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Analytical solutions for impedance of the cathode catalyst layer in PEM fuel cell: Layer parameters from impedance spectrum without fitting
Journal of Electroanalytical Chemistry 691 (2013) 13–17
Contents lists available at SciVerse ScienceDirect
Journal of Electroanalytical Chemistry journal homepage: www.elsevier.com/locate/jelechem
Short Communication
Analytical solutions for impedance of the cathode catalyst layer in PEM fuel cell: Layer parameters from impedance spectrum without fitting A.A. Kulikovsky a,⇑,1, M. Eikerling b,1 a b
Forschungszentrum Juelich GmbH, Institute of Energy and Climate Research, IEK-3: Electrochemical Process Engineering, D-52425 Jülich, Germany Department of Chemistry, Simon Fraser University, Burnaby, BC, Canada V5A 1S6
a r t i c l e
i n f o
Article history: Received 8 August 2012 Received in revised form 5 November 2012 Accepted 4 December 2012 Available online 23 December 2012 Keywords: PEM fuel cell Catalyst layer Impedance Modeling
a b s t r a c t A simple physical model for impedance of the cathode catalyst layer (CCL) in PEM fuel cell is solved analytically in the Tafel regime for small, but non-vanishing cell current densities (between 10 and 100 mA cm2). Analytical results make it possible to obtain the Tafel slope, the double layer capacitance and the CCL proton conductivity from a single impedance spectrum without fitting. The method is illustrated with the experimental spectrum taken from literature. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Electrochemical impedance spectroscopy (EIS) is a powerful diagnostic tool for studying fuel cells. Applying harmonic voltage perturbation to the cell and measuring the response current yield invaluable information on the cell transport and kinetic properties [12]. One of the key components of PEM fuel cell is cathode catalyst layer (CCL). Theory and practice of CCL impedance spectroscopy has been a subject of numerous studies. Following a classic work of de Levie [10], Lasia [8,9] obtained fundamental solutions for the impedance of the cylindrical porous electrode. Eikerling and Kornyshev published numerical and analytical study of the CCL impedance based on the macrohomogeneous model for electrode performance [1,2]. Recently, Makharia et al. used similar model for processing the experimental spectra [11]. Jaouen and Lingbergh derived equations for the CCL impedance based on a flooded agglomerate model [4]. An overview of CCL impedance studies has been published by Gomadam and Weldner [3]. In this Letter, we report a refined physical model of the CCL impedance, which leads to simple relations between the characteristic points on the spectrum and the CCL parameters (Tafel slope, double layer capacitance and ionic conductivity). We show that
⇑ Corresponding author. Address: Moscow State University, Research Computing Center, 119991 Moscow, Russia. E-mail addresses: [email protected] (A.A. Kulikovsky), [email protected] (M. Eikerling). 1 ISE member. 1572-6657/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jelechem.2012.12.002
these three fundamental parameters can easily be determined from a single impedance spectrum without fitting, provided that the spectrum is measured at a relatively small cell current. We specify the range of fuel cell current densities for which the relations obtained are valid. 2. Model 2.1. Performance model We consider the regime of low current densities of CCL operation; in this regime, oxygen transport loss can be ignored and the oxygen concentration in the CCL can be taken constant. In that case, non-stationary CCL performance can be described by a system of two equations [1,6]
g @ g @j c sinh ¼ 2i þ cref @t @x b @g j ¼ rt @x
C dl
ð1Þ ð2Þ
Here Cdl is the double layer volumetric capacitance (F cm3), g is the half-cell overpotential, t is time, j is the local proton current density, x is the distance from the membrane, i⁄ is the volumetric exchange current density (A cm3), c is the independent of x oxygen concentration in the CCL, cref is the reference oxygen concentration, b is the Tafel slope
b¼
RT aF
14
A.A. Kulikovsky, M. Eikerling / Journal of Electroanalytical Chemistry 691 (2013) 13–17
Table 1 The physical and dimensionless parameters. Tafel slope b, V Proton conductivity rt, X cm1 Exchange current density i⁄, A cm3 CL capacitance Cdl, F cm3 CL thickness lt, cm j⁄, A cm2 t⁄ , s
0.05 0.03 103 20 0.001 1.5 500 7.5 105
e2
a is the transfer coefficient, and rt is the CCL proton conductivity.2 Note that Eqs. (1) and (2) are written assuming by convention that g > 0. Eq. (1) describes the decay of proton flux entering the CCL at the membrane interface, towards the gas-diffusion layer (GDL) due to the charging of double layer (the term with Cdl) and due to the proton conversion in the ORR (the right side of this equation). Eq. (2) is the Ohm’s law relating proton current density to the gradient of overpotential. To simplify calculations we introduce dimensionless variables
~x ¼
x ; lt
~t ¼ t ; t
g j g~ ¼ ; ~j ¼ ; b
j
e ¼ Z rt ; Z lt
~c ¼
c cref
ð3Þ
Here lt is the CCL thickness and
t ¼
C dl b ; 2i
j ¼
rt b
ð4Þ
lt
~ ¼ xt is the real-valued dimensionless frequency of the where x perturbation, and the superscripts 0 and 1 mark the steady-state variables and perturbation amplitudes, respectively. ~ 1 j~x¼0 . The perturbation of the total voltage loss in the system is g By definition, the CCL impedance is
~ 1 g~ 1 e¼g Z ¼ 1 ~j1 ~ =@ ~x~x¼0 @g ~x¼0
ð10Þ
where ~j1 j0 is the perturbation of proton current at ~ x ¼ 0 (membrane interface) and we have used Ohm’s law. Solution to Eq. (9) is subject to the following boundary conditions
g~ 1 j~x¼1 ¼ g~ 11 ;
~1 @g ¼0 @ ~x ~x¼1
ð11Þ
The first of Eq. (11) establishes a small amplitude perturbation at ~x ¼ 1 (the CCL/GDL interface); the second means zero proton current at this interface. Note that the perturbation of overpotential can be applied on either side of the CCL; linearity of Eq. (9), guarantees that in both cases, the solution would be the same. The subscripts 0 and 1 mark the values at ~x ¼ 0 (membrane interface) and ~ x ¼ 1 (CCL/GDL interface), respectively. The only free parameter which appears in the boundary conditions is the amplitude of the e does not depend ~ 11 . It is easy to show that Z potential perturbation g on this parameter. 2.3. Steady-state overpotential and polarization curve
are the scaling parameters for time and current density, respectively. With these variables, the system (1) and (2) takes the form
Here, we will use the explicit solutions derived in [6] for the case of
~ @g @~j ~ ¼ ~c sinh g þ e2 ~ @ ~x @t ~ ~j ¼ @ g @ ~x
e2 ~j00
ð5Þ ð6Þ
where
e¼
sffiffiffiffiffiffiffiffiffiffi rt b
ð7Þ
2
2i lt
1
~ ~ @g @2g ~ e2 2 ¼ ~c sinh g ~ @ ~x @t
ð8Þ
2.2. Impedance equation ~ 0 ð~ xÞ be the steady-state solution to Eq. (8). Substituting Let g
~ ~tÞ; g ~1 1 g~ ¼ g~ 0 þ g~ 1 expðix ~0
into Eq. (8) and subtracting the steady-state equation for g , we get ~ ; ~xÞ in the ~ 1 ðx an equation for the complex perturbation amplitude g frequency domain
ð9Þ
2 Eq. (2) is an approximate form of the Butler–Volmer equation, valid (i) for small overpotentials (g b) if the transfer coefficient is 1/2, and (ii) for arbitrary transfer coefficients if overpotential is large (g b) ([5], pp. 9–13). The results of this work are obtained for the overpotential g J b, so that Eq. (1) can, in principle be reduced to the Tafel law, as in this range 2sinh (g/b) ’ exp (g/b). However, for generality, here we keep sinh-function, since the limits of validity of the model appear later as a result of calculations.
ð12Þ
where ~j00 ¼ ~j0 ð0Þ is the steady-state proton current density at the membrane/CCL interface (the cell current density). Estimate with the data from Table 1 shows, that in PEM fuel cells, Eq. (12) holds 0 for currents j0 > 1 mA cm2 , i.e., for all practically interesting cell current densities. Moreover, we will assume that Eq. (12) and the condition ~j00 1 are simultaneously fulfilled, yielding
1=e ~j00 1
is the dimensionless reaction penetration depth. The typical set of CCL physical parameters together with the resulting parameter e are listed in Table 1. ~: Substituting (6) into (5), we get the basic equation for g
~1 @2g ~ Þg ~1 e2 ~2 ¼ ð~c cosh g~ 0 þ ix @x
2
ð13Þ
or, in dimensional form,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2i rt b j0 rt b=lt
ð14Þ
With the typical values for PEMFC’s (Table 1), the left side of Eq. (14) is in the order of 103 A cm2, while the right side is about 1 A cm2. Thus, the model works for impedance curves measured for the cell currents between 10 and 100 mA cm2. For the currents obeying to the left side of Eq. (13), the steadystate distributions of local proton current density and overpotential are given by [6]
~j0 ¼ b tan b ð1 ~xÞ 2 ! b2 e2 b 0 2 ð1 ~xÞ g~ ¼ arcsinh ~ 1 þ tan 2 2c 2 e 2 ~0 2 ¼ arcsinh ðb þ ðj Þ Þ 2~c
ð15Þ
ð16Þ
where b is a solution to Eq. (17) resulting from setting ~ x ¼ 0 in Eq. (15)
~j0 ¼ b tan b 0 2
ð17Þ
~0 In the early work of one of us, the steady-state solution for g has been obtained in an implicit form containing the steady-state ~ 0 ð1Þ [2]. This complicates the analysis overpotential at ~ x ¼ 1; g
A.A. Kulikovsky, M. Eikerling / Journal of Electroanalytical Chemistry 691 (2013) 13–17
and leads to equations, which are more difficult for interpretation. Below we show, that the explicit solutions (15) and (16) lead to a simple analytical expression for the CCL impedance. For small cell currents, from Eq. (17) it follows that b is small and hence tan (b/2) ’ b/2. From Eq. (17) we get3
b¼
qffiffiffiffiffiffiffi 2~j00
ð18Þ
Setting in Eq. (16) ~ x ¼ 0 and using (18), we get
2 2 e g~ 00 ¼ arcsinh ~ 2~j00 þ ~j00 2c
2 Since ~j00 1, we can neglect here ~j00 and this equation takes the form
e2~j00
~ 00
g ¼ arcsinh
!
2e2~j00 ~c
! ð20Þ
which is a Tafel equation. Differentiating (20) over ~j00 , we get the CCL charge-transfer resistivity
e ct ¼ 1 R ~j0 0
ð21Þ
3. Analytical solution for impedance ~ 0 from Eq. (16), after simple manipulations, Eq. (9) takes With g the form
0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 1 ~ @ g e4 b4 Ag ~ þ ~c2 þ ~1 e2 ~2 ¼ @ix 4 @x 4 cos ðbð1 ~xÞ=2Þ
e
~1 @2g ¼ @ ~x2
e2 b2 2
e¼ Z
1 ; where u ¼ u tan u
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 ip 2
ð25Þ
where p is given below. Separating here the real and imaginary parts we obtain
1 c sinhðcÞ a sinðaÞ e re ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z coshðcÞ cosðaÞ b4 þ 4p2 1 a sinhðcÞ þ c sinðaÞ e im ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z coshðcÞ cosðaÞ b4 þ 4p2
ð26Þ
ð27Þ
rq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a¼ b4 þ 4p2 b2 rq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c¼
ð28Þ
b4 þ 4p2 þ b2
ð29Þ
2
ð30Þ
~ =e p¼x qffiffiffiffiffiffiffi b ¼ 2~j00
~ =e2 only; Note that in Eqs. (23)–(29), e appears in a combination x this is immediately seen if we divide Eq. (23) by e2. Thus, e simply re-scales the frequency and the shape of a Nyquist spectrum does not depend on e. Eqs. (26) and (27) determine the low-current impedance spectrum as a function of the dimensionless cell current density ~j00 . Examples of these spectra are shown in Fig. 1. As can be seen, the spectra are semicircles linked to the straight 45° high-frequency line on the left side. The first characteristic point of the spectrum is the right intercept with the real axis, which gives the differential CCL resistance e ccl ¼ Z e re j ~ . Setting in Eq. (26) x ~ ¼ 0 we get R x¼0
ð22Þ
As e 1 (Table 1) and using that e2~j20 1, we have 2 4 4 4 ~0 e b ¼ 4e j0 1 and hence we may neglect ~c2 under the square root in Eq. (22). Further, since b 1, we replace cos4 in Eq. (22) by unity and this equation simplifies to 2
Differentiating (24) over ~x and calculating the impedance (10), we find
where
ð19Þ
~c
2 Finally, as e~j00 1 and ~c < 1, the argument of arcsinh-function is large. Therefore, this function can be replaced by the logarithm of twice the value of the argument and for the CCL polarization curve we finally find
g~ 00 ¼ ln
15
(a)
! ~ g ~1 þ ix
ð23Þ
Physically, this approximation means that the steady-state overpo~ 0 is constant along ~x, even though spatial dependence must tential g ~1 . be accounted for in the perturbation g Solution to Eq. (23) is
0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 ~ b i x 1 1 g~ ¼ g~ 1 cos @ 2 ð1 ~xÞA 2 e
(b)
ð24Þ
3 The right side of inequality (13) can be mitigated. Indeed, expansion of tanfunction in Eq. (17) is justified if b p. From (18), for the upper limit of cell current we then get ~j0 p2 =2 and Eq. (13) takes the form
1=e ~j0 p2 =2 In dimension form, this extends the upper limit of cell current densities to ’500 mA cm2. However, at this current, the oxygen transport in the CCL can contribute to the impedance.
Fig. 1. (a) Analytical spectra for ~j00 ¼ 0:1 and 0.2 (Eqs. (26) and (27)). The points ~ max and x ~ ¼ 0 are indicated by small circles. The spectra in this corresponding to x Figure are indistinguishable with the exact numerical spectra obtained from solution of Eq. (22). (b) The high-frequency part of the spectra showing the link of e im =@ Z e 2 ¼ 0 is the semicircle with the straight HF-line. The point where @ 2 Z re indicated by arrow.
16
A.A. Kulikovsky, M. Eikerling / Journal of Electroanalytical Chemistry 691 (2013) 13–17
e ccl R
pffiffiffi pffiffiffi 2 sinh 2b pffiffiffi ¼ b cosh 2b 1
(a)
e ccl ¼ 1=3 þ 2=b2 . Since b is small, the last equation transforms to R Taking into account (18), we find
e ccl ¼ 1 þ 1 R 3 ~j00
ð31Þ
The last term in Eq. (31) coincides with (21) and it gives the ORR activation resistivity. The first term describes the contribution of the CCL ionic resistivity (see below). Another characteristic point of the spectrum is the maximal vae im (the top point of the semicircle, Fig. 1a). Parameter lue of Z ~ max =e2 , corresponding to this point is obtained from the pmax x e im =@ Z e re ¼ 0. This equation is equivalent to solution of equation @ Z
e im =@p @Z ¼ 0; or e re =@p @Z
e im @Z ¼0 @p
(b)
ð32Þ
Numerical solution of Eq. (32) shows that to a high accuracy, the following relation holds:
~ max ~0 x ¼ j0 e2
ð33Þ
e im g (Fig. 1a). ~ max is the frequency corresponding to maxf Z where x As the cell current density is small, we may neglect 1/3 in Eq. (31). In the dimension variables, Eqs. (31) and (33) take the form 0 0 Rccl ’ b=j0 and j0 ¼ xmax bC dl lt , respectively. Combining these equations we get 0
b ¼ Rccl j0 xmax Rccl C dl lt ¼ 1
ð34Þ ð35Þ
Eq. (35) means, that in the frequency domain around xmax, the CCL behaves as a parallel RC-circuit with the resistive element Rct and the capacitance Cdllt. Thus, measuring a single small-current spectrum, we immediately get the Tafel slope from Eq. (34), and the double layer capac~ =e2 does not contain the itance from Eq. (35). Note that the ratio x exchange current density, and hence this parameter cannot be determined from the spectrum. With Cdl in hand, we can determine the CCL proton conductivity rt from the slope of the high-frequency straight line Zre vs. x1/2 [1,7]. However, rt can also be determined directly from the Nyquist spectrum. The point of interest is the junction of the semicircle with the straight high-frequency line (Fig. 1b). At this point, the e im =@ Z e 2 ¼ 0 (Fig. 1b).4 Expressing this derivasecond derivative @ 2 Z re tive in terms of derivatives over p with Eq. (32), for the parameter phf corresponding to the junction we get an equation
! ! e re @ Z e im @ Z e im e re @2 Z @2 Z ¼0 2 2 @p @p @p @p
ð36Þ
Numerical solution to this equation shows that to a high accuracy, the following relation holds
phf ¼
7 ~0 1=4 j ; or ~j00 ¼ 2 0
~ hf 2x 7e2
4 ð37Þ
(Fig. 2a). Transforming Eq. (37) into dimension form and solving the resulting equation for rt, we get
4
e im =@ Z e 2 ¼ 0. This point corresponds to the minimal root of equation @ 2 Z re
Fig. 2. (a) Points – exact numerical solution to Eq. (36), line – fitting equation, Eq. (37). (b) Real part of the CCL impedance at the point corresponding to parameter ~ hf =e2 (the junction of the semicircle with the straight HF-line, see Fig. 1b). phf ¼ x
rt
!1=3 4=3 4 4 7 !1=3 7 fhf C dl lt b fhf4 C 4dl lt b 4p ¼ ’ 2:18 0 0 7 j0 j0
ð38Þ
where fhf = xhf/(2p) and xhf corresponds to the junction point phf. A simpler, though somewhat less accurate equation forffi rt is obqffiffiffiffiffiffi e re , tained, if we substitute the first of Eq. (37) and b ¼ 2~j00 into Z e re;hf ð~j0 Þ is depicted in Fig. 2b. As Eq. (26). The resulting function Z 0 can be seen, at the junction point, the real part of the CCL impedance is almost independent of ~j00 and approximately equals 1/3. e re;hf ¼ 1=3 we get the following dimension relation5: From Z
rt ¼
lt 3Z re;hf
ð39Þ
This equation has been used in [11] for the case of zero cell current. Eq. (38) gives us the last undefined parameter, the CCL proton conductivity rt. Thus, Eqs. (34) and (35) and one of Eqs. (38) and (39) allow us to characterize the CCL from a single low-current impedance curve without fitting. Unfortunately, parameter xmax is usually not reported in literature. An exclusion is [11]; Fig. 3 reprinted from this work shows that the impedance curve for the cell current density of 0 j0 ¼ 0:03 A cm2 (the largest semicircle) reaches maximum of Zim at the frequency fmax ’ 6 Hz. Further, for 0.03 A cm2, the CCL resistivity is Rccl ’ 1.65 X cm2 (Fig. 3). With these data, from Eq. (34) we get b ’ 0.0495 V, which is a typical value for the ORR catalysts. Taking into account that f = 2px and lt = 0.0013 cm, from Eq. (35) we get Cdl ’ 12.4 F cm3. This value is close to 15.4 F cm3 reported in [11] from a more complicated analysis. Fig. 10 in [11] (not shown) displays an enlarged junction of the semicircle and the straight HF-line; from this Figure it is seen, that Zre,hf ’ 0.04 X cm2. With lt = 0.0013 cm, from Eq. (39) we obtain rt ’ 0.011 X1 cm1, which agrees with the literature data. With the 5 e re;hf as ~j0 ! 0; this is also seen Calculation shows that 1/3 is the exact limit of Z 0 from Fig. 2b.
A.A. Kulikovsky, M. Eikerling / Journal of Electroanalytical Chemistry 691 (2013) 13–17
17
correspond to the model assumptions and they can be recommended for impedance studies of the CCL. 4. Conclusions We report a refined physical model for the cathode catalyst layer impedance in PEM fuel cell. The model assumes ideal oxygen transport and it takes into account voltage loss due to proton transport in the CCL. Analytical solutions for the impedance spectrum are derived. Based on these solutions, a simple relations between the characteristic points on the spectrum and fundamental CCL parameters (Tafel slope, double layer capacitance and CCL ionic conductivity) are obtained. The method is illustrated with the recently published experimental spectrum. References Fig. 3. Complex-plane impedance plots for H2/O2 operation at constant current density (A cm2) (Fig. 3 from [11]). Reprinted with permission of The Electrochemical Society.
parameters obtained from this analysis and the exchange current density from Table 1, it is easy to verify that the conditions (13) hold. Experiment [11] has been performed with pure oxygen under large stoichiometry of the cathode flow. This, together with the low cell current makes the resistivity due to oxygen transport in the channel and in the GDL vanishingly small. These conditions
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
M. Eikerling, A.A. Kornyshev, J. Electroanal. Chem. 453 (1998) 89–106. M. Eikerling, A.A. Kornyshev, J. Electroanal. Chem. 475 (1999) 107–123. P. Gomadam, J.W. Weidner, Int. J. Energy. Res. 29 (2005) 1133–1151. F. Jaouen, G. Lindbergh, J. Electrochem. Soc. 150 (2003) A1699–A1710. A.A. Kulikovsky, Analytical Modelling of Fuel Cells, Elsevier, Amsterdam, 2010. A.A. Kulikovsky, Electrochim. Acta 55 (2010) 6391–6401. A.A. Kulikovsky, J. Electroanal. Chem. 669 (2012) 28–34. A. Lasia, J. Electroanal. Chem. 397 (1995) 27–33. A. Lasia, J. Electroanal. Chem. 428 (1997) 155–164. R. de Levie, Electrochemical response of porous and rough electrodes, in: P. Delahay (Ed.), Advances in Electrochemistry and Electrochemical Engineering, vol. 6, Interscience, New-York, 1967, pp. 329–397. [11] R. Makharia, M.F. Mathias, D.R. Baker, J. Electrochem. Soc. 152 (2005) A970– A977. [12] M.E. Orazem, B. Tribollet, Electrochemical Impedance Spectroscopy, Wiley, New-York, 2008.
Contents lists available at SciVerse ScienceDirect
Journal of Electroanalytical Chemistry journal homepage: www.elsevier.com/locate/jelechem
Short Communication
Analytical solutions for impedance of the cathode catalyst layer in PEM fuel cell: Layer parameters from impedance spectrum without fitting A.A. Kulikovsky a,⇑,1, M. Eikerling b,1 a b
Forschungszentrum Juelich GmbH, Institute of Energy and Climate Research, IEK-3: Electrochemical Process Engineering, D-52425 Jülich, Germany Department of Chemistry, Simon Fraser University, Burnaby, BC, Canada V5A 1S6
a r t i c l e
i n f o
Article history: Received 8 August 2012 Received in revised form 5 November 2012 Accepted 4 December 2012 Available online 23 December 2012 Keywords: PEM fuel cell Catalyst layer Impedance Modeling
a b s t r a c t A simple physical model for impedance of the cathode catalyst layer (CCL) in PEM fuel cell is solved analytically in the Tafel regime for small, but non-vanishing cell current densities (between 10 and 100 mA cm2). Analytical results make it possible to obtain the Tafel slope, the double layer capacitance and the CCL proton conductivity from a single impedance spectrum without fitting. The method is illustrated with the experimental spectrum taken from literature. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Electrochemical impedance spectroscopy (EIS) is a powerful diagnostic tool for studying fuel cells. Applying harmonic voltage perturbation to the cell and measuring the response current yield invaluable information on the cell transport and kinetic properties [12]. One of the key components of PEM fuel cell is cathode catalyst layer (CCL). Theory and practice of CCL impedance spectroscopy has been a subject of numerous studies. Following a classic work of de Levie [10], Lasia [8,9] obtained fundamental solutions for the impedance of the cylindrical porous electrode. Eikerling and Kornyshev published numerical and analytical study of the CCL impedance based on the macrohomogeneous model for electrode performance [1,2]. Recently, Makharia et al. used similar model for processing the experimental spectra [11]. Jaouen and Lingbergh derived equations for the CCL impedance based on a flooded agglomerate model [4]. An overview of CCL impedance studies has been published by Gomadam and Weldner [3]. In this Letter, we report a refined physical model of the CCL impedance, which leads to simple relations between the characteristic points on the spectrum and the CCL parameters (Tafel slope, double layer capacitance and ionic conductivity). We show that
⇑ Corresponding author. Address: Moscow State University, Research Computing Center, 119991 Moscow, Russia. E-mail addresses: [email protected] (A.A. Kulikovsky), [email protected] (M. Eikerling). 1 ISE member. 1572-6657/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jelechem.2012.12.002
these three fundamental parameters can easily be determined from a single impedance spectrum without fitting, provided that the spectrum is measured at a relatively small cell current. We specify the range of fuel cell current densities for which the relations obtained are valid. 2. Model 2.1. Performance model We consider the regime of low current densities of CCL operation; in this regime, oxygen transport loss can be ignored and the oxygen concentration in the CCL can be taken constant. In that case, non-stationary CCL performance can be described by a system of two equations [1,6]
g @ g @j c sinh ¼ 2i þ cref @t @x b @g j ¼ rt @x
C dl
ð1Þ ð2Þ
Here Cdl is the double layer volumetric capacitance (F cm3), g is the half-cell overpotential, t is time, j is the local proton current density, x is the distance from the membrane, i⁄ is the volumetric exchange current density (A cm3), c is the independent of x oxygen concentration in the CCL, cref is the reference oxygen concentration, b is the Tafel slope
b¼
RT aF
14
A.A. Kulikovsky, M. Eikerling / Journal of Electroanalytical Chemistry 691 (2013) 13–17
Table 1 The physical and dimensionless parameters. Tafel slope b, V Proton conductivity rt, X cm1 Exchange current density i⁄, A cm3 CL capacitance Cdl, F cm3 CL thickness lt, cm j⁄, A cm2 t⁄ , s
0.05 0.03 103 20 0.001 1.5 500 7.5 105
e2
a is the transfer coefficient, and rt is the CCL proton conductivity.2 Note that Eqs. (1) and (2) are written assuming by convention that g > 0. Eq. (1) describes the decay of proton flux entering the CCL at the membrane interface, towards the gas-diffusion layer (GDL) due to the charging of double layer (the term with Cdl) and due to the proton conversion in the ORR (the right side of this equation). Eq. (2) is the Ohm’s law relating proton current density to the gradient of overpotential. To simplify calculations we introduce dimensionless variables
~x ¼
x ; lt
~t ¼ t ; t
g j g~ ¼ ; ~j ¼ ; b
j
e ¼ Z rt ; Z lt
~c ¼
c cref
ð3Þ
Here lt is the CCL thickness and
t ¼
C dl b ; 2i
j ¼
rt b
ð4Þ
lt
~ ¼ xt is the real-valued dimensionless frequency of the where x perturbation, and the superscripts 0 and 1 mark the steady-state variables and perturbation amplitudes, respectively. ~ 1 j~x¼0 . The perturbation of the total voltage loss in the system is g By definition, the CCL impedance is
~ 1 g~ 1 e¼g Z ¼ 1 ~j1 ~ =@ ~x~x¼0 @g ~x¼0
ð10Þ
where ~j1 j0 is the perturbation of proton current at ~ x ¼ 0 (membrane interface) and we have used Ohm’s law. Solution to Eq. (9) is subject to the following boundary conditions
g~ 1 j~x¼1 ¼ g~ 11 ;
~1 @g ¼0 @ ~x ~x¼1
ð11Þ
The first of Eq. (11) establishes a small amplitude perturbation at ~x ¼ 1 (the CCL/GDL interface); the second means zero proton current at this interface. Note that the perturbation of overpotential can be applied on either side of the CCL; linearity of Eq. (9), guarantees that in both cases, the solution would be the same. The subscripts 0 and 1 mark the values at ~x ¼ 0 (membrane interface) and ~ x ¼ 1 (CCL/GDL interface), respectively. The only free parameter which appears in the boundary conditions is the amplitude of the e does not depend ~ 11 . It is easy to show that Z potential perturbation g on this parameter. 2.3. Steady-state overpotential and polarization curve
are the scaling parameters for time and current density, respectively. With these variables, the system (1) and (2) takes the form
Here, we will use the explicit solutions derived in [6] for the case of
~ @g @~j ~ ¼ ~c sinh g þ e2 ~ @ ~x @t ~ ~j ¼ @ g @ ~x
e2 ~j00
ð5Þ ð6Þ
where
e¼
sffiffiffiffiffiffiffiffiffiffi rt b
ð7Þ
2
2i lt
1
~ ~ @g @2g ~ e2 2 ¼ ~c sinh g ~ @ ~x @t
ð8Þ
2.2. Impedance equation ~ 0 ð~ xÞ be the steady-state solution to Eq. (8). Substituting Let g
~ ~tÞ; g ~1 1 g~ ¼ g~ 0 þ g~ 1 expðix ~0
into Eq. (8) and subtracting the steady-state equation for g , we get ~ ; ~xÞ in the ~ 1 ðx an equation for the complex perturbation amplitude g frequency domain
ð9Þ
2 Eq. (2) is an approximate form of the Butler–Volmer equation, valid (i) for small overpotentials (g b) if the transfer coefficient is 1/2, and (ii) for arbitrary transfer coefficients if overpotential is large (g b) ([5], pp. 9–13). The results of this work are obtained for the overpotential g J b, so that Eq. (1) can, in principle be reduced to the Tafel law, as in this range 2sinh (g/b) ’ exp (g/b). However, for generality, here we keep sinh-function, since the limits of validity of the model appear later as a result of calculations.
ð12Þ
where ~j00 ¼ ~j0 ð0Þ is the steady-state proton current density at the membrane/CCL interface (the cell current density). Estimate with the data from Table 1 shows, that in PEM fuel cells, Eq. (12) holds 0 for currents j0 > 1 mA cm2 , i.e., for all practically interesting cell current densities. Moreover, we will assume that Eq. (12) and the condition ~j00 1 are simultaneously fulfilled, yielding
1=e ~j00 1
is the dimensionless reaction penetration depth. The typical set of CCL physical parameters together with the resulting parameter e are listed in Table 1. ~: Substituting (6) into (5), we get the basic equation for g
~1 @2g ~ Þg ~1 e2 ~2 ¼ ð~c cosh g~ 0 þ ix @x
2
ð13Þ
or, in dimensional form,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2i rt b j0 rt b=lt
ð14Þ
With the typical values for PEMFC’s (Table 1), the left side of Eq. (14) is in the order of 103 A cm2, while the right side is about 1 A cm2. Thus, the model works for impedance curves measured for the cell currents between 10 and 100 mA cm2. For the currents obeying to the left side of Eq. (13), the steadystate distributions of local proton current density and overpotential are given by [6]
~j0 ¼ b tan b ð1 ~xÞ 2 ! b2 e2 b 0 2 ð1 ~xÞ g~ ¼ arcsinh ~ 1 þ tan 2 2c 2 e 2 ~0 2 ¼ arcsinh ðb þ ðj Þ Þ 2~c
ð15Þ
ð16Þ
where b is a solution to Eq. (17) resulting from setting ~ x ¼ 0 in Eq. (15)
~j0 ¼ b tan b 0 2
ð17Þ
~0 In the early work of one of us, the steady-state solution for g has been obtained in an implicit form containing the steady-state ~ 0 ð1Þ [2]. This complicates the analysis overpotential at ~ x ¼ 1; g
A.A. Kulikovsky, M. Eikerling / Journal of Electroanalytical Chemistry 691 (2013) 13–17
and leads to equations, which are more difficult for interpretation. Below we show, that the explicit solutions (15) and (16) lead to a simple analytical expression for the CCL impedance. For small cell currents, from Eq. (17) it follows that b is small and hence tan (b/2) ’ b/2. From Eq. (17) we get3
b¼
qffiffiffiffiffiffiffi 2~j00
ð18Þ
Setting in Eq. (16) ~ x ¼ 0 and using (18), we get
2 2 e g~ 00 ¼ arcsinh ~ 2~j00 þ ~j00 2c
2 Since ~j00 1, we can neglect here ~j00 and this equation takes the form
e2~j00
~ 00
g ¼ arcsinh
!
2e2~j00 ~c
! ð20Þ
which is a Tafel equation. Differentiating (20) over ~j00 , we get the CCL charge-transfer resistivity
e ct ¼ 1 R ~j0 0
ð21Þ
3. Analytical solution for impedance ~ 0 from Eq. (16), after simple manipulations, Eq. (9) takes With g the form
0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 1 ~ @ g e4 b4 Ag ~ þ ~c2 þ ~1 e2 ~2 ¼ @ix 4 @x 4 cos ðbð1 ~xÞ=2Þ
e
~1 @2g ¼ @ ~x2
e2 b2 2
e¼ Z
1 ; where u ¼ u tan u
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 ip 2
ð25Þ
where p is given below. Separating here the real and imaginary parts we obtain
1 c sinhðcÞ a sinðaÞ e re ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z coshðcÞ cosðaÞ b4 þ 4p2 1 a sinhðcÞ þ c sinðaÞ e im ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z coshðcÞ cosðaÞ b4 þ 4p2
ð26Þ
ð27Þ
rq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a¼ b4 þ 4p2 b2 rq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c¼
ð28Þ
b4 þ 4p2 þ b2
ð29Þ
2
ð30Þ
~ =e p¼x qffiffiffiffiffiffiffi b ¼ 2~j00
~ =e2 only; Note that in Eqs. (23)–(29), e appears in a combination x this is immediately seen if we divide Eq. (23) by e2. Thus, e simply re-scales the frequency and the shape of a Nyquist spectrum does not depend on e. Eqs. (26) and (27) determine the low-current impedance spectrum as a function of the dimensionless cell current density ~j00 . Examples of these spectra are shown in Fig. 1. As can be seen, the spectra are semicircles linked to the straight 45° high-frequency line on the left side. The first characteristic point of the spectrum is the right intercept with the real axis, which gives the differential CCL resistance e ccl ¼ Z e re j ~ . Setting in Eq. (26) x ~ ¼ 0 we get R x¼0
ð22Þ
As e 1 (Table 1) and using that e2~j20 1, we have 2 4 4 4 ~0 e b ¼ 4e j0 1 and hence we may neglect ~c2 under the square root in Eq. (22). Further, since b 1, we replace cos4 in Eq. (22) by unity and this equation simplifies to 2
Differentiating (24) over ~x and calculating the impedance (10), we find
where
ð19Þ
~c
2 Finally, as e~j00 1 and ~c < 1, the argument of arcsinh-function is large. Therefore, this function can be replaced by the logarithm of twice the value of the argument and for the CCL polarization curve we finally find
g~ 00 ¼ ln
15
(a)
! ~ g ~1 þ ix
ð23Þ
Physically, this approximation means that the steady-state overpo~ 0 is constant along ~x, even though spatial dependence must tential g ~1 . be accounted for in the perturbation g Solution to Eq. (23) is
0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 ~ b i x 1 1 g~ ¼ g~ 1 cos @ 2 ð1 ~xÞA 2 e
(b)
ð24Þ
3 The right side of inequality (13) can be mitigated. Indeed, expansion of tanfunction in Eq. (17) is justified if b p. From (18), for the upper limit of cell current we then get ~j0 p2 =2 and Eq. (13) takes the form
1=e ~j0 p2 =2 In dimension form, this extends the upper limit of cell current densities to ’500 mA cm2. However, at this current, the oxygen transport in the CCL can contribute to the impedance.
Fig. 1. (a) Analytical spectra for ~j00 ¼ 0:1 and 0.2 (Eqs. (26) and (27)). The points ~ max and x ~ ¼ 0 are indicated by small circles. The spectra in this corresponding to x Figure are indistinguishable with the exact numerical spectra obtained from solution of Eq. (22). (b) The high-frequency part of the spectra showing the link of e im =@ Z e 2 ¼ 0 is the semicircle with the straight HF-line. The point where @ 2 Z re indicated by arrow.
16
A.A. Kulikovsky, M. Eikerling / Journal of Electroanalytical Chemistry 691 (2013) 13–17
e ccl R
pffiffiffi pffiffiffi 2 sinh 2b pffiffiffi ¼ b cosh 2b 1
(a)
e ccl ¼ 1=3 þ 2=b2 . Since b is small, the last equation transforms to R Taking into account (18), we find
e ccl ¼ 1 þ 1 R 3 ~j00
ð31Þ
The last term in Eq. (31) coincides with (21) and it gives the ORR activation resistivity. The first term describes the contribution of the CCL ionic resistivity (see below). Another characteristic point of the spectrum is the maximal vae im (the top point of the semicircle, Fig. 1a). Parameter lue of Z ~ max =e2 , corresponding to this point is obtained from the pmax x e im =@ Z e re ¼ 0. This equation is equivalent to solution of equation @ Z
e im =@p @Z ¼ 0; or e re =@p @Z
e im @Z ¼0 @p
(b)
ð32Þ
Numerical solution of Eq. (32) shows that to a high accuracy, the following relation holds:
~ max ~0 x ¼ j0 e2
ð33Þ
e im g (Fig. 1a). ~ max is the frequency corresponding to maxf Z where x As the cell current density is small, we may neglect 1/3 in Eq. (31). In the dimension variables, Eqs. (31) and (33) take the form 0 0 Rccl ’ b=j0 and j0 ¼ xmax bC dl lt , respectively. Combining these equations we get 0
b ¼ Rccl j0 xmax Rccl C dl lt ¼ 1
ð34Þ ð35Þ
Eq. (35) means, that in the frequency domain around xmax, the CCL behaves as a parallel RC-circuit with the resistive element Rct and the capacitance Cdllt. Thus, measuring a single small-current spectrum, we immediately get the Tafel slope from Eq. (34), and the double layer capac~ =e2 does not contain the itance from Eq. (35). Note that the ratio x exchange current density, and hence this parameter cannot be determined from the spectrum. With Cdl in hand, we can determine the CCL proton conductivity rt from the slope of the high-frequency straight line Zre vs. x1/2 [1,7]. However, rt can also be determined directly from the Nyquist spectrum. The point of interest is the junction of the semicircle with the straight high-frequency line (Fig. 1b). At this point, the e im =@ Z e 2 ¼ 0 (Fig. 1b).4 Expressing this derivasecond derivative @ 2 Z re tive in terms of derivatives over p with Eq. (32), for the parameter phf corresponding to the junction we get an equation
! ! e re @ Z e im @ Z e im e re @2 Z @2 Z ¼0 2 2 @p @p @p @p
ð36Þ
Numerical solution to this equation shows that to a high accuracy, the following relation holds
phf ¼
7 ~0 1=4 j ; or ~j00 ¼ 2 0
~ hf 2x 7e2
4 ð37Þ
(Fig. 2a). Transforming Eq. (37) into dimension form and solving the resulting equation for rt, we get
4
e im =@ Z e 2 ¼ 0. This point corresponds to the minimal root of equation @ 2 Z re
Fig. 2. (a) Points – exact numerical solution to Eq. (36), line – fitting equation, Eq. (37). (b) Real part of the CCL impedance at the point corresponding to parameter ~ hf =e2 (the junction of the semicircle with the straight HF-line, see Fig. 1b). phf ¼ x
rt
!1=3 4=3 4 4 7 !1=3 7 fhf C dl lt b fhf4 C 4dl lt b 4p ¼ ’ 2:18 0 0 7 j0 j0
ð38Þ
where fhf = xhf/(2p) and xhf corresponds to the junction point phf. A simpler, though somewhat less accurate equation forffi rt is obqffiffiffiffiffiffi e re , tained, if we substitute the first of Eq. (37) and b ¼ 2~j00 into Z e re;hf ð~j0 Þ is depicted in Fig. 2b. As Eq. (26). The resulting function Z 0 can be seen, at the junction point, the real part of the CCL impedance is almost independent of ~j00 and approximately equals 1/3. e re;hf ¼ 1=3 we get the following dimension relation5: From Z
rt ¼
lt 3Z re;hf
ð39Þ
This equation has been used in [11] for the case of zero cell current. Eq. (38) gives us the last undefined parameter, the CCL proton conductivity rt. Thus, Eqs. (34) and (35) and one of Eqs. (38) and (39) allow us to characterize the CCL from a single low-current impedance curve without fitting. Unfortunately, parameter xmax is usually not reported in literature. An exclusion is [11]; Fig. 3 reprinted from this work shows that the impedance curve for the cell current density of 0 j0 ¼ 0:03 A cm2 (the largest semicircle) reaches maximum of Zim at the frequency fmax ’ 6 Hz. Further, for 0.03 A cm2, the CCL resistivity is Rccl ’ 1.65 X cm2 (Fig. 3). With these data, from Eq. (34) we get b ’ 0.0495 V, which is a typical value for the ORR catalysts. Taking into account that f = 2px and lt = 0.0013 cm, from Eq. (35) we get Cdl ’ 12.4 F cm3. This value is close to 15.4 F cm3 reported in [11] from a more complicated analysis. Fig. 10 in [11] (not shown) displays an enlarged junction of the semicircle and the straight HF-line; from this Figure it is seen, that Zre,hf ’ 0.04 X cm2. With lt = 0.0013 cm, from Eq. (39) we obtain rt ’ 0.011 X1 cm1, which agrees with the literature data. With the 5 e re;hf as ~j0 ! 0; this is also seen Calculation shows that 1/3 is the exact limit of Z 0 from Fig. 2b.
A.A. Kulikovsky, M. Eikerling / Journal of Electroanalytical Chemistry 691 (2013) 13–17
17
correspond to the model assumptions and they can be recommended for impedance studies of the CCL. 4. Conclusions We report a refined physical model for the cathode catalyst layer impedance in PEM fuel cell. The model assumes ideal oxygen transport and it takes into account voltage loss due to proton transport in the CCL. Analytical solutions for the impedance spectrum are derived. Based on these solutions, a simple relations between the characteristic points on the spectrum and fundamental CCL parameters (Tafel slope, double layer capacitance and CCL ionic conductivity) are obtained. The method is illustrated with the recently published experimental spectrum. References Fig. 3. Complex-plane impedance plots for H2/O2 operation at constant current density (A cm2) (Fig. 3 from [11]). Reprinted with permission of The Electrochemical Society.
parameters obtained from this analysis and the exchange current density from Table 1, it is easy to verify that the conditions (13) hold. Experiment [11] has been performed with pure oxygen under large stoichiometry of the cathode flow. This, together with the low cell current makes the resistivity due to oxygen transport in the channel and in the GDL vanishingly small. These conditions
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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