Generalised equivalent circuits for mass and charge transport :chemical capacitance and its implications

J. Jamnik*¤ and J. Maier

Max-Planck Institute 1, 70569 Stuttgart, Germany.fu� r Festko� rperforschung, Heisenbergstra�eE-mail : jamnik=chemix.mpi-stuttgart.mpg.de

Received 4th January 2001, Accepted 23rd February 2001First published as an Advance Article on the web 30th March 2001

An exact equivalent circuit including terminal parts, which takes account of electrical and chemical controlparameters in a uniÐed way, is derived for a cell with a mixed conductor (or electrolyte) without internalsources or sinks. In one-dimensional problems electrochemical kinetics can be mapped by two-dimensionalcircuits exhibiting the spatial and the thermodynamic displacement as two independent coordinates. One mainadvantage of the exact circuits with respect to the underlying di†erential equations is the ability to simplify thedescription according to speciÐc situations. As we show in several examples in the second part of the paper, itis straightforward to select the elements relevant for the particular experimental conditions and so to makeappropriate approximations. This is most helpful for the description of electrochemical systems, such as fuelcells, membranes, pumps and batteries.

1 IntroductionIn many electrochemical processes electrical driving forcescause chemical changes inside the sample and/or vice versa.Let us consider two examples from solid state electrochem-istry : First, when a voltage is applied to a mixed conducting(i.e. electronically and ionically conducting) sample with elec-trodes which allow for only one of the two carrier species tobe transferred across the contact (e.g. sandwichedAg2Sbetween two Pt-electrodes1 or a titanate ceramic capacitorunder voltage stress at low temperatures,2) stoichiometric gra-dients are built up internally. Second, a mixed conductorunder a chemical gradient (e.g. separating two gasSrTiO3chambers with di†erent oxygen partial pressures) gives rise toambipolar Ñuxes inside. All these and related e†ects in thetransient and in the steady state can be successfully treated bywell-known transport equations combined with the continuityequations and PoissonÏs equation at least as long as werestrict the conditions to proximity to equilibrium.

The sometimes complex situation becomes much moretransparent when mapping the mathematical model by equiv-alent circuits obeying Kirchho†Ïs laws. The application ofsuch circuits is also extremely helpful in deriving relevant sim-pliÐcations. In this paper we give a rigorous treatment of theuse of such circuits as well as relevant examples. It turns outthat besides electrostatic capacitors and electrical resistorsalso non-electrostatic capacitors which describe e.g. stoichio-metric changes, also have to be included. These capacitorsstore the free energy primarily as entropy. A closer inspectionof the literature shows that non-electrostatic capacitorsappear sporadically under di†erent names and in di†erentbranches of physics and chemistry.3h24

An example par excellence is di†usion. Respective capa-citors in the transmission line describing di†usion are notrelated to the dielectric properties of the sample, but to thecarrier density c.5 Their capacitance per unit length is given

¤ On leave from the National Institute of Chemistry, Hajdrihova 19,SI-1000 Ljubljana, Slovenia.

by z being the carrierÏs charge number, the Boltz-z2c/kBT , kBmann constant and T the temperature. Further literatureexamples of non-electrostatic bulk elements are ““ redoxÏÏcapacitors6 or ““electrochemical ÏÏ capacitors7 which character-ise mixed conducting polymers. Referring to the interfacialprocess, ““pseudo-capacitors ÏÏ8,9 mapping the adsorption ofintermediates on electrode surfaces, are used. The capacitanceof such adsorption capacitors is determined by dh/d/, where hstands for the fraction of the electrode surface covered byadsorbed intermediates and / denotes the electric potential atthe electrode. Even an ordinary parallel metal-plate capacitor,if of mesoscopic size, cannot be described solely in terms ofelectrostatics.10 The electric charge Q stored in a mesoscopiccapacitor is proportional to the di†erence in the electrochemi-cal potentials of the electrons on both plates, rather*k8 eon ,than to */ alone. Thus the respective capacitance is an elec-trochemical one : (where e denotes the absoluteedQ/dk8 eonvalue of the electronic charge).

Let us Ðrst consider purely non-electrostatic capacitorsfrom a thermodynamic point of view. As pointed out byPelton11 a ““chemical capacitance ÏÏ of a thermodynamicC

jd

system with regard to an exchangeable species j can bedeÐned, by analogy to heat capacity or compressibility, as thesecond derivative of the Gibbs energy G with respect to thenumber of moles of that species Unliken

j: C

jd P (d2G/dn

j2)

T, P~1 .the Ðrst derivative, is an extensive quantity thus being pro-C

jd

portional to the thickness (volume) of the sample. An exampleof a system having high chemical capacitance is an aqueoussolution of an acidÈbase pair which bu†ers pH with respect toadditions of an acid. Another example is a solid state materialthe stoichiometry of which varies signiÐcantly upon smallchanges of the component partial pressure. In the Ðeld of solidstate electrochemistry traditionally the inverse of knownC

jd ,

as the ““ thermodynamic factor ÏÏ,1,12 is used. Is the capacitanceof ““pseudo- ÏÏ, ““ redoxÏÏ, etc. capacitors always proportional to(d2G/dn

j2)

T, P~1 ?A question of equal importance is how to connect the

circuit elements in an equivalent circuit. In order to takeaccount of di†erent carriers, spatial inhomogeneities, etc. the

1668 Phys. Chem. Chem. Phys., 2001, 3, 1668È1678 DOI: 10.1039/b100180i

This journal is The Owner Societies 2001(

circuits are often constructed intuitively, e.g. on the basis ofelectrical impedance measurements as a function of frequency.The common problems of such constructions are well known:(i) the validity range of the circuit is difficult to estimate and(ii) the dependences of the circuit elements on the materialsparameters (carrier mobilities, mass action constants, impurityconcentrations, charge numbers, interfacial rate constants andcell dimensions) are often obscure. More precise constructionrelies on analytical25,26 or numerical27 solutions of the di†er-ential equations constituting a particular physical model. Thecircuit is still constructed intuitively, but the validity of thecircuit and the meaning of the circuit elements are recast fromthe Ðtting of the circuit impedance to that predicted by themodel.28

The next step is obviously the construction of the circuitdirectly from the physical model. Such ““exact equivalentcircuits ÏÏ are a useful representation, mapping the underlyingmodels. A very simple example is dielectric relaxation ; theunderlying model comprises OhmÏs law and PoissonÏs equa-tion and yields an equivalent circuit which combines thesampleÏs resistance and its electrostatic capacitance in parallel.In electrochemistry and semiconductor physics the situation ismore complex and is usually described by driftÈdi†usionmodels. They consist of the NernstÈPlanckÈPoisson set ofequations with ChangÈJa†e or ButlerÈVolmer boundary con-ditions.25h28 Using such a model, Sah13 derived exact equiva-lent circuits for the transport in semiconductors taking intoaccount di†erent generationÈrecombination mechanisms.Independently, in the Ðeld of liquid electrochemistry Barker14found intuitively the exact equivalent circuit for a cell with abinary electrolyte and blocking electrodes. Later, Brumleveand Buck15 showed that BarkerÏs circuit is, close to equi-librium, isomorphous to the driftÈdi†usion model. It is inter-esting to note that in none of these studies (see also ref. 16and 17) was the non-electrostatic capacitance identiÐed as

and hence given on a thermodynamic basis. In(d2G/dnj2)

T, P~1contrast, in the simpliÐed equivalent circuit describing electro-chemical polarisation of mixed conductors, the chemicalcapacitance (essentially the inverse of the thermodynamicfactor) was used explicitly.18h21

Essentially, the treatments of the exact equivalent circuitsdiscussed above can be viewed from the perspective ofnetwork thermodynamics. This theory22 provides the prin-ciples and tools for a graphical representation of time-dependent irreversible thermodynamic systems in general.Horno and co-workers23,24 applied the theory for the con-struction of equivalent circuits.

In this paper we consider the exact equivalent circuit in thelight of solid state electrochemistry. We start with the deriva-tion of the circuit from the driftÈdi†usion model. We intro-duce terminal parts, which take into account electrical andchemical driving forces. During the derivation several impor-tant questions which we mentioned in the Introduction will beaddressed. In the second part the circuit is simpliÐed and tai-lored for several cases of theoretical and practical interest. Thelucidity of the circuit, when compared to underlying di†eren-tial equations, makes this procedure very e†ective.

2 Transport equationsIn the linear regime, which we refer to in this paper, thecarrier Ñuxes are proportional to driving forces. The transportequations derived from irreversible thermodynamics imposeno limitations on the carrier density,” ( j stands for thec

jcarrier type). The Ñux density, is proportional to theJj,

gradient of the corresponding electrochemical potential,

” In the dilute limit, in which Boltzmann statistics applies, the Ñuxequation derived from irreversible thermodynamics and the(linearised) NernstÈPlanck (driftÈdi†usion) equation are identical.

being the carrierÏs chemical potential, itsk8j\ k

j] z

je/, k

jzjcharge number including sign, e the absolute value of elec-

tronic charge, and / the electric potential. If we neglectOnsager cross coefficients, the Ñux densities depend on theposition, r 4 (x, y, z), and time, t, as follows

Jj(r, t) \ [

pj(r)

(zje)2

+k8j(r, t). (1)

The conductivity is determined by the carrierpj4 z

jec

jujmobility and by the carrier density At t \ 0 the initialu

jcj.

conditions of our experiments are established. In general, cj(r,

t \ 0) and thus may be position dependent, e.g. due topj(Mc

jN)

inhomogeneous doping (frozen-in proÐles), or due to spacecharge regions adjacent to phase boundaries (e.g. pÈnjunctions). We assume further the absence of internal sourcesand sinks for carriers. Thus the continuity equation

+ É Jj(r, t) \ [

dcj(r, t)dt

(2)

is valid. If we neglect magnetic Ðelds, the description is com-pleted by MaxwellÏs equation

+ É D\ e ;j

zjcj(r, t). (3)

Considering a simple material (isotropic, non-ferroelectric)and long timescales, the dielectric displacement D depends onthe electric potential / according to D(r, t).t) \[e(r)e0 +/(r,The symbols e and denote the relative permittivity and thee0permittivity of free space, respectively.

The above formulation is not the most suitable for anequivalent circuit representation. For this we need relation-ships which relate Ñuxes and potentials, as does eqn. (1). Theappropriate transformation of eqn. (3) is state of the art (seee.g. ref. 29) : Taking into account the continuity equation forthe charge density and the deÐnition of the displacementcurrent density, the ÑuxÈpotential relationshipIdis 4 dD/dt,reads The total current,Idis(r, t) \ e(r)e0+d//dt. Idis ] &

jIj,

then has a vanishing divergence.In order to recast eqn. (2) into a ÑuxÈpotential relationship

we write it as

dcj

dt\Adc

jdk

j

Bt

dkj

dt. (4)

Also here we neglect cross-coefficients, and use the(dci/dk

j)iEj

,fact that the explicit t-dependence, is zero. Although(dc

j/dt)kj ,the substitution (4) seems trivial° it has an enormous impact

on the treatment and interpretation of mass and charge trans-port in terms of equivalent circuits. The reason is that thechoice of the driving force deÐnes the meaning of the circuitelements and their links. The derivative in eqn. (4) taken atconstant t reÑects the capability of the system to accept orrelease additional charge carriers of type j on a given variationof their chemical potential Therefore it suggests the deÐni-k

j.

tion of chemical capacitance (deÐned per unit volume) allocat-ed to the carrier j as,

Cj4 (ez

j)2Adk

jdc

j

Bt/0

~1, (5)

which is in agreement with PeltonÏs deÐnition (seeintroduction) We set t \ 0 to emphasiseC

jP (d2G/dn

j2)

T, P~1 .that is a materials parameter, which is independent of theC

jdriving force in the linear regime. The thermodynamic stabil-ity theory31 demands that are all positive (more precisely,C

js

has to be positive deÐnite) in order to ensure chemicalCi, j

° Many other substitutions are formally possible as well, see forexample ref. 24.

Phys. Chem. Chem. Phys., 2001, 3, 1668È1678 1669

Fig. 1 (a) ““Electrochemical ÏÏ resistor and (b) electrochemical capa-citor. The latter element is meaningful only in situations in which thechemical and electrostatic capacitors appear in series (no additionalbranch connected to the node in between).

stability. Renormalising the chemical and electrochemicalpotentials according to and andk

j* 4 (1/ez

j)k

jk8j* 4 k

j* ] /,

replacing the Ñux densities by the electric current densitiesJjwe Ðnally obtainI

j\ z

jeJ

j,

Ij(r, t)\ [ p

j(r)+(k

j* ] /), (6a)

Idis(r, t)\ [ e(r)e0d

dt+/, (6b)

+ É Ij(r, t)\ [C

j(r)

d

dtkj*. (6c)

We recognise that only in eqn. (6a) does the Ñux follow thetime variations of the driving force without delay. In otherwords, both are ““ in phase ÏÏ, which means that eqn. (6a) refersto an energy dissipative process. Eqn. (6a) deÐnes a gener-alised ““electrochemical ÏÏ resistance via as the generalised*k8driving force : (see Fig. 1a). In contrast toRelectrochemP *k8 /Ieqn. (6a), the Ñuxes and driving forces in eqn. (6b) and (6c) areexactly ““out of phase ÏÏ, reÑecting pure energy storage. At thispoint we see that only in the process of energy dissipation dothe chemical and electrical potential appear symmetrically(eqn. (6a)), i.e. the dissipation of energy is independent of thepartition of the total driving force in the chemical and electri-cal parts.

On the other hand, the storage of energy does depend onthis partition. In contrast to the previous case there are twofundamentally di†erent proportionality constants character-ising the storage : the electrical and chemical capacitance.Naively, by analogy to the ““electrochemical ÏÏ resistance, onemight think that electrical and chemical capacitances could bealso combined to form an ““electrochemical ÏÏ capacitor withthe capacitance Q denoting theCelectrochemP (dk8 */dQ)~1,charge. Such an electrochemical capacitor as deÐned e.g. inref. 10 can be conceived as a series combination of the electri-cal and chemical capacitances (Fig. 1b) as demonstrated bysplitting into its electrical and chemical partk8 *

CelectrochemPAd/

dQ]

dk*

dQB~1

PA 1

Celectro]

1

Cchem

B~1. (7)

However, as will be explicitly shown later, the ““charges ÏÏstored in electrical and chemical capacitors are not necessarilyequal and therefore such a series representation only appliesin a few cases and is generally not adequate ; rather there maybe additional branches connected to the node between bothcapacitors, as indicated in Fig. 1b. In steady state experiments(pure energy dissipation) the only relevant potential is k8(splitting into electrical and chemical parts is meaningless).

This introductory discussion of circuit elements revealsalready an important observation : apart from interfacialrelaxations two di†erent timescales may be associated withthe transport processes : typically a short timescale involvingelectrical capacitances and a long timescale involving chemicalcapacitances.

3 Equivalent circuit

3.1 Bulk part

3.1.1 Constitutive relations of electrochemical resistance,electrical and chemical capacitances. Let us represent the set ofequations (6) by an equivalent circuit. This is especiallystraightforward if the situation is quasi one-dimensional, i.e.we consider a laterally homogeneous sample of a cross-sectionarea A extending along the x-coordinate from x \ [L tox \ L (see Fig. 2a). We divide the sample into n compartmentsof thickness s, discretise the equations (6) and transform themfrom the time domain into the frequency domain (thus thetime derivatives are replaced by iu). Although the potentialsand currents now refer to complex-valued Fourier amplitudeswe will keep the same symbols.Ò

Note that the only approximation which we make here,is due to the discretisation. The transport eqn. (6) thenread :

Ij, i(u) \

1

Rj, i

*ik8j*(u) (8a)

Idis(u) \ iuCiq *

i/(u) (8b)

*iIj(u) \ [iuC

j, id dkj, i* (u). (8c)

The subscript j denotes the type of the carrier and the sub-script i the spatial location. If the sample is entirely homoge-neous, i can be omitted. The operator calculates the*

idi†erence between the centers of two compartments sand-wiching the location i, while the operator d calculates thedeviation from the equilibrium (t \ 0) values (Fig. 2c) withinthe compartments. The resistances the electrostaticR

j, i ,capacitances Cq and the chemical capacitances are deÐnedCj, id

as

Rj, i4 p

j, i~1sA

, (9a)

Ciq4 e

ie0

As

, (9b)

Cj, id 4 (ez

j)2Adk

j, idc

j, i

Bt/0

~1As. (9c)

While the chemical capacitances are assigned to the com-partments (Fig. 2a), and refer to the ““ transitionÏÏC

iq R

i, jregions in between.p The resistances parametrise theRi, jcarrier transfer and the electrostatic coupling between theC

iq

compartments.Whilst the electrostatic capacitances are inversely pro-

portional to the compartment thickness, the chemical capac-itances are directly proportional to it. In dilute situationsBoltzmann statistics applies, thus k

j, i\ kj, io ] kBT ln(c

j, i/Nj, i)(symbol denotes the standard concentration independentkj, io

term and the density of available sites). Then the deriv-Nj, iative in eqn. (9c) can be explicitly evaluated yielding

Cj, id \

(ezj)2

kBTcj, iAs. (10)

Since is proportional to the volume, it can be very largeCj, id

compared to Cq. If the relative carrier density is 0.01%, thearea 1 cm2 and the thickness 1 mm, then the chemical capac-itance at ambient temperature amounts to about 0.1 F. This is

Ò If the time domain variables are sinusoidal (usual case in impedancespectroscopy) amplitude and phase of the Fourier transforms reÑectdirectly amplitude and phase of the sinusoidal time domain quantities.p These transition compartments/regions (not shown in Fig. 2a) are ofthe same size as the carrier residence compartments but shifted by s/2along the x-coordinate (for details see ref. 31).

1670 Phys. Chem. Chem. Phys., 2001, 3, 1668È1678

Fig. 2 (a) Discretisation of the sample, (b) coordinate system used in construction of the circuits, (c) basic circuit branches : carriers and displace-ment rails (d) Ðnal equivalent circuit and (e) Kircho†Ïs voltage law for the loop indicated in the circuit.

10 orders of magnitude more than the sampleÏs electrostaticcapacitance (if e \ 10).

Expression (10) has already been used in the context ofequivalent circuits. In ref. 17 it is termed ““NernstÈPlanckcapacitance ÏÏ. Franceschetti16 wrote two expressions for the““ shunt ÏÏ capacitance, as he called it : in the special case inwhich local electroneutrality applies, he used the shunt capac-itance equals to ours (eqn. (10)), while in the case of theNernstÈPlanckÈPoisson model it surprisingly includes e.

3.1.2 Construction of the circuit. After the three basic ele-ments, electrochemical resistances, electrostatic and chemicalcapacitors have been deÐned, they just have to be connectedsuch, that eqn. (8) are fulÐlled. Since our system is relativelysimple we do not necessarily need a bond-graph approach.22Nevertheless, di†erent potentials (/, will appear in thek

j)

circuit, thus the simple scheme displayed in Fig. 2b will beuseful : the transport coordinate which is identical with thespatial coordinate x describes the direction of transport : allthe circuit elements, which are assigned to transport steps (R

j,

Cq) have to appear aligned along the transport coordinate.The coordinate orthogonal to it is reserved for internal reac-tions. Since we assume local equilibrium we do not have toconsider any reaction resistances, just chemical capacitors. Asimple example of an internal reaction is the local dissolutionof oxygen in an oxide or intercalation of lithium in graphite.The corresponding ions and electrons are stored in chemicalcapacitors.

Since we consider three transport mechanisms, we will havethree rails aligned along the transport coordinate : the ionic,electronic and displacement rails (Fig. 2c). These three mapeqn. (8a) and (8b). The coupling among the rails is describedby eqn. (8c) (essentially part of Kircho†Ïs law for the currentsand Ñux-potential relation for chemical capacitors). The diver-

gence of the carrier currents has to be balanced by storage ofthe carriers in chemical capacitors. Thus, Cd will be attachedto the nodes within the carrier rails (Fig. 2c) and alignedorthogonally (no electric potential drop is allowed across Cd).The remaining balance for the currents,

will be obeyed if we connect the free[*iI1[ *

iI2 \ *

iIdis ,terminals of Cd to the nodes of the displacement rail. The

circuit obtained is shown in Fig. 2d. Finally we verifyKircho†Ïs law for the potentials. To do so, it is sufficient torestrict ourselves to the elementary loop indicated by thedashed line in Fig. 2e. Since the elementary loop refers to asingle carrier we skip the subscript j. The potential drop alongthe loop equals Taking intodk

i* ] *

ik8 * [ dk

i`1* [ *i/.

account that the sum obviously*ik8 * 4 dk

i`1* [ dki* ] *

i/,

vanishes. Since any other loop can be composed of such ele-mentary loops, Kircho†Ïs law for the potentials (electrical,chemical and electrochemical) applies to any part of thecircuit.

Since the chemical capacitors once built into the circuit aretreated as the electrical ones we use the same symbol (Fig. 2).In contrast to R and Cq, the two terminals of the chemicalcapacitor refer to the same locus yet to di†erent potentials(states) : one terminal is held at zero potential (i.e. at equi-librium potential) and the other at variable potential Indk

j, i* .contrast to an electrostatic capacitor there is no separation ofelectric charge in a chemical capacitor. While in an electro-static capacitor the Gibbs energy is stored elec-*G\ 12Cq*/2trostatically, in a chemical capacitor Gibbs energy *G\

is stored as entropy changes (in non-dilute cases12Cddk*2energy changes also). In general the ionic and electronic capa-citors connected to the particular node in the displacementrail are not charged equally (with the negative sign). Referringto the circuit, the currents Ñowing through them are notequal. This di†erence is compensated by the current in thedisplacement rail in the middle. In this way the electrical cur-

Phys. Chem. Chem. Phys., 2001, 3, 1668È1678 1671

rents due to the carrier Ñow (the top and the bottom rail) canswitch over to the displacement rail. Such a switching, whichis always accompanied by a local electrical charging, necessar-ily occurs e.g. at interfaces which do not allow for the transferof ions and/or electrons (e.g. Pt/AgCl or SiO2/Si).

On the other hand, in homogeneous regions (the subscript ican be dropped in Cq, far away from interfaces, localR

j, C

jd)

electroneutrality usually holds. Then the chemical capacitorsare charged and discharged evenly and no current redistri-bution takes place in the nodes of the displacement rails (Fig.2b). In such cases ionic and electronic chemical capacitorsbehave as truly connected in series. Thus it is advantageous todeÐne a component chemical capacitance,

1

Cd4 ;

j

1

Cjd, (11)

the reciprocal of which is well-known in solid state electro-chemistry under the term ““ thermodynamic factor ÏÏ1,12 (e.g. inan oxide, where the subscript O denotes the com-PdkO/dcO ,ponent, oxygen in our case). The component chemical capa-citor with the capacitance given by eqn. (11) has been provento be a useful element in approximate equivalent circuits ofmixed conductors.18h21

The ““charge ÏÏ stored according to the constitutive relation(the normalised component chemical potentialdQ\Cddkcomp*

is deÐned as refers in this case to thedkcomp* \ dkion* [ dkeon* )amount of neutral species stored : V beingdQ\ ezionV dccomp ,the volume. The ““chargingÏÏ of Cd means changing the chemi-cal composition rather than changing the electric chargedensity. The most familiar examples of this kind of ““charge ÏÏstorage are batteries.9

3.2 Terminal parts

The terminal parts of the circuit are deÐned by the boundaryconditions. Since at phase boundaries many di†erent experi-mental situations are encountered the set of possible bound-ary conditions is large. Though being a non-trivial part of theoverall problem, little attention has been devoted to this topicso far.15h17 For the following discussion it is worthwhile totreat cases with and without electrodes separately.

3.2.1 Electrode boundary conditions. A situation very oftenencountered in the solid state refers to electrochemical reac-tions in which the reaction product and/or educt is in the gasphase. The morphological changes at such interfaces can, atleast to a Ðrst approximation, be neglected. On the otherhand, the geometry of the reaction sites may be quite complexand, on the microscopic level, far from being planar (purelyelectronically conducting electrodes are usually porous). Weconsider here, as an example, a mixed conducting oxide incontact with very thin electrodes which serve as current collec-tors and are basically permeable for oxygen (Fig. 3a, top). Thesample must Ðrst be equilibrated in an atmosphere with thesame oxygen partial pressure at both sides (x \ L andx \ [L ). This procedure deÐnes the initial situation. Besidesthe di†erence of electron electrochemical potential, *k8 eon ,applied we also allow for external chemical driving forces, i.e.for deviations of the oxygen chemical potential from the initialvalues, and referring to the left- and right-dkO(gas)@ dkO(gas)A ,hand side of the sample, respectively. Note that dkO(gas) 4All three independent driving forces, */,12dkO2(gas) . dkO(gas)@ ,

are control parameters and may be time (frequency)dkO(gas)A ,dependent. Next, we have to clarify the discretisation of theterminal parts. As already mentioned the sample is dividedinto compartments of thickness s. To both sides we add anadditional compartment which stands for the electrodes (Fig.3a). The adjacent compartment refers to the core of the inter-faces, i.e. in our case to a transition region of varied structure

Fig. 3 Terminal parts of the circuit. (a) Electrode boundary condi-tions. (b) Free surface boundary conditions.

between the metal and mixed conductor. The chemistry of theelectrode compartments is often highly complicated : the elec-trodes may be porous, there may be adsorbed oxygen, reac-tions may be complex, etc. In this paper we consider them ona “black boxÏ level which does not require any particularmechanistic model, but assumes a quasi-steady state approx-imation (see remarks at the end of this section and also ref.30). Under these conditions the construction of the terminalpart of the circuit is as follows : transition of electrons betweenthe metal and core compartment is characterised by the““electrochemical ÏÏ resistance (see Fig. 3a). The displace-ReonM

ment rail is connected to the electrode via the electrostaticcapacitance, Since, in the metal compartment, the densityC

Mq .

of electrons is very high, is vanishing there. Consequent-dkeon*ly the terminals of the displacement and electronic rails haveto be at the same potential, which is satisÐed by short-circuiting the two terminals as shown in Fig. 3a, bottom.Finally we are left with connecting the ionic rail to the elec-trode. The overall reaction relevant to this reads

12O2(gas) H Ocore2~ [ 2eelectrode~ . (12)

Close to equilibrium the rate of reaction (12) will be pro-portional to the di†erence of the (electro-) chemical potentialsreferring to the left- and right-hand side of the equation. Thus,the oxygen Ñux density readsJO ,

JO \ KOMkO(gas) [ (k8 O2~ [ 2k8 e~)N. (13)

1672 Phys. Chem. Chem. Phys., 2001, 3, 1668È1678

The proportionality constant, is, in our approximation, aKO ,frequency-independent materials parameter referred to equi-librium. Eqn. (13) is actually the boundary condition we hadbeen looking for, however, it is much easier to visualise itgraphically, by introducing normalised quantities : We replace

by the equivalent electrical current densityJO Iion \ [2eJO ,deÐne resistance of reaction (12) and normalise1/RionM 4 4e2KOthe chemical potential of according toO2(gas)

dkO(gas)* 41

[2edkO(gas) .

Then eqn. (13) may be rewritten

IionRionM \ dkO(gas)* [ dkion* [ */. (14)

The electric potential di†erence, */\ /core[ /electrode ,appears, since the ions and electrons in reaction (12) refer todi†erent loci. The boundary condition (14) is implementedinto the circuit as shown in Fig. 3a. The chemical potential of

which corresponds to one of the externally appliedO2(gas) ,driving forces, is implemented as an ideal voltage source (itspotential is independent of the current). The source is dis-played in Fig. 3a as an ellipse with two terminals. One caneasily verify the terminal part of the circuit by applying eqn.(14) as Kirchho†Ïs voltage law to the loop indicated in Fig. 3aby a dashed line. Please note, that according to our rules, theelectrical potential changes only from one carrier com-partment to the other (along the x-coordinate), while alongbranches in y-direction (reaction coordinate) it remainsinvariant.

The circuit elements, and which appear as a conse-RjM C

Mq ,

quence of the boundary conditions, deserve some comments.The oxygen exchange reaction (12) is a multi-step process,thus stands for the sum of resistances of all the individualRionM

steps. If just one step is rate determining, then can beRionM

interpreted as the inverse of the exchange current density inthe ButlerÈVolmer formalism or as the inverse of the corre-sponding rate constant in the ChangÈJa†e description (seealso ref. 30). The description of the overall reaction rate with aresistance only assumes that the output Ñux instantaneouslyfollows the time-dependent driving forcesÈa quasi-steady-state approximation. Under general conditions, a transferfunction o†ers a proper description. The frequency depen-dence stems from the inclusion of energy storage mechanisms,e.g. oxygen adsorption, surface di†usion, etc. As in the bulk,these storage processes can easily be taken into account bychemical capacitances. A detailed treatment in this respectwhich highlights the power of the proposed tool for handlingelectrode kinetics will be given elsewhere.

As far as is concerned, we should stress that the relativeCMq

permittivity of the corresponding transition-compartmentlocated between electrode and core certainly di†ers from thebulk values (see in-depth studies concerning liquidelectrochemistry). In terms of liquid electrochemistry the inter-facial core refers to the inner Helmholtz plane (IHP), thus C

Mq

to the capacitance of the region between the electrode and theIHP.14 The capacitance due to the di†use (GouyÈChapmann)layer is already included in the ““bulk part ÏÏ of the transmis-sion line (see Section 4.1, also ref. 13 and 15) ; thus, there is noneed to include it explicitly in the terminal part, as has beendone in the literature.17

3.2.2 Free surface boundary conditions. In the Ðeld of solidstate electrochemistry, samples are not necessarily electricallycontacted. An important example is a mixed conducting mem-brane separating two chambers with di†erent componentpartial pressures. Here di†erent charge carriers move in anambipolar and e†ectively electroneutral fashion. If the mobi-lities of the carriers, e.g. O2~ and e~ in the case of oxygenpermeation, di†er, the membrane surface will become chargedalbeit not contacted externally (see for example ref. 31).

In the simplest case the sample ends with the surface core(no electrode compartments) (see Fig. 3b) and the only drivingforces are and The overall reaction betweendkO(gas)@ dkO(gas)A .the gas phase and the carriers in the surface core, whichoccurs in several steps, can be written as

12O2(gas) H Ocore2~ [ 2ecore@ , (15)

which is quite similar to reaction (12). The only di†erence isthat here the electrons refer to the core, not to the electrode.Using almost the same procedure as in the previous section(we denote the surface reaction resistance by we deriveRs)

^IionRs \ dkO(gas)* [ (dkion* [ dkeon* ) at x \ ^L . (16)

The source of oxygen chemical potential, is connecteddkO(gas)* ,to the terminals and (see Fig. 3b) through which, onT1 T2 Rs ,the level of the equivalent circuit, describes the internal resist-ance of the source. The division of the external driving force

into and is determined by the structure ofdkO(gas)* dkion* dkeon*the circuit. Please note that the core chemical capacitors forions and electrons are in general not Ðlled evenly, thus storingan excess surface charge, which was mentioned above. Thischarge is needed in order to enable the existence of the dragelectric Ðeld within the sample, which is characteristic forambipolar di†usion. In the literature dealing with chemicaldi†usion the surface reaction is usually characterised by aphenomenological rate constant It is straightforward tok6 s .show that is related to through the component chemi-k6 s Rs~1cal capacitance of the whole sample,30,32 viz. :

k6 s\L

RsCd. (17)

From eqn. (17) we see immediately, that plays the role ofk6 s/La time constant which characterises the transient behaviour ofthe surface controlled di†usion.

4 Applications of the equivalent circuit

4.1 A single carrier conductor polarised with blockingelectrodes

One of the simplest situations, which is met in solid state elec-trochemistry, is a single carrier system contacted by electrodeswhich do not allow for any carrier transfer across the interface(completely blocking electrodes). Such a cell is approximatelyrealised by Pt/AgCl/Pt or C/b-alumina/C (as long as weneglect electronic carriers in the solid electrolyte, i.e. we donot consider the low frequency part of the impedance spectranear the dc conditions). Another single carrier system withblocking electrodes is the MOS (metal-oxide-semiconductor)structure Here, the material under focusM/SiO2/Si/SiO2/M.is Si doped such that only electrons or holes are of impor-tance. If we neglect ions in the contacts per-SiO2 , SiO2/Sifectly block the carriers in Si. The impedance of these systemshas been treated in detail on the continuum level25 as well ason the level of intuitive33 and exact equivalent circuits.15 Ouraim here is to derive from the equivalent circuit some inter-esting properties of the electrode capacitance (Is it electro-static and/or chemical by nature?) and of MaxwellÈWagnerrelaxation.34 Before doing so, let us reproduce the well-knownimpedance relations.

4.1.1 Impedance. Since we have just one charge carrier toconsider, the general circuit simpliÐes (here we take forexample the ions as charge carriers, but the circuit would havethe same topology also for electrons) : The electronic rail aswell as the chemical capacitors due to electrons can beomitted (see Fig. 4a). In the case of blocking electrodes theinterfacial resistance is inÐnitely large, thus the ionic rail issimply disconnected from the electrodes. First we assume arti-

Phys. Chem. Chem. Phys., 2001, 3, 1668È1678 1673

Fig. 4 Single carrier material contacted with blocking electrodes.Equivalent circuits (a) for the whole cell, (b) for the di†use layer and(c) for the interfacial capacitance with core e†ects.

Ðcially that the interfacial core exhibits the same structure asthe bulk. Then the carrier concentration in the core is thesame as in the bulk and no equilibrium space charge layersare present. Correspondingly, in the equivalent circuit all theelements are spatially invariant (we can omit the subscript i).

The impedance of the circuit is deÐned as Z(u) \(the symbol * denotes the di†erence between the*k8 eon(u)/I(u)

electrodes). Since the density of electrons in macroscopicmetal electrodes is very high (mesoscopic e†ects are con-sidered later in this section), The ratio, voltage*k8 eonB */.drop along the displacement rail over the total current, can bereadily calculated15

Z\R

1 ] iuq]

1

1 ] iuq1

iuCqtanh(iL )

iL. (18)

The symbol denotes the dielectric relaxation time.q4RionCqThe decay length, describesi~1 4 2L JCq/Cd (1 ] iuq)~1@2,the decay of electrochemical perturbations (dk, +/) in thedirection towards the bulk. If we replace the resistance, andelectrostatic and chemical capacitances in eqn. (18) by the deÐ-nitions (eqn. (9)), the obtained impedance will be identical tothe well-known result derived on the continuum level.25 Thisis no surprise since the equivalent circuit is an exact represen-tation of the continuum level equations.

The Ðrst term in eqn. (18) refers to the bulk response whilethe second one includes the electrode capacitance. The latterstatement becomes particularly obvious in the low frequencyapproximation assuming in addition that (the DebyeL P 3L Dlength ThenL D 4 Jee0 kBT /(e2z2c)).

Z(u] 0)B R(1[ 3L D/L )]2

iuJCdCq. (19)

Obviously, the resistance measured at low frequencies (Ðrstterm in eqn. (19)) does not include the contributions from theinterfacial regions with a thickness of There is noB3/2L D .

faradaic current Ñowing within these boundary regions evenat the lowest us.

4.1.2 Di†use layer capacitance. As derived previously bySah13 and Brumleve and Buck15 the second term refers to thedi†use layer capacitance, Interestingly, isCdiff . Cdl4 JCdCqthe geometric mean of the sampleÏs dielectric and chemicalcapacitances. Further evaluation of the geometric meanaccording to eqn. (9b and 9c), yields the more familiar expres-sion of the ““Ñat bandÏÏ di†use layer capacitance

JCdCq \ ee0AS e2z2c

ee0 kT\ ee0

AL D

. (20)

The Debye length, i.e. the screening length, reveals aL D ,further interesting meaning. The chemical capacitance of thewhole sample increases with the sampleÏs thickness, while itselectrostatic capacitance decreases. Therefore, there must be acritical thickness at which Cq \ Cd. In single carrier systemsthis thickness is exactly the Debye length : if the sample isthinner than its electrostatic capacitance will be larger, ifL D ,the sample is thicker the chemical capacitance will predomi-nate.

As we noted, is a ““mixture ÏÏ of an electrostatic and aCdlchemical capacitance (Fig. 4b). Since the capacitors arearranged in a transmission line, the term ““di†use layercapacitance ÏÏ is very meaningful. A straightforward calculationshows that one half of the free energy in such transmissionline is stored electrostatically, and the other half chemically(conÐgurational entropy term). Although can thus be con-Cdlsidered as an element in which the energy is stored““electrochemically ÏÏ it is not an electrochemical capacitor inthe sense of its deÐnition (see Fig. 1b). The chemical and elec-trostatic capacitors are connected in a transmission line ratherthan in series. Already this shows that the concept of an elec-trochemical capacitance is not general.

4.1.3 Interfacial capacitance including core e†ects. Thedi†use layer capacitance of cells with solid ionic conductorshas never been clearly measured. The values found werealways much higher (orders of magnitude) than expected from

Although di†erent interpretations have been given in theL D .literature35,36 we propose an explanation which followsdirectly from the circuit. In the treatment above we assumedthat the sample is completely homogeneous. In reality, thecarrier density in the interfacial core will be di†erent from thebulk since there the underlying structure di†ers from theregular structure in the bulk. Thus the core chemical capac-itance should be distinguished from the chemical capacitancesof the subsequent compartments. The equivalent circuit forthe interfacial capacitance, C, is then composed of C

Mq , Ccored

and as shown in Fig. 4c and derived earlier from theCdl ,steady state considerations.37 If the core carrier density is veryhigh, then and C will appear as a series com-Ccored [ Cdlbination of the electrostatic and chemical Thus anC

Mq Ccored .

interpretation of C as an electrochemical capacitance is indeedpossible in this case (see also Fig. 1b). If a partial charge trans-fer between the metal and core takes place, will e†ectivelyC

Mq

be very large and Such a situation is met in super-CB Ccored .capacitors, in the case of underpotential deposition or theelectrosorption process of e.g. Pb on Au (see e.g. ref. 9). Ourcore defects in these cases refer to the electrosorbed speciescomprising a part of the monolayer on the top of the elec-trode. To make this point clear let us calculate the core capac-itance explicitly. According to eqn. (9c) the capacitance isdeÐned as Using the surface coverage,P(dkcore/ccore)~1. h 4

rather than the concentration, and taking intoccore/cmax ,account that the density of core defects is limited by theircmax ,chemical potential reads : ln [h/(1 [ h)].kcore\ kcore0 ] kBT

1674 Phys. Chem. Chem. Phys., 2001, 3, 1668È1678

The capacitance obtained isC\ (ezj)2/KBT )cmaxV (1[ h)h

identical to the pseudo-capacitance9 derived from the electro-chemical Langmuir isotherm.

4.1.4 Maxwell–Wagner relaxation. There is another pointworth mentioning in a single carrier system. Let us consider amaterial which exhibits a Schottky contact with the metal.Then the equilibrium carrier density in the space charge layerwill be signiÐcantly depleted and the referring to the com-C

id

partments within the space charge layer will be much smallerthan in the bulk. To a Ðrst approximation, can be omitted,C

id

and the equivalent circuit of the space charge layer will consistof the layerÏs resistance in parallel with the layerÏs electrostaticcapacitance. If, on the other hand, the carrier density changessigniÐcantly only on a scale large compared to then theL D ,38opposite limit is realised : The discretisation can be coarser

and the will be relatively large (when compared to(s [ L D), Cid

Thus, the can, to a Ðrst approximation, be short-Ciq). C

id

circuited, resulting in a series of RCq terms. This description,which is known in the literature under the term MaxwellÈWagner relaxation,34 does not assume local electroneutralitybut rather neglects the relative changes of carrier density. Aswe have shown, the MaxwellÈWagner approximation is verygood for single carrier systems on a scale much larger than

but, although used in the literature,39 is not appropriateL D ,within space charge layers.

4.2 Quantum-conÐned capacitor

So far we have neglected the changes in electron densities inmetal electrodes upon applied potential di†erence. Since thenumber of electrons in metal electrodes is very high whencompared to the number of carriers in an ionic conductor, therelative changes in the metal are indeed negligible. The situ-ation is di†erent, however, if the size of the electrodes becomesvery small (Fig. 5a). Let us brieÑy consider such a mesoscopicparallel plate capacitor10 using our equivalent circuit. If weneglect, for simplicity, the spatial variations within the plates,the equivalent circuit shown in Fig. 5b results. Since thechemical and electrostatic capacitors, unlike in the case con-sidered in Section 4.1, appear in series then the system capac-itance in this case is a proper electrochemical one, as derivedearlier.10 At T \ 0, and to a good approximation also at Ðnitetemperatures, the chemical capacitance of electrons Ceond \

is proportional to the density of(e2/kBT )V (dkeon/dceon)~1states at Fermi energy multiplied by the electrode volume. Aswe already mentioned, due to the high number of states, Cd isusually much greater than Cq and thus negligible in theoverall expression. Interesting e†ects appear at mesoscopicsizes. If the width and thickness of the electrodes are of theorder of a few nm, the electrodes behave as quantum wires

Fig. 5 (a) Mesoscopic capacitor composed of two parallel quantumwires. (b) Corresponding equivalent circuit.

(Fig. 5a). Then the chemical capacitance is not proportional tothe volume but to the length, L , of the wires, hCd \ e2L /(hvF),being the Planck constant. Assuming L \ 1 lm and thevelocity of electrons at the Fermi energy to be 108 cm s~1vFthen the chemical capacitance Cd B 0.05 fF becomes compara-ble to Cq.

4.3 Warburg di†usion and stoichiometry polarisation :General

In general, in solid materials ionic and electronic charge car-riers are mobile. Here we discuss such a mixed conductor withone ionic and one electronic charge carrier sandwichedbetween electrodes which are nearly reversible for electronsand nearly blocking for ions. We use the word ““nearly ÏÏ sincewe discuss a realistic cell. Such cells are often used in solidstate electrochemistry in order to investigate separately theionic and the electronic conductivity and to measure the com-ponent di†usion coefficient. The electrochemical cells men-tioned in the previous section but now observed over theentire frequency range are appropriate examples. Others areC/PbO/C, and also cells in which the electronic carriers areblocked. For this purpose pure ionic conductors are used aselectrode phases, e.g. Pt, (YSZ),O2/yttria-stabilised-zirconiacontacted to PbO,40 or or WeBaTiO3 ,41 YBa2Cu3O6`x

.42further assume, just for simplicity, that the mixed conductor isa single crystal and that the component partial pressures atboth sides of the sample are kept equal and constant. At highfrequencies of the applied voltage the electrode inÑuence isnegligible ; the dielectric relaxation in the bulk yields a semi-circle in the complex plane described by the ““non-blockedbulk resistance ÏÏ and the dielectricRion Reon/(Rion] Reon)capacitance of the sample. In this regime both carriers con-tribute fully to the bulk conductivity. At lower frequencies,however, the selectively blocking electrodes prevent ions fromcrossing the interface : ions periodically accumulate at oneside of the sample while becoming depleted at the other. Ofcourse, this accumulation and depletion over larger distancesis only possible if the electron density changes accordingly,such that the sample remains locally electroneutral. In thisfrequency range chemical di†usion (45¡ Warburg di†usionturning over into a semicircle) takes place : in an oxide, the““oxygenÏÏ e†ectively swings between both halves of thesample, and thus the composition changes as well.1 In ref. 20and 21 we introduced intuitively an approximate equivalentcircuit for such a cell and showed that it applies in a very-broad materials parameter window. However, we could notderive it directly from the basic equations. Here we show howthis is possible (we will use an exact equivalent circuit ratherthan di†erential equations as the starting point).

The exact equivalent circuit of a mixed conductor is shownin Fig. 3a (we omit outer chemical driving forces here). On thebasis of the previous single carrier study the following simpliÐ-cations of the circuit can be made : (i) the bulk of the sample iselectrically homogeneous which suggests that local elec-troneutrality will be a good approximation there. Thus Ciond

and will be charged evenly, and there will be no currentCeond

leakage into the displacement rail ; consequently the bulk Cqbehave as truly connected in series. We replace the Cqs fromthe bulk part of the displacement rail by a single geometricalcapacitance of the sample connected externally (see Fig. 6a).This is an approximation since the interfacial Cqs are nowcounted twice, but as long as the approximation isL A L D ,good. Now the bulk ionic and electronic chemical capacitorsallocated to the same compartments also appear to be con-nected in seriesÈthus they can be replaced by the chemicalcapacitance of the component, Cd, as derived previously. ThisÐrst simpliÐcation step yields the circuit shown in Fig. 6a.

In the interfacial regions local electroneutrality is clearlyviolated. In this region we neglect, besides all other resist-R

jM,

Phys. Chem. Chem. Phys., 2001, 3, 1668È1678 1675

Fig. 6 Mixed conductor. The exact circuit (Fig. 3a) is reduced overthe steps (a) and (b) resulting in the circuit (c) which is simple enoughto be used in routine analyses of experimental data but still generalenough to apply to a variety of materials (restriction : L A L D).

ances. The capacitive network that remains (Fig. 6b) is athree-terminal element which can in general be described bythree impedances (see for example ref. 43) and taken intoaccount explicitly in the evaluation of the overall impedance.This description is, however, beyond the scope of this paper,and we rather make further approximations as shown in Fig.6b. We replace the capacitive network by two capacitors withwell deÐned values obtained as follows : First we neglect elec-tronic carriers (the bottom part of the network). The remain-ing two-terminal transmission line (see Fig. 4b) exactly reÑectsthe di†use layer capacitance due to ions, Second, weCdlion.neglect ions and substitute the bottom part of the network by

Of course such a substitution of the three-terminalCdleon.network by that composed of and (Fig. 6b) is anCdlion Cdleonapproximation, yet, as the detailed analysis shows,21 the erroris less than 50%. This is still acceptable, especially in view ofthe fact that the impedance of the resulting circuit (Fig. 6c)can be obtained in a closed form and applies for a very broadmaterials parameter window. The obtained closed formimpedance explains, for example, a long lasting ““paradoxÏÏ inthe Ðeld of impedance spectroscopy, referring to the transitionof 45¡ Warburg impedance to 90¡ rise which we expect if westart with a mixed conductor and then neglect one of thecharge carriers. For details see ref. 20 and 21.

It is interesting to note that the interfacial capacitance (thethree terminal capacitive network) in general does not refer tothe usually deÐned parameter Only if the elec-Cdl\ ee0/L D .trode is completely blocking for both carriers does this holdtrue (in this case Under selectivelyCdl\ JCq(Ciond ] Ceond )).blocking conditions the interfacial capacitance is determinedby the blocked (not necessarily majority) carriers and thus bythe Debye length referring to one (blocked) carrier only.

4.4 Warburg di†usion and stoichiometry polarisation : SOFCkinetics

In this section we apply the circuit just derived above to theinteresting example of the electrode kinetics of solid oxide fuelcells.44,45 To be speciÐc, we consider the impedance of the cell(see Fig. 7a) From theO2(gas)/(La, Sr)CoO3~d

/YSZ/Pt, O2(gas) .point of view of the SOFC cathode this is the set-up of aWagnerÈHebb polarisation cell in which the electronic con-ductivity is blocked by YSZ electrolyte. Since (La, Sr)CoO3~d(LSC) is a mixed conductor with relatively high electronicconductivity the circuit shown in Fig. 6c can be further simpli-Ðed by short-circuiting the electronic resistors and by connect-ing the external lead to LSC directly is not necessary).(ReonM

We use highly permeable LSC instead of the manganate sincewe can then consider quasi-one-dimensional transport. Theoxygen exchange reaction is taken into account by the resist-ance in parallel with the interfacial capacitor asR

sM Cdlion,derived previously (see Fig. 6c). The right-hand side terminal

part of the circuit refers to the interface between the mixedand ionic conductor. Here the electrons are blocked, thus only

remains in the electronic rail. The ionic transfer is charac-Cdleonterised by Let us assume that a reference electrode isRionM .used, connected to YSZ close to the interface, such thatfurther transport steps are not of importance. The resultingcircuit is shown in Fig. 7b.

An interesting situation occurs, if the transport in LSC issurface controlled, then the ionic resistance of the mixed con-ductor vanishes and the chemical capacitors, which are nowall in parallel, can be replaced by a single Cd which refers tothe whole sample. The resulting circuit (Fig. 7c) becomes evenmore transparent when presented in a nested form (Fig. 7d) : ityields two impedance semicircles in the complex plane, thehigh frequency one referring to the interface LSC/YSZ whilethe low frequency one is composed of the surface exchangeresistance and the chemical capacitance of LSC. Since CdR

sM

Fig. 7 An example of SOFC kinetics. The reduction of the circuitfrom (b) to (d) explains the properties of the observed impedancespectra.44

1676 Phys. Chem. Chem. Phys., 2001, 3, 1668È1678

is, unlike proportional to the thickness, we expect theRsM,

relaxation time to scale linearly with the thickness of LSC.This was indeed observed and also properly explained.44 Thisexample is interesting, since Cd appears as a part of an RCterm just like the electrostatic capacitances which we are usedto. In general the chemical storage of energy always occurslocally and is therefore necessarily combined with the trans-port steps which supply the carriers ; thus the storage and dis-sipation of energy is inherently distributed in space with theconsequence of dispersive impedance spectra (transmissionlines). If, however, the transport steps are relatively fast, as inthe case here, the chemical storage is reÑected in its pure form,i.e. as a perfect capacitor without any frequency dispersion.

4.5 Intercalation battery

In this section we discuss two ionically and electronicallyreversible electrodes separated by an ionic conductor, e.g.

which is the set-up of aPt/LixMn2O4/Li`-electrolyte/Li

xC

modern rocking chair battery in which Li swings forth andback from one intercalation electrode to the other. From thepoint of view of the intercalation electrode (e.g. Li

xMn2O4)this is a set-up of a coulometric titration cell, in which

as the mixed conductor is sandwiched between aLixMn2O4metal electrode (blocking for ions) and an ionically reversible

electrode (Li`- being blocking for electrons.electrolyte/LixC)

A detailed description of the electrochemical impedance ofsuch cells is possible but requires numerical simulations.46Nevertheless, several basic features can also be derived fromour equivalent circuit.

The cell is depicted schematically in Fig. 8a. In reality, theintercalation electrodes are often porous, we assume here forsimplicity that the conductivity within the pores is very highand consider only a single ““grain ÏÏ of the intercalation elec-trode. The electrolyte, which is e.g. a lithium ion conductor, iscontacted with an intercalation electrode material (e.g.

being capable of accepting or releasing lithium inLidMn2O4)the form of ions and electrons : ionic transfer takes place at

the electrode/electrolyte interface and electronic transfer at thecurrent-collector/electrode interface. For this situation thecircuit shown in Fig. 7b is sufficiently general. Disconnectingthe ionic rail from the current collector and neglecting theelectronic resistances within the electrode the circuit reducesas shown in Fig. 8b. It is obvious that at low frequencies thecircuit describing a battery is blocking ; the imaginary part ofthe impedance will be determined by Cd : the charge is swing-ing between the two chemical capacitors (two electrodes). The

Fig. 8 (a) Scheme of a lithium battery (anode is not shown). (b) Sim-pliÐed equivalent circuit.

chemical capacitance Cd is a di†erential quantity by deÐnition.Thus, the overall charge stored in a battery is obtained byintegration, in which and stand for theQ\ /k1k2 Cd(k)dk, k1 k2upper and lower cut-o† voltage. The over-all capacity thenreads

C1 \1

k2[ k1

Pk1

k2Cd(k)dk, (21)

which is just the average chemical capacity of the electrodes.In other words is the integral chemical capacitance, just byC1analogy to the integral electrostatic capacitance.

4.6 Pure chemical di†usion

As the Ðnal example we discuss the appropriateness of thederived circuit for mass transport with vanishing net currentÑow. This type of transport refers to chemical di†usion of acomponent, e.g. oxygen, in the form of oxygen ions and elec-trons, through an oxide. Owing to its selectivity with respectto the species transferred, the process can be used for gasseparation e.g. to Ðlter out oxygen from air by simply apply-ing a pressure di†erence across an oxide membrane.

A properly terminated general circuit for this situation wasdeveloped in Section 3.2.2 and is shown in Fig. 3b. On a time-scale large compared to the dielectric relaxation time and witha spatial resolution well above the Debye length we mayassume electroneutrality which simpliÐes the circuit substan-tially (see Fig. 9a). Transmission lines of this form are a well-known description of di†usion transport in general.5 In ourcase, the transmission line is not only a ““black-boxÏÏ descrip-tion with a corresponding inputÈoutput relationship, butdescribes the local ionic and electronic Ñuxes as well as localvariations of stoichiometry in detail. The transmission line(Fig. 9a) consists of ionic and electronic resistances, com-ponent chemical capacitances, and an external source ofchemical potential which represents the changes in the exter-nal component partial pressure, e.g. The Ðnite rate of thepO2

.incorporation reaction is taken account of by the surfaceexchange resistance If the membrane is homogeneousR

s.

(single crystal), all the circuit elements are spatially invariant.Then the solution of the circuit is isomorphous to the solutionof FickÏs second law, the di†usion (chemical) coefficient being5

Dd 4L2

(Rion] Reon)Cd. (22)

Substituting R and Cd in eqn. (22) by their deÐning equations(9) the more familiar form1 is readilyDd \ teonDion] tionDeonreproduced (note that internal trapping reactions47 areneglected here).

Fig. 9 (a) Approximate equivalent circuit for chemical di†usion in apolycrystal with blocking grain boundaries. (b) Further reduction ofthe circuit for situations in which the grain boundary chemical resist-ance prevails over the bulk one.

Phys. Chem. Chem. Phys., 2001, 3, 1668È1678 1677

The circuit becomes a useful tool if the sample is not homo-geneous. Let us consider a polycrystal in which grain bound-aries are ““blockingÏÏ for chemical di†usion (e.g. Fe-doped

ceramics.48) We assume a brick layer model for theSrTiO3polycrystal, such that a quasi-one-dimensional approximationis reasonable. The resistances and chemical capacitanceswhich refer to the grain boundary regions will di†er from theelements within the grains. The complexity of this circuit ingeneral requires a numerical approach. However, a simpleanalytical solution can be given, if the resistances in the grainboundary regions are high when compared to the resistancesof the grains, may be neglected, and if in addition theRgraingrain size, is large compared to the boundaryL grain L D ,volume and thus the boundary chemical capacitance is negli-gible. The correspondingly tailored circuit (Fig. 9b) is then justa superstructure of a transmission line for a single crystal : thetransport on a scale larger than the grain size appears as di†u-sion in a homogeneous material but characterised withresistances and capacitances The correspondingRgb Cgraind .di†usion coefficient reads :

DMd 4

L grain2(Rgbion] Rgbeon)Cgraind

. (23)

Please, note that in eqn. (23) the resistances and thecapacitances refer to di†erent loci (boundaries and grains,respectively).

5 ConclusionsAn equivalent circuit, which exactly maps the driftÈdi†usionmodel of an e†ectively one dimensional mixed conductor con-tacted with two electrodes, has been derived. Under condi-tions of time-invariance and linear response the circuitconsists of only three basic elements : (i) ““ electrochemical ÏÏresistance, (ii) electrostatic capacitance and (iii) chemicalcapacitance, which are deÐned in terms of the chemical andelectrical properties of the cell. The connection of the elementshas to satisfy the fact that the spatial and thermodynamic dis-placement occur along two ““orthogonal ÏÏ coordinates. As weshowed in several examples in the second part of the paper, itis very easy to identify in the exact circuit the elements rele-vant for the particular experimental conditions and to neglectthe unimportant ones. This is most relevant for solid stateelectrochemistry, since it allows for the description of di†erentexperimental set-ups as well as of a variety of devices, e.g. fuelcells, oxygen membranes and pumps, batteries. As simple asthe tailored circuit Ðnally is, it has two advantages when com-pared to empirically constructed circuits : (i) the validity rangeof the circuit is well deÐned and (ii) the circuit elements areclearly related to basic materials parameters.

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