# The effect of mechanical stress on lithium distribution and geometry
optimisation for multi-material lithium-ion anodes.

Ian P. E. Roper${}^{\text{a},*}$, S. Jon Chapman${}^{\text{b,c}}$, Colin P.
Please${}^{\text{b,c}}$ ${}^{\text{a}}$ InFoMM CDT, Mathematical Institute,
Woodstock Road, Oxford, UK
${}^{\text{b}}$ OCIAM, Mathematical Institute, Woodstock Road, Oxford, UK
${}^{\text{c}}$ Faraday Institution

###### Abstract

A model is presented for predicting the open-circuit voltage (OCV) and lithium
distribution within lithium-ion anodes containing multiple materials, coupling
linear elasticity with a stress-dependent chemical potential. The model is
applied to a spherical radially-symmetric nano-particle with a silicon core
and a graphite shell, highlighting the large effect on lithium distribution
and OCV caused by the stress-coupling. Various performance measures based on
the expanded volume, the amount of lithium intercalated and the maximum stress
induced, are calculated for a silicon core with a graphite shell to enable
optimisation of the volume of the silicon core.

###### keywords:

stress-assisted diffusion , lithium-ion batteries, linear elasticity

††journal: Journal of The Electrochemical Society††∗ Corresponding author
Email addresses<EMAIL_ADDRESS>(Ian P. E. Roper)
<EMAIL_ADDRESS>(S. Jon Chapman)
<EMAIL_ADDRESS>(Colin P. Please)

##  1 Introduction

Lithium-ion batteries are ubiquitous in modern technology, powering mobile
phones, laptop computers and electric vehicles [48]. Many different materials
are being tested for use in all components of lithium-ion batteries to achieve
the largest energy capacity while remaining portable and light-weight. A key
component for maximising energy capacity is the anode; the more lithium that
can be intercalated into the anode, the more charge can be stored, and thus
the more energy can be obtained from a single charge. However, upon lithiation
many high capacity anodes increase in volume [56], due to changes in the
crystal structure. This can cause mechanical stresses because of geometrical
constraints within the anode and the rest of the battery, but also because of
concentration gradients of lithium within the anode causing non-uniform
expansion. The expansion of the anode can displace other components of the
battery and the high stresses within the anode material itself can cause it to
crack [2]. These effects can result in loss of connection between components
within the battery, leading to capacity fade after several charge/discharge
cycles.

One anode material with high expansion and a large capacity for lithium atoms
which has received a lot of recent research attention is silicon. When fully
lithiated, silicon can accommodate 3.75 lithium atoms per silicon atom,
forming the alloy Li3.75Si at room temperature [17], giving silicon anodes a
volumetric capacity of 3500 mAh/g. This is much greater than the volumetric
capacity of 372 mAh/g achieved by commonly-used graphite anodes [25], due to
graphite only being able to accommodate one lithium atom per six carbon atoms
(LiC6). This makes silicon a very inviting anode material; however, when
silicon is lithiated, it expands to around four times its original size [6],
much more than the $\sim 10$% volume increase observed in fully lithiated
graphite.

Experimental efforts to make silicon a viable anode material often involve
using nano-structures within the anode design to minimise the variation in the
lithium concentration. These designs are often based on nanoparticles [22, 27,
29, 40], nanowires [7, 18, 54] and nanotubes [41, 45]. Additionally, there are
many nano-structure designs which attempts to constrain the expansion of the
silicon, for example in a core–shell structure [40, 51, 52], or yolk–shell
structure [26, 46], or using self-healing polymers [44]. These structures are
often used in conjunction with each other, for example Wang et al. showed that
for a silicon oxide shell which is thick enough, inwards growth can be induced
in silicon nanotubes [45].

In determining which structures may be the most effective at mitigating the
adverse effects of highly expanding anode materials, it is insightful to
construct mathematical models for the mechanical behaviour of the anode and
the behaviour of the lithium within. Several mathematical models for the
expansion of anodes upon lithiation have been proposed. In general, the
expansion is assumed to be due to the intercalation of lithium, and opposed by
a mechanical response from the stiffness of the anode material, causing
stresses. Linear elasticity models [8, 36, 57] have been used to model the
mechanical response when strains are small, but more commonly finite strain
models using geometrically nonlinear elasticity are used for silicon to
capture the large strains that occur [4, 6, 9, 10, 15]. Sethuraman et al. also
showed that silicon plastically flows during cycling [38] due to the high
stresses induced. This plastic flow has been incorporated into several finite
strain models [4, 6, 9], as well as linear models [50]. Several more simple
models have neglected the elastic stresses altogether and modelled the
mechanical response of the silicon as solely plastic [21, 58]. An additional
phenomenon that has been the basis of several modelling studies is crack
formation and propagation due to the high stresses within the anode [8, 13,
15, 37, 50].

Two types of model for the lithiation are commonly used: a single phase model
in which the silicon is gradually lithiated [4, 6, 10], and two-phase models
in which there is a sharp reaction front between the lithiated phase and the
unlithiated phase [20, 50, 55]. The former can be simply modelled using
diffusion of the lithium atoms, whereas the lithium kinetics in the latter are
often described using a Cahn-Hilliard phase-field model. Experimentally, the
two-phase model has been shown to likely be more physically accurate [9, 29].
The time-dependence of these models for the evolution of the lithium
concentration often cause the main focus of these studies to be on the
stresses inside the nano-structures due to silicon non-uniformity, rather than
due to material heterogeneity [8, 9, 10, 36].

The coupling of the lithium distribution to the stress is a two-way
phenomenon, since in addition to lithiation causing expansion, lithium
diffusion is also affected by stress. Stress-assisted diffusion is typically
included into lithium transport models by considering the chemical potential
of the lithium in the anode to depend on the hydrostatic stress. This coupling
affects the diffusion of the lithium through the anode due to the stress
induced by lithiation. Several works [4, 8, 11, 14, 36] have included stress-
assisted diffusion using the model formulated by Larche and Cahn [23] using a
small-strain assumption. Works by Wu [49] and Cui et al. [10] have resulted in
a more general framework for incorporating a stress-dependence into the
chemical potential of the lithium. However, Cui’s general form of stress-
dependent chemical potential reduces to those proposed by Wu, Larche and Cahn
under the appropriate assumptions. This general stress-assisted diffusion
framework has more recently been applied to finite strain models, for example
in [6].

To simplify the mathematical models of these anode nano-structures, simplified
geometries are often used. Nano-particles are approximated by spherical
geometries [10, 50, 58], nano-wires and nano-rods are approximated to
cylindrical geometries [6], and thin films of silicon are assumed to grow
unidirectionally [5]. However, even in these simplified geometries, the use of
nonlinear elasticity or other complex models requires the use of software
packages to solve the resulting differential equations [6, 9, 10]. Despite
these systems requiring numerical solvers, the chemical potential of the
lithium is often approximated as an analytical expression of the lithium
concentration [6, 10, 39]. These approximations fail to capture the large
‘steps’ in chemical potential that occur at transitions in the structure of
the intercalated anode [3]. Using an interpolated function of the chemical
potential of the lithium in a material would capture these steps and their use
in these complicated models would only marginally increase the computational
cost.

In this work, we consider a multi-material anode, which we call a hybrid
anode. We present a linearly elastic mechanical model coupled to a stress-
assisted diffusion model using the stress-dependent chemical potential
proposed by Larche and Cahn with a stress-independent term depending on
lithium concentration. We focus on a time-scale much slower than the diffusion
of lithium through the nano-particle allowing us to use a quasi-static
approximation, focusing on the stresses induced by the different expansions of
the different materials in chemical and mechanical equilibrium. We present
this model as a general chemo-mechanical model for anode lithiation. We then
apply our model to a spherical, radially-symmetric geometry consisting of a
core and shell of different materials, representing a nano-particle with a
constraining shell.

Previous works have often modelled a constrained core–shell anode nano-
particle using a zero displacement condition at the shell interface [10, 58]
or by modelling the shell as a material that cannot be lithiated [16, 43]. One
of our main focuses is the distribution of lithium between the core and the
shell and thus we include the lithiation of both materials and the resulting
displacement of the interface. We show how the distribution of lithium between
the two materials depends heavily on both stress-assisted diffusion and the
geometry of the nano-particle.

The lithium distribution predicted by the model is used to calculate the
chemical potential of the lithium in the hybrid nano-particle, which we can
then convert to an OCV. OCVs can be used to infer the state of charge (SOC) of
a battery at a given voltage and are used in battery management systems and
other battery models [12, 47]. While experimentally measured OCVs for single
material anodes are easily found within the literature [32, 24],
experimentally obtaining the OCV curve of a new anode design with multiple
materials can be expensive and is very time-consuming. Therefore, a
theoretical technique for predicting OCVs of multi-material anodes is of
interest to manufacturers and researchers. Sethuraman showed that stresses can
significantly affect the OCVs [39], thus the OCV of a hybrid anode is not only
dependent on the individual materials but also on the stresses induced in each
material by the others at equilibrium. We show the importance of including
stress into the lithium concentration model by comparing chemical potentials
when including and excluding stress-assisted diffusion.

Lastly, while the modelling of the stresses within lithium-ion anodes has been
widely studied, very few of these models discuss optimal designs based on the
results of the models. In the second half of this work, we discuss several
different performance measures that could be used in defining optimality for
the anode design. These are i) the amount of lithium intercalated, ii) the
expanded volume of the anode, and iii) the maximum induced stress. We then use
these performance measures to suggest three different optimality conditions
for the hybrid anode design. We derive these conditions for the silicon
core–graphite shell nano-particle design and show how the objective function
varies with the volume of the silicon core and find the optimal design in each
of the three cases.

The rest of this paper is organised as follows. In Section 2.1 we formulate
the chemo-mechanical model we use to describe the lithiation of a lithium-ion
anode for a general multi-material anode. In Section 2.2, we describe how the
stress-free chemical potential of the lithium in each anode material can be
calculated from the material’s OCV. We then simplify the general model to one
for a radially symmetric nano-particle with a core of material 1 and a shell
of material 2 and solve this model, showing that for certain simple
geometries, the concentrations are uniform in each material, even with the
inclusion of stress-assisted diffusion. In Section 3, we present the lithium
concentrations and chemical potentials in a nano-particle with a silicon core
and a graphite shell to show the importance of including stress-assisted
diffusion into the lithium concentration model. In Section 4.1 we then discuss
the three performance measures we will use to then find an optimal design of
the silicon–graphite nano-particles in Section 4.2. Finally in Section 5, we
discuss the conclusions that can be drawn from this work and the validity of
the results in certain regimes.

##  2 Mathematical Model

In this section, we develop a mathematical model for the lithiation of a
multi-material anode. We begin with a general geometry and arbitrary number of
different anode materials. We then apply the model to a spherical, radially
symmetric geometry with a core of one material and a shell of another
material. The anode is the only battery component which we model in this work;
we do not model the electrolyte, the cathode, binder, current collectors or
any other components.

###  2.1 General Formulation

We denote the domain of the entire anode as $\Omega$ and the region occupied
by each anode material as $\Omega_{a}$ with $a=1,\dots,n$, where $n$ is the
number of different materials in the hybrid anode. We present boundary
conditions on the boundary between the anode and the electrolyte $\Gamma_{e}$
and on boundaries between different anode materials $\Gamma_{ab}$ where $a\neq
b$ and $a,b\in\\{1,\dots,n\\}$.

####  2.1.1 Mechanical Model

We suppose the deformation in the anode is due to the volumetric changes due
to lithiation and is counteracted by an elastic response from the material due
to its stiffness. For simplicity, we assume the strains are small enough that
we may use linear elasticity theory. Similar to previous linearly-elastic
studies of anode expansion [8, 57], we write the elastic strain tensor as

$\textbf{E}^{e}=\frac{1}{2}\Big{[}\big{(}\boldsymbol{\nabla}^{*}\boldsymbol{u}^{*}\big{)}^{\text{T}}+\boldsymbol{\nabla}^{*}\boldsymbol{u}^{*}\Big{]}-\eta_{a}V^{m}_{a}c^{*}_{a}\boldsymbol{1}\quad\text{
in }\Omega_{a},$ (1)

where $\boldsymbol{u}^{*}=(u_{1}^{*},u_{2}^{*},u_{3}^{*})$ is the displacement
(m), $c_{a}^{*}$ is the concentration of lithium (mol m-3) and
$\boldsymbol{\nabla}^{*}$ is the gradient operator. The molar volume of
material $a$, measured when zero lithium has been intercalated, is denoted as
$V^{m}_{a}$ (m3 mol-1) and $\eta_{a}$ is the coefficient of compositional
expansion (CCE) of material $a$, which is a measure of the volumetric
expansion due to lithiation [42].

We relate the stress to the strain tensor by Hooke’s law,

$\boldsymbol{\sigma}^{*}=\mathbb{C}_{a}^{*}(c_{a}^{*}):\textbf{E}^{e}\quad\text{
in }\Omega_{a},$ (2)

where $\mathbb{C}_{a}^{*}$ is the stiffness tensor of material $a$ which can
vary with the lithiation state of the material. We assume that each anode
material is isotropic, allowing us to write the stiffness tensor in suffix
notation as

$\mathbb{C}^{*}_{a}(c_{a}^{*})={c_{ijkl}^{i,*}}(c_{a}^{*})=\lambda_{a}^{*}(c_{a}^{*})\delta_{ij}\delta_{kl}+G_{a}^{*}(c_{a}^{*})\delta_{ik}\delta_{jl}+G_{a}^{*}(c_{a}^{*})\delta_{il}\delta_{jk},$
(3)

where $\lambda^{*}_{a}$ is the first Lamé parameter of material $a$ and
$G_{a}^{*}$ is the shear modulus of material $a$ [19], both of which can vary
with the lithiation state of the material. We now write (2) as

$\boldsymbol{\sigma}^{*}=\mathbb{C}^{*}_{a}(c_{a}^{*}):\big{(}\boldsymbol{\nabla}^{*}\boldsymbol{u}^{*}-\eta_{a}V^{m}_{a}c^{*}_{a}\boldsymbol{1}\big{)}\quad\text{
in }\Omega_{a},$ (4)

which in suffix notation, becomes

$\sigma^{*}_{ij}=\lambda_{a}^{*}(c_{a}^{*})\delta_{ij}\frac{\partial
u_{k}^{*}}{\partial x_{k}^{*}}+G_{a}^{*}(c_{a}^{*})\bigg{(}\frac{\partial
u_{i}^{*}}{\partial x_{j}^{*}}+\frac{\partial u_{j}^{*}}{\partial
x_{i}^{*}}\bigg{)}-\eta_{a}V^{m}_{a}c^{*}_{a}(3\lambda_{a}^{*}(c_{a}^{*})+2G_{a}^{*}(c_{a}^{*}))\delta_{ij}\quad\text{
in }\Omega_{a},$ (5)

where we use the summation convention for repeated indices. Lastly, the entire
anode is in mechanical equilibrium, which is given by

$\boldsymbol{\nabla}^{*}\cdot\boldsymbol{\sigma}^{*}=0\quad\quad\quad\text{ in
}\Omega.$ (6)

We prescribe a traction free boundary condition on $\Gamma_{e}$, giving

$\boldsymbol{\sigma}^{*}\cdot\textbf{n}_{e}=0\quad\quad\quad\text{ on
}\Gamma_{e},$ (7)

where $\textbf{n}_{e}$ is the unit vector normal to the interface between the
anode and the electrolyte. We also prescribe continuity of normal stress and
displacement between anode materials

$[\boldsymbol{\sigma}^{*}\cdot\textbf{n}_{ab}]^{+}_{-}=[\boldsymbol{u}^{*}]^{+}_{-}=0\quad\quad\quad\text{
on }\Gamma_{ab},$ (8)

where $\textbf{n}_{ab}$ is the unit vector normal to the interface between
anode materials $a$ and $b$.

####  2.1.2 Chemical Model

We model the movement of lithium within the anode as a diffusive process, with
flux $\textbf{j}^{*}$ given by

$\textbf{j}^{*}=-\frac{D_{a}}{R_{g}T}c_{a}^{*}\boldsymbol{\nabla}^{*}\mu_{a}^{*}\quad\text{
in }\Omega_{a},$ (9)

where $\mu_{a}^{*}$ is the chemical potential of the lithium intercalated into
material $a$, $D_{a}$ is the diffusion coefficient of lithium through material
$a$, $R_{g}$ is the ideal gas constant 8.314 J mol-1 K-1 and $T$ is the
temperature, which we assume to be constant, uniform and equal to 298 K. The
time-scale we focus on is much slower than any dynamics in the system, so that
we may make the quasi-steady approximation $j^{*}=0$. Thus

$c_{a}^{*}\boldsymbol{\nabla}^{*}\mu^{*}_{a}=0\quad\quad\quad\text{ in
}\Omega_{a},$ (10)

implying the chemical potential of the lithium is uniform in each anode
material for all non-trivial concentrations $c_{a}$. At the interface with the
electrolyte we have

$[\mu^{*}_{a}]^{+}_{-}=0\quad\text{ on }\Gamma_{e}.$ (11)

Similarly, at the interface between anode materials $\mu^{*}_{a}$ must be
continuous:

$[\mu^{*}_{a}]^{+}_{-}=0\quad\text{ on }\Gamma_{ab},$ (12)

and therefore $\mu^{*}_{a}$ takes the same value for $a=1,\dots n$ and we
denote this value by $\mu^{*}$.

We define the chemical potential of the lithium intercalated into material $a$
using the stress-dependent chemical potential commonly used with linear
elasticity [23], given by

$\mu^{*}=\tilde{\mu}_{a}^{\text{SF},*}(c_{a}^{*})-\eta_{a}V_{a}^{m}\text{tr}\big{(}\boldsymbol{\sigma}^{*}\big{)}\quad\text{
in }\Omega_{a},$ (13)

where, $\text{tr}(\boldsymbol{\sigma}^{*})$ denotes the trace of
$\boldsymbol{\sigma}^{*}$ and $\tilde{\mu}_{a}^{\text{SF,*}}(c_{a}^{*})$ is
the stress-free chemical potential of lithium in material $a$. We explain how
$\tilde{\mu}_{a}^{\text{SF,*}}$ is obtained in Section 2.2.

To close the system, we prescribe the total amount of intercalated lithium. We
describe this as a proportion of the maximum amount possible using the state
of charge parameter $c_{0}\in[0,1]$, where $c_{0}=0$ denotes no lithium and
$c_{0}=1$ denotes a fully lithiated anode. Thus we impose

$c_{0}\sum_{a=1}^{n}\bigg{(}\int_{\Omega_{a}}c_{a}^{\text{max}}\;\;\text{d}V\bigg{)}=\sum_{a=1}^{n}\bigg{(}\int_{\Omega_{a}}c_{a}^{*}\;\;\text{d}V\bigg{)},$
(14)

where $c_{a}^{\text{max}}$ is the maximum lithium concentration possible in
material $a$.

In summary, the full dimensional mechanical model is given by the governing
equation (6), the Cauchy stress (4) and the boundary conditions (7)-(8). The
full dimensional chemical model is given by the uniform chemical potential
(13) and state of charge condition (14). Given a value of $c_{0}\in[0,1]$, the
concentration and displacement profile can be calculated from the model.

###  2.2 Using OCVs to Calculate Chemical Potential

In this section, we outline how the stress-free chemical potential of the
lithium atoms $\tilde{\mu}_{a}^{\text{SF},*}(c_{a}^{*})$ in a single anode
material $a$ can be determined from the OCV of that material. We follow Bazant
[1] and Newman et al. [30] by equating the electrochemical potentials of the
reactants and the products of the surface reaction occurring at $\Gamma_{e}$.
The reversible reaction at the surface of each electrode during charging or
discharging is

$\text{Li}_{\text{s}}\xleftrightharpoons{}\text{Li}^{+}_{\text{aq}}+e^{-}_{\text{s}},$
(15)

where the subscript s denotes that the lithium atoms and electrons are in the
solid electrode and the subscript aq denotes that the lithium ions are
dissolved in the electrolyte.

As these reactions are in equilibrium during OCV measurement, we may equate
the total electrochemical potential of the reactants and products. Since we
are only concerned with single-material electrodes here, and the electrodes
are assumed to be unconfined, there is no stress-dependent contribution to the
chemical potential. The balances of electrochemical potentials at the surface
of the anode and the cathode are thus given by

$\displaystyle\tilde{\mu}^{\text{SF},*}_{\text{Li},a}(c_{\text{Li},a}^{*})$
$\displaystyle=\mu_{\text{Li}^{+},\text{el}}^{*}(c_{\text{Li}^{+},\text{el}}^{*})+ze\phi^{*}_{\text{el}}+\mu_{e^{-},a}^{*}(c_{e^{-},a}^{*})-ze\phi^{*}_{a},$
(16)
$\displaystyle\tilde{\mu}^{\text{SF},*}_{\text{Li},c}(c_{\text{Li},c}^{*})$
$\displaystyle=\mu_{\text{Li}^{+},\text{el}}^{*}(c_{\text{Li}^{+},\text{el}}^{*})+ze\phi^{*}_{\text{el}}+\mu_{e^{-},c}^{*}(c_{e^{-},c}^{*})-ze\phi^{*}_{c},$
(17)

respectively, where, $\phi^{*}$ represents the electrical potential. The first
subscript of the chemical potentials $\mu^{*}$ in (16)-(17) denotes the
chemical species that the chemical potential corresponds to. The subscript of
the electrical potentials and the second subscript of the chemical potentials
denote the phase which the species is in, $a$ denoting anode, $c$ denoting
cathode and el denoting electrolyte. We have used the first subscript to avoid
confusion with specifying the chemical potentials of different species,
however, it should be noted that $\tilde{\mu}^{\text{SF},*}_{a}(c_{a}^{*})$ in
the notation of Section 2.1.2 is
$\tilde{\mu}^{\text{SF},*}_{\text{Li},a}(c_{\text{Li},a}^{*})$ in (16).
Finally, the charge of the Li+ ions is denoted by $z$ and $e$ is the
fundamental charge 1.60217$\times 10^{-19}$C.

The OCV, denoted by $E^{\text{OC}}$, measures the difference in the
electrochemical potential of the electrons in the anode and the cathode,
divided by the charge of the electron $-e$ [30], and hence is given by

$E^{\text{OC}}=-\frac{\mu_{e,a}^{*}(c_{e,a}^{*})}{e}+z\phi^{*}_{a}+\frac{\mu_{e,c}^{*}(c_{e,c}^{*})}{e}-z\phi^{*}_{c}.$
(18)

We can thus subtract (17) from (16), substitute (18), and rearrange to obtain
an expression for the chemical potential of the lithium in the anode in terms
of the OCV, given by

$\displaystyle\tilde{\mu}^{\text{SF},*}_{\text{Li},a}(c_{\text{Li},a}^{*})=-eE^{\text{OC}}+\mu^{\text{SF},*}_{\text{Li},c}(c_{\text{Li},c}^{*}).$
(19)

Hence, we convert the OCV, $E^{\text{OC}}$, conventionally measured in volts,
to chemical potential $\tilde{\mu}^{\text{SF},*}_{\text{Li},a}$, measured in J
mol-1, by multiplying by the fundamental charge, $e$ (J V-1), and Avagadro’s
number, 6.02214086 $\times 10^{23}$ mol-1.

The chemical potential for the lithium in the cathode
$\mu^{\text{SF},*}_{\text{Li},c}(c_{\text{Li},c}^{*})$ in (19) does not
concern this work for two reasons. Firstly, the OCV of the anode material is
often measured against a cathode of lithium metal, thus the backwards reaction
of (15) is lithium plating and so $\mu^{\text{SF},*}_{\text{Li},c}$ is
independent of $c_{\text{Li},c}^{*}$. This makes
$\mu^{\text{SF},*}_{\text{Li},c}$ an additive constant, only shifting the
reference potential for $\tilde{\mu}^{\text{SF},*}_{\text{Li},a}$. Secondly,
we are only concerned with comparing the potentials of the lithium in
different anode materials or comparing the potentials of lithium in an anode
material at different states of charge. Therefore, as long as the cathode used
when calculating the OCV of the different materials is the same, this additive
constant will cancel when comparing them.

###  2.3 Nondimensionalisation

We now nondimensionalise the variables in our chemo-mechanical model. The
spatial coordinate $\boldsymbol{x}^{*}$, the displacement vector
$\boldsymbol{u}^{*}$, the Cauchy stress $\boldsymbol{\sigma}^{*}$, the
concentration in each material $c_{a}^{*}$ and the stress-free and stress-
dependent chemical potentials $\mu_{a}^{\text{SF,*}}$ and $\mu_{a}^{*}$ are
nondimensionalised by setting

$\displaystyle\boldsymbol{x}^{*}$
$\displaystyle=L\boldsymbol{x},\quad\quad\quad\boldsymbol{u}^{*}=L\eta_{1}V_{1}^{m}c_{1}^{\text{max}}\boldsymbol{u}\quad\quad\quad\boldsymbol{\sigma}^{*}=G_{1}^{*}(0)\eta_{1}V^{m}_{1}c_{1}^{\text{max}}\boldsymbol{\sigma},$
(20) $\displaystyle c_{a}^{*}=c_{a}^{\text{max}}c_{a},$
$\displaystyle\quad\quad\quad\tilde{\mu}_{a}^{\text{SF},*}(c_{a}^{*})=R_{g}T\mu_{a}^{\text{SF}}(c_{a}),\quad\quad\quad\mu_{a}^{*}=R_{g}T\mu_{a},\quad\text{
for }a=1\dots n,$

respectively. Here, $L$ is the representative length-scale of the anode
particle, $G_{1}(0)$ is the shear modulus of material 1 at zero lithiation and
we have chosen to scale each variable using the parameters relevant to
material 1. We note that the linear forms of the stress (4) and the chemical
potential (13) are only justified in situations where
$\eta_{a}V_{a}^{m}c_{a}^{\text{max}}\ll 1$ for $a=1,\dots,n$, since then the
linearisation of the elastic strain tensor in (1) is permitted. A full
derivation from the fully nonlinear formulation to the mechanical model we use
here can be found in [35].

Using this nondimensionalisation, the governing equation (6) is

$\displaystyle\boldsymbol{\nabla}\cdot\boldsymbol{\sigma}$
$\displaystyle=0\quad$ $\displaystyle\text{ in }\Omega,$ (21)

with

$\displaystyle\boldsymbol{\sigma}$
$\displaystyle=\mathbb{C}_{a}:\big{(}\boldsymbol{\nabla}\boldsymbol{u}-\gamma_{a}c_{a}\boldsymbol{1}\big{)}$
$\displaystyle=\lambda_{a}(c_{a})\delta_{ij}\frac{\partial u_{k}}{\partial
x_{k}}+G_{a}(c_{a})\bigg{(}\frac{\partial u_{i}}{\partial
x_{j}}+\frac{\partial u_{j}}{\partial
x_{i}}\bigg{)}-\gamma_{a}c_{a}(3\lambda_{a}(c_{a})+2G_{a}(c_{a}))\delta_{ij}\quad\text{
in }\Omega_{a},$ (22)

while the uniform chemical potential is given by

$\mu=\mu_{a}^{\text{SF}}(c_{a})-S_{a}^{d}\text{tr}\big{(}\boldsymbol{\sigma}\big{)}\quad\text{
in }\Omega_{a},$ (23)

where the dimensionless parameters are given by

$\lambda_{a}(c_{a})=\frac{\lambda_{a}^{*}(c_{a}^{*})}{G_{1}^{*}(0)},\quad
G_{a}(c_{a})=\frac{G_{a}^{*}(c_{a}^{*})}{G_{1}^{*}(0)},\quad\gamma_{a}=\frac{\eta_{a}V^{m}_{a}c_{a}^{\text{max}}}{\eta_{1}V_{1}^{m}c_{1}^{\text{max}}},\quad\text{
and }\quad
S_{a}^{d}=\frac{\eta_{a}\eta_{1}V^{m}_{a}V^{m}_{1}c_{1}^{\text{max}}G_{1}^{*}(0)}{R_{g}T}.$
(24)

Henceforth, we drop the explicitly written lithiation-dependence of the Lamé
parameters and write $\lambda_{a}\equiv\lambda(c_{a})$ and $G_{a}\equiv
G_{a}(c_{a})$. The boundary conditions are

$\displaystyle\boldsymbol{\sigma}\cdot\textbf{n}$
$\displaystyle=0\quad\quad\quad\text{ on }\Gamma_{e},$ (25)
$\displaystyle[\boldsymbol{\sigma}\cdot\textbf{n}]^{+}_{-}=[\boldsymbol{u}]^{+}_{-}$
$\displaystyle=0\quad\quad\quad\text{ on }\Gamma_{ab},$ (26)

while the SOC condition is

$c_{0}\sum_{a=1}^{n}\bigg{(}\int_{\Omega_{a}}c_{a}^{\text{max}}\;\;\text{d}V\bigg{)}=\sum_{a=1}^{n}\bigg{(}\frac{c_{a}^{\text{max}}}{c_{1}^{\text{max}}}\int_{\Omega_{a}}c_{a}\;\;\text{d}V\bigg{)}.$
(27)

###  2.4 Spherical Geometry

We now focus on the specific geometry of a single, spherical, radially
symmetric nano-particle comprising a core of anode material 1 enclosed in a
shell of anode material 2. We denote the radius of the core made of material 1
by $R_{1}$, and thus material 1 occupies the domain $\Omega_{1}=\\{r|0\leq
r<R_{1}\\}$. We denote the radius of the entire nano-particle by $R_{2}$ so
that the shell made of material 2 has thickness $R_{2}-R_{1}$ and occupies the
domain $\Omega_{2}=\\{r|R_{1}<r\leq R_{2}\\}$. This geometry is shown
schematically in Figure 2. Our representative length-scale of the anode
particle $L$ is now given by the radius, $R_{2}$.

Figure 1: Schematic of a slice through a spherical nano-particle with a
spherical central core of material 1 denoted by $\Omega_{1}$ and a shell of
material 2 denoted by $\Omega_{2}$. The outer radii of the two regions are
labelled as $R_{1}$ and $R_{2}$, respectively, and the direction of the
spatial variable $r$ is labelled.

We assume the displacement $\boldsymbol{u}$ is solely radial due to symmetry
and thus we write $\boldsymbol{u}=u(r)\boldsymbol{e}_{r}$. The Cauchy stress
and the chemical potential of lithium in material $a$, (22) and (23), are then
given by

$\displaystyle\boldsymbol{\sigma}$ $\displaystyle=\text{
diag}\begin{bmatrix}\lambda_{a}\Big{(}\frac{\text{d}u}{\text{d}r}+\frac{2u}{r}\Big{)}+2G_{a}\frac{\text{d}u}{\text{d}r}-\gamma_{a}(3\lambda_{a}+2G_{a})c_{a}\\\
\lambda_{a}\Big{(}\frac{\text{d}u}{\text{d}r}+\frac{2u}{r}\Big{)}+2G_{a}\frac{u}{r}-\gamma_{a}(3\lambda_{a}+2G_{a})c_{a}\\\
\lambda_{a}\Big{(}\frac{\text{d}u}{\text{d}r}+\frac{2u}{r}\Big{)}+2G_{a}\frac{u}{r}-\gamma_{a}(3\lambda_{a}+2G_{a})c_{a}\end{bmatrix}\quad$
$\displaystyle\text{ in }\Omega_{a},$ (28) $\displaystyle\mu$
$\displaystyle=\mu_{a}^{\text{SF}}\big{(}c_{a}(r)\big{)}-S_{a}^{d}(3\lambda_{a}+2G_{a})\bigg{(}\frac{1}{r^{2}}\frac{\text{d}}{\text{d}r}\big{(}r^{2}u\big{)}-3\gamma_{a}c_{a}\bigg{)}\quad$
$\displaystyle\text{ in }\Omega_{a},$ (29)

respectively. The mechanical equilibrium equation (21) becomes

$\displaystyle\frac{\text{d}}{\text{d}r}\bigg{(}\frac{1}{r^{2}}\frac{\text{d}}{\text{d}r}(r^{2}u)\bigg{)}-\frac{\gamma_{a}(3\lambda_{a}+2G_{a})}{\lambda_{a}+2G_{a}}\frac{\text{d}c_{a}}{\text{d}r}=0,\quad$
$\displaystyle\text{ in }\Omega_{a},$ (30)

Integrating (30) gives

$\frac{\text{d}u}{\text{d}r}+\frac{2u}{r}-\frac{\gamma_{a}(3\lambda_{a}+2G_{a})}{\lambda_{a}+2G_{a}}c_{a}=\beta_{a}\quad\text{
in }\Omega_{a},$ (31)

where the integration constant $\beta_{a}$ may be different in the two
materials. The trace of the Cauchy stress tensor (28) is given by

$\text{tr}(\boldsymbol{\sigma})=\big{(}3\lambda_{a}+2G_{a}\big{)}\bigg{(}\frac{\text{d}u}{\text{d}r}+\frac{2u}{r}-3\gamma_{a}c_{a}\bigg{)}\quad\text{
in }\Omega_{a}.$ (32)

Using (31) gives

$\text{tr}(\boldsymbol{\sigma})=(3\lambda_{a}+2G_{a})\bigg{(}\beta_{a}+\frac{\gamma_{a}(3\lambda_{a}+2G_{a})}{\lambda_{a}+2G_{a}}c_{a}-3\gamma_{a}c_{a}\bigg{)}\quad\text{
in }\Omega_{a}.$ (33)

Thus the trace of the Cauchy stress and therefore the chemical potential,
(29), can be written in terms of $c_{a}$ only. Since $\mu_{a}$ is uniform in
each material we conclude that $c_{a}$ is also uniform in each material.

Substituting a uniform $c_{a}$ into (31) and solving the resulting ODE yields

$u=A_{a}r+\frac{B_{a}}{r^{2}}\quad\text{ in }\Omega_{a},$ (34)

where
$A_{a}=3\beta_{a}+(3\gamma_{a}(3\lambda_{a}+2G_{a})c_{a})/(\lambda_{a}+2G_{a})$
is now the constant arising from the first integration, $B_{a}$ is the
constant arising from the second integration and each integration constant may
be different in the two materials. Substituting (34) into (29) gives

$\mu=\mu^{\text{SF}}_{a}(c_{a})-3S^{d}_{a}\big{(}3\lambda_{a}+2G_{a}\big{)}(A_{a}-\gamma_{a}c_{a}),\quad\text{
in }\Omega_{a}.$ (35)

We can substitute (34) into (28) and write the radial stress and hoop stresses
as

$\displaystyle\sigma_{rr}$
$\displaystyle=(3\lambda_{a}+2G_{a})(A_{a}-\gamma_{a}c_{a})-\frac{4G_{a}B_{a}}{r^{3}}\quad\text{
in }\Omega_{a},$ (36) $\displaystyle\sigma_{\theta\theta}=\sigma_{\phi\phi}$
$\displaystyle=(3\lambda_{a}+2G_{a})(A_{a}-\gamma_{a}c_{a})+\frac{2G_{a}B_{a}}{r^{3}}\quad\text{
in }\Omega_{a},$ (37)

respectively. It remains to use the boundary conditions to determine the
integration constants $A_{a}$ and $B_{a}$.

The boundary conditions (25) and (26) for the spherical geometry are written
as

$\displaystyle\sigma_{rr}$ $\displaystyle=0,\quad\text{ on }r=1,$ (38)
$\displaystyle[\sigma_{rr}]^{+}_{-}=[u]^{+}_{-}$ $\displaystyle=0,\quad\text{
on }r=R,$ (39)

respectively, where $R=R_{1}/R_{2}$. Additionally, we ensure that the
displacement is bounded at the origin by prescribing

$u=0\text{ at }r=0.$ (40)

Substituting (34) and (36) into (38)-(40) gives

$\displaystyle A_{1}$
$\displaystyle=\frac{1}{\omega}\Big{[}(3\lambda_{1}+2G_{1})\big{(}(3\lambda_{2}+2G_{2})+4G_{2}R^{3}\big{)}\gamma_{1}c_{1}+4G_{2}(1-R^{3})(3\lambda_{2}+2G_{2})\gamma_{2}c_{2}\Big{]},$
(41) $\displaystyle B_{1}$ $\displaystyle=0,$ (42) $\displaystyle A_{2}$
$\displaystyle=\frac{1}{\omega}\Big{[}(3\lambda_{2}+2G_{2})\big{(}4G_{2}(1-R^{3})+(3\lambda_{1}+2G_{1})\big{)}\gamma_{2}c_{2}+4G_{2}R^{3}(3\lambda_{1}+2G_{1})\gamma_{1}c_{1}\Big{]},$
(43) $\displaystyle B_{2}$
$\displaystyle=\frac{1}{\omega}\Big{[}(3\lambda_{1}+2G_{1})(3\lambda_{2}+2G_{2})(\gamma_{1}c_{1}-\gamma_{2}c_{2})R^{3}\Big{]},$
(44)

where

$\omega=(3\lambda_{1}+2G_{1})(3\lambda_{2}+2G_{2})+4G_{2}\big{(}(3\lambda_{2}+2G_{2})(1-R^{3})+(3\lambda_{1}+2G_{1})R^{3}\big{)}.$
(45)

Thus the displacements and stress are given once the concentrations $c_{1}$
and $c_{2}$ have been determined.

The SOC condition (27) becomes

$c_{0}\bigg{[}R^{3}+\big{(}1-R^{3}\big{)}\frac{c_{2}^{\text{max}}}{c_{1}^{\text{max}}}\bigg{]}=R^{3}c_{1}+\frac{c_{2}^{\text{max}}}{c_{1}^{\text{max}}}(1-R^{3})c_{2},$
(46)

which we can rearrange as

$c_{1}=c_{0}+\frac{c_{2}^{\text{max}}}{c_{1}^{\text{max}}}\big{(}1-R^{-3}\big{)}\big{(}c_{2}-c_{0}\big{)}.$
(47)

Finally, we use (35) to relate the concentrations $c_{1}$ and $c_{2}$.
Combining this with (47) gives a single algebraic equation in $c_{2}$ for a
given $c_{0}$ to be solved numerically222For very high values of $R$ and very
low values of $c_{0}$, this function has multiple roots. For consistency, we
take the lowest root in these cases.. As the concentrations $c_{a}$ must be
between zero and one due to the nondimensionalisation (20), we place bounds on
$c_{2}$, ensuring that both $c_{1},c_{2}\in[0,1]$. We use (47) to find that

$\max\bigg{\\{}0,\frac{(1-c_{0})c^{\text{max}}_{1}R^{3}}{c^{\text{max}}_{2}(R^{3}-1)}+c_{0}\bigg{\\}}<c_{2}<\min\bigg{\\{}1,c_{0}-\frac{c_{0}c_{1}^{\text{max}}R^{3}}{c_{2}^{\text{max}}(R^{3}-1)}\bigg{\\}},$
(48)

where the first and second elements of the maximum and minimum functions
ensure $c_{2}\in[0,1]$ and $c_{1}\in[0,1]$, respectively.

##  3 Chemical Results

We illustrate the behaviour of a nano-particle consisting of a silicon core
and a graphite shell described by the model and geometry explained in Section
2.4. The relevant mechanical and chemical parameters for these materials are
given in Table 1, including the nondimensional parameters $\gamma_{a}$ and
$S^{d}_{a}$. We calculate the Lamé parameters of silicon and graphite from the
Young’s modulus and the Poisson’s ratio. The MATLAB code to produce the
figures in this section is available publicly on GitHub [34].

We assume the Young’s modulus varies linearly with the lithiation state using
the values at zero and full lithiation as reference values. Therefore, the
dimensional Lamé parameters are given by

$\lambda_{a}^{*}=\frac{E^{0*}_{a}(1+\eta^{E}_{a}V_{a}^{m}c_{a}^{*})\nu_{a}}{(1+\nu_{a})(1-2\nu_{a})},\quad\quad
G_{a}^{*}=\frac{E^{0*}_{a}(1+\eta^{E}_{a}V_{a}^{m}c_{a}^{*})}{2(1+\nu_{a})},$
(49)

where $E^{0*}_{a}$ is the Young’s modulus of material $a$ at zero lithiation,
$\nu_{a}$ is the Poisson’s ratio of material $a$ and $\eta^{E}_{a}$ is the
linear variation of the Young’s modulus of material $a$. As the Poisson’s
ratios of these materials vary negligibly with lithiation state [10], we
assume $\nu_{a}$ to be independent of $c_{a}$.

Parameter | Description | Silicon | Graphite
---|---|---|---
$J^{c}$ | Relative expanded volume at $c_{a}=1$ | $3.8^{\text{\cite[cite]{[\@@bibref{Number}{liu2012size}{}{}]}}}$ | $1.1^{\text{\cite[cite]{[\@@bibref{Number}{qi2014lithium}{}{}]}}}$
$x^{\text{max}}$ | Maximum stoichiometric ratio of Li | 3.75${}^{\text{\cite[cite]{[\@@bibref{Number}{Hatchard2004}{}{}]}}}$ | 0.167${}^{\text{\cite[cite]{[\@@bibref{Number}{chakraborty2015combining}{}{}]}}}$
$V^{m}$ | Molar volume (m3mol-1) | $1.205\times 10^{-5\text{\cite[cite]{[\@@bibref{Number}{cui2012finite}{}{}]}}}$ | $8.69\times 10^{-6\text{\cite[cite]{[\@@bibref{Number}{yu1999determination}{}{}]}}}$
$c^{\text{max}}$ | Maximum Li concentration (mol m-3) | $3.11\times 10^{5\text{\cite[cite]{[\@@bibref{Number}{Roper2019silicon}{}{}]}}}$ | $1.92\times 10^{4\text{\cite[cite]{[\@@bibref{Number}{Roper2019silicon}{}{}]}}}$
$\eta$ | Coefficient of compositional expansion | 0.2489${}^{\text{\cite[cite]{[\@@bibref{Number}{Roper2019silicon}{}{}]}}}$ | 0.2${}^{\text{\cite[cite]{[\@@bibref{Number}{Roper2019silicon}{}{}]}}}$
$\nu$ | Poisson’s ratio | 0.29${}^{\text{\cite[cite]{[\@@bibref{Number}{qi2014lithium}{}{}]}}}$ | 0.32${}^{\text{\cite[cite]{[\@@bibref{Number}{qi2014lithium}{}{}]}}}$
$E_{0}^{*}$ | Young’s modulus at $c_{a}=0$ (GPa) | 96.0${}^{\text{\cite[cite]{[\@@bibref{Number}{qi2014lithium}{}{}]}}}$ | 32.0${}^{\text{\cite[cite]{[\@@bibref{Number}{qi2014lithium}{}{}]}}}$
$\eta_{a}^{E}$ | Linear conc. dependence of Young’s Modulus | $-0.1302^{1}$ | $14.4375^{1}$
$\gamma$ | Nondim. expansion coefficient (Eq. 24) | 1.0 | 0.0357
$S^{d}$ | Stress-assisted diffusion param. (Eq. 24) | 42.046 | 1.502

1 Calculated from the Young’s modulus at $c_{a}=1$ from [31]

Table 1: Material parameters of silicon and graphite.

We plot the non-dimensionalised stress-free chemical potentials of the lithium
intercalated into silicon and graphite, $\mu_{1}^{\text{SF}}$ and
$\mu_{2}^{\text{SF}}$, in Figure 2. It can be seen that there are several
regions where a small change in lithium concentration causes a large change in
the chemical potential, as noted in [3].

In Figure 3 we plot the lithium concentration in each material, $c_{1}$ and
$c_{2}$, against $c_{0}$ for different values of the volume fraction
$\psi=R^{3}$ of the silicon core. It can be seen that the graphite is
saturated ($c_{2}=1$) at fairly low $c_{0}$ values, especially for large
silicon cores, whereas the lithium concentration in the silicon, $c_{1}$,
remains relatively low until the graphite is saturated. After this point, all
the lithium being intercalated must be intercalated into the silicon and thus
$c_{1}$ is linear in $c_{0}$.

Figure 2: Nondimensional stress-free chemical potential $\mu^{\text{SF}}_{a}$
of the lithium in each material as a function of nondimensional concentration
of lithium for silicon and graphite, $c_{1}$ and $c_{2}$. Data captured with
permission using grabit software for MATLAB from Figure 4c in [24] (silicon)
and Figure 1 in [32] (graphite), and linearly interpolated. Figure 3:
Nondimensional lithium concentrations within each material, $c_{1}$ (solid
lines) and $c_{2}$ (dashed lines), against state of charge, $c_{0}$, for
different core volume fractions, $\psi$. We also plot $c_{1}=c_{2}=c_{0}$
labelled as “Single Material” to highlight the difference in concentration
caused by the presence of a second material. Figure 4: Nondimensional trace of
the Cauchy stress tensor, $\text{tr}(\boldsymbol{\sigma})$ for each material
(solid lines for material 1, dashed lines for material 2), against state of
charge, $c_{0}$, and for different core volume fractions, $\psi$.

The saturation of the graphite, and the low $c_{1}$ values, at low SOC are due
to the hydrostatic stresses induced in the silicon core and graphite shell.
The trace of the Cauchy stress in each material, given by (33), is plotted
against $c_{0}$ in Figure 4 for different silicon volume fractions, $\psi$.
Since $\gamma_{1}\gg\gamma_{2}$, the expansion of the silicon is much greater
than that of the graphite. Therefore, when lithiated, the expansion of the
silicon is being constrained by the graphite shell, inducing a compressive
stress in the silicon, causing $\text{tr}(\boldsymbol{\sigma})$ to be negative
in $\Omega_{1}$. Conversely, the graphite is being stretched by the large
expansion of silicon, inducing a tensile stress, causing
$\text{tr}(\boldsymbol{\sigma})$ to be positive in $\Omega_{2}$. The tensile
stress in the graphite lowers the chemical potential of the lithium whereas
the compressive stress in the silicon increases the chemical potential of the
lithium. Therefore, for the chemical potentials of the lithium in each
material to be equal, the stress-free potential of the lithium in the silicon
must therefore decrease and the stress-free potential of the lithium in the
graphite must increase. The stress-free potential of the lithium in each
material is a monotonically increasing function of the concentration,
therefore $c_{1}$ must decrease and $c_{2}$ must increase.

Figure 5: Nondimensional chemical potential, $\mu$, against state of charge,
$c_{0}$, for different core volume fractions, $\psi$. The chemical potentials
of lithium in graphite and silicon from Figure 2 are also plotted as
‘Graphite’ and ‘Silicon’, respectively.

The low $c_{1}$ values are present for a larger range of SOC for small silicon
cores than for large ones since a thicker graphite shell is stronger, inducing
a larger compressive stress on the silicon. Likewise, a small silicon core
does not stretch the graphite shell as much and thus, for example, the $c_{2}$
plot for $\psi=0.05$ is more similar to the single material case than that for
$\psi=0.5$.

In Figure 5 we plot the chemical potential of the lithium in the hybrid nano-
particle against SOC for different values of $\psi$. It can be seen that for
SOC values for which the graphite is saturated, the chemical potential is much
greater than that of the individual materials. This is because if the graphite
is saturated but more lithium is intercalated, the lithium concentration in
the silicon must increase. This increases $\mu_{a}^{\text{SF}}$ but also
decreases $\text{tr}(\boldsymbol{\sigma})$ further as seen in Figure 4,
causing the chemical potential to increase rapidly with increasing SOC.

In Figures 6 and 7 we replicate Figures 3 and 5, respectively, but we do not
account for the effect of stress-assisted diffusion (we set $S_{a}^{d}=0$) to
show the importance of including the stress of the anode materials in the
chemical potential of the lithium. It can be seen in Figure 6 that
$c_{1}>c_{2}$ for $c_{0}\lessapprox 0.8$ when $S^{d}_{a}=0$, which is very
different to the result found in Figure 3 in which $c_{2}>c_{1}$ for
$c_{0}\gtrapprox 0.05$. This is solely due to the chemical potential of the
lithium in the silicon being less than that for graphite for $c_{a}\lessapprox
0.8$, seen in Figure 2. In Figure 7, the chemical potential of the lithium in
the hybrid nano-particle is approximately an interpolation between the
chemical potentials of the lithium within the individual materials, which is
also very different to the case in which $S_{a}^{d}\neq 0$.

Figure 6: Nondimensional lithium concentrations within each material, $c_{1}$
(solid lines) and $c_{2}$ (dashed lines), against state of charge, $c_{0}$,
for different core volume fractions, $\psi$, without stress-assisted diffusion
($S^{d}_{a}=0$). Figure 7: Nondimensional chemical potential, $\mu$, against
state of charge, $c_{0}$, for different core volume fractions, $\psi$, without
stress-assisted diffusion ($S^{d}_{a}=0$).

The large differences in $c_{1}$ and $c_{2}$ between Figures 3 and 6 caused by
the exclusion of stress-coupling demonstrates how expansion due to lithium
will be dramatically affected by this mechanism. Furthermore, the large
differences in the chemical potential of the lithium in the hybrid particle
between Figures 5 and 7 show that stress-modelling is essential in predicting
OCVs of hybrid anodes as a function of SOC.

##  4 Optimal Size of the Silicon Core

###  4.1 Performance Measures

We now discuss the performance measures that might be considered to find an
optimal anode design, applying them to the nano-particle geometry shown in
Figure 1. The three properties we consider as these performance measures are
i) the total amount of lithium absorbed, ii) the expanded volume of the anode
and iii) the maximum stress induced. We first derive each of these measures
using the results of our model and then investigate how these might be applied
in practice to determine the optimal volume fraction of the silicon core in
the spherical core–shell nano-particle geometry.

####  4.1.1 Amount of Lithium

The capacity of a lithium-ion battery is closely related to the amount of
lithium that the anode can accommodate and thus this is a very important
performance measure of a lithium-ion battery anode. For a general anode with
domain $\Omega$ and different anode materials $i=1,\dots,n$, each with domain
$\Omega_{i}$, we measure the total amount of intercalated lithium relative to
a fully lithiated anode of material 1 and domain $\Omega$. We denote this
relative amount of lithium as

$Q=\frac{\sum_{i=1}^{n}\int_{\Omega_{i}}c_{i}^{*}\;\text{d}V}{\int_{\Omega}c_{1}^{\text{max}}\;\text{d}V}.$
(50)

For the core–shell geometry in Figure 1, $Q$ is given by

$Q=c_{1}R^{3}+c_{2}\frac{c_{2}^{\text{max}}}{c_{1}^{\text{max}}}(1-R^{3}),$
(51)

where we recall $R=R_{1}/R_{2}$.

####  4.1.2 Relative Expanded Volume

Figure 8: Relative expanded volume, $V$, given by (52), against core volume
fraction, $\psi$, for different states of charge, $c_{0}$.

The expansion of the anode can have serious adverse effects on the battery
performance and so we consider this expansion by calculating the volume of the
expanded nano-particle compared to the original volume, denoting this as the
relative expanded volume, $V$. For a general geometry of anode, $V$ would
usually have to be calculated numerically, however, for the spherical nano-
particle geometry in Figure 1, $V$ is given by

$V=\frac{(R_{2}+u^{*}(R_{2}))^{3}}{R_{2}^{3}}=(1+\eta_{1}V_{1}^{m}c_{1}^{\text{max}}u(1))^{3}.$
(52)

Here, we use the dimensional displacement
$u^{*}(R_{2})=\eta_{1}V_{1}^{m}c_{1}^{\text{max}}R_{2}u(1)$ as in (20).

In Figure 8, we plot $V$ against $\psi$ for different states of charge. It can
be seen that $V$ increases with the volume of the silicon core with this
effect becoming more prominent for larger $\psi$. Additionally, a greater SOC
causes a greater $V$ for all volumes of silicon core. The calculated $V$ value
for a fully lithiated nano-particle with a large silicon core, is much greater
than the observed expanded volume for a solely silicon nano-particle, given by
$V=J^{c}_{1}=3.8$. This over expansion for large silicon cores is due to our
adoption of linear elasticity in Section 2.1.1. This assumes that
$\eta_{1}V_{1}^{m}c_{1}^{\text{max}}\ll 1$, whereas for silicon,
$\eta_{1}V_{1}^{m}c_{1}^{\text{max}}=0.933375\sim 1$. Thus the nonlinear
elasticity formulation should be retained to produce quantitatively accurate
results for silicon at high concentrations.

####  4.1.3 Maximum Induced Stress

Capacity fade exhibited by expanding anode materials after cycling is often
attributed to cracks in the anode material caused by high stresses. Thus the
final performance measure we use to optimise anode performance is the maximum
induced stress. We use the von Mises stress [19] as a scalar effective stress
measure. For a general anode geometry, this is given by

$\sigma_{\text{eff}}^{*}=\bigg{(}\frac{(\sigma_{11}^{*}-\sigma_{22}^{*})^{2}+(\sigma_{22}^{*}-\sigma_{33}^{*})^{2}+(\sigma_{33}^{*}-\sigma_{11}^{*})^{2}+6(\sigma_{12}^{*2}+\sigma_{23}^{*2}+\sigma_{31}^{*2})}{2}\bigg{)}^{\frac{1}{2}}.$
(53)

In radial symmetry, the von Mises stress (53) can be written as

$\sigma^{*}_{\text{eff}}=|\sigma_{rr}^{*}-\sigma_{\theta\theta}^{*}|,$ (54)

so that from (20), (36) and (37), we have

$\sigma^{*}_{\text{eff}}=\frac{6G_{1}^{*}\eta_{1}V^{m}_{1}c_{1}^{\text{max}}G_{a}|B_{a}|}{r^{3}},$
(55)

for $a=1,2$. The condition at $r=0$ (40) gives $B_{1}=0$. Therefore, the
effective stress is only non-zero in $\Omega_{2}$. The $r^{-3}$ dependence
shows that this effective stress is the greatest at the minimum value of $r$
in $\Omega_{2}$, which is $r=R=R_{1}/R_{2}$. Therefore, the maximum induced
stress is given by

$\sigma^{*}_{\text{eff}}(R_{1})=\frac{6\eta_{1}V^{m}_{1}c_{1}^{\text{max}}G_{2}^{*}|B_{2}|}{R^{3}}.$
(56)

In Figure 9, we plot $\sigma_{\text{eff}}^{*}(R_{1})$ against $\psi$ for
different SOC. For all states of charge, $\sigma_{\text{eff}}^{*}(R_{1})$ is
minimised as $\psi\rightarrow 0$; therefore, a very small silicon core is
optimal. Of course, a single-material anode nano-particle will not induce any
stress in equilibrium and thus the two single-material designs in this case
are globally optimal for this measure with $\sigma_{\text{eff}}^{*}(R_{1})=0$.

Figure 9: Maximum induced stress, $\sigma_{\text{eff}}^{*}(R_{1})$, given by
(56), against core volume fraction, $\psi$ for different states of charge,
$c_{0}$.

###  4.2 Optimisation

We now suggest objective functions and constraints that could be used to
optimise the design of an anode using the three performance measures defined
above. We derive these objective functions and constraints for the core–shell
spherical nano-particle shown in Figure 1, and plot how the objective function
varies with the volume of the silicon core with the aim of finding the optimal
size of silicon core for each optimisation problem.

####  4.2.1 Amount of Lithium per Relative Expanded Volume

The performance of anode materials are often measured by their volumetric or
gravimetric capacity. The first measure we use to optimise the anode geometry
is the amount of lithium per expanded volume which is equivalent to the
volumetric capacity. We calculate the amount of lithium per expanded volume by
dividing the relative amount of lithium, $Q$ in (51), by the relative expanded
volume, $V$ in (52), giving

$\frac{Q}{V}=\frac{c_{1}R^{3}+c_{2}\frac{c_{2}^{\text{max}}}{c_{1}^{\text{max}}}(1-R^{3})}{(1+\eta_{1}V_{1}^{m}c_{1}^{\text{max}}u(1))^{3}}.$
(57)

Figure 10: Relative amount of lithium per relative expanded volume, $Q/V$,
given by (57), against core volume fraction, $\psi$, for different states of
charge, $c_{0}$.

In Figure 10, we plot $Q/V$ against $\psi$ for the same states of charge as in
Figures 8-9. We can see from this plot that $Q/V$ is maximised by a fully
lithiated nano-particle with a silicon core of volume fraction $\approx 0.45$.
The reason that $Q/V$ is not optimised by a nano-particle of a single material
is that the amount of lithium $Q$ is linear in $\psi$ ($=R^{3}$), whereas the
expanded volume $V$ increases much more rapidly with $\psi$ at large values of
$\psi$ than for smaller values. Therefore, as $\psi$ becomes larger, the extra
lithium that can be intercalated is out-weighed by the increased expansion and
$Q/V$ begins to decrease with $\psi$. However, as explained in Section 4.1.2,
the linear elasticity model overestimates $V$ at high SOC and high $\psi$ so
this prediction needs to be validated with a nonlinear model. For small SOC
values, $Q/V$ has a much more linear relationship with $\psi$, and thus if the
nano-particle is only lithiated a small amount, a fully silicon nano-particle
is the optimal design according to the $Q/V$ measure. Lastly, we observe that
for large $\psi$, $Q/V$ is not monotonic with $c_{0}$ and so having a
partially lithiated nano-particle yields a higher volumetric capacity than a
fully lithiated one.

####  4.2.2 Maximising Amount of Lithium Constrained by Maximum Expanded
Volume

We now consider the problem of maximising the amount of lithium the anode can
hold subject to constraints on the expanded volume and the maximum effective
stress. We begin with the expanded volume constraint. Thus, we want to
maximise $Q$, given by (50), subject to $V\leq V_{\text{max}}$, for some
prescribed $V_{\text{max}}$.

For a given $\psi$, $Q$ is maximised by maximising the SOC, $c_{0}$. From
Figure 8, we see that for a given $\psi$ value, the expansion $V$ is
monotonically increasing with $c_{0}$. Thus, the maximum viable $c_{0}$ value,
which we denote as $\check{c}_{0}$ occurs when $V=V_{\text{max}}$, unless the
fully lithiated volume is less than the constraint, in which case
$\check{c}_{0}=1$. Therefore, for each $\psi\in[0,1]$, we find $\check{c}_{0}$
such that

$V_{c_{0}=\check{c}_{0}}=\min\big{[}V_{c_{0}=1},V_{\text{max}}\big{]}.$ (58)

We then maximise the corresponding $Q_{\text{max}}=Q(\check{c}_{0})$ over
$\psi$.

Figure 11: Maximum amount of lithium $Q_{\text{max}}$ against core volume
fraction $\psi$ for different values of the maximum permitted expanded volume
$(V/V_{0})_{\text{max}}$. Figure 12: Maximum induced stress,
$\sigma_{\text{eff}}^{*}(R_{1})$, against state of charge, $c_{0}$, for
different core volume fractions $\psi$. The inset plot is focused on
$0<\sigma_{\text{eff}}^{*}(R_{1})<20.0$ GPa to show more clearly that it is
monotonically increasing in $c_{0}$.

The value of $\psi$ at which $V_{c_{0}=\check{c}_{0}}$ changes from
$V_{c_{0}=1}$ to $V_{\text{max}}$ in (58) can be calculated by substituting
(34) into (52) for $c_{1}=c_{2}=1$ and equating to the prescribed
$V_{\text{max}}$, giving

$\big{[}1+\eta_{1}V_{1}^{m}c_{1}^{\text{max}}\big{(}\hat{A}_{2}+\hat{B}_{2}\big{)}\big{]}^{3}=V_{\text{max}},$
(59)

where $\hat{A}_{2}$ and $\hat{B}_{2}$ are given by (43) and (44) with
$c_{1}=c_{2}=1$. This can be solved to find a critical value of $R$, which we
denote $\hat{R}$, which corresponds to the $\psi$ value at which
$V_{c_{0}=1}=V_{\text{max}}$. For all values of $\psi$ such that $R<\hat{R}$,
$Q_{\text{max}}$ is given by (51) with $c_{1}=c_{2}=1$ and $R=\psi^{1/3}$.
However, for $R>\hat{R}$, we must numerically solve

$\big{[}1+\eta_{1}V_{1}^{m}c_{1}^{\text{max}}(A_{2}+B_{2})\big{]}^{3}=\bigg{(}\frac{V}{V_{0}}\bigg{)}_{\text{max}},$
(60)

to find $\check{c}_{0}$ and $Q_{\text{max}}$. We both derive the expression
for $\hat{R}$ and the bounds on $V_{\text{max}}$ such that $0<\hat{R}<1$ in A.

In Figure 11 we plot $Q_{\text{max}}$ against $\psi$ for four different values
of $V_{\text{max}}$. The $\psi$ values for which $R<\hat{R}$ can be seen by
the straight line on the left side of the plot. This is because for
$c_{1}=c_{2}=1$, $Q$ is a linear function of $R^{3}=\psi$ in (51). Immediately
to the right of this region, $Q_{\text{max}}$ decreases for $R>\hat{R}$. The
$\psi$ value which gives the largest $Q_{\text{max}}$ is $\psi=\hat{R}^{3}$ as
can be seen by the peaks in $Q_{\text{max}}$ after the linear region for small
$\psi$.

####  4.2.3 Maximising Amount of Lithium Constrained by Maximum Stress

We now wish to maximise the amount of intercalated lithium under a constraint
on the maximum induced stress, defined in (53). We wish to find the anode
geometry which maximises (50) subject to the maximum induced stress in the
anode (56) being less than some prescribed maximum $\sigma_{\text{max}}$.

As with the expanded volume constraint in Section 4.2.2, for each volume
fraction of silicon, $\psi$, we must find the maximum $c_{0}$ such that the
maximum induced stress $\sigma_{\text{eff}}^{*}(R_{1})$ is less than
$\sigma_{\text{max}}$. In Figure 12, we plot $\sigma_{\text{eff}}^{*}(R_{1})$
against SOC and it can be seen that $\sigma_{\text{eff}}^{*}(R_{1})$ is a
monotonically increasing function of $c_{0}$. Therefore, the maximal $c_{0}$
value, denoted by $\check{c}_{0}$, under the constraint
$\sigma_{\text{eff}}^{*}(R_{1})<\sigma_{\text{max}}$ will be such that
$\sigma_{\text{eff}}^{*}(R_{1})=\sigma_{\text{max}}$. We then calculate the
maximum amount of lithium, $Q_{\text{max}}=Q(\check{c}_{0})$, for that $\psi$
value by substituting $\check{c}_{0}$ into (51).

Figure 13: Maximum amount of lithium, $Q_{\text{max}}$, against core volume
fraction $\psi$ for different values of the maximum permitted effective stress
$\sigma_{\text{max}}$.

In Figure 13, we plot $Q_{\text{max}}$ against $\psi$ for several different
values of $\sigma_{\text{max}}$. It can be seen that $\psi=1$ gives the
greatest $Q_{\text{max}}$ value for each value of $\sigma_{\text{max}}$.
However, for $\sigma_{\text{max}}=4.0$ and 5.0 GPa, $Q_{\text{max}}$ is not
monotonic in $\psi$, thus there are local maxima of $Q_{\text{max}}$. This is
useful for design purposes given other constraints. For example if the maximum
stress that is allowed is $\sigma_{\text{max}}=4.0$ GPa, but we must also
restrict $\psi$ to be less than 0.25, having $\psi=0.05$ would allow a larger
amount of lithium to be intercalated than with $\psi=0.2$.

As in Section 4.2.2, for sufficiently large $\sigma_{\text{max}}$, there are
$\psi$ values for which a fully lithiated nano-particle ($\check{c}_{0}=1$)
does not produce stresses as large as $\sigma_{\text{max}}$. Therefore, the
the linear relationship for small $\psi$ in Figure 11 is also seen for large
values of $\sigma_{\text{max}}$. However, the $\sigma_{\text{max}}$ values
illustrated in Figure 13 are too small to see this behaviour; all of the
curves there have $\check{c}_{0}<1$. We give the value of $\hat{R}$ (the core
radius for which the maximum induced stress at $c_{0}=1$ is
$\sigma_{\text{max}}$) and the bounds on $\sigma_{\text{max}}$ such that
$0<\hat{R}<1$ (thus the linear relationship for small $\psi$ is observed) in
A.

The $\sigma_{\text{max}}$ values plotted for here are much greater than the
tensile strength of graphite, $\sigma_{\text{C}}^{Y}$, typically between 8 and
12 MPa [28]. The results for the constraint
$\sigma_{\text{max}}=\sigma_{\text{C}}^{Y}$ are qualitatively similar to that
of $\sigma_{\text{max}}=0.5$ GPa. Therefore, to avoid the yielding or cracking
of the graphite at the interface, the optimal design is as large a silicon
core as possible. For $\psi=0.99$, the silicon can only be lithiated to around
$c_{1}=2.2\times 10^{-4}$ for the interfacial stress (56) to remain under the
yield stress, while the graphite remains unlithiated. This achieves a
$Q_{\text{max}}$ value also of around $2.2\times 10^{-4}$ (as $c_{2}=0$ in
this case).

##  5 Conclusions and Discussion

In this paper, we present a model for the inclusion of mechanical stress in
calculating lithium distribution for multi-material lithium-ion batteries. We
show how the stress can be included into the concentration model for simple
geometries; for example spheres, cylinders and 1-dimensional plates, and
determine that the lithium concentration is uniform in each material when
diffusion is fast compared to the charging rate. By applying the model to a
radially symmetric spherical nano-particle with a silicon core and a graphite
shell, we show there are large changes in lithium distribution if the chemo-
mechanical coupling effects are neglected. Finally, we present a framework for
finding the optimal design for the geometry of multi-material anodes and
present three performance measures and three optimality conditions. We present
results from optimising the core volume for a silicon core, graphite shell
geometry as an example of the insight that can be obtained from using these
measures.

There are several limitations to the static linear elasticity model we derive
in Section 2. Firstly, while the quasistatic assumption is valid for small
currents, the non-uniformity in the lithium concentration caused by diffusion
through the anode during more strenuous battery function causes substantial
stresses that are not captured by this model. Secondly, there are mechanical
phenomena commonly observed experimentally, such as plasticity and cracking,
that are also not included in this model. Our solely elastic mechanical model
assumes that the materials are able to act elastically regardless of the
stress induced by the lithiation and there is no yield stress at which the
material either begins to plasticise or crack. Lastly, as noted in Section
2.3, the linear elasticity assumptions used to derive the mechanical model in
Section 2.1.1 relies on the parameter $\eta_{a}V_{a}^{m}c_{a}^{\text{max}}\ll
1$ for $a=1,\dots,n$, thus restricting the materials this linear model is
applicable to.

We also make several assumptions about the materials in our model which are
significant simplifications for many anode materials. We assume the anode
materials are isotropic to allow us to write the stiffness tensor in terms of
two Lamé parameters, $\lambda$ and $G$, and this ignores the anisotropy of the
crystal structure of the anode. While we have taken the lithium-concentration-
dependence of the Young’s modulus into account, we still assume that the
Poisson’s ratio and molar density are lithium-concentration-independent.
Finally, we use parameters and a model suitable for bulk materials, while we
are imagining nano-structures in our example. It has been experimentally shown
that nano-sized materials have very different mechanical properties from bulk
materials [59] and accounting for surface effects in nano-structures becomes
increasingly important as the size of the material decreases [33].

We use the example of a silicon core and a graphite shell to show the results
in Figures 3-13; however, for silicon,
$\eta_{1}V_{1}^{m}c_{1}^{\text{max}}=0.933375$. Therefore, our application of
this model to a silicon core surrounded by a graphite shell needs to be
interpreted with some care. The expanded volume of the hybrid particle with a
silicon core and graphite shell, shown in Figure 8, is larger than the
expected values because of the linear assumption being violated. Additionally,
the disparity between $\eta_{1}$ and $\eta_{2}$ in Table 1 means that the
graphite shell is being stretched by a large amount, causing very large
tensile stresses. As we have neglected cracking and plasticity from the model,
this means the tensile stresses of the graphite are most likely exaggerated in
the results shown here, which will not only affect the optimisation results in
Section 4 but also the lithium distributions within the anode. Although the
model is somewhat outside its range of validity when applied to silicon
anodes, we use silicon as material 1 in our example to clearly show the
necessity of including stress into the static lithium concentration model due
to its large expansion. This model can still be applicable to silicon if we
were to restrict the lithiation of the silicon such that
$V^{m}_{1}c_{1}^{\text{max}}<3.75$, thus the results shown here are still
accurate for small $c_{1}$ levels.

Despite the limitations of the model, the simplifications made in this work
allow us to analytically show the consequences of including a stress-
dependence into the chemical potential. One key result of this is to show that
stress is induced solely by the presence of different materials, independently
of the non-uniformity of the lithium concentration due to diffusion. While the
differences in stress-dependent lithium concentrations and stress-independent
concentrations will be less pronounced with materials with lower expansions
than silicon, these stresses are still important to include for accurate OCV
prediction and stress modelling. While several anode materials have non-
isotropic crystal structures, causing the isotropy assumption to be
inappropriate, the crystalline structure of anode materials usually become
amorphous after the first charge, especially if the stresses are large.
Additionally, the introduction of $\lambda$ and $G$ does not hugely simplify
the model for general geometries and so removing this assumption for these
cases would only slightly increase the complexity of the model. Finally, the
simplifications made to the model have allowed to provide analytical insight
into the key performance indicators of anode nano-particles that we define in
Section 4. Therefore, this model can be used for insight into optimal anode
design for lithium-ion batteries, but it must be adapted for more
quantitatively accurate results and for more challenging design geometries to
be used in practice.

##  6 Acknowledgements

This publication is based on work supported by the EPSRC Centre For Doctoral
Training in Industrially Focused Mathematical Modelling (EP/L015803/1) in
collaboration with Nexeon. Special acknowledgement goes to the industrial
collaborator, Bill Macklin, for very insightful discussions and motivations
for this work. JC and CP acknowledge funding from the Faraday Institution
(EP/S003053/1).

##  Appendix A Calculation of $\hat{R}$ for Maximum Expansion and Maximum
Stress Constraints

We present expressions for the critical radii $\hat{R}$ for the maximum
expansion constraint and the maximum stress constraint where $\hat{R}$ is the
radius of the silicon core such that the constraint is met when $c_{0}=1$. We
also present the inequalities that the constraints $V_{\text{max}}$ and
$\sigma_{\text{max}}$ must satisfy such that $0<\hat{R}<1$.

###  A.1 Maximum Expansion Constraint

Substituting (43)-(44) into (59) yields

$\displaystyle\frac{\bar{\eta}_{1}\big{[}\Lambda_{1}\Lambda_{2}\big{(}\hat{R}^{3}\gamma_{1}+(1-\hat{R}^{3})\gamma_{2}\big{)}+4G_{2}\big{(}\Lambda_{1}\hat{R}^{3}\gamma_{1}+\Lambda_{2}(1-\hat{R}^{3})\gamma_{2}\big{)}\big{]}}{\Lambda_{1}\Lambda_{2}+4G_{2}\big{(}\Lambda_{2}(1-\hat{R}^{3})+\Lambda_{1}\hat{R}^{3}\big{)}}=V^{\frac{1}{3}}_{\text{max}}-1,$
(61)

where $\Lambda_{a}=3\lambda_{a}+2G_{a}$ for $a=1,2$ and
$\bar{\eta}_{1}=\eta_{1}V_{1}^{m}c_{1}^{\text{max}}$. This can be rearranged
to give

$\hat{R}=\Bigg{[}\frac{(\Lambda_{1}\Lambda_{2}+4G_{2}\Lambda_{2})\Big{(}V_{\text{max}}^{1/3}-1\Big{)}-\bar{\eta}_{1}\Lambda_{2}\gamma_{2}(\Lambda_{1}+4G_{2})}{\bar{\eta}_{1}\big{[}\Lambda_{1}\Lambda_{2}(\gamma_{1}-\gamma_{2})+4G_{2}(\Lambda_{1}\gamma_{1}-\Lambda_{2}\gamma_{2})\big{]}-4G_{2}(\Lambda_{1}-\Lambda_{2})\Big{(}V_{\text{max}}^{1/3}-1\Big{)}}\Bigg{]}^{\frac{1}{3}}.$
(62)

We rearrange the inequality $0<\hat{R}<1$ with $\hat{R}$ given by (62) to find

$\bigg{(}\frac{\bar{\eta}_{1}\Lambda_{2}\gamma_{2}(\Lambda_{1}+4G_{2})}{\Lambda_{1}\Lambda_{2}+4G_{2}\Lambda_{2}}+1\bigg{)}^{3}<V_{\text{max}}<(1+\bar{\eta}_{1}\gamma_{1})^{3}.$
(63)

###  A.2 Maximum Stress Constraint

Substituting (44) into (56) with $c_{1}=c_{2}=c_{0}=1$ yields

$\sigma_{\text{max}}=\frac{6\eta_{1}V_{1}^{m}c_{1}^{\text{max}}G_{2}^{*}\Lambda_{1}\Lambda_{2}|\gamma_{1}-\gamma_{2}|}{\Lambda_{1}\Lambda_{2}+4G_{2}\big{(}\Lambda_{2}(1-\hat{R}^{3})+\Lambda_{1}\hat{R}^{3}\big{)}},$
(64)

which can be rearranged to give

$\hat{R}=\Bigg{[}\frac{\Lambda_{1}\Lambda_{2}(6\eta_{1}V_{1}^{m}c_{1}^{\text{max}}G_{2}^{*}|\gamma_{1}-\gamma_{2}|-\sigma_{\text{max}})-4G_{2}\Lambda_{2}\sigma_{\text{max}}}{4G_{2}(\Lambda_{1}-\Lambda_{2})\sigma_{\text{max}}}\Bigg{]}^{\frac{1}{3}}.$
(65)

We rearrange $0<\hat{R}<1$ with $\hat{R}$ now given by (65) to find

$\frac{6\eta_{1}V_{1}^{m}c_{1}^{\text{max}}G_{2}^{*}\Lambda_{1}\Lambda_{2}|\gamma_{1}-\gamma_{2}|}{\Lambda_{1}\Lambda_{2}+4G_{2}\Lambda_{2}}<\sigma_{\text{max}}<\frac{6\eta_{1}V_{1}^{m}c_{1}^{\text{max}}G_{2}^{*}\Lambda_{1}\Lambda_{2}|\gamma_{1}-\gamma_{2}|-4G_{2}(\Lambda_{1}-\Lambda_{2})}{\Lambda_{1}\Lambda_{2}+4G_{2}\Lambda_{2}},$
(66)

must be satisfied to satisfy $0<\hat{R}<1$.

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