Effectiveness factor of fast (Fe³⁺/Fe²⁺), moderate (Cl₂/Cl⁻) and slow (O₂/H₂O) redox couples using IrO₂-based electrodes of different loading

Abstract

The effectiveness factor, E _f, (fraction of the electrode surface that participates effectively in the investigated reaction) of fast (Fe³⁺/Fe²⁺), moderate (Cl₂/Cl⁻) and slow (O₂/H₂O) redox couples has been estimated using IrO₂-based electrodes with different loading. The method of choice was linear sweep voltammetry (measurement of the anodic peak current) for the Fe³⁺/Fe²⁺ redox couple and steady-state polarization (determination of the exchange current) for the O₂ and Cl₂ evolution reactions. The results have shown that the effectiveness factor depends strongly on the kinetics of the investigated redox reaction. For the Fe³⁺/Fe²⁺ redox couple, effectiveness factors close to zero (max 4%) have been obtained contrary to the O₂ evolution reaction where effectiveness factors close to 100% can be achieved, all being independent of IrO₂ loading. For the Cl₂ evolution reaction, intermediate values of the effectiveness factor have been found and they decrease strongly, from 100% down to about 60%, with increasing loading.

Appendix: Analytical solution of the effectiveness factor

Coeuret et al. [2] attributed the observed decrease of effectiveness with increasing electrode thickness to an overpotential distribution in the electrolyte within the coating. In order to determine the effectiveness of a DSA^® electrode, we have adopted this approach proposed originally for the design of a porous medium for the recovery of heavy metals. As illustrated in Fig. 6, we consider a differential element of thickness, Δx, and cross-section, A _g, and we suppose no concentration gradient in the pores.

Fig. 6

Schematic representation of the DSA^® electrode according to a spherical particle configuration (SPC) consisting of IrO₂ spherical particles with diameter, d _p, showing a differential element of thickness, Δx, and cross-section, A _g. L is the thickness of the coating and X is the dimensionless distance. (a) Ti substrate; (b) electrolyte

Considering that the electronically conductive phase of the coating is equipotential, one can obtain for the overpotential, η, as a function of the dimensionless distance, X, the following second order linear ordinary differential equation:

$$ \frac{{d^{2} \eta (X)}}{{dX^{2} }} - K^{2} \eta (X) = 0 $$

(12)

whose solution is written as

$$ \frac{\eta (X) - \eta (0)}{\eta (1) - \eta (0)} = \frac{\cosh (KX) - 1}{\cosh (K) - 1} $$

(13)

The dimensionless number K ² includes the morphological parameters like specific particle area, a _p (cm⁻¹), void fraction of the DSA^® coating, porosity, ε (–), thickness of the coating, L (cm), the apparent electrolyte conductivity (γ) and the kinetics of the investigated electrochemical reaction (j ₀). For spherical particles, the specific area, a _p, is equal to 6/d _p, where d _p is the particle diameter (cm). The apparent electrolyte conductivity within the DSA^® coating, γ, depends on the void fraction, ε, as follows [2]:

$$ \gamma = \gamma_{0} \left( {\frac{2\varepsilon }{3 - \varepsilon }} \right) $$

(14)

where γ is the apparent electrolyte conductivity inside the DSA^® coating (Ω⁻¹ cm⁻¹), and γ₀ is the bulk electrolyte conductivity (Ω⁻¹ cm⁻¹).

Using the low-field approximation of the Butler-Volmer relationship,

$$ j = j_{0} \frac{F}{RT} \cdot \eta $$

(15)

where j is the current density (A cm⁻²), j ₀ is the exchange current density (A cm⁻²), F is the Faraday constant 96485 (C mol⁻¹), R is the molar gas constant 8.31 (J mol⁻¹ K⁻¹), T is the temperature (K) and η is the overpotential (V), the parameter K is given by Eq. 16:

$$ K^{2} = \frac{1}{\gamma }L^{2} j_{ 0} \frac{F}{RT}(1 - \varepsilon )a_{\text{p}} $$

(16)

Finally, the effectiveness factor, E _f, of the DSA^® coating can be calculated from K with Eq. 17:

$$ E_{\text{f}} = \frac{\tanh (K)}{K} $$

(17)