{\mathrm{C_{11}}}}[(\epsilon_{\alpha}+\epsilon_{\beta}-2E_{s}]+\epsilon_{\gamma}.$
Note, the values of $\epsilon_{\alpha},\epsilon_{\beta},\epsilon_{\gamma}$
change as a function of Li-content($c$). Consequently the in-plane stresses
evolve as a function of Li-content as:

$\displaystyle\sigma_{11}$
$\displaystyle=\mathrm{C_{11}}\mathrm{[E_{s}-\epsilon_{\alpha}}]+\mathrm{\frac{C_{12}^{2}}{C_{11}}}[\epsilon_{\alpha}+\epsilon_{\beta}-2\mathrm{E_{s}}]+\mathrm{C_{12}}[\mathrm{E_{s}-\epsilon_{\beta}}]$
$\displaystyle=(\mathrm{C_{11}+C_{12}-2\frac{C_{12}^{2}}{C_{11}}})\mathrm{E_{s}+(-C_{11}}+\mathrm{\frac{C_{12}^{2}}{C_{11}})\epsilon_{\alpha}}+(\mathrm{-C_{12}+\frac{C_{12}^{2}}{C_{11}}})\epsilon_{\beta}$
$\displaystyle\sigma_{22}$
$\displaystyle=\mathrm{C_{11}}[\mathrm{E_{s}-\epsilon_{\beta}}]+\mathrm{\frac{C_{12}^{2}}{C_{11}}}[\epsilon_{\alpha}+\epsilon_{\beta}-2\mathrm{E_{s}}]+\mathrm{C_{12}}[\mathrm{E_{s}-\epsilon_{\alpha}}]$
$\displaystyle=\mathrm{(C_{11}+C_{12}-2\frac{C_{12}^{2}}{C_{11}})}\mathrm{E_{s}+\mathrm{(-C_{12}}+\frac{C_{12}^{2}}{C_{11}})\epsilon_{\alpha}}+(\mathrm{-C_{11}+\frac{C_{12}^{2}}{C_{11}})\epsilon_{\beta}}$
$\displaystyle\sigma_{33}$
$\displaystyle=\mathrm{C_{12}}[\epsilon_{\alpha}+\epsilon_{\beta}-2\mathrm{E_{s}}]-\mathrm{C_{11}}[\epsilon_{\alpha}+\epsilon_{\beta}-\mathrm{2E_{s}}]$
$\displaystyle=0$ (21)

The analytical stresses in Eq. 21 are homogeneous in-plane stresses that
evolve as a function of Li-content in $\mathrm{Li_{x}V_{2}O_{5}}$. These
analytical expressions could be applied in a computational framework in which
a spatially resolved Li-content $c(\mathbf{x})$ (i.e., composition
microstructure) would be used to predict heterogeneous stress distribution in
the thin-film $\mathrm{Li_{x}V_{2}O_{5}}$. We propose to study the
heterogeneous stress distribution in a future work. For current purposes we
analytically model the in-plane stress evolution during phase transformations
and compare it with the stresses measured experimentally using the MOSS
technique.

###  7.2 Thin-film experiment

In this section we give details of the thin-film preparation,
characterization, electrochemical cycling and measurements.

#### Film Preparation

Fig. 1(a) shows a schematic illustration of the thin-film electrode we
prepared for the experiment. We deposited $\mathrm{50nm}$ film of
$\mathrm{V_{2}O_{5}}$ electrode on two different substrates, namely, an
Aluminium-coated-quartz substrate and a Gold-coated-quartz substrate. The
substrates were $\mathrm{250\mu m}$ thick and $25.4\mathrm{mm}$ in diameter.
The substrates were thoroughly cleaned in acetone, ethanol, and deionized
water, and were mounted in an electron-beam evaporation system in order to
deposit the metallic current collectors (Al or Au). For the Au specimen, 10nm
of Titanium was first deposited on quartz ($\mathrm{SiO_{2}}$) to improve
adhesion between the Gold and quartz substrate. Finally, the deposited
$\mathrm{V_{2}O_{5}}$ layers were annealed at $550^{\circ}\mathrm{C}$ in the
air for three hours, in order to crystallize and oxidize them. Further details
in preparing the thin-film substrate are provided by Sheth (2017).

#### Film characterization

We characterized the thin films in two ways: First, we used scanning electron
microscopy (SEM) to characterize the microstructures of the thin-film
electrodes. Figs. 6(a-b) shows the SEM images of the $\mathrm{V_{2}O_{5}}$
surface on Al and Au substrates before electrochemical cycling.
$\mathrm{V_{2}O_{5}}$ films on both Al and Au substrates are crystalline in
the initial state. The crystallinity of these films was also verified by
characterizing them using transmission electron microscopy (TEM), see Fig. 11.
Second, we carried out X-ray diffraction (XRD) with a diffractometer using Cu
$K_{\alpha}$ radiation ($\lambda=1.5406\mbox{\AA}$) at glancing angles with
the films. Figs. 6(c) show the XRD measurements of $\mathrm{V_{2}O_{5}}$ films
deposited on Al and Au substrates respectively. In the initial state,
$\mathrm{V_{2}O_{5}}$ films on both Al and Au substrates crystallize into the
orthorhombic structure, with a dominant peak corresponding to the
$\mathrm{V_{2}O_{5}}(001)$ crystallographic direction. Other peaks correspond
to the current collectors (Al or Au). The presence of a single dominant peak
for $\mathrm{V_{2}O_{5}}(001)$ suggests a polycrystalline texture of the thin-
film electrode. Note, there is a bump in the XRD measurement on the Al
substrate—we suspect that this arises from the quartz substrate present
beneath the Al layer. In the case of Au, an in-between Ti layer shields the
XRD peak corresponding to the quartz substrate. Further details and
specifications of the instruments used in the thin-film characterization is
provided by Sheth (2017).

Figure 11: TEM image of the $\mathrm{V_{2}O_{5}}$ thin-film electrode on Au
substrate showing its lattice fringes and the corresponding fast Fourier
transformation of the TEM image confirms the crystallinity of
$\mathrm{V_{2}O_{5}}$ before electrochemical cycling.

#### Electrochemical Cycling

We designed four custom-made electrochemical cells (labelled A,B,C,D)
consisting of $\mathrm{V_{2}O_{5}}$ thin-film cathode, Li-metal anode, and a
liquid electrolyte (equimolar mixture of Ethylene Carbonate, Dimethyl
Carbonate and 1M $\mathrm{LiPF_{6}}$ salt. In two of these cells (labelled
A,B), the $\mathrm{V_{2}O_{5}}$ was deposited on Al-coated-quartz substrate,
and in the other two cells (labeled C,D) $\mathrm{V_{2}O_{5}}$ was deposited
on Au-coated-quartz substrate. We divided the four electrochemical cells into
two studies: (a) Galvanostatic experiment: electrochemical cells A and C were
charged/discharged under constant current density ($\mathrm{0.2\mu A/cm^{2}}$)
and at a rate of C/20, and (b) Potentiostatic experiment: electrochemical
cells B and D were cycled under constant voltage (defined here as current
$<4\times 10^{-8}\mathrm{A})$. Next, we give an overview of the
electrochemical and stress measurements made in-situ in the two studies.

#### Electrochemical and Stress Measurements

Voltage measurement: During each cycle, the electrochemical response (namely
the voltage) was recorded. Fig. 7 shows the voltage plateaus for
$\mathrm{V_{2}O_{5}}$ thin-film electrodes deposited on Al and Au current
collectors, respectively, and cycled galvanostatically. These voltage plateaus
resemble the characteristic features of the voltage curve for
$\mathrm{V_{2}O_{5}}$ electrode measured in previous works (Christensen et
al., 2018), however, the exact voltage values at which these plateaus appear
are shifted to lower values. As predicted the film strain imposed on the thin-
film electrode plays a key role in shifting the voltage curves and we discuss
this in detail in Sections 2 and 3.

Stress measurement: During each cycle, the in-plane stresses generated during
Li insertion and extraction were measured in-situ using the multi-beam optical
stress sensor (MOSS) technique. This technique employs an array of parallel
laser beams that are focussed on the underside of the quartz substrate and
help monitor the changes in substrate curvature during electrochemical
cycling. The substrate curvature $\kappa$ is used as a measure to estimate
average in-plane stresses $<\sigma>$ in the thin-film using Stoney’s equation,
$<\sigma>h_{\mathrm{film}}=\frac{M_{\mathrm{sub}}h_{\mathrm{sub}}^{2}\kappa}{6}.$
Here, $h_{\mathrm{film}}$ and $h_{\mathrm{sub}}$ are the thickness of the film
and substrate respectively, and $M_{\mathrm{sub}}$ is the biaxial modulus of
the substrate. Specific details of measuring stresses using the MOSS technique
are described by Sheth (2017); Sethuraman et al. (2010); Tripuraneni et al.
(2018). The in-plane stresses measured during the Galvanostatic experiment are
shown in blue in Fig. 8. These stresses are sensitive to the chemo-mechanical
strains in $\mathrm{V_{2}O_{5}}$ electrode during electrochemical cycling and
evolve as a function of Li-content.

XRD and SEM measurements: We conducted post-cycling XRD measurements after
three charge/
discharge potentiostatic cycles of the $\mathrm{V_{2}O_{5}}$ electrode. The
corresponding XRD peaks are shown in Fig. 6(c). For $\mathrm{V_{2}O_{5}}$
deposited on Al-coated-quartz substrate the diffraction peak corresponding to
$\mathrm{V_{2}O_{5}}(001)$ broadens with repeated cycling but does not
disappear. This shows that $\mathrm{V_{2}O_{5}}$ undergoes structural
transformations during lithiation/delithiation, but retains its crystallinity
upon repeated cycling. However, for $\mathrm{V_{2}O_{5}}$ deposited on Au-
coated-quartz substrate $\mathrm{V_{2}O_{5}}(001)$ peak disappears after 3
cycled indicating a loss of crystallinity (or amorphization) in the
$\mathrm{V_{2}O_{5}}$ thin-film. This was further confirmed by SEM images that
show a powdery microstructure after 3 cycles on $\mathrm{V_{2}O_{5}}$
deposited on gold substrate, see Fig. 6(b).

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