Finite but Small Current An Analytical Solution

In this section, the superscript 0 at will be omitted, that is,yo = is the undisturbed cell current density. The results of this section are valid if the relations 1 and 

With the typical values for PEFCs (Table 5.7), the left-hand side of Equation 5.114 is in the order of 1 mA cm , while the right-hand side is about 1 A cm . Thus, for PEECs, the model of this section works for impedance curves measured under the cell currents between 10 and 100 mA cm .  

Approximate solutions to Equation 5.111 can be obtained using the following arguments(KulikovskyandEikerling, 2013). Ase 1 (Table 5.6), it can be assumed that ejo 1, therefore, unity may be neglected under the square root in Equation 

Small jo means that the parameter p (see Equation 4.126) is also small 

Differentiating Equation 5.116 over x and calculating the impedance (Equation 5.93), gives 

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The term Ici/crp in Equation 5.110 is the proton resistivity of the CCL. This will be discussed in the section Finite but Small Current An Analytical Solution. Interestingly, if e is small. Red does not contain this term. Physically, if the reaction penetration depth is small (hydrogen electrode), the reaction runs close to the membrane and, therefore, protons do not need to be transported deep into the catalyst layer. Thus, the contribution of the proton transport to the cell resistivity does not appear as a separate term. The proton transport represented by ap is included into the total CL resistivity bf 2hap). 

Inthecaseofe 1, the term 1/3 in Equation 5.110 can be neglected and we obtain Red — b/ 2ii,lcL)- As IcL is usually known, the right intercept of the semicircle with the real axis gives the ratio //. If one of the parameters / , / is known (see the section Finite but Small Current An Analytical Solution on how to measure b using EIS), the other can be directly calculated. Note that this result holds only for EIS evaluated close to the OCR, when currents in the system are vanishingly small. 

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