Impedance modelling of air electrodes in alkaline media

Electrochimica Acta 53 (2008) 7483–7490

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Impedance modelling of air electrodes in alkaline media Ann Mari Svensson a,∗ , Helge Weydahl b,c , Svein Sunde c a

SINTEF Materials and Chemistry, 7465 Trondheim, Norway Department of Materials Technology, NTNU, 7491 Trondheim, Norway c Prototech AS, 5862 Bergen, Norway b

a r t i c l e

i n f o

Article history: Received 3 August 2007 Received in revised form 25 February 2008 Accepted 26 February 2008 Available online 12 March 2008 Keywords: Impedance Model Alkaline Porous Electrode

a b s t r a c t A mathematical model of the impedance response of porous air electrodes in alkaline solutions, based on flooded agglomerate theory, was developed. The model results provides insight into characteristic features of the impedance spectra according to the relative rate of the various reactions and mass transfer processes occurring. The modelled spectra where dissolution, transport and reaction of oxygen dominates the impedance are close to ideal semicircles. With increasing contribution from ionic transport in the solution inside the pores, two distinct features occur, namely the appearance of a 45◦ branch at high frequencies, and an inductive loop at low frequenices. Examples from the literature where inductive loops occurs are given. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Successful development of a number of electrochemical devices, like metal air batteries, fuel cells and electrolysis, rely on the development of stable and efficient gas diffusion electrodes (GDE). GDEs are, however, complex structures composed of several phases, with a number of processes (transport, chemical and electrochemical reactions) occurring in each phase, and it is difficult to distinguish the impacts of the various coupled phenomena on the overall performance. Impedance spectroscopy is a powerful characterization technique which can resolve contributions from various processes, provided that the time constants differ sufficiently. The technique is widely used for experimental investigations of the oxygen reduction reaction on metallic surfaces in alkaline solutions [1–3], as well as porous gas diffusion electrodes in alkaline media [3–7]. Impedance spectroscopy has proven to be a powerful technique for investigation of reaction mechanisms for a number of electrochemical reaction sequences, an extensive study of oxygen reduction on Pt in 6 M KOH is provided in ref. [2]. Successful interpretation of data could, however, only be achieved in combination with extensive model developments. For porous electrodes, the interpretation of impedance spectra is even more difficult due to the complex interdependencies of the geometry/morphology, transport properties of the species involved as well as material and kinetic parameters. Mathematical

∗ Corresponding author. Tel.: +47 98230450; fax: +47 73591105. E-mail address: [email protected] (A.M. Svensson). 0013-4686/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.electacta.2008.02.111

modelling of impedance response of porous electrodes has been performed along two main paths. One approach is transmission line models, where the structure of the electrode is idealized, for example, by considering pores to be cylindrical. One of the first theoretical treatments of the impedance response of a porous electrode was provided by de Levie [8]. In this work, an expression for the impedance response was derived for a metallic electrode consisting of identical, cylindrical pores with a homogeneous solution. A later example of an extensive transmission line model for porous oxygen electrodes, with detailed considerations of geometry (the surface of the carbon electrode was modelled as a plane covered with hexagonally oriented conducting spheres), is provided in ref. [9]. The other approach is to apply porous electrode theory [10], where the actual geometry of the pores is neglected, and reaction rates are averaged over the interfacial area between solid phase and the pores. For porous oxygen electrodes, like fuel cell cathodes, where three phases are present, the so called flooded-agglomerate model has been frequently applied. Here, the oxygen reduction reaction is assumed to occur inside agglomerate particles flooded with liquid electrolyte, and the agglomerates are incorporated into porous electrode models. Stationary models for liquid and polymer fuel cell electrodes based on flooded agglomerate model are presented in ref. [11]. Impedance models of active layers of the PEM fuel cell cathode based on flooded-agglomerate models have been developed by Jaouen and Lindbergh [12]. In the present work, a phenomenological impedance model is developed for a porous oxygen electrode in combination with an alkaline electrolyte based on the flooded agglomerate model. The oxygen reduction reaction is assumed to occur by a simple one-

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step reaction mechanism, and the purpose of the modelling work is to resolve the contributions to the impedance response from the various processes (oxygen reduction reaction and transport of the various species). The model can be easily transferred to other porous electrode systems with a binary electrolyte and gaseous reactants. In another work, the corresponding transient models have been solved, providing results for the response of such electrodes to potential steps and other transients of the potential [13]. 2. Mathematical model As mentioned above, a key challenge when modelling GDEs is the geometric structure of the catalyst layer. The highly porous, multi-phase structure cannot be modelled in geometric detail without causing inadequate model complexity. Perry et al. [11] developed a stationary model of the cathode of liquid-electrolyte fuel cells (including AFC) based on agglomerate structure theory. In their model, the coupled effects of reactant transport, electrode potential and faradaic reaction were accounted for. In this work, the stationary model presented by Perry et al. [11], is further developed into an impedance model. A schematic view of the structure of the modelled electrode is given in Fig. 1. It is assumed that the cathode consists of spherical, porous carbon agglomerates mixed with Pt. For convenience, the radius of agglomerates is assumed to be uniform. The open space between the agglomerates is filled with air. The pores inside the porous agglomerates are hydrophilic and filled with liquid electrolyte. The electrode thus consists of three phases: (i) a solid phase (carbon and Pt in the porous agglomerates), (ii) a liquid phase (electrolyte-filled pores in agglomerates) and (iii) a gaseous phase (air or oxygen in the open space between the agglomerates). The electrode model couples a 1D model porous electrode with a 3D (spherical) agglomerate model through a source term. The finite extension of the agglomerates is thus accounted for, and the radius of the agglomerate is one of the parameters determining the rate of the oxygen reduction reaction. 2.1. Basic equation and assumptions Before presenting the mathematical models, a brief introduction to the basic equations and assumptions is given. The electrochemical reaction is assumed to occur at the catalyst sites as a one-step transfer of four electrons, as also assumed by Kimble and White [14] and Perry et al. [11]: 2H2 O + O2 + 4e− → 4OH−

−RORR = ka c0 exp −

b

(2)

where ka is the reaction rate constant, c0 the concentration of dissolved oxygen in the liquid phase and b is the Tafel slope. The local overpotential, , is defined in accordance with Perry et al. [11], with respect to a reference electrode placed just outside the electrode:  = ˚1 − ˚2 − U 

Si = −

Adl Cdl dqi ∂(˚1 − ˚2 ) zi F dq ∂t

(4)

where Adl is the surface area of carbon/Pt in the electrolyte filled agglomerate pores per unit agglomerate volume, Cdl the double layer capacitance per unit surface area of carbon/Pt and q is the surface charge of the solid phase. For an incremental change in surface charge of the solid phase, dq, the incremental change in ionic double layer charge due to K+ ions, dq+ , and OH− ions, dq− , are related by −dq = dq+ + dq−

(5)

In accordance with Verbrugge and Liu [16], it is assumed that dq+ /dq = dq− /dq = 1/2. The flux of species i in the electrolyte phase is given by the Nernst–Planck equation, which implies that the dilute solution approximation is assumed: Ni = −zi

εl ∂˚2 ∂c ε − l Di i u Fc l i i ∂x l ∂x

(6)

where zi is the charge number of the species i, respectively, and ui and Di are the mobility and free stream diffusivity of species i, respectively. Mobilities and diffusivities are corrected for the porosity and tortuosity of the electrolyte phase, by multiplying with εl / l . In addition to the assumptions stated above, in order to be able to formulate the mathematical model with reasonable complexity, the agglomerates are assumed to be spherically symmetric and of uniform size. Furthermore, the following assumptions were made:  Temperature gradients within the electrode are neglected.  Gradients in the oxygen partial pressure in the gas-filled pores are neglected.  Gaseous oxygen is in equilibrium with dissolved oxygen at the agglomerate surface.  Transport of dissolved oxygen to the active sites occurs by diffusion. For most of the results presented, the resistivity of the solid phase is assumed to be negligible compared to the electrolyte phase. However, we also show how the finite conductivity of the solid phase can be included in the model, and some results for this case are also included.

(1)

The reaction rate of the oxygen reduction reaction is assumed to obey Tafel kinetics, and be first-order in the oxygen concentration of the electrolyte [15]:

 ln(10) 

electrode are moderate. A discussion of Eq. (3) is provided in ref. [13].The charging/decharging of the double layers are related to the production/consumption Si of electrolyte species through [16]

2.2. Impedance model neglecting the resistivity of the solid phase Material balances are formulated within the framework of the porous electrode theory [10], which implies that concentrations are averaged over the pore volume, and reactions over the interfacial area of the pore walls. Mass balances for positive (K+ ) and negative (OH− ) ions in the electrolyte phase can thus be written as: ∂ce ε ∂ εl = z+ u+ F l l ∂x ∂t −

(3)

where ˚1 is the potential of the solid phase and ˚2 is the potential in the electrolyte phase. The local overpotential is assumed to be constant within each agglomerate. The assumption of Tafel kinetics limits the validity of the model to moderate and high overpotentials ( should be larger than 0.2 V). This definition of the overpotential is reasonable as long as the variations in concentration inside the

εl

∂˚2 ce ∂x



+ D+

εl ∂2 ce l ∂x2

Adl Cdl (1 − εg ) dq+ ∂(˚1 − ˚2 ) z+ + F dq ∂t

∂ce ε ∂ = z− u− F l l ∂x ∂t −



 ce

∂˚2 ∂x

 + D−

(7)

εl ∂2 ce l ∂x2

Adl Cdl (1 − εg ) dq− ∂(˚1 − ˚2 ) s− − a Ia z+ + F − nF dq ∂t

(8)

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Fig. 1. Schematic view of the structure of the electrode.

where ce is the concentration of ions in the solution (cK+ /+ = cOH /− = ce due to electroneutrality, and + and − are the number of K+ and OH− ions, respectively, that each KOH molecule dissociates into), D+ and D− the (dilute) diffusion coefficients of positive and negative ions, respectively, εg the porosity of the gas phase, s− the stoichiometric coefficient of OH− ions and a Ia denotes the specific reduction rate of oxygen within the agglomerates (same notation as in ref. [11]). The calculation of this term is shown below. As can be seen from Eqs. (7) and (8), charging/discharging of the electrochemical double layer is taken into account in the mass balance for both ions, whereas only the negative ions take part in the faradaic reaction (the oxygen reduction reaction). The corresponding boundary conditions are: ce =

ceb

at x = 0

∂ce = 0 at x = L ∂x ˚2 = U  at x = 0 ∂˚2 =0 ∂x

(9)

at x = L

The initial condition ˚2 = U is simply related to an arbitrary selection of zero potential [11]. The density of agglomerates, a (number of agglomerates per electrode volume), is given as: a =

3(1 − εl )

Ra



Ra

Aa ka c0 exp

= −nF4 0

r2

∂c0 ∂r

 + Aa RORR

(12)

c0 = c0s ∂c0 =0 ∂r

at r = Ra , all t (13)

at r = 0, all t

where c0s is the concentration of dissolved oxygen in equilibrium with the gas phase.Since potential variations in the solid phase of the electrolyte are neglected, and ∂˚2 ∂ =− ∂x ∂x

(14)

Eqs. (7) and (8) may be rearranged to the following set of equations for the variables ce and  by elimination of the potential gradient: εl

Adl Cdl (1 − εg ) ∂ ∂ce ε ∂2 ce s− − a Ia t+ = De l − 2F l ∂x2 nF ∂t ∂t p εl

+

l



∂ce ∂ ∂2  + ce 2 ∂x ∂x ∂x

 + Dem

(15)

εl ∂2 ce l ∂x2

Adl Cdl (1 − εg ) ∂ s− a Ia = 0 + F nF ∂t

(16)

where t+ is the transport number of the positive ions, given as [10]

Aa ifar r 2 dr 0



where the rate of the ORR is given by Eq. (2), and the corresponding boundary conditions are

−De

where Ia is the total faradaic current produced in one agglomerate, found by integrating the faradaic current over the agglomerate volume: Ia = 4

εa 1 ∂ ∂c0 = D0 a r 2 ∂r ∂t

(10)

4 Ra3



agglomerate, which requires solution of the material balance for dissolved oxygen:

 − ln(10)  b

2

r dr

t+ = (11)

where n is the number of electrons transferred in the reaction and ifar denotes the local faradaic current density. Ia can only be determined from the oxygen concentration profiles within each

z+ D+ z+ D+ − z− D−

(17) p

and De (binary diffusion coefficient) and De and Dem are given as De =

z+ u+ D− − z− u− D+ z+ u+ − z− u−

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De =

FDe 2RT

Dem =

De 2

p

1 t+

1

t−

+ −

1 t−

1 t+





The boundary conditions that apply to the local overpotential, , are:  = app

at x = 0, all t

∂ =0 ∂x

at x = L, all t

(18)

(i) Any variable X can be expressed as the sum of one term which corresponds to the stationary (dc) response of the system, and another term which is governed by the perturbing signal, i.e.: X = X ss + X˜ sin(ωt + ) where ω is the frequency, the phase angle (arbitrary) and X˜ is the amplitude. For convenience, the complex representation is used, implying that variables are expressed as X=X

+ X exp(jωt + )

(19)

where √ X is the complex amplitude of the perturbation and j = −1. (ii) The perturbation is assumed to be sufficiently small, such that the response of the system can be considered linear (i.e. Taylor expansion can be applied). Insertion of Eq. (19) for the variables ce and  into Eqs. (15) and (16) gives the following set of equations for the corresponding complex amplitudes: Adl Cdl (1 − εg )jω d2 ce  jω s− t+ = l ce +

 −

(a Ia ) De 2F nF dx2

(20)

(21) ss

Here, and are the stationary solutions for the electrolyte concentration and the local overpotential, respectively, which are obtained according to the model of Perry et al. [11]. Since Eq. (15) is linear, the stationary terms will cancel, such that only the complex amplitudes take part in Eq. (20). Eqs. (20) and (21) must be solved together with the following boundary conditions:

d ce =0 dx

at x = 0 at x = L

 = app ∂  =0 ∂x

at x = 0

L

( (a Ia ) + Cdl (1 − εg )Adl iω)  dx

(23)

and the impedance response of the oxygen electrode can thus be determined as a function of frequency from Z=

app

icell

(24)

2.3. Impedance model including a finite conductivity of the solid phase Accounting for a finite conductivity of the solid phase in the impedance model requires also development of a new stationary model, since the impedance model is linearized around steady state. It can be shown that for the stationary case, the potential of the solid phase, ˚ss , obeys the equation: l d2 ˚ss l dx2

=

1 di2 ε F De d2 cess = l

dx l 1 − t− dx2

(25)

Further details of the derivation and solution of this equation is given in Appendix B. In the dynamic formulation of the problem, the current transferred between the matrix and the solution is involved in either double-layer charging or faradaic electrode reaction, such that

∇ · i2 = a Ia + Adl Cdl (1 − εg )

∂ ∂t

(26)

When linearized, and put into Eq. (25), this gives the following equation for the complex amplitude, ˚l −

d2 ˚l = (a Ia ) + Adl Cdl (1 − εg )jω  dx2

˚l = app

adl Cdl (1 − εg )l jω Dm 1 d2 ce s− l + ep ss +

 −

(a Ia ) p nF Dep ce εl De ce dx2 De cess εl

ce = 0

icell =

(27)

which has to be solved simultaneously with Eq. (20) for ce and Eq. (21) for , together with the boundary condition:

d2  1 dss d ce 1 dc ss d  d2 ss 1 = − ss

ce − ss e − 2 ce dx dx ce dx dx dx dx2 cess

cess



0

The condition that the local overpotential at x = 0 is equal to the overpotential applied to the electrode, app , is related to the choice of zero potential. Based on the dynamic description of the electrode processes, the impedance response is obtained by assuming that:

ss

concentration must first be obtained. The solution for the oxygen concentration profile can be obtained analytically, equations and solutions are provided in Appendix A for c0 (r) and (a Ia ). Once the perturbation of all concentration profiles are determined, the perturbation of the cell current density can be determined by integration of the local current density along the thickness of the active layer:

(22)

at x = L

For the determination of the complex amplitude of the source term, a Ia , a solution of the complex amplitude for the oxygen

at x = 0

 εl FDe d ce  d ˚l at x = L =  dx l (1 − t− ) dx x=0

(28)

The set of non-linear differential equations were solved in the MATLAB environment by means of MATLABs bvp solver, bvp4c, which is based on a finite difference code implementing a collocation technique [17]. All the results were obtained assuming that the modulation of the overpotential has the complex value:

app = 0.01 + 0.01j

(29) √ where j is still equal to −1 as above. This implies an arbitrary selection of a phase angle, which has no influence on the calculated impedance. 3. Results and discussion Examples of modelled impedance spectra obtained at various cathodic overpotentials are shown in Fig. 2(a)–(c). The spectra are obtained from solutions of the coupled set of Eqs. (20) and (21), and all parameters related to the oxygen reduction reaction, including also the geometric parameters of the agglomerates, are given

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Fig. 2. Impedance spectra obtained for various overpotentials for (a) base case set of parameters, (b) base case set of parameters, but with De = 0.25De (base case) and (c) base case set of parameters, but with De = 0.04De (base case).

in Table 1. A more detailed description on determination of the parameters is provided in ref. [13]. Fig. 2(a) represents a case where the transport of the ionic species in the electrolyte is assumed to occur relatively fast (corresponding to the base case set of parameters), and the impedance response is mainly determined by the ORR, including also dissolution and transport of oxygen in the liquid phase of the agglomerates. The shape of the impedance spectra are nearly ideal semi-circles, and the potential dependence of the polarization resistance reflects the assumed Tafel behaviour of the ORR, Eq. (2). As has been discussed in ref. [11], however, at higher currents, when transport of dissolved oxygen is important, the potential dependence of the spectra reflects the increase in the apparent Tafel slope, see, e.g. refs. [11,12] and the impedance response can be attributed to the oxygen reduction reaction. Fig. 2(b) and (c) represent cases where the transport of the ionic species in the electrolyte are relatively slow (4 and 25 times lower

than for the base case parameters, respectively). As can be inferred from Fig. 2(b) and (c) (see insert of Fig. 2b)), slow transport of ions in the liquid phase of the porous electrode leads to a 45◦ line in the spectra at high frequencies. The 45◦ branch extends as the ionic transport is reduced, and it’s relative contribution to the spectrum increases with increasing current density. With an increasing dominance of ionic transport, the decay of the polarization resistance with increasing potential is less than for the case of fast ionic transport, dominated by the assumed Tafel behaviour of the ORR. It should be noted that the features characteristic of the spectra shown in Fig. 2(a)–(c) could be obtained by several sets of parameters, depending on the relationship between parameters of oxygen transport and reduction, ionic transport and geometry of the active layer (thickness and morphology). Another feature observed in the impedance spectra is the inductive loop observed at low frequencies. This loop is observed for

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Table 1 Base case parameter values for 6.6 M KOH at 1 atm and 25 ◦ C Parameter

Value

εa εl εg a D0 (m2 s−1 ) Ra (cm) Aa (cm−1 ) Adl (cm−1 ) n s− F (C mol−1 ) R (J K−1 mol−1 ) Cdl (F cm−2 ) T (K) ka (cm s−1 ) c0 (mol cm−3 ) b (V/dec) t− t+ L (cm) l De (cm2 s−1 ) ceb (mol cm−3 )

0.375 0.3 0.2 4.0 6.0 × 10−6 5.0 × 10−4 5.0 × 104 5.0 × 105 4 4 96,487 8.314 3.0 × 10−6 313 4.5 × 10−5 1.0 × 10−6 0.12 0.7283 1 − t− 0.01 8.0 3.3 × 10−5 0.0066

the spectra which also exhibit the 45◦ line related to ionic transport in the porous structure, and the real axis is crossed at around 1 Hz for the set of parameters chosen (Table 1). Inductive loops are frequently observed for multi-step electrochemical reactions, and there are numerous examples in the literature of reaction sequences/intermediate species which are shown to exhibit low frequency inductive loops. For Pt electrodes in alkaline media, reaction sequences for low and high overpotentials were identified by means of impedance spectroscopy together with detailed kinetic models of various reaction schemes in ref. [2], as was also mentioned in Section 1 above. In the development of the present model, however, the ORR was assumed to follow a simple one-step mechanism, so the inductive loop must be related to some transport mechanism in the porous electrode structure. Fig. 3 shows

Fig. 4. (a) Contributions of the complex amplitudes of the various fluxes to the total flux for the spectrum in Fig. 3 where t− = 0.73. (b) Phase angle of the total flux as well as the migration flux and diffusion flux.

the impedance spectrum of Fig. 2(b) at  = 0.3 V together with the corresponding spectrum obtained with a value of t− close to one (t− = 0.999). As can be seen from Fig. 3, immobilization of the positive ion (not participating in the electrochemical reaction) causes the inductive loop to disappear. This is further discussed below. Inductive loops appear when Im(Z) > 0, i.e. when both the real and imaginary parts of the overpotential modulation are positive, i.e. an inductive behaviour occurs when imag( app ) · real( icell ) > real( app ) · imag( icell ) which can be realised from the definition of the impedance (Eq. (24)): Z = =

real( app ) + j · imag( app ) real( icell ) + j · imag( icell ) real( app ) · real( icell ) + imag( app ) · imag( icell )

+j ·

Fig. 3. Comparison of impedance spectra obtained at  = 0.3 V with the same parameter set as in Fig. 2(b), and the corresponding spectrum obtained when the transport number of the negative OH− ions is close to 1 (t− = 0.999).

2

real( icell ) + imag( icell )

2

imag( app ) · real( icell ) − real( app ) · imag( app ) 2

real( icell ) + imag( icell )

2

In order to obtain some insight into the underlying mechanism, the complex amplitudes of the fluxes of the OH− and K+ ions were plotted against the frequencies for the base case set of parameters at  = 0.3 V (see Fig. 4(a)). It appears that for frequencies up to around 1 Hz, the real part of the amplitude of the total flux (proportional to the amplitude of the current) is larger than the imaginary part, which will cause an inductive behaviour (due to the assumption of positive real and imaginary parts of app , see Eq. (29)). As can

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4. Conclusions The modelled spectra where dissolution, transport and reaction of oxygen dominates the impedance are close to ideal semicircles. With increasing contribution from ionic transport in the solution inside the pores, two distinct features occur, namely the appearance of a 45◦ branch at high frequencies, and an inductive loop at low frequenices. The 45◦ branch may be partly masked by the finite conductivity of the solid phase. The inductive loop originates from transport of ions induced by concentration gradients, which implies that a diffusion flux of OH− causes a phase change of the total current, and a corresponding positive impedance at low frequencies.

Appendix A The linearized equation for the complex amplitude of the oscillating oxygen concentration within the agglomerates is given as

 d c  0 2

1 d r 2 dr

εa jω c0 = D0

r



2.3 ss c  b 0

× c0 − Fig. 5. Impedance spectra obtained for various values of the effective conductivity of the carbon phase, (i.e. is corrected for porosity and tortuosity). The spectra are obtained at  = 0.3 V, for the set of parameters corresponding to Fig. 2(b).

also be inferred from Fig. 4(a), this is related to the amplitudes of the diffusion fluxes, in particular the flux of OH− ions. Obviously, at low frequencies, concentration gradients arise which causes a motion of ions due to diffusion with a phase differing from the motion caused by migration, which determines the total flux at higher frequencies. As is illustrated in Fig. 4(b), the phase of the diffusion flux (similar for K+ and OH− ) deviates significantly from the phase of the migration flux and the total flux. If the positive ion is immobilized, as was demonstrated in Fig. 3, the dominant transport mechanism is migration, and the inductive loop disappears. Inductive loops were observed experimentally for porous MnO2 electrodes in 37% KOH at intermediate frequencies (20–2 Hz) [6]. With the basis of a simple pore model, they proposed that the inductive behaviour could be explained by lateral movement (diffusion) of OH− ions out of pores to compensate for depletion of the electrolyte (H2 O consumption) close to the solid/electrolyte interface. In ref. [3], an inductive loop was observed in the impedance spectra at low frequencies for gas diffusion electrodes with active layers fabricated from Pt/C (Pt nanoparticles) together with PTFE (40%) in alkaline solutions. The inductive loops were attributed to a two-electron pathway followed by chemical decomposition of OH2 − on the Pt catalyst particle. However, it is interesting to notice that the inductive loop was not observed for the impedance response of the same active layers deposited on a disc electrode. For the impedance spectra presented in Fig. 5, a finite conductivity of the solid (carbon) phase of the active layer was also included, based on solutions of Eqs. (20), (21) and (28). As can be seen from Fig. 5, lowering the effective conductivity of the porous carbon phase leads to an increase of the total resistance of the electrode, as well as a slight distortion of the response at high frequencies. A low electronic conductivity of the solid phase can thus partly mask the 45◦ response at high frequencies related to ionic transport in the porous structure.

dr

 ln(10)ss 

− Aa ka exp −



b (A1)

With the boundary conditions given as ∂ c =0 ∂r

at r = 0

c0 = 0

at r = Ra

(A2)

Eq. (A1) together with boundary conditions (A2) may be solved analytically to give K2

 · sinh

c0 (r) = r





K1 D0 r



D0 /K1 ka c0s Ra exp(− ln(10)ss /b) ln(10)

+ 

⎡ ·⎣

2D0 br sinh(Ra )



sinh(r) + sinh( ( +









K1 /D0 r)

K1 /D0 )

⎤



sinh(r) − sinh(

−

( −



K1 /D0 r)

K1 /D0 )

⎦

(A3)

The corresponding complex amplitude of the faradaic reaction term (obtained by linearization of Eq. (2), neglecting second-order terms and higher), is then obtained by integration over the agglomerate volume:

(a Ia ) = −

3(1 − εl )



4 Ra3

× 0

Ra



 ln(10)ss 

4 nFAa ka exp −

c0 −

b



2.3 ss c  r 2 dr b 0

(A4)

The integral in Eq. (A4) can be solved analytically, which gives for the complex amplitude of the faradaic reaction term

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(a Ia ) = −

K2

0

K1



(−2.3ss /b) Ra3

3Aa nFka exp

   D D0 K1 ka Aa c s Ra

K1 D 0 Ra

exp

cosh



−



Ra cosh(Ra )−sinh(Ra ) 2

−

√

K1 /D0

ln(10)c s Ra2 coth(Ra ) 0 

− sinh

2D0 b

Ra cosh(Ra )− sinh(Ra ) (D0 /K1 )( √ + 2 (+ K1 /D0 )





K1 D0 R a

b

−

+

Ra 2



sinh(Ra )

K1 /D0 Ra cosh(



coth

·

P=−

K1 /D0 )



Aa ka exp D0

=

K1 = Aa ka exp



=−

 ·

2b

 − ln(10)ss 

+ εa jω

b

dcess

K1 /D0 Ra )D0 b

sinh(Ra ) sinh(

dx



sinh(Ra ) + sinh(



K1 /D0 )

sinh(Ra ) − sinh( 2( −



K1 /D0 Ra )

d˚ss l dx



K1 /D0 )

Following the notation from Newman and Thomas-Alyea [10], with i1 and i2 denoting the superficial current densities in the solid phase and the liquid phase, respectively, the charge transport in the solid phase obeys Ohm’s law: ∂˚1 ∂x

(B1)

where is an effective conductivity, corrected for porosity and tortuosity of the solid phase. As a consequence of the assumption of electroneutrality, the divergence of the total current is zero

∇ · ii + ∇ · i2 = 0

(B2)

With the superficial current density in the liquid phase given as i2 =

εl De ∂cess F l 1 − t− ∂x

(B3)

the following equation is obtained for the stationary potential, ˚ss : l d2 ˚ss l dx2

=

b



Aa ka Ra2 −1 D0

(B5)

(B6)

Ra2 De

1 di2 ε F De d2 cess = l

dx l 1 − t− dx2

2.3RT 2bF

(B7)

at x = 0

=0

at x = L

app

Appendix B

i1 = −



3n(1 − t− )(1 − εg )D0 c0s

˚ss = ˚l l

K1 /D0 Ra )



−



b

− ln(10)˚ss l

cess = ceb

D0 /K1 Aa ka c0s Ra exp(− ln(10)ss /b)

2( +

exp

− ln(10)˚ss l

Notice that in the stationary case, the variable ˚2 can be eliminated from the set of equations, due to the coupling to the concentration profile, as relation (13) in ref. [11] still holds. The stationary profiles of concentration and potential can thus be obtained by simultaneous solution of Eqs. (B4) and (B5) together with the boundary equations:

 − ln(10)ss 



K2 = −

 cess,



and

where



Aa ka Ra2 exp D0

where

√ √ K /D R )−sinh( K1 /D0 Ra )) √ 1 0 a

(−

ce



K1 /D0 )

K1 /D0 Ra cosh(



+

√ √ K /D R )− sinh( K1 /D0 Ra )) √ 1 0 a

(+

√

K1 D0 Ra



√

(D0 /K1 )(

d2 cess =P dx2

· 

 − ln(10)ss  ln(10)

0

−



(B4)

which has to be solved together with the equation given for cess in the stationary model [11]:

=



εl FDe dcess   l (1 − t− ) dx x=0

at x = 0

(B8)

at x = L

at x = L expresses the fact that The boundary condition for ˚ss l current is conserved within the electrode: at x = 0, all the current is carried by ions, whereas at x = L, all the current is carried by app electrons. The boundary condition for ˚ss at x = 0, ˚ss = ˚l is l l chosen for convenience, and implies that the actual applied potenapp tial is equal to ˚l plus the ohmic drop through the solid phase. This must be kept in mind when results are compared to the case neglecting the conductivity of the solid phase. References [1] M. Itagaki, H. Hasegawa, K. Watanabe, T. Hachiya, J. Electroanal. Chem. 557 (2003) 59. [2] D.B. Zhou, H. Van der Porten, J. Electrochem. Soc. 145 (1998) 936. [3] L. Genies, Y. Bultel, R. Faure, R. Durand, Electrochim. Acta 48 (2003) 3879. [4] F. Alcaide, E. Brillas, P.-L. Cabot, Electrochem. Commun. 4 (2002) 838. [5] H. Huang, W. Zhang, M. Li, Y. Gan, J. Chen, Y. Kuang, J. Colloid Int. Sci. 284 (2005) 593. [6] J.B. Arnott, G.J. Browning, S.W. Donne, J. Electrochem. Soc. 153 (2006) A1332. [7] C.-C. Yang, S.-T. Hsu, W.-C. Chien, M.C. Shih, S.-J. Chiu, K.-T. Lee, C.L. Wang, Int. J. Hydrogen Energy 31 (2006) 2076. [8] R. de Levie, Adv. Electrochem. Eng. 6 (1967) 329. [9] C.C. Waraksa, G. Chen, D.D. Macdonald, J. Electrochem. Soc. 150 (2003) E429. [10] J. Newman, K.E. Thomas-Alyea, Electrochemical Systems, third ed., Wiley Interscience, NJ, 2004. [11] M.L. Perry, E. Cairns, J. Newman, J. Electrochem. Soc. 145 (1998) 5. [12] F. Jaouen, G. Lindbergh, J. Electrochem. Soc. 150 (2003) A1699. [13] H. Weydal, A.M. Svensson, S. Sunde, J. Electrochem. Soc., submitted. [14] M.C. Kimble, R.E. White, J. Electrochem. Soc. 138 (1991) 3370. [15] D.B. Sepa, M.V. Vojnovic, A. Damjanovic, Electrochim. Acta 25 (1980) 1491. [16] M. Verbrugge, P. Liu, J. Electrochem. Soc. 152 (2005) D79. [17] Using MATLAB, The Mathworks, Inc., 2002.