Short communication
From 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

https://doi.org/10.1016/j.jelechem.2020.114835Get rights and content

Highlights

  • Anomalous diffusion in thin-film materials under restricted conditions.

  • Voltammetric response of the direct insertion reaction to a potential ramp.

  • Discrimination of anomalous diffusion models from the LSV behaviour.

  • Closed-form approximations of the voltammetric peak coordinates.

  • Zone diagram for the conditions of observation of the limiting behaviours.

Abstract

The voltammetric response of one-step reversible electrochemical reactions (involving dissolved redox species) to a linear potential ramp can be modelled using the calculation method proposed in our previous work (J. Electroanal. Chem. 818 (2018) 84) provided that the mass transfer function for the species involved in the electrochemical reaction is known in closed form under one-dimensional mass transport conditions. The above approach applies in this short communication to the direct insertion reaction in thin-film planar materials under both anomalous and spatially restricted (confined) diffusion conditions. The Faradaic current is obtained first as an alternating infinite series well suited to formal analysis, and then as an integral formulation well suited to numerical computation. Finally, closed-form expressions of the voltammograms are derived at very low or very high values of the potential sweep rate, respectively. Two anomalous diffusion models taken from the electrochemical literature are investigated and compared, based on their voltammetric responses. The voltammetric peak coordinates obtained from the best model are thoroughly examined, and closed-form approximations are proposed for the voltammetric peak current and potential at very low or very high potential sweep rates. A zone diagram is plotted to illustrate the conditions for observation of those limiting situations. Finally, following a referee's request, an alternative modelling approach is proposed in Appendix, which does not require the use of scientific computation software. This approach involves both the Integral Equation method formalism and numerical inversion of Laplace transform.

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:limσψ=n=11nnβexp=Liβexpξwhere Lis(y) = ∑k=1yk/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)

  • T. Pajkossy et al.

    Diffusion to fractal surfaces-II. Verification of theory

    Electrochim. Acta

    (1989)
  • T. Pajkossy et al.

    Diffusion to fractal surfaces-III. Linear sweep and cyclic voltammograms

    Electrochim. Acta

    (1989)
  • B.A. Boukamp

    Derivation of a distribution function of relaxation times for the (fractal) finite length Warburg

    Electrochim. Acta

    (2017)
  • S. Malmgren et al.

    Anomalous diffusion of ions in electrochromic tungsten oxide films

    Electrochim. Acta

    (2017)
  • C. Montella et al.

    New approach of electrochemical systems dynamics in the time domain under small-signal conditions: III – discrimination between nine candidate models for analysis of PITT experimental data from LixCoO2 film electrodes

    J. Electroanal. Chem.

    (2009)
  • A. Sharifi-Viand et al.

    Investigation of anomalous diffusion and multifractal dimensions in polypyrrole film

    J. Electroanal. Chem.

    (2012)
  • J. Mocák et al.

    Use of MATHEMATICA software for theoretical analysis of linear sweep voltammograms

    J. Electroanal. Chem.

    (2004)
  • C. Montella

    LSV modelling of electrochemical systems through numerical inversion of Laplace transforms. I- the GS-LSV algorithm

    J. Electroanal. Chem.

    (2008)
  • C. Montella

    Cinétique Formelle et Analyse Harmonique. Contribution à l’étude Formelle de Modèles de Systèmes Électrochimiques, Thèse d’Etat, Grenoble

    (1989)
  • G.H. Hardy

    Divergent Series

    (1949)
  • E.J. Weniger

    Nonlinear Sequence Transformations: Computational Tools for the Acceleration of Convergence and the Summation of Divergent Series, arXiv:math/0107080v1 [math.CA]

    (2001)
  • P.L. Butzer et al.

    The summation formulae of Euler–Maclaurin, Abel–Plana, Poisson, and their interconnections with the approximate sampling formula of signal analysis

    RM

    (2011)
  • G. Dahlquist

    On Summation Formulas due to Plana, Lindelöf and Abel, and related Gauss-Christoffel rules, II, BIT

    (1997)
  • B. Candelpergher

    Ramanujan summation of divergent series

  • Cited by (2)

    View full text