(Invited) Estimation of the Onset of Mass Transfer Limitations in Pulsed Electrochemical Processes
Authors: Skinn Brian 
Published: January 2022

The contributions of the physical phenomena governing the distribution of current across an electrode in an electrochemical process are conventionally categorized as primary, secondary, and/or tertiary current distribution effects, which respectively embody geometric/ohmic, kinetic polarization, and concentration polarization effects. On virtually all non-trivial workpieces of interest to industrial electrochemical practice, it is important to be able to control the areas affected by the process; viz., preferentially adding or removing material to some regions over others. Two of the most significant phenomena contributing to the tertiary current distribution in electrochemical processes are depletion (for electrodeposition) and saturation (for electrodissolution) of the active soluble metal species at the workpiece surface. Both of these phenomena lead to mass-transfer limitations: taking electrodissolution as an example, if material is being dissolved at a particular point on the electrode surface at a rate greater than diffusion can carry the products away from the surface, then mass-transfer limitations will result. The tertiary current distribution effects arising from these limitations will tend to disfavor further increases in the local electrodissolution current density at that point, thus shifting the current density distribution to other locations on the workpiece surface, to other reactions at the same location, or both. Thus, exerting control over these tertiary current distribution effects can be highly valuable for developing an efficient and accurate electrochemical process.
An interesting feature of these mass-transfer-limiting phenomena is that they are almost entirely inactive for a short time (generally < 1 s for processes of practical interest) after the electrical voltage is applied, even if the applied current density is sufficiently high that significant mass transfer limitations will result after this initial interval. Thus, it follows that pulsing the applied potential/current at sufficiently high frequencies has the potential to enable significant control of these tertiary current distribution effects, by allowing the physicochemical conditions contributing to mass-transfer limitations at the electrode surface to “relax” while the potential is turned off. This “relaxation” behavior is schematized in Figure 1 for a generic pulse-electrodissolution process under steady-periodic conditions, where the orange and blue traces represent the concentration profiles at the end of the on-time and off-time, respectively, under conditions where no mass-transfer limitations are active at any point in time. For the purposes of electrochemical process optimization, the ability to estimate the maximum concentration of dissolved species at the electrode surface for a given system and applied waveform would provide guidance as to whether and when a particular mode of mass-transfer limitation is likely to be active. In particular, evaluation of the “transition time,” the value of the waveform on-time above which mass-transfer limitations become appreciable, is of significant practical interest.
Methods for transition time estimation based on linearized approximation of the boundary-layer concentration dynamics under a number of simplifying assumptions are available in the literature; e.g., Ref. [1]. However, the transition times calculated using these methods were found to deviate from COMSOL Multiphysics® simulation results by anywhere between –80% to +2780%, depending on the form of the estimation used and the particular waveform under consideration. This talk summarizes a method developed to provide appreciably more accurate predictions of transition times, under a similar set of simplifying assumptions as in Ref. 1. Separate on-time and off-time analytical solutions of the time-dependent steady-periodic mass transport behavior in a one-dimensional boundary layer were developed via the ‘finite Fourier transform’ (FFT) technique [[2]] and used to generate transition time estimates. Optimal values of the FFT model parameters were separately identified for fifty-three pairs of two pulsed-waveform timing parameters, period and duty cycle, spanning substantially the entire parameter space of practical industrial relevance. When compared to COMSOL® simulation results, the deviation of the transition time predictions (equivalently, predictions of the maximum surface concentration, in the electrodissolution paradigm of the model) was within 9% for all of the examined sets of timing parameters, with most deviating less than 5%. This FFT method thus provides a highly accurate method for estimation of transition times, within the approximations made in constructing the model.

References

[[1]] Ibl, N. “Some Theoretical Aspects of Pulse Electrolysis.” Surface Technology
10: 81-104 (1980).
[[2]] Deen, W.M. “Analysis of Transport Phenomena,” 2nd ed., Ch. 5. New York: Oxford University Press, 2012.

Figure 1