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A computational homogenization approach for Li-ion battery cells.
Part 1 - Formulation
Salvadori, A., Bosco, E., Grazioli, D.
DICATAM - Dipartimento di Ingegneria Civile, Architettura, Territorio, Ambiente e Matematica
Universita di Brescia, via Branze 43, 25123 Brescia, Italy
Abstract
Very large mechanical stresses and huge volume changes emerged during intercalation and extraction
of Lithium in battery electrodes. Mechanical failure is responsible for poor cyclic behaviors and quick
fading of electrical performance, especially in energy storage materials for the next generation of Li-ion
batteries. A multi scale modeling of the phenomena that lead to mechanical degradation and failure
in electrodes is the concern of the present publication. The computational homogenization technique
is tailored to model the multi physics events that coexist during batteries charging and discharging
cycles. At the macroscale, diffusion-advection equations model the coupling between electrochemistry
and mechanics in the whole cell. The multi-component porous electrode, diffusion and intercalation of
Lithium in the active particles, the swelling and fracturing of the latter are modeled at the micro-scale.
A rigorous thermodynamics setting is stated and scale transitions are formulated.
1 Notation
1.1 Symbols and operators
- the symbol 1 denotes the identity matrix- the symbol Tr[−] denotes the trace operator- the symbol Dev[−] denotes the deviator operator- the symbol div [− ] denotes the divergence operator- the symbol ∇ [− ] denotes the gradient operator- the symbol ∆ [− ] denotes the Laplace operator- the symbol ||x||2 = x · x denotes the squared norm of vector x, with · scalar product- the symbol ej with j ≤ n denotes the jth unit vector of the canonical basis for Rn
1.2 Subscripts and apexes
- the subscript a identifies quantities in the active particles, thus at the micro scale only- the subscript c identifies quantities in the conductive particles, thus at the micro scale only- the subscript e identifies quantities in the electrolyte, at both scales- the subscript e− identifies quantities for electrons, at both scales- the subscript Li+ identifies quantities for Lithium cations, at both scales- the subscript X− identifies quantities for anions, at both scales- the subscript Li identifies quantities for neutral Lithium, at both scales- the subscript E identifies quantities in the electrode, thus at the macro scale only- the apex m identifies quantities at the micro scale only.- the apex M identifies quantities at the macro scale only.
1.3 Variables and fields
- b denotes the body force vector- c denotes the concentration measure called molarity, moles per unit volume
1
- cmaxa denotes the saturation limit for Lithium in the active particles material
- D denotes the diffusivity coefficient- Dij denote the Maxwell-Stefan diffusivities- E denotes electric field- fα denotes the activity coefficient of species α- f± denotes the mean activity coefficient for a dissolved salt- G denotes the shear modulus- hα denotes the the mass flux of species α, the number of moles of species per unit area per unit time- hα denotes the mass supply of species α, the number of moles of species α per unit volume per unit time- i denotes the electric current density (in amperes per unit area)- i denotes the charge supply (in coulombs per unit volume per unit time)- i0, denotes the exchange current density in the Butler-Volmer interface conditions- I steady given current density in a galvanostatic process- K denotes the bulk modulus- M denotes the mobility tensor- M0 denotes the mobility constant- n denotes the outward normal direction on a surface- q denotes the heat supplied by external agencies- q denotes the heat flux vector- t denotes time- tLi+ denotes the transference (or transport) number of Li+ ions in the solution- u denotes the specific internal energy- uα denotes the α-ion mobility- V denotes a domain at the microscale- ∂V denotes a boundary or an interface between domains at the micro scale- y denotes the set of variables that model driving forces- zα denotes the valency of ion α- β denotes the symmetry factor in the Butler-Volmer interface conditions- γ denotes the thermodynamic imbalance at electrode-electrolyte interface- Γ denotes a boundary or an interface between domains at the macro scale- ǫ denotes the porosity- ε denotes the small strain tensor- ζ denotes the charge density, in coulombs per unit volume- η denotes the specific entropy- ϑ denotes the absolute temperature- κ denotes the conductivity- λ denotes the Lame elastic modulus- µ denotes the chemical potential, also known as partial molar free energy- µ denotes the electrochemical potential- µ0 denotes the reference value of the chemical potential of the diffusing species- ρ denotes the mass density- ξ denotes the discontinuity for the electrostatic potential- σ denotes the Cauchy stress tensor- φ denotes the electric potential- Φ denotes the Grand or Landau potential- χ denotes denotes the surface over-potential- ψ denotes the Helmholtz free energy density- ωa denotes the coefficient of chemical expansion of neutral Lithium in the host material- Ω denotes a domain at the macro scale
1.4 Constants and parameters
- NA is Avogadro’s number, NA = 6.02214129(27)× 1023mol−1
- e is elementary charge, e = 1.602176565(35)× 10−19C- F = NA e is Faraday’s constant, F = 96485.3383C mol−1
2
- R is the universal gas constant 8.3144621 JK−1mol−1
- ε0 is the vacuum permittivity or electric constant, 8.85418782× 10−12Fm−1
2 Introduction
Batteries are by far the most common form of storing energy [1]. Li-ion batteries, particularly the nextgeneration silicon based technology [2], have the potential to span from several megawatt huge batteryinstallations used for ”spinning reserves” to ensure grid reliability, to automotive, aerospace, medical, andmilitary industries.
There is being great interest in developing next generation of Li-ion battery for higher capacity and longerlife of cycling, in order to develop significantly more demanding energy storage requirements for humanityexisting and future inventories of power-generation and energy-management systems [3, 4]. Industry andacademic are looking for alternative materials and Silicon is one of the most promising candidates for theactive material in anodes, because it has a high theoretical specific energy capacity. At the same time, hugeefforts are devoted to cathode materials investigations, as for rechargeable lithium-oxygen or Li-S batteries[5].
Li-ion batteries are based on the classical intercalation reaction during which Li is inserted into orextracted from electrodes [6, 7]. Large volume changes are associated with this process. They induceexperimentally observed inelastic effects [8, 9], phase-segregation [10], micro-cracks and particle fracture[11], decrepitation or pulverization, loss of integrity and of electric contact with current collector. Thismajor liability is obstructing the way toward widespread application of the next generation technology andlimited the use of lithium batteries to relatively low power applications such as portable devices.
Stress evolution, diffusion, and fracture in representative battery microstructures have been investigatedin a long series of recent publications. Most of them are relevant to single particle analysis, nano-wires, orthin films. Some dealt with the multi-scale and multi-physics modeling of the whole battery cell [12, 13, 14,15, 16, 17, 18, 19, 20, 21, 22]. The electrochemical and mechanical performance of Li-ion batteries stronglydepends on the interaction between micro and nano-scale phenomena, in particular within the electrodes.Intercalation, swelling and the consequent mechanical failures originate at a scale three order of magnitudessmaller than the battery cell scale, at which ion mobility is usually modeled. Directly resolving all scalesand modeling all particles in the electrodes is not practical. Instead, the nano-scale effects are incorporatedinto the micro-scale problem through homogenization approaches and constitutive models that are derivedfrom homogenization methods. This idea has been pursued by several authors, among others [15, 19, 20], viaMori-Tanaka, Hashin-Strickman, and Wiener bounds. Moving from the state of art of the field, the presentcontribution aims at tailoring the Computational Homogenization technique [23], using a multiscale schemewith a complex multi-particle representative volume element (RVE).
An electrochemical cell necessarily consists of several phases [24]. They must include two electrodemetals, a separator, and an electrolytic solution. An electrode is a material in which electrons are the mobilespecies. An electrolyte is a material in which the mobile species are ions and free movement of electrons isblocked. Accordingly, mass must be transported through the electrolyte from one electrode to the other tobring reactants to the interfaces, whereas electrons flow in the external conductor and finally travel throughthe electrodes, doing work or requiring energy. A binary ionic compound is a salt consisting of only twoelements in which both elements are ions, a cation (which has a positive charge) and an anion (which has anegative charge). An electrolyte that contains only one solvent and one salt is called a binary electrolyte.
The difference in electric potential between the two electrodes when they are not connected by an externalcircuit and are in electrochemical equilibrium is called open-circuit potential. When such a potential differenceis kept constant during charging, the process is termed potentiostatic. Its dual, when steady battery currentis applied, is termed galvanostatic.
Mass transfer in an electrolytic solution requires a description of the movement of mobile ionic species.In the absence of convection, as assumed henceforth, movement of species is governed by diffusion, drivenby gradients of concentration, and by migration due to an electric field for the species that are charged.In addition to a standard balance for the species concentration, the virtual-power approach [25] is used todevelop the other balances in the formulation.
3
In section 5 a rigorous thermodynamic analysis is performed, to provide a set of thermodynamic restric-tions that shall be satisfied by constitutive specifications. Besides framing the theory in a rigorous setting,thermodynamics provides a firm basis for the multi scale homogenization scheme developed in section 7.The micro to macro scale transition which stems from the so-called Hill-Mandel condition, moves from theenergy and entropy balances developed in section 5 in view of the potential-based formulation here adopted.
In order to focus on the novelties of the two-scale formulation, some simple hypothesis will be madethroughout the paper. The battery cell is supposed to have a single binary electrolyte which is a solutionof a binary salt, say LiX, plus solvent, say a polymer, in which the concentrations of ionic species vary withlocation in the cell. Active particles in the composite electrodes are idealized as network solids followingLarche and Cahn [26], with the lattice material assumed as insoluble in the electrolyte. The concentrationof ions in the solid phases is neglected. Magnetic fields and polarization are neglected as well.
There are no reasons to restrict ourselves to simple elastic materials and small displacements and strainsapart from simplicity. Whereas small deformations is still an adopted hypothesis in simulations (see forinstance [22]), extension to general kinematics and dissipative materials may be carried out following [18, 27].Special care must be taken however to the proper configuration - either current or reference - in which thescale transitions shall be formulated (see for instance [28, 29]) for a broader discussion on the matter). Toavoid adding technical complexities that may further involve the already complicated scenario here depicted,small displacements and strains have been taken.
The computational homogenization technique is essentially based on the solution of nested boundary valueproblems, one for each scale. A complete set of equations and boundary conditions governing the stress,electric, chemical, and electrochemical potentials (eventually the concentration field as well) are derivedfor the whole battery cell for both scales in sections 7.1, 7.2 following nonequilibrium thermodynamics ofporous electrodes [24, 20]. Two macroscopic phases (solid and electrolyte) and three phases at the microscale (electrolyte, active particles, conductive particles) are modeled. The binder has not been taken intoaccount in the RVE. At the micro-scale, the ionic fluxes in all phases are related to electrochemical potentialgradients, consistent with non-equilibrium thermodynamics. Intercalation kinetics, expressed by Butler-Volmer equations, model the flux of Lithium between the particles and the electrolyte, which at the macroscale is modeled as a bulk supply. At such a scale, transport and stress evolution are modeled via volumeaveraged conservation equations.
The solution of nested boundary value problems can be only approximated numerically. This task is notattempted here, yet discretized weak forms and the Newton-Raphson procedure are provided in section 8.The numerical solution is typically achieved with two nested finite element simulations (whence the acronymFE2) but in principle there is no reason to exclude other numerical techniques. The implementation and theconsequent numerical solution will be the subject of forthcoming publications.
3 Local balance laws for diffusing species
3.1 Mass balance
The mass balance equation may be written as
∂cα∂t
+ div [hα ] = hα (1)
where: cα is the molarity (i.e. the number of moles per unit volume) of a generic species α; hα is the massflux in terms of moles, i.e. the number of moles of species α measured per unit area per unit time; hα isthe mass supply in terms of moles, i.e. the number of moles of species α measured per unit volume per unittime. It applies to ions Li+, X−, electrons e−, as well as neutral Li.
3.2 Gauss law
Gauss’s law relates the distribution of electric charge ζ (in Coulombs per unit volume) to the resultingelectric field E (in Volts per meter).
div [E ] =1
ε0ζ (2)
4
The factor ε0 is termed vacuum permittivity or electric constant. Its value is 8.85418782× 10−12 CVm .
3.3 Faraday’s law
In a solution charge is transported by ions and the flux of mass of each species contributes to a currentdensity in view of Faraday’s Laws of Electrolysis
i = F∑
α
zα hα , i = F∑
α
zα hα (3)
where F = 96485.3383C mol−1 is Faraday’s constant and zα is the valency1 of ion α. The continuity lawin terms of charge density ζ, electric current density i (in Amperes per unit area) and charge supply i (inCoulombs per unit volume per unit time) reads
∂ζ
∂t+ div [ i ] = i (4)
4 Principle of virtual power. Balance of forces. Hill-Mandel con-
dition.
As in [27] the fundamental assumption will be made that the power expended by each “rate-like” kinematicaldescriptor can be expressed in terms of an associated force system consistent with its own balance. In thesimplistic assumption of small displacements and strains, the conjugated pairs within the internal expenditureof virtual power Wint are the Cauchy stress σ and the small strain tensor ε, whose hatted amount ε is takenhere as a virtual velocity field. The principle of virtual power (with the two requirements: Power balanceWext = Wint on any part, frame indifference for Wint on any part for any virtual velocity) leads to the usualbalance of forces:
div [σ ] + b = 0 (5)
and to the symmetry of tensor σ. Among bulk forces b the electrostatic forces of interaction of a chargedensity ζ in an electric field E will be denoted by bζ = ζ E. Balance equation (5) states that electricand stress fields are unavoidably related in the presence of charge densities and such a relationship hasnot a constitutive nature. Electroneutrality assumption, see appendix 0, uncouples the fields in the balanceequation.
In homogenization theory, it is generally assumed that the internal expenditure of virtual power Wint ispreserved in the scale transition:
WMint = Wm
int (6)
with apexes M and m denoting macro and micro scale, respectively. Such a condition is usually knownas Hill-Mandel condition [30]. In the present work, assumption (6) will be extended, so that the internalexpenditure of virtual power of mechanical forces, of charge and mass fluxes is preserved in the scale transition- see section 7.3.4.
5 Thermodynamics
5.1 Energy balance
As usual in continuum mechanics, one can write the local form of the first principle in terms of rates of thespecific internal energy u :
ρ u = σ · ε+ q − div [q ] +∑
α
µα hα − div [µα hα ] (7)
One recognizes three contributions in the variation of internal energy: a mechanical contribution σ · ε inwhich ε is the small strain tensor and σ the Cauchy stress tensor; a heat flux contribution q− div [q ] where
1i.e. the number of electrons transferred per ion, typically +1 for Li+ cations and −1 for X− anions.
5
the scalar q is the heat supplied by external agencies and q is the heat flux vector; a mass flux contributionµh−div [µh ] with the scalar µ denoting the electrochemical potential, the scalar h the species supply2 andh the mass flux vector. Mass flux and supply refer to electrons, to neutral and ionic species and it is hereleft unsaid that α ranges from 1 to the total number of species.
In the electrodes, charge flux is due to electrons by definition. Lithium intercalation is neutral and takesplace in active particles merely. In view of the hypothesis made, the energy balance in the active particlesreads:
ρ u = σ · ε+ q − div [q ] + µLi hLi − div [µLi hLi ] + µe− he− − div [µe− he− ] (8)
In conductive particles and in the binder within electrodes no intercalation takes place. Balance equation(7) becomes
ρ u = σ · ε+ q − div [q ] + µe− he− − div [µe− he− ] (9)
In the electrolyte charge is transported by ions and the flux of mass of each species contributes to a currentdensity in view of Faraday’s Laws of Electrolysis.
ρ u = σ · ε+ q − div [q ] +∑
α
µαhα − div [µα hα ] (10)
5.2 Entropy imbalance
The entropy imbalance can be given a local form in terms of the specific entropy η and of the absolutetemperature ϑ:
ρ η −q
ϑ+ div
[ q
ϑ
]
≥ 0 (11)
By noting that div[
q
ϑ
]
= 1ϑ
div [q ]− 1ϑ2 q·∇ [ ϑ ] the entropy imbalance can be expressed in terms of internal
energy:
ρ ϑ η −
ρ u− σ · ε−∑
α
µα hα − div [µα hα ]
−1
ϑq · ∇ [ϑ ] ≥ 0 (12)
For simple elastic materials one may consider the internal energy to be function of the entropy η, of kinematicvariables here represented by the small strain tensor ε, of the concentrations cα. One has therefore
u =∂u
∂ηη +
∂u
∂ε· ε+
∑
α
∂u
∂cαcα (13)
to be inserted in (12).
For the active particles the entropy imbalance (12) specifies as:
ρ ϑ η − ρ u− σ · ε− (µLi hLi − div [µLi hLi ]) − (µe− he− − div [µe− he− ]) −1
ϑq · ∇ [ϑ ] ≥ 0 (14)
in view of (8). Accounting for mass and charge balances (1-4):
µLi hLi − div [µLi hLi ] = µLi cLi − hLi · ∇ [µLi ] (15)
µe− he− − div [µe− he− ] = µe− ce− − he− · ∇ [µe− ] (16)
it finally becomes
ρ η
(
ϑ−∂u
∂η
)
+ ε ·
(
σ − ρ∂u
∂ε
)
+ cLi
(
µLi − ρ∂u
∂cLi
)
+ ce−
(
µe− − ρ∂u
∂ce−
)
+
−hLi · ∇ [µLi ] − he− · ∇ [µe− ] −1
ϑq · ∇ [ϑ ] ≥ 0 (17)
By applying the Coleman-Noll procedure above, one requires that the latter inequality must be satisfied forall constitutive processes [31, 32]. This leads to the following thermodynamic restrictions:
ϑ−∂u
∂η= 0 , σ − ρ
∂u
∂ε= 0 , µLi − ρ
∂u
∂cLi= 0 , µe− − ρ
∂u
∂ce−= 0 ,
2tipically by intercalation
6
hLi · ∇ [µLi ] ≤ 0 , he− · ∇ [µe− ] ≤ 0 ,1
ϑq · ∇ [ϑ ] ≤ 0 (18)
on any active particle.
Following the same path of reasoning for the conductive particles, where no ionic mass transport takesplace, the entropy imbalance (14) is obtained in the form
ρ η
(
ϑ−∂u
∂η
)
+ ε ·
(
σ − ρ∂u
∂ε
)
+ ce−
(
µe− − ρ∂u
∂ce−
)
− he− · ∇ [µe− ] −1
ϑq · ∇ [ϑ ] ≥ 0 (19)
whence the thermodynamic restrictions:
ϑ−∂u
∂η= 0 , σ − ρ
∂u
∂ε= 0 , µe− − ρ
∂u
∂ce−= 0 ,
he− · ∇ [µe− ] ≤ 0 ,1
ϑq · ∇ [ϑ ] ≤ 0 (20)
In the electrolyte, in view of the balance of energy (10), the entropy imbalance can be written as
ρ ϑ η −
ρ u− σ · ε−∑
α
µαhα − div [µα hα ]
−1
ϑq · ∇ [ϑ ] ≥ 0 (21)
which, in view of (13) turns out to
ρ η
(
ϑ−∂u
∂η
)
+ ε ·
(
σ − ρ∂u
∂ε
)
+∑
α
(
µαhα − div [µα hα ] − ρ∂u
∂cαcα
)
−1
ϑq · ∇ [ϑ ] ≥ 0
as no electron flux takes place in the electrolyte, ce− = 0. Accounting for mass balance (1)
µα hα − div [µα hα ] = µα cα − hα · ∇ [µα ] (22)
the entropy imbalance finally becomes
ρ η
(
ϑ−∂u
∂η
)
+ ε ·
(
σ − ρ∂u
∂ε
)
+∑
α
cα
(
µα − ρ∂u
∂cα
)
−∑
α
hα · ∇ [µα ] −1
ϑq · ∇ [ϑ ] ≥ 0 (23)
and the thermodynamic restrictions
ϑ−∂u
∂η= 0 , σ − ρ
∂u
∂ε= 0 , µα − ρ
∂u
∂cα= 0 ,
2∑
α=1
hα · ∇ [µα ] ≤ 0 ,1
ϑq · ∇ [ϑ ] ≤ 0 (24)
come out for all α.
5.2.1 Remarks
According to Hatsopoulos and Gyftopoulos, [33] section 7.6, an important feature of electrostatic field E isthat the net electric charge ζ is independent on the electrostatic potential φ. Therefore, the energy of thesystem can be written as a sum of two terms, one representing the energy in the absence of the force fieldand the other representing the energy due to force field. Accordingly,
ρ e = ρ u+ div[
φE]
−1
ε0ζ φ
In view of Gauss’ law (2), and denoting with τ = ∇ [φ ] it comes out:
ρ e = ρ u+ E · τ (25)
7
If the total energy e(η, ε, cα, φ) is assumed to be dependent on φ in terms of τ only, one has
e = u+∂e
∂τ· τ (26)
Accordingly, entropy imbalances (17), (19), and (23) contain a further term
τ ·
(
E− ρ∂e
∂τ
)
(27)
which vanishes after applying the Coleman-Noll procedure. Thermodynamic restriction
E− ρ∂e
∂τ= 0 (28)
corresponds. This approach - with the modification of considering the nominal electric displacement ratherthan the electric field because of the electric dipole moments in the material - is also customary in dielectricmaterials, see among others [34].
Different thermodynamic potentials can be considered rather than the internal energy u. A classicalone is the Helmholtz free energy ψ(ϑ, ε, cα, φ) = e(η, ε, cα, φ) − ϑ η. The Grand or Landau potential Φ forsimple elastic materials it is a function of the temperature ϑ, of kinematic variables ε, of the electrochemicalpotentials µα and of the electric potential φ. It is related to the specific energy e by the Legendre transform:
Φ(ϑ, ε, µα, φ) = e(η, ε, cα, φ) − ϑ η −∑
α
ρ−1 cα µα (29)
It is straightforward to show that the following thermodynamic restriction applies:
cα + ρ∂Φ
∂µα
= 0 (30)
By the Legendre transform
r(η, ε, cα,E) = e(η, ε, cα, φ) − ρ−1τ ·E (31)
the new thermodynamic restriction
τ + ρ∂r
∂E= 0 (32)
descends.
6 Electrode kinetics
Lithium deposition or dissolutionLi Li+ + e− (33)
cannot be treated as a simple chemical reaction in a single phase since the process must generate or consumeelectrical charge, and this will always be accompanied by the formation of an electrostatic potential differencebetween the two phases. If the chemical potential µLi exceeds the electrochemical potential of the Lithiumions in the solution µLi+ and electrons in the metal µe− then Li dissolution takes place and the solution nextto the electrode will become positively charged with respect to the electrode itself. A so-called double layeris set3 whose modeling is not carried out here. By assuming the double layer as a zero-thickness interface,during charge and discharge a discontinuity in electrical potential ξ arises between an electrode and itssurrounding solution which in turn depends on the velocity of the battery operation.
3The modeling of the double layer is mainly due to the work of Stern, Gouy and Chapman, see [35] for details.
8
The potential jump may inhibit further dissolution. The value of the electrical potential discontinuity atthe thermodynamic equilibrium for reaction (33) is termed ξNernst and is related to the activity coefficientof Li brought in contact with a solution of its ions. The potential difference
χ = ξ − ξNernst (34)
is called surface over-potential. A net current passing between electrode and solution is related to χ.
As stated above, the departure from thermodynamic equilibrium is due to the imbalance of electrochem-ical potentials in reaction (33), which is the driving force for the electrode reaction. Imbalance is measuredby the amount γ, defined as
γ = µLi − µe− − µLi+ (35)
and thermodynamic equilibrium is clearly reached if and only if γ = 0. A relation between γ and χ isconstitutive in nature and will be discussed in appendix 0.
The mass flux in the outward normal direction n at the surface of an electrode is related by Faraday’slaw to the current density i · n in the same direction. The macroscopic relationship between i · n and χand the composition adjacent to the electrode surface is usually described by means of the Butler-Volmerequation [24, 35]:
i · n = i0
exp
[
(1 − β)
RϑF χ
]
− exp
[
−β
RϑF χ
]
(36)
Factor i0, termed exchange current density, depends generally on the composition of the solution adjacentto the electrode. It is often assumed to be constant at a metallic anode. Parameter β is called symmetryfactor and represents the fraction of the surface over potential that promotes cathodic reaction. It is widelyassumed β = 1/2. The reader may refer to specialized literature [24, 36] for further details.
As observed in [18], by using the Butler-Volmer equation stress can only influence reaction by means ofthe electrochemical potentials, i.e. by altering the concentrations of the reacting species, because mechanicfields does not enter equation (36) directly. A more refined model of the interaction of mechanics andelectrochemical reactions involve the mechanical characterization of the Stern and Gouey-Chapman layers,the former of which has atomistic size. Such an attempt is not carried out in the present note, the readermay refer to [18, 35].
7 Multiscale formulation.
A schematization of a Li-Ion battery is shown in Figure 1 for a half-cell with a metallic Lithium electrode(Figure 1-a) as well as for a whole cell with porous electrodes (Figure 1-b). The behavior of the batterycell is intrinsically multi-scale, as the multi-physics phenomena involving diffusion, migration, intercalation,and their mechanical effects take place at the length scale proper of the electrode compound. A numericalsimulation must incorporate these effects. Unfortunately, discretizing the whole battery cell at the particlecharacteristic size is currently computationally unfeasible. Recourse is thus made to homogenization schemes.Golmon et. al [15, 19] considered the Mori-Tanaka approach at a meso-scale; Hashin-Strickman as well asWiener bounds have been considered by Ferguson et al [20]. In this paper the computational homogenizationframework [37, 38, 39, 40, 41, 42] is pursued. Computational homogenization is essentially based on thesolution of nested boundary value problems, one for each scale. A first order theory is adopted for both themechanical and electrochemical homogenization procedure which hinges on the principles of local action andof scales separation [43]. Deformation localizations are excluded.
Two scales will be considered to model composite electrodes and porous separators ΩP = ΩAn∪ΩS∪ΩCa- see Figure 1 for notation, whereas a single macroscopic scale is adequate to model the metallic electrodes,the current collectors, as well as all other parts that are homogeneous.
The macroscopic scale modeling is framed into porous materials theory, taking into account the pore-filling electrolyte4 (from now on with apex e) and the porous solid material (apex E). At the microscopic
4The electrolyte in the current technology of Li-ion batteries [7] might be a solid, a liquid or a gel. An up to date discussionon the subject can be found in [44].
9
(Composite) CathodeAnode Separator Current collector
ΩAn ΩS ΩCa ΩCc
ΓAS ΓSC ΓCC
ΓA ΓC
a)
CathodeAnode Separator Current collector
ΩAn ΩS ΩCa
ΓAS ΓSCΓCC
Current collector
ΩA
C
ΓCA
ΩC
C
ΓC
CΓ
A
C
b)
Figure 1: a) A schematic of a lithium ion battery half cell and of the notation used for domains and interfacesat the macro-scale. A metallic anode is here depicted. b) A schematic of a lithium ion battery whole cell witha porous anode considered in the proposed multi scale scheme. Current collector for anodes are generallycopper foils, whereas aluminum foils are used for cathodes current collectors.
level, the electrode is separated into its particles compound (active particles, conducting particles, binder):the representative volume element (RVE) contains them all and the pore-filling electrolyte, as in Figure 2.
At the macroscopic scale, the intercalation of Lithium into the particle is described by a volume supplyhLi and no mechanical coupling is provided explicitly. The amount of the intercalated Lithium hLi is notgiven, as unknown are the constitutive laws for all phenomena involved (diffusion, migration, intercalation,and the mechanical coupling). Those quantities are upscaled from the underlying microstructure.
At the micro-scale all fundamental mechanisms and their coupling are modeled in full. Scale transitionsare invoked to: i) allow the solution of the micro scale BVP (the macro to micro scale transition - see section7.3.1 - that provides boundary condition at the micro scale) by linking it to the set of macroscopic variablesthat drive the whole problem; ii) provide tangent matrices and updated values of the (dual) macroscopic fields(the micro to macro scale transition - see section 7.3.4), namely the average stress field, ionic mass fluxes,electron current density, concentration of ions in the electrolyte and of Lithium in the electrode, intercalation
10
supply of neutral Lithium in the electrode. In order to upscale all these quantities, the Hill-Mandel conditionand the mass conservation across the scales will be imposed.
7.1 Macroscale
At each point x ∈ ΩP two continuous phases superpose as for saturated soils. The electrolyte (subscripte) fills the pores of the electrodes compound and of the separator. Electrodes compound include the activematerial, binders and conductive additives (subscript E). The separator includes the solid phase (againsubscript E). Superscript M distinguishes the macroscopic amounts form the same ones at the micro scale.
For the sake of shortness, a composite cathode ΩCa only will be described in full. The formulation foranodes and separator can be derived straightforwardly. The porosity ǫe of the electrode is defined as thefraction of void space in the material, where the voids are filled by the electrolyte. It is defined by the ratio:
ǫe =VV
VT
where VV is the volume of void-space and VT is the total volume of material, including the solid and voidcomponents. The complementary ratio is denoted by ǫE = 1− ǫe. Both porosities will be related to the RVEat the micro scale in section 7.2.
7.1.1 Balance equations
The macroscopic mass balance equations characterize the species transport in the phases of composite cath-odes, namely: i) the transport of neutral Lithium in the active particles, macroscopically undistinguishedfrom the other particles in the porous electrode; ii) the transport of Li+ cations and X− anions in theelectrolyte.
∂cMLi
∂t+ div
[
hMLi
]
= hLi x ∈ ΩCa × [0, T ] (37)
∂cMLi+
∂t+ div
[
hMLi+
]
= hLi+ x ∈ ΩCa × [0, T ] (38)
∂cMX−
∂t+ div
[
hMX−
]
= 0 x ∈ ΩCa × [0, T ] (39)
The supply hLi denotes the average amount at x ∈ ΩCa of neutral Lithium that intercalates in the activeparticles, leaving the electrolyte because of chemical reactions at the particle/electrolyte interface. In viewof the two scale formulation, hLi will be estimated by a scale transition in the homogenization framework -see section 7.3.4; in any case, stoichiometry of the chemical reaction (33) requires
ǫe hLi+ = −ǫE hLi
in equation (38). The concentrations, the mass flux of species hMα , and of neutral Lithium hM
Li will beupscaled within the homogenization procedure.
The flux of charge in the electrode is caused by the transport of electrons hMe− in the active and conductive
particles. For each ion that leaves the electrolyte an electron corresponds that leaves the flux he− = −hLi.The balance of charges thus reads:
−F∂cMe−
∂t+ div
[
iMe−]
= −F he− x ∈ ΩCa × [0, T ] (40)
Hypothesis has been made that in the active and conductive particles the positively charged cores are steady,i.e. hM
+ = 0. According the (given) initial concentration cM+ is maintained during the whole process.Two separated electric fields are considered at each point, one pertaining to the electrolyte EM
e and oneto the electrode EM
E . Gauss law is imposed in both phases.
div[
EME
]
=F
ε0(cM+ − cMe−) x ∈ ΩCa × [0, T ]
div[
EMe
]
=F
ε0(cMLi+ − cMX−
) x ∈ ΩCa × [0, T ] (41)
11
It is convenient to express Gauss law in an incremental form. By means of the mass balance equations, easyalgebra leads from (41) to
div[
EMe
]
=F
ε0
(
hLi+ − div[
hMLi+
]
+ div[
hMX−
])
x ∈ ΩCa × [0, T ]
whence
div
[
EMe +
1
ε0iMe
]
=F
ε0hLi+ x ∈ ΩCa × [0, T ] (42)
The counterpart in the electrode reads
div
[
EME +
1
ε0iMe−
]
= −F
ε0he− x ∈ ΩCa × [0, T ] (43)
Equilibrium holds as in (5). Bulk forces are here neglected but the electrostatic forces of interactionbζ = ζ E.
div[
σME
]
+ F (cM+ − cMe−)EME = 0 x ∈ ΩCa × [0, T ] (44)
skw[σME ] = 0 x ∈ ΩCa × [0, T ] (45)
div[
σMe
]
+ F (cMLi+ − cMX−)EM
e = 0 x ∈ ΩCa × [0, T ] (46)
skw[σMe ] = 0 x ∈ ΩCa × [0, T ] (47)
7.1.2 Boundary and initial conditions
Initial conditions are suitably taken, depending on the process that is simulated - galvanostatic/potentiostatic,charge/discharge. The battery cell is initially undeformed in any case.
At the anode current collector interface ΓA the electric potential is set to zero, according with the selectionof the Li-anode as the reference electrode. A Li-metal anode is here assumed, but extension to porous anodeis straightforward: it will be sufficient to apply at the anode the path of reasoning that will be here appliedto the porous cathode. During a galvanostatic discharge process with steady current density I, the boundarycondition iA · n = −I holds at a generic point ΩA ∋ x → ΓA.
At interface ΓAS three conditions are enforced. Anions cannot leave the separator, therefore hX− ·n = 0at a generic point ΩS ∋ x → ΓAS . Moreover, for every electron that leaves the anode, a Li+ ion enters theelectrolyte: thus F ǫe hLi+ ·n = −I at point ΩS ∋ x → ΓAS. Here the normal is outward with respect to theseparator, coherently with the choice of ΩS ∋ x. Finally, a boundary condition of an influx of Li+ ions isstated in terms of the surface over potential5 χM . The Butler-Volmer equation (36) relates the flux of Li+
ions in the outward normal direction n on the surface ΓAS to χ:
F ǫe hLi+ · n = i0
exp
[
(1 − β)F
Rϑχ
]
− exp
[
−βF
Rϑχ
]
(48)
The open circuit potential ξNernst in equation (34) can be taken as zero if the anode is used as referenceelectrode.
At interface ΓSC continuity conditions for electric potential and electrochemical potentials are imposedbecause anions and cations are free to pass from the separator to the electrolyte in the porous cathode ina thermodynamic equilibrium condition at the micro scale. Electrons and neutral Lithium cannot exit thecathode, therefore iLi = 0 as well as hLi · n = 0 at ΩCa ∋ x → ΓSC .
At the cathode-current collector interface ΓCC , neither Lithium ions, nor counterions, nor neutral Lithiumcan leave the battery hLi+ · n = hX− · n = hLi · n = 0 at ΩCa ∋ x → ΓCC . The electrode carries all thecurrent iE = I. Such a current is constant in the current collector. In case of potentiostatic process, theelectric potential at the current collector boundary ΓC is given.
Mechanical boundary conditions consider no displacements at both ends or an external pressure appliedat the boundaries ΓA and ΓC .
5Such a condition is expected at the micro scale only. This is correct for porous materials. However, as the anode foil ishere described as homogeneous, the Butler-Volmer interface condition is expressed macroscopically, too.
12
7.1.3 Weak form
The weak form for the macroscale balance equations and boundary conditions6 can be given in terms of thepotentials, displacements, and concentrations in a time interval [0, T ] as
Find yM ∈ V [0,T ] such that
d
dtb(
zM (t), yM)
+ a(yM (t), yM ) = f(yM ) ∀yM ∈ V (49)
The identification of the functional spaces V [0,T ],V falls beyond the scope of the present note. The formerspace includes time dependent functions, whereas space V does not (see [45], chapter 12). Vectors zM andyM collect the time-dependent unknown fields. The former contains the concentration fields
zM = cMLi , cMe− , c
MLi+ , c
MX−
whereas the latter contains all the potentials and displacements
yM = µMLi , µ
Me− , µ
MLi+ , µ
MX−, φM
e , φME , uM
Vector yM collects the steady-state test functions that correspond to the unknown fields in yM . Vector zM
should be dependent on yM by some constitutive equations, as the ones derived from the Landau potential(30). It is well known that in the computational homogenization technique such a constitutive dependencyis imposed only incrementally, with tangent modulus upscaled from the micro scale. Bilinear forms a(·, ·)and b(·, ·) are defined as
b(
yM , zM)
= −
∫
ΩCa
ǫE
(
µMLi c
MLi + µ
M
e− cMe−
)
+ ǫe
(
µM
Li+ cMLi+ + µ
M
X− cMX−
)
dΩ (50)
a(
yM ,yM)
=
∫
ΩCa
ǫE
(
∇[
µMLi
]
· hMLi + ∇
[
µM
e−
]
· hMe−
)
+ ǫe
(
∇[
µM
Li+
]
· hMLi+ + ∇
[
µM
X−
]
· hMX−
)
dΩ
+
∫
ΩCa
ǫE
∇[
φME
]
·
(
EME +
1
ε0iME
)
+ ǫe
∇[
φMe
]
·
(
EMe +
1
ε0iMe
)
dΩ
+
∫
ΩCa
σM · ˆεM
+ ǫE F cMe− EM
E · ˆuM
− ǫe F (cMLi+ − cMX−)EM
e · ˆuM
dΩ
+
∫
ΩCa
hLi
(
γM +F
ε0ξM
)
dΩ (51)
with hLi = ǫE hLi = ǫE he− = −ǫe hLi+ , γM = µMLi − µ
M
e− − µM
Li+ , and ξM = φME − φM
e . The given terms aredue to the electrostatic interactions in the bulk and to the eventual tractions on the Neumann part of theboundary of the electrode ∂NΩCa.
f(yM ) =
∫
ΩCa
ǫE F cM+ EM
E · ˆuM
dΩ +
∫
∂N ΩCa
t · ˆuM
dΓ (52)
The weak form here proposed is remarkably different from other forms in the literature [15, 19, 22] forbeing written in terms of electrochemical potentials rather than concentrations. By this choice the weakform maintains the usual physical meaning of power expenditure, what is here mandatory to perform thecorrect scale transitions. A similar path of reasoning has been recently followed for the problem of hydrogenembrittlement [46], that indeed shows several similarities with the one at hand. As an eventual drawback,this formulation might make the numerical solution more involved. Such an eventuality was not pointed outin [46], at any rate.
7.2 Microscale
The evolution of electrochemical and mechanical fields at the finest scale is defined and monitored on anRVE of volume V and boundary ∂V . It is provided with the essential physical and geometrical information
6As no constitutive laws are assumed at the macro-scale the notion of governing equations is vacuous at such a scale.
13
Active particles
Conductive particles
Binder
Electrolyte
Figure 2: A schematic of a microscale RVE for a porous cathode, that resembles figure 5 in [47].
about the microstructural components, that are the active particles, the binder, the conductive particles,and the pore filling electrolyte. The choice of the RVE is a delicate task, as the RVE should be statisticallyrepresentative of all microstructural particles and phenomena, and at the same time should remain smallenough so that the principle of scale separation is not violated. Theoretically, the principle of scale separationassumes an infinitesimally small underlying fine scale structure (infinitesimal neighborhood of a materialpoint) though finite sized RVEs are used in real predictive computations.
Two kinds of particles are modeled within the RVE. The active particles (collectively occupying domainVa) and the conductive particles that define domain Vc. The binder, that could be modeled as a cohesiveinterface between particles, currently has not been considered. Accordingly, the solid phase will occupyvolume VE = Va ∪ Vc. The electrolyte will fill all the remaining parts of the RVE, in domain Ve such thatV = VE ∪ Ve.
7.2.1 Balance equations
The microscopic mass balance equations characterize the species transport in two phases, namely the trans-port of neutral Lithium in the active particles and the transport of Li+ and X− ions in the electrolyte.
∂cmLi
∂t+ div [hm
Li ] = 0 x ∈ Va (53)
∂cmLi+
∂t+div
[
hmLi+
]
= 0 x ∈ Ve (54)
∂cmX−
∂t+ div [hm
X−] = 0 x ∈ Ve (55)
There is no supply of species at the micro scale, as intercalation phenomena are analytically described alonginterfaces without any homogenization assumption. Current density in active and conductive particles isdue to electron transport, and the local form of mass and charge conservation read:
∂cme−
∂t+div [hm
e− ] = −1
F(∂cme−
∂t+ div [ ime− ]) = 0 x ∈ VE (56)
Charge and mass equations for electrons are equivalent in the assumption that the positively charged coresare steady.
Gauss law governs the electric field Em.
div [Em ] =F
ε0(cm+ − cme−) x ∈ VE (57)
div [Em ] =F
ε0(cmLi+ − cmX−
) x ∈ Ve (58)
Incrementally, they hold
div
[
Eme +
1
ε0ime
]
= 0 x ∈ Ve (59)
14
div
[
EmE +
1
ε0ime−
]
= 0 x ∈ VE (60)
Inertia effects as well as non electrostatic bulk forces are neglected, so that the force balance (5) specializesas:
div [σm ] + F (cm+ − cme−)Em = 0 , x ∈ VE (61)
div [σm ] + F (cmLi+ − cmX−)Em = 0 , x ∈ Ve (62)
skw[σm ] = 0 x ∈ V (63)
7.2.2 Interface conditions
Li+
Li+
Li
Li
X−
X−
e−
e−
Li
hm
Li· n|
a= 0
hm
e−· n|
a= h
m
e−· n|
c
µm
e−|a
= µm
e−|c
Em · n|
a= E
m · n|c
φm|a
= φm|c
um|
a= u
m|c
σm
n|a
= σm
n|c
hm
Li+· n|
e= 0
hm
X−· n|
e= 0
hm
e−· n|
e= 0
Em · n|
e= E
m · n|c
φm|e
= φm|c
um|
e= u
m|c
σm
n|e
= σm
n|c
hm
Li+· n
∣
∣
e= −hBV
hm
Li· n|
a= hBV
hm
e−· n|
a= −hBV
hm
X−· n|
e= 0
(Em +1
ε0
im) · n
∣
∣
∣
∣
a
=iBV
ε0
(Em +1
ε0
im) · n
∣
∣
∣
∣
e
= −
iBV
ε0
um|
e= u
m|a
σmn|
e= σ
mn|
a
Figure 3: Interface phenomena. The active (conductive) particle is depicted in light (dark) gray whereas theelectrolyte ideally fills the picture background.
In the present multiscale framework, continuity of displacements and surface tractions are imposed atall interfaces. This hypothesis is indeed strong and implies essentially that the binder is modeled as aperfect adhesive with zero thickness. A cohesive interface model would definitely be more adequate butsuch a formulation, which overall does not make the problem much more involved, is useless until materialparameters are measured, especially in terms of capability of transferring mass and charge against a criticaldetachment of particles. Such a crucial aspect is related to the failure mode of electrodes, topic that goesbeyond the scope of the present work and will constitute subject of further developments.
To the sake of clearness, it will be denoted with:
π|w = π(x) Vw ∋ x → ∂Vw
15
for π a generic field (either potential or stress and displacements) and for w = e, a, c. The outward normaldirection on surface ∂Vw will be denoted with nw.
At the interface between active particles and conductive particles ∂Vc ∩ ∂Va, electrons are free to flowwithout causing discontinuities in the electric field and in potentials φ, µe− . Neutral Lithium does notintercalate into conductive particles. Continuity of displacements and surface tractions are imposed.
hmLi · n|a = 0 x ∈ ∂Vc ∩ ∂Va
hme− · n|
a= hm
e− · n|c
x ∈ ∂Vc ∩ ∂Va
µme− |a = µm
e− |c x ∈ ∂Vc ∩ ∂Va
Em · n|a = Em · n|c x ∈ ∂Vc ∩ ∂Va
φm|a = φm|c x ∈ ∂Vc ∩ ∂Va
um|a = um|c x ∈ ∂Vc ∩ ∂Va
σmn|a = σmn|c x ∈ ∂Vc ∩ ∂Va
At the interface between electrolyte and conductive particles ∂Ve ∩ ∂Vc, there is no intercalation neitherof Li-ions nor of counterions. Electrons do not flow through the interface. Continuity of displacements andsurface tractions are imposed.
hmLi+ · n
∣
∣
e= 0 x ∈ ∂Vc ∩ ∂Ve
hmX−
· n|e
= 0 x ∈ ∂Vc ∩ ∂Ve
hme− · n|
e= 0 x ∈ ∂Vc ∩ ∂Ve
Em · n|e = Em · n|c x ∈ ∂Vc ∩ ∂Ve
φm|e = φm|c x ∈ ∂Vc ∩ ∂Ve
um|e = um|c x ∈ ∂Vc ∩ ∂Ve
σmn|e = σmn|c x ∈ ∂Vc ∩ ∂Ve
At the interface between electrolyte and active particles ∂Ve ∩ ∂Va, there is no intercalation of X-anions,whereas a Faradaic reaction converts the oxidized Lithium to its neutral state before its diffusion into theactive particles lattice. Electrode kinetics is here modeled via the Butler-Volmer equation (36), in terms ofthermodynamic excess γ and surface over potential χ defined in equations (34, 35). It will be denoted with
hBV =i0F
exp
[
(1 − β)
Rϑ
γ
F
]
− exp
[
−β
Rϑ
γ
F
]
iBV = i0
exp
[
(1 − β)
Rϑχ
]
− exp
[
−β
Rϑχ
]
Continuity of displacements and surface tractions are imposed.
hmLi+ · n
∣
∣
e= −hBV x ∈ ∂Va ∩ ∂Ve (64)
hmLi · n|a = hBV x ∈ ∂Va ∩ ∂Ve
hme− · n|
a= −hBV x ∈ ∂Va ∩ ∂Ve
hmX−
· n|e
= 0 x ∈ ∂Va ∩ ∂Ve
(Em +1
ε0im) · n
∣
∣
∣
∣
a
=iBV
ε0x ∈ ∂Va ∩ ∂Ve
(Em +1
ε0im) · n
∣
∣
∣
∣
e
= −iBV
ε0x ∈ ∂Va ∩ ∂Ve
um|e = um|a x ∈ ∂Va ∩ ∂Ve
σmn|e = σmn|a x ∈ ∂Va ∩ ∂Ve
16
7.3 Scales transitions and weak form
Define the volume ratios of each microscopic phase with respect to the overall microscopic volume V of theRVE.
ǫe =Ve
V, ǫa =
Va
V, ǫc =
Va
V(65)
ǫe is usually termed the porosity of the electrode material. It will be denoted also with ǫE = ǫa + ǫc = VE/V ,where VE = Va + Vc is the volume of the electrode at the micro scale. Porosity can actually be considered amacroscopic property, being uniquely defined at each point of the macroscopic domain ΩCa on the basis ofthe underlying microstructure. For the sake of readiness however apex will be omitted on ǫ.
In the computational homogenization scheme at hand two main assumptions will be made, namelythesteady-state mass and charge transport hypothesis at the microscale and theelectroneutrality. The formeris a usual assumption in multi scale time-dependent problems [28, 29] in order to perform a computationalhomogenization scheme without time-scale transitions. A steady-state mass and charge transport is justifiedat the micro level in view of the fast diffusive response7 and the negligibly small size of the RVE. In fact whatis meant by steady-state is instantaneous changes for concentration and potentials at the micro scale, whichare dictated through the boundary conditions that originate from the time-dependent macro scale problem.
Electroneutrality is a generally accepted assumption in battery cell modeling, according to Newman’stheory [24]. It is here invoked to circumvent the issues due to the body forces and bulk charges distributions.As well known, the presence of bulk terms in homogenization theory pose still unsolved problems.
7.3.1 Macro to micro scales transitions
A first order theory is adopted for mechanical and electrochemical homogenization procedures, stemmingfrom the principle of scales separation. As for the weak forms, the problem will be formulated in terms of anindependent variable set y that includes the following fields: displacements um, chemical µm
Li, electrochemicalµm
Li+ , µmX−
, µme− , and electric φm
E , φme potentials. They can be decomposed without loss of generality at the
micro scale in a macroscopic contribution linearized over the microscale domain and in a microfluctuationfield, denoted with the symbol ˜
πm(x) − πm(xref ) = ∇[
πM]
· (x − xref ) + π(x) , x ∈ Vπ (66)
denoting with the symbol π a generic field in y. Microscopic fields in (66) refer to their values at a referencepoint xref .
The displacement field at reference point xref is set to zero in order to eliminate the rigid body translation;the rigid body rotation will be suppressed through the microscopic boundary conditions introduced furtherin section 7.3.2. A scale transition of “order zero” must be enforced for jumps γm = µm
Li − µme− − µm
Li+ ,ξm = φm
E − φme .
The following average condition is enforced for γ:
γM = µMLi − µM
e− − µMLi+ =
1
Va
∫
Va
µmLidV −
1
VE
∫
VE
µme−dV −
1
Ve
∫
Ve
µmLi+dV (67)
together with the assumptionγm(xref ) = γM (68)
By the linearization (66) the integral constraint
1
Va
∫
Va
µLi(x)dV −1
VE
∫
VE
µe−(x)dV −1
Ve
∫
Ve
µLi+(x)dV =
1
Va
∫
Va
∇[
µMLi
]
· (x − xref )dV −1
VE
∫
VE
∇[
µMe−
]
· (x − xref )dV −1
Ve
∫
Ve
∇[
µMLi+
]
· (x − xref )dV
7Diffusive processes scales as 1/L2, being L the length scale [48]. The time for a particle to reach the steady state whenconcentration is changed scales as 1/L2.
17
has to be satisfied. One could for instance enforce∫
Va
µLi(x)dV = 0 ,
∫
VE
µe−(x)dV = 0 ,
∫
Ve
µLi+(x)dV = 0
whence the constraint on the position of the reference point xref
1
Va
∇[
µMLi
]
·
∫
Va
x − xrefdV −1
VE
∇[
µMe−
]
·
∫
VE
x − xrefdV −1
Ve
∇[
µMLi+
]
·
∫
Ve
x − xrefdV = 0
and the constraints on the microscopic fields∫
Va
µmLi(x) − µm
Li(xref ) −∇
[
µMLi
]
· (x − xref )dV = 0 (69a)
∫
VE
µme−(x) − µm
e−(xref ) −∇[
µMe−
]
· (x − xref )dV = 0 (69b)
∫
Ve
µmLi+(x) − µm
Li+(xref ) −∇[
µMLi+
]
· (x − xref )dV = 0 (69c)
The same path of reasoning can be applied to ξ, under the assumption
ξm(xref ) = ξM (70)
“Order one” scale transition conditions on the average values of the gradients of the independent variablefields must be enforced
∇[
πM]
=1
Vπ
∫
Vπ
∇ [πm ] dVπ (71)
leading to a set of integral constraints on the microfluctuation fields:∫
Vπ
∇ [ πm ] dVπ =
∫
∂Vπ
πm ⊗ n = 0 (72)
which will be satisfied through the definition of proper boundary conditions for fields π along ∂Vπ. They arediscussed in the next Section.
7.3.2 Micro scale boundary conditions
Different kinds of microscopic boundary conditions may arise from constraint (72). Classical ones are Taylor,Linear displacements, Minimal Kinematical or Tractions, Periodic [37, 38]. In this contribution periodicboundary conditions will be applied on the microfluctuation displacement field u. Denoting with l the sidelength of the square RVE, and with ∂LV and ∂BV its left and bottom sides, the periodic boundary conditionsread:
u(x) = u(x + le1) x ∈ ∂LV (73a)
u(x) = u(x + le2) x ∈ ∂BV (73b)
or, equivalently, in terms of the displacement field um
um(x + le1) − um(x) = l εMe1 x ∈ ∂LV (74a)
um(x + le2) − um(x) = l εMe2 x ∈ ∂BV (74b)
Extension to three dimensional RVEs is trivial.
The boundary conditions for the electrochemical and electric potentials have to be formulated in adifferent way because the boundary of each phase does not coincide with the external RVE boundary.Constraint (72) specifies as
∫
∂Ve
µLi+ n dΓ = 0 ,
∫
∂Ve
µX− n dΓ = 0 ,
∫
∂Va
µLi n dΓ = 0 ,
∫
∂VE
µe− n dΓ = 0 (75)
18
∫
∂Ve
φe n dΓ = 0 ,
∫
∂VE
φE n dΓ = 0 (76)
The constraints are imposed along the interfaces ∂Vj ∩ ∂Vi, i, j = a, e, E as well as along the sides of theRVE, ∂Vj ∩ ∂V , j = a, e, E. The easiest way to satisfy (75-76) is the so called Taylor approach, settingvanishing fluctuations along the interfaces and the RVE boundary
µLi+ |∂Ve= µX− |∂Ve
= µLi|∂Va= µe− |∂VE
= φE
∣
∣
∣
∂VE
= φe
∣
∣
∣
∂Ve
= 0 (77)
In terms of microscopic fields
µmLi+(x) = ∇
[
µMLi+
]
· (x − xref ) + µmLi+(xref ) x ∈ ∂Ve (78a)
µmX−
(x) = ∇[
µMX−
]
· (x − xref ) + µmX−
(xref ) x ∈ ∂Ve (78b)
µmLi(x) = ∇
[
µMLi
]
· (x − xref ) + µmLi(x
ref ) x ∈ ∂Va (78c)
µme−(x) = ∇
[
µMe−
]
· (x − xref ) + µme−(xref ) x ∈ ∂VE (78d)
φmE (x) = ∇
[
φME
]
· (x − xref ) + φmE (xref ) x ∈ ∂VE (78e)
φme (x) = ∇
[
φMe
]
· (x − xref ) + φme (xref ) x ∈ ∂Ve (78f)
7.3.3 Weak form
The weak form for the microscale problem can be given in terms of the potentials, displacements, andconcentrations in a time interval [0, T ] as
Find ym ∈ V [0,T ] such that
d
dtb (zm(t), ym) + a(ym(t), ym) = f(ym) ∀ym ∈ V (79)
As for the macro scale, the identification of the functional spaces V [0,T ],V falls beyond the scope of thepresent note. Vectors zm and ym collect the time-dependent unknown fields and are defined as for themacro scale with reference to the micro scale fields. Vector ym collects the steady-state test functions thatcorrespond to the unknown fields in ym. Vector zm is now dependent on ym by constitutive equations,as the ones derived from the Landau potential (30). The governing equations are properly formulated instrong and weak form. Ellipticity of operators, functional and numerical properties of the solution and ofits approximation depend on the constitutive assumptions and on the choice of the correct functional spacesV [0,T ],V . Bilinear forms a(·, ·) and b(·, ·) are defined as
b (ym, zm) = −
∫
Va
µmLi c
mLi dΩ −
∫
Va∪Vc
µm
e− cme− dΩ −
∫
Ve
µm
Li+ cmLi+ + µ
m
X− cmX−dΩ (80)
a (ym,ym) =
∫
Va
∇ [ µmLi ] · h
mLi dΩ +
∫
Va∪Vc
∇[
µm
e−
]
· hme− dΩ +
∫
Ve
∇[
µm
Li+
]
· hmLi+ + ∇
[
µm
X−
]
· hmX−
dΩ
+
∫
Va∪Vc
∇[
φmE
]
·
(
EmE +
1
ε0imE
)
dΩ +
∫
Ve
∇[
φme
]
·
(
Eme +
1
ε0ime
)
dΩ
+
∫
V
σm · ˆεm
dΩ +
∫
Va∪Vc
F cme− EmE · ˆu
mdΩ +
∫
Ve
F (cmLi+ − cmX−)Em
e · ˆum
dΩ
−
∫
∂Va∩∂Ve
hBV γmdΓ −
∫
∂VE∩∂Ve
1
ε0iBV ξmdΓ (81)
The given terms are due to the electrostatic interactions in the bulk
f(yM ) =
∫
Va∪Vc
F cm+ EmE · ˆu
mdΩ (82)
19
7.3.4 Micro to macro scales transitions
The homogenized macroscopic quantities are extracted from the solution of the microscale problem andupscaled. The micro to macro scale transition is achieved by extending the Hill-Mandel condition (6),namely the balance between microscopic volume average of the virtual power on the RVE and the “pointwise” one at the macroscale. In view of the choice made in the present note of assuming as independentvariable fields the potentials rather than the concentrations, the power expenditure can be easily recognizedwithin the weak form. The time-dependent part of the expenditure is enclosed in bilinear form b(·, ·). Itwill be assumed that the micro to macro scale transition pertain to the time independent power expenditureonly, as if the micro scale problem would be steady state in the sense explained already. Therefore onlybilinear form a(·, ·) is of interest in this regard. The assumption of electroneutrality will be embraced. Suchan assumption is customary in battery modeling, for the charge distribution effects on the power expenditureare usually taken as secondary. Accordingly, in bilinear form a(·, ·) all contributions due to bulk interactionsmust be neglected. Denoting with hatted quantities the virtual velocity fields, the extended Hill-Mandelcondition thus reads
∫
Va
∇ [ µmLi ] · h
mLi dΩ +
∫
Va∪Vc
∇[
µm
e−
]
· hme− dΩ +
∫
Ve
∇[
µm
Li+
]
· hmLi+ + ∇
[
µm
X−
]
· hmX−
dΩ +
∫
Va∪Vc
∇[
φmE
]
· ℵmE dΩ +
∫
Ve
∇[
φme
]
· ℵme dΩ +
∫
V
σm · ˆεm
dΩ −
∫
∂Va∩∂Ve
hBV γmdΓ +
−
∫
∂VE∩∂Ve
1
ε0iBV ξmdΓ = ǫE ∇
[
φME
]
· ℵME + ǫe ∇
[
φMe
]
· ℵMe + σM · ˆε
M+ hLi
(
γM +F
ε0ξM
)
+
ǫE
(
∇[
µMLi
]
· hMLi + ∇
[
µM
e−
]
· hMe−
)
+ ǫe
(
∇[
µM
Li+
]
· hMLi+ + ∇
[
µM
X−
]
· hMX−
)
(83)
having set ℵ = E + 1ε0
i for the sake of conciseness. By means of linearization (66), applying the divergencetheorem and the chain rule, finally taking into account electroneutrality and equilibrium equation (62), themechanical contribution in identity (83) can be written as
1
V
∫
V
σm · εm dV =1
V
∫
V
σm dV · εM +1
V
∫
∂V
tm · ˆu dΓ =1
V
∫
V
σm dV · εM (84)
due to the anti-periodicity of the tractions tm = σ n over the boundary of the RVE. The virtual powerdue to the electrode kinetics at the interface ∂Ve ∩ ∂Va between active particles and the electrolyte can berestated noting that γ is by definition a linear combination of potentials, to which linearization (66) applies.Therefore, the same linearization holds for γ, if taken in the sense of limit of internal linearized fields. Itholds then
∫
∂Va∩∂Ve
hBV γmdΓ =
∫
∂Va∩∂Ve
hBV dΓ γM +
∫
∂Va∩∂Ve
hBV (x − xref )dΓ · ∇[
γM]
(85)
in view of Taylor boundary conditions on the micro fluctuation and of assumption (68). By the same pathof reasoning, it holds
∫
∂VE∩∂Ve
iBV ξmdΓ =
∫
∂VE∩∂Ve
iBV dΓ ξM +
∫
∂VE∩∂Ve
iBV (x − xref )dΓ · ∇[
ξM]
(86)
owing to (70). Again taking into account the Taylor boundary conditions on the micro fluctuation it comesout
∫
Va
∇ [µmLi ] · h
mLi dV =
∫
Va
hmLi dV · ∇
[
µMLi
]
+
∫
Va
∇ [ µ ] · hmLi dV =
∫
Va
hmLi dV · ∇
[
µMLi
]
(87)
in view of the (steady-state) mass balance equation (53). The previous path of reasoning applies to theelectrochemical and to the electrostatic potentials contributions in equation (83) as well.
By comparing the corresponding power contributions into the Hill-Mandel condition (83), the followingmicro to macro scale transitions are finally obtained:
σM =
1
V
∫
V
σm dV (88a)
20
ǫEhMLi =
∫
Va
hmLi dVa −
∫
∂Va∩∂Ve
hBV (x − xref ) dΓ (88b)
ǫEhMe− =
∫
VE
hme− dVE +
∫
∂Va∩∂Ve
hBV (x − xref ) dΓ (88c)
ǫehMLi+ =
∫
Ve
hmLi+ dVe +
∫
∂Va∩∂Ve
hBV (x − xref ) dΓ (88d)
ǫehMX−
=
∫
Ve
hmX−
dVe (88e)
ǫEℵME =
∫
VE
ℵmE dVE −
∫
∂Ve∩∂VE
iBV
ε0(x − xref ) dΓ (88f)
ǫeℵMe =
∫
Ve
ℵme dVe +
∫
∂Ve∩∂VE
iBV
ε0(x − xref ) dΓ (88g)
hLi = −
∫
∂Va∩∂Ve
hBV dΓ (88h)
F hLi = −
∫
∂Ve∩∂VE
iBV dΓ (88i)
Average identities (88h, 88i) envisage a further constraint on hBV and iBV . Such a constraint must havea constitutive origin because relates electrochemical and electric potentials. After imposing constitutiveprescriptions, it is well known from the literature that identity
χ =1
Fγ
comes out ( see [49] and also appendix 0, formula (102) ). The latter, together with the interface conditionξ = 0 that has been assumed along ∂Vc ∩ ∂Ve are the required constraints on hBV and iBV .
For the resolution of the macroscopic boundary value problem the concentration is needed in bilinearform b(·, ·) at each increment: an additional micro-to-macro scale transition is required in order to evaluatecM , based on the conservation of mass through the scales and of the Landau potential.
cMLi =1
Va
∫
Va
cmLi dVa (89a)
cMLi+ =1
Ve
∫
Ve
cmLi+ dVe (89b)
cMX−=
1
Ve
∫
Ve
cmX−dVe (89c)
cMe− =1
VE
∫
VE
cme− dVE (89d)
8 Hints on the solution procedure
8.1 Discretization
The macroscale weak form (49) can be written in the following semi-discrete (continuous in time) approxi-mated form
Find yMh ∈ V
[0,T ]h such that
d
dtb(
zMh (t), yM
h
)
+ a(yMh (t), yM
h ) = f(t, yMh ) ∀yM
h ∈ Vh , t ∈ (0, T ) (90)
with yMh (0) = y0,h an approximation of the initial datum. The Galerkin approximation for the steady
problem using as Vh the classical finite element space is not satisfactory when advection is much stronger
21
than diffusion, as the approximate solution is highly oscillatory unless the mesh size h is sufficiently small.Advection-dominated problems require special care which is deferred to devoted investigations.
As motivated in the previous section, homogenization issues suggest to restrict the micro scale approx-imation to steady-state mass and charge transport under the hypothesis of electroneutrality. The discreteweak form of (79) becomes:
Find ym ∈ Vh such that a(ymh , y
mh ) = f(ym
h ) ∀ym ∈ Vh (91)
with Vh the classical finite element space.
8.2 Time advancing by finite differences
A family of time-advancing methods based on the so-called θ-scheme can be built for the macroscale discreteproblem (90). For the sake of readability the apex M will be omitted without inducing any confusion, as themicro scale problem is taken as time-independent.
1
∆tb(
zn+1h − zn
h, yh
)
+ a(θ yn+1h + (1 − θ)yn
h , yh) = θf(tn+1, yh) + (1 − θ)f(tn, yh) ∀yh ∈ Vh (92)
for each n = 0, 1, ..., N − 1, where 0 ≤ θ ≤ 1, ∆t = T/N is the time step, N is a positive integer and y0h ∈ Vh
is a suitable approximation of the initial datum. This includes the forward Euler scheme (θ = 0), backwardEuler (θ = 1), and Crank-Nicolson (θ = 1/2).
8.3 Newton-Raphson scheme
Depending on the constitutive assumptions, equations (91) and (92) can be non linear in ymh , yn+1
h respec-tively, which is the usual case. In particular equation (92) may be solved by means of the Newton Raphsoniterative method. Define function g(yn+1
h ) as
g(yn+1h ) =
1
∆tb(
zn+1h − zn
h, ϕj
)
+ a(θ yn+1h + (1 − θ)yn
h , ϕj) − [ θf(tn+1, ϕj) + (1 − θ)f(tn, ϕj) ]
ϕj |j = 1, ..., Nh being a basis for Vh and pose x = yn+1h for the sake of readability. Equation (92) reads
g(x) = 0 (93)
and the Newton Raphson iterative scheme in an abstract formalism can be obtained by making recourse tothe notion of Gateaux derivative. Taking ǫ ∈ R it reads:
g(xk+1) = g(xk) +d
dǫg(xk + ǫ∆xk)
∣
∣
∣
∣
ǫ=0
= 0 (94)
to be solved for increment ∆xk iteratively until its norm is sufficiently small. Within the computationalhomogenization procedure, the macroscopic Cauchy stress, mass fluxes, and electric fields are recovered fromthe numerical solution of the micro-scale problem (91) defined on the underlying microstructure (RVE) bymeans of micro to micro scale transitions (88). The Newton-Raphson iteration (94) requires macroscopictangents. Their consistent derivation can be achieved in analogy with the scheme proposed in [28, 29] forthermo-mechanical analysis of heterogeneous solids.
9 Conclusions
A two-scale modeling of several electrochemical and mechanical processes that take place during charg-ing/discharging cycles in Li-ion battery electrodes has been investigated in the present note. The perfor-mance of batteries relies on the interaction between micro and nano-scale phenomena, in particular withinthe electrodes. Intercalation, swelling and the consequent eventual mechanical failure originate at the nano-scale, where neutral Lithium is inserted or extracted from active particles. Modeling all particles in theelectrodes is unfeasible, because of the computational cost of the numerical approximation. Therefore, thenano-scale effects are usually incorporated into a micro-scale modelization through averages techniques and
22
constitutive models that are derived from homogenization methods. In the present contribution the compu-tational homogenization has been formulated for the multi physics problem at hand. The key componentsof the scheme, i.e. the formulation of the microstructural boundary value problem and the coupling betweenthe micro and macrolevel based on the averaging theorems, are addressed.
At the macro-scale, the electrode is considered as a porous continuum, whereas at the micro level itis heterogeneous and multiphase. The microscopic length scale is still much larger than the moleculardimensions, so that a continuum approach is justified for every constituent. The microscopic length scale,say the particle characteristic size, is three order of magnitudes smaller than the characteristic macroscopiclength, that can be identified with the cell size. Therefore, the model is well framed into the principle ofseparation of scales.
The basic concepts of the proposed scheme, as established by Suquet [37], consist in the formulation of themicroscopic boundary conditions from the macroscopic input variables (macro-to-micro transition) as wellas in the calculation of the macroscopic output variables from the analysis of the deformed microstructuralRVE (micro-to-macro transition). This two steps stem from a rigorous thermodynamic analysis, that firmlydistinguished the balance and the constitutive contributions to the formulation. Furthermore, the Hill-Mandel condition [30] has been extended, so that the internal expenditure of virtual power of mechanicalforces, of charge and mass fluxes is preserved in the scale transition. The variational framework here outlinedgreatly benefited from choosing potentials rather than concentration as independent field in the modelization.Weak forms keep the usual significance of power expenditure, key concept in a multi scale approach. Theprocedure itself is independent on the constitutive laws at the micro-scale, provided that thermodynamicrestrictions discussed in section 5 are satisfied and that the microscopic boundary value problem admits aunique solution. In fact, constitutive theory was not provided so far. Supplementary material is provided inappendices, where simple constitutive laws are applied.
Processes in Li-ion batteries electrodes are modeled with the partial differential equations setting ofdiffusion-migration-intercalation problems and their mechanical coupling. It appears a new challenge forthe computational homogenization technique. The (numerical) relation between the macroscopic input andoutput variables, particularly the tangent “stiffnesses” to be inserted in each incremental step of the Newton-Raphson scheme (94), have not been provided here but can be easily determined following the usual approachof the computational homogenization scheme: the reader may for instance refer to the scheme proposed in[28, 29] for thermo-mechanical analysis of heterogeneous solids.
Companion papers will be devoted to the implementation of the proposed scheme as well as to itsextension to finite strain, taking special care to the proper configuration - either current or reference -in which the scale transitions shall be formulated. Being independent on the constitutive assumptions atthe micro-scale, the most relevant ideas in the current impetuous development of constitutive modeling ofthe mechanical phenomena that take place during lithiation and delithiation, including plastic flow, phasesegregation, damage, can be included in the solution scheme. In particular, degradation and growth ofcracks, that have been largely observed in charging/discharging cycles in Li-ion battery electrodes, will beconsidered eventually following the approach in [50, 51, 52]. The binder, that has not been considered sofar, could be included as a cohesive interface between particles and may play a fundamental role in modelingmass and charge transfer against a critical detachment of particles.
The computational homogenization scheme presented may provide a significant contribute to the multiphysics simulation in the fundamental area of the mechanics of energy storage materials. If it will reacha sufficient maturity and robustness, the computational homogenization scheme can be attractive even forbattery makers companies, interested in capturing a significant part of the expected strong increase indemand of Li-Ion batteries.
AcknowledgementsThe present paper is the outcome of long time of training and discussions with several colleagues, pro-
fessors, students. We are gratefully indebted with Prof. L. Anand, T. Wierzbicki, and Dr. C. Di Leo, withwhom authors had long and fruitful discussions. The same applies to Prof. M. Geers, V. Kouznetsova whotrained us on the computational homogenization technique. Ideas and remarks have been shared with Prof.A. Bower and his team at Brown University, whose work was extremely inspiring.
23
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Appendix 1 - Constitutive theory
26
The computational homogenization framework described in this paper applies to the most general con-stitutive theory, provided that thermodynamic restrictions discussed in section 5 are satisfied. The readermay feel unsatisfied though, because of the absence of constitutive specifications. She may doubt for intensethat suitable form of the Landau potential can be written. To avoid such a disappointment, supplementarymaterial is here provided. There are no novelties with respect to the large literature on the field. Further-more, some assumptions are consciously simplistic. Exhaustive analysis on constitutive assumptions will becoupled with the implementation in further publications.
1.1 - Constitutive theory for the electrolyte
Following [18], the electrolyte will be idealized as an inviscid, incompressible fluid (i.e. a perfect fluid).Since the electrolyte fluid is incompressible, the small strain tensor must satisfy Tr[ε] = 0 and because of nullviscosity the stress tensor has vanishing deviatoric part, i.e. σ = p1. The pressure p does not expend powerinternally and does not enter the entropy imbalance (12). In other words, the pressure is irrelevant to theinternal thermodynamic structure of the theory and following [31] the pressure is not specified constitutively.
In the absence of charged species, only diffusion takes place; the relevant constitutive theory can be foundin [31], section 66. In its dual way, in the absence of gradients of chemical potential diffusion cannot proceed.Current is thus driven by migration only and one expects the charge flux to be related to the electrostaticpotential via Ohm’s law. When diffusion is present, a current density rises due to Faraday’s law and bothprocesses contribute to the charge flux. It is therefore clear that concentration gradients and electric fieldacts contemporarily to generate ion mobility. This is the intimate nature of the electrochemical potentialµα.
The chemical potential in a mixture depends on the composition of the mixture. It is convention to definead “ideal” dependence of µα on composition, and then to define a parameter called activity coefficient todescribe the deviation of the actual electrochemical potential from the ideal. Ideal conditions are convention-ally set in infinitely dilute solutions [24]. Therefore, in writing constitutive theories one firmly distinguishesthe items of dilute and concentrated solutions in order to specify the thermodynamic potentials (either ψ orΦ) an to express the dependency of fluxes hα in terms of appropriate driving forces for each ionic movementmechanism.
1.1.1 - Dilute solutions .
One moves from restrictions (24) and assumes that the species flux hα is given by a constitutive equationin terms of the electrochemical potential µα and of concentration cα:
hα = −M(cα) ∇ [µα ] (95)
i.e. one assumes a linear dependency of the mass flux of species α from the gradient of the electrochemicalpotential as usual in Fickian-diffusion. A classical specialization of mobility tensor M(cα) is the isotropic,linear choice
M(cα) = uα cα 1 (96)
Definition (96) represents the physical requirement that the pure phase cα = 0 has vanishing mobility andthere is no saturation limit for species α in the electrolyte. The amount uα > 0 is usually termed the ionmobility and represents the average velocity of species α in the solution when acted upon by a force of 1N/mol independent of the origin of the force.
Infinitely dilute solutions. In “ideal”, infinitely diluted conditions the interactions between solutes areneglected. Only friction forces between the solute and the solvent are considered. The chemical interactionsbeing ignored, an idealized electric potential that is the result of moving idealized electric charges from oneelectrode to the other can be taken as the electrostatic potential φ. Ignoring the chemical environment thatelectrons experience, allows the complete separation of the phenomena of diffusion and migration in the masstransfer:
ρ udiff =∑
α
µα hα − div [µα hα ]
27
ρ umigration =∑
α
φ iα − div [φ iα ]
In view of Faraday’s law (3) one writes
ρ (udiff + umigration) =∑
α
(µα + zαF φ)hα − div [ (µα + zαF φ)hα ] (97)
and by comparison with the mass flux contribution in (21) one splits the electrochemical potential of speciesα in the sum
µα = µα + zα F φ (98)
Such a splitting, with chemical potential µα merely depending on cα, holds merely in ideal conditions.
To relate cα and µα suitable specifications must be selected for the thermodynamic potentials. A freeenergy density ψ(cα) = ψdiff (cα) + ψmigr(cα, φ) is given by a regular solution model for the continuumapproximation to mixing
ψdiff (cα) = ρ−1(
µ0α cα +Rϑ cα log cα
)
(99)
ψmigr(cα, φ) = ρ−1 (zα F cα φ) (100)
withR the universal gas constant. Specialization (99) of Helmholtz’ free energy density represents the entropyof mixing for an ideal solution, with no energetic interactions. µ0
α is a reference value of the chemical potentialof diffusing species α, taken as independent on the electric state, whereas it may depend on temperatureand stress. Specialization (100) describes the migration contribution. Fick’s law (95) then becomes
hα = −uαRϑ ∇ [ cα ] − zα F uα cα ∇ [φ ] (101)
which is the usual form for the flux density of species α in absence of convection (see for instance [24],formula (11.1)) in dilute solutions. Defining the diffusion coefficient Dα with Dα = uαRϑ is nothing butthe so called Nernst-Einstein equation.
In the Helmholtz free energy decomposition (99-100) no coupling with mechanical effects has been con-sidered. This might be incorrect at high amounts of pressure in the electrolyte. Such a fundamentalthermodynamical analysis goes beyond the scope of the present work.
In view of constitutive assumptions (99, 100) the potential imbalance γ defined in equation (35) reads
γ = µLi − µe− − µLi+ = µ0Li − µ0
e− − µ0Li+ −Rϑ+Rϑ log
cLi
ce− + cLi++ Fφe− − FφLi+
At thermodynamic equilibrium γ = 0 and ξ = ξNernst, which implies
ξNernst = −1
F
(
µ0Li − µ0
e− − µ0Li+ −Rϑ+Rϑ log
cLi
ce− + cLi+
)
thusγ = F (ξ − ξNernst) = F χ (102)
Moderately diluted solutions. In non ideal but moderately diluted solutions it is assumed again alinear dependency of the mass flux of species α upon the gradient of the electrochemical potential of thesame species, as in (95), ignoring cross-effects.
The main advantage of splitting (98), that once again holds in ideal conditions only, stands in its abilityto characterize the electric state of a phase by means of an electric potential φ, that has the usual meaningof an electrostatic potential. The free energy density
ψ(cα) = ψdiff (cα) + ψdev(cα) + ψmigr(cα, φ)
28
is enriched with a a term ψdev(cα) that expresses a deviation from ideality in terms of the so called ionicactivity coefficient fα that represents the energetic interactions of mixing8. One would like to relate fα tothe yet undefined electric potential φ in some way.
Modulo some unavoidable element of arbitrariness, see [24] chapter 3, an electric field potential can beproperly defined. In order to assess the electrical state of a part of the system one usually refers to thereference electrode. But other choices are possible. A so called quasi-electrostatic potential φ is defined byarbitrarily selecting an ionic species β and defining φ as if in ideal conditions for β:
µβ = µβ + zβ F φ
where µβ , termed the quasi-chemical potential of species β, is defined as to be the “ideal” chemical potentialof species β by means of the usual thermodynamics restrictions without any activity coefficients, as fromspecialization (99). Then one has for any species α 6= β:
µα =
(
µα −zα
zβ
µβ
)
+zα
zβ
µβ + zα F φ
As the term in parentheses corresponds to a neutral combination of ions due to the electroneutrality condition,it must be independent on any definition of electric field. By taking specializations (99-100) for the free energydensities of species, and assuming as customary the following definition of activity coefficients:
∂ψdev
∂cα= Rϑ log fα
easy algebra leads to:
µα = Rϑ log cα + zα F φ+Rϑ log f± +
(
µ0α −
zα
zβ
µ0β
)
(103)
with f± = fLi+ fX− usually termed mean activity coefficient - see [24], formula (2.25). At constant temper-ature, Fick’s law then becomes
hα = −uαRϑ ∇ [ cα ] − zα F uα cα ∇ [φ ] − uα cα Rϑ ∇ [ log f± ] (104)
that extends (101) to concentrated solutions. The mean activity coefficient is independent on the electricstate and expresses the deviation from ideality of the chemical interactions.
The arbitrariness of the definition of the quasi-electrostatic potential stands in the need to choose anionic species, but it conforms to the usual concept of electrostatic potential. In the limit of infinitely dilutedsolutions, the definition of φ thus becomes independent on β, consistently with the theory of diluted solutions.
Equation (103) is a sound extension of splitting (98) in non ideal cases. It requires a proper definitionof the free energy ψdev to define the activity coefficients and their dependency upon the concentration.Equation (103) can also be easily inverted
cα = exp
[
1
Rϑ
(
µα − µ0α − zα F φ
)
]
(105)
having set µ0α =
(
µ0α − zα
zβµ0
β
)
+Rϑ log f±.
1.1.2 - Concentrated solutions .
Ficks law (95) states that the flux of phase α is proportional to the gradient of the concentration ofthis species, directed against the gradient. There is no influence of the other phases, i.e. cross-effects are
8Activity coefficients must approach unit when the solution tends to an infinitely dilute solution for an electrically neutralcombinations of ions: in such a case in fact the influence of the electric state vanishes. For the LiX salt at hand, it implies:
fLi+
fX− → 1 as (c
Li++ c
X−) → 0
29
ignored although well-known to appear in reality. To account for energetic interactions between phases, amulticomponent diffusion approach is required for realistic models. The standard approach [53] within thetheory of Irreversible Thermodynamics replaces Fickian fluxes by linear combinations of the gradients of allinvolved electrochemical potentials. In the case of binary electrodes:
hα = −
2∑
β=1
Mαβ(c1, c2) ∇[
µβ
]
α = 1, 2 (106)
Mobility tensors Mαβ(c1, c2) on turn depend on the concentration of all phases. A classical specialization ofsuch mobility tensors is an isotropic choice
Mαβ(c1, c2) = Mαβ(c1, c2) cα 1
whereas linearity is not usually assumed for Mαβ(c1, c2). The full matrix of mobility coefficients Mαβ has tobe positive semi-definite in order to be consistent with thermodynamic restriction (24) and symmetric dueto the Onsager reciprocal relations. The approach that provides specifications for Mαβ is usually known asMaxwell-Stefan approach [54] and is here described for the case of a binary electrolyte under the assumptionof absence of convective motion of the solvent.
The Maxwell-Stefan approach assumes that the thermodynamical driving forces of the i-th species, namelyci
Rϑ∇ [µi ] is in local equilibrium with the total friction force. The mutual friction force between species i
and j is assumed to be proportional to the mass fluxes. The Maxwell-Stefan equations of multicomponentdiffusion thus read:
ciRϑ
∇ [µi ] = −∑
j 6=i
cj hi − ci hj
cT Dij
i = 0, 1, 2 (107)
with cT = c0 + c1 + c2. Factors Dij are termed Maxwell-Stefan diffusivities and the symmetry propertyDij = Dji holds. In equations (107) one notices that the solvent is explicitly involved (the phase termed“zero”) and that only 2 out of 3 equations are linearly independent. In fact, the sum of the three equationslead the null identity, as the left hand side becomes the Gibbs-Duhem equation whereas the right is zero forthe symmetry of Dij -s. In the assumption of absence of convective motion of the solvent, h0 = 0, trivialalgebra from equations (107) leads to
ciRϑ
∇ [µi ] = −∑
j = 0, 1, 2j 6= i
(
cjcT Dij
−ci
cT Di0
)
hi +∑
j = 1, 2j 6= i
(
cicT Dij
)
hj i = 1, 2 (108)
The system of equation can be inverted (see [54] for a detailed complete justification: for the binary case athand it is trivial to prove that the system matrix is positive definite) thus leading to constitutive equations(106) that shows to be non linear in the concentrations.
Specialization for the free energy densities of species can still be assumed as in moderately dilutedsolutions.
Summarizing thus, for diluted solutions assuming the Fickian representation for mass flux (95) andequation (96) as a specialization of mobility tensor, one has
hmLi+(µm
Li+ , φm) = −uLi+ c
mLi+ ∇
[
µmLi+
]
x ∈ Ve (109)
hmX−
(µmX−, φm) = −uX− cmX−
∇ [µmX−
] x ∈ Ve (110)
with cmLi+
(µmLi+ , φ
m) and cmX−(µm
X−, φm) derived from constitutive identity (105). For concentrated solutions
one makes recourse to the theory of Irreversible Thermodynamics, replacing Fickian fluxes hmα by linear
combinations of the gradients of Li+, X− electrochemical potentials as in (106). Equation (105) holds forboth species.
1.2 - Constitutive theory for the active particles
30
Thermodynamics restrictions (18) for active particles suggest that the Li flux hLi is given by Fick’s law,i.e. a constitutive equations of the form
hLi = −M(cLi) ∇ [µLi ] (111)
with the positive definite tensor M(cLi) termed mobility tensor. Diffusion processes in active particles is thuscompletely described by the chemical potential µLi and its constitutive relationship in terms of the internalenergy u (or equivalently the free energy ψ)
µLi − ρ∂u(y)
∂cLi= µLi − ρ
∂ψ(y)
∂cLi= 0
and of the set of variables y that model driving forces. Specific literature in this regard is nowadays broadand several choices for y have been considered.
The deviation from stoichiometry in a solid composition due to atomic diffusion may cause large swelling.If it is not accommodated properly, mechanical stresses arise, which in turn affect intercalation. A classicalmodel that takes volumetric changes into account is the network model of Larche and Cahn [26]. It assumesthat the lattice sites of the hosting material form a network within which neutral Lithium can diffuse. Thenetwork lattice material, if Si or C for instance, is taken to be insoluble in the electrolyte. Stress dependentfamilies of chemical potential µLi have been recently considered by several authors [27, 55, 56, 57, 58, 49]considering large eigenstrain and including non convex double-well potentials to model phase-segregation aswell as higher order kinematical descriptors9.
Here simpler specializations will be considered to the sake of this supplementary material appendix. TheHelmholtz free energy density ψ is widely assumed to consist in three separated factors. One, here termedψdiff , describes the pure diffusive process, usually adopting the concentration as the state variable. Thesecond, ψmech, models the elastic energy density, whereas the last one, ψcoupling, couples the mechanical anddiffusive processes. A classical specialization is the isotropic one that follows
ψdiff (cLi) = µ0 cLi +Rϑ
(
cLi log cLi + (cmaxLi − cLi) log
cmaxLi − cLi
cmaxLi
)
(112a)
ψmech(ε) =1
2ε ·Cε =
1
2
(
K Tr2[ε] + 2G ||Dev[ε] ||2)
(112b)
ψcoupling(cLi, ε) = −3K ωa cLi Tr[ε] (112c)
with R the universal gas constant and K, G are the bulk and shear modulus respectively. ConcentrationcmaxLi is reached in the active particles when all accommodating sites are filled, i.e. cmax
Li is the saturationlimit for Lithium in the active particles material. ωa is the coefficient of chemical expansion of neutralLithium in the active host material and µ0 is a reference value of the chemical potential of the diffusingspecies. Symbol Tr[−] denotes the trace operator whereas Dev[−] is the deviator operator. Mobility tensorM(cLi) is assumed to be isotropic, in the form
M(cα) = M0 cLi
(
1 −cLi
cmaxLi
)
1 (113)
Definition (113) represents the physical requirement that the pure phases cLi = 0 and cLi = cmaxLi have
vanishing mobility. The amount M0 > 0 is the mobility constant. Thermodynamics restrictions (18) imply:
σ = K Tr[ε]1 + 2GDev[ε] − 3K ωa cLi 1 (114)
µLi = µ0 + Rϑ log cmaxLi − Rϑ log
[
cmaxLi
cLi− 1
]
− 3K ωa Tr[ε] (115)
The latter constitutive equation can be inverted, thus obtaining:
cLi = cmaxLi
(
1 + exp
[
µ0 + Rϑ log cmaxLi − µLi − 3K ωa Tr[ε]
Rϑ
])−1
(116)
9The first order theory adopted in the present work cannot deal with high order material models, which take into accountmicroforces at both scales. To extend the multi scale framework a second order homogenization scheme is required.
31
that can be seen as a specification for the Grand potential Φ in thermodynamic restriction (30) for neutralLithium.
Fick’s law (111) becomes
hLi(cLi, ε) = −Da ∇ [ cLi ] + 3 Da cLi
(
1 −cLi
cmaxLi
)
∇ [ Tr[ε] ] x ∈ Va (117)
with Da = M0Rϑ the diffusivity coefficient and Da = M0Ka ωa a chemical expansion coefficient. In termsof chemical potential, Fick’s law is given in (111) with cLi from (116).
The flow of electrons in the active particles is modeled similarly to the flow of a single charged specie αin a dilute solution. By defining the Helmholtz free energy as
ψ(ce− , φ) = µ0e− ce− +Rϑ ce− log ce− + ze− F ce− φ
it comes out:
ce−(µe− , φ) = exp
[
1
Rϑ(µe− − µ0
e− −Rϑ+ F φ)
]
(118)
and finally:he−(µe− , φ) = −M(ce−(µe− , φ)) ∇ [µe− ]
In terms of concentration and electrical potential, making recourse to equations (95, 96, 99, 100) a Fick’slaw of the following kind arises:
he−(ce− , φ) = −ue− Rϑ ∇ [ ce− ] − ze− F ue− ce− ∇ [φ ] (119)
1.3 - Constitutive theory for the conductive particles
The flow of electrons in conductive particles is modeled as for the active particles, even though elec-troneutrality conditions are usually respected. No ionic mass transport is present. Conductive particlesare generally made of graphite and/or carbon black [47]. Natural and crystalline graphites are known forbrittleness. In this regard the classical strain energy function of infinitesimal isotropic elasticity, eventuallyextended to large deformations in terms of the logarithmic elastic strain, might be the most proper choice.
1.4 - Constitutive theory for electrode foils
In the same spirit of [18] and following Larche and Cahn [59], an homogeneous electrode can be idealizedas a network solid. Li will be assumed to form an interstitial solution in this network. Then Lithium maydiffuse through the interior of the electrode, driven by variations in concentration or stress according forinstance to the isotropic specializations (112). In a half-cell, the electrode can be taken in the form ofmetallic lithium as in [22], so that no diffusion processes take place as Li is removed/deposited directlyfrom at the electrolyte interface and no current collector is needed. This option is likely more interesting toreproduce experimental setups [5] rather than industrial batteries.
1.5 - Constitutive theory for separators
Not much attention has been put so far to the mechanical constitutive assumptions on the separators.They clearly depend on the manufacture of the separator and its interactions with the electrolyte, that in thecurrent technology of Li-ion batteries [7] might be a solid, a liquid or a gel. This crucial aspect of materialmodeling falls beyond the scope of the present paper, in which the isotropic linear elastic constitutive law
σe = λe Tr[εe]1 + 2Ge εe (120)
is taken.
1.6 - Constitutive theory for the electric field
32
The constitutive assumption
e(η, ε, cα, ce− , φ) = u(η, ε, cα, ce−) +1
2 ρτ · τ (121)
will be taken. The thermodynamic restriction on the electric field thus becomes
∇ [φ ] = E (122)
Appendix 2 - Microscale governing PDEs
In view of balance equations (54, 55) and of the constitutive assumptions (109, 110), the governing PDEsfor mass transport in the electrolyte read:
div[
cmLi+(µmLi+ , φ
m) ∇[
µmLi+
] ]
= 0 x ∈ Ve
div [ cmX−(µm
X−, φm) ∇ [µm
X−] ] = 0 x ∈ Ve
Gauss law (58) and the thermodynamic restriction (122) lead to:
ε0 ∆[φm ] = F (cmLi+(µmLi+ , φ
m) − cmX−(µm
X−, φm)) x ∈ Ve
In force of the assumption of electrolyte as a perfect fluid, the stress tensor σ = p1 can be evaluated via themomentum balance, that reads:
∇ [ p ] + F (cmLi+ − cmX−)∇ [φm ] = 0 , x ∈ Ve (123)
Gauss law (57) and the thermodynamic restriction (122) lead to:
ε0 ∆[φm ] = F (cm+ − cme−(µme− , φ
m)) x ∈ Vc
whereas current density is governed by
div [ cme−(µme− , φ
m) ∇ [µme− ] ] = 0 x ∈ Vc
Having assumed a linear elastic isotropic constitutive behavior for the conductive particles, balance equation(61) becomes
λc ∇ [ Tr[εm] ] + 2Gc div [ εm ] − F (cm+ − cme−(µme− , φ
m))∇ [φm ] = 0 , x ∈ Vc
Force balance equation (61) and mass balance equation (53) become coupled in view of constitutiveequations (114), (117):
Ka ∇ [ Tr[εm] ] + 2Ga div [ Dev[εm] ] − 3Ka ωa ∇ [ cmLi ] − F (cm+ − cme−)∇ [φm ] = 0 x ∈ Va (124)
−Da ∆[ cmLi ] + 3 Da (1 − 2cmLi
cmaxLi
)∇ [ cmLi ] · ∇ [ Tr[εm] ] + 3 Da (1 −cmLi
cmaxLi
) cmLi ∆[ Tr[εm] ] = 0 x ∈ Va(125)
Equations (124), (125) are stated in terms of electrochemical potentials µe− , µLi in view of equations (116),(118). Gauss law (57) and the thermodynamic restriction (122) lead to:
ε0 ∆[φm ] = F (cm+ − cme−(µme− , φ
m)) x ∈ Va (126)
whereas current density is governed by
div [ cme−(µme− , φ
m) ∇ [µme− ] ] = 0 x ∈ Va (127)
33
Appendix 3 - The electroneutrality assumption
Electroneutrality∑
α
zαcα = 0 (128)
is often assumed in the electrolyte and in electrodes [24]. For the solution of binary salt LiX object of studyit holds
cLi+ − cX− = 0 (129)
In this appendix the impact and the applicability of electroneutrality assumption will be analyzed.
3.1 - Range of validity
The electrochemical part of the problem in the electrolyte is formulated in terms of three potentialsµLi+ , µX− , φ and the balance laws of mass and Gauss law properly describe their evolution. May electroneu-trality over constrain the problem?
To investigate this issue, electroneutrality will be assumed and its impact evaluated in the framework ofdiluted electrolyte solutions. Gauss law reads:
div [E ] =F
ε0(cLi+ − cX−) = 0
and by means of thermodynamic restriction (27)
∆ [φm ] = 0 x ∈ Ve (130)
The conservation of charge in the electrolyte is linearly dependent upon equations (54) and (55) in view ofelectroneutrality and mass balances:
div [ ime ] = 0 x ∈ Ve (131)
Owing to Fick’s law (101), the flux of Li-ions is constitutively linked to concentration and electrostaticpotential. Applying Faraday’s law (3) and electroneutrality condition (129), the current density constitutiveequation follows:
ime = −le ∇[
cmLi+
]
− κme ∇ [φm ] x ∈ Ve (132)
having set: κe = F 2 (uLi+ + uX−) cmLi+
, le = F (DLi+ −DX−), and DX− = uX− Rϑ. Parameter κe is termedconductivity of the solution.
Furthermore, by substitution of (101) into a linear combination of mass balance equations (54-55) thestrong form of the steady-state problem of ion-transport reads:
∆[
cmLi+
]
= 0 x ∈ Ve (133)
Inserting (132) into (131), the requirement:
∇[
cmLi+
]
· ∇ [φm ] = 0 x ∈ Ve (134)
comes out in view of (130) and (133). Such an identity is usually not satisfied, because ∇[
cmLi+
]
is neithervanishing nor orthogonal to the electric field. However, the effect of charge on electrochemical potential ishugely greater than the effect of concentration: this predominance of migration with respect to diffusionjustifies electroneutrality assumption in the presence of an electric field across the battery cell, at least awayfrom the Stern and Gouey-Chapman layers.
Once the steady state is reached in the electrolyte, if the mass flux of anions is zero the requirement(134) becomes
∇ [φm ] · ∇ [φm ] = 0 x ∈ Ve (135)
thus requiring a small variation of the electrostatic potential in the bulk.
34
3.2 - Constitutive theory
In view of Fick’s law (101), Faraday’s law (3) and electroneutrality (128), for the binary salt at hand onehas:
i =
2∑
α=1
zα F hα = −
2∑
α=1
zα F Dα ∇ [ cLi+ ] −
2∑
α=1
z2α F
2 uα cLi+ ∇ [φ ] (136)
in the absence of gradients of concentration (and thus of chemical potential in view of (99)) current is drivenby migration only and it holds:
i = −
2∑
α=1
z2α F
2 uα cLi+ ∇ [φ ] = κe ∇ [φ ] (137)
i.e. Ohm’s law with κe termed electrolytic conductance. Linearity of κe with respect to concentration hasbeen experimentally assessed at concentrations far from the saturation limit in several binary salt solutions.
3.3 - A property of binary electrolyte .
For the binary salt LiX under consideration, and defining terms by comparison with (103), one writes:
µLi+ = µ0Li+X−
(fLi+ , fX−) +Rϑ log cLi+ + Fφ
µX− = µ0X−Li+(fLi+ , fX−) +Rϑ log cX− − Fφ
and by subtraction
φ(µLi+ , µX−) =µLi+ − µX− − µ0
Li+X−(fLi+ , fX−) + µ0
X−Li+(fLi+ , fX−)
2F(138)
in view of electroneutrality. In view of property (138) the electric state of the electrolyte can be expressedin terms of the electrochemical potential of the two ions in the solution. The concentration of Li-ions can bethus written as:
cLi+(µLi+ , µX−) = exp3µLi+ − µX− − 3µ0
Li+X−(fLi+ , fX−) + µ0
X−Li+(fLi+ , fX−)
2Rϑ(139)
in the same terms. By comparison with (105) it can be observed that in view of electroneutrality cLi+
becomes independent on φ.
3.4 - Two scale formulation
3.4.1 - Macroscale
The conservation of charge in the electrolyte under the assumption of electroneutrality is provided bythe linear combination of equations (38) and (39) for anions and cations.
div[
iMe]
= −F hLi x ∈ ΩCa × [0, T ] (140)
In view of the assumption of electroneutrality Gauss laws become homogeneous and only one macroscopicelectric field E suffices. Similarly, as the electrostatic forces of interaction bζ vanishes, one single stress fieldonly is required in (44-46).
3.4.2 - Microscale
Body forces bζ are insignificant as far as the electrode and electrolyte are concerned if they remainelectrically neutral. Inertia effects as well as bulk forces are neglected too, so that the force balance (5)specializes as:
div [ σm ] = 0 x ∈ V (141)
35
skw [ σm ] = 0 x ∈ V (142)
Constitutive theories for the mass transport can be expressed in terms of electrochemical potentialsparticularizing appendix 0. For diluted solutions, assuming the Fickian representation for mass flux (95) andequation (96) as a specialization of mobility tensor M(cα), one has
hmLi+(µm
Li+ , µmX−
) = −uLi+ cmLi+ ∇
[
µmLi+
]
x ∈ Ve (143)
hmX−
(µmLi+ , µ
mX−
) = −uX− cmX−∇ [µm
X−] x ∈ Ve (144)
with cmLi+
= cmX−derived from constitutive identity (139) in terms of the two distinct electrochemical poten-
tials. For concentrated solutions one makes recourse to the theory of Irreversible Thermodynamics, replacingFickian fluxes (143-144) by linear combinations of the gradients of Li+, X− electrochemical potentials as in(106). Again electroneutrality allows to take advantage of (139) for both species.
In terms of concentrations, constitutive theories read as follows. For infinitely diluted solutions oneconsiders Fick’s law (101) together with (132). In the absence of concentration gradients, one defines
imLi+ = −F 2 uLi+ cmLi+ ∇ [φ ] =
uLi+
uLi+ + uX−
ime = tLi+ ime , x ∈ Ve (145)
An analogous equation applies for imX−. Parameter tLi+ = uLi+ (uLi+ +uX−)−1 is thus a material property for
the salt in the solution. It is termed transference (or transport) number. It must be stressed that equation(145) holds in the absence of concentration gradients only. In such a case, constitutive equation (101) canbe rewritten rigorously as:
hmLi+(cmLi+ , φ) = −DLi+ ∇
[
cmLi+
]
−tLi+
Fime , x ∈ Ve (146)
For concentrated solutions one should make recourse to the theory of Irreversible Thermodynamics,replacing Fickian flux (101) by linear combinations of the gradients of all involved electrochemical potentialsas in (106). To have again the mass flux of each phase to be constitutively independent on the electrochemicalpotential of other phases, the notion of transference number has been used by Newman - see [24], chapter12 - also in the case of concentrated solutions:
hmLi+(cmLi+ , φ) = −De(c
mLi+) ∇
[
cmLi+
]
−tLi+(cm
Li+)
Fime , x ∈ Ve (147)
Parameter De represents the usually measured diffusion coefficient of the binary salt which is not providedby the Nernst-Einstein equation anymore. It is related to the diffusion coefficient of the electrolyte by anactivity coefficient (see [24], chapter 12 formula (12.12) or [22], formula (2)) thus becoming a function ofconcentration cLi+ . Furthermore, the transference number tLi+(cLi+) has now to be taken as an unknownfunction. It is no longer a material property and it depends upon concentration. It is a measurable quantityhowever, and outcomes from tailored experiments that provided accurate estimation of tLi+(cLi+) resultedin an abundant and still increasing scientific literature.
The current density ime was derived in [24] as a function of the usually measured conductivity of thesolution, here again denoted with κe with a small abuse of notation10, and of the reference electrode potentialφ with respect to an actual electrode, usually a Lithium reference electrode.
ie(cmLi+ , φ) = −κm
e ∇ [φm ] − χme (cmLi+) ∇
[
cmLi+
]
x ∈ Ve (148)
with
χme (cmLi+) = −2RϑF (uLi+ + uX−)
(
1 +∂ log f±∂ log cm
Li+
)
(
1 − tLi+(cmLi+))
(149)
The flow of electrons in the active particles is modeled similarly to the flow of a single charged speciein a dilute solution. The assumption will be made that in steady state conditions charge transport in the
10In fact it does not coincide with the one for ideal solutions and it depends upon the concentration cm
Li+.
36
active particles involves electrons only, whereas the flux of positive charges is null at all x ∈ Va. By such anassumption and electroneutrality from the counterpart of (119) it comes out:
Rϑ1
ce−∇ [ ce− ] = −F ∇ [φ ] (150)
andhe− = 2F ue− ce− ∇ [φ ] (151)
Constitutive law (118) applies. As charge movement is due to electron flux he− only, the current densityamounts at ime− = −F he− and Ohm’s law relates current density to the electric potential:
ime− = −κa ∇ [φm ] x ∈ Va (152)
with κa = 2F 2 ue− ce− . A similar outcome applies to conductive particles.
By substitution of (101) into the mass balance (54) and of (132) into charge balance (131) the strongform of the steady-state problem of ion-transport in the electrolyte comes out in the assumptions of elec-troneutrality and dilute solutions:
∆[
cmLi+
]
= 0 , x ∈ Ve (153)
κme ∆[φm ] + ∇ [κm
e ] · ∇ [φm ] = 0 , x ∈ Ve (154)
Equation (153) comes out after inserting (101) for anions X− into the mass balance (55) and subtracting.Equation (153) is in terms of cm
Li+and the system of differential equations becomes uncoupled.
By substitution of (147) into the mass balance (54) and of (148) into charge balance (131) the strongform of the problem of ion-transport in the electrolyte comes out in the assumptions of electroneutrality andconcentrated solutions:
−De ∆[
cmLi+
]
−∇ [De ] · ∇[
cmLi+
]
+1
F(κe ∇ [φm ] + χm
e ∇[
cmLi+)]
· ∇[
tLi+(cmLi+)]
= 0 , x ∈ Ve(155)
χme ∆
[
cmLi+
]
+ κme ∆[φm ] + ∇ [κm
e ] · ∇ [φm ] + ∇ [χme ] · ∇
[
cmLi+
]
= 0 , x ∈ Ve (156)
with χme as in equation (149). Balance of mass is equivalent to equation (153) if De and tLi+(cm
Li+) are
constant in the solution, as in case of diluted solutions. Assuming that those two parameters are constantis a strong statement that however simplifies the formulation of the governing PDEs significantly. In such acase, equation (156) reduces to (154) in view of the definition of χm
e .
The governing PDEs can be written in terms of the electrochemical potentials µmLi+ , µm
X−rather than
concentration cmLi+
and φ by exploiting identities (138- 139) into (153-156).
The current density in the conductive particles is related to the electric potential by the classical Ohm’slaw (152) with κc in place of κa. Conservation of charge and mass (56) lead to
∆ [ cme− ] = 0 x ∈ Vc (157)
κmc ∆[φm ] + ∇ [κm
c ] · ∇ [φm ] = 0 x ∈ Vc (158)
These equations are uncoupled to the balance equation (141), that in view of linear isotropic elasticityassumption becomes
λc ∇ [ Tr[ε] ] + 2Gc div [ ε ] = 0 x ∈ Vc (159)
Analogously, conservation of charge and mass (56) lead to
∆ [ cme− ] = 0 x ∈ Va (160)
κma ∆[φm ] + ∇ [κm
a ] · ∇ [φm ] = 0 x ∈ Va (161)
37
