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4 EIS at carbon fiber cylindrical microelectrodes The dc diffusion field between two concentric cylinders has a logarithmic dependence on the cylinder radii, Eq. (1), as known for a long time, e.g., from the analogous current distribution problem [147]. π(π )βπ(π )=βππ½(π)lnππΏ (4.1) πΏ0 π· π0 Here π and π are the radii of the outer and inner cylinders, π· is the diffusivity, π½ (π) is the πΏ0 radial flux (mol mβ2 s β1), and the product ππ½ (π) is independent of π. The case of interest here isalargesolutionvolumeπ ββ.Inthatcase,thelogarithmicfactortendstoinfinityandso the system does not reach a true steady state. However, as in the 1-D semi-infinite case, we proceed to solve the ac problem as though there were a steady state, recognizing that the impedance will go to infinity at low frequencies. Fleischmann et al. [146] derived the impedance for a quasi-reversible redox couple at a cylindrical microelectrode. The solution was given as a complicated expression involving the Kelvin functions Kei and Ker. They stated that the real and imaginary parts increase to infinity because there is no steady state, but their complex-plane plot suggests that the imaginary part tends to a constant value at low frequencies. Later, Jacobsen and West [148] gave the solution for the dimensionless mass transportimpedanceintermsofBesselfunctionsasEq.(2),whereπΎ andπΎ arethemodified 01 πΏ Bessel functions of the second kind of order 0 and 1, and ππ is a diffusion time constant. π§ = πΜπ· = πΎ0(βππππ) βπππππΎ1(βππππ) ππ = 0 π· (4.2) (4.3) Μ π½π 0 They correctly pointed out that the imaginary part tends to a constant value of βπ/4 at zero frequency. (The similar case of insertion and diffusion into a cylindrical electrode was earlier solved in terms of Bessel functions by Barral et al. [149]). The mass transport impedance in usual units is given by ππ = Οβ²βπππ§, which replaces ππ = πβ²/βππ for the more usual 1-D semi-infinite case. 42 π2PDF Image | Electron Transfer Kinetics in Redox Flow Batteries
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