Uniform theory for second-order methods

Both the double-layer charging and the faradaic charge transfer are non-linear processes, i.e. the charging current density, jc, and the faradaic 

Obviously, our first task is to find the relation between SF and Sc and the parameters kf, a, Cd, etc. defining the interfacial processes. 

As before, we start from the implicit relationship jF = f(F, cQ, cR) or preferably jF = f( , cQ, cR ) where = (nF/RT) (E — E°). It is convenient to introduce short-hand notations F, 0, R for the first-order partial derivatives of jF and FF, OF, RF, 00, OR, RR for the second-order partial derivatives. Definitions of these parameters are given in Table 4 together with the appropriate explicit expressions that are easily derived from eqn. (18). 

In order to relate AjF to Aip, AcQ and AcR, the Taylor expansion is written including first- and second-order terms. 

As the second-order terms all contain products of two small quantities (Atp, Ac0 or AcR) it is clear that it was justified to neglect them in the treatments of Sects. 2.1.3, 2.2.6, and 2.3 provided that the dictated A F or A p values were sufficiently small. Here, abandoning this condition, we must accept that both AjF and A p contain a significant contribution of a second-order nature. Therefore, we write 

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