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Electrochimica Acta 56 (2011) 4369–4377

Contents lists available at ScienceDirect

Electrochimica Acta

journa l homepage: www.e lsev ier .com/ locate /e lec tac ta

athematical modeling of porous battery electrodes—Revisit of Newman’s model

ei Laia,∗, Francesco Ciuccib

Department of Chemical Engineering and Materials Science, Michigan State University, East Lansing, MI 48824, USAHeidelberg Graduate School of Mathematical and Computational Methods for the Sciences, University of Heidelberg, INF 368, D – 69120 Heidelberg, Germany

r t i c l e i n f o

rticle history:eceived 13 August 2010eceived in revised form5 November 2010ccepted 14 January 2011

a b s t r a c t

The most established mathematical description of porous battery electrodes is Newman’s model,which accounts for the behavior of both solid and liquid phases via concentrated solution theory andporous electrode theory. In the present work, formal volume averaging of reformulated generalizedPoisson–Nernst–Planck (PNP) equations in the form of concentration and electrochemical potential, orin the form of equivalent circuit on a representative porous microstructure, lead to “upscaled” equa-

vailable online 31 January 2011

eywords:ewman’s modelolume averagingorous electrode

tions similar to those in Newman’s model. The similarities and differences of the results from the twoapproaches are discussed throughout the paper.

© 2011 Elsevier Ltd. All rights reserved.

oisson–Nernst–Planckithium ion batteries

. Introduction

Almost all commercial lithium batteries utilize a porous com-osite electrode design, with the exception of solid state thin filmatteries. In the composite electrode, a porous solid network is

nfiltrated with the liquid electrolyte and usually covered with con-ucting additive as the current collector. A schematic presentationf the porous electrode with interpenetrating electrode particles,iquid electrolyte and current collector is shown in Fig. 1(a). The rel-vant charge carriers are the cation and anion (e.g. Li+ and X− for LiXalt) in the liquid electrolyte, lithium cation, Li+, and electron, e−,n the solid electrode, and e− in the current collector. At the triplehase junction where the liquid electrolyte (l), solid electrode (s)nd current collector (CC) meet together, Li+ in the electrolyte and− in the current collector insert into the solid electrode and followglobal electrochemical reaction of this kind

i+(l) + e−(CC) → Li+(s) + e−(s) (1)

his intercalation process is shown schematically in Fig. 1(b). Inhe solid electrode, Li+ occupies the interstitial site while e− goesnto a band or sits on the transitional metal as a polaron. These two

harge carriers can be considered as being in equilibrium with airtual neutral Li species as

i+(s) + e−(s) ↔ Li(s) (2)

∗ Corresponding author.E-mail address: laiwei@msu.edu (W. Lai).

013-4686/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.oi:10.1016/j.electacta.2011.01.012

The combination of Eqs. (1) and (2) leads to the usual perceptionof the charge transfer reaction for Li insertion

Li+(l) + e−(CC) → Li(s) (3)

It is obvious that all these electrochemical processes (transportof charge carriers in the bulk and across the interface, etc.) andmicrostructural features (distribution of triple phase junction, par-ticle size, etc.) have to be taken into account to completely describethe system.

The most established mathematical model of batteries withporous electrodes has been developed by Newman et al. [1–5].It includes both concentrated solution theory and porous elec-trode theory. The concentrated solution theory is used to accountfor the transport of Li+ and X− in the liquid electrolyte as a non-ideal solution, based on equations similar to Stefan–Maxwell ones[5]. The porous electrode theory is used to account for the porousmicrostructure. Although this model has since been widely usedby various researchers [6–9], either in original or modified form,a complete derivation of the equations in the model seems to bemissing and discussions are scattered in several literature articles[1–4] and an electrochemistry book [5].

In the field of electrochemistry, multicomponent transportof charge carriers is generally described by the Poisson–Nernst–

Planck (PNP) equation set, which is composed of Nernst–Planckequation, continuity equation, and Poisson equation. The PNPequations have been widely used in liquid electrochemistry, semi-conductors, ionic channels in biological membranes, etc. [10–13].Although batteries are also electrochemical devices, surprisingly,

4370 W. Lai, F. Ciucci / Electrochimica

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tt

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2e

c

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−

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i

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ig. 1. Schematic presentation of (a) porous electrode with interpenetrating elec-rode particles, liquid electrolyte and current collector, and (b) electrochemicalrocesses at the triple phase junction of liquid electrolyte, solid particle and currentollector.

here are few applications of PNP equation to the study of chargeransport in batteries.

In the present paper, volume averaging of generalizedoisson–Nernst–Planck equations on a de Levie straight pore model14] is performed rigorously to present a set of equations, similaro those in Newman’s model, that can be used to model porousattery electrodes. This work starts with a review of generalizedoisson–Nernst–Planck equations, Section 2; then these equationsre separately applied to the relevant charge carriers in the liquidlectrolyte, solid electrode particle, and current collector employedn porous battery electrodes, Section 3; afterwards, a review of vol-me averaging as the upscale method on a de Levie pore model

s presented, Section 4; subsequently the volume averaging ispplied to the microscopic equations in Section 3 to obtain a setf micro-macroscopic coupled equations, Section 5; finally, it isemonstrated that a different form of PNP equations, equivalentircuit approach, can be upscaled to yield the same results, Sec-ion 6. Comparison with Newman’s model on the microscopic and

acroscopic level is made throughout the present work.

. Short review of generalized Poisson–Nernst–Planckquations

The generalized Poisson–Nernst–Planck equations in electro-hemical systems (ignoring the convection term) are [15–18]

i = −�i∇�∗i = −�i∇(�∗

i + �) (4)

∂ci

∂t+ ∇ · Ji

zie= 0 (5)

εrε0∂2�(x, t)

∂x2=

∑i

zieci(x, t) (6)

q. (4) is the generalized Nernst–Planck transport equation thatelates the driving force (reduced electrochemical potential) to theurrent density, based on linear irreversible thermodynamics. Thelectrochemical potential is the sum of chemical potential �i andlectrical energy zie�, in which zi is the charge number, e is thelementary electron charge, and � is the electrical potential.

˜ i = �i + zie� (7)

The conductivity �i is the product of concentration ci and mobil-ty bi as

i = (zie)2cibi (8)

It is to be noted that the current density Ji, reduced electro-hemical potential �∗

i, and reduced chemical potential �∗

iare

sed instead of the conventional mass flux Ni and electrochemical

Acta 56 (2011) 4369–4377

potential �i, and chemical potential �i, respectively. The relationis given by

J i = zieNi, �i = zie�∗i , �i = zie�∗

i (9)

Eqs. (5) and (6) are the continuity and Poisson equations, respec-tively.

The above PNP equations are coupled differential equations withcurrent densities Ji, concentration ci, reduced chemical potential�∗

i, and electrical potential �. While the concentration ci is corre-

lated with electrical potential � by the Poisson equation, Eq. (6),the correlation of concentration ci with reduced chemical potential�∗

ican be expressed as either the volumetric chemical capacitance

Cchemi

or thermodynamic factor � i [15–19]. The volumetric chemicalcapacitance represents the change of volumetric electrical chargeqi = zieci upon the change of chemical potential and is defined as

Cchemi = ∂qi

∂�∗i

= zie∂ci

∂�∗i

(10)

This is analogous to the definition of dielectric capacitance asthe change of electrical charge qi upon change of electrical poten-tial � as ∂qi/∂�. It is to be emphasized that chemical capacitanceis equivalent to the thermodynamic factor or activity coefficient tobe discussed shortly and it is a term that appears recently in theliteratures. The use of chemical capacitance is especially relevantto batteries since they store electrical energy in the form of chem-icals. The use of chemical capacitance allows easy mapping of Eqs.(4)–(6), applicable to systems beyond batteries, to an equivalentcircuit [15–18,20,21].

With chemical capacitance (10), the generalized Nernst–Planckequation (4) becomes

J i = −zieDi∇ci − �i∇� (11)

with the definition of chemical diffusivity Di as

Di = �i

Cchemi

(12)

In the solution theory, the dependence of chemical potential onconcentration is explicitly written as a form with activity coefficientfi(ci)

�i = �0i + kBT ln

(fi

ci

c0i

)(13)

where �0i

is the standard chemical potential, c0i

is the referenceconcentration, kB is the Boltzmann constant and T is the absolutetemperature. Then the chemical diffusivity in Eq. (12) can also bewritten as

Di = �iDi (14)

where � i is the thermodynamic factor

�i = 1 + ∂ ln fi∂ ln ci

(15)

and Di is the self-diffusivity defined as, i.e., the Nernst–Einsteinrelation

�i = Dici(zie)2

kBT(16)

Then Eq. (11) becomes

J i = −zieDi

(1 + ∂ ln fi

)∇ci − �i∇� (17)

i

In the case of an ideal solution, the thermodynamic factor is 1and Eq. (17) becomes

J i = −zieDi∇ci − �i∇� (18)

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W. Lai, F. Ciucci / Electroch

his ideal solution version of Eq. (11) or (17) has been calledhe Nernst–Planck, diffusion–drift, diffusion–migration in theiteratures and the whole equation set with continuity and Pois-on equations, Eqs. (18), (5) and (6), are called PNP equations10–13,15–17,20–25]. On the other hand, generalized PNP equa-ions, Eq. (4), (11) or (17), Eqs. (5) and (6), take the non-idealolution thermodynamics into account [18,19].

Either the non-generalized, Eqs. (18), (5) and (6), or generalized,q. (11) or (17), Eqs. (5) and (6), PNP equations employ concentra-ion ci and electrical potential � as variables of interest. It is possibleo reformulate the PNP equations in two alternative ways by:

Choosing concentration ci, electrochemical potential �∗i, and

electrical potential � as variables of interest, i.e., previously listedEqs. (4)–(6).Choosing reduced chemical potential �∗

i, reduced electrochemi-

cal potential �∗i, and electrical potential � as variables of interest.

This involves removing concentration ci from Eq. (5) by the useof Eq. (10), and removing concentration ci from Eq. (6) by the useof Eq. (5). The details can be found in Refs. [15–18,20,21].

The first choice is the focus of this work and it will be shownhat it leads to equations similar to those in Newman’s model. Thedvantage of the second choice is that all the variables of interestave units of voltage and equations can be mapped to an equivalentircuit so that physical meanings of different charge transport cane viewed pictorially [15–18,20,21]. This choice will be discussed

n Section 6.For the present study of lithium (-ion) batteries, the electroneu-

rality condition is used instead of Poisson’s equation Eq. (6) inrder to simplify the mathematical treatment. This means the rightand side of Eq. (6) goes to zero. This approximation is valid inegions that are much thicker than the Debye length but it will breakown near interfaces. For a two charge carrier (+ and −) monovalentystem, c = c+ = c−. The PNP equations, e.g. Eqs. (4) and (5), become

+ = −�+∇�∗+ = −eD+∇c − �+∇� (19)

− = −�−∇�∗− = eD−∇c − �−∇� (20)

∂c

∂t+ ∇ · J+

e= 0 (21)

∂c

∂t+ ∇ · J−

−e= 0 (22)

The individual conductivities �+, �− and individual chemicaliffusivities D+, D− can be related to total conductivity � andotal/ambipolar chemical diffusivity D through transference num-er of cation t+

� = �+ + �−,

t+ = �+�

,

t− = �−�

= 1 − t+

(23)

˜ = t+D− + (1 − t+)D+ (24)

These are the well-known definitions in the literature [5,26].oth total conductivity � and chemical diffusivity D are kineticroperties, they can be related to thermodynamic properties suchs chemical capacitance/thermodynamic factor/activity coefficientiscussed before, i.e., Eqs. (10), (14) and (15). The total chemicalapacitance Cchem± , thermodynamic factor �±, activity f± can beefined as

chem± = 1

1/Cchem+ + 1/Cchem−= t+(1 − t+)

�

D(25)

± = �+ + �−2

= 1 + ∂ ln f±∂ ln c

= ce2

2kBT

1t+(1 − t+)

D

�(26)

Acta 56 (2011) 4369–4377 4371

f± =√

f+f− (27)

Under the assumption of electroneutrality, the left hand side ofEq. (6) is dropped and the electrical potential � can be eliminatedfrom (19) and (20) to get

J+ = −t+�∇�∗+ (28)

J− = 1t+

eD∇c − (1 − t+)�∇�∗+ (29)

In order to relate individual current densities to the total currentdensity, Eqs. (28) and (29) can be rewritten as

J+ = −eD∇c + t+(J+ + J−) (30)

J− = eD∇c + (1 − t+)(J+ + J−) (31)

J+ + J− = −t+�∇�∗+ + 1

t+eD∇c (32)

It should be noted that only two out of these three equations areindependent. If the activity coefficient f±, Eqs. (26) and (27), is usedinstead of diffusivity D, then Eq. (32) becomes

J+ + J− = 2kBT�

e(1 − t+)

(1 + ∂ ln f±

∂ ln c

)∇ ln c − t+�∇�∗

+ (33)

Both alternatives to Eqs. (19) and (20), i.e. Eqs. (28) and (29) orEqs. (30)–(32), employ concentration c and reduced electrochem-ical potential of cation �∗+ as variables of interest. Combined withcontinuity Eqs. (21) and (22), they will be applied to the individualcomponents in the porous composite electrodes as shown in Fig. 1.

Comparison with Newman’s modelIn Newman’s model, starting equations equivalent to general-

ized Nernst–Planck equations in the present work are used, alsowith electrochemical potential as the driving force [5]. However,electrochemical potentials of charge carriers were removed torelate individual current densities to the total current density,Eqs. (30) and (31) [1–5]. Later, a virtual reference electrode wasintroduced to add a virtual potential back to the equations. In thepresent work, the electrochemical potential of charge carrier is keptthroughout, e.g. �∗+ in Eqs. (28)–(32). This allows easy tracking ofcharge transport of lithium ion in both the liquid electrolyte andsolid electrode and across the liquid–solid interface, since electro-chemical potential of lithium ion is consistently used as the drivingforce. The similar argument applies to electrons in both the solidelectrode and current collector.

3. Microscopic equations in porous composite electrodes

3.1. Liquid electrolyte

In the liquid electrolyte, as discussed in the introduction, thetwo relevant charge carriers are Li+ and X−. Eqs. (28) and (29) orEqs. (30)–(32) can be combined with continuity equations Eqs. (21)and (22) to form the microscopic equations in the liquid electrolyte.

Comparison with Newman’s modelEq. (33), if �∗+ is relabeled as ˚2, and (31) are basically the same

equations used in Newman’s model [1–5]. ˚2 was called potentialin the solution in Newman’s model and was the potential of a virtualreference electrode to another fixed reference electrode [1–5]. Inthe present work, �∗+ is used instead. This is the parameter of phys-ical importance since it is the driving force that determines how theLi ion is inserted or extracted from the liquid–solid interface.

3.2. Solid electrode particle

For the solid electrode, the cation and anion in the equationsrefer to Li+ and e−. Eqs. (30) and (31), and (21) and (22) will be

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372 W. Lai, F. Ciucci / Electroch

sed. If one supposes that the solid electrode is composed by aollection of independent particles, then the total inflow of currentensity is zero, due to the charge neutrality requirement. Thiseans that the ionic current and electronic current cancel each

ther throughout the particle:

+ + J− = 0 (34)

This suggests that the two continuity Eqs. (21) and (22) arequivalent. Then Eq. (30) becomes

+ = −eD∇c (35)

With the continuity Eq. (21), this becomes

∂c

∂t+ ∇ · (−D∇c) = 0 (36)

nd this has the form of classical diffusion equation.

Comparison with Newman’s modelEq. (36) appears to be the same equation that was used in New-

an’s model [1–5]. However, it was not clear how the diffusivityhould be related to other thermodynamic and kinetic properties ofhe material in this model. Actually the diffusion coefficient D wasften treated as a constant even for phase transformation materialuch as graphite, LiMn2O4, and LiCoO2 [9,27,28], which is an overimplification. In the present work, as usual, the electronic trans-erence number t− can be taken to close to 1 since the mobility oflectron is larger than that of lithium ion. Thus D is determined byithium ionic diffusivity D+ from Eq. (24). The diffusion coefficient˜ + is the ratio of ionic conductivity �+ to the chemical capacitance

chem+ in the solid, as shown in Eq. (12). It should be noted that:

The conductivity �+ is determined by the concentration c andmobility b+ of the Li ion in the lattice. For a vacancy transportmechanism, the mobility b+ depends on whether the potentialjump site is empty, so it should scale with (1 − c/c0) from a meanfield approach, where c0 is the maximum number of sites. For adi-vacancy mechanism [29], a term (1 − c/c0)2 can be used.The chemical capacitance Cchem+ is proportional to the inverseof thermodynamic factor and is the derivative of concentrationc with respect to the reduced chemical potential �∗+. Thischemical potential can come from atomistic modeling [30,31]or from a mean field approach such as regular solution model[18,32]. For the phase transformation, a Cahn–Hilliard interfacialenergy penalty term, the diffuse interface phase field method[33–35], can be employed to account for the chemical potentialin the phase coexistence region. It is to be noted an alternativeapproach to phase field in dealing with phase transformation isthe shrinking core model [36] but the sharp interface model isnumerically more challenging since it involves abrupt change ofvalues [37,38].

It can be seen that, in the present work, the diffusion coefficient˜ arises “internally” from the thermodynamics and linear irre-ersible thermodynamics. This is a thermodynamically consistentpproach to treat diffusion in solids.

.3. Current collector

Generally, the current collector is purely electronic conductoro Eqs. (20) and (22) will be used. The electron concentration inhe current collector such as carbon can be considered high andonstant so the continuity equation becomes

· J− = ∇ · (−�−∇�∗−) = 0 (37)

Comparison with Newman’s modelThis is basically the same equation as that in Newman’s model,

ith �1 used instead of �∗−. �1 was called potential in the solid.

Acta 56 (2011) 4369–4377

Again, it is believed that use of electrochemical potential �∗− in thepresent work is more physically intuitive and consistent with thenotion of driving force.

3.4. Triple phase junction

The interfacial charge transfer reaction at the triple phase junc-tion is in the form of Eq. (1) or (3) for simplicity. The drivingforce is the difference in the electrochemical chemical poten-tial �+(l) + �−(CC) − �Li(s) or e

[�∗+(l) − �∗−(CC) − �∗

Li(s)]

in thereduced form. According to Eq. (2), the chemical potential �Li(s)of virtual Li satisfies

�Li(s) = e[�∗

+(s) − �∗−(s)

]= e�∗

Li(s) (38)

Since the driving force is usually large at the interface, linearirreversible thermodynamics will not be applied. If the jump rate isassumed to depend on the exponential of the driving force instead,the transport from the liquid to the solid can be written as

J l−s = Jl−s0 exp

[�∗+(l) − �∗−(CC) − �∗

Li(s)

kBT/e

]�n (39)

where Jl−s0 is the pre-exponential term and �n is unity vector nor-

mal to the solid/liquid interface and directed from the liquid to thesolid. When the driving force is small, Eq. (39) is reduced to Eq.(4) and the pre-exponential term Jl−s

0 goes into the conductivity,so Eq. (39) can be considered as a nonlinear version of irreversiblethermodynamics. Jl−s

0 depends on how many sites are available toinitiate the jump in the liquid and how many sites are empty in thesolid to accommodate the jump

Jl−s0 = c+(l)

[1 − c+(s)

c0+(s)

]Jl−s00 (40)

where c0+(s) indicates the maximum number of sites available. Sim-ilarly, the jump from the solid to liquid can be written as

Js−l = −Js−l0 exp

[− �∗+(l) − �∗−(CC) − �∗(s)

kBT/e

]�n (41)

with

Js−l0 = c+(s)Js−l

00 (42)

Both Jl−s00 and Js−l

00 are concentration independent constants. Thetotal charge transfer current density JCT then becomes

JCT = J l−s + Js−l ={

Jl−s0 exp

[�∗+(l) − �∗−(CC) − �∗

Li(s)

kBT/e

]− Js−l

0

exp

[− �∗+(l) − �∗−(CC) − �∗

Li(s)

kBT/e

]}�n (43)

As mentioned above, the reduced electrochemical potential ofLi+ in the liquid �∗+(l) and the reduced electrochemical potential ofelectron in the current collector �∗−(CC) can be relabeled as �2 and�1 at the interface, respectively. The question is how to deal with�Li(s) or �∗+(s) − �∗−(s) which can be written as

�∗+(s) − �∗

−(s) = �∗+(s) − �∗

−(s) = �∗Li(s) (44)

In other words, the contribution from the electrical potential iscancelled. It is a pure chemical term that depends on the surfaceconcentration c+(s). The effect of equilibrium or non-equilibrium(zero or non-zero current) condition only affects c+(s) but not the

form of the chemical potential. Then an equilibrium condition witha concentration of c+(s) can be used to obtain �Li(s). Under thisequilibrium of Eq. (3), then

�∗,eq+ (l) − �∗,eq

− (CC) − �∗,eqLi (s) = 0 (45)

imica Acta 56 (2011) 4369–4377 4373

op

�

tdc

�

E

�

olt

wt

ˇ

√

wafthtTsofitttsd

(a) (b)

Fig. 2. (a) A representative volume element (RVE) of a porous composite electrode.The liquid electrolyte is surrounded by the solid while the current collector at theliquid–solid interface is not shown for simplicity. The geometry is taken to be pseudo

W. Lai, F. Ciucci / Electroch

If this interface is put against a Li reference electrode under thepen circuit condition, �∗,eq

Li (s) can be converted an experimentalarameter that can be measured. At the reference electrode

˜ ∗,eq+ (Li) − �∗,eq

− (Li) − �∗,eqLi (Li) = 0 (46)

Under the open circuit condition, there is no driving force forhe electrochemical potential of Li ion, �∗,eq

+ (l) − �∗,eq+ (Li) = 0. The

ifference of electrochemical potential of electron gives the openircuit voltage Voc(s)

˜ ∗,eq− (CC) − �∗,eq

− (Li) = Voc(s) (47)

The chemical potential of bulk Li reference electrode is 0. Thenqs. (45)–(47) become∗,eqLi (s) = −Voc(s) (48)

This is basically the Nernst equation that relates the differencef chemical potential to the voltage. As discussed above, the equi-ibrium �∗,eq

Li (s) and non-equilibrium values �∗Li(s) are the same for

he same surface solid concentration, so Eq. (43) becomes

JCT ={

Jl−s0 exp

[�∗+(l) − �∗−(CC) + Voc(s)

kBT/e

]− Js−l

0

exp

[− �∗+(l) − �∗−(CC) + Voc(s)

kBT/e

]}�n (49)

Alternatively, Eq. (49) can be written as

JCT =√

Jl−s0 Js−l

0

{exp

[ˇ

�∗+(l) − �∗−(CC) + Voc(s)kBT/e

]−

exp

[−ˇ

�∗+(l) − �∗−(CC) + Voc(s)kBT/e

]}�n (50)

here ˇ is not a constant and it depends on the current and poten-ial terms

= 1 + 12

kBT/e

�∗+(l) − �∗−(CC) + Voc(s)ln

Jl−s0

Js−l0

(51)

The pre-exponential term can be written as

Jl−s0 Js−l

0 = J00

√c+(l)c+(s)[c0+(s) − c+(s)] (52)

Comparison with Newman’s modelIn Newman’s model, the interfacial rate is described as

JCT = J0c+(s)˛c c+(l)˛a [c0+(s)−c+(s)]

˛a{

exp[

˛a˚1 − ˚2 − Voc(s)

kBT/e

]

− exp[−˛c

˚1 − ˚2 − Voc(s)kBT/e

]}�n (53)

hich is based on the classical Butler–Volmer equation [26] and ˛a

nd ˛c are the symmetry factors. J0 is a constant. If the symmetryactor ˛a and ˛c are taken to be 1/2, the pre-exponential term hashe identical form as Eq. (52). The overpotential term in Eq. (50)as a similar form as that in Newman’s model although the poten-ial terms have slightly different meanings in the present paper.he symmetry factor in the present work is not a constant. It washown that the symmetry factor has to be allowed to change inrder to generate a good fit with the experimental data [9]. The dif-erence in the present approach is that “electrochemical kinetics”nstead of “chemical kinetics” is used and electrochemical poten-

ial instead of electrical potential is the driving force, which leadso different meanings of symmetry factor ˇ and pre-exponentialerm. However, Eqs. (50)–(52) have almost the same form as theymmetrized form of transport equation under large gradient ofriving force [39].

one-dimensional with the dashed line as the z trajectory while x is the directionalong the electrode thickness. The surface between the liquid and the solid is S andthe outward normal unit vector is �n. (b) Equivalent volume element with straightpores which is also known as de Levie model.

All of these microscopic equations, corresponding to the liquidelectrolyte, solid electrode, current collector, and triple phase junc-tion, can be used to model systems with any kind of complicatedmicrostructures if all the microstructural parameters are known[40,41].

4. Short review of volume averaging and de Levie poremodel

For the complex microstructure, the application of the abovemicroscopic equations is expected to be computationally intensive.Generally, upscaling/homogenization methods such as volumeaveraging [2,42–44] can be used to convert the microscopic equa-tions to the macroscopic scale. The idea of volume averaging is tointegrate the differential equations over a representative volumeelement (RVE). The RVE is small enough to include all the micro-scopic phenomena (solid, liquid and the interfacial processes, etc.)and big enough to represent the macroscopic properties. If thefluctuation of a parameter value in this RVE is small, the volumeaveraged value can be used in order to describe its mesoscopicbehavior. In the present work, a RVE is shown schematically inFig. 2(a) with the liquid surrounded by the solid. The current col-lector (not shown for simplicity) is taken to be at the liquid–solidinterface. The geometry is taken to be pseudo one dimensional withthe dashed line as the z trajectory while x is the direction along theelectrode thickness. The surface between the liquid and the solidis S and the outward normal unit vector is �n. The relation betweenthe z and x directions is the tortuosity

= dz

dx(54)

For the following discussion, the RVE in Fig. 2(b) is used whichis the stretched version of Fig. 2(a) and is a widely used modelfor porous microstructure, known as de Levie model [2,14]. Thecross section area of the RVE and liquid is taken to be A0 and Al,respectively. Similarly, the volume of RVE and liquid is taken to beV0 and Vl, respectively.

The definition of volume averaging of parameter a over the vol-ume V0 is given by

〈a〉 = 1V0

∫V

adV (55)

0

If a only has the definition in the liquid volume Vl

〈a〉 = 1V0

∫Vl

adV = εl〈a〉l (56)

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ε

wa

a

d⟨

⟨⟨

wtvdsid

∇

⟨

ai

5

5

cHnt

〈⟨

ws⟨⟨w

−

jflflae

374 W. Lai, F. Ciucci / Electroch

ith

l = Vl

V0, 〈a〉l = 1

Vl

∫Vl

cdV (57)

here εl is the liquid volume fraction and 〈a〉l is the liquid phaseverage of a. The fluctuation of a is

ˆ = a − 〈a〉l (58)

The averaging theorem for the temporal derivative, gradient andivergence [44] is

∂a

∂t

⟩= ∂ 〈a〉

∂t− 1

V0

∫S

av · ndS (59)

∇a⟩

= ∇ 〈a〉 + 1V0

∫S

andS (60)

∇ · a⟩

= ∇ · 〈a〉 + 1V0

∫S

a · ndS (61)

here v is the velocity of interface. For Eq. (59), it is assumed thathere is no volume change so v becomes zero and this leads to theanishing of the integral term. For Eq. (60), due to the pseudo-one-imensional symmetry along the directional vector of z, a has theame values at the cross sectional area of z direction. Then thentegral in Eq. (60) also vanishes. The integral in Eq. (61) will beiscussed later. Also under this pseudo-1D approximation,

〈a〉 = ∇ · 〈a〉 = ∂ 〈a〉∂z

(62)

The volume average of the product of two quantities is

a · b⟩

= εl

⟨a · b

⟩l = εl〈a〉l⟨

b⟩l + εl

⟨a · b

⟩(63)

If in the latter the order of magnitude of the fluctuation isssumed to be smaller than the average, then the cross term

⟨a · b

⟩s smaller than the average.

. Macroscopic equations in porous composite electrodes

.1. Liquid electrolyte

As discussed in the microscopic equations, Section 3.1, the twoouples (28) and (29), and (21) and (22) are the relevant equations.owever, for averaging purposes, Eqs. (30) and (31) can be conve-iently used lieu of Eqs. (28) and (29). First, for the concentration,he above averaging considerations lead to

c〉 = ε〈c〉l (64)

∂c

∂t

⟩= ε

∂〈c〉l

∂t(65)

here 〈c〉l is the liquid phase averaged concentration. For the flux,imilarly

∇ · J+⟩

= ∇ ·⟨

J+⟩

− Sajloc (66)

∇ · J−⟩

= ∇ ·⟨

J−⟩

(67)

ith

Sajloc = 1V0

∫S

J+ · �ndS (68)

loc is the average value of J+ or JCT over the interface (pore wallux) and Sa is the volumetric interface area [2–4]. For the cationux, there is exchange between the liquid and solid at the interfacend this gives the extra term in Eq. (66). On the other hand, thexchange for the anion is zero and this gives Eq. (67). Using the

Acta 56 (2011) 4369–4377

above averaged quantity, the volume averaging of Eqs. (21) and(22) along z direction gives

ε∂〈c〉l

∂t+ 1

e

∂⟨

J+⟩

∂z= Sajloc

e(69)

ε∂〈c〉l

∂t− 1

e

∂⟨

J−⟩

∂z= 0 (70)

Combining Eqs. (69) and (70), Eq. (69) can be rewritten as

∂⟨

J+ + J−⟩

∂z= Sajloc (71)

Similarly, the volume averaging of Eqs. (32) and (31) gives

⟨J+ + J−

⟩= −ε〈�〉l

∂⟨

�∗+⟩l

∂z+ e

t+ε⟨

D⟩l ∂〈c〉l

∂z(72)

⟨J−

⟩= εe

⟨D⟩l∇〈c〉l + (1 − t+)

⟨J+ + J−

⟩(73)

where the cationic transference number t+ is assumed to be con-stant. If x direction is used instead and the notation for the intrinsicphase average is dropped to make the expressions clean, Eqs.(70)–(73) become

∂

∂x

[− ε

2�

∂�∗+∂x

+ e

t+ε

2D

∂c

∂x

]= Sajloc (74)

ε∂c

∂t+ ∂

∂x

[− ε

2D

∂c

∂x

]= (1 − t+)

Sajloc

e(75)

It is obvious that the porosity or liquid volume fraction ε showsup since the current is normalized to the apparent section areaA0 instead of the liquid section area Al; the tortuosity shows upsince the spatial derivative is with respect to the apparent elec-trode thickness, i.e., ∂/∂ z = (1/)( ∂/∂ x). The square of tortuosity, 2,is called tortuosity factor and sometimes is also called tortuosityas well [45,46]. A generalization of these concepts to the generalthree-dimensional case will be given elsewhere. Eqs. (74) and (75)are listed under the Macroscopic column of Table 1.

On the other hand, if Eq. (33) is used instead of (32), volumeaveraging gives

∂

∂x

[−�

ε

2

∂�∗+∂x

+ 2kBT

e�

ε

2(1 − t+)

(1 + ∂ ln f±

∂ ln c

)∂ ln c

∂x

]= Sajloc

(76)

This is equivalent to Eq. (74).

Comparison with Newman’s modelIn Newman’s model, the following two macroscopic equations

are used

∂

∂x

[−�eff

∂˚2

∂x+ 2kBT

e�eff (1 − t+)

(1 + ∂ ln f±

∂ ln c

)∂ ln c

∂x

]= Sajloc

(77)

ε∂c

∂t+ ∂

∂x

[−Deff

∂c

∂x

]= (1 − t+)

Sajloc

e(78)

They are the same as Eqs. (76) and (75) in the present work ifthe effective conductivity and diffusivity are defined as

�eff = �ε

2, Deff = D

ε

2(79)

In Newman’s model, these two parameters are generally defined

as

�eff = �ε˛B , Deff = Dε˛B (80)

where ˛B is called the empirical Bruggeman coefficient. Here thedependence of macroscopic values on the fractions of components

W. Lai, F. Ciucci / Electrochimica Acta 56 (2011) 4369–4377 4375

Table 1Comparison of volume-averaged micro-macroscopic equations in the present work and those in Newman’s model for individual components in the composite electrode.

Macroscopic Newman

Liquid∂

∂x

[− ε

2�

∂�∗+

∂x+ e

t+

ε

2D

∂c

∂x

]= Sajloc (74)

∂

∂x

[−�eff

∂˚2

∂x+ 2kBT

e�eff (1 − t+)

(1 + ∂ ln f±

∂ ln c

)∂ ln c

∂x

]= Sajloc (77)

ε∂c

∂t+ ∂

∂x

[− ε

2D

∂c

∂x

]= (1 − t+)

Sajloc

e(75) ε

∂c

∂t+ ∂

∂x

[−Deff

∂c

∂x

]= (1 − t+)

Sajloc

e(78)

Solid∂c

∂t+ ∇ · (−D∇c) = 0 (36)

∂c

∂t+ ∇ · (−D∇c) = 0 (36)

CC∂

∂x

(− ε

2�−

∂�∗−

∂x

)= −Sajloc (84)

∂

∂x

(−�eff

∂˚1

∂x

)= −Sajloc (85)

Interface −jloc = jl−s0 exp

[�∗

+(l) − �∗−(s) + Voc(s)

kBT/e

]jloc = j0c+(s)˛c c+(l)˛a

[c0

+(s) − c+(s)]˛a

xp

[˛

iotmmw

�

caεabTp

˛

sctHftttpaa

5

stosdoetatotma

− js−l0 exp

[− �∗

+(l) − �∗−(s) + Voc(s)

kBT/e

](87) ×

{e

n a composite is being investigated so this falls into the fieldf effective-medium theory [47,48]. The liquid–solid mixture inhe composite can be considered analogously as metal–insulator

ixture. Then based on the asymmetrical Bruggeman effective-edium model [47,48], the effective liquid conductivity can beritten as

eff = �ε1.5 (81)

In fact, a value of 1.5 is most commonly used for the empiri-al Bruggeman coefficient [3,4,8,9]. However, it is well-known thatsymmetrical Bruggeman model only works at dilute solution, i.e.,∼ 0 [47]. For the porosity of 20–40% that is of interest for practicalpplications, it is expected that ˛B will deviate from 1.5, which haseen suggested from several experimental measurements [49,50].he empirical Bruggeman coefficient can be expressed in terms oforosity and tortuosity as

B = 1 − 2ln

ln ε(82)

In addition, Eqs. (77) and (78) in Newman’s model involves threeets of thermodynamic and kinetic data, conductivity �, activityoefficient f±, and diffusivity D. Using this form, f± is either simplyreated as 1 [3,27,28,49], or comes from a separate set of data [9,51].owever, as discussed before in the microscopic equation section

or the solid, chemical diffusivity of the liquid should also be linkedo conductivity by activity (equivalent to chemical capacitance orhermodynamic factor). In fact, it can be clearly seen from Eq. (26)hat �, f±, and D are not independent. For Eqs. (74) and (75) in theresent work, only conductivity � and chemical diffusivity D datare needed for input. Not only Eq. (74) has a simpler form, it canlso avoid the possibility of mis-implementation of data.

.2. Solid electrode

Theoretically speaking, volume averaging can be applied to theolid particle as well. However, since the diffusion in the solid par-icle is much slower than that in the liquid solution, the hat termf Eq. (58) is relatively large and the second term on the right handide of Eq. (63) cannot be eliminated. Hence, the concentration gra-ient in the solid particle is an important feature of the process. Inther words, the volume averaging will not lead to a close formxpression as in the liquid phase. Thus all the microscopic equa-ions in the intercalating solid particle will be kept and this formsmicro-macroscopic coupled model, as shown in Table 1. In par-

icular as the diffusion of lithium cation is slow in the solid, it willnly extend to the microscopic level or conversely it will take a longime to show effects at the macroscale. This allows one to write a

icro-only equation for Li+ that is coupled to the macroscale byppropriate boundary conditions. The justification of keeping the

a˚1 − ˚2 − Voc(s)

kBT/e

]− exp

[−˛c

˚1 − ˚2 − Voc(s)kBT/e

]}(88)

solid microscopic equations in this hybrid model can be provenwith homogenization [52].

5.3. Current collector

In the current collector, the relevant microscopic equation to bevolume averaged is Eq. (37). Analogous to the cationic flux betweenthe liquid and solid interface, the electronic flux between the cur-rent collector and solid interface has the following relation⟨∇ · J−

⟩= ∇ ·

⟨J−

⟩+ Sajloc (83)

The extra term has a different sign from Eq. (66) since the Li ioniccurrent and electronic current have opposite directions. Again, ifthe simple notation is used

∂

∂x

(− ε

2�−

∂�∗−∂x

)= −Sajloc (84)

This is equation is listed under the Macroscopic column ofTable 1.

Comparison with Newman’s modelIn Newman’s model, the following equation for the current col-

lector is used

∂

∂x

(−�eff

∂˚1

∂x

)= −Sajloc (85)

It is noted that ε in Eq. (84) of the present work refers to thefraction of current collector while �eff in Eq. (85) is using an empir-ical Bruggeman relation with volume fraction of the solid plus thecurrent collector.

5.4. Triple phase junction

At the interface, the volume averaging of Eq. (49) leads to

− Sajloc = 1V0

∫S

Jl−s0 exp

[�∗+(l) − �∗−(s) + Voc(s)

kBT/e

]− Js−l

0

exp

[− �∗+(l) − �∗−(s) + Voc(s)

kBT/e

]dS (86)

Due to the 1D symmetry along z direction, the result is

− jloc = jl−s0 exp

[�∗+(l) − �∗−(s) + Voc(s)

kBT/e

]− js−l

0

exp

[− �∗+(l) − �∗−(s) + Voc(s)

kBT/e

](87)

where jl−s0 and js−l

0 are the surface average values of Jl−s0 and Js−l

0 .

4376 W. Lai, F. Ciucci / Electrochimica Acta 56 (2011) 4369–4377

F aightc erpend

j

w

6

tocIlmtatr

attcIemztej

ig. 3. Upscaled equivalent circuit for the representative volume element with strollector along with the interfacial elements. z and y are the directions along and p

Comparison with Newman’s modelIn Newman’s model, the surface average of Eq. (53) is

loc = j0c+(s)˛c c+(l)˛a[c0+(s)−c+(s)

]˛a{

exp[

˛a˚1 − ˚2 − Voc(s)

kBT/e

]

− exp[−˛c

˚1 − ˚2 − Voc(s)kBT/e

]}(88)

here j0 is a constant. Both Eqs. (87) and (88) are listed in Table 1.

. Equivalent circuit approach

As discussed above, there are three equivalent formulations ofhe PNP equations. It has been shown above that the upscalingf PNP equations in the formulation of concentration and electro-hemical potential lead to equations similar to Newman’s model.t was also shown recently that the PNP equations in the formu-ation of chemical potential and electrochemical potential can be

apped to a hierarchical equivalent circuit [18]. It is expected thathe upscaling of this equivalent circuit will yield the same resultss in Section 5. In this section, volume averaging is performed pic-orially with the equivalent circuit in Ref. [18] and it is shown thatesults same to Section 5 can be obtained.

For the RVE in the present work, Fig. 2(b), the upscaled equiv-lent circuit is shown in Fig. 3. The resistors dR are coming fromransport of charge carriers under the generalized transport equa-ion, Eq. (4). The capacitors dC± are coming from the chemicalapacitance under the continuity equation, Eq. (5) [15–18,20,21].n both the liquid and solid particle, there are two charge carri-rs. In the current collector, there is only electronic carrier. The big

inus terminal corresponds to the physical current collector. At

= 0, which is the interface between the composite electrode andhe physical current collector, both cation and anion in the liquidlectrolyte are blocked. The interfacial reaction at the triple phaseunction is represented by the interfacial element INT.

pores. It includes the circuit for the liquid electrolyte, solid electrode and currenticular to the pores, respectively.

The upscaled equivalent circuit in the liquid will be used as anexample to show the equivalence of results from this approachand what is used in Section 5. The current I–voltage relation of thecircuit in the liquid reads

I+ = −∂�∗+∂R+

(89)

I− = −∂�∗+∂R−

(90)

dI− = dC±∂(�∗+ − �∗−)

∂t(91)

dI+ = dC±∂(�∗− − �∗+)

∂t− Iint (92)

�∗+ and �∗− are the voltages at the junction points of dR+

and dR− rails, respectively. The resistor dR+ in the liquid is theupscaling of all the microscopic resistors in the liquid and is char-acterized by the cross sectional area and the thickness of thewhole liquid in RVE. The resistor and capacitor circuit elementsare

dRi = dz

�iAl(93)

dCi = ciAldz (94)

1dC±

= 1dC+

+ 1dC−

(95)

The relation between the current and current densitiesare

Ii = JiAl =⟨

Ji⟩

A0 (96)

Iint = SajlocA0dz (97)

If Eqs. (93)–(97) are put back into Eqs. (89)–(92), the same equa-tions as Eqs. (69) and (70) can be obtained and this suggests thatthe upscaling of equivalent circuit also yield the same results. The

imica

ct

7

PiutsiueotsaTanaesnbm

A

p

A

AAbccCDDefIjJknqttTRSvVVz

G˛˛˛

[

[[[[[[[[[[[[[[

[[

[[

[

[[[[[[

[[[

[[[[[[[[[

[

W. Lai, F. Ciucci / Electroch

ircuit for the current collector behaves similarly. The circuit forhe solid particle is kept unchanged.

. Conclusions

In the present paper, volume averaging of generalizedoisson–Nernst–Planck equations on a de Levie straight pore models performed rigorously to present a set of equations, that can besed to model porous battery electrodes composed of liquid elec-rolytes, solid particles, and current collectors. These equations areimilar to those established in Newman’s model but they are notdentical. First, electrochemical potentials of charge carriers aresed throughout of the work instead of the physically less intuitivelectrical potentials in Newman’s model. This allows easy trackingf transport of different charge carriers in different components ofhe composite. Second, the equations in the liquid electrolyte andolid particle have innate correlation between the thermodynamicnd kinetic properties of materials of interest in the present work.he diffusivity in either liquid or solid is related to the conductivitynd thermodynamic property (chemical capacitance, thermody-amic factor, or activity coefficient) in the same material. Third,n interfacial charge transfer equation under the driving force oflectrochemical potential is presented as an alternative to the clas-ical Butler–Volmer equation. While offering the same level ofumerical easiness, the equations in the present work appear toe more thermodynamically consistent than those in Newman’sodel.

cknowledgement

W.L. would like to acknowledge Michigan State University forroviding the start-up package.

ppendix A. Nomenclature

0 cross section area of representative volume elementl cross section area of liquid

mobilityconcentration

0 reference or maximum concentrationchem chemical capacitance

diffusivity˜ chemical diffusivity

electron chargeactivity coefficientcurrent

loc volumetric interfacial current densitycurrent density

B Boltzmann constant� direction vector

volumetric charge densitytime

+ transference number of positive chargeabsolute temperatureresistance

a volumetric surface areavelocity

0 volume of representative volume elementl volume of liquid

charge number

reek lettersa anodic symmetry factorc cathodic symmetry factorB Bruggeman coefficient

[

[[

[

Acta 56 (2011) 4369–4377 4377

ˇ symmetry parameterε volume fractionε0 vacuum permittivityεr relative permittivity� electrical potential˚1 potential in the solid˚2 potential in the liquid� thermodynamic factor� chemical potential�* reduced chemical potential� electrochemical potential�∗ reduced electrochemical potential� conductivity tortuosity

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