Short communicationFrom the anomalous diffusion impedance to the closed-form, infinite-series and integral formulations of the voltammetric response of thin-film insertion materials under restricted diffusion conditions. A modelling contribution based on the anomalous mass transfer function
Introduction
In a recent work [1], we focused on the Faradaic response (Faradaic current versus controlled potential) of one-step reversible electrochemical reactions to the Linear potential Sweep Voltammetry (LSV) technique. We showed that the mass transfer function [[2], [3], [4]] of the electroactive-dissolved species is the key point for derivation of the infinite series solution of this problem under one-dimensional mass transport conditions [1]. The above series is an alternating infinite series, which is convergent or divergent depending on the domain explored for the electrode potential. Although summation of alternating infinite series is possible, even in their domain of divergence, by using appropriate nonlinear sequence transformations [[5], [6], [7]] together with the multi-precision computing mode of computer algebra systems, it is more convenient for computation purpose to transform the infinite series into at least one of the equivalent integral formulations derived in Ref. [1] using the summation formulae of Abel [8], Lindelöf [9] and Euler-Ramanujan [10]. The objective of this short communication is to apply the above approach to the voltammetric response of the direct (one-step) insertion reaction occurring in a thin-film planar electrode under both anomalous and spatially restricted diffusion conditions in the electrode material, within some simplification of the physical model considered.
Section snippets
Anomalous diffusion
The generic term anomalous diffusion covers an increasing variety of experimental scenarios for physical systems. As far as we are concerned with thin films of insertion materials here, deviation from ordinary diffusion conditions may be due to different physical processes, e.g. distribution of diffusion length [11,12], trapping of diffusing species on structural defects of the electrode material [[13], [14], [15], [16]], distribution of waiting times between successive jumps in a continuous
Anomalous mass transfer function
The key point for modelling the electrochemical response of a thin-film insertion material to a current/potential step, pulse or ramp is the derivation of the relevant mass transfer function for the inserted species. In a general way, the mass transfer function of electroactive-dissolved species depends on the electrode geometry, the mass transport process involved and the boundary condition for mass transport written away from the electroactive interface [2,3]. This function is a useful link
Asymptotic formulations at high dimensionless sweep rate
The limiting formulation of the Faradaic current derived at very large values of the dimensionless potential sweep rate σ, Eq. (13), pertains to semi-infinite diffusion conditions in the host material. Its dimensionless formulation, derived from Eq. (12), is given by the following equation involving the polylogarithm (or Jonquière's) function:where Lis(y) = ∑k=1∞yk/ks [36]. The asymptotic formulation of the voltammetric response follows directly from Eqs. (11)
Conclusion
Starting from the relevant expression of the mass transfer function for anomalous diffusion of inserted species in a thin-film planar host material under restricted diffusion conditions, and using the general calculation method proposed in our previous work [1], we have derived in this short communication the theoretical formulation of the voltammetric response of such an electrode to a linear potential ramp, under Langmuir isotherm conditions. The Faradaic current has been calculated as a
Author Contributors
Claude Montella: Conceptualization, Methodology, Formal analysis, Visualization, Writing original draft.
Declaration of competing interest
The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References (48)
A family of integral representations for the voltammetric responses of reversible electrochemical reactions, arising from Abel, Lindelöf and Euler-Ramanujan summations of the alternating series solution
J. Electroanal. Chem.
(2018)A unified approach to linear electrochemical systems. I. The formalism
J. Electroanal. Chem.
(1974)- et al.
Diffusion–convection impedance using an efficient analytical approximation of the mass transfer function for a rotating disk
J. Electroanal. Chem.
(2015) - et al.
From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions
Phys. Rep.
(2007) - et al.
Impedance investigation of the charge transport in film-modified electrodes
J. Electroanal. Chem.
(1991) - et al.
Influence of particle size distribution on insertion processes in composite electrodes. Potential step and EIS theory. Part I. Linear diffusion
J. Electroanal. Chem.
(2001) Analysis of the kinetics of ion intercalation. Ion trapping approach to solid-state relaxation processes
Electrochim. Acta
(2002)- et al.
Diffusion-trapping impedance under restricted linear diffusion conditions
J. Electroanal. Chem.
(2003) - et al.
The impedance related to the electrochemical hydrogen insertion into WO3 films – on the applicability of the diffusion-trapping model
Electrochem. Commun.
(2008) - et al.
Theory of the electrochemical impedance of anomalous diffusion
J. Electroanal. Chem.
(2001)