The Theory of Non‐Cottrellian Diffusion on the Surface of a Sphere or Truncated Sphere

Abstract

A method is developed whereby spherical and other particles can be derivatised with electroactive species on their surface and then immobilised on the surface of an electrode. The chronoamperometric and voltammetric responses in the limit of reversible electrode kinetics are modelled using a theory of charge movement over the surface of the spheres where this movement is considered as a diffusional process. The model is extended to include different distributions of sphere radii and to model the scenario of truncated spheres resting on the electrode surface. It is found that a good estimation of the truncation angle can be found by fitting the experimental data with theoretical predictions.

Solid‐State Electrochemical Reactions of Electroactive Microparticles and Nanoparticles in a Liquid Electrolyte Environment

“…For many other cases of functionalized carbon nanotubes (or modified graphite electrodes), Compton et al also identified a square root of scan rate dependency of peak currents [31,32]. These findings have been explained on a theoretical basis by the (non-Cottrellian) propagation of charge on the carbon particle surface [33,34], whereby electrons are assumed to hop from one chemical moiety to another, all being attached to the particle surface, as depicted in Figure 6.7.…”

“…The charge movement can be treated diffusion-like as electron hopping with coupled solution phase ion motion, contributing to the effective diffusion coefficient [34] (as in the studies cited for the general treatment of a three-phase electrode in Section 6.3.1). Thompson and Compton investigated, from a theoretical standpoint, the case of a spherical microparticle with an electroactive compound on its surface and attached to a solid electrode surface [33,34].…”

“…Thompson and Compton investigated, from a theoretical standpoint, the case of a spherical microparticle with an electroactive compound on its surface and attached to a solid electrode surface [33,34]. The movement of charge was assumed to start exclusively from the contact point (or line) between the microparticle and electrode (i.e., at the three-phase boundary, if an electrolyte phase is considered) and to proceed over the particle surface only (see also Section 6.3.1).…”

“…The hemispherical microparticle exhibits the most Cottrell-like response for the voltammetric (and chronoamperometric; see Ref. [34]) behavior. The peak-to-peak separation tends towards zero for the slowest scan rates, as the immobilized microparticle acts like an adsorbed redox species in a thin layer-like diffusion regime.…”

“…34 Pt particles on a surface can form hemispherical or raft-like particles. (a) The two geometrical models, with r representing the radius of contact of the particle and h its height; (b) Height versus radius graph of the Pt particles obtained from the analysis of sections across 25 particles.…”