Nonlinear coupled equations for electrochemical cells as developed by the general equation for...

Nonlinear coupled equations for electrochemical cells as developed by the generalequation for nonequilibrium reversible-irreversible couplingDick Bedeaux, Signe Kjelstrup, and Hans Christian Öttinger Citation: The Journal of Chemical Physics 141, 124102 (2014); doi: 10.1063/1.4894759 View online: http://dx.doi.org/10.1063/1.4894759 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The role of non-equilibrium fluxes in the relaxation processes of the linear chemical master equation J. Chem. Phys. 141, 065102 (2014); 10.1063/1.4891515 Mechano-chemical coupling in Belousov-Zhabotinskii reactions J. Chem. Phys. 140, 124110 (2014); 10.1063/1.4869195 Superfast oxygen exchange kinetics on highly epitaxial LaBaCo2O5+δ thin films for intermediate temperaturesolid oxide fuel cells APL Mat. 1, 031101 (2013); 10.1063/1.4820363 Computational fluid dynamics model development on transport phenomena coupling with reactions inintermediate temperature solid oxide fuel cells J. Renewable Sustainable Energy 5, 021420 (2013); 10.1063/1.4798789 Evaluation of Sr 2 CoMoO 6 − δ as anode material in solid-oxide fuel cells: A neutron diffraction study J. Appl. Phys. 109, 034907 (2011); 10.1063/1.3544068

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.241.87.96 On: Fri, 10 Oct 2014 09:28:09

http://scitation.aip.org/content/aip/journal/jcp?ver=pdfcov

http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/1017745580/x01/AIP-PT/JCP_ArticleDL_1014/AIP-2293_Chaos_Call_for_EIC_1640x440.jpg/47344656396c504a5a37344142416b75?x

http://scitation.aip.org/search?value1=Dick+Bedeaux&option1=author

http://scitation.aip.org/search?value1=Signe+Kjelstrup&option1=author

http://scitation.aip.org/search?value1=Hans+Christian+�ttinger&option1=author

http://scitation.aip.org/content/aip/journal/jcp?ver=pdfcov

http://dx.doi.org/10.1063/1.4894759

http://scitation.aip.org/content/aip/journal/jcp/141/12?ver=pdfcov

http://scitation.aip.org/content/aip?ver=pdfcov

http://scitation.aip.org/content/aip/journal/jcp/141/6/10.1063/1.4891515?ver=pdfcov

http://scitation.aip.org/content/aip/journal/jcp/140/12/10.1063/1.4869195?ver=pdfcov

http://scitation.aip.org/content/aip/journal/aplmater/1/3/10.1063/1.4820363?ver=pdfcov

http://scitation.aip.org/content/aip/journal/aplmater/1/3/10.1063/1.4820363?ver=pdfcov

http://scitation.aip.org/content/aip/journal/jrse/5/2/10.1063/1.4798789?ver=pdfcov

http://scitation.aip.org/content/aip/journal/jrse/5/2/10.1063/1.4798789?ver=pdfcov

http://scitation.aip.org/content/aip/journal/jap/109/3/10.1063/1.3544068?ver=pdfcov

THE JOURNAL OF CHEMICAL PHYSICS 141, 124102 (2014)

Nonlinear coupled equations for electrochemical cells as developedby the general equation for nonequilibrium reversible-irreversible coupling

Dick Bedeaux,1,2,a) Signe Kjelstrup,1,2,b) and Hans Christian Öttinger1,c)

1Department of Material Science, Swiss Federal Institute of Technology, Zurich, Switzerland2Department of Chemistry, Norwegian University of Science and Technology, Trondheim, Norway

(Received 22 May 2014; accepted 25 August 2014; published online 22 September 2014)

We show how the Butler-Volmer and Nernst equations, as well as Peltier effects, are contained in thegeneral equation for nonequilibrium reversible and irreversible coupling, GENERIC, with a uniquedefinition of the overpotential. Linear flux-force relations are used to describe the transport in thehomogeneous parts of the electrochemical system. For the electrode interface, we choose nonlinearflux-force relationships. We give the general thermodynamic basis for an example cell with oxygenelectrodes and electrolyte from the solid oxide fuel cell. In the example cell, there are two acti-vated chemical steps coupled also to thermal driving forces at the surface. The equilibrium exchangecurrent density obtains contributions from both rate-limiting steps. The measured overpotential isidentified at constant temperature and stationary states, in terms of the difference in electrochem-ical potential of products and reactants. Away from these conditions, new terms appear. The ac-companying energy flux out of the surface, as well as the heat generation at the surface are for-mulated, adding to the general thermodynamic basis. © 2014 Author(s). All article content, exceptwhere otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License.[http://dx.doi.org/10.1063/1.4894759]

I. INTRODUCTION

The aim of this work is to show how the expressions thatare in use for the surface overpotential of an electrode canbe connected via a broader thermodynamic basis. The surfaceoverpotential is a central concept in electrochemistry, and itis of utmost importance to have a link between its theoreticaldescription and its various measurements. Before we explainwhy a better basis is needed, we briefly recapitulate how thesurface overpotential, η, is measured and interpreted. The aimis to specify in detail, conditions for which the well-knownexplicit expressions for η are valid, by providing more generalexpressions.

The surface overpotential is often obtained1–3 from athree-electrode set up, see Fig. 1. An electric current is pass-ing the external circuit between the working- (W) and counter(C) electrode. The potential difference V is measured betweenW and a currentless reference electrode (R) of the same kindas W. The overpotential is calculated as the difference be-tween the measurement of V at current density j and j = 0.Ohmic potential drops that contribute to the measurement inFig. 1 are corrected for, for instance, by current interruptiontechniques. This gives

η = Vj − Vj=0. (1)

It is well known that the measurement is sensitive to the posi-tioning of the reference electrode R.4–6 The measurement hasbeen interpreted as departure of the surface potential jumpfrom the equilibrium potential jump.1, 2 The overpotential is

a)Electronic mail: [email protected])Electronic mail: [email protected])Electronic mail: [email protected]

also routinely determined by impedance experiments, whichgive the surface resistance Rp and η = Rpj. The method is usedwith small amplitudes of the oscillating field, in the presenceand absence of a direct current.

The relation between j and η for large, steady state valuesof j is expressed by the Butler-Volmer equation. The equationgives j as the difference between two exponential functionswhich can be interpreted as unidirectional rates. For the an-ode, we have2

j = j0

[exp

(n(1 − β)Fη

RT

)− exp

(−nβFη

RT

)]. (2)

Here β is the symmetry factor or the apparent transfer coeffi-cient, while R, T, n, and F are the gas constant, the absolutetemperature, the number of electrons involved in the electrodereaction, and Faraday’s constant, respectively. In the deriva-tion of (2), β has been used to indicate the constant positionin space of the activation energy barrier in front of the sur-face. It is 0.5, if the peak is symmetric. The parameter j0 isthe equilibrium exchange current density, expressing the two-way traffic in the absence of an overpotential. The j0 has anArrhenius-type behavior. Equation (2) owes its success to thenumerous experimental results that follow the predicted rela-tionship.

The electrochemical literature deals with the product ηjas a pure heat source at the electrode in question.1 A scalarchemical reaction rate does not couple to vectorial forces ina homogeneous phase.7–10 With the word coupling we referto the direct impact one driving force has on other fluxes ina system.9 But such coupling is possible at electrode sur-faces, leading to work, not dissipation. At the surface, thescalar component of a vectorial force, directed normal to the

0021-9606/2014/141(12)/124102/17 © Author(s) 2014141, 124102-1


129.241.87.96 On: Fri, 10 Oct 2014 09:28:09

http://dx.doi.org/10.1063/1.4894759

http://dx.doi.org/10.1063/1.4894759

http://dx.doi.org/10.1063/1.4894759

mailto: [email protected]



http://crossmark.crossref.org/dialog/?doi=10.1063/1.4894759&domain=pdf&date_stamp=2014-09-22

124102-2 Bedeaux, Kjelstrup, and Öttinger J. Chem. Phys. 141, 124102 (2014)

FIG. 1. A sketch of the overpotential measurement. The working electrodeW is an anode. Current is passed between W and the counter electrode C.The reference electrode is R (with zero electric current). There is an Ohmicpotential drop in the electrolyte to be corrected for. The arrow indicates thedirection of the electron current.

surface, can couple to the scalar chemical force.10, 11 In otherwords, chemical or thermal energy can here be used to pro-duce electrical work, or vice versa.

Equation (2) includes, however, no explicit coupling toother processes. In spite of a predicted large dissipation of en-ergy as heat in the surface due to the product ηj, one normallyassumes isothermal conditions at the electrode surface. There-fore, one may question whether there is a broader thermody-namic basis for the Butler-Volmer equation that include cou-pling and can distinguish between reversible and irreversibleeffects.

Classical nonequilibrium thermodynamics has been usedto formulate the coupling of the electric potential at anelectrode interface to interface temperature jumps.7–9 Lin-ear flux-force relations were used, however, while Eq. (2) isclearly nonlinear. A nonlinear current-potential relationshiphas been derived for an isothermal electrode a using meso-scopic nonequilibrium thermodynamics.12 In the mesoscopicbranch of nonequilibrium thermodynamics, internal variablesare introduced. In order to obtain an expression that applies tothe macroscopic level, like the form (2), one has to integrateover the internal variables. The integration depends on the as-sumption of a quasi-stationary state;13 an assumption relatedto the mechanism of transport which may not always hold.14

These facts have motivated the present work. We areseeking to give the nonlinear form of (2) a more generalthermodynamic basis, avoiding assumptions that restrict itsapplication. We are exploring the possibility for relations toother driving forces, and conditions under which they apply.From this, we aim to derive a well defined expression for theoverpotential which can provide insight into the experimentalsituation.

We shall use a cell with identical electrodes in the firstattempt to analyze the problems posed. The electrode is thetechnically important oxygen electrode of the solid oxide fuelcell, see, e.g., Shao and Haile.5, 15 The symmetrical cell canfor convenience be divided into two halves. One half cell maylater be combined with another half cell to produce a forma-

FIG. 2. A sketch the cell with two oxygen electrodes. Subsystems are num-bered 1-4. The flux of the oxygen is shown at the anode, a and cathode, c,in the stationary state. The electrode surface is denoted by s, the electrolyteby e.

tion cell, using the same procedures. We present a procedure,using this example for illustration, and show that we can gainnew general insight in this manner.

The system is illustrated in Fig. 2. On the anode side (a),electrons and oxygen are produced from oxygen ions comingin this case from the solid state electrolyte:

2O2−(e) ⇀↽ O2(a) + 4e−(a). (3)

The reaction at the cathode side (c) is the reverse. The fluxesinto and out of the anode, corresponding to this reaction areillustrated in Fig. 2. The five subsystems in series are: (1) aporous electron conducting material as anode, a, (2) the sur-face of the anode, i.e., the material(s) between the electronicconductor and the electrolyte, s, a, (3) the electrolyte, e, (4)the surface of the cathode, i.e., the material(s) between theelectrolyte and the electron conducting materials, s, c, and (5)a porous electron conducting material as cathode, c. The cellcan, in spite of being relatively simple, convert thermal energyinto electrical energy, from reversible heat exchange with thesurroundings.16

II. PROPERTIES OF GENERIC

GENERIC17 (general equation for nonequilibrium re-versible and irreversible coupling) is designed to test the com-patibility of a model with the laws of thermodynamics. Re-versible and irreversible contributions to the equation of mo-tion are identified, making it possible to answer questions ondissipative and other phenomena. GENERIC provides expres-sions for the time-evolution of system variables, grouped invectors x. It moreover is a powerful tool for thermodynamicmodeling activities, allowing us to digest experimental infor-mation, empirical equations, statistical mechanical models,symmetries, and intuition into thermodynamically consistentequations. In compact notation we write

dx

dt= L(x) · δE

δx+ M(x) · δS

δx, (4)

where x is the (sub)system variable vector, t is the time, Eis the system total energy, and S is the system total entropy.By writing δ/δx in front of E or S, we mean to perform afunctional derivative with respect to all independent variablesof the system. Different choices of the vector x are possible.The symbols L(x) and M(x) are also operators on the system


129.241.87.96 On: Fri, 10 Oct 2014 09:28:09


variables called the Poisson and the friction matrix operators,respectively.

The Poisson matrix is an operator which gives the re-versible contributions to the time evolution of the system vari-ables. The application to δS/δx is zero because there are noentropy changes associated with reversible processes.

The Poisson matrix is always antisymmetric, while thesymmetry property of the M-matrix is not clear from the out-set. A symmetric M-matrix can be expected17 for ensemblesthat remain essentially unchanged on the time scale of theobservation. When the ensembles differ from that, typicallywhen the process progresses by rare events, the symmetry canbe lost. A nonsymmetric M-matrix was previously found forthe GENERIC formulation of the Boltzmann equation.17 Wedo not expect a symmetric M- matrix for the activated pro-cesses behind Eq. (2). Also when the M-matrix is not sym-metric, it must be positive semi-definite.

The friction matrix gives the irreversible contributions tothe time evolution of the system variables. Energy is con-served, also during dissipation, so the application to δE/δxgives also zero. Taking the possible symmetry properties intoaccount, we write

δS

δx· L(x) = 0 = L(x) · δS

δx, (5)

δE

δx· M(x) = 0. (6)

These equations are called the degeneracy conditions. Theyhave a predictive power in the way that they can be used torule out forms which do not comply with the laws of thermo-dynamics, and shall be used here. The derivatives δE/δx andδS/δx in the degeneracy conditions are transposed. We do notexplicitly indicate this, following Ref. 17.

GENERIC can be used with several variable sets, includ-ing sets that do not belong to classical irreversible thermody-namics. The method can, for instance, handle the variable setproposed by extended irreversible thermodynamics, which in-cludes fluxes as variables. The form of the entropy productionwill depend on the set. For the set of classical thermodynamicvariables used here, the resulting equations obey, as we shallsee, local equilibrium. In the context of nonlinear flux-forcerelations, this is noteworthy.

In this work we develop for the first time GENERIC (4)for an electrochemical system. GENERIC is used in the out-set for isolated systems. This may appear as a problem, aselectrochemical cells are always open. We can proceed to liftthe restriction, once the results from a time-dependent iso-lated system are obtained. We proceed to follow the prescrip-tions of GENERIC for the five subsystems of the cell sketchedin Fig. 2. Each subsystem will first be characterized by a setof independent variables. We next give the contributions toE and S for each subsystem. The functional derivatives of Eand S follow, leading to construction of the operators L(x) andM(x). The degeneracy requirements (5) and (6) are used toconfirm the correctness of the equations of motion obtained.The GENERIC description of bulk phase phenomena is to alarge extent taken from the literature,17 while the GENERICdescription of surface phenomena is new.

The aim is at the end to make connections to the experi-mental situation, and to see how Eq. (2), the Nernst equation,and thermal coupling (Peltier) effects all can be derived fromthe same basis.

III. THERMODYNAMIC VARIABLES OF CELLSUBSYSTEMS

The first step is to specify sets of variables for each of thefive subsystems of the total cell. We choose concentrations(molar densities), momentum densities (mass densities timesvelocities), and internal energy densities as independent vari-ables for all subsystems. As an example take a hydrodynamicflow problem. This set of variables will then render the Eulerequations from the L(x)-operator.17, 18

The choice of a frame of reference for velocities is simplefor the cell in Fig. 2. The interface between a set of the non-moving solid state materials is chosen (see below for furtherdescriptions). Gradients and fluxes along the surfaces play norole for the systems considered, so the transport processes canbe regarded as one-dimensional. The z-axis is taken as the co-ordinate axis. All density-like variables and fluxes are thenaverage values over the cross-section of the cell and dependonly on z. All but one of the fluxes are in the z-direction. Theexception being the exchange of heat between the pores andthe metal in the electrodes, cf. Sec. III B. We start with thesimplest phases, the homogeneous phases. The electrochem-ical energy conversion takes place at the anode and cathodesurfaces, and the descriptions of these are more complicated.Analogies with the description of the homogeneous phasesare pointed out.

A. Subsystem 3: The electrolyte

The electrolyte, subsystem 3, is a homogeneous ceramicmaterial, some hundred μm thick.19 Yttria stabilized zirco-nium oxide has a number of oxygen ion vacancies in the lat-tice (typically 60 mol/m3,20) and oxygen ions can move in thelattice, jumping between vacancies. The distributions of Y3 +

over the Zr2 + lattice matches the vacancy distribution, mak-ing the material electroneutral. The vacancy concentration isa function of the oxygen pressure. The electrolyte does notallow oxygen molecules to short-circuit the cell.

As thermodynamic variables of the electrolyte, we takethe oxygen ion concentration, cO2− , the momentum, MO2− , ofthe moving oxygen ions, and the internal energy density, ue.The internal energy of the material can be split into contribu-tions from the cation lattice, ulat, and its oxygen ion part, uO2− ,where ue = uO2− + ulat. The state of subsystem 3 is describedby the variable vector x = (cO2− , MO2− , ue).

B. Subsystems 1 and 5: The electron conductorand gas supply channel

The bulk of the anode and cathode contains a current col-lector, normally a metal, and supply channels (pores) for oxy-gen in the form of air. This part of the system can be described


129.241.87.96 On: Fri, 10 Oct 2014 09:28:09


as two parallel transport systems, one for electrons and theother for oxygen molecules.

The density of conduction electrons in the metal is cme−(z),

while the oxygen density in the pores is cp

O2(z). These densi-

ties refer here to the total cross section of the anode or cath-ode volumes. The momentum density of electrons is Mm

e−(z)≡ me−cm

e−(z)ve − (z), while the momentum density of oxygenin the pores is Mp

O2(z) ≡ mO2

cp

O2(z)vO2

(z). Both momenta re-fer also to the total cross section of the anode or cathodevolumes.

The internal energy density in the metal, um(z) in J/m3,is the sum of the internal energy density of the metal atomsum

m(z) and the electrons ume−(z). The quantities um

m(z) andum

e−(z) are given in terms of the partial internal energies ofthe metal by

umm(z) = cm

m(z)∂um(z)

∂cmm(z)

and ume−(z) = cm

e− (z)∂um(z)

∂cme− (z)

. (7)

In the channels, the internal energy density up(z) of the air hascontributions from nitrogen and oxygen, u

pN2

(z) and up

O2(z),

respectively. They are given in terms of the partial internal en-ergies of the air by a relation similar to Eq. (7). All internal en-ergies refer to the total anode or cathode volumes. The metaland the pores have their own temperature profiles, Tm(z) andTp(z), respectively. There is heat exchange between the metaland the pores. We will not consider the possible adsorptionand decomposition of oxygen molecules on the surface of thepores.

On this background we choose as variables for themetal phase, the electron concentration, cm

e− , its momentumdensity Mm

e− ≡ me−cme−ve− = ρm

e−ve− and the internal energydensity um = um

m + ume− . The variable vector of the metal

in subsystems 1 and 5 is thus xm = (cme− , Mm

e− , um). Forthe gas phase, the corresponding variables are c

p

O2, Mp

O2

≡ mO2cp

O2vO2

= ρp

O2vO2

and up = upN2

+ up

O2. The variable

vector for the oxygen channels in subsystems 1 and 5 is thusxp = (cp

O2, Mp

O2, up). We assume that nitrogen is present in

the pores with no velocity. It only contributes to the energydensity.

C. Subsystems 2 and 4: The electrode surfaces

The most important subsystems are the electrode sur-faces. The material of the electrode surfaces, the catalystfor the electrochemical reaction, is in this case typically ananoporous perovskite, which adsorbs gas, and conducts elec-trons (holes) as well as oxygen ions. The material is in con-tact with the electron conductor, the gas channels (subsystems1 or 5), and with the electrolyte (subsystem 3). The materialadsorbs oxygen gas and contains a large fraction of oxygenvacancies.20

The perovskite has a certain thickness, say 10 μm, butwe can deal with it as a two-dimensional system, and use asvariables the integrals over the variable profiles in the ma-terial. Each layer with a finite thickness can be integratedand replaced by a surface in this manner. The alternative isto describe the perovskite layer as 2 surfaces and one homo-geneous layer. Coupling between the reaction and the heat

flux will then occur at the 2 surfaces. The outcome wouldnot change, but the analysis becomes considerably morecomplicated. The perovskite layer is the thinnest layer. It istreated as a single surface because the electrode reaction takesplace in this region only. It is impossible to locate the re-action in three-dimensional space. By locating it to a two-dimensional surface we can better describe its interaction withother forces. When densities have their peaks in the interfacialregion, the excess densities are in good approximation inde-pendent of the precise location of the dividing surface. Theequimolar surface of the electrolyte was chosen as the divid-ing surface for the Gibbs excess variables, see Fig. 3. All sur-face excess densities are defined relative to this non-movingsurface. Figure 3 shows how the oxygen excess density is ob-tained for the given dividing surface. The densities of the ad-joining bulk phases are extrapolated to the dividing surface.The excess density in the plot is the big area to the left minusthe small area to the right.

The excess momentum normal to the non-moving sur-face (which is a scalar quantity) is zero, as the excess massof the surface is not moving in the chosen frame of reference.This is compatible with a nonzero flux of matter through thesurface in this frame of reference. Excess momenta along thesurface are not relevant in the one-dimensional formulation ofthe problem. The only other variable is the excess internal en-ergy. The variable vector for the electrode surface (a or c) istherefore xs = (cs

e− , csO2

, cs

O2− , us). Superscript s denotes thesurface.

We rewrite the electrochemical reaction for the anodewith more precise reference to the sub-processes involved

2O2−(e) ⇀↽ 2O2−(s,a), (8)

2O2−(s,a) ⇀↽ O2(s,a) + 4e−(s,a), (9)

FIG. 3. The surface excess density of oxygen in the anode. The dividing sur-face is shown on the electrolyte side of the figure. The density is the differencebetween the big and the small area, obtained by extrapolating the oxygen con-centration in the pores to the left to the dividing surface. The electrolyte hasno oxygen gas e.


129.241.87.96 On: Fri, 10 Oct 2014 09:28:09


O2(s,a) ⇀↽ O2(a), (10)

4e−(s,a) ⇀↽ 4e−(a). (11)

Here s,a means the surface of the anode. The first equation de-scribes the adsorption of oxygen ions in the perovskite. Thenext equation is the reaction in the surface. The transfer ofoxygen to the channel, and the transfer of the electron to theconduction band follows. According to the literature,15, 19, 20

the chemical reaction (the second step) and the oxygen sur-face exchange (the third step) are slower than the other two.The second step is an activated electrochemical reaction. Thethird step, may also be activated, as the oxygen molecule atthe surface possibly is dissociated into 2 oxygen atoms. Theoverall reaction, (3), obtains the combined rate of (9) and (10).On the cathode side, all steps are reversed.

IV. THE SYSTEM TOTAL ENERGY AND ENTROPYAND THEIR FUNCTIONAL DERIVATIVES

The next step of GENERIC is to give the system’s to-tal energy and entropy. The total energy is found by straightforward summing over the internal, electrostatic, and kineticenergies of each subsystem:

E =∫

a

⎛⎝ua + ze−cmae− Fψa +

∣∣Mmae−

∣∣2

2me−cmae−

+∣∣Mpa

O2

∣∣2

2mO2cpa

O2

⎞⎠ dz

+us,a + (ze−c

s,ae− + zO2−c

s,a

O2−)Fψs,a

+∫

e

(ue + zO2−ce

O2−Fψe +∣∣Me

O2−∣∣2

2mO2−ce

O2−

)dz

+us,c + (ze−c

s,ae− + zO2−c

s,a

O2−)Fψs,c

+∫

c

⎛⎝uc + ze−cmce− Fψc +

∣∣Mmce−

∣∣2

2me−cmce−

+∣∣Mpc

O2

∣∣2

2mO2cpc

O2

⎞⎠ dz.

(12)

Charged particles add electrostatic energy zkFψ , where zk isthe charge number, and F is Faraday’s constant, and ψ isthe electrostatic potential. We have ze− = −1, zO2− = −2 andzO2

= 0. The kinetic energy of the two non-moving surfacesare zero. As said before, all densities are averages over thetotal cross section of the whole volume in the anode, elec-trolyte, and cathode, and over the whole surface for the anodeand cathode surfaces.

The functional derivatives with respect to the variables inthe electrolyte are

δE

δxe=

⎛⎜⎜⎜⎝δE

δce

O2−δE

δMe

O2−δEδue

⎞⎟⎟⎟⎠ =

⎛⎜⎜⎜⎝zO2−Fψe − 1

2

(ve

O2−)2

ve

O2−

1

⎞⎟⎟⎟⎠ . (13)

The functional derivatives of E with respect to the variables inthe anode are the same with the superscript e replaced by main the anode metal and by pa in the pores. The symbol zO2−

is replaced by ze− or zO2in the metal or the pores, respec-

tively. The functional derivatives with respect to the variablesin the cathode are the same with superscript a replaced by c.The functional derivatives with respect to the variables of theanode surface are

δE

δxs,a=

⎛⎜⎜⎜⎜⎜⎜⎝

δEδc

s,a

e−δE

δcs,aO2

δEδc

s,a

O2−δE

δus,a

⎞⎟⎟⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎜⎜⎝ze−Fψs,a

0

zO2−Fψs,a

1

⎞⎟⎟⎟⎟⎟⎟⎠ . (14)

For the cathode surface the functional derivatives are the samewith the superscript s, a replaced by s, c.

The entropy of the total system is the sum of the entropiesof the three phases and the surfaces:

S =∫

a

sadz + ss,a +∫

e

sedz + ss,c +∫

c

scdz. (15)

We assume local equilibrium (validity of Gibbs equation)in order to find the functional derivatives of S with respect tothe variables in the electrolyte:

δS

δxe=

⎛⎜⎜⎜⎝δS

δce

O2−δS

δMe

O2−δSδue

⎞⎟⎟⎟⎠ =

⎛⎜⎜⎝−μe

O2−/T e

0

1/T e

⎞⎟⎟⎠ , (16)

where μk are chemical potentials. The functional derivativesof S with respect to the variables in the anode are the samewith the superscript e replaced by ma in the anode metal andby pa in the pores. The functional derivatives with respect tothe variables in the cathode are the same with superscript areplaced by c. The functional derivatives with respect to thevariables of the anode surface are

δS

δxs,a=

⎛⎜⎜⎜⎜⎜⎜⎝

δSδc

s,a

e−δS

δcs,aO2

δSδc

s,a

O2−δS

δus,a

⎞⎟⎟⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎜⎜⎝−μ

s,ae− /T s,a

−μs,aO2

/T s,a

−μs,a

O2−/T s,a

1/T s,a

⎞⎟⎟⎟⎟⎟⎟⎠ . (17)

For the cathode surface the functional derivatives are the samewith the superscript s, a replaced by s, c.

V. THE POISSON MATRICES

The Poisson matrix is the operator that produces the re-versible contributions to the equations of motion from theenergy gradient. Taking again hydrodynamic flow as an exam-ple, the outcome will lead to the Euler equations. When oper-ating on the entropy gradient the result is zero, as the entropyis not affected by reversible operations. We will first discusssubsystem 3, the electrolyte. With only one moving compo-nent this is easier than in the bulk electrodes where there aretwo moving components.


129.241.87.96 On: Fri, 10 Oct 2014 09:28:09



The variables in the electrolyte are xe = (ce

O2− , Me

O2− , ue). The Poisson matrix is17

Le(z, z′) =

⎛⎜⎜⎜⎝0 ce

O2−(z′) ∂δ∂z′ 0

− ∂δ∂z

ce

O2−(z) Me

O2−(z′) ∂δ∂z′ − ∂δ

∂zMe

O2−(z) − ∂δ∂z

[ue(z) + pe(z′)]

0 [ue(z′) + pe(z)] ∂δ∂z′ 0

⎞⎟⎟⎟⎠ , (18)

where pe is the pressure of the electrolyte and δ = δ(z − z′) is the Dirac delta function. By integrating over the z′ coordinate, wecan write Le as a differential operator

Le(z) =

⎛⎜⎜⎜⎝0 − ∂

∂zce

O2−(z) 0

−ce

O2−(z) ∂∂z

− ∂∂z

Me

O2−(z) − Me

O2−(z) ∂∂z

− [ue(z) ∂

∂z+ ∂

∂zpe(z)

]0 − [

∂∂z

ue(z) + pe(z) ∂∂z

]0

⎞⎟⎟⎟⎠ . (19)

The product Le(x) · δEδxe gives the reversible contribution to the

time derivatives of the variables:

∂ce

O2−(z)

∂t= − ∂

∂zce

O2−(z)ve

O2− (z), (20)

∂Me

O2−(z)

∂t= −zO2−ce

O2−(z)F∂

∂zψe(z)

− ∂

∂zMe

O2−(z)ve

O2−(z) − ∂

∂zpe(z), (21)

∂ue(z)

∂t= −

[∂

∂zue(z) + pe(z)

∂

∂z

]ve

O2−(z). (22)

The first two equations are simply the well known balanceequations for mass and momentum. These equations corre-spond to the Euler equations in hydrodynamics. The last equa-tion gives the flow of internal energy density.

The degeneracy requirement is satisfied. We recover theGibbs-Duhem equation:

ce

O2− (z)∂

∂z

μe

O2−(z)

T e(z)− ue(z)

∂

∂z

1

T e(z)− ∂

∂z

pe(z)

T e(z)= 0. (23)

This equation applies for a system in local thermodynamicequilibrium. We have assumed that (μ/T) is constant for theelectrolyte lattice.

B. Subsystems 1 and 5: The electron conductorand gas supply channels

The Poisson matrix in the anode metal and in the poresare the same with the superscript e replaced by ma in the an-ode metal and by pa in the pores. In the expression for the re-versible contribution to the time derivative one should use ze−

and zO2in the metal and the pores, respectively. The Gibbs-

Duhem relation is also the same, when one uses the propersuperscripts, and where we assume that μ/T is constant foratoms in the metal and for the nitrogen in the pores.


The L operators for the anode and the cathode surfacesare zero. There is no reversible contribution to the time rate ofchange of the excess variables that we use for the surface.

D. Remarks

We have seen above that the Poisson operator generatesequations for the bulk phases that are familiar or taken forgranted; namely, the balance equations for mass, momentum,and internal energy. The corresponding Gibbs-Duhem’s equa-tions are consequences of the degeneracy requirement. Forthe surface, there are no reversible contributions. A completeset of equations for reversible transformations of energy cantherefore be given using GENERIC.

The appearance of these equations means that any vol-ume element of the cell obeys local equilibrium.21, 23 This ap-plies also to the two-dimensional system, the surface, when itis defined with Gibbs excess variables. In electrochemistry theassumptions of local thermodynamic equilibrium and elec-troneutrality are commonly used, see, e.g., Ref. 21.

Momentum balances are frequently not considered inelectrochemical modeling of solid state cells. Equations like(21) express that the pressure gradient balances the gradient inthe electric potential at steady state. We return to these issuesin Sec. IX.

VI. THE FRICTION MATRICES FOR LINEAR-FLUXFORCE RELATIONS

The friction matrix is the operator that produces the ir-reversible contributions to the equations of motion from theentropy gradient. Taking again hydrodynamic flow as an ex-ample, the outcome gives, together with the reversible contri-butions, the Navier-Stokes equation. The generalized frictionof a system is the system’s total entropy production:∫

δS

δx(z)· M · δS

δx(z)dz =

∫σ (z)dz. (24)


129.241.87.96 On: Fri, 10 Oct 2014 09:28:09


Classical nonequilibrium thermodynamics gives the localvalue of the entropy production as a product sum of indepen-dent fluxes and forces:

σ =∑

i

JiXi. (25)

The bilinear form of the entropy production applies to lin-ear, as well as nonlinear processes.17, 22, 24 GENERIC does notprovide a particular form of the flux-force relations, but canprovide restrictions via the form of the chosen friction ma-trix, by requiring that the degeneracy conditions are obeyed.17

We shall choose the classical form, given by irreversible ther-modynamics for the homogeneous phases and for a first ap-proach to the interfaces. The degeneracy requirement of thefriction matrix, δE/δx · M = 0, means that the total energy isnot changed by irreversible operations. This condition shall beshown to be true for all subsystems in the example cell, firstfor linear and eventually for nonlinear flux-force relations.


The entropy production has two fluxes and conjugatedriving forces.9 With the chosen set of independent variableswe have contributions from the total heat flux, J e

q , multipliedby the gradient in the inverse temperature of the electrolyte∂∂z

1T e , and the oxygen ion flux, J e

O2− , multiplied with its driv-

ing force, ( ∂∂z

μe

O2−T e +zO2− F

T e

∂ψe

∂z):

σ e = J eq

∂

∂z

1

T e− J e

O2−

(∂

∂z

μe

O2−

T e+zO2−

F

T e

∂ψe

∂z

). (26)

The total entropy production in the electrolyte is equal to theintegral of σ e(z) over the electrolyte. The linear flux-force re-lations are accordingly:

J e

O2− = −�e11

(∂

∂z

μe

O2−

T e+zO2−

F

T e

∂ψe

∂z

)+ �e

12∂

∂z

1

T e,

(27)

J eq = −�e

21

(∂

∂z

μe

O2−

T e+zO2−

F

T e

∂ψe

∂z

)+ �e

22∂

∂z

1

T e,

where the Onsager reciprocal relation is �e21 = �e

12. The On-sager coefficients can depend on the variables and dependtherefore indirectly on z. By introducing these relations into

the entropy production, we obtain

σ e(z) =(

−(

∂

∂z

μe

O 2−

T e+zO2−

F

T e

∂ψe

∂z

),

∂

∂z

1

T e

)· �e

·(

−(

∂

∂z

μe

O2−

T e+zO2−

F

T e

∂ψe

∂z

),

∂

∂z

1

T e

), (28)

where the �e matrix is

�e =(

�e11 �e

12

�e21 �e

22

). (29)

The resulting Me-matrix relates to the number of independentvariables of the system, while �e-matrix relates to the numberof processes. GENERIC connects the two via the C-matrix:

Me = [Ce]T · �e · Ce, (30)

where T as a superscript indicates the transpose of a matrix.The chosen Me operator is symmetric in view of the symme-try of the Onsager matrix. One choice of the Ce operator is

such that Ce · δS/δxe = (−( ∂∂z

μe

O2−T e +zO2− F

T e

∂ψe

∂z), ∂

∂z1T e ) and

δE/δxe · Ce = 0. The operation Ce · δS/δxe will then producethe driving forces of the system, familiar to classical nonequi-librium thermodynamics. The resulting choice can be writtenas

Ce =⎛⎝ ∂

∂zve

O2−∂∂z

−zO2−F∂ψe

∂z

0 0 ∂∂z

⎞⎠ . (31)

We see that Ce has three columns (one for each variable) andtwo rows (one for each process). We verify, using partial inte-gration, that∫

el

δS

δxe(z)· Me · δS

δxe(z)dz =

∫el

σ e(z)dz (32)

gives the total entropy production. The degeneracy require-ment is obeyed, because

δE

δxe(z)· (Ce)T =(

zO2−Fψe − 12

(ve

O2−)2

, ve

O2− , 1) · (Ce)T =0.

(33)The operator Me has now all properties asked for byGENERIC. We can then calculate the irreversible contribu-tion to the time rate of change of the variables

Me · δS

δxe= −[Ce]T · �e · Ce ·

⎛⎜⎜⎝−μe

O2−/T e

0

1/T e

⎞⎟⎟⎠= [Ce]T · �e ·

⎛⎝−(

∂∂z

μe

O2−T e +zO2− F

T e

∂ψe

∂z

)∂∂z

1T e

⎞⎠

= −

⎛⎜⎜⎜⎝∂∂z

0

∂∂z

ve

O2− 0

zO2−F∂ψe

∂z∂∂z

⎞⎟⎟⎟⎠ ·

⎛⎜⎝−�e11

(∂∂z

μe

O2−T e +zO2− F

T e

∂ψe

∂z

)+ �e

12∂∂z

1T e

−�e21

(∂∂z

μe

O2−T e +zO2− F

T e

∂ψe

∂z

)+ �e

22∂∂z

1T e

⎞⎟⎠


129.241.87.96 On: Fri, 10 Oct 2014 09:28:09


=

⎛⎜⎜⎜⎜⎝∂∂z

�e11

(∂∂z

μe

O2−T e +zO2− F

T e

∂ψe

∂z

)− ∂

∂z�e

12∂∂z

1T e

∂∂z

ve

O2−

{�e

11

(∂∂z

μe

O2−T e +zO2− F

T e

∂ψe

∂z

)− �e

12∂∂z

1T e

}∂∂z

�e21

∂∂z

μe

O2−T e + zO2− F

T e∂∂z

�e21

∂ψe

∂z− ∂

∂z�e

22∂∂z

1T e + zO2−F

∂ψe

∂z�e

11

(∂∂z

μe

O2−T e +zO2− F

T e

∂ψe

∂z

)⎞⎟⎟⎟⎟⎠ . (34)

B. Subsystems 1 and 5: The electron conductorand gas supply channels

We do the analysis for the anode. The results for the cath-ode are the same as for the anode with superscript a replacedby c. We remember that δS/δxma = (−μa

e−/T ma, 0, 1/T ma)and δS/δxpa = (−μa

O2/T pa, 0, 1/T pa). The entropy produc-

tion is then

σa = σma + σpa + σmpa, (35)

where9

σma = Jmaq

∂

∂z

1

T ma− J a

e−

(∂

∂z

μae−

T ma+ze−

F

T ma

∂ψa

∂z

),

σ pa = Jpaq

∂

∂z

1

T pa− J a

O2

∂

∂z

μaO2

T pa, (36)

σmpa = Jmpq

(1

T pa− 1

T ma

).

The first contribution σ ma contains the fluxes of heat and elec-trons in the z direction in the metal, multiplied with their ap-propriate driving forces, the second contribution σ pa describescontains the fluxes of heat and oxygen gas in the z directionin the pores multiplied with their appropriate driving forces,and the third contribution σ mpa contains the heat flux betweenthe metal and the pore multiplied with the difference in theinverse temperature between the phases.

These three contributions describe processes which donot couple directly since they take place in different locations.Because of this we can discuss their contributions to the fric-tion operator separately. We did not add m as a superscript toμa

e− and ψa as these quantities are not affected by the averag-ing over the cross section of the anode. Similarly, we did notadd p as a superscript to μa

O2. Neither did we add m and p as

superscript to J ae− and J a

O2as these are both averaged over the

cross section of the anode.The friction matrix and all the other relations in the anode

metal and in the pores are the same as for the electrolyte inSec. VI A with the superscript e replaced by ma in the anodemetal and by pa in the pores. In the metal and the pores, weuse ze− and zO2

, respectively.The heat exchange between the metal and the pores is

governed by

Jmpaq = �

mpaq

(1

T pa− 1

T ma

), (37)

By introducing this into the entropy production, we obtain

σmpa =(

1

T pa− 1

T ma

)�

mpaq

(1

T pa− 1

T ma

). (38)

For this contribution to the friction matrix we useδS/δxa = (δS/δxma, δS/δxpa) = (−μa

e−/T ma, 0, 1/T ma,

−μaO2

/T pa, 0, 1/T pa). The friction matrix can be written as

Mmpa = (Cmpa)T · �mpaq · Cmpa (39)

with

Cmpa = (0, 0,−1, 0, 0, 1). (40)

It is now easy to verify that∫a

δS

δxa· Mmpa · δS

δxadz =

∫a

σmpa(z)dz (41)

gives the total entropy production for heat exchange betweenthe metal and the pores. We verify also that Mmpa satisfies thedegeneracy requirement:

δE

δxa(z)· (Cmpa)T

= (−Fψa − 12

(va

e−)2

, vae− , 1,− 1

2

(va

O2

)2, va

O2, 1

)· (Cmpa)T = 0. (42)

After having constructed the Mmpa operator, we calculatethe irreversible contribution to the time rate of change of thevariables

Mmpa · δS

δxa= −[Cmpa] T · �mpa · Cmpa ·

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−μae−/T ma

0

1/T ma

−μaO2

/T pa

0

1/T pa

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠= [Cmpa]T · �mpa

(1

T pa− 1

T ma

)

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0

0

−�mpa(

1T pa − 1

T ma

)0

0

�mpa(

1T pa − 1

T ma

)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (43)

This is exactly the contribution to the time rate of change ofthe internal energy of the metal and of the pores due to theexchange of heat between the metal and the pores.


129.241.87.96 On: Fri, 10 Oct 2014 09:28:09



We do the analysis for the anode surface. The resultsfor the cathode surface are then obtained by replacing super-script s, a by s, c. The entropy production is obtained fromclassical nonequilibrium thermodynamics for surfaces.9 Withthe independent variables used here, we obtain seven contri-butions: the flux of electrons, J

a,ee− , multiplied with its driv-

ing force, −(μ

s,a

e−T s,a − μ

a,e

e−T a,e + ze−F

�as

ψ

T s,a ), the flux of oxygen gas,

Ja,eO2

, multiplied with its driving force, (μ

s,aO2

T s,a − μa,eO2

T a,e ), the flux

of the oxygen ion, Je,a

O2− , multiplied with its driving force,

−(μ

e,a

O2−T e,a − μ

s,a

O2−T s,a + zO2−F

�se

ψ

T s,a ), the chemical reaction rs, a mul-tiplied by its driving force, −�Gs,a

T s,a , the heat flux into thesurface from the metal in the anode, J

ma,eq , multiplied with

the difference in the inverse temperature between the rele-vant phases, the heat flux into the surface from the pores ofthe anode, J

pa,eq , multiplied with the difference in the inverse

temperature between the relevant phases, and the heat fluxfrom the surface into the electrolyte, J

e,aq , multiplied with

the difference in the inverse temperature between the relevantphases:

σ s,a = −Ja,ee−

(μ

s,ae−

T s,a− μ

a,ee−

T a,e+ ze−F

�asψ

T s,a

)

− Ja,eO2

(μ

s,aO2

T s,a−

μa,eO2

T a,e

)

− Je,a

O2−

(μ

e,a

O2−

T e,a− μ

s,a

O2−

T s,a+ zO2−F

�seψ

T s,a

)− rs,a �Gs,a

T s,a

+ Jma,eq

(1

T s,a− 1

T ma,e

)+ J

pa,eq

(1

T s,a− 1

T pa,e

)+ J

e,aq

(1

T e,a− 1

T s,a

), (44)

where �asψ ≡ ψ s, a − ψa, e, �seψ ≡ ψe, a − ψ s, a, and�Gs,a ≡ 4μ

s,ae− + μ

s,aO2

− 2μs,a

O2− .

1. The isothermal linear regime

In order to elucidate the essential physics of the pro-cesses, we shall first use some drastic simplifications. We as-sume that the system is unable to maintain temperature differ-ences, in spite of the porous gas filled structure, and that allmass fluxes are related through the stationary state condition:

rs,a ≡ −1

4J

e,ae− ≡ −1

2J

e,a

O 2− ≡ JO2≡ 1

4Fj. (45)

The only term left in the entropy production is the productof the flux, j , and the conjugate driving force X

s,aj (σ s,a

= jXs,aj ), giving

Xs,aj ≡ − 1

F

[μ

a,ee− + 1

4μ

a,eO2

− 1

2μ

e,a

O2− + F�aeψ

]. (46)

By introducing the electrochemical potentials of component k

μ̃k ≡ μk + zkFψ, (47)

the driving force can be written as the difference in electro-chemical potentials of the species that enter the reaction:

Xs,aj = − 1

F

[μ̃

a,ee− + 1

4μ

a,eO2

− 1

2μ̃

e,a

O2−

]≡ − 1

nF�G̃. (48)

For reversible conditions σ s, a = 0, Xs,aj = 0. The common

form of the Nernst equation1–3 follows from the last parenthe-sis:

�aeψj=0 = − 1

F

[μ

a,ee− + 1

4μ

a,eO2

− 1

2μ

e,a

O2−

]j=0

. (49)

For small current densities, we have a linear flux-forcerelationship between the driving force and the flux from σ s,a

= jXs,aj :

Xs,aj = Rpj, (50)

where Rp is the Ohmic resistance of the interface. This is ex-actly the relation used in the impedance measurement, if weidentify η with X

s,aj :

η = − 1

F

[μ

a,ee− + 1

4μ

a,eO2

− 1

2μ

e,a

O2− + F�aeψ

]. (51)

The overpotential describes therefore a deviation from Nernstbehavior. It is the net driving force of the electrochemical re-action when divided by T/F. The relations (50) and (51) con-firm the determination of the overpotential from impedancemeasurements. In more general terms we now have

η = − 1

nF�G̃ = − 1

nF[�G + nF�a,eψ], (52)

where �G̃ was defined above, �G is the normal reactionGibbs energy of any electrochemical reaction, and n is thenumber of electrons in the electrode reaction (here n = 4).This expression was given by Newman and Thomas-Alyea,1

but not connected to σ s, a.

2. The general linear regime

Consider next a less restrictive regime where the excessoxygen concentration at the surface is allowed to vary. Thesurface is not isothermal, but still electroneutral, meaning thatthe electron and oxygen ion fluxes are uniquely related to theelectric current density:

rs,a ≡ −1

4J

a,ee− ≡ −1

2J

e,a

O 2− ≡ 1

4Fj. (53)

On the time scale we consider the exchange of electrons andoxygen ions with the surface is in equilibrium. This impliesthat

μs,ae−

T s,a− μ

a,ee−

T a,e− F

�asψ

T s,a= 0,

(54)μ

e,a

O2−

T e,a− μ

s,a

O2−

T s,a− 2F

�seψ

T s,a= 0.

The reaction and the exchange of oxygen molecules with thesurface, cf. Sec. III C, can both be rate-limiting.


129.241.87.96 On: Fri, 10 Oct 2014 09:28:09


By using Eq. (53) or (54), the number of terms in theentropy production becomes smaller

σ s,a = −Ja,eO2

(μ

s,aO2

T s,a−

μa,eO2

T pa,e

)

− rs,a

(4μ

a,ee−

T ma,e+

μs,aO2

T s,a− 2μ

e,a

O2−

T e,a+ 4F

�aeψ

T s,a

)

+ Jma,eq

(1

T s,a− 1

T ma,e

)+ J

pa,eq

(1

T s,a− 1

T pa,e

)+ J

e,aq

(1

T e,a− 1

T s,a

)(55)

where �aeψ = �asψ + �seψ = ψe, a − ψa, e. It is known thatthe exchange of oxygen molecules with the surface and thereaction are both activated processes15 showing an Arrheniusbehavior. For ease of notation, we introduce

Xs,ar ≡ −

[4μ

a,ee−

T ma,e+

μs,aO2

T s,a− 2μ

e,a

O2−

T e,a+ 4F

�aeψ

T s,a

]. (56)

We now have five fluxes and conjugate forces. The nor-mal components of the fluxes are scalar, like the reactionrate rs, a. Therefore, coupling is possible between all fluxesand forces. The more general linear flux-force relations havetherefore, using Onsager’s reciprocal relations, (5 × 6)/2= 15 independent transport coefficients. This is still verymany, with regard to experimental determination, so wewould like to reduce the number of unknowns further. Fluxesin the metal and the pores take place at different locations, andwe expect that their coupling can be neglected. Similarly, wewill neglect the coupling of fluxes on one side of the surfaceto fluxes on the other side. These approximations, �13 = �15= �34 = �35 = �45 = 0, reduce the number of independentcoefficients to 10.

The resulting linear flux-force relations are

Ja,eO2

= −�s,a11

(μ

s,aO2

T s,a−

μa,eO2

T pa,e

)+ �

s,a12 X

s,ar

+ �s,a14

(1

T s,a− 1

T pa,e

),

rs,a = −�s,a21

(μ

s,aO2

T s,a−

μa,eO2

T a,e

)+ �

s,a22 X

s,ar

+ �s,a23

(1

T s,a− 1

T ma,e

)

+�s,a24

(1

T s,a− 1

T pa,e

)+ �

s,a25

(1

T e,a− 1

T s,a

),

Jma,eq = �

s,a32 X

s,ar + �

s,a33

(1

T s,a− 1

T ma,e

),

Jpa,eq = −�

s,a41

(μ

s,aO2

T s,a−

μa,eO2

T pa,e

)+ �

s,a42 X

s,ar

+ �s,a45

(1

T s,a− 1

T pa,e

),

Je,aq = �

s,a52 X

s,ar + �

s,a55

(1

T e,a− 1

T s,a

), (57)

where the Onsager reciprocal relations, �s,aij = �

s,aji for i, j

= 1, . . . , 5, refer to an �s, a-matrix which is a 5 × 5 matrix. Theentropy production can be written as the double contraction

of the 5 force elements (−(μ

s,aO2

T s,a − μa,eO2

T pa,e ), Xs,ar , 1

T s,a − 1T ma,e ,

1T s,a − 1

T pa,e , 1T e,a − 1

T s,a ) with the Onsager matrix. The entropyproduction can also be written with different choices of vari-ables. Linear relations were written with thermal and otherdriving forces, using (52) as the driving force of the electro-chemical reaction.7, 8 The impact of temperature jumps on theelectrode potential drop was estimated.8

In the present case, the main chemical reaction is coupledto adsorption of oxygen at the surface, and to thermal drivingforces from all supply channels. The equations can be usedto explain, say, impedance measurements, where the electricfield oscillates with small amplitudes, and the condition ofelectroneutrality applies to the surface.

3. Electrode Peltier heats

The linear laws from the entropy production are basis fordefinition of thermoelectric phenomena.9 The Peltier coeffi-cient of a conductor, π i, is the measurable heat flux through aconductor, i, J ′i

q , divided by the electric current density overFaraday’s constant:

πi ≡[

J ′iq

j/F

]�μ

j=0,�T =0,η=0

. (58)

The heat flux here is the sum of heat fluxes of parallel conduc-tors J ′a

q = J′paq + J ′ma

q . The definition prescribes reversibleconditions for heat transfer.

The measurable heat flux was not a variable in the setchosen here. The energy flux, which is continuous and con-stant through all layers, was more convenient. The energy fluxis the measurable heat flux plus the sum of the enthalpy trans-ported by the species. We introduce this definition for the en-ergy flux in the homogeneous phases and obtain

dus

dt= 0 = J

pa,eq + J

ma,eq + jψa,e − J

e,aq − jψe,a (59)

= J′a,eq + J

a,eO2

Ha,eO2

+ Ja,ee− H

a,ee− + jψa,e

− J′e,aq − J

e,a

O2−He,a

O2− − jψe,a. (60)

We can determine the heat produced at the electrode interfaceat reversible conditions, �a, from the jump in the measurableheat fluxes in the equation above:

�a = πe,a − πa,e =[J

′e,aq − J

′a,eq

j/F

]�μ

j=0,�T =0,η=0

= �a,eH − F�a,eψ = T �a,eS. (61)


129.241.87.96 On: Fri, 10 Oct 2014 09:28:09


In the last step we used the expression for the Nernst potential(49). The electrode Peltier heat is equal to the entropy changeof the single electrode reaction. Away from reversible condi-tions, we can still use Eq. (59), but now with Eqs. (50) and(51). This gives for the stationary state (in the linear regime)

[J

′e,aq − J

′a,eq

j/F

]= �a,eH − F�a,eψ = T �a,eS + Fη.

(62)This equation is the same as that derived by Newman andThomas-Alyea1 from the energy balance. They use Fourier’slaw for the heat fluxes, however. This is probably not a goodassumption in view of Eq. (57). More detailed descriptionsof thermal effects based on the measurable heat flux can bemade, but will presently carry to far.

4. The friction matrix for the linear case

From GENERIC we deduce a symmetric Ms, a operatorfor these sets of conditions:

Ms,a = [Cs,a]T · Ls,a · Cs,a. (63)

By combining δS/δxs, a and δS/δxma, δS/δxpa, δS/δxe

close to the surface into one vector we have (δS/δxs, a,δS/δxma, e, δS/δxpa, e, δS/δxe, a). Remember that δS/δxs,a

= (−μs,ae− /T s,a,−μ

s,aO2

/T s,a,−μs,a

O2−/T s,a, 1/T s,a), δS/δxma,e

= (−μa,ee− /T ma,e, 0, 1/T ma,e), δS/δxpa,e = (−μ

a,eO2

/T pa,e, 0,

1/T pa,e), δS/δxe,a = (−μe,a

O2−/T e,a, 0, 1/T e,a). The Cs, a

operator is again chosen such that Cs, a · (δS/δxs, a, δS/δxma, e,δS/δxpa, e, δS/δxe, a) gives the thermodynamic forces whileCs, a · (δE/δxs, a, δE/δxma, e, δE/δxpa, e, δE/δxe, a) = 0.

This gives

Cs,a =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1 0 0 0 0 0 −1 − 12 va

O20 0 0 0

0 1 0 −2F�aeψ 4 2vae− 0 0 0 0 −2 −ve

O2− 0

0 0 0 1 0 0 −1 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 −1 0 0 0

0 0 0 −1 0 0 0 0 0 0 0 0 1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠. (64)

The form of the C-matrix reflects that there are 13 variables and 5 processes in the surface. The vertical bars are put in as a guideto the eye, to separate between interactions within the surface, between the surface and the metal, between the surface and theoxygen pore, and between the surface and the electrolyte. One can verify that the entropy production is given by

δS

δxs,a· Ms,a · δS

δxs,a= σ s,a, (65)

and that

( δE/δxs,a, δE/δxma,e, δE/δxpa,e, δE/δxe,a ) · (Cs,a)T = 0, (66)

so that Ms, a satisfies the degeneracy requirement.After having constructed the Ms, a operator, we can verify that the fluxes are given by⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

Ja,eO2

rs,a

Jma,eq

Jpa,eq

Je,aq

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠= Ls,a · Cs,a ·

⎛⎜⎜⎜⎜⎜⎜⎝δS/δxs,a

δS/δxma,e

δS/δxpa,e

δS/δxe,a

⎞⎟⎟⎟⎟⎟⎟⎠ = Ls,a ·

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

−(

μs,aO2

T s,a − μa,eO2

T pa,e

)−X

s,ar

1T s,a − 1

T ma,e

1T s,a − 1

T pa,e

1T e,a − 1

T s,a

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (67)

This operation confirms the consistency of the linear forms of the equations of motions presented above for the electrode surface.

VII. NONLINEAR CONTRIBUTIONS TO THEEQUATION OF MOTION

We can soon address one of the questions that motivatedthis work: Can we give the Butler-Volmer equation a betterbasis in thermodynamics? We first need to discuss how to ex-pand in a consistent manner the linear relations (67) to thenonlinear regime. This will be done here.

An electrochemical reaction has often large drivingforces; |�G| being larger than 100 kJ/mol. The electric po-tential difference and the reaction Gibbs energy are both largein the surface. This can explain why the electrochemical reac-tion is nonlinear in its driving forces, like common chemicalreactions are.25 As we argued above, not only the reaction,also the surface adsorption of oxygen can be rate-limiting,15


129.241.87.96 On: Fri, 10 Oct 2014 09:28:09


possibly because the molecule has to split into atoms, beforethe ion is formed. The task is therefore to give a proper de-scription of (at least) two nonlinear coupled processes withinthe framework of GENERIC.

For simplicity, consider again a matrix with a single di-agonal coefficient. The only contribution in Eq. (67) to theelectrochemical reaction rate is then

rs,a = �s,a22 X

s,ar . (68)

This is the case described above for the linear regime, cf. (49).If �

s,a22 depends only on state variables, this equation is linear

in Xs,ar . A more complicated expression for �

s,a22 is used in

GENERIC, see Ref. 18,

�s,a22 = −R�

s,a22,0

Xs,ar

[exp

(4μ

a,ee−

RT ma,e+

μs,aO2

RT s,a− 4F

ψa,e

RT s,a

)

− exp

(2μ

e,a

O2−

RT e,a− 4F

ψe,a

RT s,a

)]. (69)

This expression for �s,a22 keeps M positive definite as required,

and leads to a form which is similar to the law of mass action,which we expect

rs,a = −R�s,a22,0

[exp

(4μ

a,ee−

RT ma,e+

μs,aO2

RT s,a− 4F

ψa,e

RT s,a

)

− exp

(2μ

e,a

O2−

RT e,a− 4F

ψe,a

RT s,a

)]. (70)

The term R�s,a22,0 exp(

4μa,e

e−RT ma,e + μ

s,aO2

RT s,a − 4Fψa,e

RT s,a ) can be under-stood as the reverse reaction rate in the electric field (fromoxygen molecules and electrons to oxygen ions). The pref-actor �

s,a22,0 contains an Arrhenius-type factor, as the law of

mass action does. Close to equilibrium, the nonlinear equa-tion (70) reduces to the linear equation (68) with �

s,a22 = �

s,a22,0

as expected. By construction we have then obtained a non-linear flux-force relation for the diagonal contribution. In thepresent case, there are two activated processes, the oxygenadsorption and the electrochemical reaction.15 The same pro-cedure must then be used for the diagonal coefficients of theOnsager matrix which are related to oxygen adsorption.

The next important question is how to deal with the off-diagonal coefficients in the matrix. One possibility is to mul-tiply the off-diagonal coefficients also, with the factor

− R

Xs,ar

[exp

(4μ

a,ee−

RT ma,e+

μs,aO2

RT s,a− 4F

ψa,e

RT s,a

)

− exp

(2μ

e,a

O2−

RT e,a− 4F

ψe,a

RT s,a

)](71)

and similar factors from the other diagonal elements. Close toequilibrium these factors reduce to unity, as they should. Sucha procedure will make the M-matrix asymmetric. The multi-plication of the columns in the �-matrix with the suggestedfactors from the diagonal elements, systematically replaces,e.g., X

s,ar in the expressions for the fluxes by the nonlinear

“force”

− R

[exp

(4μ

a,ee−

RT ma,e+

μs,aO2

RT s,a− 4F

ψa,e

RT s,a

)

− exp

(2μ

e,a

O2−

RT e,a− 4F

ψe,a

RT s,a

)](72)

and similarly for the other columns. Following this procedure,we obtain for the diagonal and off-diagonal coefficients of thefirst two columns and rows:

�s,aj1 = R�

s,aj1,0

expμ

s,aO2

RT s,a − expμ

a,eO2

RT pa,e

μs,aO2

T s,a − μa,eO2

T pa,e

,

�s,a1k = �

s,ak1,0 = �

s,a1k,0 for k = 3, 4, 5,

�s,aj2 = −R�

s,aj2,0

Xs,ar

[exp

(4μ

a,ee−

RT ma,e+

μs,aO2

RT s,a− 4F

ψa,e

RT s,a

)

− exp

(2μ

e,a

O2−

RT e,a− 4F

ψe,a

RT s,a

)],

�s,a2k = �

s,ak2,0 = �

s,a2k,0 for k = 3, 4, 5. (73)

Close to equilibrium, this reduces to �s,aj1 = �

s,a1j = �

s,aj1,0

= �s,a1j,0 and �

s,aj2 = �

s,aj2,0 = �

s,a2j,0 = �

s,a2j . The symmetry is re-

stored in this limit, but does not exist beyond this limit. Theother 3×3 coefficients have their normal symmetric form. Wesee in Eq. (73) that the columns in the nonlinear �-matrix ob-tain the same nonlinear factor as the diagonal element in thecolumn. As these nonlinear factors are all positive definite,the �-matrix remains positive semi-definite. This ensures thatthe second law of thermodynamics is valid for the nonlinearcase. Similar expressions were obtained in the study of ac-tivated transport processes using mesoscopic nonequilibriumthermodynamics.26 The flux-force relations were then linearon the mesoscopic level, but not after integration to the macro-scopic level.

GENERIC makes it possible to use an asymmetric formof the �-matrix, a form which remains positive semi-definiteas it should. In this form, all columns in the symmetric, pos-itive semi-definite �0-matrix are multiplied with the samenonlinear factor. In the mesoscopic analysis of a transportproblem,26 the corresponding �0 matrix had small deviationsin the symmetry. From GENERIC, we learn that this shouldbe corrected.

To summarize this section: An asymmetric friction ma-trix arises in the nonlinear case. It can be constructedwith help from mesoscopic nonequilibrium thermodynam-ics, and relates fluxes to driving forces in a nonlinear man-ner. An asymmetric friction matrix has also been found whenGENERIC was used for the Boltzmann equation.17

VIII. GENERALIZED BUTLER-VOLMER EQUATIONS

The nonlinear equations presented in Sec. VII gives themost general contributions to the equations of motion for


129.241.87.96 On: Fri, 10 Oct 2014 09:28:09


the electrode surface, under the assumption of local equi-librium. We chose to consider flux equations related to tworate-limiting chemical steps plus the associated transport ofenergy. The equations describe also nonstationary conditions.

The Butler-Volmer equation applies to stationary states,which allows for a further reduction. In a stationary state,there is also no accumulation of oxygen molecules at the sur-face. This implies that

rs,a = −Ja,eO2

. (74)

This relation will lead to an equation set with coupling be-tween the reaction rate (the current density) and thermal driv-ing forces. The second activated chemical reaction has a rateequal to the first activated reaction. As a consequence thereremains only one equation for the rate in addition to the equa-tions for the energy fluxes. This represents a generalization ofthe common Butler-Volmer equation (2). The expression forthe energy fluxes that accompany the Butler-Volmer equationare new in electrochemistry.

Equation (67) together with Eq. (73) imply for the non-linear case that

Ja,eO2

= −R�s,a11,0

(exp

μs,aO 2

RT s,a− exp

μa,eO2

RT pa,e

)

−R�s,a12,0

[exp

(4μ

a,ee−

RT ma,e+

μs,aO2

RT s,a− 4F

ψa,e

RT s,a

)

− exp

(2μ

e,a

O2−

RT e,a− 4F

ψe,a

RT s,a

)]

+�s,a13

(1

T s,a− 1

T ma,e

)+ �

s,a14

(1

T s,a− 1

T pa,e

)

+ �s,a15

(1

T e,a− 1

T s,a

)

rs,a = −R�s,a21,0

(exp

μs,aO2

RT s,a− exp

μa,eO2

RT pa,e

)

−R�s,a22,0

[exp

(4μ

a,ee−

RT ma,e+

μs,aO2

RT s,a− 4F

ψa,e

RT s,a

)

− exp

(2μ

e,a

O2−

RT e,a− 4F

ψe,a

RT s,a

)]

+�s,a23

(1

T s,a− 1

T ma,e

)+ �

s,a24

(1

T s,a− 1

T pa,e

)

+ �s,a25

(1

T e,a− 1

T s,a

). (75)

In the stationary state the forces have adjusted them-selves such that Eq. (74) is true. We use this to eliminateexp(μs,a

O2/RT s,a) in the reaction rate. We also apply the as-

sumptions used before, that certain coupling coefficients are

small (zero). This results in

j = 4Frs,a = −4FR(�

s,a22,0 − B2

(�

s,a12,0 + �

s,a22,0

))×

[exp

(4μ

a,ee−

RT ma,e+

μa,eO2

RT pa,e− 4F

ψa,e

RT s,a

)

− exp

(2μ

e,a

O2−

RT e,a− 4F

ψe,a

RT s,a

)]

+ 4F�s,a23 (1 − B2)

(1

T s,a− 1

T ma,e

)+ (

�s,a24 − B2

(�

s,a14 + �

s,a24

)) (1

T s,a− 1

T pa,e

)+ 4F�

s,a25 (1 − B2)

(1

T e,a− 1

T s,a

), (76)

where

Bj =�

s,aj1,0 + �

s,aj2,0 exp

(4μ

a,e

e−RT ma,e − 4F

ψa,e

RT s,a

)�

s,a11,0 + �

s,a21,0 + (

�s,a12,0 + �

s,a22,0

)exp

(4μ

a,e

e−RT ma,e − 4F

ψa,e

RT s,a

) .

(77)

For the energy fluxes, three expressions similar to Eq. (76)can be givenas

Jq,j = −R(�

s,aj2,0 − Bj

(�

s,a12,0 + �

s,a22,0

))×

[exp

(4μ

a,ee−

RT ma,e+

μa,eO2

RT pa,e− 4F

ψa,e

RT s,a

)

− exp

(2μ

e,a

O2−

RT e,a− 4F

ψe,a

RT s,a

)]

+ (�

s,aj3 − Bj�

s,a23

) (1

T s,a− 1

T ma,e

)+ (

�s,aj4 − Bj

(�

s,a14 + �

s,a24

)) (1

T s,a− 1

T pa,e

)+ (

�s,aj5 − Bj�

s,a25

) (1

T e,a− 1

T s,a

), (78)

where Jq,3 = Jma,eq , Jq,4 = J

pa,eq , Jq,5 = J

e,aq .

We find that the reaction has in addition to the usualcontribution three additional contributions due to tempera-ture differences in Eq. (76). The same is true for the energyfluxes. Due to the stationary state conditions, we were ableto eliminate the thermodynamic force for adsorption of oxy-gen molecules. The resulting expressions become, on the onehand, simpler in the sense that they contain only four termsdue to four driving forces. On the other hand, they obtainmore complicated effective coefficients. The coefficients be-come combinations of the elements in the original 5×5 L ma-trix. In particular the coefficient Bj depends on coefficients,some of which contain Arrhenius factors in addition to de-pending on the density of the electrons in the anode, i.e., onexp((μe,a

e− /RT e,a) − (Fψe,a/RT s,a)).The expressions for j and Jq, j can be said to constitute

a generalized set of Butler-Volmer equations. This set hasa basis in nonequilibrium thermodynamics, as confirmed by


129.241.87.96 On: Fri, 10 Oct 2014 09:28:09


GENERIC, and can therefore be said to provide a broaderthermodynamic basis for the common Butler-Volmer equa-tion (2), as it also contains thermal driving forces. It can beshown in detail below how the common Butler-Volmer equa-tion is contained in the generalized form (Sec. VIII A).

We first note that isothermal conditions at the electrodesurface give

j = −A1

(exp

(4μ̃

a,ee− + μ

a,eO2

RT

)− exp

(2μ̃

e,a

O2−

RT

))(79)

and

Jq,j = −Aj

(exp

(4μ̃

a,ee− + μ

a,eO2

RT

)− exp

(2μ̃

e,a

O2−

RT

)).

(80)The coefficients Ai contain the prefactors of the other equa-tions. The energy flux has clearly the same form as the rateof the chemical reaction. It is not possible to obtain the resultwithout starting from the more general basis. This equation isnew in electrochemistry.

A. The common Butler-Volmer equation

We return to the empirical form of Butler-Volmer’s equa-tion (2). Again, we note that the Butler-Volmer equation ap-plies at isothermal, stationary state conditions (no charge ormass accumulation at or in the electrode surface). This meansthat we should be able to recover the equation from (76),by neglecting contributions from thermal driving forces. Weobtain

j = j ′[ exp[(2μ̃

e,a

O2−)/RT] − exp

[(4μ̃

a,ee− + μ

a,eO2

)/RT]]

,

(81)where j ′/F = 4R(�s,a

22,0 + B2(�s,a12,0 + �

s,a22,0)), and the electro-

chemical potentials were defined in (47). We label reactantswith (r) and products with (p) and write more generally:

G̃r = 2μ̃e,a

O2− ,(82)

G̃p = 4μ̃a,ee− + μ

a,eO2

,

where �G̃ = G̃p − G̃r = �G + nF�a,eψ as in Eq. (52)with n = 4. The last identity gives again the Nernst equation,at reversible conditions, when the net driving force is zero.We can now write Eq. (81) in the form

j = j ′[exp[G̃r/RT ] − exp[G̃p/RT ]]. (83)

In the same manner as before,12 we subtract the value ofthe electrochemical potential at reversible conditions (equi-librium), G̃j=0 = G̃r,eq = G̃p,eq from the chemical potentialsof reactants (r) and products (p) and define

G̃r − G̃j=0 ≡ 4(1 − β)Fη,(84)

G̃p − G̃j=0 ≡ −4βFη.

The difference of these equations gives the identification usedbefore �G̃ = −nFη, cf Eq. (51). The overpotential of theoxygen anode becomes

η = − 1

F

(μ̃

a,ee− + 1

4μ

a,eO2

− 1

2μ̃

e,a

O2−

)= X

s,aj . (85)

This is the same expression as derived in the linear regime,demonstrating the robustness of the definition.

By introducing the identities (84) into the flux equation(83), we obtain the Butler-Volmer equation that was givenin Eq. (2). We can identify the exchange current density byj0 = j ′ exp(G̃j=0/RT ), which contains an Arrhenius factoras expected. In this case, it gives the overall result of twoslow steps. We have seen that it is restricted to isothermalconditions.

In the small current regime, we obtain using n = 4

j = j04ηF

RT(86)

giving the identification of the resistance Rp = RT/(4Fj0) ob-tained from impedance measurements at small current densi-ties.

For a single rate-limiting reaction at isothermal condi-tions the product jη will describe pure dissipation of theelectrochemical energy as heat into the surroundings. In thepresence of temperature differences or chemical potentialdifferences (electrolysis conditions) this is no longer true.The general expression (76) showed that work terms areinvolved.7, 8

We can now return to the interpretation of the measure-ments in Fig. 1.1, 2 We write for the two experiments

�a,eψj= − 1

F

[μ

a,ee− + 1

4μ

a,eO2

− 1

2μ

e,a

O2−

]j

+ Rpj,

(87)

�a,eψj=0= − 1

F

[μ

a,ee− + 1

4μ

a,eO2

− 1

2μ

e,a

O2−

]j=0

.

We subtract the last from the first, introduce Rpj = η, andobtain

�a,eψj− �a,eψj=0

= η + 1

F

[μ

a,ee− + 1

4μ

a,eO2

− 1

2μ

e,a

O2−

]j=0

− 1

F

[μ

a,ee− + 1

4μ

a,eO2

− 1

2μ

e,a

O 2−

]j

.

(88)

If we make the identification Vj − Vj=0 = �a,eψj

− �a,eψj=0there is an assumption involved in the in-

terpretation of the measurement in terms of a the jumps in theMaxwell potential at the surface. The two last terms, both arelarge, must cancel in order to obtain the wanted interpretationof η.

IX. DISCUSSION

A. GENERIC’s contributions

This is the first time GENERIC has been used to developequations of motion for an electrochemical cell.

1. The reversible contributions

It is comforting to first see in Sec. V that the familiarbalance equations appear. The energy balance for the inter-face, has the same form as that used by Newman1 to describe


129.241.87.96 On: Fri, 10 Oct 2014 09:28:09


the energy flux between the surface and the bulk. The resultsare presented mainly to show how these equations can be de-veloped using GENERIC. There are interesting details to thepicture given by the balance equations. One of them gives themomentum balance. It may be useful to have a momentumbalance on the same footing as the other equations for an elec-trochemical cell. A balance of forces, created at all interfaces,might well have an impact on the lifetime of electrochemicalsystems, e.g., of Li-batteries.

2. The irreversible contributions

In addition to the conservation laws, nonequilibrium ther-modynamics offers yet an equation for the energy flux for ourexample, here represented by Eq. (80) and its forerunners.GENERIC confirms that this extra equation is essential foragreement with the second law of thermodynamics. This isa new theoretical addition to electrochemistry; obtained fromthe search for a more general thermodynamic basis for theButler-Volmer equation. It becomes clear that other drivingforces than the electrochemical one, can contribute to the de-scription in the linear as well as the nonlinear regime.

In our search for contributions from the friction opera-tor, we always use linear flux-force relations for the homoge-neous phases. The mechanism for transport of energy, mass,and charge is by diffusion in these materials, and classicalnonequilibrium thermodynamics therefore applies. This is nolonger true in the electrode regions. The electrochemical reac-tions are activated (rare events), motivating the use of nonlin-ear flux-force relations. We construct a plausible form of thisnonlinearity. The relevant friction matrix becomes asymmet-ric, but remains positive semi-definite. This is crucial for thevalidity of the second law. The constructed form of the non-linearity is supported by results from mesoscopic nonequilib-rium thermodynamics and its form is restricted by GENERIC.(The asymmetric friction matrix satisfies the degeneracy re-quirement of GENERIC.) We were not able to find this resultin the mesoscopic analysis. GENERIC adds therefore a firmbasis to all equations.

In the linear regime (classical nonequilibrium thermody-namics), we associate Onsager symmetry with time-reversalinvariance. The nonlinear flux-force relations do not disagreewith time-reversal invariance. In the mesoscopic analysis weconsider a finer scale (a mesoscopic scale). By integratingacross this scale, which has linear relations, we find nonlinearrelations.12, 26 An asymmetric M-matrix, which is also foundwhen one formulates the Boltzmann equation in the contextof GENERIC, is therefore not in contradiction with time-reversal invariance. This point will be further discussed later.

B. Assumptions

Several types of assumptions have been made in the de-velopment. We distinguish between assumptions specific toGENERIC, to nonequilibrium thermodynamic theory, and tothe particular problem that is used as an example.

Input to GENERIC are the sets of independent variablesfor each subsystem. Different sets can be chosen, but no as-sumption about their relationships are needed. We have used

validity of normal thermodynamic relations in the construc-tion of the expressions for E and S, in Sec. IV. In this sense wehave used the assumption of local equilibrium. To then findthe Gibbs-Duhem equation for the bulk phases by GENERIC,is in agreement with this postulate. Validity of linear flux-force relations has also been taken as a sign that the hypothe-sis of local equilibrium applies.

Nonlinear flux-force relations have been taken as a signof being far from equilibrium. It is necessary to specify thesystem in this context. Interfaces, which are autonomous ther-modynamic systems, have been found to be in local equi-librium even under severe gradients.23, 25 We expect this tobe true also here, arguing that this assumption is essentialin mesoscopic nonequilibrium thermodynamics, which alsogives nonlinear flux-force relations.

Assumptions used with the example cell, inspired by thesolid oxide fuel cell materials, are of a different character.Based on literature reports, we have assumed a set of foursteps connected with the oxidation of oxygen in the anode.The steps can be simplified or changed. Changing the assump-tions behind the model will have an impact on the preciseform of the equations obtained in Sec. VIII. The equationspresented here are only “general” as far as our descriptionis a good model of reality. When this is said, it is clear thatother “general” sets of equations can be found for other de-scriptions. The results are general in the sense that they in-clude all possible coupling effects. It remains to be seen byexperiments, whether these coupling effects are large or canbe neglected.

One motivation for the choice of an example with aporous electrode was that temperature gradients are morelikely around this electrode than in metal electrodes in con-tact with liquid electrolytes. This makes an elaborate modelpossibly more relevant. We have seen in Sec. VIII A how thegeneral set of equations simplify drastically by introducingisothermal conditions, an assumption that is always used inconnection with overpotential measurements. The equationspresented in Sec. VIII A are still more general than Eq. (2),because they derive from a more general basis.

C. Coupled transport processes

Nonequilibrium thermodynamics predicts coupling oftransport processes in the bulk phases and at the interface. Thecoupling terms are well established for transport in the bulkphase. Special for the interface is the possibility for couplingof the reaction with scalar normal components of vectorialfluxes.9 For example, a chemical reaction, which cannot cou-ple to heat transport in the bulk phase, will have a coupling co-efficient with this flux at the interface. For the interface, thesecoupling terms are not yet in regular use, and it is difficult topredict their magnitude in the absence of experiments. In thesimpler case of (energy) heat and mass transport, coupling co-efficients have been shown to be essential.11 GENERIC sup-ports the existence of these coefficients and gives the properform of the asymmetric friction matrix for the nonlineardescription.


129.241.87.96 On: Fri, 10 Oct 2014 09:28:09


We observe that the existence and inclusion of such termslead to a generalized form of the Butler-Volmer equation, rep-resented by Eq. (76). The results can be said to expand onearlier results.7, 8, 12 In addition to the electrochemical driv-ing force, also thermal driving forces contribute to the reac-tion rate, cf. Eq. (76). The new equations for the interfacecan, both in the linear and nonlinear form, be used to de-scribe impedance measurements, thermoelectric phenomena,concentration or formation cell potentials.

The generalized Butler-Volmer equation contains possi-bilities for work through the coupling terms. This means thatthe equations do not describe purely dissipative phenomena.Given a set of n thermodynamic forces and fluxes, it is alwayspossible to diagonalize the �-matrix. This gives n eigenvaluesand eigenvectors corresponding to n independent, purely dis-sipative processes. But given the symmetry of the linear case,there are n(n + 1)/2 independent matrix elements. The n(n− 1)/2 degrees of freedom that arise, can give transfer of en-ergy from one form to another. Electric work can, for instance,be taken from a concentration cell (in reverse electrodialysiscells). Another example is the Soret effect, where a tempera-ture gradient creates a concentration gradient which can alsobe used to do separation work.

D. The overpotential and the energy flux

This study reinforces the definition (52) used for the over-potential in the electrochemical literature, and its determina-tion via impedance measurements. There is a clear connec-tion of this expression to the expression in the Butler-Volmerequation, via Eqs. (84) and (85). This means that one can alsotrust the generalized forms given in Sec. VIII. The equationsin this section give new expressions for the overall reaction,where all rate-limiting steps contribute, in addition to ther-mal driving forces. Moreover, the same definition can be usedfor the overpotential in the linear as well as the nonlinearcase. As shown by (88), the interpretation of the measuredoverpotential in terms of electrostatic potential differences isnot straight forward. The determination from impedance mea-surements is to be preferred, and can be done in the linear aswell as the nonlinear regime.

The energy flux (80) has not been described in con-nection with the overpotential before and deserves a spe-cial comment. It gives additional information on the en-ergy flux into and out of the electrode surface, additional tothe energy conservation equation obtained from the Poissonoperator.

Three contributions were ascribed to the energy flux inthe z-direction in the present case. There were energy fluxcontributions from the oxygen pore, from the metal, and fromthe electrolyte. The coupling terms are specific to each flux.In the case of good thermal conductivity at the interface (nothermal driving forces), the expression reduces and we obtaina Butler-Volmer-like expression also for the energy flux. It isin all directions driven by the same effective electrochemicaldriving force for the two superimposed activated processes.The equation provides a strong link between the overpotentialand the energy flux.

X. CONCLUSIONS

We have developed a description of a simple electro-chemical cell using GENERIC.17, 18 We find that it gives amore general thermodynamic basis to the Butler-Volmer andNernst equations, including also Peltier effects. A multitudeof possibilities arise for coupling between the electrochemi-cal driving force, chemical driving forces and thermal drivingforces in the nonlinear regime, extending the postulate of thelinear to the nonlinear regime.7, 8 GENERIC, which containsNET at its center, ensures that the more general expressionobeys the laws of thermodynamics. The friction matrix is nolonger symmetric, a result anticipated from the mesoscopicderivation of analogous equations.26

This is the first time GENERIC is used for an electro-chemical cell. Formation cells have different electrodes; a de-scription of them will follow the same procedure as used here.GENERIC can set all relevant phenomena on the same basis,and is therefore suited to deal with all relevant effects, includ-ing hydrodynamic effects. Like in classical nonequilibriumthermodynamics, a prescription is found on how to integrateacross the cell. The local dissipation can then be found, whichis essential for electrode optimization discussions. Electro-chemical science is a vast field, with numerous applications.Experimental evidence for the ideas proposed here may helpa further development.

ACKNOWLEDGMENTS

ETH Zurich is thanked for awarding guest professorshipsto D.B. and S.K.

1J. Newman and K. E. Thomas-Alyea, Electrochemical Systems, 3rd ed.(Wiley, Hoboken, 2004).

2J. O’M. Bockris, A. K. N. Reddy, and M. Gamboa-Aldeco, Modern Elec-trochemistry 2A (Klüwer-Academic Publishers, New York, 2002).

3A. J. Bard and L. R. Faulkner, Electrochemical Methods. Fundamentalsand Applications, 2nd ed. (Wiley, Hoboken, 2001).

4S. H. Chan, X. J. Chen, and K. A. Khor, “Reliability and accuracy of mea-sured overpotential in a three-electrode fuel cell system,” J. Appl. Elec-trochem. 31, 1163–1170 (2001).

5V. Krishnan and S. McIntosh, “Measurement of electrode overpotentials infor direct hydrocarbon conversion fuels,” Solid State Ionics 166, 191–197(2004).

6S. B. Adler, “Reference electrode placement in thin solid electrolytes,” J.Electrochem. Soc. 149, E166–E172 (2002).

7S. Kjelstrup Ratkje and D. Bedeaux, “The overpotential as a surface singu-larity described by nonequilibrium thermodynamics,” J. Electrochem. Soc.143, 779–789 (1996).

8D. Bedeaux and S. Kjelstrup Ratkje, “The dissipated energy of electrodesurfaces. Temperature jumps from coupled transport processes,” J. Elec-trochem. Soc. 143, 767–779 (1996).

9S. Kjelstrup and D. Bedeaux, Nonequilibrium Thermodynamics of Hetero-geneous Systems. Series on Advances in Statistical Mechanics (World Sci-entific, 2008), Vol. 16.

10D. Bedeaux, A. M. Albano, and P. Mazur, “Boundary conditions andnonequilibrium thermodynamics,” Physica A 82, 438–462 (1976).

11J. Ge, S. Kjelstrup, D. Bedeaux, J. M. Simon, and B. Rousseau, “Transfercoefficients for evaporation of a system with a Lennard-Jones long-rangespline potential,” Phys. Rev. E 75, 061604 (2007).

12J. M. Rubi and S. Kjelstrup, “Mesoscopic nonequilibrium thermodynamicsgives the same thermodynamic basis to Butler-Volmer and Nernst equa-tions,” J. Phys. Chem. B 107, 13471–13477 (2003).

13H. A. Kramers, Physica 7, 284–304 (1940).14T. van Erp, T. Trinh, S. Kjelstrup, and K. Glavatskiy, “On the relation

between the Langmuir and thermodynamic flux equations,” Front. Phys.Phys. Chem. Chem. Phys. 1, 1–14 (2014).


129.241.87.96 On: Fri, 10 Oct 2014 09:28:09

http://dx.doi.org/10.1023/A:1012232301349

http://dx.doi.org/10.1023/A:1012232301349

http://dx.doi.org/10.1016/j.ssi.2003.10.007

http://dx.doi.org/10.1149/1.1467368

http://dx.doi.org/10.1149/1.1467368

http://dx.doi.org/10.1149/1.1836538

http://dx.doi.org/10.1149/1.1836537

http://dx.doi.org/10.1149/1.1836537

http://dx.doi.org/10.1016/0378-4371(76)90017-0

http://dx.doi.org/10.1103/PhysRevE.75.061604

http://dx.doi.org/10.1021/jp030572g

http://dx.doi.org/10.1016/S0031-8914(40)90098-2

http://dx.doi.org/10.3389/fphy.2013.00036

http://dx.doi.org/10.3389/fphy.2013.00036


15Z. Shao and S. M. Haile, “A high-performance cathode for the next gener-ation of solid-oxide fuel cells,” Nature 431, 170–173 (2004).

16S. Kjelstrup Ratkje and Y. Tomii, J. Electrochem. Soc. 140, 59 (1993).17H. C. Öttinger, Beyond Equilibrium Thermodynamics (Wiley-Interscience,

Hoboken, 2005).18H. C. Öttinger and M. Grmela, “Dynamics and thermodynamics of com-

plex fluids. Illustration of a general formalism,” Phys. Rev. E 56, 6633–6655 (1997).

19S. B. Adler, “Mechanism and kinetics of oxygen reduction onporous La1 − xSrxCoO3 − δ

electrodes,” Solid State Ionics 111, 125–134(1998).

20A. M. Svensson, S. Sunde, and K. Nisancioglu, “Mathematical modelingof oxygen exchange and transport in air-perovskite-YSZ interface region.II. Direct exchange of oxygen vacancies,” J. Electrochem. Soc. 145, 1390–1400 (1998).

21T. Kawada, J. Suzuki, M. Sase, A. Kamai, K. Yashiro, Y. Nigara, J.Mizusaki, K. Kawamura, and H. Yugami, J. Electrochem. Soc. 149, E252–E259 (2002).

22S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (North-Holland, 1962).

23T. Savin, K. Glavatskiy, D. Bedeaux, S. Kjelstrup, and H. C. Öttinger, Eur.Phys. Lett. 97, 40002 (2012).

24J. Ross and P. Mazur, “Some deductions from a statistical mechanical the-ory of chemical kinetics,” J. Chem. Phys. 35, 19–28 (1961).

25J. Xu, S. Kjelstrup, and D. Bedeaux, “Molecular dynamics simulations ofa chemical reaction, conditions for local equilibrium in a temperature gra-dient,” Phys. Chem. Chem. Phys. 8, 2017–2027 (2006).

26D. Bedeaux, and S. Kjelstrup, “The measurable heat flux that accompaniesactive transport by Ca2 +-ATPase,” Phys. Chem. Chem. Phys. 10, 7304–7317 (2008).


129.241.87.96 On: Fri, 10 Oct 2014 09:28:09

http://dx.doi.org/10.1038/nature02863

http://dx.doi.org/10.1149/1.2056110

http://dx.doi.org/10.1103/PhysRevE.56.6633

http://dx.doi.org/10.1016/S0167-2738(98)00179-9

http://dx.doi.org/10.1149/1.1838471

http://dx.doi.org/10.1149/1.1479728

http://dx.doi.org/10.1209/0295-5075/97/40002

http://dx.doi.org/10.1209/0295-5075/97/40002

http://dx.doi.org/10.1063/1.1731889

http://dx.doi.org/10.1039/b516704c

http://dx.doi.org/10.1039/b810374g

Nonlinear coupled equations for electrochemical cells as developed by the general equation for...

Documents

Transcript of Nonlinear coupled equations for electrochemical cells as developed by the general equation for...