Breakthrough in Modeling of Electrodiffusion Processes: Continuation and Extensions of the Classical Work of Richard Buck
Authors: Filipek Robert , Danielewski Marek , Fausek Janusz , Jasielec Jerzy , Kucza Witold , Lewenstam Andrzej , Sokalski Tomasz , Szyszkiewicz Krzysztof 
Published: February 2020

The first really successful numerical modeling of electrodiffusion problems was presented in 1978 in the important paper by Timothy Brumleve and Richard Buck [1]. Novelty of their approach consisted of fully implicit method, non-uniform grids, and two different types of finite differences (one for concentrations and second for fluxes) allowed to “create an efficient computer algorithm which permits treatment of multi-ion systems, thick or thin cells (membranes), and interfacial kinetics” [1].
In spite of the rapid increase of computational power at the time this approach was not immediately widely recognized and followed by the electrochemical community. However, it has changed to some degree at the beginning of 21stcentury e.g. [2-5,7,8]. But still the majority of present numerical modeling of electrodiffusion processes, e.g. interpretations of ion-sensor response, determination of transient behavior of a system to electrical and chemical perturbations (electrochemical impedance spectra) focus on simplified models based on electroneutrality assumption, equivalent circuit models, equilibrium or steady-state, thus ignoring electrochemical migration and time-dependent effects, respectively [1]. These theoretical approaches, due to their idealizations, make theorizing on ion distributions and electrical potentials in space and time domains impossible.
For the above reasons, the approach of Brumleve and Buck based on Nernst-Planck and Poisson (NPP) equations is utilized here to model the transient behavior of various electrochemical processes [11]. Additionally, including several layers and reaction terms in mass balance equation allows extending the applications to such areas as: selectivity and the low detection limit with variability over time, influence of parameters, such as ion diffusivity, membrane thickness, permittivity, rate constants and primary to interfering ion concentration ratios on ion-sensor responses. Moreover solution of the NPP inverse problem allows searching for optimal sensor properties and measurement conditions. The conditions under which experimentally measured selectivity coefficients are true (unbiased) and detection limits are optimized are demonstrated, and practical conclusions relevant to clinical measurements and bioassays are derived [7].
Another important field of application includes modeling of durability and diagnosis of reinforced concrete [9]. Based on extended NPP model electrochemical impedance spectra (EIS) can be obtained and analyzed to predict steel and concrete corrosion and finally utilize to prevent material failure.
NPP model can be also the useful tool in description of biological systems [10]. The NPP model has the potential to reopen frontiers in the study of variety of the problems related to the electrochemistry of biological membranes. Challenges in modeling of ionic channels in biological systems are discussed.
References
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Acknowledgements
Financial support from INNOTECH project no. K1/IN1/25/153217/NCBR/12 is acknowledged.