Analytical Pore Scale Modeling of the Reactive Regions of Polymer Electrolyte Fuel Cells

Analytical Pore Scale Modeling of the Reactive Regionsof Polymer Electrolyte Fuel CellsL. Pisani,z M. Valentini, and G. Murgia

Center for Advanced Studies, Research and Development in Sardinia, 09010 Uta (Cagliari), Sardinia, Italy

This paper analyzes the effects of the catalyst layer porous structure on the performances of polymer electrolyte membrane fuelcells. Comparing the characteristic lengths of the porous structure with the characteristic lengths of the diffusion phenomenashows that the oxygen and hydrogen concentrations in the electrolyte phase change significantly at the pore scale level; therefore,the related diffusion phenomena need a nonhomogeneous description. These rapidly varying concentrations are coupled to the cellpotentials through the reaction rate expression,i.e., the Butler-Volmer equation. Thus, to employ a macrohomogeneous descriptionof the fuel cell without loss of accuracy, it is necessary to find an effective expression for the reaction rate which does not dependexplicitly on the rapidly varying concentrations. This is done here through an analytical averaging procedure and results in aneffective Butler-Volmer expression that includes implicitly the effects of nonhomogeneity of the porous structure. This expressionis compared with the ordinary Butler-Volmer expression and with the agglomerate models in the literature. The former turns outto be valid only in the limit of low current densities, and the latter only in the high porosity limit. Finally, the effectiveButler-Volmer expression is inserted in the framework of macrohomogeneous models. From the analysis of the model results, onecan conclude that the effects of the porous structure on the cell performances are crucial for the correct description of the cellconcentration polarization and the estimation of the effective Tafel slope at high current densities.© 2003 The Electrochemical Society.@DOI: 10.1149/1.1621876# All rights reserved.

Manuscript submitted December 16, 2002; revised manuscript received May 5, 2003. Available electronically October 9, 2003.

The main goal of fuel cell modeling is the description of thedevice performances starting from the underlying physical phenom-ena, material parameters, and operating conditions. The modelsshould be as simple as possible to reduce the numerical complexitybut accurate enough to describe correctly the fuel cell operation.Some assumptions are often used to simplify the mechanistic mod-els such as:1 one-dimensional~1D! geometry, constant gas porosity,fully hydrated membrane, isothermal conditions, steady-state opera-tion, and homogeneity of the media. Qualitative considerations canbe used to estimate the applicability range of such assumptions. Forexample, the 1D geometry approximation can be applied when thegas concentrations do not vary too much along the flow channels, asin the case of high stoichiometric flow ratio, and when channels andribs are sufficiently thin to render homogeneous the delivery of elec-trons and reactants. Where the approximations are no longer appli-cable, the underlying assumptions must be relaxed, and the modelsbecome more complex. In the literature, several models have beenpresented with the aim of going beyond the following approxima-tions: 1D geometry,2-4 constant gas porosity in the diffusiveregion,4-6 fully hydrated electrolyte membrane,7,8 isothermalconditions,8,2 and steady-state operation.9,4 Although these exten-sions are straightforward, relaxation of the homogeneity approxima-tion requires a clever strategy to handle the complex porous struc-ture of the media.

The reactive region of a polymer electrolyte membrane fuel cell~PEMFC! has a complicated structure:10-12a matrix of electronicallyconductive 20-40 nm carbon grains forms agglomerates of 200-300nm with platinum islands of 2-3 nm supported on them. This solidporous structure has a bimodal pore size distribution. Smaller, 20-40nm pores exist inside the agglomerates between the carbon grains,and larger pores~40-200 nm! constitute the void space betweenagglomerates. The ionic conductive electrolyte fills part of the largerpores, possibly together with Teflon, which can be added as a hy-drophobizing agent. The smaller pores are available for the transportof the gas species when they are not flooded with water.

In the literature, several models have been published based onsimplified descriptions of the porous reactive region: Giner andHunter13 and Iczkowski and Cutlip14 consider cylindrical agglomer-ates consisting of a homogeneous mixture of carbon, platinum, andelectrolyte, surrounded by gas pores; Perryet al.15 consider sphericagglomerates, and Gloaguenet al.16 slab geometry agglomerates.Despite the various geometries, the effects of the nonhomogeneity

on the fuel cell performances, as described by these models, can besummarized by a value of the Tafel slope at high current densitiestwice the value of the Tafel slope at lower current densities. Someauthors, such as Giner and Hunter,13 Broka and Ekdunge,17 andJaouenet al.,18 consider three-phase reactive regions made by solidagglomerates covered by an electrolyte layer and separated by gaspores. The presence of the electrolyte layer limits the maximumcurrent density achievable.

In a previous work,6 we considered cylindrical gas pores sepa-rated by a homogeneous mixture of carbon, platinum, and electro-lyte, and we have found a very good agreement with the experimen-tal results.

The models from all the preceding papers have been presentedwith insufficient analysis on the influence of the employed porousstructure geometry on the model results. This lack of analysis re-stricts the model reliability. The main goal of this paper is to elimi-nate this shortcoming and, consequently, to reach a deeper under-standing of the phenomenology associated with the diffusionreaction on a two-phase nonhomogeneous medium.

To reach this aim, we first ascertain which phenomena need apore scale description; subsequently, we apply a volume averagingprocedure to decouple the variables varying at the pore level fromthe constant ones. This allows the preservation of a macrohomoge-neous level description of the fuel cell without loss of accuracy. Byconsidering five different simplified geometrical descriptions of theporous structure, the volume averaging leads to five analytical ex-pressions for the effective reaction rate. The five expressions arecompared among themselves and with the expressions used withinthe homogeneous and agglomerate models to get a better under-standing of the phenomenology associated with reactive transportwithin a porous structure. In the macrohomogeneous model section,the use of these effective reaction rate expressions in the frameworkof macrohomogeneous models is discussed. In the results section,we show the effects of the porous structure on the polarization curveof air and methanol electrodes.

Pore Scale Model

The relevant transport phenomena inside the reactive region of aPEMFC are proton transport in the electrolyte phase, electron trans-port in the solid~carbon! phase, and reactant diffusion in the gas,liquid, and electrolyte phases. To decide the extent of details re-quired for the description of the phenomena, we must compare thecharacteristic lengths of the region~i.e., thickness of the region andpore-agglomerate lengths! with the diffusion lengths~i.e., the dis-tances over which the physical variables related to the transportz E-mail: [email protected]

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A1558

phenomena change significantly!. For example, when the diffusionlengths are much larger than the thickness of the region, we canconsider the related physical variables as constants inside the region.In this case, for the sake of the phenomena description, the wholeregion can be reduced to a single interface. If, instead, the diffusionlengths are larger than the pore scale lengths but smaller than theregion thickness, we must solve the diffusion equations inside theregion, but we can still consider the physical variables as constantsat the pore scale level. In this case, a homogeneous model leads toan accurate solution. Finally, if the diffusion lengths are smaller thanthe pore scale lengths, we cannot neglect the nonhomogeneity of theregion, and we must solve the diffusion equations at the pore scalelevel. In Table I, we report an estimate of the diffusion lengths of thepotentialsf and concentrationsc for methanol, oxygen, and hydro-gen electrodes.

The thickness of the catalyst region of a PEMFC is around1025 m, while the characteristic pore scale lengths, as just de-scribed, are in the range 1027 to 1028 m. By comparing these valueswith the diffusion lengths of Table I, we observe that ionic potential,electronic potential, and reactant concentration in the gas phase canbe considered approximately constant at the pore scale level,whereas oxygen and hydrogen concentrations in the electrolytephase should not.

The reactive region, as described in the introduction, is much toocomplicated to be treated analytically. To give the simplest nonho-mogeneous description of the reactant transport, we consider themedium as composed by only two homogeneous phases: the porephase~superscript P!, which is characterized by a constant reactantconcentration at the pore scale level and by the absence of electro-catalytical reactions

¹creacP 5 0 @1#

¹ • iP 5 0 @2#

and a mixed phase~superscript M!, where the reaction does takeplace, which is characterized by a nonconstant reactant concentra-tion at the pore scale level

¹creacM Þ 0 @3#

¹ • iM Þ 0 @4#

The condition of constant reactant concentration at the pore scalelevel, expressed by Eq. 1, is verified when the reactant diffusionlength is larger than the characteristic pore scale length,i.e., 1027 to1028 m. According to the diffusion length values given in Table I,this condition is met when the pore phase coincides with the gasphase, for oxygen or hydrogen electrodes, and with the liquid phase

for liquid feed methanol. Consequently, the electrolyte, the solidcarbon, and the platinum supported on the carbon are constituents ofthe mixed phase for all electrodes. For oxygen or hydrogen elec-trodes, the liquid phase is also a component of the mixed phase. Inthe following, we talk about porosity~e! referring to the volumefraction of the pore phase.

To model the reactant diffusion inside the mixed phase, we sup-pose that the overpotentialh 5 fE 2 fS is constant at the porescale level ~see Table I!; moreover, we use a Butler-Volmerequation19 for the reaction kinetics, and we assume a first-orderdependence on the reactant concentration

¹ • i 5ai0,ref

creac,refFexpcreac

M @5#

where

Fexp 5 @eaabFh 2 e2acbFh# @6#

and a is the effective catalyst area per unit volume,i 0,ref the ex-change current density,creac,refthe reactant concentration at the ref-erence state,a i the electrode transfer coefficients, andbF the Fara-day constant in units ofRT.

Then, by combining the material balance

¹ • NreacM 5

sreac

nF¹ • i @7#

where N is the molar flux,s the stoichiometric coefficient,n thenumber of electrons, andF the Faraday constant, and the diffusionequation

NreacM 5 2D reac

M ¹creacM @8#

whereD is the diffusion coefficient, with the Butler-Volmer equa-tion ~Eq. 5!, we get the second-order differential equation

~dM!2¹2creacM 5 creac

M @9#

wheredM is

dM 5 Acreac,ref

ai0,ref

nFDreacM

sreacFexp@10#

and it can be interpreted as the reactant diffusion length in the mixedphase.

Table I. Estimated diffusion lengths of the potentialsf and concentrationsc at a cell current density I of 1 AÕcm2 for methanol, oxygen, andhydrogen electrodes.

Variable Expression

Length ~m!

Oxygen Hydrogen Methanol

fE kh

I1024 1025 1024

fS sh

I1023 1024 1023

creacE nFDreaccreac

sreacI1028 to 1029 1027 to 1029 1025 to 1027

creacL nFDreaccreac

sreacI1024 to 1026

creacG nFPDreac,inertxreac

RTsreacxinertIe1.5 >1026 >1024

Superscripts E, S, L, and G indicate electrolyte, solid, liquid, and gas phase, respectively; subscript reac indicates O2 , H2 , or by CH3OH, and subscript inertindicates N2 or CO2 according to the electrode. The other symbols are listed at the end of this paper.

Journal of The Electrochemical Society, 150 ~12! A1558-A1568~2003! A1559

Finally, the transfer current density expression that can be used ina macrohomogeneous model is obtained by the volume averaging ofEq. 5

¹ • iav 5*V¹ • idV

V@11#

which, by using Eq. 7 and 8 and the Gauss theorem, can also bewritten as

¹ • iav 5nFDreac

M

sreac

*S¹creacM

• dS

V@12#

whereV is the integration volume andS the surface between thepore phase and the mixed phase.

Unfortunately, the analytical solutions of Eq. 9 and 12 can onlybe obtained for simple geometries.

In Fig. 1, we give a schematic representation of the geometry ofthe porous structure. In conditions of low gas porosity~Fig. 1a!, thegas pores are well separated by the mixed phase, and the influenceof one pore to the neighboring ones can be restricted to the conditionof zero reactant flux in the middle planes between them~dashedlines!. But, in conditions of high gas porosity~Fig. 1b!, a continuumof gas surrounds the agglomerates and the electrolyte. These consid-erations lead to two different model topologies, as depicted on theright side of Fig. 1a and b; in a, a continuum of mixed phase~darkgray! surrounds the isolated pore, whereas in b, a continuum of gasphase surrounds the isolated agglomerate. The characteristic lengthsare the mean pore radiusr p ~internal circle! and the mean pore-poredistancedpp ~external circle! in a and the mean agglomerate radiusr a in b.

The analytical solutions of Eq. 12 for each model topology arefound in both cylindrical and spherical symmetry approximations.For completeness, the planar symmetry approximation~planar ag-glomerates of thickness 2r a) also has been employed. The boundaryconditions and the additional equations required to seek the solu-tions of Eq. 12 are shown in Table II, The analytical results corre-sponding to the just identified five different geometries are schemati-cally presented as follows.

1. Separated pores, cylindrical symmetry

¹ • iav 5Sav

r p

nFDreacM

sreaccreac

P FSPcylS r p

dM ,dpp

dM D @13#

where

FSPcyl~x,y! 5 x

K1~y!I 1~x! 2 I 1~y!K1~x!

K1~y!I 0~x! 1 I 1~y!K0~x!@14#

andK0 , K1 , I 0 and I 1 , are modified Bessel functions.20

2. Separated pores, spherical symmetry

¹ • iav 5Sav

r p

nFDreacM

sreaccreac

P FSPsphS r p

dM ,dpp

dM D @15#

where

FSPsph~x,y! 5

1

y Fy 2 x 1 x1 2 y2

1 2 y coth~y 2 x!G @16#

3. Separated agglomerates, cylindrical symmetry

¹ • iav 5Sav

r a

nFDreacM

sreaccreac

P FSAcylS r a

dMD @17#

where

FSAcyl~x! 5 x

I 1~x!

I 0~x!@18#

4. Separated agglomerates, spherical symmetry

¹ • iav 5Sav

r a

nFDreacM

sreaccreac

P FSAsphS r a

dMD @19#

where

FSAsph~x! 5 x coth~x! 2 1 @20#

5. Planar symmetry

¹ • iav 5Sav

r a

nFDreacM

sreaccreac

P FSAplaS r a

dMD @21#

where

Figure 1. Geometry of the porous structure. The agglomerates are repre-sented in black, the electrolyte in dark gray, the water in light gray, and thegas in white.

Journal of The Electrochemical Society, 150 ~12! A1558-A1568~2003!A1560

FSApla~x! 5 x tanh~x! @22#

Some of these expressions can be found, in different forms, in theliterature. In particular, Eq. 13 has been derived by us in Ref. 6;furthermore, the agglomerate models in the literature use expres-sions for the reaction rate which correspond to the separated-agglomerates solutions Eq. 17,13,14 Eq. 19,21,15,14,18 and Eq.21.13,17,16

The five expressions for the averaged reaction rate given by Eq.13, 15, 17, 19, and 21 show strong similarities as the differencesrelated to geometries are enclosed in the shape functionsF. Notethat the argumentsx andy of the shape functions correspond to theratio between the characteristic lengths of the porous structurer a,r p , dpp and the diffusion lengthdM; therefore, the limits for largeand small values ofx and y correspond to the physical regimes ofreactant diffusion.

Next, we analyze and compare the five expressions for the aver-aged reaction rate given by Eq. 13, 15, 17, 19, and 21.

Separated-agglomerates expressions.—Elementary analysisshows that the functionsFSA(x) given by Eq. 18, 20, and 22 havethe limiting expressionsFSA(x) ' x2/2 , x2/3, andx2, respectively,for x ! 1 andFSA(x) ' x for x @ 1. By inserting these expres-sions into Eq. 17, 19, and 21, respectively, we get, for the threecases, the same limiting expressions

¹ • iav 5 creacP ~1 2 e!

ai0,ref

creac,refFexp dM @ r a

¹ • iav 5 creacP SavA ai0,ref

creac,ref

nFDreacM

sreacFexp dM ! r a @23#

The expression appearing in the first line of Eq. 23 is almostidentical to the Butler-Volmer equation~Eq. 5!. When the diffusionlength becomes much larger than the agglomerate radius, the reac-tant concentration becomes approximately constant and the effect ofvolume averaging reduces to the elimination of the void volumefraction ~factor 1 2 e!.

For lower values ofdM ~second line in Eq. 23!, we notice twoimportant differences.

1. The geometrical factor, instead of (12 e), becomesSav. Inthis limit, the volume reached by the reactant becomes proportionalto the pore surface.

2. The overpotential exponentials (Fexp) are now under a squareroot.

Separated pores, cylindrical symmetry.—In Fig. 2, Eq. 14 isplotted as a function ofx and for a set of differenty values andcompared with elementary functions. Three different regimes can berecognized. For lowx, FSP

cyl(x,y) coincides with the quadratic func-

tion 0.5(y2 2 x2); for higher x, it coincides with the function„ln(1 1 1/x)…21, which, for x @ 1, becomes approximately linear.The quadratic and logarithmic functions cross at aroundy5 A22/ln(x/y). By inserting these expressions into Eq. 13, we getthe following limiting expressions

¹ • iav 5 creacP ~1 2 e!

ai0,ref

creac,refFexp dM . 2

dppAln~e!

2

¹ • iav 5 creacP

Sav

r p

nFDreacM

sreaclnF1 1

dM

r pG21

dM , 2dppAln~e!

2

¹ • iav 5 creacP SavA ai0,ref

creac,ref

nFDreacM

sreacFexp dM ! r p @24#

We see that in the limit of large and smalldM ~first and third linein Eq. 24! we obtain expressions identical to those of Eq. 23; onlythe expression for the pore surface per unit volume,Sav, changesdue to the different model geometries~see Table II!. Unlike the caseof separated agglomerates, in this case there exists a third regime~second line in Eq. 24! which appears at intermediate values ofdM.

Separated pores, spherical symmetry.—In the limit of small xand y, the functionFSP

sph(x,y) becomes equal to (y3 2 x3)/(3x),whereas for high values ofy it becomes equal to 11 x. The two

Table II. Expressions for Laplace’s operator, pore surface per unit volume„Sav…, porosity, and boundary conditions used in the five analyticalcalculations.

Separated pores Separated agglomerates

Cylindrical sym. Spherical sym. Cylindrical sym. Spherical sym. Planar sym.

¹2 1

r

]

]rr

]

]r

1

r2

]

]rr2

]

]r

1

r

]

]rr

]

]r

1

r2

]

]rr2

]

]r

]2

]r2

Sav 2e/r p 3e/r p 2(1 2 e)/r a 3(1 2 e)/r a (1 2 e)/r a

erp

2

dpp2

r p3

dpp3 Independent Independent Independent

BC1 creacM (r p) 5 creac

P creacM (r p) 5 creac

P creacM (r a) 5 creac

P creacM (r a) 5 creac

P creacM (r a) 5 creac

P

BC2 ¹creacM (dpp) 5 0 ¹creac

M (dpp) 5 0 ¹creacM (0) 5 0 ¹creac

M (0) 5 0 ¹creacM (0) 5 0

Figure 2. Analysis of the separated-pores, cylindrical symmetry solution.FSP

cyl ~Eq. 14! is shown for three different porosity values and compared withelementary functions.

Journal of The Electrochemical Society, 150 ~12! A1558-A1568~2003! A1561

approximate functions cross aty3 5 (1 1 x)3 2 1. By insertingthese expressions into Eq. 15, we get the following limiting expres-sions

¹ • iav 5 creacP ~1 2 e!

ai0,ref

creac,refFexp

dM~dM 1 r p! . 2dpp

3 2 r p3

3r p

¹ • iav 5 creacP SavSA ai0,ref

creac,ref

nFDreacM

sreacFexp 1

1

r p

nFDreacM

sreacD

dM~dM 1 r p! , 2dpp

3 2 r p3

3r p

¹ • iav 5 creacP SavA ai0,ref

creac,ref

nFDreacM

sreacFexp dM ! r p @25#

Again, in the limit of large and smalldM ~first and third line inEq. 25! we obtain expressions identical to the ones of Eq. 23 and 24and, again, as in the case of separated pores, cylindrical symmetry, athird regime~second line in Eq. 25! appears at intermediate valuesof dM.

In Fig. 3, Eq. 16 is shown as a function ofx for two differentporosity values and is compared with Eq. 14. We see that the behav-ior of the spherical and cylindrical symmetry solutions is essentiallythe same.

1. We have a highx regime and a lowx regime with differentslopes.

2. In the low porosity curves (e 5 0.01), we observe the pres-ence of an intermediate regime that disappears at high porosity (e5 0.64).

By comparing the three validity ranges in Eq. 24, we see that theintermediate regime disappears when 2r p 5 dppAln(e), i.e., atarounde . 0.3; likewise, the intermediate regime of Eq. 25 disap-pears at arounde . 0.15. Then, when used in regimes of high po-rosity, the separated-pores solutions of Eq. 24 and 25 reduce exactlyto the separated-agglomerates solution of Eq. 23. This observationgains a particular interest if we recall that the separated-pores topol-ogy is appropriate for describing low porosity conditions, whereasthe separated-agglomerates topology is more appropriate in condi-

tions of high porosity. This confers a sort of continuity between thehigh and low porosity limits and suggests the possibility of usingEq. 13 and 15 in the whole porosity range.

However, the separated-agglomerates solutions, when used inlow porosity conditions, are unable to describe the intermediate re-gime.

Effects of porosity on the Tafel slope.—When the exponent inFexp ~Eq. 6! is much larger than one, according to its sign, one of thetwo exponential terms becomes negligible and we can write

Fexp . euhu/b @26#

Eq. 5 in this approximation becomes

¹ • i 5ai0,ref

creac,refcreac

M euhu/b @27#

This is known as the Tafel equation, andb 5 1/abF is the Tafelslope. This approximation is very good for oxygen and methanolelectrodes, because of the large value ofh, but is not suitable todescribe hydrogen electrodes. Because the Tafel slope is one of themost widely used parameters in electrochemistry, it can be interest-ing to see how it is affected by the nonhomogeneity. To reach thisgoal, we define a pseudo Tafel equation

G 5 Keuhu/bTMF @28#

where K is an arbitrary constant and the Tafel slope multiplyingfactor ~TMF! is a variable that accounts for possible changes of theTafel slope. From Eq. 10 we see that

dM 5 AnFDreacM

sreac

creac,ref

ai0,refe2uhu/2b @29#

and

uhu 5 22b ln~dMB! @30#

whereB 5 Asreac/nFDreacM ai0,ref/creac,ref. By substituting Eq. 30 in

Eq. 28, we get

G 5 Ke22 ln(dMB)/TMF 5 K~dMB!22/TMF @31#

from which we obtain

TMF 5 22G

dM]G

]dM

@32#

WhenG is given by Eq. 27,TMF is constant and equal to one. InFig. 4 we show what happens toTMF whenG is given by Eqs. 13,15, 17, 19, and 21. When, as in Fig. 4a, Eq. 13 and 15 are calculatedfor high values of the gas porosity (e 5 0.5), the five curves have avery similar behavior.

1. For dM . 1028 m, the Tafel slope is unchanged (TMF5 1).

2. For dM , 1029 m, due to the square root appearing on thelow dM limit expressions of Eq. 23, 24, and 25, the Tafel slope ismultiplied by a factor of two.

In Fig. 4b, theTMF of the two separated-pores expressions, Eq.13 and 15, is plotted as a function ofdM and for three differentvalues of porosity. At low porosity values, a third regime appears: inthe intermediate range ofdM, the Tafel slope becomes even largerand depends on the porosity value.

The analysis that we have done thus far can be rationalized bythe following points.

1. The choice of the topology is crucial for the correct descrip-tion of the porous media. (i ) The homogeneous model, which doesnot have any characteristic length, has only one diffusion regime

Figure 3. Comparison between the separated-pores, cylindrical and spheri-cal symmetry solutions.FSP

cyl(x,y) ~Eq. 14! andFSPsph(x,y) ~Eq. 16! are shown

for two different porosity values.

Journal of The Electrochemical Society, 150 ~12! A1558-A1568~2003!A1562

(TMF 5 1). (i i ) The separated-agglomerates topology, which hasonly one characteristic length (r a), can have only two diffusionregimes:dM @ r a (TMF 5 1) and dM ! r a (TMF 5 2). (i i i )The separated-pores topology, which has two characteristic lengths(r p anddpp), has a three-regime behavior:dM @ dpp (TMF 5 1),dM ! r p (TMF 5 2), and r p ! dM ! dpp (TMF . 2). Clearly,when the two characteristic lengths become approximately equal~asin high porosity conditions!, the third regime disappears.

2. By comparing the three topologies and the corresponding dif-fusion regimes, it can be established that the homogeneous model iscorrect only in low current density conditions and the agglomeratemodels are correct only in high porosity conditions. However, theseparated-pores models describe correctly both the low and highporosity limits.

3. The choice of the specific geometry~cylindrical, spherical, orplanar! is much less important, and the porous structure can bedescribed in terms of generic physical quantities as the porositye orthe pore surface per unit volumeSav.

Macrohomogeneous Models

This section aims to (i ) study the use of the reaction rate expres-sions, Eq. 13, 15, 17, 19, and 21, in the context of macrohomoge-

neous models and (i i ) provide analytical expressions for the cellcurrent density to easily analyze the effects of the porous structureon the fuel cell performances.

Numerical.—The proton transport in the membrane phase can bedescribed by the phenomenological equation22

i 5 2k¹f 1 Fcfv @33#

where k is the ionic conductivity,cf is the concentration of themembrane charges, andv is the water velocity.

The reactant transport in the gas phase can be described by theStefan-Maxwell equation21

¹xi 5 (j51

nRT

PDijeff ~xiNj 2 xjNi! for i , j 5 reac,inert,water

@34#

in which D ijeff is thei j -pair effective diffusion coefficient in a binary

mixture, andxi is the molar fraction of speciesi . The dissolved-reactant concentration in the membrane phase just outside the poreis related to the gas-phase molar fraction inside the pore by a Hen-ry’s law constantKh

xi 5Kh

Pci @35#

When the main reactant transport mechanism is in the liquidphase~as for a liquid feed direct methanol fuel cell!, the Nernst-Planck equation19 without the migration term can be used to de-scribe the flux of the neutral chemical species

Ni 5 2D i¹ci 1 civ @36#

To describe the water transport, we can use the Schlo¨glequation23,24

v 5kf

mzf cf F¹f 2

kP

m¹P @37#

wherekf andkP are the electrokinetic and hydraulic permeabilities,m is the water viscosity, andzf is the valence of the membranecharges.

The material balance for water and reactant can be written as

¹ • Nreac5sreac

nF¹ • i

¹ • v 5sw

rnF¹ • i @38#

wherer is the water density.Now, the three phenomenological flux equations, Eq. 33, 34~or

36!, and 37, together with the two material balance equations~Eq.38!, and with the source term¹ • iav chosen among Eq. 13, 15, 17,19, or 21, form a closed set that can be solved numerically byfollowing the same procedure outlined in Ref. 1 or 25. As alreadypointed out in Ref. 25, the numerical solution of these coupled equa-tions can be very difficult due to the strong nonlinearity of thesource term. However, the use of¹ • iav instead of¹ • i does notadd any numerical difficulty, while also bringing the important ad-vantage of considering the effects of the nonhomogeneous porousstructure which are introduced in the averaging procedure. The ap-plication of this approach to the study of a hydrogen electrode is tobe published elsewhere.

Analytical.—The aforementioned numerical difficulties push usto search for analytical solutions. To accomplish this goal, we mustapply some simplifying assumptions. Due to the complicated depen-dence of the reaction rate expressions, Eq. 13, 15, 17, 19, and 21, on

Figure 4. TMF of Eq. 13, 15, 17, 19, and 21 as a function of the reactantdiffusion distance in the mixed phase.~a! Separated-pores expressions evalu-ated in conditions of high porosity.~b! Separated-pores expressions evalu-ated for three different porosity values.

Journal of The Electrochemical Society, 150 ~12! A1558-A1568~2003! A1563

the overpotential, analytical solutions can be found only when theoverpotential can be considered as a constant. In this case,dM alsobecomes constant, and the only variable left in the reaction rateexpressions is the reactant concentration in the pore phase. FromTable I, we see thatfS 2 fE . const. is a good approximation forboth oxygen and methanol~but not for hydrogen!. However, thispoint is controversial, as the value of the proton conductivity in thereactive region depends on the humidification of the electrolyte.26,27

Therefore, if humidification is poor, the following analytical deriva-tions are not strictly valid, and a numerical approach, as outlined inthe former section, becomes more reliable.

Equation 34 at low reactant concentrations becomes an ordinarydiffusion equation;6 clearly the same is true for Eq. 36 when theconvective term is negligible. Within these approximations, we canwrite

NreacP 5 2D reac

P ¹creacP @39#

Then, by combining Eq. 39 and 7 with any reaction rate expressionamong Eq. 5, 13, 15, 17, 19, or 21, we get the second-order differ-ential equation

~dP!2¹2creacP 5 creac

P @40#

wheredP can be interpreted as the reactant diffusion length in thepore phase, and its expression depends on the reaction rate chosen.For a homogeneous electrode~Eq. 5!

~dP!22 5ai0,ref

creac,ref

sreac

nFDreacP Fexp @41#

For separated pores~Eq. 13 and 15!

~dP!22 5D reac

M

D reacP

Sav

r pFSP

symS r p

dM ,dpp

dM D @42#

For separated agglomerates~Eq. 17, 19, and 21!

~dP!22 5D reac

M

D reacP

Sav

r aFSA

symS r a

dMD @43#

where the superscript sym stands for cylindrical, spherical, or planarsymmetry.

The solution of Eq. 40 in one dimension (¹2 5 ]2/]z2) andwith the boundary condition of zero reactant flux through the mem-brane can be found in Ref. 6, and the resulting equation is

creacP ~z! 5 creac

P ~zR/D!

coshS LR 2 z 1 zR/D

dP DcoshS 2

LR

dPD @44#

where LR is the thickness of the reactive region andzR/D is thecoordinate of the interface between the diffusive and reactive regionof the electrode.

To get the cell current density, we use the global material bal-ance, which, under conditions of zero reactant flux through themembrane, writes as

I 5nF

sreacNreac

P ~zR/D! @45#

By using Eq. 39 and 44, we get

I 5 creacP ~zR/D!

nFDreacP

sreacdP tanhS LR

dPD @46#

To get polarization curves for the electrodes, we still need toexpress the interface valuecreac

P (zR/D) in Eq. 46 as a function of thecell current densityI . This expression should depend on the reactantdiffusion through the diffusive region of the electrodes. In Ref. 28an accurate, although qualitative, expression was derived by takinginto account the effects of electrode flooding

creacP ~zR/D,I ! 5 creac

CH F1 2I

I lSm(I /I l 21)G @47#

wherecreacCH is the reactant concentration in the flow channel,I l is the

limiting current density, andSm represents the ratio between theeffective diffusion coefficients atI 5 0 and atI 5 I l . When the gasporosity does not change significantly with the current density, asimplified expression holds

creacP ~zR/D,I ! 5 creac

CH F1 2I

I lG @48#

which can be derived by settingSm 5 1 in Eq. 47~see also Ref. 29for an alternative analytical derivation!. Although this assumptiondoes not hold for the whole range of cell operative conditions, itallows us to express explicitly the cell current density as a functionof the electrode overpotential; consequently, it brings us to astraightforward analysis of the effects of the geometrical structureon the electrode polarization curves. By using Eq. 48 and Eq. 46, weget

I 5 I l

creacCH

nFDreacP

sreacdP tanhS LCL

dP DI l 1 creac

CHnFDreac

P

sreacdP tanhS LCL

dP D @49#

In the results section, this expression is applied to evaluate thepolarization curves of oxygen and methanol electrodes.

Results

In this section, we apply the expressions for the reaction rate andfor the cell current density derived in the previous sections to thepractical cases of an oxygen and a methanol electrode. The constantsused to relatedM anddP to the overpotentialh are given in Table III.For the oxygen electrode, we use a smaller porosity value becausethe water produced inside the electrode is expected to fill most of thevoid space. In Fig. 5, theTMF of Eq. 13, 15, 17, 19, 21, and 27 forthe oxygen and methanol electrodes are shown and compared withthe typical ranges of activation and concentration overpotentials. Wesee that, for the oxygen electrode, the separated-pores solutions in-dicate a strong increase of the Tafel slope in the whole concentrationoverpotential range, whereas the separated-agglomerates solutions

Table III. Base case parameters.

Reactantai0ref

cref

~A/mol!

2n

sreac

D reacM

~m2/s!D reac

P

~m2/s!r p

~m!r a

~m!a e

Oxygen 8 3 102 4 1.2 3 10210 1027 1028 1027 1.2 0.01Methanol 4 6 2.6 3 10210 1.3 3 1029 1028 1027 1.0 0.1

Journal of The Electrochemical Society, 150 ~12! A1558-A1568~2003!A1564

indicate considerably lower values ofTMF. However, for a metha-nol electrode, all the solutions indicate that the Tafel slope changesonly at very high overpotential values.

In Fig. 6, we show theTMF of Eq. 46. The effects of nonhomo-geneity, as depicted in Fig. 5, are enhanced by the effects of limitedreactant diffusion through the reactive region. This causes a strongincrease of the Tafel slope in the concentration overpotential currentdensity range of an oxygen electrode. However, for a methanol elec-trode, all the solutions coincide with the homogeneous model solu-tion up until very high overpotential values.

For these reasons, note that the conventional Tafel equation~Eq.27! can be safely used in the framework of a macrohomogeneousapproach for the anode of a methanol fuel cell. However, for oxygenelectrodes, we must take into account the effects of nonhomogeneityto get a correct description of the polarization curve in the concen-tration overpotential current density range.

In Fig. 7, the results of the application of Eq. 49 to an oxygenelectrode are shown for two values of gas porosity. At high porosity(e 5 0.1), the separated-agglomerates, separated-pores, and homo-geneous models are indistinguishable. At lower porosity values (e5 0.01), the separated-agglomerates and homogeneous models areagain indistinguishable, but the separated-pores polarization curves

Figure 5. TMF of Eq. 13, 15, 17, 19, 21, and 27 as a function of theoverpotentialh. ~a! is obtained for an oxygen electrode,~b! for a methanolelectrode.

Figure 6. TMF of Eq. 46 as a function of the overpotentialh. ~a! is ob-tained for an oxygen electrode,~b! for a methanol electrode.

Figure 7. Oxygen electrode overpotential obtained by using the three differ-ent topologies in high and low porosity conditions. Beside the parametersreported in Table III, we have imposedI l 5 1 A/cm2 andLR 5 1023 cm.

Journal of The Electrochemical Society, 150 ~12! A1558-A1568~2003! A1565

are considerably different. These results are in agreement with theones shown in Fig. 6a, when comparing the ranges of values of theelectrode overpotential at which the different topologies give differ-ent results.

Note that the two separated-pores curves are very close to eachother. A simple analysis of Eq. 24 and 25 shows that the same valueof Sav must be used to get the same behavior in the high overpoten-tial limit. This requires, as seen from Table II, the use of two differ-ent pore radius values,i.e., r p 5 1.5 3 1028 m andr p 5 1028 m inthe spherical and cylindrical symmetry curves, respectively. Tomatch the high current density behavior of the separated-agglomerates and separated-pores topology curves, a value ofr a

5 1026 m should be used. Beside the fact that such a high value ofthe agglomerate radius is outside the expected range for the realmaterial, its use would also produce a polarization curve with awrong shape in the intermediate range of overpotential values, asshown by the dot-dashed curve in Fig. 7.

These results confirm the three important points itemized at theend of the section on the pore scale model.

In Fig. 8, the results of the application of Eq. 49 to an oxygenelectrode are shown for different values of the gas porositye and byusing the separated-pores expression fordP given by Eq. 42. Figure8a is obtained by imposingr p 5 1028 m, which corresponds

roughly to the radius of the micropores existing inside the agglom-erates. Figure 8b is obtained by imposingr p 5 1027 m, which cor-responds roughly to the radius of the macropores constituting thevoid spaces between the agglomerates. The shape of the concentra-tion overpotential is strongly influenced by the value of the gasporosity. Because the gas porosity is influenced by water flooding,and the extent of water flooding depends on the current density, weexpect that the gas porosity also depends on the current density.Therefore, to have a more realistic view of what should resemble anoxygen electrode overpotential, in Fig. 8 we represent with a solidline the overpotential obtained by using a variable gas porosity,calculated by using the following equation

e 5e0

10I/I l@50#

wheree0 is equal to 0.1 and 0.3 in Fig. 8a and b, respectively. Thejustification for the use of this equation is given elsewhere.28

Figure 8a represents the case when oxygen is transported preva-lently through the micropores inside the agglomerates, where Fig.8b represents the case when the oxygen transport takes place preva-lently through the macropores constituting the void spaces betweenthe agglomerates. By comparing Fig. 8a and b, we observe that inthe former case, the effects of nonhomogeneity affect the polariza-tion curves at lower gas porosity and higher current density valuesthan in the latter case. Consequently, the shape of the polarizationcurves can give useful information on the actual porous structure ofthe reactive region.

In Fig. 9, the results of the application of Eq. 49 to an oxygenand a methanol electrode are shown. The porosity of the methanolelectrode is kept fixed toe 5 0.1, and the porosity of the oxygenelectrode is given by Eq. 50 withe0 5 0.1. The continuous linecurves are obtained by using Eq. 42, whereas the symbols curves areobtained by using Eq. 41 and therefore represent homogeneous elec-trodes. Note that the homogeneous description of the air electrodegives a wrong shape of the concentration overpotential fall. How-ever, the methanol electrode is very well described at the homoge-neous level.

To support and strengthen the preceding analysis and results, wegive here some experimental data for comparison.

If we write I /creacP (zR/D) as a pseudo Tafel equation

I

creacP ~zR/D!

5 Keuhu/bTMF @51#

it follows that

Figure 8. Oxygen electrode overpotential obtained by using the separated-pores topology at different porosity values.~a! is obtained by imposingr p

5 1028 m and ~b! by imposing r p 5 1027 m. Other parameters are re-ported in Table III and in the caption of Fig. 7.

Figure 9. Oxygen and methanol electrode overpotentials.

Journal of The Electrochemical Society, 150 ~12! A1558-A1568~2003!A1566

h 5 bTMF~ ln~ I ! 2 ln~creacP ~zR/D!! 2 ln~K !! @52#

The first term in the parentheses is dominant at low current den-sities and gives an expression for the activation overpotential. Thesecond term becomes dominant when the reactant concentration issmall and expresses the concentration overpotential. It follows that,when this equation is used as a fitting equation, the coefficient val-ues of the first and second logarithm terms are extracted from theexperimental behaviors at low and highI , respectively. In Ref. 28,an expression equivalent to Eq. 52 has been deeply investigated andused to fit the experimental polarization curves for a number ofair-hydrogen fuel cells at varying pressures, temperatures, and oxy-gen dilutions. The results of the fitting are collected in the histogramof Fig. 10, where the abscissa represents theTMF fitted at highI .Because both the concentration and activation overpotentials of theair-hydrogen fuel cells derive from the oxygen electrode, the experi-mentalTMF of Fig. 10 can be compared with the modelTMF ofFig. 6a in the concentration overpotential range. We see the follow-ing. (i ) All the experiments give aTMF . 2; this result is clearlyincompatible with the homogeneous model for whichTMF < 2.( i i ) Ten curves out of 16 haveTMF . 4; this result also is incom-patible with the agglomerate models for whichTMF < 4. (i i i ) Therange ofTMF values is in very good agreement with the modelresults obtained by using the separated-pores topology.

The same kind of analysis can be done for the methanol elec-trode. By fitting the experimental anodic polarization curves pre-sented in Ref. 30, the high current densityTMF values are veryclose to one. Again, the agreement with the model results presentedin Fig. 6b is excellent.

Conclusions

Comparing the characteristic lengths of the porous structure withthe characteristic lengths of the diffusion phenomena shows thatoxygen and hydrogen concentrations in the electrolyte phase changesignificantly at the pore scale level; therefore, their transport in thereactive region must be described by a nonhomogeneous model.

By analyzing the analytical solutions of five different pore scalenonhomogeneous models, three common characteristic behaviorshave been found which do not depend on the specific geometry andseem to be intrinsic in the phenomenon of diffusion reaction inporous media.

1. For small current densities, the reaction rate is the same as inhomogeneous models.

2. For very high current densities, a Tafel slope twice the valueof that for the low current densities has been found, and the reactionrate becomes proportional to the pore surface per unit volume.

3. In the intermediate range, the Tafel slope becomes even largerand depends on the porosity value. This regime disappears at highporosity values.

An analysis of the model results presented in the pore scalemodel section shows that the homogeneous model describes cor-rectly only the low current density regime, whereas the agglomeratemodels are effective both at low and high current density regimes,provided the gas porosity of the reactive region is high. These limi-tations have been overtaken by employing the effective Butler-Volmer expression given by Eq. 13~or Eq. 15! which was shown todescribe correctly the cell performance in the whole reactive regionporosity and current density ranges. Moreover, the use of Eq. 13instead of the normal Butler-Volmer equation in the framework ofmacrohomogeneous models does not add any numerical difficulty,yet also brings the important advantage of taking into account theeffects of the nonhomogeneous porous structure. Therefore, westrongly suggest the use of Eq. 13~or Eq. 15! in place of Eq. 5 as aneffective Butler-Volmer equation for the reaction rates in porouselectrodes.

By comparing analytical polarization curves for air and methanolelectrodes obtained by using the homogeneous model, the agglom-erate model, and the separated-pores model reaction rates, we seethe following.

1. The homogeneous model of the air electrode gives a wrongshape of the concentration overpotential fall.

2. The methanol electrode is very well described at the homoge-neous level.

3. At low porosity conditions, the agglomerate model correctsonly partially the misbehavior of the homogeneous model descrip-tion.

Acknowledgments

The authors thank Dr. Bruno D’Aguanno for helpful discussionsand comments on the manuscript. This work has been carried outwith the financial contribution of the Sardinia Regional Authorities.

The Center for Advanced Studies, Research and Development in Sardiniaassisted in meeting the publication costs of this article.

List of symbols

a effective catalyst area per unit volumeb Tafel slopecf concentration of the membrane chargesci concentration of speciesiD i diffusion coefficient of speciesiD ij diffusion coefficient of the pairi j in a gas binary mixturedpp mean pore-pore distance

F Faraday constantI cell current densityI l limiting cell current densityi ionic current density

i 0,ref exchange current densityKh Henry’s law constantkf electrokinetic permeabilitykP hydraulic permeabilityLR thickness of the reactive regionNi mass flux of speciesin number of electrons of the rate-determining stepP gas pressureR gas constantr transversal distance

r a mean agglomerate radiusr p mean pore radiusS surface

Sav pore surface per unit volumesi stoichiometric coefficient of speciesiT temperature

TMF Tafel multipying factorv water velocityxi molar fraction of speciesiV volumez distance

Figure 10. Frequency distribution of experimental curves corresponding tovalue intervals of the high currentTMF.

Journal of The Electrochemical Society, 150 ~12! A1558-A1568~2003! A1567

zR/D coordinate of the reactive region/diffusive region boundaryzf valence of the membrane charges

Greek

aa anode transfer coefficientac cathode transfer coefficientbF Faraday constant in units ofRT

e porosityk ionic conductivitym pore-fluid viscosityh overpotentialf electric potentialr water densitys electronic conductivity

Subscripts and superscripts

CH flow channelD diffusive regionE electrolyte

eff effective in materialG gas phase

inert inert speciesL liquid phase

M mixed phaseP pore phaseR reactive region

reac reactant speciesref at the reference stateS solid phase

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Journal of The Electrochemical Society, 150 ~12! A1558-A1568~2003!A1568