4.3 Brief Discussion of Equivalent Circuits in PEM Fuel Cells
4.3.4 Conductive Polymers
184 X-Z. Yuan, C. Song, H. Wang and J. Zhang
A typical profiling of the catalyst layer is shown in Figure 4.36, which compares a Nafion®-impregnated cathode and a cathode without Nafion® impregnation. The former has a much higher ionic conductivity than the latter.
Figure 4.36. Normalized resistivity (open points) and conductivity (solid points) profiles for Pt/C cathodes with (triangles) and without (circles) Nafion® impregnation [8]. (Reproduced by permission of the authors and of ECS—The Electrochemical Society, from Lefebvre MC, Martin RB, Pickup PG. Characterization of ionic conductivity within proton exchange membrane fuel cell gas diffusion electrodes by impedance spectroscopy.)
conductive polymers are based on a transmission line model. There are two types of electric circuit for conductive polymers, depending on the negligibility of the charge-transfer resistance and the double-layer capacitance. If these two factors are negligible, the electric circuit can be constructed by a finite transmission lines as shown in Figure 4.37.
Figure 4.37. Equivalent circuit of conducting polymers [10]. (Albery WJ, Mount AR. Dual transmission line with charge-transfer resistance for conducting polymers. J Chem Soc Faraday Trans 1994;90:1115–9. Reproduced by permission of The Royal Society of Chemistry.)
For this electric circuit, the resistance can be expressed as [11]
Z/R∑=ρ+2ρf1+(1−2ρ)f2 (4.111)
where Z is the impedance, R∑ =R1+R2, R1 is the electric resistance, R2 is the ionic resistance, ρ=R1R2/R∑2:
f1=sinhθcosθ−coshθsinθ−i[sinhθcosθ+coshθsinθ]
2θ[sinh2θcos2θ+cosh2θsin2θ] (4.112) and
f2=sinhθcoshθ−cosθsinθ−i[sinhθcoshθ+cosθsinθ]
2θ[sinh2θcos2θ+cosh2θsin2θ] (4.113) where
θ=(ωR∑C∑/ 2)1/ 2 (4.114)
and C∑ is the total capacitance of the polymer.
186 X-Z. Yuan, C. Song, H. Wang and J. Zhang
Several important conclusions can be obtained from these equations. At low frequencies, the impedance is expressed approximately as
Z=R∑/ 3−i/ωC∑ (4.115)
The impedance is dominated by the capacitance, and the real impedance is a constant, R∑/ 3.
At high frequencies
Z−1=R1−1+R2−1 (4.116)
which is consistent with what is clearly observable in the equivalent circuit. At high frequencies there is no charging and the capacitance is effectively like that of an open circuit; the transmission line is then only two resistances connected in parallel.
If charge-transfer resistance and double-layer capacitance are considered, there are two scenarios: (a) the RctCdl circuit is at the interface of the electrode and the polymer, or (b) the RctCdlcircuit is at the interface of the polymer and electrolyte solution, as shown in Figure 4.38 [12].
Figure 4.38. Equivalent circuits of conducting polymers with a Randles circuit [12].
(Reprinted from Journal of Electroanalytical Chemistry, 420, Ren X, Pickup PG. An impedance study of electron transport and electron transfer in composite polypyrrole plus polystyrenesulphonate films, 251–7, ©1997 with permission from Elsevier and from the authors.)
The high-to-medium impedance of both cases is given by the following equations [10, 11]:
Z=R1R2/R∑+R22/[R∑(sR∑C∑)]1/ 2+Z1 (4.117) where
1
Z1 =1+sCdl(Rct+[c/(a+c)]Z2)
Rct+Z2 (4.118)
and
Z2=R12/[R∑(sR∑C∑)1/ 2] (4.119) and where c is the concentration of redox sites in the polymer film, a is the concentration of oxidized redox sites, s=iω, and ω is the frequency.
When R2>>R1, the polymer is more conducting than the pores. The Randles circuit, which is located at the polymer/electrolyte interface (case b), shunts the resistive ionic rail through the polymer. At high frequencies, the equation can be simplified to
Z≈R1+(R2/C∑ω)1/ 2−i/Cdlω (4.120)
The real impedance is dominated by the transmission line, and the imaginary part is controlled by the double-layer capacitance. Both the double-layer and the transmission-line capacitances must be involved.
In the case R1>>R2, the equation can be simplified to
Z≈R2+1/[RctCdl2ω2]−i/Cdlω (4.121)
The Randles circuit is at the electrode/polymer interface (case a). The imaginary component of the impedance is dominated by the double-layer capacitance, and the real one is controlled by the double layer. The capacitance of the transmission line is shunted and the transmission line is not involved.
Conductive polymers have been investigated extensively using AC impedance.
A typical example, shown here, is from an AC impedance study of reduced polypyrrole polystyrene sulfonate (PPY-PSS), which exhibits different behaviours at different potential ranges [12, 13]. In the potential range 0.1 to –0.6V, the PPY- PSS film shows a simple transfer line, in which the electronic resistance is negligible [13]. Figure 4.39 shows the Nyquist plot of a PPY-PSS film coated on a Pt electrode in aqueous NaClO4 solution. The solution resistance, intercepted at high frequencies, is the same at all investigated potentials. The low-frequency limiting real impedance, Rlow, is the sum of Rs and Rion/3. The ionic resistance decreases with decreasing electrode potential.
188 X-Z. Yuan, C. Song, H. Wang and J. Zhang
Figure 4.39. Nyquist plot of a Pt and Pt/PPY|PPY/PSS bilayer electrode in 0.2 M NaClO4
solution. Potential: Pt: OCV; conducting polymer: ○, 0.1; □, –0.2; ∆, –0.3; ◊, –0.4, *, –0.5 V [13]. (Reprinted with permission from J Phys Chem 1993;97:3941–3. ©1993 American Chemical Society.)
In the potential range –0.6 to –0.69 V, the high-frequency intercept changes with potential, indicating that the electronic resistance becomes comparable with the ionic resistance [12]. The electronic and ionic resistances can be calculated by the following equations:
1/(Rhigh – Rs) = 1/Re + 1/Rion (4.122)
3(Rlow – Rs) = Rion + Re (4.123)
where Rhigh is the real impedance at a high-frequency intercept, Rs is the solution resistance, Re is the electronic resistance, and Rion is the ionic resistance.
Further decrease in electrode potential leads to the appearance of semicircles as shown in Figures 4.40 and 4.41, indicating that the interfacial charge transfer becomes slow. The plot of the imaginary impedance versus 1/ω is shown in Figure 4.42. A linear relationship is observable, and the capacitance of the double layer is obtained from the slope.
Figure 4.40. Nyquist plot of PPY-PSS film at different potentials [12]. (Reprinted from Journal of Electroanalytical Chemistry, 420, Ren X, Pickup PG. An impedance study of electron transport and electron transfer in composite polypyrrole plus polystyrenesulphonate films, 251–7, ©1997 with permission from Elsevier and from the authors.)
Figure 4.41. Impedance of PPY-PSS in the potential range –0.75 V to –0.84 V [12].
(Reprinted from Journal of Electroanalytical Chemistry, 420, Ren X, Pickup PG. An impedance study of electron transport and electron transfer in composite polypyrrole plus polystyrenesulphonate films, 251–7, ©1997 with permission from Elsevier and from the authors.)