Nonequilibrium Thermodynamics of Porous Electrodes

Todd R. Ferguson1 and Martin Z. Bazant1, 2

1Department of Chemical Engineering, Massachusetts Institute of Technology2Department of Mathematics, Massachusetts Institute of Technology

(Dated: August 24, 2012)

We reformulate and extend porous electrode theory for non-ideal active materials, including thosecapable of phase transformations. Using principles of non-equilibrium thermodynamics, we relatethe cell voltage, ionic fluxes, and Faradaic charge-transfer kinetics to the variational electrochemicalpotentials of ions and electrons. The Butler-Volmer exchange current is consistently expressed interms of the activities of the reduced, oxidized and transition states, and the activation overpotentialis defined relative to the local Nernst potential. We also apply mathematical bounds on effectivediffusivity to estimate porosity and tortuosity corrections. The theory is illustrated for a Li-ionbattery with active solid particles described by a Cahn-Hilliard phase-field model. Depending onthe applied current and porous electrode properties, the dynamics can be limited by electrolytetransport, solid diffusion and phase separation, or intercalation kinetics. In phase-separating porouselectrodes, the model predicts narrow reaction fronts, mosaic instabilities and voltage fluctuationsat low current, consistent with recent experiments, which could not be described by existing porouselectrode models.

I. INTRODUCTION

Modeling is a key component of any design process. Anaccurate model allows one to interpret experimental data,identify rate limiting steps and predict system behavior,while providing a deeper understanding of the underly-ing physical processes. In systems engineering, empiricalmodels with fitted parameters are often used for designand control, but it is preferable, whenever possible, toemploy models based on microscopic physical or geomet-rical parameters, which can be more easily interpretedand optimized.

In the case of electrochemical energy storage devices,such as batteries, fuel cells, and supercapacitors, the sys-tems approach is illustrated by equivalent circuit models,which are widely used in conjunction with impedancespectroscopy to fit and predict cell performance anddegradation. This approach is limited, however, by thedifficulty in unambiguously interpreting fitted circuit el-ements and in making predictions for the nonlinear re-sponse to large operating currents. There is growing in-terest, therefore, in developing physics-based porous elec-trode models and applying them for battery optimiza-tion and control [1]. Quantum mechanical computationalmethods have demonstrated the possibility of predictingbulk material properties, such as open circuit potentialand solid diffusivity, from first principles [2], but coarse-grained continuum models are needed to describe themany length and time scales of interfacial reactions andmultiphase, multicomponent transport phenomena.

Mathematical models could play a crucial role in guid-ing the development of new intercalation materials, elec-trode microstructures, and battery architectures, in or-der to meet the competing demands in power density andenergy density for different envisioned applications, suchas electric vehicles or renewable (e.g. solar, wind) en-ergy storage. Porous electrode theory, pioneered by J.Newman and collaborators, provides the standard mod-

eling framework for battery simulations today [3]. Asreviewed in the next section, this approach has been de-veloped for over half a century and applied successfullyto many battery systems. The treatment of the activematerial, however, remains rather simple, and numerousparameters are often needed to fit experimental data.

In porous electrode theory for Li-ion batteries, trans-port is modeled via volume averaged conservation equa-tions [4]. The solid active particles are modeled asspheres, where intercalated lithium undergoes isotropiclinear diffusion [5, 6]. For phase separating materials,such as LixFePO4 (LFP), each particle is assumed to havea spherical phase boundary that moves as a “shrinkingcore”, as one phase displaces the other [7–9]. In thesemodels, the local Nernst equilibrium potential is fittedto the global open circuit voltage of the cell, but this ne-glects non-uniform composition, which makes the volt-age plateau an emergent property of the porous elec-trode [10–13]. For thermodynamic consistency, all ofthese phenomena should derive from common thermo-dynamic principles and cannot be independently fittedto experimental data. The open circuit voltage reflectsthe activity of intercalated ions, which in turn affectsion transport in the solid phase and Faradaic reactionsinvolving ions in the electrolyte phase [14, 15].

In this paper, we extend porous electrode theory tonon-ideal active materials, including those capable ofphase transformations. Our starting point is a generalphase-field theory of ion intercalation kinetics developedby our group over the past five years [12, 15–19], whichhas recently led to a quantitative understanding of phaseseparation dynamics in LFP nanoparticles [13]. The ionicfluxes in all phases are related to electrochemical poten-tial gradients [18, 20], consistent with non-equilibriumthermodynamics [21, 22]. For thermodynamic consis-tency, the Faradaic reaction rate is also related to elec-trochemical potential differences between the oxidized,reduced, and transition states, leading to a generalized

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Butler-Volmer equation [15] suitable for phase-separatingmaterials. These elements are integrated in a generalporous electrode theory, where the active material is de-scribed by a Cahn-Hilliard phase-field model [22, 23],as in nanoscale simulations of Li-ion battery materi-als [12, 13, 16, 17, 19, 24–27]. This allows us to describethe non-equilibrium thermodynamics of porous batteryelectrodes in terms of well established physical principlesfor ion intercalation in nanoparticles.

II. BACKGROUND

A. Mathematical Modeling of Porous Electrodes

We begin by briefly reviewing volume-averaged porouselectrode theory, which has been the standard approachin battery modeling for the past 50 years, in orderto highlight similarities and differences with our ap-proach. The earliest attempts to formulate porous elec-trode models [28, 29] related current density distribu-tions to macroscopic properties such as porosity, aver-age surface area per volume, and effective conductivity,and capacitive charging was added in transmission linemodels [30]. Sixty years ago, the seminal work Newmanand Tobias [31] first described the effects of concentra-tion variations on kinetics and introduced the well-knownmass conservation equations for porous electrodes, whichform the basis for modern battery modeling. Extensiveliterature surveys are available by Newman and coau-thors [3, 32] for work up to the 1990s.

Here, we only draw attention to some specific papersand recent developments that set the stage for our theo-retical approach. Perhaps the earliest use of conceptsfrom non-equilibrium thermodynamics in porous elec-trode theory was by Ksenzhek, who incorporated concen-trated solution theory in the transport equations insidea porous electrode, and referred to gradients in electro-chemical potential as the driving force for transport [33].This is the fundamental postulate of linear irreversiblethermodynamics in chemical physics [21] and materialsscience [22], and it has also recently been applied to elec-trochemical systems [20, 34–42] and electrokinetic phe-nomena [18, 43–45]. Although concentrated solution the-ory is widely applied to batteries [3], the thermodynamicdriving force for transport has only recently been con-nected to the battery voltage [20, 36, 37] and Faradaicreaction kinetics [12, 13, 15].

Porous electrode theories make a number of underly-ing assumptions regarding properties of the cell that canbe critical to performance. For example, an early paperof Grens [46] showed that the assumption of constantconductivity for the electron conducting phase is usuallyvalid, while the assumption of constant electrolyte con-centration, often used for mathematical convenience, isonly valid over a narrow range of operating conditions.These concepts are extended here to volume averagingover solid reaction products undergoing phase transfor-

mations (which further narrows the range of validity ofporous electrode models to exclude mosaic instabilitiesamong discrete particles in a representative continuumvolume element).

Our work also focuses on the nonlinear dynamicsof porous electrodes, which could only be addressedas computer power improved. Early work focused onsteady state [31, 47], mostly at small (linearized Butler-Volmer) or large (Tafel regime) overpotentials [48], ortransient response for small sinusoidal perturbations(impedance) [49] or fast kinetics [50]. Similar to our moti-vation below, Atlung et al.[51] investigated the dynamicsof solid solution (i.e. intercalation) electrodes for differ-ent time scales with respect to the limiting current, al-though without considering configurational entropy andchemical potentials as in this work.

As computers and numerical methods advanced, sodid simulations of porous electrodes, taking into accountvarious nonlinearities in transport and reaction kinet-ics. West et al. first demonstrated the use of numericalmethods to simulate discharge of a porous TiS2 electrode(without the separator) in the typical case of electrolytetransport limitation [52]. Doyle, Fuller and Newman firstsimulated Li-ion batteries under constant current dis-charge with full Butler-Volmer kinetics for two porouselectrodes and a porous separator [5, 6, 53]. These pa-pers are of great importance in the field, as they devel-oped the first complete simulations of lithium-ion bat-teries and solidified the role of porous electrode theoryin modeling these systems. The same theoretical frame-work has been applied to many other types of cells, suchas lithium-sulfur [54] and LFP [7, 8] batteries, with par-ticular success for lithium polymer batteries at high dis-charge rates

Battery models invariably assume electroneutrality,but diffuse charge in porous electrodes has received in-creasing attention over the past decade, driven by appli-cations in energy storage and desalination. The effects ofdouble-layer capacitance in a porous electrode were orig-inally considered using only linearized low-voltage mod-els [55, 56], which are equivalent to transmission linecircuits [30, 57]. Recently, the full nonlinear dynamicsof capacitive charging and salt depletion have been an-alyzed and simulated in both flat [41, 58] and porous[59] electrodes. The combined effects of electrostatic ca-pacitance and pseudo-capacitance due to Faradaic reac-tions have also been incorporated in porous electrode the-ory [60, 61], using Frumkin-Butler-Volmer kinetics [62].These models have been successfully used to predict thenonlinear dynamics of capacitive desalination by porouscarbon electrodes [63, 64]. Although we do not considerdouble layers in our examples below (as is typical for bat-tery discharge), it would be straightforward to integratethese recent models into our theoretical framework basedon non-equilibrium thermodynamics [15, 42].

Computational and experimental advances have alsobeen made to study porous electrodes at the microstruc-tural level and thus test the formal volume-averaging,

3

which underlies macroscopic continuum models. Garciaet al. performed finite-element simulations of ion trans-port in typical porous microstructures for Li-ion batter-ies [25], and Garcia and Chang simulated hypotheticalinter-penetrating 3D battery architectures at the particlelevel [65]. Recently, Smith, Garcia and Horn analyzed theeffects of microstructure on battery performance for var-ious sizes and shapes of particles in a Li1−xC6/LixCoO2

cell [66]. The study used 3D image reconstruction of areal battery microstructure by focused ion beam milling,which has led to detailed studies of microstructural ef-fects in porous electrodes [67–69]. In this paper, we willdiscuss mathematical bounds on effective diffusivities inporous media, which could be compared to results foractual battery microstructures. Recently, it has also be-come possible to observe lithium ion transport at thescale in individual particles in porous Li-ion battery elec-trodes [70, 71], which could be invaluable in testing thedynamical predictions of new mathematical models.

B. Lithium Iron Phosphate

The discovery of LFP as a cathode material by theGoodenough group in 1997 has had a large and unex-pected impact on the battery field, which provides themotivation for our work. LFP was first thought to bea low-power material, and it demonstrated poor capac-ity at room temperature. [72] The capacity has sincebeen improved via conductive coatings and the forma-tion of nanoparticles. [73, 74], and the rate capabilityhas been improved in similar ways [75, 76]. With highcarbon loading to circumvent electronic conductivity lim-itations, LFP nanoparticles can now be discharged in 10seconds [27]. Off-stoichiometric phosphate glass coatingscontribute to this high rate, not only in LFP, but also inLiCoO2 [77].

It has been known since its discovery that LFP is aphase separating material, as evidenced by a flat voltageplateau in the open circuit voltage [72, 78]. There area wide variety of battery materials with multiple stablephases at different states of charge [79], but LixFePO4

has a particularly strong tendency for phase separation,with a miscibility gap (voltage plateau) spanning acrossmost of the range from x = 0 to x = 1 at room temper-ature. Padhi et al. first depicted phase separation insideLFP particles schematically as a “shrinking core” of onephase being replaced by an outer shell of the other phaseduring charge/discharge cycles [72]. Srinivasan and New-man encoded this concept in a porous electrode theoryof the LFP cathode with spherical active particles, con-taining spherical shrinking cores. [7] Recently, Dargavilleand Farrell have expanded this approach to predict activematerial utilization in LFP electrodes. [8] Thorat et al.have also used the model to gain insight into rate-limitingmechanisms inside LFP cathodes. [9]

To date, the shrinking-core porous electrode modelis the only model to successfully fit the galvanostatic

discharge of an LFP electrode, but the results are notfully satisfactory. Besides neglecting the microscopicphysics of phase separation, the model relies on fitting aconcentration-dependent solid diffusivity, whose inferredvalues are orders of magnitude smaller than ab initiosimulations [76, 80] or impedance measurements [81].More consistent values of the solid diffusivity have sincebeen obtained by different models attempting to accountfor anisotropic phase separation with elastic coherencystrain. [82] Most troubling for the shrinking core picture,however, is the direct observation of phase boundarieswith very different orientations. In 2006, Chen, Song,and Richardson published images showing the orienta-tion of the phase interface aligned with iron phosphateplanes and reaching the active facet of the particle. [83]This observation was supported by experiments of Del-mas et al., who suggested a “domino-cascade model” forthe intercalation process inside LFP [84]. With furtherexperimental evidence for anisotropic phase morpholo-gies [71, 85], it has become clear that a new approach isneeded to capture the non-equilibrium thermodynamicsof this material.

C. Phase-Field Models

Phase-field models are widely used to describe phasetransformations and microstructural evolution in mate-rials science [22, 86], but they are relatively new toelectrochemistry. In 2004, Guyer, Boettinger, Warrenand McFadden [87, 88] first modeled the sharp elec-trode/electrolyte interface with a continuous phase fieldvarying between stable values 0 and 1, representing theliquid electrolyte and solid metal phases. As in phase-field models of dendritic solidification [89–92], they useda simple quartic function to model a double-welled homo-geneous free energy. They described the kinetics of elec-trodeposition [88] (converting ions in the electrolyte tosolid metal) by Allen-Cahn-type kinetics [86, 93], linearin the thermodynamic driving force, but did not makeconnections with the Butler-Volmer equation. Severalgroups have used this approach to model dendritic elec-trodeposition and related processes [94–96]. Also in 2004,Han, Van der Ven and Ceder [24] first applied the Cahn-Hilliard equation[22, 86, 97–100] to the diffusion of in-tercalated lithium ions in LFP, albeit without modelingreaction kinetics.

Building on these advances, Bazant developed a gen-eral theory of charge-transfer and Faradaic reaction ki-netics in concentrated solutions and solids based on non-equilibrium thermodynamics [14, 15, 101], suitable foruse with phase-field models. The exponential Tafel de-pendence of the current on the overpotential, defined interms of the variational chemical potentials, was first re-ported in 2007 by Singh, Ceder and Bazant [16, 102], butwith spurious pre-factor, corrected by Burch [19, 103].The model was used to predict “intercalation waves”in small, reaction-limited LFP nanoparticles in 1D [16],

4

2D [17], and 3D [26], thus providing a mathematical de-scription of the domino cascade phenomenon [84]. Thecomplete electrochemical phase-field theory, combiningthe Cahn-Hilliard with Butler-Volmer kinetics and thecell voltage, appeared in 2009 lectures notes [14, 101]and was applied to LFP nanoparticles [12, 13].

The new theory has led to a quantitative understand-ing of intercalation dynamics in single nanoparticles ofLFP. Bai, Cogswell and Bazant [12] generalized theButler-Volmer equation using variational chemical poten-tials (as derived in the supporting information) and usedit to develop a mathematical theory of the suppressionof phase separation in LFP nanoparticles with increas-ing current. This phenomenon, which helps to explainthe remarkable performance of nano-LFP, was also sug-gested by Malik and Ceder based on bulk free energycalculations [104], but the theory shows that it is en-tirely controlled by Faradaic reactions at the particle sur-face [12, 13]. Cogswell and Bazant [13] have shown thatincluding elastic coherency strain in the model leads toa quantitative theory of phase morphology and lithiumsolubility. Experimental data for different particles sizesand temperatures can be fitted with only two parameters(the gradient penalty and regular solution parameter, de-fined below).

The goal of the present work is to combine the phase-field theory of ion intercalation in nanoparticles with clas-sical porous electrode theory to arrive at a general math-ematical framework for non-equilibrium thermodynam-ics of porous electrodes. Our work was first presentedat the Fall Meeting of the Materials Research Society in2010 and again at the Electrochemical Society Meetingsin Montreal and Boston in 2011. Around the same time,Lai and Ciucci were thinking along similar lines [36, 37]and published an important reformulation of Newman’sporous electrode theory based non-equilibrium thermo-dynamics [20], but they did not make any connectionswith phase-field models or phase transformations at themacroscopic electrode scale. Their treatment of reactionsalso differs from Bazant’s theory of generalized Butler-Volmer or Marcus kinetics [14, 15, 101], with a thermo-dynamically consistent description of the transition statein charge transfer.

In this paper, we develop a variational thermodynamicdescription of electrolyte transport, electron transport,electrochemical kinetics, and phase separation, and weapply to Li-ion batteries in what appears to be the firstmathematical theory and computer simulations of macro-scopic phase transformations in porous electrodes. Simu-lations of discharge into a cathode consisting of multiplephase-separating particles interacting via an electrolytereservoir at constant chemical potential were reported byBurch [103], who observed “mosaic instabilities”, whereparticles transform one-by-one at low current. This phe-nomenon was elegantly described by Dreyer et al. interms of a (theoretical and experimental) balloon model,which helps to explain the noisy voltage plateau andzero-current voltage gap in slow charge/discharge cycles

of porous LFP electrodes [10, 11]. These studies, how-ever, did not account for electrolyte transport and associ-ated macroscopic gradients in porous electrodes undergo-ing phase transformations, which are the subject of thiswork. To do this, we must reformulate Faradaic reac-tion kinetics for concentrated solutions, consistent withthe Cahn-Hilliard equation for ion intercalation and New-man’s porous electrode theory for the electrolyte.

III. GENERAL THEORY OF REACTIONS ANDTRANSPORT IN CONCENTRATED SOLUTIONS

In this section, we begin with a general theory of re-action rates based on non-equilibrium thermodynamicsand transition state theory. We then expand the modelto treat transport in concentrated solutions (i.e. solids).Finally, we show that this concentrated solution modelcollapses to Fickian diffusion in the dilute limit. For moredetails and examples, see Refs. [14, 15].

A. General Theory of Reaction Rates

The theory begins with the diffusional chemical poten-tial of species i,

µi = kBT ln ci + µexi = kBT ln ai (1)

where ci is the concentration, ai is the absolute chemicalactivity, µexi = kBT ln γi is the excess chemical potentialin a concentrated solution, and γi is the activity coeffi-cient (ai = γici). In linear irreversible thermodynamics(LIT) [21, 22, 105], the flux of species i is proportionalto its chemical potential gradient, as discussed below.

In a thermodynamically consistent formulation of re-action kinetics [14, 106], therefore, the reaction complexexplores a landscape of excess chemical potential µex(x)between local minima µex1 and µex2 with transitions overan activation barrier µex‡ , as shown in Figure 1. For long-

lived states with rare transitions (µex‡ −µexi kBT ), thenet reaction rate is given by

R = R1→2 −R2→1

= ν[e−(µex‡ −µ1)/kBT − e−(µex‡ −µ2)/kBT

](2)

=ν(a1 − a2)

γ‡

which automatically satisfies the De Donder rela-tion [106],

µ1 − µ2 = kBT ln

(R1→2

R2→1

). (3)

The frequency prefactor ν depends on generalized forceconstants at the saddle point and in one minimum (e.g.state 1, with a suitable shift of µex‡ ) as in Kramers’ es-

cape formula [107, 108] and classical transition state the-ory [109, 110].

5

μ1

μ2

μTS

FIG. 1. Typical reaction energy landscape. The set ofatoms involved in the reaction travels through a transitionstate as it passes from one state to the other in a landscapeof total excess chemical potential as a function of the atomiccoordinates.

For the general reaction,

S1 =∑i

siMi →∑j

sjMj = S2, (4)

the activities, a1 =∏i asii and a2 =

∏j a

sjj , are equal

in equilibrium, and the forward and backward reactionsare in detailed balance (R = 0). The equilibrium con-stant is thus the ratio of the reactant to product activitycoefficients:

K =c2c1

=

∏j csjj∏

i csii

=

∏i γ

sii∏

j γsjj

=γ1

γ2= e(µex1 −µ

ex2 )/kBT = e−∆Gex/kBT (5)

where ∆Gex is the excess free energy change per reaction.In order to describe reaction kinetics, however, we alsoneed a model for the transition state activity coefficientγ‡, in (2).

The subtle difference between total and excess chem-ical potential is often overlooked in chemical kinetics.Lai and Ciucci [20, 36, 37], who also recently appliednon-equilibrium thermodynamics to batteries, postulatea Faradaic reaction rate based on a barrier of total (notexcess) chemical potential. The equilibrium condition(Nernst equation) is the same, but the rate (exchangecurrent) is different and does not consistently treat thetransition state. We illustrate this point by deriving soliddiffusion and Butler-Volmer kinetics from the same reac-tion formalism.

B. General Theory of Transport in Solids andConcentrated Solutions

In solids, atoms (or more generally, molecules) fluc-tuate in long-lived states near local free energy minimaand occasionally move through a transition state to aneighboring well of similar free energy. In a crystal, thewells correspond to lattice sites, but similar concepts alsoapply to amorphous solids. Figure 2 demonstrates thispicture of diffusion and shows an energy (or excess chem-ical potential) landscape for an atom moving through a

medium. Tracer diffusion of individual atoms consists ofthermally activated jumps over some distance betweensites with an average “first passage time” [108] betweenthese transitions, τ , which is the inverse of the mean tran-sition rate per reaction event above. Using the generalthermodynamic theory of reaction rates above for theactivated diffusion process, the time between transitionsscales as

τ = τo exp

(µex‡ − µex

kBT

). (6)

The tracer diffusivity, D, is then the mean square dis-tance divided by the mean transition time,

D =(∆x)

2

2τ=

(∆x)2

2τo

(γ

γ‡

)= Do

(γ

γ‡

), (7)

where Do is the tracer diffusivity in the dilute-solutionlimit.

μTS

EX

μEX

FIG. 2. Typical diffusion energy landscape. The sameprinciples for reactions can also be applied to solid diffusion,where the diffusing molecule explores a landscape of excesschemical potential, hopping by thermal activation betweennearly equivalent local minima.

1. Diffusivity of an Ideal Solid Solution

To model an ideal solid solution, we consider a latticegas model for the configurational entropy, which accountsfor finite volume effects in the medium, and neglect anydirect atom-atom interactions which contribute to theenthalpy. Figure 3 illustrates this model. The chemicalpotential for an atom in an ideal solid solution is

µ = kBT ln

(c

1− c

)+ µo, (8)

where µo is the chemical potential of the reference stateand c = c/cmax is the dimensionless concentration. Theexcluded volume of an atom is one lattice site. How-ever, the transition state requires two available sites, ef-fectively doubling the excluded volume contribution tothe chemical potential. Using the definition of the activ-ity coefficient, µ = kBT ln a = kBT ln (cγ), we obtain theactivity coefficients of the atom in the site, and in the

6

FIG. 3. Lattice gas model for diffusion. The atoms areassigned a constant excluded volume by occupying sites on agrid. Atoms can only jump to an open space, and the transi-tion state (red dashed circle) requires two empty spaces.

activated state,

γ =

(1− c

cmax

)−1

exp

(−µminkBT

), (9)

γ‡ =

(1− c

cmax

)−2

exp

(− µ‡kBT

). (10)

Inserting these two activity coefficients into Equation 7,the diffusivity, D, is

D = Do

(1− c

cmax

). (11)

This diffusivity is for an ideal solid solution with a finitenumber of lattice sites available for atoms [23]. As thelattice sites fill, the diffusivity of an atom goes to zero,since the atom is unable to move as it is blocked by otheratoms on the lattice.

2. Concentrated Solution Theory Derivation

Here we will derive the general form of concentratedsolution theory, which postulates that the flux can bemodeled as

F = −Mc∇µ, (12)

where M is the mobility. Let us consider the scenarioin Figure 3, where an atom is sitting in an energy well.This atom’s energy fluctuates on the order of kBT until ithas enough energy to overcome some energy barrier thatexists between the two states. Figure 4 demonstrates thisin one dimension. The flux, F, is

Fi =R

Aei, (13)

where ei is a coordinate vector in the i direction and Fiis the flux in the i direction.

We see that the atom’s chemical potential is a functionof location, as concentrations and therefore chemical po-tentials, will vary with position. Let’s define the rightside of the page as the positive x-direction. Using ourpreviously defined form of the reaction rate in Equation(2), we can substitute this into Equation (13). However,we need an expression for the barrier-less reaction rate.

Δx/2

Acell

FIG. 4. Diffusion through a solid. The flux is given by thereaction rate across the area of the cell, Acell. In this latticemodel, atoms move between available sites.

This comes from the barrier-less diffusion time in Equa-tion (6). The barrier-less reaction rate should be equiva-lent to the inverse of two times the barrier-less diffusiontime,

Ro =1

2τo. (14)

The one half comes from the probability the atom travelsin the positive x direction. Plugging this into Equation(13) along with Equation (2), and considering the factthat our chemical potential is a function of position, weobtain

Fx =1

2τoAcellγ‡

[exp

(µ(x)− ∆x

2

∂µ(x)

∂x

)− exp

(µ(x) +

∆x

2

∂µ(x)

∂x

)], (15)

where µ(x) denotes the chemical potential scaled by thethermal voltage, kBT . Next, we assume that the atom isclose to equilibrium. That is, the difference in chemicalpotential between the states is small. This allows us tolinearize Equation (15). Linearizing the equation yields

Fx = − a(x)

τoAcellγ‡

(∆x

2

)∂µ(x)

∂x, (16)

where a(x) is the activity as a function of position. Thiscan be simplified to a(x) = V γ(x)c(x). Plugging thisinto Equation (16), using our definition of the diffusivity,D, from Equation (7), and the Einstein relation, whichstates that M = D/kBT , we obtain the flux as predictedby concentrated solution theory in the x-dimension. Wecan easily expand this to other dimensions. Doing so,we obtain the form of the flux proposed by concentratedsolution theory,

F = −Mc∇µ, (17)

where c = c(x, y, z). Taking the dilute limit, as c →0, and using the definition of chemical potential, µ =kBT ln a, where a = γc and γ = 1 (dilute limit), weobtain Fick’s Law from Equation (17),

F = −D∇c. (18)

7

IV. CHARACTERIZATION OF POROUSMEDIA

In batteries, the electrodes are typically compositesconsisting of active material (e.g. graphite in the anode,iron phosphate in the cathode), conducting material (e.g.carbon black), and binder. The electrolyte penetrates thepores of this solid matrix. This porous electrode is advan-tageous because it substantially increases the availableactive area of the electrode. However, this type of sys-tem, which can have variations in porosity (i.e. volume ofelectrolyte per volume of the electrode) and loading per-cent of active material throughout the volume, presentsdifficulty in modeling. To account for the variation inelectrode properties, various volume averaging methodsfor the electrical conductivity and transport propertiesin the electrode are employed. In this section, we willgive a brief overview of modeling the conductivity andtransport of a heterogeneous material, consisting of twoor more materials with different properties [111–114]

A. Electrical Conductivity of the Porous Media

To characterize the electrical conductivity of theporous media, we will consider rigorous mathematicalbounds over all possible microstructures with the samevolume fractions of each component. First we considera general anisotropic material as shown in Figure 5, inwhich case the conductivity bounds, due to Wiener, areattained by simple microstructures with parallel stripesof the different materials [112]. The left image in Figure 5represents the different materials as resistors in parallel,which produces the lowest possible resistance and the up-per limit of the conductivity of the heterogeneous mate-rial. The right image represents the materials as resistorsin series, which produces the highest possible resistance,or lower limit of the conductivity. These limits are re-

σ1

σ2

σ3 σ1 σ2 σ3E, j E, j

FIG. 5. Wiener bounds on the effective conductiv-ity of a two-phase anisotropic material. The left figuredemonstrates the upper conductivity limit achieved by stripesaligned with the field, which act like resistors in parallel. Theright figure demonstrates the lower bound with the materialsarranged in transverse stripes to act like resistors in series.

ferred to as the upper and lower Wiener bounds, respec-tively. Let Φi be the volume fraction of material i. Forthe upper Wiener bound, attained by stripes parallel to

the current, the effective conductivity is simply the arith-metic mean of the individual conductivities, weighted bytheir volume fractions,

σmax = 〈σ〉 =∑i

Φiσi. (19)

The lower Wiener bound is attained by stripes perpen-dicular to the current, and the effective conductivity is aweighted harmonic mean of the individual conductivities,as for resistors in parallel,

σmin = 〈σ−1〉−1 =1∑i

Φiσi

. (20)

For a general anisotropic material, the effective conduc-tivity, σ, must lie within the Wiener bounds,

〈σ−1〉−1 ≤ σ ≤ 〈σ〉. (21)

There are tighter bounds on the possible effective con-ductivity of isotropic media, which have no preferred di-rection, due to Hashin and Shtrikman (HS) [112]. Thereare a number of microstructures which attain the HSbounds, such as a space-filling set of concentric circles orspheres, whose radii are chosen to set the given volumefractions of each material. The case of two components isshown in Figure 6. The HS lower bound on conductivity

12

FIG. 6. Hashin-Shtrikman bounds on the effec-tive conductivity of a two-phase isotropic material.Isotropic random composite of space-filling coated sphereswhich attain the bounds. The white represents materialwith conductivity σ1 and the black represents material withconductivity σ2. Maximum conductivity is achieved whenσ1 > σ2 and minimum conductivity is obtained when σ2 > σ1.The volume fractions Φ1 and Φ2 are the same.

is attained by ordering the individual materials so as toplace the highest conductivity at the core and the lowest

8

conductivity in the outer shell, of each particle. For theHS upper bound, the ordering is reversed, and the lowestconductivity material is buried in the core of each parti-cle, while the highest conductivity is in the outer shell,forming a percolating network across the system.

For the case of two components, where σ1 > σ2, theHS conductivity bounds for an isotropic two-componentmaterial in d dimensions are

〈σ〉 − (σ1 − σ2)2

Φ1Φ2

〈σ〉+ σ2 (d− 1)≤ σ ≤ 〈σ〉 − (σ1 − σ2)

2Φ1Φ2

〈σ〉+ σ1 (d− 1),

(22)where

〈σ〉 = Φ1σ1 + Φ2σ2

and

〈σ〉 = Φ1σ2 + Φ2σ1.

The Wiener and Hashin-Shtrikman bounds above pro-vide us with possible ranges for isotropic and anisotropicmedia with two components. Figure 7 gives the Wienerand Hashin-Shtrikman bounds for two materials, withconductivities of 1.0 and 0.1.

0 0.2 0.4 0.6 0.8 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Φ1

σ eff

Isotropic(Hashin−Shtrikman)

Anisotropic(Wiener)

FIG. 7. Conductivity bounds for two-phase compos-ites versus volume fraction. The above figure shows theWiener bounds (blue) for an anisotropic two component ma-terial and Hashin-Shtrikman bounds (red) for an isotropic twocomponent material versus the volume fraction of material 1.The conductivities used to produce the figure are σ1 = 1 andσ2 = 0.1.

Next, we consider ion transport in porous media. Iontransport in porous media often consists of a solid phase,which has little to no ionic conductivity (i.e. slow orno diffusion) permeated by an electrolyte phase whichhas very high ionic conductivity (i.e. fast diffusion). Inthe next section, we will compare different models foreffective porous media properties.

B. Conduction in Porous Media

For the case of ion transport in porous media, thereis an electrolyte phase, which has a non-zero diffusivity,

and the solid phase, through which transport is very slow(essentially zero compared to the electrolyte diffusivity).Here, we consider the pores (electrolyte phase) and givethe solid matrix a zero conductivity. The volume fractionof phase 1 (the pores), Φ1, is the porosity:

Φ1 = εp, σ1 = σp.

The conductivity for all other phases is zero. This re-duces the Wiener (anisotropic) and Hashin-Shtrikman(isotropic) lower bounds to zero. Figure (8) demonstratesa typical volume of a porous medium.

εp

FIG. 8. Example of a porous volume. This is an exampleof a typical porous volume. A mixture of solid particles ispermeated by an electrolyte. The porosity, εp, is the volumeof electrolyte as a fraction of the volume of the cube.

In porous electrode models for batteries [5, 7, 53], theempirical Bruggeman formula is used to relate the con-ductivity to the porosity,

σB = ε3/2p σp (23)

although it is not clear what mathematical approxima-tion is being made. As shown in Figure 9, the Bruggemanformula turns out to be close to (and fortunately, below)the HS upper bound, so we can see that it correspondsto a highly conducting isotropic material, similar to acore-shell microstructure with solid cores and conductingshells. This makes sense for ionic conductivity in liquid-electrolyte-soaked porous media, but not for electronicconductivity based on networks of touching particles.

To understand the possible range of conductivity, weconsider the rigorous bounds above. If we assume themedia consists of two phases (Φ2 = 1− εp, σ2 = 0), thenthe Wiener and Hashin-Shtrikman upper bounds can besimplified to

σWienermax = Φ1σ1 = εpσp, (24)

and

σHSmax = σpεp

(d− 1

d− εp

). (25)

9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

εp

σ eff/σ

p

WienerHashin−ShtrikmanPercolationBruggeman

FIG. 9. Various models for effective conductivityin 3D. This figure demonstrates the effective conductiv-ity (scaled by the pore conductivity) using Wiener bounds,Hashin-Shtrikman bounds, a percolation model, and theBruggeman formula. The percolation model uses a criticalporosity of εc = 0.25.

where again d is the embedding dimension. The HS up-per bound is attained by spherical core-shell particleswith the conducting pore phase spanning the system viaconducting shells on non-conducting solid cores, similarto electron-conducting coatings on active battery parti-cles [115].

The lower bounds vanish because it is always possi-ble that the conducting phase does not “percolate”, orform a continuous path, across the system. Equivalently,the non-conducting matrix phase can percolate and blockconduction. In such situations, however, the bounds areof little use, since they give no sense of the probability offinding percolating paths through a random microstruc-ture. For ionic conduction through the electrolyte, whichpermeates the matrix, percolation may not be a majorissue, but for electron conduction it is essential to main-tain a network of touching conducting particles (such ascarbon black in a typical battery electrode) [115].

In statistical physics, percolation models serve to quan-tify the conductivity of random media due to geomet-rical connectivity of particles [113, 114]. The simplestpercolation models corresponds to randomly coloring alattice of sites or bonds with a probability equal to themean porosity and measuring the statistics of conductionthrough clusters of connected sites or bonds. Continuumpercolation models, such as the “swiss cheese model”,correspond to randomly placing or removing overlappingparticles of given shapes to form clusters. The strikinggeneral feature of such models is the existence of a criticalporosity εc in the thermodynamic limit of an infinite sys-tem, below which the probability of a spanning infinitecluster is zero, and above which it is one. The criticalpoint depends on the specific percolation model, and forlattice models and decreases with increasing coordina-tion number (mean number of connected neighbors), as

more paths across the system are opened. Just above thecritical point, the effective conductivity scales as a powerlaw

σperc ∼ (εp − εc)tp (26)

where the exponent is believed to be universal for allpercolation models in the same embedding dimensionsand equal to tp = 2 in three dimensions. A simple formto capture this behavior is

σperc ∼=

σp(εp−εc1−εc

)2

εc ≤ εp ≤ 1

0 0 ≤ εp ≤ εc. (27)

C. Diffusion in Porous Media

We now relate the conductivity to the effective diffu-sivity of the porous medium. The porosity is the volumeof the electrolyte as a fraction of the total volume. Ifthe porosity is assumed to be constant throughout thevolume, then the area of each face of the volume is pro-portional to the porosity. Also, the total mass inside thevolume is given by the volume averaged concentration,c = εpc. We begin with a mass balance on the volume,

∂c

∂t+∇ · F = 0, (28)

where F is the flux at the surfaces of the volume. Thenet flux is

F = −σd∇c, (29)

where c is the concentration in the pores and σd is themean diffusive conductivity of the porous medium (withthe same units as diffusivity, m2/s), which, as the nota-tion suggests, can be approximated or bounded by theconductivity formulae in the previous section, with σpreplaced by the “free-solution” diffusivity Dp within thepores. It is important to recognize that fluxes are drivenby gradients in the microscopic concentration within thepores, c, and not the macroscopic, volume-averaged con-centration, c. Regardless of porosity fluctuations inspace, at equilibrium the concentration within the pores,which determines the local chemical potential, is constantthroughout the volume.

Combining Equations (28) and (29), we get

∂c

∂t= D∇2c, (30)

where the effective diffusivity in a porous medium, D, isgiven by

D =σdεp. (31)

The reduction of the diffusivity inside a porous mediumcan be interpreted as a reduction of the mean free path.

10

The tortuosity, τp, is often used to related the effec-tive macroscopic diffusivity to the microscopic diffusivitywithin the pores,

D =Dp

τp, (32)

as suggested long ago by Peterson [116]. One must keepin mind, however, that the tortuosity is just a way ofinterpreting the effective diffusivity in a porous medium,which is not rigorously related to any geometrical prop-erty of the microstructure. In Fick’s Law, which in-volves one spatial derivative, the tortuosity can be inter-preted as the ratio of an effective microscopic diffusionpath length Lp to the macroscopic geometrical length:Lp = τpL, although it is usually not clear exactly whatkind of averaging is performed over all possible paths.Indeed, other definitions of tortuosity are also used [117].(In particular, if the length rescaling concept is appliedto the diffusion equation, which has two spatial deriva-tives, then the definition D = Dp/τ

2 is more natural, butequally arbitrary.)

In any case, using the definition above, the effectiveconductivity can be expressed as

σd =Dpεpτp

(33)

which allows us to interpret all the models and boundsabove in terms of Peterson’s tortuosity τp. The upperbounds on conductivity become lower bounds on tortu-osity. The Wiener lower bound tortuosity for anisotropicpores is

τWienerp = 1. (34)

For the Hashin-Shtrikman model, the lower bound of thetortuosity is

τHSp =d− εpd− 1

(35)

in d dimensions. The percolation model produces a piece-wise function for the tortuosity, above and below the crit-ical porosity, which is given by

τpercp∼=

εp(

1−εcεp−εc

)2

εc ≤ εp ≤ 1

∞ 0 ≤ εp ≤ εc(36)

Note that, as the conductivity approaches zero, the tor-tuosity makes no physical sense as it no longer repre-sents the extra path length. Instead it represents thedecreasing number of available percolating paths, whichare the cause of the lowered conductivity. Finally, fromthe Bruggeman empirical relation we get the empiricaltortuosity formula,

τBp = ε−1/2, (37)

which is widely used in porous electrode models for bat-teries, stemming from the work of J. Newman and col-laborators. The different tortuosity models are plotted

in Figure 10, and we note again the close comparison ofthe Bruggeman-Newman formula to the rigorous Hashin-Shtrikman upper bound for an isotropic porous medium.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

εp

τ p

WienerHashin−ShtrikmanPercolationBruggeman

εc

FIG. 10. Tortuosity versus porosity for different effec-tive conductivity models. This plot gives the tortuosityfor different porosity values. While the Wiener and Hashin-Shtrikman models produce finite tortuosities, the percolationand Bruggeman models diverge as porosity goes to zero.

V. POROUS ELECTRODE THEORY

A. Conservation Equations

Using the principles laid out in the first section of thispaper on concentrated solution theory, the Porous Elec-trode Theory equations will be derived using mass andcharge conservation combined with the Nernst-PlanckEquation and a modified form of the Butler-VolmerEquation. The derivation will present the equations andhow their properties have deep ties to the thermody-namics of the system. Then, the equations will be non-dimensionalized and scaled appropriately using charac-teristic time and length scales in the system.

1. Mass and Charge Conservation

We begin with the definition of flux based on concen-trated solution theory. Assuming the system is close toequilibrium, the mass flux is

Ni = −Mici∇µi, (38)

where Mi is the mobility of species i, ci is the concen-tration of species i, and µi is the chemical potential ofspecies i. The conservation equation for concentration isgiven by the divergence of the flux,

∂ci∂t

= −∇ ·Ni −Ri. (39)

11

It is important to note that Ri is the volumetric con-sumption of species i. In order to express this conserva-tion equation in a form that is relevant to electrochem-ical systems, we must first postulate a suitable form ofthe chemical potential. We begin with the standard def-inition of the chemical potential including the activitycontribution, then include electrostatic effects to obtain

µi = kBT ln (ai) + zieφ. (40)

This chemical potential can be inserted into Equation(38). If the activity of the electrolyte is available fromexperimental values, then this form of the flux facilitatesits use. However, diffusivities are typically given as afunction of concentration. Simplifying Equation (38) us-ing Equation (40) for the chemical potential yields theNernst-Planck Equation,

Ni,± = −Dchem,i∇ci ∓ezikBT

Dici∇φ, (41)

where Dchem,i is the chemical diffusivity of species i,which is defined as

Dchem,i = Di

(1 +

∂ ln γi∂ ln ci

). (42)

The dilute limit diffusivity, Di, can also have concentra-tion dependence. Above, γi is the activity coefficient,and φ is the potential. The charge of the species is zi,which is treated as the absolute value.

For the bulk electrolyte, the electroneutrality approx-imation will be used. This approximation assumes thatthe double layers are thin, which is a reasonable approxi-mation when there is no depletion in the electrolyte. (Forporous electrode modeling including double layer effects,see Refs. [59–61].) The electroneutrality approximationassumes

ρ = z+ec+ − z−ec− ≈ 0, (43)

where z+ and z− are defined as the absolute values ofthe charge of the cation and anion, respectively. We willderive the ambipolar diffusivity, which assumes we havea binary z : z electrolyte.

For porous electrodes, we also need to account for theporosity of the medium. The porosity affects the inter-facial area between volumes of the porous electrode. Italso affects the concentration of a given volume of theelectrode. Accounting for porosity, Equations (39) and(38) become

ε∂ci∂t

= −∇ ·Ni −Ri (44)

and

Ni = −εMici∇µi, (45)

where ε is the porosity, which is the volume of electrolyteper volume of the electrode. This value may change with

position, but this derivation assumes porosity is constantwith respect to time. With this assumption, the Nernst-Planck Equation can be defined for the positive and neg-ative species in the electrolyte. This yields the cationand anion fluxes,

N+ = −εDchem,+∇c+ − εz+e

kBTD+c+∇φ, and (46)

N− = −εDchem,−∇c− + εz−e

kBTD−c−∇φ. (47)

Next, the flux equations for the cation and anion in Equa-tions (46) and (47) are inserted into Equation (44) andcombined with the electroneutrality assumption in Equa-tion (43) to eliminate the potential. The mass conserva-tion equation is

ε∂c

∂t= ∇ · (εDamb∇c)−∇ ·

((t+ − t−

2

)i

)−(

z+R+

2+z−R−

2

), (48)

where t+ and t− are the cation and anion transferencenumbers, respectively, and Damb is the ambipolar diffu-sivity. These values are defined as

t± ≡z±D±

z+D+ + z−D−, (49)

and

Damb ≡z+D+Dchem,− + z−D−Dchem,+

z+D+ + z−D−. (50)

In equation (48), i is the current density in the elec-trolyte, which is given by the sum of the cation and anionfluxes multiplied by their charge,

i = ez+N+ − ez−N−. (51)

Furthermore, the concentration c, using the electroneu-trality assumption, is defined as

c ≡ z+c+ = z−c−. (52)

Next, it is necessary to relate the charge conservation tothe mass conservation to simplify Equation (48).

The electroneutrality approximation puts a restrictionon the charge accumulation in the electrolyte. Since thecations and anions must balance, the divergence of thecurrent density must balance with the ions being pro-duced/consumed via Faradaic reaction in the volume.To determine the charge balance in some volume of theelectrode, we begin with the current density as given byEquation (51). Simplifying this expression and combin-ing it with the definition of c based on the electroneu-trality assumption, the current density is

i = −e (Dchem,+ −Dchem,−) ε∇c−e2

kBT(z+D+ + z−D−) εc∇φ. (53)

12

The divergence of the current density gives the accumu-lation of charge within a given volume. As stated above,this value must equal the charge produced or consumedby the reactions within the given volume, therefore

ez+R+−ez−R− = eap,+jin,+−eap,−jin,− = −∇·i, (54)

where ap,i is the area per unit volume of the active inter-calation particles, and jin,i is the flux into the particlesdue to Faradaic reactions of species i. For the remainderof the derivation, the term apjin will imply the sum of thereaction rates of the species. Substituting this expressioninto Equation (48) and using the definition t+ + t− = 1,the conservation equation is

ε∂c

∂t= ∇ · (εDamb∇c) +∇ ·

((1− t+) i

e

). (55)

Substituting Equation (54) into Equation (55), the fa-miliar Porous Electrode Theory equation,

ε∂c

∂t+ apjin = ∇ · (εDamb∇c)−∇ ·

(t+i

e

), (56)

is derived. Since the potential was eliminated in the am-bipolar derivation, and the potential gradient is depen-dent on the current density via Equation (53), Equations(53) and (54) can be used to formulate an expression forthe local electrolyte potential,

apjin = ∇ · [(Dchem,+ −Dchem,−) ε∇c+e2

kBT(z+D+ + z−D−) εc∇φ

]. (57)

Finally, an expression for jin is required to complete theset of equations. This can be modeled via the Butler-Volmer Equation.

For phase transforming materials, the activity of theatoms and energy of the transition state can have a dra-matic effect on the reaction rate. To account for this,a modified form of the Butler-Volmer Equation, whichaccounts for the energy of the transition state, will bederived.

2. Faradaic Reaction Kinetics

The reader is referred to Bazant [14, 15] for detailed,pedagogical derivations of Faradaic reaction rates in con-centrated solutions and solids, generalizing both the phe-nomenological Butler-Volmer equation [118] and the mi-croscopic Marcus theory of charge transfer [119–121].Here we summarize the basic derivation and focus ap-plications to the case of lithium intercalation in a solidsolution.

In the most general Faradaic reaction, there are n elec-trons transferred from the electrode to the oxidized stateO to produce the reduced state R:

O + ne− R.

Typically, one electron transfer is favored [118–120], butfor now let us keep the derivation as general as possible.The reaction goes through a transition state, which in-volves solvent reorganization and charge transfer. Thenet reaction rate, Rnet, is the sum of the forward andreverse reaction rates,

Rnet = k

[exp

(−µex‡ − µ1

kBT

)− exp

(−µex‡ − µ2

kBT

)].

(58)Once again, for an isothermal process (which is reason-able at the microscopic scale) the concentration of thetransition state is constant and can be factored into therate constant.

It is first necessary to postulate forms of the electro-chemical potentials in the generic Faradaic reactionabove. Here it is assumed that both the oxidant andreductant are charged species, and that the electron isat a potential φM , which is the potential of the metal-lic electron-conducting phase (e.g. carbon black). Theelectrochemical potentials of the oxidant and reductantare broken into chemical and electrostatic contributionsas follows:

µO = kBT ln aO + eqOφ− neφM + EO (59)

and

µR = kBT ln aR + eqRφ+ ER, (60)

where EO and ER are the reference energies of the ox-idant and reductant, respectively. The excess chemicalpotential of the transition state is assumed to consist ofan activity coefficient contribution and some linear com-bination of the potentials of the oxidant and reductant,

µex‡ = kBT ln γ‡ + αeqRφ+ (1− α)e (qOφ− nφM ) + E‡,(61)

where α, also known as the transfer coefficient, denotesthe symmetry of the transition state. This value is typi-cally between 0 and 1. Charge conservation in the reac-tion is given by

qO + n = qR (62)

At equilibrium, µO = µR, and the Nernst potential,

∆φeq = V o +kBT

neln

(aOaR

), (63)

is obtained, where V o = (EO − ER) /ne. Equations (59),(60), and (61) can be substituted directly into the gen-eration reaction rate, (58), to obtain

R =koγ‡

[aO exp

(EO − E‡

)exp

(−αn∆φ

)−

aR exp(ER − E‡

)exp

((1− α)n∆φ

)], (64)

where the energy is scaled by the thermal energy andthe voltage is scaled by the thermal voltage. Next, the

13

definition of overpotential is substituted into Equation(64). The overpotential is defined as

η ≡ ∆φ−∆φeq. (65)

Combining the definition of the overpotential with theNernst equation and substituting into Equation (64), af-ter simplifying we obtain the Modified Butler-VolmerEquation,

ejin = io [exp (−αη)− exp ((1− α) η)] , (66)

where io, the exchange current density, is defined as

io =neko (aO)

(1−α)n(aR)

αn

γ‡, (67)

and ko, the rate constant, is given by

ko = ko exp(αnER + (1− α)nEO − E‡

)(68)

The main difference is that the overpotential and ex-change current are defined in terms of the activities ofthe oxidized, reduced and transition states, each of whichcan be expressed variationally in terms of the total freeenergy functional of the system (below).

Using the Butler-Volmer Equation, the value of jin(the flux into the particles due to Faradaic reactions)can be modeled. The overpotential is calculated via thedefinition given in Equation (65), and the equilibriumpotential is given by the Nernst Equation, where the ac-tivity of the surface of the active material is used.

3. Potential Drop in the Conducting Solid Phase

The reaction rate at the surface of the particles is de-pendent on the potential of the electron as well as thepotential of lithium in the electrolyte. This is expressedas ∆φ, which contributes to the overpotential in Equation(65). The potential difference is the difference betweenthe electron and lithium-ion potential,

∆φ = φM − φ,

where φM is the potential of the metallic electron-conducting phase (e.g. carbon black) phase and φ is thepotential of the electrolyte. The potential of the elec-trolyte is determined by the charge conservation equationin Equation (54). To determine the potential drop in theconducting phase, we use current conservation which oc-curs throughout the entire electrode, given by

i + iM = I/Asep, (69)

where iM is the current density in the carbon black phase.For constant current discharge, the relation between thelocal reaction rate and the divergence of the current den-sity in the conducting phase is

eap,+jin,+ − eap,−jin,− = ∇ · iM . (70)

The current density in the conducting phase can be ex-pressed using Ohm’s Law. For a given conductivity ofthe conducting phase, the current density is

iM = −σm∇φ. (71)

The conductivity of the conducting phase can be mod-eled or fit to experiment based on porosity, the loadingpercent of the carbon black, and/or the lithium concen-tration in the solid,

σm = σm (cs, Lp, ε) .

As lithium concentration increases in the particles, thereare more electrons available for conduction. These are afew of the cell properties that can have a large impacton the conductivity of the solid matrix in the porouselectrode.

4. Diffusion in the Solid

Proper handling of diffusion in the solid particles re-quires the use of concentrated solution theory. Diffu-sion inside solids is often non-linear, and diffusivities varywith local concentration due to finite volume and otherinteractions inside the solid. The first section on con-centrated solution theory laid the groundwork for propermodeling of diffusion inside the solid. Here, we beginwith the flux defined in Equation (38),

Ni = −Mici∇µi,

where Ni is the flux of species i, Mi is the mobility, ciis the concentration, and µi is the chemical potential.With no sink or source terms inside the particles, themass conservation equation from Equation (39) is

∂ci∂t

= −∇ ·Ni. (72)

There are many different models which can be used forthe chemical potential. For solid diffusion, one modelthat is typically used is the regular solution model, whichincorporates entropic and enthalpic effects.[22, 97, 122].The regular solution model free energy is

g = kBT [c ln cs + (1− cs) ln (1− cs)] + Ωcs (1− cs) ,(73)

where cs is the dimensionless solid concentration (cs =cs/cs,max). Figure 11 demonstrates the effect of the reg-ular solution parameter (i.e. the pairwise interaction) onthe free energy of the system. The model is capable ofcapturing the physics of homogeneous and phase sepa-rating systems.

Homogeneous particles demonstrate solid solution be-havior, as all filling fractions are accessible. This behav-ior is typically indicated by a monotonically decreasingopen circuit voltage curve. In terms of the regular solu-tion model, a material that demonstrates solid solution

14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Filling Fraction

Dim

ensi

onle

ss F

ree

Ene

rgy,

g/k

BT

Ω=1kBT

Ω=2kBT

Ω=3kBT

Ω=4kBT

FIG. 11. Regular solution model for the free energy ofa homogeneous mixture. This figure shows the effect of theregular solution parameter Ω (mean pair interaction energy)and temperature T on the free energy versus composition c ofa regular solution of atoms and vacancies on a lattice. For Ω <2kBT , there is a single minimum. For Ω > 2kBT , there aretwo minima. This produces phase separation, as the system isunstable with respect to infinitesimal perturbations near thespinodal concentration, which is where the curvature of thefree energy changes.

behavior has a regular solution parameter of less than2kBT , that is Ω < 2kBT . This is related to the free en-ergy curve. When Ω ≤ 2kBT , there is a single minimumin the free energy curve over the range of concentrations.However, for Ω > 2kBT , there are two minima, result-ing in phase separation and a common tangent, whichcorresponds to changing fractions of each phase.

The common tangent construction arises from the factthat phases in equilibrium have the same chemical po-tential (i.e. slope). The chemical potential of the regularsolution model is

µ =∂gi∂cs,i

= kBT ln

(cs

1− cs

)+ Ω (1− 2cs) . (74)

To obtain an analogous equation to Fick’s First Law,Equation (38) can be expressed as

Ni = −Do (1− cs)(

1 +∂ ln γi∂ ln cs,i

)∇cs,i = −Dchem∇cs,i,

(75)where Do is the diffusivity of species i in the solid in theinfinitely dilute limit and Dchem is the chemical diffu-sivity in a concentrated solution. It is important to notethat Do can still be a function of concentration. The reg-ular solution model in Equation (74) can be substitutedinto Equation (75) using the definition of the chemicalpotential, µ = kBT ln(cγ), to obtain the chemical diffu-sivity,

Dchem = Do

(1− 2Ωcs + 2Ωc2s

), (76)

where Ω = Ω/kBT , the dimensionless interaction energy.When the interaction parameter, Ω, is zero, the dilute

limit diffusivity (Fick’s Law) is recovered. The mass con-servation equation using the effective diffusivity is

∂cs∂t

= ∇ · (Dchem∇cs) . (77)

Phase separating materials (e.g. LiFePO4) can be de-scribed by the Cahn-Hilliard free energy functional,[97]

G[c(x)] =

∫V

[ρsg(c) +

1

2κ (∇c)2

]dV +

∫A

γs (c) da,

(78)where g (c) is the homogeneous bulk free energy, ρs isthe site density, κ is the gradient energy (generally, atensor for an anisotropic crystal), with units of energyper length, and γs (c) is the surface tension, which is in-tegrated over the surface area A to obtain the total sur-face energy. The “gradient penalty” (second term) can beviewed as the first correction to the free energy for hetero-geneous composition, in a perturbation expansion aboutthe homogeneous state. When phase separation occurs,the gradient penalty controls the structure and energy ofthe phase boundary between stable phases (near the min-ima of g(c)). For example, balancing terms in (78) in thecase of the regular solution model, the phase boundarywidth scales as λi ≈

√κ/Ω, and the interphasial tension

as γi ≈√κΩρs [19, 22, 97].

More complicated phase-field models of the total freeenergy can also be used in our general porous electrodetheory. For example, elastic coherency strain can beincluded with additional bulk stress-strain terms [13,123, 124], as described below. It is also possible to ac-count for diffuse charge and double layers by incorpo-rating electrostatic energy in the total free energy func-tional [15, 42, 87, 88, 123], although we neglect such ef-fects here and assume quasi-neutrality in the electrolyteand active solid particles.

Once the total free energy functional is defined, thechemical potential of a given species is defined by theEuler-Lagrange variational derivative with respect toconcentration, which is the continuum equivalent of thechange in free energy to “add an atom” to the system.The chemical potential per site is thus

µ =1

ρs

δG

δc= µ (c)−∇ ·

(κ

ρs∇c), (79)

where µ is the homogeneous chemical potential. UsingEquation (38), the flux is based on the gradient of thechemical potential, and the conservation equation is

∂c

∂t= ∇ · (Mc∇µ) . (80)

For typical second-order diffusion equations, the bound-ary condition relates the normal flux to the reaction rateof each species. When the Cahn-Hilliard chemical poten-tial is used in Equation (79), however, the conservationequation contains a fourth derivative of concentration,requiring the use of another boundary condition. The

15

calculus of variations provides the additional “variationalboundary condition”,

n · κ∇ci =∂γs∂ci

(81)

which ensures continuity of the chemical potential [19]and controls surface wetting and nucleation [12].

The choice of the gradient and divergence operatorsis dependent upon the selected geometry of the particles.To complete the modeling of the particles, we impose twoflux conditions: one at the surface and the other at theinterior of the particle. For example, consider a sphericalparticle with a radius of 1. The boundary conditions are

∂c

∂r

∣∣∣∣r=0

= 0 (82)

and

−Ds∂c

∂r

∣∣∣∣r=1

= jin, (83)

where Ds is the solid diffusivity (can be a function of con-centration). These equations demonstrate the symmetrycondition at the interior of the particle, and the relationto the reaction rate at the surface of the particle, whichcomes from the modified Butler-Volmer Equation.

5. Modeling the Equilibrium Potential

To complete the model, a form of the open circuitpotential (OCP) is required. While traditional batterymodels fit the OCP to discharge data, the OCP is actu-ally a function of the thermodynamics of the material.The OCP can be modeled using the Nernst Equationgiven in Equation (63),

∆φeq = V o − kBT

neln

(aRaO

),

where V o is the standard potential. Typically, we takelithium metal as the reference potential for the anodeand cathode materials. For the cathode material, thisallows us to treat the activity of the oxidant as a constant.Let’s again consider the regular solution model. Usingthe definition for chemical potential, µ ≡ kBT ln a, wesubstitute in our regular solution chemical potential toget

∆φeq = V o − kBT

eln

(cs

1− cs

)− Ω

e(1− cs) . (84)

Figure 12 shows open circuit potential curves for differentregular solution parameter values. For Ω > 2kBT , thesystem is phase separating. This corresponds to a non-monotonic voltage diagram.

Since the reaction occurs at the surface, and the con-centration inside the solid is not necessarily uniform, then

0 0.2 0.4 0.6 0.8 1−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Filling Fraction

Vol

tage

(V o−

Vo )

Ω=0kBT

Ω=2kBT

Ω=4kBT

FIG. 12. Open circuit potential for different regularsolution parameter values. The battery voltage is thechange in free energy per electron transferred. In this model,the homogeneous voltage curve is non-monotonic when thesystem has a tendency for phase separation.

surface concentration determines the local OCP. Thisin turn affects the overpotential and the reaction rate.Larger overpotentials are required when the solid has aslow diffusivity. As lithium builds up at the surface ofthe particle, a higher overpotential is required to drivethe intercalation reaction.

B. Non-Dimensionalization and Scaling

In this section, the equations are non-dimensionalizedfor the full three dimensional case. Here we assumethe anode is lithium metal with fast kinetics. This al-lows us to model the separator and cathode. This non-dimensionalization can easily be expanded to model theanode as well. The electrode is assumed to have a con-stant cross sectional area, which is typical in rolled elec-trodes where the area of the separator is much largerthan the electrode thickness. The total current is thesum of the fluxes into the particles in the electrode. Thisis represented by the integral equation

I =

∫As

ejindAs =

∫Vs

eapjindVs, (85)

where ap is the area per volume of the particles. The solidvolume, Vs, can be expressed as (1− ε)LpV , where ε isthe porosity, Lp is the volume fraction of active material,and V is the volume of the cell. Scaling the time by thediffusive time (in the dilute limit), td = L2/Damb,o, andthe charge by the capacity of the entire electrode, thedimensionless current is

I =Itd

e (1− ε)PLV cs,max=

∫V

˜jindV , (86)

16

where the dimensionless reaction flux, jin, is defined as

jin =apjintdcs,max

. (87)

The non-dimensional current density in the electrolyteis

i = −(Dchem,+ − Dchem,−

)∇c−

(z+D+ + z−D−

)c∇φ,(88)

where the dimensionless current density i is defined as

i =tdi

Leco. (89)

The diffusivities in the dimensionless current densityequation above are scaled by the dilute limit ambipolardiffusivity. Similarly, the non-dimensional charge conser-vation equation becomes

βjin = −∇ · i, (90)

where β = Vscs,max/Veco is the ratio of lithium capacityin the solid to initial lithium in the electrolyte. This pa-rameter is important, as it determines the type of cell.For β 1, the system has essentially no storage ca-pability, and the equations are typically used to modelcapacitors. At β ≈ 1, the system has comparable stor-age in the electrolyte and solid. This is typically seen inpseudocapacitors. The equations for systems like thesetypically include a term for double layer charge storageas well. For β 1, there is a large storage capacity inthe solid, which is typically found in batteries.

Next, a mass balance on the electrolyte and solid areperformed. Equation (56) is non-dimensionalized forsome control volume inside the electrode. In this con-trol volume, the electrolyte and solid volumes are rep-resented by Ve and Vs, respectively. It is assumed thatthe electrode has the same properties throughout (e.g.porosity, loading percent, area per volume, etc.). Thedimensionless mass balance is

∂c

∂t+ βjin = ∇ ·

(Damb∇c

)− ∇ ·

(t+ i), (91)

where the time is scaled by the diffusive time scale, td,the gradients are scaled by the electrode length, L, thediffusivity is scaled by the dilute limit ambipolar diffu-sivity, Damb,o, the electrolyte concentration is scaled bythe initial electrolyte concentration, co, and the currentdensity, jin, is scaled as in Equation (87).

Next, we need to find the dimensionless boundary con-ditions for the system. This can be done via integratingthe equations over the volume of the cell (in this case theseparator and cathode, but this can easily be extended toinclude the anode). Integrating Equation (91) over thevolume yields∫

V

[∂c

∂t+ βjin = ∇ ·

(Damb∇c

)− ∇ ·

(t+ i)]dV . (92)

First, we deal with the left most term. Given the elec-troneutrality constraint, this term becomes zero becausethe amount of anions in the system remains constant.This assumes no SEI growth. If SEI growth is modeled,then this term will be related to the time integral of theanion reaction rate. Integrating the second term, for con-stant β, reduces to βI. The two terms on the right handside of the equation facilitate the use of the FundamentalTheorem of Calculus. Simplifying, we obtain

βI =(Damb∇c− t+ i

)∣∣∣10. (93)

Given the no flux conditions in y and z, and the no fluxcondition at x = 1, the flux into the separator is

−Damb∇c∣∣∣x=−xs

= (1− t+)βI. (94)

This set of dimensionless equations and boundary condi-tions are used in the simulations presented in the resultssection. Table I lists the equations used in the simula-tions.

VI. MODEL RESULTS

To characterize the properties of the model, we willdemonstrate some results from the non-dimensionalmodel. Again it is assumed that the anode is lithiummetal with fast kinetics, allowing us to model the sepa-rator and cathode. Results for monotonic (i.e. homoge-neous) and non-monotonic (i.e. phase separating) opencircuit potential profiles for particles demonstrating solidsolution behavior will be given for constant current dis-charge.

The electrolyte concentration, electrolyte potential,and solid concentration are all coupled via the mass andcharge conservation equations listed above. Solving theseequations is often done via Crank-Nicholson and use ofthe BAND subroutine, which is used to solve the systemof equations. [3] Botte et al. have reviewed the numeri-cal methods typically used to solve the porous electrodeequations. [125] The system of equations presented inthis paper was solved using MATLAB and its ode15sdifferential algebraic equation (DAE) solver. This codeutilizes the backwards differentiation formula (BDF) fortime stepping and a dogleg trust-region method for itsimplicit solution. The spatial equations were discretizedusing a finite volume method. Constant current dischargeinvolves an integral constraint on the system. This inte-gral constraint makes the system ideal for formulating thesystem of equations as a DAE. Formulation of the systemof equations as well as some basic numerical methods em-ployed in solving these types of DAE’s will be the focusof a future paper.

These results will highlight the range of physics in themodel, which include electrolyte diffusion limited dis-charge and solid diffusion limited discharge. These two

17

Equation Boundary Conditions

ε ∂c∂t

+ apjin = ∇ · (εDamb∇c)−∇ ·(t+i

e

)i|x=−δs = I/Asep

i = −e (D+ −D−) ε∇c− e2

kBT(z+D+ + z−D−) εc∇φ

jin = −∇·ieap

= io[exp

(− αeηkBT

)− exp

((1−α)eηkBT

)]

io =e(kocaO)1−α(koaaR)α

γ‡

η ≡ ∆φ−∆φeq

∆φeq = V o − kBTne

ln(aRaO

)∂cs∂t

= ∇ ·(DscskBT∇µ)

−DscskBT

∂µ∂r

∣∣∣r=R

= jin

TABLE I. Dimensional set of equations. A list of the set of dimensional equations for Modified Porous Electrode Theory.

limitations represent the most common situations in acell. Another common limitation is electron conductivityin the solid matrix. This limitation is often suppressedvia increasing the amount of conductive additive used.Furthermore, some active materials naturally conductelectrons, alleviating this effect.

The electrolyte diffusion limitation can also be alle-viated with proper cell design (i.e. thinner electrode),but this comes at the cost of capacity of the cell. Todemonstrate the effect of electrolyte diffusivity limita-tions and solid diffusivity limitations, different dischargerates were selected and different solid diffusivities weremodeled. First, we consider the case of homogeneousparticles. Then we demonstrate phase separating parti-cles using the Cahn-Hilliard free energy functional withand without approximated stress effects.

A. Simulation Values

The ambipolar diffusivity (given by Equation (50)) istaken from literature values for the diffusivity of Li+ andPF−6 in an EC/EMC non-aqueous electrolyte. Using lit-erature values for the diffusivities, a value of 1.9× 10−10

m2s−1 was calculated for Damb,o. [126, 127] Suitable cellsize parameters were used, including a cross sectionalarea of 1 cm2, separator thickness of 25µm, and an elec-trode length of 50µm. A porosity value of 0.4 was used,which is a little larger than typical cell values. While

cell dimensions are typically fixed, the ambipolar diffu-sivity and porosity values are flexible, and can be varied(within reason) to fit experimental data.

Using these cell dimensions and ambipolar diffusivity,the diffusive time scale for the system is 13.125 seconds.This value is important, as it affects the non-dimensionaltotal current (which is scaled by the electrode capacityand the diffusive time), the non-dimensional current den-sity, and the non-dimensional exchange current density(i.e. rate constant). Using this value of the ambipolar

diffusivity, a dimensionless current of I = 0.00364 cor-responds to approximately a 1C discharge. The soliddiffusivity is incorporated in a dimensionless parameter,

δd =L2sDamb

L2Ds(95)

which is the ratio of the diffusion time in thesolid (L2

s/Ds) to the diffusion time in the electrolyte(L2/Damb). This parameter, which is typically typicallylarger than one, can vary by orders of magnitude fordifferent materials. Typically, solid diffusivities are un-known, and this parameter needs to be fit to data.

The rate constant, which directly affects the exchangecurrent density, is another value that is unknown in thesystem. The dimensionless value of the exchange currentdensity is scaled to the diffusive time. It also dependson the average particle size, as this gives the surface areato volume ratio. For 50 nm particles, using the ambipo-lar diffusivity above, a dimensionless exchange current

18

density of one corresponds to approximately 1.38 A/m2.This is a relatively high exchange current density. Forthe simulations below, a dimensionless exchange currentdensity of 0.01 is used. It is important to note that thisvalue must be fit to data, though.

B. Homogeneous Particles

Homogeneous particles can access all filling fractions asthey are discharged. Here we consider homogeneous par-ticles using the regular solution model for the open circuitpotential and diffusivity inside the solid, as in Equation76. A value of Ω = 1kBT was used. Figures 14, 16,and 17 demonstrate the effect of various discharge ratesand solid diffusivities on the voltage profile. Each figurecontains three different voltage plots. The red dots onthe voltage curves indicate the filling fraction of the solidconcentration contours below. The contour plots are ar-ranged in the same order as the red dots, going from leftto right, top to bottom. Figure 13 gives the axes for thesimulations. Each particle is modeled in 1D, with theintercalation reaction at the top and diffusion into thebulk of the particle. The xs axis is the depth into theparticle.

x/L

x s/Lx,

s

Electrolyte diusion

FIG. 13. Plot axes for diffusion-limited solid-solutionparticles. This figure shows how the simulation results beloware plotted for porous electrodes with isotropic solid solutionparticles. The y-axis of the contour plots represent the depthof the particles while the x-axis represents the depth into theelectrode. The particles are modeled in 1D.

The contour plots give the solid concentration profile ofeach volume of particles along the length of the electrode.The y-axis is the depth in the solid particle, with the top(y = 1) denoting the interface between the particle andthe electrolyte. The x-axis denotes the depth into theelectrode, with the left side representing the separator-electrode interface and the right side representing thecurrent collector. It is important to note that in order forlithium to travel horizontally, it must first diffuse throughthe solid, undergo a Faradaic reaction to leave the solid,diffuse through the electrolyte, then intercalate into an-other particle and diffuse. Therefore sharp concentrationgradients in the x-direction are stable, especially for the

case of non-monotonic voltage profiles, as is seen in phaseseparating materials.

Figure 14 demonstrates the effect of various dischargerates on the voltage. At I = 0.001 (C/3), the dischargeis slow and the solid in the electrode fills homogeneouslythroughout. As the discharge rate is increased, increasedoverpotential follows. Furthermore, gradients in solidconcentration down the length of the electrode begin toemerge. Concentration gradients within the solid are notpresent because of the high solid diffusivity (δd = 1, in-dicating the solid and electrolyte diffusive time scales arethe same).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 13.05

3.1

3.15

3.2

3.25

3.3

3.35

3.4

3.45

3.5

3.55

Filling Fraction

Vol

tage

i=0.001i=0.01i=0.05

x s / L x,

s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

x s / L x,

s

x / Lx

0 0.2 0.4 0.6 0.8 10

0.5

1x s /

L x,s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

x s / L x,

s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

x s / L x,

s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

x s / L x,

s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

FIG. 14. Effect of current on homogeneous particles.This figure demonstrates the effect of different discharge rateson the voltage profile. The non-dimensional currents corre-spond to roughly C/3, 3C, and 15C. The solid diffusion isfast, with δd = 1.

As the current is increased, gradients in solid concen-tration across the electrode begin to become prevalent.At the same time, transport limitations in the electrolytelead to a capacity limitation, as the electrolyte is inca-pable of delivering lithium quickly enough deeper into theelectrode. Figure 15 demonstrates the electrolyte deple-tion leading to the concentration polarization in the 15Cdischarge curve. While the voltages appear to stop, theseare actually points where it drops off sharply. Tighter tol-erances, which can significantly increase the computationtime, are needed to get the voltage down to zero.

19

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 13

3.05

3.1

3.15

3.2

3.25

3.3

3.35

3.4

3.45

3.5

Filling Fraction

Vol

tage

x s / L x,

s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

−0.5 0 0.5 10

1

2

3c

/ c0

x / L

x s / L x,

s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

−0.5 0 0.5 10

1

2

3

c / c

0

x / L

x s / L x,

s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

−0.5 0 0.5 10

1

2

3

c / c

0

x / L

FIG. 15. Depletion of the electrolyte at higher cur-rent. This figure shows the depletion of the electrolyte ac-companying Figure 14 for the 15C discharge. The left figureshows the solid concentration while the right figure demon-strates the electrolyte concentration profile in the separatorand electrode.

It is important to note that δd is not the ratio of dif-fusivities, but the ratio of diffusive times. Therefore, asparticle size increases, the diffusive time scales as thesquare of the particle size. Solid diffusivities are typi-cally much slower than in the electrolyte. To demonstratethe effect of increased current with slower solid diffusion,Figure 16 demonstrates the same discharge rates as theprevious figure, except the solid diffusive time scale hasbeen increased to 100 times the electrolyte diffusive timescale.

For decreased solid diffusivity, concentration gradientsin the depth direction of the particles are more prevalent.At low current (i.e. slow discharge), the gradients in theelectrode and particles are minimal. As the current isincreased, gradients in the particles begin to emerge. Atthe fastest discharge rate, these solid concentration gra-dients become very large. Finite volume effects at thesurface of the particles increase the overpotential sub-stantially, producing a sharp voltage drop-off and lowutilization. This effect is caused by the slow solid dif-fusion only. Despite plenty of lithium being available inthe electrolyte, high surface concentrations block avail-

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 13.05

3.1

3.15

3.2

3.25

3.3

3.35

3.4

3.45

3.5

3.55

Filling Fraction

Vol

tage

i=0.001i=0.01i=0.05

x s / L x,

s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

x s / L x,

s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

x s / L x,

s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

x s / L x,

s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

x s / L x,

s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

x s / L x,

s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

FIG. 16. Effect of current on homogeneous particleswith slower solid diffusion. This figure demonstrates theeffect of different discharge rates on the voltage profile. Thenon-dimensional currents correspond to roughly C/3, 3C, and15C. The solid diffusion is slower than the electrolyte diffusion(δd = 100).

able sites for intercalation.

To show the effect of solid diffusion alone, Figure 17demonstrates the effect of decreasing solid diffusivity ata constant discharge rate. When the diffusive time scalesof the solid and electrolyte are comparable, each parti-cle fills homogeneously. There are small variations alongthe length of the electrode, but these do not affect theutilization, as almost 100% of the electrode is utilized.

As the solid diffusivity is decreased, and the diffusivetime scale approaches 50 times the electrolyte diffusivetime scale, we see over a 10% drop is capacity. Concen-tration gradients in the solid particles begin to emerge.As the solid diffusivity is further decreased, and the soliddiffusive time scale approaches 100 times the electrolytediffusive time scale, the solid concentration gradients be-come quite large, leading to a 50% drop in capacity.While these changes in δd seem significant, they repre-sent approximately a two order of magnitude change indiffusivity, and a one order of magnitude change in par-ticle size.

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 13.05

3.1

3.15

3.2

3.25

3.3

3.35

3.4

3.45

Filling Fraction

Vol

tage

δ=1δ=50δ=100

x s / L x,

s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1x s /

L x,s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

x s / L x,

s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

x s / L x,

s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

x s / L x,

s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

x s / L x,

s

x / L0 0.2 0.4 0.6 0.8 1

0

0.5

1

FIG. 17. Effect of solid diffusivity on homogeneousparticles. This figure demonstrates the effect of decreasingsolid diffusivity on the voltage profile. Each of these simula-tions was run at a dimensionless exchange current density of0.01 and a dimensionless current of 0.01.

C. Phase Separating Particles

For the case of phase separating materials, the equilib-rium homogeneous voltage curve is non-monotonic. Thisis demonstrated in Figure 12, for regular solution pa-rameters greater than 2kBT . For these materials, thefree energy curve has two local minima. When the sec-ond derivative of the free energy with respect to fillingfraction changes sign (positive to negative), the systemis unstable for infinitesimal perturbations, resulting inphase separation. A tie line represents the free energy ofthe system, and the proportion of the two phases changesas the system fills.

Modeling phase separating materials requires the useof the Cahn-Hilliard free energy functional as given inEquation (78), and the Cahn-Hilliard diffusional chem-ical potential, given in Equation (79). When we insertthe chemical potential into the modified Butler-VolmerEquation, we obtain a forced Allen-Cahn type equation.Here, we present the first solution of multiple phase sep-arating particles in a porous electrode.

For phase separating particles, values of Ω = 4kBT andκ = 0.001 were used along with a regular solution model

to model the homogeneous chemical potential, µ. Thesame exchange current as above was used. The figuresare similar to those of the homogeneous plots, but insteadof the depth direction, we now plot along the surface.Figure 18 depicts the axes plotted. This assumes that thediffusion into the particle is fast, and that the process isessentially surface reaction limited. This is a reasonableapproximation for LiFePO4. [12] Figure 19 demonstratesslow discharge (approx. C/30).

x/L

ys/Ly,s

Electrolyte diusion

FIG. 18. Plot axes for reaction-limited phase separat-ing particles. This figure shows how the results are plottedbelow for porous electrodes with reaction-limited phase sepa-rating nanoparticles. The y-axis of the contour plots representthe length along the surface of the particle, since diffusion isassumed to be fast in the depth direction. The x-axis repre-sents the depth in the electrode.

Initially, the discrete filling of the electrode suppressesphase separation inside the particles. Towards of theend of the discharge, decreased electrolyte diffusion (fromlonger path length) allow for particles to phase separate.Another important feature of the simulation is the volt-age spikes towards the end of the simulation. These volt-age spikes, which are on the order of the thermal volt-age, are an artifact of the discrete nature of the model.Towards the end of the simulation, only a few particlesremain to fill, therefore the voltage is dominated by effec-tively the single particle response. Dreyer et al. demon-strated this previously for phase separating particles fill-ing homogeneously. [10, 11]

The kinetics of phase separating particles can also beheavily influenced by stress effects, as demonstrated re-cently by Cogswell and Bazant. [13] Including stress in-volves the addition of energy terms in the free energymodel. With stress included, the full form of the bulkfree energy functional is

G[c(x)] =

∫V

[ρsf(c) +

1

2κ (∇c)2

+1

2Cijklεijεkl − σijεij

]dV,

(96)where the additional terms represent the elastic strain en-ergy and the homogeneous component of the total strain,respectively. (Here, we neglect the surface term, whichmainly affects nucleation of phase separation via surfacewetting [12].) The effects of coherency strain on phaseseparation can be approximated by a volume averaged

21

FIG. 19. Phase separating particles slowly discharged.This figure shows slow discharge (approx. C/30) of phase sep-arating particles. Adequate electrolyte diffusion and discretefilling don’t allow time for the particles to phase separateearly on. At the end of the discharge, sufficient time allowsthe particles to phase separate.

stress term [100, 128, 129] The homogeneous componentof the total strain is then

1

2Cijklεijεkl ≈

1

2B (c−X)

2, (97)

where X is the volume averaged concentration. This ap-proximation limits local fluctuations and promotes homo-geneous filling depending on the value of the constant B(which generally depends on orientation [13]). Includingthis term the chemical potential we obtain

µ = µ−∇ ·(κ

ρs∇c)

+B

ρs(c−X) . (98)

As the difference between the local and average globalparticle concentration increases, the overpotential re-quired to drive the intercalation reaction increases. Thispromotes homogeneous filling of the particles. Figure 20demonstrates how this additional term suppresses phaseseparation. However, the discrete filling still producesthe voltage plateau and spikes in voltage.

While these spikes appear to be large, they are actuallyon the order of the thermal voltage or smaller. At typi-cal voltage scales (2.0V-3.5V) these spikes are not seen,

FIG. 20. Phase separating particles including coherentstress effects slowly discharged. This figure shows slowlydischarge (approx. C/30) phase separating particles. Theinclusion of the coherent stress effects suppresses phase sepa-ration inside the particles. This figure is the same as Figure19, with an additional coherent stress term.

resulting in a flat voltage profile as seen in experimen-tal data for LiFePO4. This demonstrates that a phaseseparating material’s flat voltage profile can be modeledwithout modeling phase transformation itself. The volt-age spikes depend on the value of the Damkohler number,or ratio of the diffusion time across the porous electrodeto the typical reaction time to fill an active particle.

Figure 21 shows a faster (3C) discharge of the phaseseparating particles. The voltage spikes are suppressedand the voltage curve resembles “solid solution” behav-ior. There are three small voltages fluctuations present inthe simulation which are caused by the discrete filling ef-fect. However, instead of individual particles filling, nowlarger clusters of particles fill to alleviate the current (i.e.the number of active particles, or particles undergoing in-tercalation, has increased). To explain this, consider theequivalent circuit for a porous electrode in Figure 22.

The particles are represented by equivalent circuits.Each particle (which could also be considered to be acluster of particles with similar properties) has a chargetransfer resistance, Rct, and capacitance Cp. These val-ues can be non-linear, and vary depending on the particlefilling fraction and/or local potential. For each particleor cluster of particles, there is a charging time, τc, which

22

FIG. 21. Effect of current on phase separating parti-cles. When discharged at a higher C-rate (in this example,3C), the size of the discrete particle filling is larger, leading tomore particles filling simultaneously and a voltage curve thatresembles solid solution behavior.

scales as

τc ∼ RctCp. (99)

For a given discharge rate at constant current, particlesin the electrode must alleviate a given amount of lithiumper time in the electrode.

FIG. 22. Equivalent circuit model for a porous elec-trode. This equivalent circuit represents a typical porouselectrode in cases without significant electrolyte depletion,where the pore phase maintains nearly uniform conductiv-ity. Resistors represent the contact, transport, and chargetransfer resistances, and the capacitance of the particles isrepresented by a capacitor. All elements are not necessarilylinear.

The number of active particles scales as

nap ∼ τcI . (100)

As the discharge rate is increased, the number of activeparticles increases until it spans the electrode, resultingin the electrode filling homogeneously. For fast kineticsor slow discharge rates, the number of active particles issmall, which produces the discrete filling effect. For thenon-monotonic OCP of homogeneous phase separatingparticles, the voltage plateau has three filling fractionsthat can exist in equilibrium: the left miscibility gap fill-ing fraction, half filling fraction, and right miscibility gapfilling fraction. As the particles fill, if the kinetics aresufficiently fast, then other particles close to the activeparticle will empty to reach the equilibrium voltage (theplateau voltage). This increase in voltage for each par-ticle as it deviates from the voltage at the spinodal con-centration leads to an increase in cell voltage, producingthe voltage spikes.

For slower kinetics, this effect is suppressed by twomechanisms. First, the charge transfer resistance islarger, leading to higher charging times and subsequentlya larger number of active particles. Also, slower kinet-ics hinders the ability of particles to easily insert/removelithium, which prevents the particles from emptying andincreasing the voltage, leading to the spikes.

VII. SUMMARY

In this paper, we have generalized porous electrodetheory using principles of non-equilibrium thermodynam-ics. A unique feature is the use of the variational formu-lation of reaction kinetics [14, 15], which allows the use ofphase field models to describe macroscopic phase trans-formations in porous electrodes for the first time. Thethermodynamic consistency of all aspects of the modelis crucial. Unlike existing battery simulation models,the open circuit voltage, reaction rate, and solid trans-port properties are not left free to be independently fitto experimental data. Instead, these properties are alllinked consistently to the electrochemical potentials ofions and electrons in the different components of theporous electrode. Moreover, emergent properties of aphase-separating porous electrode, such as its voltageplateau at low current, are not fitted to empirical func-tional forms, but rather follow from the microscopicphysics of the material. This allows the model to capturestochastic, discrete phase transformation events, whichare beyond the reach of traditional diffusion-based porouselectrode theory.

In a companion paper [130], we will apply the modelto predict the electrochemical behavior of composite,porous graphite anodes [131] and LFP cathodes [10], eachof which have multiple stable phases. Complex nonlin-ear phenomena, such as narrow reaction fronts, mosaicinstabilities, zero current voltage gap, and voltage fluctu-ations, naturally follow from the simple physics contained

23

in the model. The model is able to fit experimental datafor phase transformations in porous electrodes under verydifferent conditions, limited either by electrolyte diffu-sion [131] or by reaction kinetics [10].

This work was supported by the National ScienceFoundation under Contracts DMS-0842504 and DMS-0948071 (H. Warchall) and by a seed grant from the MITEnergy Initiative.

VIII. LIST OF SYMBOLS USED

NOTE: unless explicitly noted, all quantities with atilde denote dimensionless quantities. Energies are scaledby the thermal energy, kBT , and potentials are scaled bythe thermal voltage, kBT/eSymbols used:

a activity (dimensionless)ap pore area per volume [1/m]A area [m2]Acell area of unit cell (CST derivation) [m2]Asep area of separator [m2]B volume averaged elastic strain energy [J/m3]c number concentration [1/m3]c dimensionless concentrationc volume averaged number concentration [1/m3]cmax maximum number concentration (solubility limit) [1/m3]Cp capacitance [C/V]Cijkl elastic stiffness tensor [J/m3]d dimensionalityD diffusivity [m2/s]Damb ambipolar diffusivity [m2/s]Dchem chemical diffusivity [m2/s]Do tracer diffusivity [m2/s]Dp diffusivity inside a pore [m2/s]D effective diffusivity [m2/s]e elementary charge [C]ei coordinate vectorEO reference energy of oxidant [J]ER reference energy of reductant [J]E‡ reference energy of transition state [J]f homogeneous free energy per volume [J/m3]F number flux [1/m2s]g free energy per lattice site [J]G total free energy [J]i current density [C/m2s]io exchange current density [C/m2s]I total current [C/s]jin reaction flux [1/m2s]ko rate constant [1/s]ko modified rate constant [1/s]kB Boltzmann’s constant [J/K]L characteristic length [m]

Lp characteristic pore lengthM mobility [m2/Js]Mi chemical symbol of species in number electrons transferrednap number of active particlesN number species flux [1/m2s]PL loading percent of active material by volumeq species charge numberr radial direction [m]R reaction rate [1/m3s]Rct charge transfer resistanceS1 stoichiometric sum of reactantsS2 stoichiometric sum of productssi stoichiometric coefficientst time [s]td characteristic diffusion time [s]tp percolation exponentt± transference number of positive/negative speciesT temperature [K]V volume [m3]x spatial direction [m]X average dimensionless concentrationzi charge number of species iGreek symbols:

α transfer coefficientβ ratio of solid:electrolyte lithium capacityδd ratio of characteristic solid:electrolyte diffusive timesε porosity (pore volume per total volume)εij strainεij homogeneous component of elastic strainη overpotential [V]η dimensionless overpotentialγ activity coefficient [m3]γ‡ activity coefficient of transition state [m3]κ gradient energy [J/m]κ dimensionless gradient energyµ chemical potential [J]µ dimensionless chemical potentialµex excess chemical potential [J]µ homogeneous chemical potential [J]µo reference chemical potential [J]ν attempt frequency [1/s]Ω regular solution interaction parameter [J]φ electrolyte potential [V]

φ dimensionless potentialΦ volume fractionρs site density [1/m3]σ conductivity [S/m]σ effective conductivity [S/m]σd diffusive mean conductivity [m2/s]σij applied external stress tensor [N/m2]τ time between transitions [s]τc charging time [s]τp tortuosity (pore length per total length)

24

τo barrier-less transition time [s]Subscripts:

+ positive species

− negative species

B Bruggeman model

c critical (percolation model)

eq equilibrium

i species i

O oxidant

p pore phase

perc percolation model

M metal/electron conducting phase

max maximum

min minimum

R reductant

s solid (intercalation) phaseSuperscripts:

B Bruggeman modelHS Hashin-Shtrikman modelperc Percolation modelWiener Wiener model

25

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