A Discussion of the Cell Voltage during Discharge of an Intercalation Electrode for Various C-Rates Based on Non-Equilibrium Thermodynamics and Numerical Simulations

Society

OPEN ACCESS

A Discussion of the Cell Voltage during Discharge of an Intercalation Electrode for Various C-Rates Based on Non-Equilibrium

Thermodynamics and Numerical Simulations

To cite this article: Manuel Landstorfer 2020 J. Electrochem. Soc. 167 013518

View the article online for updates and enhancements.

This content was downloaded from IP address 91.65.164.95 on 16/04/2020 at 09:29

A Discussion of the Cell Voltage during Discharge of an Intercalation Electrode for Various C-Rates Based on

Non-Equilibrium Thermodynamics and Numerical Simulations

Manuel Landstorfer ^z

Weierstrass Institute for Applied Analysis and Stochastics (WIAS) 10117 Berlin, Germany

In this work we discuss the modeling procedure and validation of a non-porous intercalation half-cell during galvanostatic discharge.

The modeling is based on continuum thermodynamics with non-equilibrium processes in the active intercalation particle, the elec- trolyte, and the common interface where the intercalation reaction Li⁺+e⁻Li occurs. The model is in detail investigated and discussed in terms of scalings of the non-equilibrium parameters, i.e. the diffusion coefficientsDAandDEof the active phase and the electrolyte, conductivityσ^Aandσ^Eof both phases, and the exchange current densitye0L

s, with numerical solutions of the underlying PDE system. The current densityias well as all non-equilibrium parameters are scaled with respect to the 1-C current density i^C_A of the intercalation electrode. We compute then numerically the cell voltageE as function of the capacityQand the C-rate Ch. Within a hierarchy of approximations we provide computations ofE(Q) for various scalings of the diffusion coefficients, the conductivities and the exchange current density. For the later we provide finally a discussion for possible concentration dependencies.

© The Author(s) 2019. Published by ECS. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 License (CC BY,http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse of the work in any medium, provided the original work is properly cited. [DOI:10.1149/2.0182001JES]

Manuscript submitted July 17, 2019; revised manuscript received October 17, 2019. Published November 19, 2019.This paper is part of the JES Focus Issue on Mathematical Modeling of Electrochemical Systems at Multiple Scales in Honor of Richard Alkire.

Lithium ion batteries (LIBs) are vital today for many branches of modern society and especially for electro-mobility. The german na- tional platform electro-mobility aims one million electric vehicles by 2020, as well as the U.S., while China targets about five million zero emission cars. To achieve these goals, substantial knowledge on the effectively non-linear behavior of LiBs is required in order to reduce cost, increase their efficiency, safety, durability and further. The in- terpretation of experimental data requires a versatile and predictive mathematical model of a LIB, which accounts for the many physico- chemical processes occurring simultaneously during charge and dis- charge, e.g. Li⁺diffusion in the electrolyte, surface reactions at the electrode/electrolyte interface, solid state diffusion in the active parti- cles, and electrical conductivity.

First academic steps to model the functional principle of LIBs with the purpose of simulating their charge/discharge behavior were carried out by Newman et al. around 1993.¹ This electrochemical model became a central tool to interpret measured data of interca- lation batteries. One of the central ingredients of the Newman model is the Butler–Volmer-type reaction rateR

s for the intercalation reaction Li⁺+e⁻Li occurring at the interfaceA,Ebetween an intercalation electrode (particle)Aand the electrolyteE. The actual functional dependency ofR

s =R

s(nE,ϕE,nA,ϕA) on the different variables of the equation system, e.g. the electrolyte concentrationnE, the electrostatic potentialϕEin the electrolyte, the concentrationnAof intercalated ions, and the electrostatic potentialϕAof the active phase, is, however, rather stated then derived. Especially the so called exchange current density and its functional relationship to the cation concentration is doubtable.

From a non-equilibrium thermodynamics (NET) point of view, the functional dependency R

s = R

s(nE,ϕE,nA,ϕA) can be consis- tently derived and NET restricts this functional dependency in a very specific manner. We discuss in this work the modeling procedure of a single transfer reaction at the interface between an active interca- lation phase and some electrolyte based on the framework of NET for volumes and surfaces and draw some conclusions regarding ther- modynamic consistent models of the reaction rate. We account also for diffusion processes in the adjacent active particle and the elec- trolyte, as well es electrical conductivity, and state the corresponding balance equations. Then we consider galvanostatic discharge in half cell of some cathode intercalation material, electrolyte, and a lithium reference electrode, which is considered as ideally polarizable counter electrode.

zE-mail:Manuel.Landstorfer@wias-berlin.de

We introduce the C₁-current density, i.e. the current at which the electrode is completely discharged during one hour, and scale all non- equilibrium parameters based on the C-rateCh, i.e. multiples of the C1current density. It is then possible to derive a general relation be- tween the measured cell voltageE, the capacityQ, and the C-rateCh

based on the reaction rateR

s =R

s(nE,ϕE,nA,ϕA). Since, however, ac- tually the concentrations at the interfaceA,Eof intercalated cationsnA

and electrolytic cationsnEenter the surface reaction rateR

s, we need to solve necessarily the diffusion equations in the adjacent phases.

We discuss various approximation regimes and parameter scalings of the non-equilibrium parameters which allows us to compare numeri- cal simulations of cell voltageE =E(Q,Ch) to some representative experimental examples, especially of Lix(Ni1/3Mn1/3Co1/3O2(NMC).

Fig.1shows the measured cell voltageE as function of the capacity (or status of charge) for various discharge rates of thin of NMC half cell.²

We show that a rathersimple(but thermodynamically consistent) model of the surface reaction rateR

s, or more precise of the exchange current density, is sufficient to understand and predict the complex non-linear behavior of the cell voltage as function of the capacityQ and the C-RateCh. We provide also computations ofE = E(Q,Ch) for the exchange current density introduced byNewmanet al., draw

0 20 40 60 80 100 120 140 160 180 2.4

2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4

200C

MX-6

Potential (V) vs. Li

Capacity (mAh/g) 100C 50C 10C

1C C/25 C/25 1C 10C 100C 50C

Figure 1. Discharge curves (lower part) for various C-rates (Data of Fig 1b from Ref.2, reprinted with permission of The Electrochemical Society).

some conclusions regarding thermodynamic consistency, and compare computations based on this expression to the cell voltage based on our simpleexpression of the current density.

Modeling

We consider an active intercalation particleA in contact with some electrolyteE. The interfaceA,E = A ∩E captures the actual surfaceAof the active particle as well as the electrochemical double layer forming at the interface, i.e.A,E=^SCLA ∪A∪^SCLE . The domainsEandAare thus electro-neutral, and we refer to Refs.

3–5for details on the derivation. The electrolyte is on the right side in contact to some metallic counter electrodeR, where the interface E,Ccaptures also the double layer forming at the interface between the electrolyte and the counter electrodeC.

We consider a 1D approximation, where the electrode-electrolyte interfaceA,E is positioned atx = xAE, the left boundary ofA is denoted byx = 0 and the right boundary ofE isx = xEC, with dA = |xAE|anddE= |xEC−xAE|. The counter electrode is positioned atx=xECand spans tox=xC.

For some quantityu(x,t), we denote with u|^±AE=u^∓

x=xAE and u|^±EC=u^∓

x=xEC [1]

the evaluation at the respective side of the interfaceA,EandE,C. If uis present only on one phase, we drop the superscript^±.

The active particleA is a mixture of electrons e⁻, intercalated cations C and lattice ions M⁺, and the electrolyte a mixture of sol- vated cations C⁺, solvated anions A⁻and solvent molecules S. The respective species densities are denoted withn_α(x,t),x ∈ i. We denote with

μ_α:= ∂ψ

∂n_α, α=EA,EC,ES,AC,Ae,AM, [2]

the chemical potential of the constituents, which are derived from a free energy density^6,7ψ=ψA+ψEwithψA=ψˆA(nAe,nA_C,nA_M) of the active particle andψE=ψˆ(nE_S,nE_A,nE_C) of the electrolyte phase.

For the surfacewe have surface chemical potentials^4,6,8,9 μ

sα:=

∂ψ

s

∂nsα, α=EA,EC,ES,AC,Ae,AM, [3]

which are derived from some general surface free energy densityψ

s

. Material functions.—For the electrolyte we consider exclusively the material model^9–11of an incompressible liquid electrolyte account- ing for solvation effects, i.e.

μα=g^R_α+kBTln (y_α)+v^R_α(p−p^R), α=ES,EA,EC, [4]

with mole fraction

y_α= n_α

n^tot_E , [5]

molar concentrationn_α, and total molar concentration of the mixture (with respect to the number of mixing particles⁹)

n^tot_E =nE_S+nE_A+nE_C. [6]

In (4)T denotes temperature,kB the Boltzmann constant,g^R_αdenotes the reference molar Gibbs free energy (or chemical potential of the pure substance),p^R the reference pressure andv_α^R the partial molar volume of constituentαin the mixture. Throughout this manuscript we assume an isothermal temperature ofT =298.15 [K].

Note that nE_S denotes the number of free solvent molecules, whereasnE_A andnE_C are the densities of the solvated ions. This is crucial for various aspects of the thermodynamic model, and we re- fer to Refs.9,10,12,13for details. Overall, the material model for the electrolyte corresponds to an incompressible mixture with solvation

effects. We assume further v^R_EC v_E^R

S

= mE_C

mE_S

and v^R_EA v^R_E

S

= mE_A

mE_S

[7]

whereby the incompressibility constraint^9–11implies also a conserva- tion of mass, i.e.

α

v_α^Rn_α=1 ⇔

α

m_αn_α=ρ= mE_S

v^R_E

S

=const.. [8]

The molar volume of the solvent is related to the mole densityn^R_ESof the pure solvent as

v^R_ES=(n^R_ES)⁻¹. [9]

Note further that the partial molar volumesv^R_αand the molar masses m_αof the cation and anion are related to the solvation numberκE_Cand κA_C, respectively.

We assume that the partial molar volume of the ionic species is mainly determined by the solvation shell, which seems reasonable for large solvents like DMC in comparison to the small ions like Li⁺. We proceed thus with the assumption

vE_C =κE·vE_S and vE_A=κE·vA_C. [10]

For the active particle, we consider an extension of a classical lattice mixture model^14–21which accounts for occupation numbersωA>1 as well as a Redlich–Kister type enthalpy term^22,23for the intercalation material Liy(Ni1/3Mn1/3Co1/3)O2 (NMC). We refer to Ref.24for a detailed discussion and derivation based on a free energyψ^A. The chemical potential of intercalated lithium is derived as

μA_C =kBT

ln ₁

ωAyA_C

1+^1−ω_ωA^AyA_C

−ωA·ln

1−yA_C

1+^1−ω_ωA^AyA_C

+γA·hA(yA_C)

[11]

with

hA(y) :=(2y−1)+ 1 2

6y(1−y)−1

−1 3

8y(1−y)−1 (2y−1)

[12]

and mole fraction

yA_C= nA_C

nA

=:yA [13]

of intercalated cations in the active phase. The number densitynA of lattice sites is constant, which corresponds to an incompressible lattice, and the enthalpy parameterγA < 2.5. Note thatγA > 2.5 entails a phase separation²⁰and requires an additional termγAdiv∇yA_C in the chemical potential. However, we assume throughout this work that no phase separation occurs, whereby in diffusional equilibrium of the intercalation phase the concentration is homogeneous. An extension of this discussion toward phase separating materials will given in a subsequent work.

For the electrons we consider^9,25 μAe =

3 8π

2

3 h²

2mA_e

n

2 3

A_e and μ

s Ae =g

s R

A_e =const. [14]

and for the lattice ions μA_M =g^R_A

M +kBTln 1−yA_C

+v^R_M(pM−p^R_M), [15]

wherev^R_M = (n^R_M)⁻¹ is the molar volume of the lattice ions, pM the partial pressure andg^R_AMthe constant molar Gibbs energy. The material functions of the active intercalation electrode is essentially model an incompressible solid with a sub-lattice for the intercalated cationsAC.

The explicit surface chemical potentials μ

sα=

∂ψ

s

∂nsα, α=EA,EC,ES,AC,AM, [16]

are not required throughout this work since we will assume that the double layer is in equilibrium and that the double layer capacity (and thus also adsorption), is negligible for the sake of this work. However, we refer to Ref.9for the explicit functions ofμ

sαand the surface free energy of a surface lattice mixture with solvation effects.

Electroneutrality condition.—The electroneutrality condition of A,EandCcan be obtained by an asymptotic expansion of the balance equations in the electrochemical double layer at the respec- tive surface. We only briefly recapture the central conclusions and refer to Refs.3–5,9,26for details on the modeling, validation and the asymptotics. Most importantly, we have that

• the double layer is in thermodynamic equilibrium, i.e.∇μα+ e0z_α∇ϕ=0 in^SCLA and^SCLE , wheree0is the elementary charge,z_α the charge number andϕthe electrostatic potential

• there exists a potential drop between the active particle surface and the hyper-surfaceA^±,Eoutside of the respective space charge layers which is denoted by

U_i^SCL:=ϕ

s

−ϕ|^±AE [17]

whereϕ|^±AEis the electrostatic potentialright outsidethe space charge layer in the electrolyte or the active particle, respectively, andϕ

s

the (continuous) potential at the surface^a. The whole potential drop across the double layer atA,Eis denoted by

U_AE^DL:=U_E^SCL−U_A^SCL=ϕ|⁺AE−ϕ|⁻AE [18]

• the chemical potential at the surface can bepulled backthrough the double layer, i.e.μ

sα=μⁱ_α−e0z_αU_i^SCL,i=A,E

• the conditionμ

se=const. entails that the potential dropU_A^SCLis constant (with respect to some applied voltage) and determined by

U_A^SCL= 1 e0

(μ

s

A_e−μA_e

AE). [19]

• the charge density in the electrolyte vanishes and that for mono- valent electrolytes the cation mole fraction (or number density) is equal to the anion mole fraction, i.e.

yE_C=yA_C =:yE. [20]

• in the active phase the electroneutrality entails

nAe=nA_M =const. [21]

whereby we abbreviate

g^R_Ae :=μAe(nA_M) [22]

which is basically the Fermi energy of the solid material.

Transport equations.—In the electrolyteEwe have two balance equations determining the concentrationnE_C(x,t) (or the mole fraction yE_C(x,t)) and the electrostatic potentialϕE(x,t) in the electrolyte,^27–32 i.e.

∂nE_C

∂t = −∂xJE_C with JE_C = −DE·n^tot_E E^tf·∂xyE_C+^tE_e^C

0JE,q [23]

0= −∂xJE,q with JE,q= −SE·n^tot_E ^tf∂xyE−EnE∂xϕE [24]

with (dimensionless) thermodynamic factor ^tfE := yE_C

kBT

∂μˆE_C

∂yE_C

=1+2κE

yE

1−2yE_C

=^tfE(yE). [25]

where

μˆE_C:=μE_C− mE_C

mE_S

μE_S =k_BT(ln yE_C

−κEln yE_S

) [26]

aNote that the continuity ofϕacrossis an assumption.

is the thermodynamic driving force for diffusion,¹¹andDEthe chem- ical diffusion coefficient,tE_C the cation transfer number, andE the molar conductivity.

Note that we assumed ^v

REC v_E^R

S

= ^m_m^EC

ES andv^R_EC =κE·v_E^R_Swhich yields the representation26. Note further that the total number densityn^tot_E = nE_S+nE_C+nE_Ain the electrolyte writes as

n^tot_E =n^R_ES· 1 1+2(κE−1)yE

=n^tot_E (yE) [27]

which is determined from the incompressibility constraint8 v^R_ESnE_S+v^R_EAnE_A+v^R_ECn^R_EC =1 [28]

and the electrolyte concentrationnE_C in terms ofyEas nE_C =yE_C·n=n^R_ES yE_C

1+2(κE−1)yE

=nE_C(yE). [29]

If we consider a simple Nernst–Planck-flux relation for the cation and anion fluxes,^11,33respectively, i.e.

J_α=D^NP_α n_α

kBT (∇μα−m_α m0

∇μE_S+e0z_αn_α∇ϕE) α=EA,EC, [30]

with constant diffusion coefficientsD^NP_EAfor the anion andD^NP_EC for the cation, we obtain (in the electroneutral electrolyte)

DE= 2D_E^NP

C ·D^NP_E

A

D^NP_EA+D^NP_EC tE_C = D^NP_E

C

D^NP_EA+D^NP_EC [31]

E= e²₀ kBT (D^NP_E

A+D^NP_E

C) SE=e0(D^NP_E

C−D^NP_E

A) [32]

Note, however, for general Maxwell-Stefan type diffusion^29–32,34 or cross-diffusion coefficients^7,24,35 in the cation and anion fluxes lead to more complexrepresentations of the transport parameters (tE_C,SE,DE, E). In general, three of the transport parameters arein- dependent, andSE,tE_CandEare related to each other via

kBT e0

(2tC−1)= SE

E

. [33]

Further, (tE_C,SE,DE, E) depend in general non-linearly on the elec- trolyte concentration nE_C. However, it is sufficient for the sake of this work to assume constant values for the transport parameters (tE_C,SE,DE, E), together with relation33.

In the active particleAwe have two balance equations determin- ing the concentrationnA_C(x,t) (or mole fractionyA_C) and the electro- static potentialϕA(x,t) in the active particle, i.e.

∂nA_C

∂t = −∂xJA_C with JA_C = −DA·nAA^tf·∂xyA_C [34]

0= −∂xJA,q with JA,q= −σA∂xϕA [35]

and (dimensionless) thermodynamic factor ^tfA = yA

kBT

∂μA

∂yA

=1+ yA

1−yA

−2γAyA=^tfA(yA). [36]

Note that in principleσAcan be dependent on the amount of interca- lated ions, i.e.σA=σA(yA).

Reaction rate based on surface thermodynamics.—We want to investigate the non-equilibrium thermodynamic modeling of the in- tercalation reaction

Li⁺

E+e⁻

ALi

A+κE·S

E. [37]

Surface thermodynamics dictates that the reaction rateR

sof this process can in general be written as4,5,13,36,37

Rs =L

s·

e^α·kB1Tλs −e^{−(1−α)·kB1Tλs}

with λs =μ

s

A_C+κE·μ

s E_S−μ

s E_C−μ

s

A_e, [38]

Figure 2. Sketch of an active intercalation phaseAin contact with some electrolyteE. The electrode-electrolyte interfaceA,Ecovers the space charge layer ^SCLE of the electrolyte and^SCLA of the electrode as well as the actual electrode surface. Several processes occur simultaneously, i.e. the intercalation reaction, electrolyte diffusion and solid state diffusion as well es electrical conductivity.

withα∈[0,1]. Note that a non-negative functionL

s in38ensures a non-negative entropy productionr

sσ,Rdue to reactions on the surface, i.e.r

sσ,R=λ

s ·R

s >0.

The quantityλ

scan be considered as surface affinity of the Reaction 37. The surface reaction rateR

s vanishes when the affinity vanishes, which is the actually the thermodynamic equilibrium condition of37, i.e.λ

s =0⇔r_σ,R=0.

Since the electrochemical double layer is in equilibrium, we can pull backthe surface chemical potentialsμ

sαthrough the double layer to the respectivepoints(in an asymptotic sense) outside of the double layer, whereby we obtain for the surface affinity

λs =μA_C⁻

AE+κE·μE_S⁺

AE−μE_C⁺

AE+e0U_A^DL_,E−μAe⁻

AE. [39]

With the material models4and 11 we can rewrite the surface affinity as

λs =e0(U_AE^DL−E_A^T_,E)+kBT

fA(yA_C|AE)− fE(yE_C|AE)

[40]

with

E_A^T_,E:= 1 e0

(g^R_E

C+g^R_A

e−g^R_A

C−κEg^R_E

S) [41]

and

fE(yE_C) :=ln

yE_C

yˆE_S(yE_C)_κE

, [42]

fA(yA_C) :=ln ₁

ωAyA_C

1+^1−ω_ω ^A

A yA_C

−ωA·ln

1−yA_C

1+^1−ω_ω ^A

A yA_C

+γA·hA(yA_C) [43]

withhAaccording to12. Note again thatyA_C|AEdenotes the evaluation ofyA_C at the interfaceA,Eand that the surface affinity40is depen- dent on the chemical potential (or the mole fraction) evaluated at the interface.

Cell Voltage.—We consider the cell voltage in a half cell with metallic lithium as counter electrode, denoted byCand position at x=xEC(see Fig.2. The cell voltage in such a cell is

E =ϕ| x=0−ϕ|⁺AE

=:−U_A^bulk

+ϕ| ⁺AE−ϕ|⁻AE

=UAE^DL

+ϕ| ⁺AE−ϕ|⁻EC

=:U_E^bulk

+ϕ| ⁻EC−ϕ|⁺EC U_EC^DL

+ϕ|⁺x=xEC−ϕ|x=xC

=:U_C^bulk

, [44]

Figure 3. Reaction rate functiong(x)=e^α·x −e^{−(1−α)·x} and its inverseg⁻¹for various values ofα.

whereU_A^bulkis the potential drop in the bulk active particle due to the electron transport,U_A^DL_,Eis the potential drop across the double layer at the interface between the active particle and the electrolyte, andU_E^bulk the bulk potential drop due to cation electric current.

We assume that the counter electrodeCisideally polarizable,²⁸ whereby the reaction

Li⁺

C+κE·S

ELi⁺

E [45]

at the the interfaceE,Cpositioned atx = xECis in thermodynamic equilibrium andU_C^bulk = ϕ|⁻x=xEC −ϕ|x=xC = 0. The equilibrium condition of45entails

U_EC^DL=ϕ|⁻x=xEC−ϕ|⁺x=xEC= 1 e0

μC_C−μE_C⁻_EC+κEμE_S⁻_EC [46]

= 1 e0

(μC_C−g^R_A

C−κEg^R_E

S)−kBT e0

fE(yE_C

EC) [47]

whereμC_C =const. is the chemical potential of the metallic lithium.

For the surface affinity40we obtain the compact typeface λs =e0(E+U_A^bulk−U_E^bulk−EA,C)+kBT

fA−fE|AE+fE|EC

[48]

with

EA,C= 1 e0

(μC_C−g^R_A

C+g^R_Ae). [49]

and

fE|AE= fE(yE_C|AE) and fE|EC= fE(yE_C|EC). [50]

Current–Voltage relation.—For the single intercalation reaction we have the following expression⁴

i= −e0R

s +C_E^DL·dU_E^SCL

dt [51]

for the current densityiflowing out of the electrodeA, whereC_E^DLis the double layer capacity. Note that the reaction rate is

Rs =L

s ·g( 1 kBT λ

s) with g(x)=

e^α·x −e^{−(1−α)·x}

. [52]

Sinceg(x) is a strictly monotone function, we can introduce the in- verse of g, i.e.g⁻¹. Forα = ¹₂ we have g(x) = 2sinh₁

2x and g⁻¹(y) = 2g⁻¹₁

2x

. For valuesα = 0.5 the inverse functiong⁻¹ is only implicitly given, however, can easily be calculated numerically.

Fig.3display the functionsgand g⁻¹ for various values ofα. We callg(x) the reaction rate function andg⁻¹the inverse reaction rate function.

Note that in the Tafel approximationg(_k¹

BTλ

s) ≈ _kB¹_Tλ

s Eq. 51 yields^b

e0

kBT U_E^DL− 1 e0L

s

C_E^DL·dU_E^DL dt = e0

kBT E_A^T_,E− fA− fE

− 1 e0L

s

i [53]

The terme0L

s can be considered as theexchange current density.²⁸ Onsager coefficient of the intercalation reaction.—The Onsager coefficientL

s (or the exchange current densitye0L

s) of the surface re- action37could in principle be a function of the surface chemical po- tentials (or surface concentrations), i.e.L

s =L

s(μ

s A_C,μ

s

E_C,μE_S,μ

s A_e) or L

s =L

s(λ

s) or the surface affinity, i.e.L

s =L

s(λ

s), as long as the condition Ls > 0 is ensured.^4,8,26Note, however, that surface thermodynamics dictates the dependency ofL

s on the surface chemical potentialsμ

sα

and not the bulk chemical potentialsμα.

bNote again thatU_AE^DL=U_E^SCL−U_A^SCLand that the space charge layer dropUA^SCLis constant due to the material model^μA_s^e =const. whereby^dUE_dt^SCL =^dUdt^AEDL.

For a general relationL

s =L

s(μ

s A_C,μ

s

E_C,μE_S) we canpull backthe surface chemical potentialsμ

sαthrough the double layer to obtain Ls =L

s

μA_C(yA_C|AE),μE_C(yE_C|⁺AE)−e0U_E^SCL,μE_S(yE_S|⁺AE) . [54]

Note that this necessarily restricts the functional dependency ofL

s on the mole fractionsy_α|^A,Eat the interfaceA,E.

Consider, for example a modelL

s = L

sE(yE_C|⁺AE), where the ex- change current density is dependent on the electrolyte concentration at the interface. This would be, however, thermodynamically incon- sistent since the general functional dependency of54requires for the electrolyte concentration at the interface

Ls =L

sE(μE_C(yE_C|⁺AE)−e0U_E^SCL)=Lˆ

sE(yE_C|⁺AE·e⁻

e0 kBTU_E^SCL

). [55]

Another commonly used model is a functional dependency ofL

s

on the concentrationyA_C|AEof intercalated ions at the interface, i.e.

L

s = L

sA(yA_C|AE). Since the space charge layer in the active particle U_A^SCLis essentially constant (becauseμ

s

A_e is constant), we can indeed write

Ls =L

sA

μA_C(yA_C|AE)

=Lˆ

sA(yA_C|AE). [56]

We discuss this aspect as well as various models for Ls(μ

s A_C,μ

s

E_C,μE_S,μ

s

A_e) in Discussion of the exchange current density section. Meanwhile we assumeL

s = const. and proceed the following derivation and the discussion based on this assumption since it turns out to be very reasonable.

Discussion of the model parameters.—At this stage, it is illustra- tive to discuss the explicit value of the parameters.

• For the electrode geometry we consider forA,Ea planar surface of areaAand a thicknessdA = 10 [μm] which yieldsVA = A·dA

and xAE = 10 [μm]. The electrolyte is considered with a thickness ofdE=50 [μm]. This corresponds to the cell dimensions of the cell MX-6 in Ref.2.

• Throughout this work we consider DMC as solvent withn^R_ES = 11.91_mol

L

and assume for the solvation numberκE= 4. The refer- ence electrolyte concentration isn^R_E=1_mol

L

and average amount of electrolyte isnEand a parameter of the model.

• Average concentrations (or mole fractions) are abbreviated as y_α= 1

VE

E

y_αdV α=E_C,E_A,E_S [57]

for the electrolyte species and y_AC= 1

VA

A

yA_CdV [58]

for the amount of intercalated ions in the active phase.

• For the active particle phase we consider Li(Ni1/3Mn1/3Co1/3)O2

(NMC) whereby q^V_A^,NMC=1294

mA h cm³

and q^M_A^,NMC=318

mA h g⁻¹ [59]

which is simply computed from the density and stoichiometry of the bulk material.³⁸As parameters for the chemical potentialμA_Cwe con- sider an occupation number ofωA =10 and a Redlich–Kister inter- action energy ofγA=13.²⁴

• The differential capacityC_E^DLhas a prescribed value (actuallyC_E^DL is a function ofU_E^SCL, but we proceed here with a constant approxima- tion for the sake of simplicity⁹of about

C_E^DL=100 μF

cm²

[60]

• The electrode capacityQis Q=

A

q^V_A ·yA_CdV =Q^V_A·yA_C with Q^V_A :=VA·q^V_A [61]

This yields the non-dimensionalcapacity Q

Q^V_A =y_AC ∈(0,1) [62]

which is sometimes also calledstatus of charge(SOC) ordepth of discharge(DOD).

Note that duringdischargeof a complete battery the cathode is actually filled up with lithium. In a half cell with metallic lithium as counter electrode,dischargethus actually meansfilling upthe interca- lation electrode, here the NMC cathode material. HenceQ/Q^V_A →0 corresponds to a fully charged cathode (i.e. no lithium in the intercala- tion compound,y_AC →0) whileQ/Q^VA →1 corresponds to a fully dis- charged cathode (i.e. the intercalation compound is completely filled with lithium,y_AC→1).

• From the charge balance35of the active particle we can deduce Q=Q⁰+

t 0

I(t)dt with Q⁰=

A

q^V_A·yA_C(x,t=0)dV [63]

whereI is the current flowing into the intercalation electrode during dischargeandQ(t = 0) the initial charge state. For a galvanostatic dischargeI >0 we obtain thus

Q=Q⁰+I·t. [64]

• The C-RateCh[1] defines (implicitly) the current at which after h-hours the intercalation cathode is completely filled during galvano- static discharge.C1 is thus the rate at which the battery is charged within one hour and commonly abbreviated just as C-rateC, i.e.

IC= Q^V_A

1 [h]=AdA·q^V_A

1 [h] . [65]

We can hence express the currentIin multiples of the C-rate, i.e.

I =Ch·IC [66]

which yields

Q=Q⁰+I·t=Q⁰+Ch·IC·t=Q⁰+Ch· Q^V_A

1 [h]·t=Q^V_A(y⁰_A

C+Ch

t [h]) [67]

The only parameter for the current densityi=I/Ais thusCh.

• For the timetwe consider the interval of one discharge cycle, i.e.t∈[0,tend] with

tend= 1 [h]

C_h [68]

We can thus introduce the non-dimensional time τ:=Ch

t

3600 [s] ∈[0,1] [69]

whereby the capacity rewrites as Q/Q^V_A =(y⁰_A

C+τ). [70]

• For the current densityiat the planar electrode we have thus i= I

A= Ch·IC

A =i^C_A·C_h with i^C_A := dA·q^V_A

1 [h] . [71]

Discussion of the scaling.—Consider the non-dimensional voltage U= e₀

k_BT U_E^SCL [72]

and abbreviate

H= e0

kBT EA,R,E− fA+ fE [73]

which yields

U−c₁·Ch

L ·dU

dτ =H(1 −τ)−Ch

L [74]

with

c1:= 1

d·q^V_AC_E^DLkBT e0

[75]

The parametersdA=0.01 [cm] andq^V_A =1294

mA h cm⁻³ yield dA·q^V_A =0.01 [cm]·1294

mA h cm⁻³

· 1

[h]=12.94 mA h

cm²

[76]

and

C_E^DLkBT e₀ =100

μF cm²

·0.0257 [V]=2.568 μC

cm²

[77]

whereby

c1=5.51·10⁻⁸. [78]

The double layer contribution in Eq.51is thus almost negligible whereby51reduces to

i= −e0L

sg 1 kBT λ

s

. [79]

We consider for the exchange current density the rescaling e0L

s =L·i^C_A =LdA·q^V_A

1 [h] . [80]

This is the crucial decomposition throughout this work andLthe pa- rameter of the surface reaction rateR

s.

For the current densityi=i^C_A·Chand the inverse functiong⁻¹we obtain thus with Eq.48for the surface affinityλ

sthe general expression E =EA,C−kBT

e0

fA− fE|AE+fE|EC

+kBT e0

g⁻¹

−Ch

L

−U_A^bulk+U_E^bulk [81]

for the cell voltageE.

Discussion

Throughout the manuscript, we assume that the initial state is com- pletely uncharged, i.e.Q⁰ =0 andy⁰_A

C = 0. If not stated otherwise, we abbreviate

yA_C =yA and yE_C=yE [82]

as well as the respective densitiesnA_C =nA,nE_C =nE, fluxesJA_C = JA,JE_C =JE, and chemical potentialμA_C =μAin the following.

We seek to discuss the general relation81of the cell voltageEas function of the capacity

Q

Q^V_A =y_A∈(0,1) [83]

during discharge of an intercalation electrode. Note that necessarily Ch >0 (discharge) andL > 0 (Onsager constraint of38), whereby g⁻¹

−¹₂^C_L^h

< 0, which entails that any current decreases the cell voltageEduring discharge.

We will discuss consecutively the following hierarchy of approxi- mations:

BV 0: infinite slow discharge - the open circuit potential

BV 1: infinite fast diffusion and conductivity in the active particle and the electrolyte

BV 2: finite conductivity in the active particle, infinite diffusion in the active particle, infinite fast diffusion and conductivity the electrolyte

BV 3: finite conductivity and diffusion in the active particle, infinite fast diffusion and conductivity the electrolyte