Impedance of porous IT-SOFC LSCF:CGO composite cathodes

Electrochimica Acta 56 (2011) 7963–7974

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Impedance of porous IT-SOFC LSCF:CGO composite cathodes Jimmi Nielsen a,∗ , Torben Jacobsen b , Marie Wandel a a b

Fuel Cells and Solid State Chemistry Division, Risø National Laboratory for Sustainable Energy, Technical University of Denmark, DK-4000, Denmark Department of Chemistry, Technical University of Denmark, DK-2800, Denmark

a r t i c l e

i n f o

Article history: Received 22 July 2010 Received in revised form 13 May 2011 Accepted 13 May 2011 Available online 20 May 2011 Keywords: LSCF:CGO IT-SOFC cathodes Gerischer impedance Finite-Length-Gerischer impedance

a b s t r a c t The impedance of technological relevant LSCF:CGO composite IT-SOFC cathodes was studied over a very wide performance range. This was experimentally achieved by impedance measurements on symmetrical cells with three different microstructures in the temperature range 550–850 ◦ C. In order to account for the impedance spectra of the poor performing cathodes the Finite-Length-Gerischer (FLG) impedance was derived and applied to the impedance data. The FLG impedance describes for a given microstructure the situation where the cathode is made too thin from a cathode development point of view. The moderate performing cathodes showed a slightly suppressed Gerischer impedance, while the impedance spectra of the well performing cathodes showed the presence of an arc due to oxygen gas diffusion. The overall impedance of the well performing cathodes could be described with a slightly suppressed Gerischer impedance element in series with a Finite-Length-Warburg (FLW) impedance element. Finally, the origin to a suppression or distortion of the FLG and the Gerischer impedance was discussed and explored in relation to e.g. numerical simulations on the effect of a slightly distributed diffusion length in the FLG, due to different torturous diffusion pathways within the cathodes. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Solid oxide fuel cells (SOFCs) are regarded as a promising technology for energy conversion. In order to reduce the material costs and increase the durability a considerable research effort is conducted on a lowering of the operating temperature to intermediate temperature 600–800 ◦ C (IT-SOFC) or an even lower temperature of 400 ◦ C. For compensation of the lower performance of pure electronic conducting LSM based cathodes with decreasing temperature, mixed oxide ion and electronic conductors (MIECs) are introduced as cathode materials. This increases the electrochemical active part of the cathode substantially and hence the overall performance of the cathode. The performances of such mixed conducting cathodes are co-limited by the oxygen chemical diffusion coefficient Dchem and the oxygen chemical exchange coefficient Kchem . A rough measure for the extension of the active region out in the cathodes is the characteristic length Lc = Dchem /Kchem [1]. Lc values for various different technological relevant materials have been listed in Ref. [2]. The values range from 2 nm for La0.5 Sr0.5 MnO3−ı to 200 ␮m for La0.6 Sr0.4 Co0.8 Ni0.2 O3−ı at 700 ◦ C and PO2 = 70 kPa. However, it is important to emphasize that the actual extension of the active region in porous cathodes also depends on micro structural parameters such as porosity, particle size, necking between

∗ Corresponding author. E-mail address: [email protected] (J. Nielsen). 0013-4686/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.electacta.2011.05.042

particles, tortuousity, etc. For further extension of the active region and a better thermal expansion coefficient (TEC) match, good oxide ion conducting particles Ce0.9 Gd0.1 O1.95 (CGO) are often introduced to form a composite cathode consisting of a network of MIEC and CGO particles. For further improvement or optimization of porous IT-SOFC cathodes a basic understanding of the kinetics is important and hence the impedance spectra, which can provide such valuable information. The Gerischer impedance type element, describing the situation with co-limited diffusion and reaction, has been reported for the most promising IT-SOFC cathode materials such as La1−x Cax Co1−y Fey O3−ı [1], La1−x Srx CoO3−ı [3], and La1−x Srx Co1−y Niy O3−ı [4]. Thus, a Gerischer type impedance response can apparently, in a relatively simple way, describe the complex nature of mixed conducting porous SOFC cathodes and hence serve as a valuable diagnostic tool in cathode development. In the derivation of the Gerischer impedance an infinite diffusion length is assumed, which corresponds to an infinite or in practice a thick electrode. However, an unambiguous Gerischer impedance response for MIEC or composite MIEC:CGO cathodes are not always observed (see e.g. Refs. [5–9]), which gives rise to a confusing and blurred picture. Here, we report a detailed impedance study over a very wide performance range of technological relevant LSCF:CGO cathodes in the hope of acquiring an overview and relate the possible different impedance responses with the performance of the cathodes. The wide performance range was achieved by measurements on three different microstructures a coarse,

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a moderate coarse and a fine LSCF:CGO microstructure, respectively. For each microstructure impedance spectra were acquired in the temperature range 550–850 ◦ C. In the impedance evaluation of the coarse LSCF:CGO microstructure at low temperature, the Finite-Length-Gerischer (FLG) impedance is introduced. The FLG impedance is more general than the Gerischer type impedance typically employed, since it is able to describe the situation with an arbitrary diffusion length and leads to the typical Gerischer type impedance in the case of a thick electrode. In other words, the FLG describes the situation where the electrode thickness is below the optimal value from a cathode development point of view. The extraction of material properties from the Gerischer impedance response has been meaningful [1,3,4]. However, suppressed or distorted Gerischer impedance responses are in most cases observed. This is not surprising since e.g. an ideal capacitive behavior is very rarely seen in impedance spectroscopy of solid state chemical systems [10]. A by far more applicable description is the constant phase element (CPE). The common interpretation of CPEs is a distribution of relaxation times introduced by a distribution in permittivity values [11]. This interpretation corresponds to a distribution of ideal capacitance values, which make sense when pure capacitive elements are involved. However, the corresponding interpretation of a suppressed or distorted Gerischer impedance response is not clear. Boukamp and Bouwmeester has proposed various ways of taking the suppressed nature into account by the introduction of a fractal Gerischer expression [12]. Nonetheless, there are several ways of making the Gerischer fractal and it is not clear which approach should be chosen and what the corresponding interpretation is. Here, we explore and discuss the discrepancy between experimental data and the derived FLG in relation to e.g. numerical simulations on the effect of a distribution in the diffusion length in the derived FLG. 2. Theory

In a solid oxide fuel cell the electrode reactions require simultaneous contact between a gaseous, an ionic and an electronic conducting phase. Thus, if a mixed conductor (MIEC) combining ionic and electronic conductivity is applied, the reaction is no longer confined to a triple phase boundary but can take place all over the gas–solid surface of the material. As illustrated in Fig. 1 the porous electrode structure is modeled as columns of MIEC material. Further it is illustrated that the charge transferring step in the oxygen electrode reaction can be considered as a formation of an electroneutral pseudo component, VO ≡ VO·· + 2e in the MIEC phase, at the MIEC–electrolyte interface  VO (MIEC)

(2)

The vacancy transport is considered as a one-dimensional diffusion process along a tortuous path of interconnected MIEC particles coupled to the surface reaction. It is described by Fick’s second law including the surface reaction as a drain, i.e. ∂cV = ∂t

ADV ∂2 cV  ∂y2

−v

RT ln



aOO

= −r G

1/2

pO aVO

(4)

2

which, under the assumption of a low concentration of vacancies, i.e. aOO  1, gives A≡

∂ ln(aVO ) ∂ ln(cVO )

=−

1 ∂ ln(pO2 ) 2 ∂ ln(cVO )

(5)

In general, due to electroneutrality conditions, DV is a combination of transport properties for the charged vacancies, VO·· , and electrons [13]. In the case of a MIEC with a dominating electronic conductivity, gradients in the electrochemical potential are insignificant and DV will be equal to the self diffusion coefficient of the vacancies VO·· . It should be noted that a condition for the homogeneous modeling used is that the thickness of the electrode is more than a few particle dimensions, although the averaging over the electrode surface will improve the statistics and thus reduce the requirements to the thickness. For thinner electrodes a detailed model, taking into account the actual microstructure of the electrode is required. At present the kinetics of the surface reaction is not known in details. Therefore, a simple kinetics is assumed, and if transport limitations for gaseous oxygen are ignored the response to a small eq perturbation of the vacancy concentration, cV = cV − cV , at the gas–solid interface can be written as

v = k0 cV

(6)

where k0 is a concentration based rate constant. Eq. (3) can now be written as:

(3)

where A is the thermodynamic factor, DV is the self diffusion coefficient of the VO species,  is the tortuousity and v is the rate of reaction (2) per unit length of the path projection on the y-axis in a unit area of the electrode.

(7)

After Laplace transform of the response from the equilibrium (t, cV ) = (0, 0) the equation can be written as the ordinary second order differential equation  d2 c V − (k0 + s)c V = 0 ADV dy2 with the general solution c V = A1 exp

 y  (s)

+ A2 exp

(8)

 −y  (s)

(9)

where the complex frequency dependent length parameter



(1)

followed by diffusion of VO to the gas–MIEC interface, where it reacts with oxygen. 1/2 O2 + VO  OO



ADV ∂2 cV ∂cV = − k0 cV  ∂t ∂y2

2.1. Porous electrode

VO·· (CGO) + 2e

The thermodynamic factor for the VO component can be determined experimentally from the equilibrium condition for reaction (2)

(s) =

ADV k0 (1 + s/k0 )

(10)

is determined by the competition between the diffusion process and the oxygen surface reaction. The higher the reaction rate, the shorter is the penetration of the reaction zone from the electrolyte interface into the porous electrode. As outlined in Fig. 2, the boundary condition at the electrode–electrolyte interface is 0 jV



dc V  = jV (y = 0) = −ADV dy 

(11) y=0

At the interface between the porous electrode and the gas phase the MIEC particles, y = L, have a somewhat larger area available for the oxygen reaction, due to the lack of further grain–grain connections. For mathematical convenience this additional active area is

J. Nielsen et al. / Electrochimica Acta 56 (2011) 7963–7974

7965

Fig. 1. Electrode model and reaction.

ignored and accordingly the vacancy flux is considered blocked at the interface

where

 dc V  jV (y = L) = −ADV dy 

0 = =0

(12)

y=L

Combining Eq. (9) with the boundary conditions gives the concentration profile through the MIEC electrode



0



jV (s) cosh (L − y)/(s) c V (y) = ADV sinh(L/(s))

(13)

and the flux impedance [14]

 c V  Zj = j  V

 = y=0

0 ADV



coth (L/0 )



1 + s/k0

1 + s/k0

 (14)



ADV k0

(15)

is a characteristic length expressing the DC penetration of the concentration perturbation from the electrode–electrolyte interface into the mixed conducting electrode. Fig. 3 shows a complex plane plot of Eq. (14) for different electrode thicknesses. For thick electrodes (L/0 > 0) the well known

Gerischer impedance [15] ∝ 1/ 1 + jω/k0 starting with a Warburg impedance at high frequencies and turning smoothly into a circle arc at low frequencies is obtained. For thinner electrodes the Warburg region is reduced and in the limiting case of very thin electrodes where no concentration gradient inside the MIEC develops, the response is that of a R–C parallel combination. Had the boundary condition in Eq. (12) been that of a nernstian diffusion layer c V (y = L) = 0

Fig. 2. Vacancy transport in a porous mixed ionic conducting electrode.

(16)

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J. Nielsen et al. / Electrochimica Acta 56 (2011) 7963–7974

−3

−2 L/λo=0.2

Z Im

ω/ko=1 ω/ko=10

−1

L/λo=0.5 ω/ko=0.1

L/λo=1 L/λo=10

0 0

1

2

3

4

5

ZRe Fig.



(coth



3. Dimensionless



(L/0 )

1 + s/k0

)/(

flux

impedance

Zj /(0 /ADV ) =

1 + s/k0 ) for different electrode thicknesses

RT E=E + ln 2F

the corresponding solution to Eq. (8) would be

V



= y=0

0 ADV



tanh (L/0 )



1 + s/k0

 (17)

1 + s/k0

⎧  

⎨ tanh (L/0 ) 1 + s/k0 → 1 L   →∞⇒

0 ⎩ coth (L/0 ) 1 + s/k0 → 1

(18)

(19)

Although the nernstian boundary condition does not appear justified for porous electrodes limited by kinetics, an impedance expression equivalent to Eq. (17), originally developed for the electronic conduction in mixed conducting oxides between ionically blocking electrodes [12], has been applied to porous SOFC anodes [16,17] and shown to fit the experimental data very well. −0.4

ZIm

L/λo=10 ω/ko=10

L/λo=1 L/λo=0.5

L/λo=2 ω/ko=1

L/λo=0.2

ω/ko=0.1

0.1 0 0

0.2

0.4

0.6

0.8

1

ZRe Fig. (tanh





4. Dimensionless



(L/0 )

1 + s/k0

)/(

flux

impedance

Zj /(0 /ADV ) =

1 + s/k0 ) for different electrode thicknesses

L. The dotted lines are equi-frequency curves.

aV ·· (CGO)



O

(21)

aVO

Since the vacancy concentration in the CGO electrolyte is high compared to that in the MIEC and fixed by the Gadolinium substitution, the vacancy activity in the CGO phase is considered as a constant For a perturbation within the linear range, the overvoltage,  = E − Eeq , can be written as =−

 RT RT ∂ ln(aVO )  ln aVO = −  ln(cVO ) 2F 2F ∂ ln(cVO )

(22)

and after introduction of the thermodynamic factor as defined in Eq. (5). =−

RT cVO A 0 2F c

(23)

Combining Eq. (23) with Faraday’s law at the y = 0 and the flux impedance expression (14) gives the current impedance



 ADV k0 (1 + s/k0 )

−0.2

(20)

VO

leading to a slightly modified version of the well known Gerischer impedance Zj,G =



0

Comparing the plot of Eq. (17) in Fig. 4 with Fig. 3 the main difference is, that for thin electrodes the nernstian boundary the impedance approaches that of ordinary nernstian diffusion impedance – in contrast to the planar electrode behavior resulting from the blocking boundary. For thick electrodes where the concentration wave does not penetrate through the electrode to reach the gas interface, the impedance does not depend on the boundary condition here. In both cases the limiting case is



2e + VO·· (CGO)  VO (MIEC)

is in equilibrium, and the electrode potential can be expressed in terms of the vacancy activities

L. The dotted lines are equi-frequency curves.

 c V  Zj = j 

Normally, optimal fuel cell electrodes are manufactured with a thickness exceeding the utilization depth considerably and the impedance expressions in Eqs. (14) and (17) will be more or less equivalent. However, it is interesting to note, that if the transition from infinite to finite thickness is within the range of an experiment – as will be demonstrated in the present work – important parameters like the effective vacancy diffusion coefficient and surface reaction constant can be determined. To obtain the current–overvoltage impedance from the flux–concentration impedance in Eq. (14) the relation between the concentration perturbation and the overvoltage at the electrode–electrolyte interface, y = 0 is needed. If concentration overvoltage is dominating, it seems reasonable to assume local equilibrium for the charge transfer reaction at this interface. Thus, for y = 0

Z=

RT 0  = i 4F 2 DV cV0



coth (L/0 )



1 + s/k0

1 + s/k0



(24)

It is interesting to note that an expression with a similar form as (24) has been derived by de Levie and Delahay [18] for porous electrodes controlled by charge transfer in a liquid electrolyte. The extensive reported work in the literature on porous electrodes in liquid electrolytes has recently been reviewed by Lasia [19]. In the de Levie case the interfacial reaction resistance and the double layer capacity/CPE on the pore wall is interconnected by the pore electrolyte resistance to form a transmissionline as shown in Fig. 5. The equivalence of these two – apparently very different systems – is due to the fact that the diffusion equation can be represented as an R–C transmissionline. In the solid state electrode the chemical surface acts as a drain in the diffusion process shunting the “diffusion capacity”, thus playing the same role as the charge transfer resistance in the liquid electrolyte system. The replacement of the capacitance C with a constant phase element Q with the impedance expression ZCPE = 1/Q(jω)˛ in the transmission line representations in Fig. 5 imply the following suppressed Gerischer expression G: ZG =

RT 4F 2 cV0



A

˛

DV k0 (1 + (s/k0 ) )

(25)

That this is the case can be seen from the impedance expression for an infinite transmission line [12]. To improve the ionic conductivity of the porous electrode and thereby extending the effective reaction zone, the electrode can

J. Nielsen et al. / Electrochimica Acta 56 (2011) 7963–7974

Gas

Gerischer

MIEC

½O 2(g)

VO

rP

rP

OO

VO

VO

OO

VO

OO

rs

Q

rs

rs

rP

rP rs

Pore

rs

Electrolyte Parameter

Gerischer

de Levie

rs

Resistance associated with oxygen vacancy diffusion in the MIEC phase.

rp

MIEC/gas phase interfacial chemical reaction resistance.

Redox reaction resistance at the electrode/electrolyte solution interface

Q

Chemical capacitance associated with changes in MIEC oxygen stoichiometry.

Electrode/electrolyte solution double layer capacitance

Electrolyte solution resistance

Fig. 5. Distributed surface reaction in a porous MIEC electrode and a porous electrode in a liquid electrolyte (de Levie) along with the transmission line equivalents.

be a composite of MIEC material and a highly conducting ionic conductor, I, as YSZ or Gadolinium substituted ceria, CGO. In this case we assume a local equilibrium between the vacancies in the MIEC phase and those in the ionic conductor, I, i.e.  ¯ VO ,MIEC (y) =  ¯ VO ,I (y)

(26)

Furthermore, if the electronic conductivity of the MIEC phase is high compared to the ionic conductivities, as it is normally the case, the electrochemical potential of the electrons inside the MIEC phase is constant. Thus, it follows that the driving force for the oxide vacancies in the I-phase is the same as that for the VO species in the MIEC-phase. Within these assumptions the overall vacancy diffusion can be described by a modification of Eq. (3) ∂cV ∂2 cV +v = ADVeff ∂t ∂y2

(27)

with the effective self diffusion coefficient DVeff combining the properties of the MIEC and the I phase according to [20] DVeff =

I cV,I DI DM + M M cV,M I

(28)

where M and I are the respective volume fractions of the MIEC and the I phase and cV , D and  have their previous meaning. Finally, the impedance of the composite MIEC–I electrode is obtained from Eqs. (15) and (24) by substituting D/ with DVeff Z=

RT



4F 2 cV0



0 =

ADeff k0



A DVeff k0



coth (L/0 )



1 + s/k0

1 + s/k0

4F 2 cV0



A DVeff k0

coth(L/0 )

(31)

3. Experimental

rP

rP Q

½O 2(g)

Q

rs

RT

Different limiting situations of (24) and (29) has been reported in the literature. The semi infinite diffusion case for a pure MIEC cathode (19) has been reported in [1], while a similar expression for the DC case (31) has been reported from continuum considerations in Ref. [21].

rs

rP

rP

OO

Q ½O 2(g)

Q

rs

Q ½O 2(g)

ZDC =

Q

rs

Q

The DC limit, s = jω → 0, of the impedance expression is of interest for the optimization of the electrode thickness

de Levie

Electrode

Gerischer

7967



(29)

(30)

The electrodes investigated were composite SOFC cathodes consisting of 50/50 wt% (La0.6 Sr0.4 )0.99 Co0.2 Fe0.8 O3−ı (LSCF) and Ce0.9 Gd0.1 O1.95 (CGO). The symmetrical cells were prepared by screen printing LSCF/CGO cathodes onto both sides of a 5 cm × 5 cm 200 ␮m thick CGO tape. Three different cathode structures were obtained by sintering such identical deposited cathodes at the temperatures 900 ◦ C, 1000 ◦ C and 1100 ◦ C for 2 h, respectively. The cells were after sintering cut into 4 mm × 4 mm symmetrical cells on to which Pt-Paste was applied as a current collector for electrochemical impedance spectroscopy (EIS) measurements. The performance of the cells was evaluated by EIS using a Hioki impedance analyzer in the frequency range 100 kHz to 0.08 Hz with 6 points pr. decade and a generator amplitude of 0.05 V. The cell temperature was varied according to the following sequence 750 ◦ C → 850 ◦ C → 800 ◦ C → 750 ◦ C → 700 ◦ C → 650 ◦ C → 600 ◦ C. After each change in temperature, a waiting period of 2 h was introduced prior to measurement. Numerical simulations were performed in MATLABR2008b, while the fitting of the impedance spectra was done using the complexnon-least-square (CNLS) fitting routine in a laboratory developed program. The cell samples for scanning electron microscopy (SEM) characterization were prepared by vacuum embedding pieces of the cells in epoxy (EpoFix from Struers A/S, Denmark) followed by grinding and polishing. The samples were subsequently coated with carbon to eliminate charging. A Zeiss Supra 35 FE-SEM was used for the micro structural characterization. 4. Results and discussion 4.1. Coarse LSCF:CGO microstructure Fig. 6 shows a cross sectional SEM image of the electrode microstructure along with the corresponding impedance spectra of the symmetrical cell in air in the temperature range 550–850 ◦ C. The impedance spectra have been fitted with the FLG expression Eq. (29) using the dimensionless electrode thickness L/0 , k0 and the prefactor as adjustable parameters. It is seen from the figure that very good fits are achieved at high temperatures of 800 ◦ C and 850 ◦ C, where the complex plane plot of the data consists of a high frequency line at 45◦ followed by a skewed semicircle. Such an impedance spectrum shape is characteristic for the typical Gerischer type impedance with infinite diffusion and represents the case of Eq. (29) with L > 0 . It is the situation where the ac signal penetration length is smaller than the electrode thickness. However, as the temperature is lowered from 800 ◦ C to 600 ◦ C a gradual change in the impedance shape is observed. At e.g. 700 ◦ C the impedance spectrum has a shape that represents the transition zone of Eq. (29) where the ac signal penetration length is comparable to the electrode thickness L ∼ 0 . The spectrum can actually be fitted quite well with a Finite-Length-Warburg (FLW) impedance element [11]. Thus, if the impedance of the cathode was studied in a narrow performance range it might lead to false conclusions. On a further lowering of the temperature to 600 ◦ C and 550 ◦ C the

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Fig. 6. SEM image of the cathode with a coarse microstructure and corresponding impedance data and fits in the temperature range 550–850 ◦ C. The impedance data has been fitted with the equivalent circuit L–Rs –FLG, where L represents an inductance, Rs the serial resistance and FLG the expression given by (29).

ac signal penetration length becomes smaller than the electrode thickness L < 0 . This situation is characterized by an impedance spectrum shape with a high frequency line at ∼45◦ followed by a transition zone and a semicircle at low frequencies. The observed temperature dependency of the shape of the impedance spectra in Fig. 6 is due to a temperature dependency of the utilization length 0 as given by Eq. (15), hence a difference in the activation energy of k0 and DVeff , respectively. In the present case where the electrode

thickness is comparable to the utilization length and the electrode thickness is known, important electrode optimization parameters can be determined. From impedance measurements the parameters L/0 and k0 can be determined. With knowledge about the electrode thickness L from e.g. SEM it is possible to determine the effective diffusion coefficient DVeff and the ratio between the thermodynamic factor A and the oxygen vacancy concentration cV0 from the value of the prefactor in Eq. (29).

J. Nielsen et al. / Electrochimica Acta 56 (2011) 7963–7974

1E-5

10

1000

k0

Ea=126KJ/mol

Rp /Ω cm

10

1

2

1E-6

0.1

Dv

eff

K 0 /S

-1

/cm s

-1

1

Ea=60.8KJ/mol

SEM determined cathode thickness

100

eff

2

Dv

7969

0.01

0.1

0

5

10

15

20

25

30

35

40

μ Fig. 9. Calculated polarization resistance Rp as a function of the cathode thickness L according to Eq. (31) with the use of fitted parameters from impedance data fits of the cathode with a coarse microstructure in Fig. 6.

0.01 0.85

0.90

0.95

1.00

1.05 3

-1

1.10

1.15

1.20

1E-7 1.25

-1

10 T /K

Fig. 7. Arrhenius plot of k0 and DVeff from the impedance fits with (29) of the cathode with a coarse microstructure in Fig. 6.

In Fig. 7 Arrhenius plots are shown of the parameters k0 and DVeff extracted from the fits with (29) in Fig. 6, where an activation energy Ea of 126 kJ mol−1 and 60.8 kJ mol−1 can be found, respectively. In Fig. 8 Arrhenius plots are shown of the in Fig. 6 impedance determined serial resistance Rs with an Ea = 55 kJ mol−1 and the polarization resistance Rp with an Ea = 168 kJ mol−1 . From LSCF conductivity relaxation measurements oxygen vacancy self diffusion activation energy values of 69 ± 11 kJ mol−1 , 89 kJ mol−1 and 93 kJ mol−1 have been reported [22,23], while a value of 94 kJ mol−1 has been reported from coulometric titration measurements [22]. The experimental determined activation energy 60.8 kJ mol−1 for 1

100

RS RP

Ea=55KJ/mol

2

0.1

1

Ea=168KJ/mol

Rp /Ω cm

Rs /Ω cm

2

10

0.1

0.01 0.85

0.90

0.95

1.00

1.05 3

-1

1.10

1.15

1.20

0.01 1.25

-1

10 T /K

Fig. 8. Arrhenius plot of the serial resistance Rs and the polarization resistance Rp of the impedance data of the cathode with a coarse microstructure in Fig. 6.

DVeff is lower than the reported value for pure LSCF. This is reasonable since CGO with a lower Ea (Dv ) than LSCF also is present. The Rs temperature dependency shown in Fig. 8 reflects the temperature dependence of the oxygen vacancy self diffusion coefficient Dv (CGO), since the number of oxygen vacancies in CGO is primarily determined by the alliovalent substitution with Gd3+ and not by thermal generation. This means that the activation energy for the self diffusion coefficient is the same as that for the oxygen vacancy diffusion coefficient [24]. An activation energy of 55 kJ mol−1 for the CGO oxygen ionic conductivity is also reported in Ref. [25]. Thus, the close Ea (Dveff ) = 60.8 kJ mol−1 value to that of the oxygen ionic conductivity in CGO suggest that CGO enhances the oxygen ionic conductivity in the composite MIEC:CGO considerable in accordance with the original idea behind the use of composite cathodes. The Ea for k0 is expected to be the same as the surface exchange coefficient Kchem . From isotope exchange depth profiling (IEDP) with secondary ion mass spectroscopy (SIMS) measurements on a 70/30 wt% LSCF:CGO composite cathode a Ea (Kchem ) value of 134 kJ mol−1 was found [26], which is in good agreement with the Ea (k0 ) value of 126 kJ mol−1 extracted from Fig. 7. It was also noted in this study that the LSCF:CGO composite cathode seem to behave as one phase in agreement with the observations of the present study. Further, the tracer diffusion coefficient value and activation energy was found to have values in between that of the pure LSCF and CGO phase, which supports the assumption made by Eq. (26) of oxygen vacancy equilibrium between the LSCF and the CGO phase in the composite cathode. Fig. 9 shows simulations of the polarization resistance Rp as a function of cathode thickness L at different temperatures. The simulations have been made using Eq. (31) and the parameters determined from the fits in Fig. 6. From the simulations it is seen that at a temperature of 550 ◦ C the electrode reaction extents, if allowed, far beyond the cathode thickness of 12 ␮m to a cathode thickness of 35–40 ␮m. In contrast to this the electrode reaction utilization length at 850 ◦ C is approximately just within the thickness of the cathode. As already mentioned, the fits in Fig. 6 at a high temperature of 800 ◦ C and 850 ◦ C are very good. However, it can be observed in Fig. 6 that the fits become poorer as the temperature is decreased. It is particularly evident that there is a discrepancy between experimental data and the fits with Eq. (29) at low temperature in the transition zone from the high frequency line at 45◦ region to the low frequency semicircle region. Eq. (29) predicts a more abrupt transition than the experimentally observations at e.g. 550 ◦ C and

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a change of the slope of the high frequency line from 45◦ to slightly below 45◦ just before the semicircle region. In contrast, the experimental data shows a more smooth transition and a transition zone line with a slope at all frequencies slightly above 45◦ . As mentioned in the theory section a somewhat similar expression as Eq. (29) has been derived for the case of a porous electronic conducting electrode in an electrolyte solution. In this context the effect on the impedance of different pore shapes has been considered along with the effect of a distribution in pore parameters [27–33]. In those studies it was clearly shown that such effects can smear out the impedance response and cause a more smooth transition from the high frequency line at 45◦ to the semicircle at low frequencies. Further, it was also shown that such effects can result in a slope of the high frequency line that is slightly above 45◦ . Such dispersion effects are illustrated in the following. It seems reasonable to assume that the diffusion length L in Eq. (29) is slightly distributed due to the different tortuous diffusion pathways in the porous electrode. Since the total impedance of the porous electrode is the summation of the impedance of each ensemble of pathways with a distribution density f(L), the total impedance can be written as: (ZTotal )

−1



=

∞

(ZFLG (L))

−1

f (L)dL

(32)

0

ZFLG (L) is Eq. (29) and f(L) is a normalized density distribution function of the diffusion length L so that f(L)dL is the number of diffusion pathways between the diffusion length L and L + dL. Fig. 10 shows simulations with the fitted parameters at 550 ◦ C in Fig. 6 with various normalized distributions of L around the SEM determined electrode thickness of 12 ␮m. From the simulations it is clearly seen that a small distribution of the diffusion length L results in a more smooth impedance response and can result in a high frequency line with a slope above 45◦ in accordance with the experimental observations in Fig. 6. A closer inspection of the low frequency semicircle, which becomes increasingly apparent as the temperature is lowered in Fig. 6, reveals that the semicircle is suppressed and hence a description as a constant phase element is more applicable. The CPE exponent (˛) in the CPE impedance expression ZCPE = 1/Q(jω)˛ can be found to be around 0.9 at a temperature of 550 ◦ C and 600 ◦ C. This observation is discussed further in Section 4.3. The FLG impedance has most likely been seen in previous studies of MIEC cathodes where the oxygen partial pressure has been varied [1,9]. A change of the oxygen partial pressure P(O2 ) does not only change the resistance associated with oxygen gas diffusion, it also changes the rate at which oxygen is exchanged between the gas phase and the solid LSCF phase. In the formulation of the surface reaction rate as k0 cv the free LSCF surface sites are represented by cv while the oxygen partial pressure dependency is put into the constant k0 . Thus, the reaction rate expression is only valid when the oxygen partial pressure is kept constant. This statement is experimentally well supported within the literature where numerous studies show a clear dependency of the oxygen exchange coefficient Kchem on the oxygen partial pressure [22,25,34–36]. Thus, a lowering of P(O2 ) also extents the utilization length , which at sufficient low oxygen partial pressures can result in a larger utilization length  than the cathode thickness L. In such a case where the impedance is measured at various sufficient low P(O2 ) s it appears reasonable that the high frequency part is roughly the same for all the spectra, but distinguished by the resistance of the low frequency semicircle. The high frequency part is expected to be roughly the same since the ac perturbation signal penetration depth at sufficient high frequencies is smaller than the cathode thickness L. Such a series of impedance spectra is exactly what is observed in the previously mentioned studies, where the oxygen partial pressure has been varied. However, it is inevitable when

Fig. 10. Illustration of the effect of a distributed diffusion length by simulations of various distributions around the cathode thickness of 12 ␮m using the fitted parameter values of the fit at 550 ◦ C in Fig. 6.

going to very low oxygen pressures to avoid a resistance contribution from gas diffusion. This is also shown in the mentioned studies where it is shown that the same oxygen partial pressure, but diluted in different gasses such as Ar and He gives rise to a difference in the resistance of the low frequency FLG semicircle as predicted by binary gas diffusion theory. 4.2. Moderate coarse LSCF:CGO microstructure Fig. 11 shows a cross sectional SEM image of the cathode with a moderate coarse microstructure along with the corresponding impedance spectra of the symmetrical cell in air in the temperature range 550–850 ◦ C. The impedance spectra up to 800 ◦ C have been fitted with an equivalent circuit L − Rserial − G, which consists of a series combination of an inductor, a serial resistance Rserial and the Gerischer impedance ZG (25). At 850 ◦ C a FiniteLength-Warburg impedance has been added to account for oxygen gas diffusion in the cathode pores and the stagnant gas layer at the electrode surface at low frequencies. This will be justified in more detail in Section 4.3. At a low temperature of 550 ◦ C the presence of an apparently small high frequency arc is clear. A thorough inspection of the similar impedance spectrum at 550 ◦ C for the coarse microstructure in Fig. 6 also reveals such an arc.

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Fig. 11. SEM image of the cathode with a moderate coarse microstructure and corresponding impedance data and fits in the temperature range 550–850 ◦ C. The impedance data 550–800 ◦ C has been fitted with the equivalent circuit L–Rs –G, while the impedance data at 850 ◦ C has been fitted with the equivalent circuit L–Rs –G–FLW where L represents an inductance, Rs the serial resistance, G the typical Gerischer (19) and FLW the Finite-Length-Warburg impedance element [37].

One possible interpretation of the arc could be that it represents the resistance associated with oxide ion charge transfer between either LSCF/CGO or CGO/CGO grains. A high frequency arc at low temperature has been observed before for LSCF:CGO cathodes, where the arc was speculated to originate from the CGO grain boundaries in the electrolyte [8].

The data and fits from 550 ◦ C to 800 ◦ C in Fig. 10 show clearly that the cathode is in a performance region, where the Gerischer type impedance is valid. At 850 ◦ C the presence of two contributions is evident, which will as already mentioned be discussed in detail in the next Section 4.3.

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Fig. 12. SEM image of the cathode with a fine microstructure and corresponding impedance data and fits in the temperature range 550–850 ◦ C. The impedance data 550–650 ◦ C has been fitted with the equivalent circuit L–Rs –G, while the impedance data 700–850 ◦ C has been fitted with the equivalent circuit L–Rs –G–FLW where L represents an inductance, Rs the serial resistance, G the typical Gerischer (19) and FLW the Finite-Length-Warburg impedance element [37].

4.3. Fine LSCF:CGO microstructure A cross sectional SEM image of the cathode with a fine microstructure along with the corresponding impedance spectra of the symmetrical cell in air in the temperature range 550–850 ◦ C is shown in Fig. 12. The 550–650 ◦ C spectra have been fitted with the equivalent circuit L − Rserial − G, while the 700–850 ◦ C spectra have been fitted with the equivalent circuit L − Rserial − G − FLW,

where L represents an inductance, Rs the serial resistance, G the typical Gerischer (19) and FLW the Finite-Length-Warburg impedance element [37]. As the electrochemical cathode reaction resistance is lowered the overall cathode resistance will at some point be limited by the mass transport of oxygen to the electrochemical reaction zone, that is, e.g. by oxygen diffusion in the pores of the cathode. It has been analytically shown that gas diffusion in the pores or in a stagnant

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gas layer at the electrode surface results in a Finite-Length-Warburg impedance response [37]. Taking the tortuousity  and the porosity (ε) into account, the FLW impedance provide, in the limiting case of ω → 0 with the use of the ideal gas law, the following expression for the resistance associated with gas diffusion: Rgas

diffusion

=

R2 T 2 L 16F 2 DP 

(33)

L is the diffusion layer thickness, P is the pressure and D is the diffusion coefficient. For the gasses typically used in fuel cells, bulk diffusion dominates at pore diameters greater than ∼10 ␮m, while Knudsen diffusion dominates at pore diameters less than 50–100 nm [38]. The pore diameter for the cathode microstructure in Fig. 12 is roughly 200 nm. Thus, the present case is located within the transition zone between the two extremes of either pure bulk or Knudsen diffusion. A calculation of the binary diffusion coefficient provides a value of 9.07 cm2 s−1 [39], while a calculation of the Knudsen diffusion coefficient, assuming the cathode consist of a collection of straight cylindrical capillaries, gives 0.72 cm2 s−1 at 850 ◦ C [38]. If the resistance of the low frequency impedance arc 3–4 m cm2 in Fig. 12 at 700–850 ◦ C and in Fig. 11 at 850 ◦ C has to be accounted for by gas diffusion according to Eq. (33) with  = 1 and ε = 0.5 a gas diffusion layer thickness L of more than 100 times the cathode thickness of 12 ␮m is necessary. Thus, the main contribution to the resistance associated with gas diffusion comes from the bulk diffusion in the stagnant gas layer at the electrode surface. The oxygen partial pressure variation within the porous cathode structure is therefore neglectable, which means, that the separation of the total impedance into a serial combination of a Gerischer and a Warburg impedance response is a very good approximation. Gas diffusion is somewhat special in the sense that it has a nonArrhenius like temperature dependency unlike chemical processes or solid state ion conduction, etc. where an energy barrier has to be overcome for the process to proceed. The binary bulk diffusion coefficient has a temperature dependency proportional to T3/2 which means according to Eq. (33) that Rgas diffusion ∝ T1/2 . A weak temperature dependency is therefore expected in accordance with the data in Fig. 12. Since it seems reasonable to observe oxygen gas diffusion impedance for well performing cathodes and the observed resistance temperature dependency is weak, the observed low frequency arc from 700 to 850 ◦ C in Fig. 12 and the low frequency arc in the spectrum at 850 ◦ C in Fig. 11 is interpreted as a gas diffusion resistance. As was the case for the impedance spectra at 550 ◦ C for the coarse and moderate coarse microstructure, the presence of a high frequency arc is also evident in the impedance spectra at 550 ◦ C and 600 ◦ C for the cathode with a fine microstructure in Fig. 12. From Figs. 11 and 12 it is clearly seen that as the temperature is increased and hence the performance of cathodes, inductive effects become increasingly important. At 800 ◦ C and 850 ◦ C the high frequency part of the two clearly resolved impedance contributions has an apparent shape, which resembles more that of a semicircle than a Gerischer impedance response. It is obvious that a continuous increase in the temperature and hence the performance will at some point lead to an utilization length (), which is smaller than the cathode particles that constitute the porous cathode. When this is the case the treatment as a 1-dimensional problem is no longer valid. Whether 2D effects are present in the spectra at 800 ◦ C and 850 ◦ C in Fig. 12 is difficult to judge. It is inherently difficult to measure such small impedances since small phase errors from the impedance analyzer, current distribution effects and inductive effects become very important. However, since fairly reasonable fits still are obtained it seems safe to assume that the 1-dimensional Gerischer impedance model is still applicable and conclude that the

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Table 1 Suppression exponent ˛ from fits with (25) or (29) in Figs. 6, 11 and 12 depending on temperature and particle size. t (◦ C)

Particle size

550 600 650 700 750 800 850 Average

Coarse 1 1 1 1 1 1 1 1

Medium 0.95 0.97 0.95 0.96 0.94 0.92 0.88 0.94

Fine 0.92 0.89 0.85 0.87 0.86 0.86 0.90 0.88

gas diffusion becomes important before a complete breakdown of the FLG/Gerischer impedance model. A high frequency slope larger than 45◦ is seen for the impedance at 550 ◦ C for the cathode with a coarse microstructure in Fig. 6. In comparison an exact 45◦ slope is seen for the cathode with a moderate coarse microstructure in Fig. 11, while a slope clearly below 45◦ is seen for the cathode with a fine microstructure in Fig. 12. In Table 1 the fitted ˛ values are shown for the fits in Figs. 6, 11 and 12. From Table 1 a trend can be observed towards a lower ˛ value as the microstructure of the cathode becomes finer. As already mentioned in Section 4.1 the appearing semicircle as the temperature is lowered in Fig. 6 is on closer inspection suppressed with a ˛ value around 0.9 at a temperature of 550 ◦ C and 600 ◦ C. The low frequency suppressed semicircle represents the situation where the whole porous cathode is active and hence the situation where the oxygen exchange reaction between the gas and solid phase limits the cathode performance. Thus, it seems tempting to relate the suppressed impedance nature with ˛ slightly below 0.9 of the cathode with a fine microstructure with the gas/solid oxygen exchange reaction. However, it is not clear what mechanism that can account for the somewhat more ideal impedance behavior of the cathodes with a moderate coarse and coarse microstructure.

5. Conclusions The Finite-Length-Gerischer impedance expression was derived for pure MIEC cathodes and composite cathodes consisting of MIEC particles and purely ion conducting material particles. The derived FLG expression was used to describe the situation where the cathode is made too thin from a cathode development point of view, which also provided a description on how the cathode polarization resistance Rp changes as a function of the cathode thickness L. Further an overview of the impedance for technological relevant LSCF:CGO composite cathodes was provided over a very wide performance range. The FLG impedance expression was used to describe the impedance of the poorly performing cathode with a coarse microstructure. The intermediate performing cathode with a moderate coarse microstructure was appropriately described by a Gerischer type impedance, while the well performing cathode with a fine microstructure showed mass transport limitation with the presence of a resistance due to oxygen gas diffusion. The impedance of the well performing cathode consisted of two impedance contributions, which could be fairly described with a Gerischer type impedance in series with Finite-Length-Warburg impedance. Finally, the smoothing effect of a distributed diffusion length L on the FLG impedance expression was clearly illustrated by simulations. Further, it was experimentally shown that there seems to a correlation between the suppression of the Gerischer type impedance and the size of the particles which constitute the porous cathode.

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Acknowledgements This work was supported financially by The Programme Commission on Sustainable Energy and Environment, The Danish Council for Strategic Research, via the Strategic Electrochemistry Research Center (www.serc.dk), contract no. 2104-06-0011, and by Energinet.dk under the project PSO 2008-1-0065 SOFC R&D II. References [1] S.B. Adler, J.A. Lane, B.C. Steele, J. Electrochem. Soc. 143 (111) (1996) 3554. [2] H.J.M. Bouwmeester, H. Kruidhof, A.J. Burggraaf, Solid State Ionics 75 (1994) 185. [3] S.B. Adler, Solid State Ionics 111 (1998) 15. [4] P. Hjalmarson, M. Søgaard, M. Mogensen, Solid State Ionics 180 (2009) 1290. [5] Y. Kim, S. Pyun, J. Kim, G. Lee, J. Electrochem. Soc. 154 (8) (2007) B802. [6] P. Hjalmarson, M. Søgaard, M. Mogensen, Solid State Ionics 179 (2008) 1422. [7] E.P. Murray, M.J. Sever, S.A. Barnett, Solid State Ionics 148 (2002) 27. [8] W.G. Wang, M. Mogensen, Solid State Ionics 176 (2005) 457. [9] N. Grunbaum, L. Dessemond, J. Fouletier, F. Prado, A. Caneiro, Solid State Ionics 177 (2006) 907. [10] J.B. Jorcin, M.E. Orazem, N. Pebere, B. Tribollet, Electrohim. Acta 51 (2006) 1473. [11] E. Barsoukov, J.R. Macdonald, Impedance Spectroscopy Theory, Experiment, and Applications, John Wiley & Sons, 2005. [12] B.A. Boukamp, H.J.M. Bouwmeester, Solid State Ionics 157 (2003) 29. [13] W. Weppner, R.A. Huggins, J. Electrochem. Soc 124 (1977) 1569. [14] T. Jacobsen, K. West, Electrochim. Acta 40 (1995) 255. [15] H. Gerischer, Z. Phys. Chem. 198 (1951) 286. [16] B.A. Boukamp, M. Verbraeken, D.H.A. Blank, P. Holtappels, Solid State Ionics 177 (2006) 2539.

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