Origins of finite transmission lines for exact representations of transport by the Nernst-Planck equations for each charge carrier

The Barker/Brumleve/Buck (BBB) circuits and-sub-circuit units, are derived and illustrated for the generation of the conventional two-feature mass transport-controlled impedance. This finite 2-port, 4-terminal network for a single salt system, replaces the classical I-port, 2-terminal cable transmission line analog because the latter describes only the lower-frequency Warburg feature. Symmetric or asymmetric cell results depend only on the ion or electron terminals used. Three feature impedance plane plots for slow, potential-dependent ion and/or electron transfers with activation resistance and relaxed double layer capacitance, are generated and illustrated. Residual solution resistance and total cell geometric capacitance are included to generate the expected four-feature impedance plots for cells without special features such as ion pairing and adsorption-reaction processes. This paper presents new results for a circuit description of electron hopping in redox polymers with concomitant counter ion motion. The forms of single species flux equation: Barker, Nernst-Planck, and electron-hopping are compared. The latter two are converted into responses I or j vs phi functions that define "T" circuit elements. Two forms arise that define the implicit R(i) or R(e), and C(i) or C(e) per unit length (i = ion, e = electron), and the corresponding explicit R and C. The analysis uses as examples (I) a single dissolved, inert electrolyte salt M(+) X(-), and (2) a redox polymer electrolyte radical cation and anion, also M(+),X(-), undergoing second order electron hopping from M to M(+). The corresponding forms for the Rs and Cs are derived. A second case is illustrated to show how the d.c. bias changes the polymer cation/anion compositions (assuming Nernstian behavior), and so determines different R(e) and C(e) at each composition through the effective second order, concentration-dependent electron hopping. (C) 1999 Elsevier Science Ltd. All rights reserved.