# Nanoscale domain patterns and a concept for an energy harvester

Ananya Renuka Balakrishna  Department of Engineering Science, University of
Oxford, Oxford, OX1 3PJ, United Kingdom  John E. Huber  Department of
Engineering Science, University of Oxford, Oxford, OX1 3PJ, United Kingdom

###### Abstract

Grain boundary migration is driven by the boundary’s curvature and external
loads such as temperature and stress. In intercalation electrodes an
additional driving force results from Li-diffusion. That is, Li-intercalation
induces volume expansion of the host-electrode, which is stored as elastic
energy in the system. This stored energy is hypothesized as an additional
driving force for grain boundaries and edge dislocations. Here, we apply the
2D Cahn-Hilliard – phase-field-crystal (CH-PFC) model to investigate the
coupled interactions between highly mobile Li-ions and host-electrode lattice
structure, during an electrochemical cycle. We use a polycrystalline FePO4 /
LiFePO4 electrode particle as a model system. We compute grain growth in the
FePO4 electrode in two parallel studies: In the first study, we
electrochemically cycle the electrode and calculate Li-diffusion assisted
grain growth. In the second study, we do not cycle the electrode and calculate
the curvature-driven grain growth. External loads, such as temperature and
stress, did not differ across studies. We find the mean grain-size increases
by $\sim 11\%$ in the electrochemically cycled electrode particle. By
contrast, in the absence of electrochemical cycling, we find the mean grain-
size increases by $\sim 2\%$ in the electrode particle. These CH-PFC
computations suggest that Li-intercalation accelerates grain-boundary
migration in the host-electrode particle. The CH-PFC simulations provide
atomistic insights on diffusion-induced grain-boundary migration, edge
dislocation movement and triple-junction drag-effect in the host-electrode
microstructure.

Key words: grain growth, phase-field crystal model, intercalation electrodes,
batteries

Li-diffusion accelerates grain growth in intercalation electrodes: a phase-
field study

Ananya Renuka Balakrishna*, Yet-Ming Chiang and W. Craig Carter

Department of Materials Science and Engineering, Massachusetts Institute of
Technology, Cambridge, MA02139, USA

∗Email<EMAIL_ADDRESS>

## Introduction

An ion storage electrode used in rechargeable batteries typically consists of
polycrystalline (secondary) particles. The electrode’s volume can
substantially expand (e.g., up to $\sim 10\%)$ as the working ion
concentration is varied [1]. This expansion is often anisotropic [2], and
sometimes leads to mechanical fracture of the electrode [3]. Mechanical
fracture impacts a battery’s electrochemical properties, such as accessible
capacity, rate capability, and lifetime [4]. An electrode’s physical
properties, such as fracture toughness and electrochemical potential,
correlate with its grain size in polycrystalline particles [5],[6]. Thermal
loads, as well as dopants, are known to affect grain size [7, 8]. At elevated
temperatures, in-situ electric currents also cause grain growth in
electrochemical systems [7][9]. To date, we are not aware of any work that
explores how cyclic intercalation affects grain growth in a battery electrode
at ambient temperatures. Here we examine whether – and, if so, to what extent
– electrochemical cycling enhances grain growth, using FePO4 / LiFePO4 as a
model system.

During an electrochemical cycle, Li-ions intercalate into a host electrode,
such as FePO4 / LiFePO4 [10]. This insertion and extraction of Li-ions
(diffusing species) produces lattice strains in the host material, and
transforms lattice structure at critical values of Li-compositions. In a
polycrystalline electrode, these lattice transformations generate an
inhomogeneous strain field [2]. In phase separating materials, such as the
FePO${}_{4},$ an additional strain arises between the lithiated and
delithiated phases. Handwerker et al. [11, 12] propose that these internal
strain fields drive grain boundaries in the host electrode. In the present
work, we apply a Cahn-Hilliard – phase-field-crystal model to demonstrate this
diffusion-induced grain-boundary migration in an FePO4 host-electrode.

Researchers have developed several theoretical models to investigate electrode
microstructures in lithium batteries [13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25]. Based on length scales, these theroetical models can be broadly
clustered into continuum and atomistic methods. The continuum methods, such as
the phase field models, describe the time evolution of electrode
microstructures. The phase field models have been applied to investigate the
role of electrode-particle size on miscibility gap [15], phase separation [16]
and stress generation [26]. A crystallographic phase-field model builds on the
classic phase-field approach by introducing grain orientation as an additional
order parameter [27]. The continuum models are developed to consider
electrochemistry [19], surface morphologies [23], and coherency strains [14,
17]. These methods, however, do not model individual lattice distortions. The
microstructural defects such as grain boundaries are empirical parameters in
these continuum methods.

Atomistic models such as the first principle methods [21] and molecular
dynamics models [24, 25], provide insights on electrochemical stability of
ionic compounds [21], lattice distortions [20, 22] and ionic diffusion in
electrodes [24]. However, these approaches are limited in length and time
scales. For example, the molecular dynamics method can typically model few
hundreds of atoms. Consequently, microstructural features such as grain
boundaries are computationally expensive to simulate using these atomistic
approaches. Furthermore, the time scales involved in molecular dynamics
computations are over thermal-vibrations. These time scales limits us to model
continuum Li-diffusion. Accelerated molecular dynamics methods use the
transition state theory to overcome the time scale limitation [28], and have
been applied to simulate crystal growth [29].

Alternatively in our recent work [30], we combined a Cahn-Hilliard model with
a phase-field-crystal model (CH-PFC). This model describes phase transitions
and interstitial diffusion in a 2D theoretical framework. In this multiscale
approach, the PFC equation models the coarse-grained lattice structure of the
host-electrode material (atomistic scale). The CH equation models Li-
intercalation in the electrode (continuum scale). The coupled CH-PFC model
captures the host-electrode lattice transformations induced from Li-diffusion.
This modeling approach enables us to investigate the time evolution of lattice
arrangements in electrode particles. For example, the model simulates
dislocation movement and grain-boundary migration in electrodes during an
electrochemical cycle.

In this paper, we apply the 2D CH-PFC methods to model Li-intercalation in an
FePO4 electrode particle. First, we calibrate the CH-PFC model to describe the
lattice geometries of FePO4 and LiFePO4. Using the model, we simulate a
polycrystalline FePO4 electrode particle. We next compute two parallel
simulations: In the first, we electrochemically cycle the electrode particle
where Li-ions intercalate into FePO4 lattices. In the second study, we do not
cycle the FePO4 electrode particle. We compare the grain boundary migration in
the two studies. We hypothesize that Li-intercalation accelerates grain-
boundary migration in the cycled electrode. Throughout, we compare the CH-PFC
results with the existing literature on microstructures in electrode
materials.

## Cahn-Hilliard – phase-field-crystal model

In this section, we briefly explain the CH-PFC model applied to FePO4
electrodes. The details of the model are given in the appendix and in Ref.
[30]. For our purposes, we note that the CH-PFC model has two order
parameters: First, the peak-density field $\psi(\vec{x})$ that represents
coarse-grained lattices of the host electrode [31]. The lattice points of the
host electrode are constrained in space and do not hop or migrate. These
lattices, however, undergo displacive transformations to represent lattice
strains. Second, the composition field $c(\vec{x})$ that represents Li-
intercalation (mobile species) through the host electrode.

During electrochemical cycling, the FePO4 electrode typically phase-separates
into lithium-rich (LiFePO4) and lithium-deficient (FePO4) phases [10]. This
FePO4 / LiFePO4 phase transition is first order (with an abrupt change of bulk
lattice constants). In experiments, however, an interphase region LiXFePO4
with $0<X<1$ has been observed to form [32, 33]. This interphase region
reduces the lattice misfit at the FePO4 / LiFePO4 contact [32]. In the current
work, we approximate the interphase region to correspond to a diffuse phase
boundary. We model the lattices in the interphase region to have averaged
lattice parameters intermediate to FePO4 and LiFePO4 lattices. Both the
interstitial Li-composition $c(\vec{x})$ and the host-electrode lattice
structure $\psi(\vec{x})$ continuously change across the diffuse interface.
The CH-PFC model describes the normalized total free energy $\mathcal{F}$ of a
two-phase system as:

$\displaystyle\mathcal{F}$ $\displaystyle=\int\\{g(c)+|\nabla
c|^{2}+\gamma[f(r,\psi)+\frac{\psi}{2}G(\nabla_{c}^{2})\psi]\\}d\vec{x.}$ (1)

In Eq. 1, $c(\vec{x})$ is the fraction of interstitial sites in FePO4
electrode occupied by Li per unit volume. The polynomial $g(c)$ represents a
regular solution model [34, 13] that describes a double well potential. The
coefficients in $g(c)$ are normalized such that the double wells are at equal
heights with minima at $c(\vec{x})=0$ and $c(\vec{x})=1$. The two minimas
correspond to the delithiated (FePO4) and lithiated (LiFePO4) phases,
respectively.111The $X$ in LiXFePO4 and $c(\vec{x})$ are related by
normalization coefficients that rescale the double well potential described by
$g(c)$.  The gradient term $|\nabla c|^{2}$ is the penalty for the changing
Li-composition field across the FePO4 / LiFePO4 diffuse phase boundary. The
peak density field $\psi(\vec{x})$ is a time-averaged density field [31],
which in its periodic state describes the coarse-grained lattice geometry of
the electrode material. The term $\frac{\psi}{2}G(\nabla_{c}^{2})\psi$ is the
energy penalty resulting from the changing host-lattice structure across the
diffuse phase boundary. The difference in lattice geometries between the FePO4
/ LiFePO4 phases gives rise to a coherency strain across the phase boundary.
The operator $G(\nabla_{c}^{2})$ introduces the coupling between Li-
composition $c(\vec{x})$ and host-electrode lattice symmetry $\psi(\vec{x})$.
The constant $\gamma$ relates the free energy normalizations of the Cahn-
Hilliard and phase field crystal models, and is discussed in the appendix.The
value of $r(\vec{x})$ in $f(r,\psi)$ affects the type of solutions for the
peak density field $\psi(\vec{x})$. In the PFC literature, researchers have
used $r(\vec{x})$ as a proxy for temperature to model glassy to non-glassy
(crystalline) transition during crystallization [35, 36]. In this paper, we
use $r(\vec{x})$ as a proxy for the amorphous Li-reservoir in our model. For
example, with $r(\vec{x})=-0.2$, $\psi(\vec{x})$ is a constant at equilibrium
and Eq. 1 models an amorphous state. While for $r(\vec{x})=+0.2$,
$\psi(\vec{x})$ has a periodic wave-form solution at equilibrium that
describes a crystalline state for the electrode material. The value of
$r(\vec{x})$ is locally defined in the coordinate space to model the
electrode-reservoir system. There is a surface energy contribution resulting
from the difference between the crystalline electrode and amorphous reservoir
regions. In this paper, the CH-PFC model does not account for the surface
energy term. This surface energy contribution is important to investigate the
wetting effects on phase transitions [37, 38] and is a subject of future
study.

The Li-composition field is coupled to the electrode lattice geometry via the
coordinate transformation coefficients of the Laplace operator
$\nabla_{c}^{2}$. Both the FePO4 (FP) and LiFePO4 (LFP) lattices have an
orthorhombic symmetry. There are two independent variables, $\alpha$ and
$\beta$, which control FP / LFP lattice transformation. The variables,
$\alpha$ and $\beta$, interpolate the lattice parameters as a function of the
composition field,
$\alpha(c)=\alpha_{\mathrm{FP}}+(\alpha_{\mathrm{LFP}}-\alpha_{\mathrm{FP}})c$
and
$\beta(c)=\beta_{\mathrm{FP}}+(\beta_{\mathrm{LFP}}-\beta_{\mathrm{FP}})c$.
Here, $(\alpha_{\mathrm{FP}},\beta_{\mathrm{FP}})$ and
$(\alpha_{\mathrm{LFP}},\beta_{\mathrm{LFP}})$ correspond to the geometric
measurements of FePO4 and LiFePO4 lattices from Table 1. With Li-composition
$c=0$ and $c=1$, the transformation coefficients correspond to the lattice
parameters of FePO4 and LiFePO4 respectively. The Li-composition across the
diffuse phase boundary is a function $c(\vec{x})$ and the lattice is
transformed according to $\alpha(c(\vec{x}))$ and $\beta(c(\vec{x}))$. The
width of the diffuse phase boundary $c(\vec{x})$ is numerically calibrated
such that it spans over $\sim 4$ lattice spacings described by
$\psi(\vec{x})$.

| $\mathrm{a}_{2}(\textrm{\r{A}})$ | $\mathrm{a}_{3}(\textrm{\r{A}})$ | $\alpha=\frac{\mathrm{a_{2}}}{\mathrm{a}_{0}}$ | $\beta=\frac{\mathrm{a_{3}}}{\mathrm{a}_{0}}$
---|---|---|---|---
FePO4 (FP) | $9.821$ | $\mathrm{a_{0}}=4.788$ | $2.0512$ | $1$
LiFePO4 (LFP) | $10.334$ | $4.693$ | $2.1583$ | $0.9802$
Table 1: A list of lattice parameters $(\mathrm{a}_{2},\mathrm{a}_{3})$ for
FePO4 and LiFePO4 used in the CH-PFC simulations [39]. The variable
$\alpha,\beta$ are calculated using
$\mathrm{a_{0}}=\mathrm{a_{3}{}_{(FePO_{4})}=4.788\mathring{A}}$ as reference.
These coefficients describe rectangular geometries of FePO4 and LiFePO4
lattices at $c=0$ and $c=1$ respectively.

The time-dependent equations in the model are the Cahn-Hillliard equation for
Li-diffusion:

$\displaystyle\frac{\partial c}{\partial\tau}$
$\displaystyle=\nabla^{2}\frac{\delta\mathcal{F}}{\delta c},$ (2)

and the elastic-relaxation equation for the host lattice structure:

$\displaystyle\frac{\partial\psi}{\partial n}$
$\displaystyle=-\frac{\delta\mathcal{F}}{\delta\psi}+\frac{1}{L^{2}}\int\frac{\delta\mathcal{F}}{\delta\psi}d\vec{x}.$
(3)

In Eqs. 2 – 3, we assume that the elastic relaxation (equilibrating the peak
density field) is infinitely faster than the evolution of the composition
field. Consequently, we model the equilibrium lattice arrangements by
maintaining $\frac{\delta\mathcal{F}}{\delta\psi}\approx 0$ throughout the
phase transition. The CH-PFC model is solved using an Euler discretization
scheme in a 2D finite-difference framework. The computational grids have
spacings of $\delta x=\delta y=\frac{4\pi}{q_{0}6\sqrt{3}}$ and have periodic
boundary conditions.222The length scale of the CH-PFC model is
$\frac{1}{q_{0}}$. At each grid point the values of the composition field
$c(\vec{x})$, the peak density field $\psi(\vec{x})$, and the constant
$r(\vec{x})$ are represented in discrete form, as $c_{ij}$, $\psi_{ij}$ and
$r_{ij}$ respectively. The time-derivative in Eq. 2 is iterated at regular
time intervals of $\Delta\tau=1$. At each time step, the composition field
$c_{ij}(\tau)$ is used to update the lattice transformation coefficients,
$\alpha(c_{ij}(\tau))$ and $\beta(c_{ij}(\tau))$. The coupled Laplacian
$\nabla_{c}^{2}$ is next updated and the equilibrium lattice arrangements
$\psi_{ij}$ are computed following Eq. 3. This general numerical procedure
produces microstructural evolution and phase transitions.

## Application to electrode microstructures

In this section, we model a representative electrode-reservoir system. The
system comprises a polycrystalline FePO4 electrode particle surrounded by an
amorphous Li-reservoir. We compute an electrochemical cycle by
inserting/extracting Li-ions into interstitial sites of the FePO4 lattice
structure. Here, we study how lattices distort across a diffuse phase-
boundary, and investigate grain-boundary migration.

Figure 1: (a) A CH-PFC simulation of a polycrystalline FePO4 electrode
particle surrounded by an amorphous-Li-reservoir. The inset box (below) shows
the host lattice points (ellipsoidal peaks) describing the FePO4 lattice
motif. The color bar indicates Li-composition $c_{ij}$ in the electrode’s
interstitial sites. (b) An enlarged image of the electrode lattice symmetry is
shown in the inset box BB. In the electrode region, the peak density field
$\psi_{ij}$ has a wave form where the coordination symmetry of ellipsoidal
peaks represent an FePO4 lattice. In the Li-reservoir region, $\psi_{ij}$ is a
constant and describes an amorphous state. At the electrode-reservoir
interface, $\psi_{ij}$ changes as a function of $r_{ij}$ according to Eq. 1.
(c) “Peak marker image” illustrating the centroids of the ellipsoidal peaks
described by $\psi_{ij}$ (in Fig. 1a). The inset box highlights the lattice
structure (solid-blue lines) and grain boundaries (dashed-green lines) in the
FePO4 electrode. Details on the crystallographic features of the
polycrystalline electrode are shown in Fig. 2.

### Electrode-reservoir system

A circular electrode of diameter $d=290\delta x$ is modeled on a periodic grid
of size $300\delta x\times 300\delta y$. The electrode is surrounded by a
reservoir-region, see Fig. 1a. On the computation grid $r_{ij}$ is locally
defined to model a crystalline-region for the electrode $(r_{ij}=-0.2)$ and an
amorphous-region for the reservoir $(r_{ij}=0.2)$. A homogeneous composition
$c_{ij}=0$ is described on the computation grid. The transformation
coefficients define the coarse-grained bond lengths and bond angles of the
FePO4 lattice geometry (see Table 1) as inputs to the CH-PFC simulation. At
the initial state of the computation, a random peak density field value,
$-0.1\leq\psi_{ij}\leq 0.5$, is assigned at each grid point. Using Eq. 3, the
peak density field for the electrode-reservoir system is computed. The peak
density field, in the electrode region, nucleates into FePO4 crystallites with
different orientations. The initial condition with $-0.1\leq\psi_{ij}\leq 0.5$
allows grains to evolve independently and form a polycrystalline electrode as
shown in Fig. 1a.

In Fig. 1(a-b), the peak density field $\psi_{ij}$ has a wave-form in the
electrode region and represents the FePO4 lattice motif, see inset-box. The
composition field for Li interstitials is $c_{ij}=0$. In the reservoir region,
$\psi_{ij}$ is a constant and represents an amorphous state. At the electrode-
reservoir interface, fading bands of the peak density field are observed, see
inset box in Fig. 1b. These bands show a gradual change in $\psi_{ij}$, from
its periodic wave-form in the crystalline electrode, to a constant value in
the amorphous reservoir. This change in $\psi_{ij}$ results from the abrupt
change of $r_{ij}$ values modeled across the electrode-reservoir interface. In
this paper, the fading bands of $\psi_{ij}$ represent the amorphous phase
gradually taking on a crystalline character.

In Fig. 1a the peaks are of ellipsoidal shape. This shape results from the
coordinate transformation coefficients in the Laplace operator
$\nabla_{c}^{2}$, which shears the peaks anisotropically. Fig. 1a shows
multiple grain boundaries in the FePO4 electrode particle. The peaks at the
grain boundaries, appear smeared and deviate from the conventional ellipsoidal
shape. We interpret this smearing of peaks as a coarse-grained lattice-
structure distortion, which maintains coherency between neighboring grains.

The centroids of the ellipsoidal peaks describe the rectangular coordination
of the FePO4 lattice. To highlight this symmetry, we substitute the
ellipsoidal peaks with a Gaussian distribution – we refer to this image as a
“peak marker image”, see Fig. 1c. Here, the centroids of the peaks,
$(x_{0},y_{0})$ are identified, and a 2D Gaussian distribution,
$g(x,y)=\frac{2}{\pi}\exp[-2\\{(x-x_{0})^{2}-(y-y_{0})^{2}\\}]$ is modeled
around each centroid. In Fig. 1c, these Gaussian distributions are shown by
white solid dots. The fading bands of $\psi_{ij}$ at the electrode-reservoir
interface are removed in the peak marker image. The FePO4 lattice structure
and a representative grain boundary are traced by solid blue and dashed green
lines respectively, see the inset box in Fig. 1c. Fig. 2 highlights the
crystallographic features in the FePO4 electrode particle. Note, the peak-
positions in Fig. 1c does not correspond to the atomic sites. The arrangement
of peaks illustrates the coarse-grained lattice-symmetry of the electrode
particle. The grain boundaries arise from the coarse-grained lattice-
distortions in the underlying atomic arrangements. Further details on the
interpretation of the coarse-grained lattices in CH-PFC simulations is
provided in the appendix of Ref. [30].

Figure 2: An illustration of the crystallographic features in the
polycrystralline FePO4 electrode particle (shown in Fig. 1a). (A) A coherent
grain boundary with a perfect match between the crystallographic planes. (B) A
triple junction formed at the intersection of three grains. The dashed yellow
lines highlight the grain boundary system. (C) A chain of defects, highlighted
by yellow circles, form a high angle grain boundary $(>15^{\circ})$. (D) An
edge-dislocation defect illustrated by a ‘T’.

In Fig. 2, we identify four representative crystallographic features ‘A-D’ of
the FePO4 electrode particle (simulated in Fig. 1a). These crystallographic
features arise naturally from computing the time derivative of $\psi_{ij}$ in
Eq. 3. Label ‘A’, highlighted by the green and blue crystallographic planes,
marks a coherent interface. These coherent grain boundaries are commonly
observed in our CH-PFC simulations. Label ‘B’ marks three grain boundaries
that intersect to form a triple junction. The dihedral angle(s) between these
grain boundaries influence(s) the migration of the grain boundary system. For
example, if the grain boundaries are equally inclined at $120^{\circ}$, the
triple-junction tends not to move. However, if the grain boundaries are
unequally inclined, they drag the triple-junction towards the centre of a
single grain. This “triple-junction drag-effect” is discussed in the next
section (Electrochemical cycle). Labels ‘C’ and ‘D’ indicate, respectively,
the high-angle and low-angle grain boundaries in the electrode particle. Grain
boundaries with larger misorientation angle $>15^{\circ}$ are incoherent and
have a chain of edge-dislocations as shown by label ‘C’. Grain boundaries with
a misorientation angle less than $15^{\circ}$ tend to be more coherent and
have fewer defects. For example, in Fig. 2, ‘D’ shows an edge-dislocation
defect that is highlighted by a pair of short yellow lines. We next lithiate
and delithiate this polycrystalline electrode particle to investigate how its
crystallographic features evolve during an electrochemical cycle.

### Electrochemical cycle

In the following simulations, we specify a chemical potential to the Li-
reservoir, which is held fixed during the lithiation / delithiation cycle.
This boundary condition is a proxy for our assumption that the Li-diffusion
rate is greater in the reservoir than in the electrode particle. For example,
we specify the Li-reservoir with $c_{ij}=1$ that is fixed during lithiation.
The composition gradient between the FePO4 electrode $(c_{ij}=0)$ and the Li-
reservoir creates a boundary condition, which causes Li-ions to diffuse into
the electrode; see Fig. 3.

Figure 3: (a) Lattice arrangements in a partially lithiated FePO4 electrode
particle. The color bar illustrates Li-composition in FePO4 (blue $c_{ij}=0$)
and LiFePO4 (red $c_{ij}=1$) phases. A diffuse phase boundary (yellow
$0<c_{ij}<1$) separates the two phases. (b) The distribution of Li-composition
field $c_{ij}$ across a diffuse phase boundary (for example, along AA). (c)
The peak density field $\psi_{ij}$ across the diffuse phase-boundary. The
width of the diffuse interface (highlighted by vertical dashed-lines) is
numerically calibrated to span over $\sim 4$ lattice points.

Fig. 3a is a CH-PFC simulation of a partially lithiated electrode particle.
Fig. 3b shows the composition variation across the diffuse FePO4 / LiFePO4
phase boundary. The inset image in Fig. 3c illustrates the coarse-grained
lattice distortions across the phase boundary. This phase boundary is
analogous to the ‘staging structure’ observed in some intercalation compounds
[32]. For example, the interfacial region between the FePO4 and LiFePO4 phases
has non-stoichiometric Li-composition (i.e., LiXFePO4 where $0<X<1$).

Lithiation

Fig. 4 shows the temporal evolution of the electrode microstructure during
lithiation, as a function of $c_{ij}$ and $\psi_{ij}$. The subfigures in row 1
illustrate the lattice arrangements in the host-electrode particle as a
function of the interstitial Li-composition. The corresponding peak marker
images of the host-electrode are shown in row 2. The approximate positions of
three representative grain boundaries ‘A-C’ are marked on the electrode’s
lattice structure. The positions and orientations of these grain boundaries
are tracked during the electrochemical cycle. Subfigures in row 3 are the
distortion maps that indicate the absolute difference in electrode’s peak
positions with a reference state. That is, the host-electrode lattices undergo
displacive transformations during an electrochemical cycle, which are tracked
on the distortion maps. In the lithiation cycle, the reference state is the
initial FePO4 particle shown in Fig. 4a. The distortion at each grid point is
calculated as
$\delta_{ij}=\frac{\left\|x_{ij}(\tau)-x_{ij}(\tau=0)\right\|}{\delta_{0}}$ .
Here, $x_{ij}(\tau)$ is the discrete representation of the peak position at
time $\tau$. The $\left\|\mathbf{...}\right\|$ represents the euclidean
distance between a peak position at time $\tau$ with reference to its initial
position at $\tau=0$ in Fig. 4a. The normalizing constant,
$\delta_{0}=\beta_{\mathrm{FP}}=\frac{4\pi}{q_{0}\sqrt{3}}$, is the
equilibrium separation between two adjacent Gaussian-peaks. The distortion
maps illustrate the lattice deformations in the host-electrode particle that
are induced by Li-diffusion.

As the Li-intercalation wave propagates into the electrode as in Fig. 4a – 4c,
the shape and size of grains ‘A-C’ change. In the grain-boundary system ‘A’,
the smallest grain ‘a1’ near the electrode edge shrinks until it disappears in
Fig. 4d. This relieves the inhomogeneous strain field in the larger grain ‘A’.
The larger grain ‘A’ grows in size during lithiation. The grain boundary ‘C’
tends to move towards its center of curvature. This is consistent with the
motion by curvature observed in material microstructures [40, 41, 42]. In Fig.
4b row 3, lattice distortions $\delta_{ij}\thickapprox 0.12$ are observed in
the lithiated phase of the electrode particle. We interpret that these
distortions correspond with the ferroelastic lattice strains that accompany
FePO4 / LiFePO4 phase transition. The lattice arrangements in the electrode
core show negligible variation in peak positions, $\delta_{ij}\approx 0$. In
Fig. 4c row 3, relatively large distortions $\delta_{ij}>0.5$ are observed
along grain boundaries in the electrode. We interpret these distortions
$\delta_{ij}>0.5$ to correspond with the grain-boundary migration. At the end
of lithiation, a LiFePO4 phase forms in the host electrode, see Fig. 4d. The
distortion map in Fig. 4d row 3, indicates ferroelastic lattice strains
throughout the electrode particle.

Figure 4: (a-d): Lithiation of the polycrystalline FePO4 electrode particle.
Starting from an initial FePO4 phase (a), Li-ions intercalate into the
electrode particle (b-c). A LiFePO4 phase is formed at the end of lithiation
(d). The subfigures in row 1 show the temporal evolution of the Li-composition
$c_{ij}$ in the electrode particle. The subfigures on row 2 show the
structural transformation of lattices in the host-electrode during lithiation.
The labels ‘A–D’ highlight representative crystallographic features in the
electrode particle. The subfigures in row 3 illustrate the distortion maps
$\delta_{ij}$ corresponding to each stage of lithiation. We interpret
$\delta_{ij}$ to show host lattice distortions induced from Li-intercalation.

Delithiation

We next delithiate the electrode particle starting from Fig. 4d. We model the
Li-reservoir with a composition field $c_{ij}=0$, which is held fixed
throughout the delithiation process. Fig. 5 shows the temporal evolution of
the electrode microstructure as Li-ions are extracted from the host lattices.
The subfigures in row 1 show the microstructures as a function of the
interstitial Li-composition. The host-electrode lattice arrangements and
distortion maps are illustrated in row 2 and row 3, respectively. Note, the
distortion maps in Fig. 5 plot the absolute difference in peak positions,
using the lattice arrangements in Fig. 5a as the reference.

Figure 5: (a-d): Delithiation of a LiFePO4 electrode particle. Starting from
the initial LiFePO4 phase (a), Li-ions are extracted from the electrode
particle (b-c). An FePO4 phase is formed at the end of delithiation (d). The
subfigures in row 1 show the temporal evolution of Li-composition $c_{ij}$ in
the electrode particle. The subfigures on row 2 show the structural
transformation of lattices in the host-electrode during delithiation. The
labels ‘A–D’ highlight representative grain boundaries. The subfigures in row
3 illustrate the lattice distortion maps $\delta_{ij}$ corresponding to each
stage of delithiation. The distortions in Fig. 5 are calculated as the
absolute difference in peak positions using the lattice arrangements in
subfigure 5a as reference.

During delithiation, LiFePO4 lattices undergo displacive transformation to
describe FePO4 lattice structure. Grain ‘B’ shrinks in size and disappears at
the end of delithiation (Fig. 5, row 2). The grain-boundary system ‘B’, with
unequal dihedral angles, drags the triple junction as shown in Fig. 5(a-d).
Note that grains ‘a1’ and ‘B’ are of comparable sizes. However, Fig. 4 – 5
qualitatively illustrates that grain ‘a1’ shrinks faster than grain ‘B’. We
interpret that the slow migration of the grain-boundary system ‘B’ results
from the triple-junction drag effect. This drag effect has been observed by
Shvindlerman et al. [43, 44] and has also been reported in electrode
microstructures [45]. The edge dislocation ‘D’ moves towards the electrode
particle’s surface during delithiation. Extraction of Li-ions from the host-
electrode induces this dislocation movement. In Fig. 5d, the electrode
particle is returned to the FePO4 phase at the end of delithiation. The
electrode particle in Fig. 5d has a different lattice-arrangement from its
initial state in Fig. 4a. Similar dislocation dynamics have been observed in
experiments [46],[47]. For example, Ulvestad et al. [46] reported that an
applied current induced dislocation movement in an intercalation cathode
particle (LiNi0.5Mn1.5O4). The dislocation was stable at room temperature and
migrated to the particle surface under an electrical load [46]. In hydrogen-
palladium nanocubes [47], Narayan et al. [47] reported that the diffusion of
hydrogen-atoms in host-palladium particle mobilized its edge-dislocations.

In Figs. 4 – 5 grain-boundary migration $(\delta_{ij}\approx 1)$ is primarily
observed in electrode regions that are swept across by the Li-intercalation
wave. We hypothesize that the grain-boundary migration is induced by Li-
intercalation [12, 11]. That is, insertion or extraction of Li-ions from the
host lattice is accompanied by a change in the host-lattice geometry. The
difference in lattice parameters between the lithiated and delithiated phases
generates lattice misfit strains. This lattice misfit results in a stored
elastic energy, which acts as an additional driving force for grain-boundary
migration beyond its curvature [12, 11]. We next test this hypothesis by
simulating grain growth in the same electrode particle (Fig. 4a), without Li-
intercalation. In this non-cycled electrode particle, grain growth has no
additional driving force.

Figure 6:  Lattice rearrangements in the non-cycled polycrystalline electrode
particle. The electrode-reservoir system is modeled with a homogeneous Li-
composition $c_{ij}=0$, and computed for the same time
$(\tau_{\mathrm{total}})$ and temperature as in Figs. 4 – 5. The host-
electrode lattice arrangements (a) before $\tau/\tau_{\mathrm{total}}=0$, and
(b) after $\tau/\tau_{\mathrm{total}}=1$ the non-cycled grain growth. (c) The
lattice distortion map $\delta_{ij}$ calculated as the absolute difference in
peak positions between the ‘before’ and ‘after’ stages of the non-cycled
electrode. The distortion map illustrates lattice rearrangements near grain-
boundaries.

Non-cycled electrode

We model the electrode-reservoir system as shown in Fig. 4a. We define a
homogeneous Li-composition field $c_{ij}=0$ on the computation grid. We
compute the time evolution of lattice arrangements in the electrode following
Eq. 3 [30]. We iterate Eq. 3 for the same computation time
$\tau_{\mathrm{total}}$ as required to electrochemically cycle the electrode
in Figs. 4 – 5. We refer to this computation as the “non-cycled grain growth”
of the electrode particle.

Fig. 6a and Fig. 6b respectively illustrate the lattice arrangements in the
host-electrode particle before and after the computation. The lattice
distortion map in Fig. 6c illustrates structural rearrangements in the host
electrode particle. Lattice distortions $\delta_{ij}\approx 1$ are primarily
observed along grain boundaries with a non-zero radius of curvature. This
suggests that grain-boundary migration in Fig. 6b – 6c is curvature-driven.
These migrations are however small in comparison to Figs. 4 – 5. For example
in Figs. 4 – 5, the grain ‘B’ shrinks at the end of lithiation / delithiation
cycles. However in Fig. 6, with no Li-intercalation, grain ‘B’ only slightly
reduces in size. Similarly grains ‘A’ and ‘C’ show a small variation in their
shape and size. Fig. 6 is consistent with our hypothesis that Li-intercalation
assists (or accelerates) grain-boundary migration.

## Discussion

We compare grain growth in electrode particles that were electrochemically
cycled and non-cycled in Fig. 7(a-b), respectively. The schematics, at the top
of the plots show Li-composition $c$ in the electrode particle at each
computation time step, $\tau/\tau_{\mathrm{total}}$. Note,
$\tau_{\mathrm{total}}$ is the computation time required for one
electrochemical cycle. The grain size $\frac{\mathcal{A}}{\mathcal{A}_{e}}$ is
normalized by the electrode area $\mathcal{A}_{e}$. Fig. 7a illustrates that
the mean grain-size, averaged over all the grains in the electrode particle,
grows by $11\%$ upon electrochemical cycling. Smaller grains ‘a1’ and ‘B’
shrink, and the larger grain ‘C’ has an approximately constant grain-size
throughout the electrochemical cycle.

In Fig. 7b, the mean grain-size grows by $\sim 2\%$ in the non-cycled
electrode particle. The grain sizes of A, B and C are approximately constant
throughout the computation. In the absence of Li-intercalation, the grain-
boundary migration is curvature-driven. In Fig. 7b, the grain growth (mean)
follows a relationship of the form $\frac{\mathcal{A}}{\mathcal{A}_{e}}\approx
g_{0}+g(\frac{\tau}{\tau_{\mathrm{total}}})^{n}$, where $n\approx
0.5$.333$g_{0}=9.23,g=1.12$ This is consistent with the curvature-driven grain
growth observed in experiments [40]. In Fig. 7a the grain growth follows a
similar relationship, however, with the exponent $n>0.5$. This suggests an
accelerated grain growth (mean) in Fig. 7a. Fig. 7(a-b) indicates that Li-
intercalation accelerates grain-boundary migration in the electrode particle.
This observation is consistent with the diffusion-induced grain-boundary
migration discussed in the works of Handwerker et al. [11, 12].

Figure 7: Grain growth $\mathcal{A}/\mathcal{A}_{e}$ in an electrode particle
when it is (a) electrochemically cycled and (b) non-cycled. The schematics
above the plots illustrate Li-composition $c$ in the electrode particle at the
corresponding computation times.

We find the work of Bates et al. [6] on sputter-deposited thin film cathodes,
as the closest experimental comparison to the diffusion-induced grain growth
effect suggested by our present work. Bates et al. [6] described the cathodes
as undergoing grain growth during electrochemical cycling at near-ambient
temperature [6]. Another related phenomena are the observations of accelerated
grain growth in fluorite structure solid electrolytes at elevated temperature.
In these solid electrolytes electrical load has been observed to accelerate
grain growth kinetics [7, 9]. We suggest that systematic experimental
investigation of battery electrodes after extensive cycling will reveal
lattice-strain-induced grain growth as observed in this paper.

Figure 8: Grain-size distributions in the electrochemically cycled (left) and
non-cycled (right) electrode particle. The subfigures correspond to the
electrode microstructures at (a) $\tau/\tau_{\mathrm{total}}=0$ (b)
$\tau/\tau_{\mathrm{total}}=0.5$ and (c) $\tau/\tau_{\mathrm{total}}=1.$ The
position of the red lines corresponds to the mean grain-size values on the
vertical axes in Fig. 7. The grain-size distributions indicate that
electrochemical cycling accelarates grain-growth in the electrode particle.

Fig. 8(a-c) shows the grain-size distributions for the electrochemically
cycled and non-cycled electrode particle. Histograms of grain-size
distributions at three stages of the CH-PFC computation, namely
$\tau/\tau_{\mathrm{total}}=0,0.5\ \mathrm{and\ }1$, are plotted. They show
the mean grain-size ($\bar{\mathcal{A}}/\mathcal{A}_{e}$) and number of grains
(frequency) in the electrode-particle at each stage. The histograms in Fig.
8(a-c) show a good fit to the log-normal size distributions, which are
observed in experiments [7, 9, 48]. In the electrochemically cycled electrode,
the variance (width) of the grain-size distribution decreases upon Li-
intercalation. In the non-cycled electrode particle, see Fig. 8(right), the
variance of its size-distribution is approximately constant throughout the
computation. These results suggest that smaller grains in the
electrochemically cycled electrode particle, with
$\log(\mathcal{A}/\bar{\mathcal{A}})<-0.5$, disappear at the expense of bigger
grains. Electrode particles with enhanced grain sizes tend to be less tough in
comparison to samples with smaller grains [5, 3]. Fig. 8 suggests that Li-
intercalation is an additional process that affects the mechanical reliability
in batteries.

We next discuss a few issues that arise in interpreting the CH-PFC simulations
in Figs. 4 to 6. For example, the electrode-particle volume (or area) is held
constant during the computation. However, the number density of peaks in the
electrode-particle is not conserved.444Please note, the addition/removal of
peaks does not affect the coarse-grained lattice structure described by
$\psi_{ij}$ [30].  That is, the peaks move in/out at the electrode-reservoir
interface to accommodate for the lattice-area change during phase
transformation. This “artificial boundary condition” applied to the electode-
reservoir interface does not impose volume-confinement stresses on the
electrode-particle. In future works, we propose to introduce numerical
correction terms [49] to Eq. (6) to conserve the total number of peaks in the
electrode region.

Another issue is related to the isotropic Li-diffusion in the elecrode-
particle. In olivine materials, such as FePO4, Li-ions preferentially diffuse
through the host’s one-dimensional channels [50]. In previous continuum models
[14, 13, 26, 16], researchers have modeled this preferential diffusion by
using an anisotropic diffusion coefficient. In these methods the
crystallographic planes of FePO4 were typically oriented relative to a
cartesian basis. The anisotopic Li-diffusion was modeled with reference to the
coordinate axes. In the CH-PFC model, however, the crystallographic planes of
FePO4 / LiFePO4 are a natural outcome in the simulations. We assume an
isotropic bulk-diffusion coefficient $D$ in Eq. (2) to model
lithiation/delithiation; see Figs. 4 and 5. We do, however, see anisotropy in
Li-diffusion arising from the coordinate-transformation-coefficients in the
coupled term $\nabla^{2}\gamma\frac{\psi}{2}\frac{\partial
G(\nabla_{c}^{2})}{\partial c}\psi$.555Please see Eq. (8) in the appendix.
This contribution is negligible as a result of the diffuse width ($\sim 4$
coarse-grained lattices) of the phase boundary. The value of the constant
$\gamma$ also influences the anisotropy contribution from the coupled term. In
this paper we set $\gamma=\frac{\lambda^{2}q_{0}^{5}}{uF_{0}}=1$ for
computational expediency. That is, the energy contribution from the PFC
equation is negligible (i.e., $\sim 10^{-19}F_{0})$. Consequently, the
isotropic diffusion of the Li-intercalation wave is relatively unaffected by
the anisotropy in $\psi_{ij}$. However, if $\gamma$ is large the effects from
material crystallography and lattice defects are expected to appear in the
simulations.

Overall, the CH-PFC methods has three specific advantages in modeling phase
transitions in intercalation electrodes: First, the CH-PFC simulations provide
qualitative insights on phenomenological mechanisms across different length
and time scales. For example, the simulations describe Li-diffusion at a
continuum scale and grain-boundary migration at an atomistic scale. Second,
the crystallographic planes of FP/LFP lattices in the electrode particle are
an emergent phenomenon of the CH-PFC simulations [51]. The crystallographic
misorientations at grain boundaries follow naturally from the simulations.
Third, in the CH-PFC model the coarse-grained lattices distort independently
as a function of the interstitial Li-composition. That is, lattice geometries
are different in the lithiated and delithiated phases. This difference
generates coherency strains across the diffuse phase boundary. The coherency
strains in the CH-PFC simulations are a natural outcome resulting from the
different lattice geometries, and are not modeled as an approximation from the
composition field [13, 14, 17, 26]. This results in inhomogeneous strains
across grain boundaries and phase boundaries in a polycrystalline electrode.

Summary

We demonstrate that Li-diffusion accelerates grain growth in intercalation
compounds using FePO4 / LiFePO4 as a model system. We present an application
of the Cahn-Hilliard – phase-field-crystal methods to model phase transitions
in a polycrystalline FePO4 electrode particle. This modeling approach
illustrates multiscale interactions between Li-diffusion (continuum parameter)
and structural transformations of host-lattices (atomistic) during an
electrochemical cycle. While grain growth from cyclic intercalation has not
been experimentally confirmed, our study suggests that volume changes in
electrode upon Li-intercalation accelerates grain growth. The results
qualitatively describe phenomenological mechanisms in intercalation
electrodes, such as edge-dislocation movement and the triple-junction-drag
effect. The CH-PFC model could be applied to investigate chemo-mechanically
coupled problems that involve solute-induced phase transitions.

Acknowledgements

The authors gratefully acknowledge the support by the grant DE-SC0002633
funded by the U.S. Department of Energy, Office of Science, in carrying out
this work. A. Renuka Balakrishna also acknowledges the support of the
Lindemann postdoctoral fellowship.

Appendix

A.1 CH-PFC model details

In this section we provide a brief explanation of the CH-PFC model. For the
derivation of the CH-PFC model please refer to Ref. [30]. The CH-PFC model
describes the total free energy $F$, as a function of the composition field
$\overline{c}$ and the peak density field $\phi$:

$\displaystyle F$
$\displaystyle=\int\\{g(\overline{c})+\kappa|\nabla\overline{c}|^{2}+f(\phi)+\frac{\phi}{2}G(\nabla_{c}^{2})\phi\\}d\vec{\mathbf{r}}$
$\displaystyle=\int\\{RT[\overline{c}\mathrm{ln}(\overline{c})+(1-\overline{c})\mathrm{ln}(1-\overline{c})]+\Omega\overline{c}(1-\overline{c})$
(4) $\displaystyle~{}~{}+\kappa|\nabla\overline{c}|^{2}+\frac{\phi}{2}(a\Delta
T_{0}+\lambda(q_{0}^{2}+\nabla_{c}^{2})^{2})\phi+u\frac{\phi^{4}}{4}\\}d\vec{\mathbf{r}}.$

Eq. 4 in its normalized form is:

$\displaystyle\mathcal{F}=\frac{F}{F_{0}}$ $\displaystyle=\int\\{g(c)+|\nabla
c|^{2}+f(\psi)+\frac{\psi}{2}G(\nabla_{c}^{2})\psi\\}d\vec{x}$
$\displaystyle=\int\\{c\mathrm{ln}(c)+(1-c)\mathrm{ln}(1-c)+\Omega_{0}c(1-c)$
(5) $\displaystyle~{}~{}+|\nabla
c|^{2}+\gamma\left(\frac{\psi}{2}(r+(1+\nabla_{c}^{2})^{2})\psi+\frac{\psi^{4}}{4}\right)\\}d\vec{x},$

where
$c=\frac{\overline{c}_{a}-\overline{c}}{\overline{c}_{a}-\overline{c}_{b}}$,
and $\psi=\phi\sqrt{\frac{u}{\lambda q_{0}^{4}}}$. The gradient energy
coefficient
$\kappa=\frac{F_{0}}{(\overline{c}_{a}-\overline{c}_{b})^{2}}\left(\frac{16\pi\xi}{q_{0}\sqrt{3}}\right)^{2}$,
is numerically calibrated such that the width of the diffuse composition
interface spans over $\sim 4$ lattice spacings described by the peak density
field, $\psi$. Note, $\frac{1}{q_{0}}$ is the length scale of the CH-PFC model
and $\xi$ is the scale factor that coarse-grains the lattice units. The form
of Eq. 4-5 is similar to that in the work of Renuka-Balakrishna and Carter
[30] where detailed explanations of the specific terms, constants
$(\overline{c}_{a},\overline{c}_{b},\lambda,q_{0},u,\xi)$ and normalizations
are provided. Note, the coefficients in the regular solution model are the
same as in Ref. [14]. In Eq. 5, $g(c)$ and $f(\psi)$ describe the homogeneous
energy contributions from the Cahn-Hilliard and phase-field-crystal equations
respectively. The composition gradient-energy is given by $|\nabla c|^{2}$.
The parameter $r$, controls the second-order phase transition of the PFC
model. The constant $\gamma=\frac{\lambda^{2}q_{0}^{5}}{uF_{0}}$ relates the
free energy normalizations of the Cahn-Hilliard and the PFC model. In this
paper we set $\gamma=1$ and $\xi=1$ for computational expediency.

The Cahn-Hilliard and the phase-field-crystal models are coupled via the
operator $G(\nabla_{c}^{2})=(1+\nabla_{c}^{2})^{2}$. The composition field is
coupled to the lattice symmetry via the Laplace operator:

$\nabla_{c}^{2}=\xi^{2}((\mathrm{A}_{11}^{2}+\mathrm{A}_{12}^{2})\frac{\partial}{\partial
x^{2}}+\mathrm{A}_{22}^{2}\frac{\partial}{\partial
y^{2}}+2\mathrm{A}_{12}\mathrm{A}_{22}\frac{\partial}{\partial x\partial y}).$
(6)

Here, $\mathrm{A_{\mathit{kl}}}$ are the elements of the transformation matrix
and are described as functions of the composition field:

$\mathbf{A\mathrm{(\mathit{c})}}=\left[{\begin{array}[]{cc}\alpha(c)&\frac{-\alpha(c)}{\sqrt{3}}\\\
0&\frac{2\beta(c)}{\sqrt{3}}\end{array}}\right].$ (7)

The matrix $\mathbf{A(\mathit{c})}$, describes affine lattice transformations
using hexagonal symmetry as the reference structure. The transformation
coefficients in Eq. 7 transform a hexagonal symmetry (of unit size) to
rectangular geometries of FePO4 (with $c=0)$ and LiFePO4 (with $c=1$) in 2D
[39], see Fig. 9. The transformation coefficients are given by
$\alpha(c)=\alpha_{\mathrm{FP}}+(\alpha_{\mathrm{LFP}}-\alpha_{\mathrm{FP}})c$
and
$\beta(c)=\beta_{\mathrm{FP}}+(\beta_{\mathrm{LFP}}-\beta_{\mathrm{FP}})c$.
The values of $(\alpha_{\mathrm{FP}},\beta_{\mathrm{FP}})$ and
$(\alpha_{\mathrm{LFP}},\beta_{\mathrm{LFP}})$ correspond to FePO4 and LiFePO4
lattices and are obtained from Table 1. During electrochemical cycling, the
composition field in a two-phase FP/LFP microstructure is a function
$c(\vec{x})$. The lattices in this microstructure are transformed according to
$\mathbf{A}(c(\vec{x}))$.

Figure 9: Schematic representations of the coupling between lattice symmetry
and composition field in the CH–PFC model. The transformation matrix
$\mathbf{A}(c),$ is described as a function of the composition field, $c$. For
$c=0$, $\mathbf{A}(c)$ deforms a hexagonal symmetry to rectanglular FePO4
lattice (in blue, far left). For $c=1$, $\mathbf{A}(c)$ describes the lattice
motif of LiFePO4 (in red, far-right). For $0<c<1$, intermediate lattice
geometries of the interphase LiXFePO4 (dashed box) is modeled.

Next, we list the detailed equations of motion used in the CH-PFC model. The
evolution of the composition field follows a generalized Cahn-Hilliard
equation:

$\displaystyle\frac{\partial c}{\partial\tau}$
$\displaystyle=\nabla^{2}\frac{\delta\mathcal{F}}{\delta c}$
$\displaystyle=\nabla^{2}(\frac{\partial g(c)}{\partial
c}-\nabla^{2}c+\gamma\frac{\psi}{2}\frac{\partial(\nabla_{c}^{4}+2\nabla_{c}^{2})}{\partial
c}\psi).$ (8)

where, $\gamma=1$ and $\tau$ is the dimensionless time variable
$\tau=t\frac{D}{L^{2}}$. Here, $D$ is the isotropic diffusion coefficient and
$L$ is the size of the computational grid. Eq. (8) introduces two Laplace
operators, $\nabla^{2}$ and $\nabla_{c}^{2}$, respectively. The Laplacian
$\nabla^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}$ describes an isotropic Li-diffusion. We do, however, see anisotopy in
$\frac{\partial c}{\partial\tau}$ because of the coordinate transformation
coefficients in $\nabla_{c}^{2}$ and from grain boundaries in the host-
electrode system. The coupled Laplacian $\nabla_{c}^{2}$ computes the
derivatives of the peak density field in a transformed coordinate space. The
term
$\nabla^{2}\gamma\frac{\psi}{2}\frac{\partial(\nabla_{c}^{4}+2\nabla_{c}^{2})}{\partial
c}\psi$ is anisotropic and influences the Li-intercalation. This anisotropic
effect is negligible in the CH-PFC simulations of this paper, because the
diffuse phase boundary spans over $\sim 4$ lattice spacings. The lattice
arrangement at each composition evolution step is computed as:

$\displaystyle\frac{\partial\psi}{\partial n}$
$\displaystyle=-\frac{\delta\mathcal{F}}{\delta\psi}+\frac{1}{L^{2}}\int\frac{\delta\mathcal{F}}{\delta\psi}d\vec{x}$
$\displaystyle=-\gamma[(r+(1+\nabla_{c}^{2})^{2})\psi+\psi^{3}]$ (9)
$\displaystyle~{}~{}+\frac{1}{L^{2}}\int\gamma[(r+(1+\nabla_{c}^{2})^{2})\psi+\psi^{3}]d\vec{x}.$

Here, we assume that the elastic relaxation (equilibrating the peak density
field) is infinitely faster than the evolution of the composition field.
Consequently, we model the equilibrium lattice arrangements by maintaining
$\frac{\delta\mathcal{F}}{\delta\psi}\approx 0$ throughout the phase
transition. In Eq. (9), $n$ is a fictive time-like variable that is rapidly
changing in comparison to the dimensionless time, $\tau$. Further details on
Eq. (4) – (5) can be found in Ref. [30].

A.2 Modeling lattice symmetries using PFC methods

The coordinate transformation coefficients introduced in the CH-PFC model [30]
resemble the lattice stretch and shear factors used in the anisotropic PFC
methods [52],[53]. The stretch/shear factors deform a lattice symmetry and are
used to calculate the anisotropic model coefficients. The transformation
coefficients in the CH-PFC model, however, differ from the anisotropic
coefficients in two ways: First, the CH-PFC model computes the Laplacian in a
transformed coordinate space. The coefficients correspond to the elements of a
transformation matrix, which relates lattice symmetries in 2D point groups via
affine transformations. Second, the transformation coefficients are coupled to
a composition field, which influences the underlying lattice symmetry of the
host material.

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