Computing the inverse Butler–Volmer relationship

Abstract

A detailed discussion is presented, of the problem of computing the inverse Butler-Volmer (electrode potential vs. current) relationship efficiently and with a high accuracy. A number of limiting cases of this problem is identified, and approximations valid for these cases are deduced. A hybrid algorithm is constructed, that combines the various approximations and improves them by Newton iterations. The algorithm is implemented in C++, employing extended precision variables. It is found that the modulus of the relative error of the calculated electrode potential is predominantly close to ca. $10^{-18}$ and locally increases to ca. $10^{-15}$. However, there are also well defined and unavoidable (albeit rather unlikely from the physical point of view) special situations in which the relative error may approach 100%, independently of the numerical algorithm used. The C++ code elaborated is made available. The code can be used for simulations of controlled current experiments, and for the analysis of diverse experimental data in electroanalytical and electrochemical engineering studies. An example application of the code, to the calculation of chronopotentiograms for a quasi-reversible charge transfer at a planar electrode, is also presented.