Kinetic calculations of the Ni anodic dissolution from EIS

Abstract

Anodic nickel dissolution in acid media has been analyzed by means of electrochemical quartz crystal microbalance and electrochemical impedance spectroscopy techniques. The experimental impedance spectra have been fitted to an equivalent circuit which is related to a mechanism of two consecutive electron transfers followed by a Ni²⁺ mass transfer step. That way, an estimation for values of rate constants and surface concentrations of the Ni(0), Ni(I) and Ni(II) species has been obtained.

Appendices

Appendix 1

A mathematical method, similar to that used by Diard et al. [40, 41] for the Volmer–Heyrovsky reduction mechanism, was used for obtaining the theoretical impedance function of the anodic dissolution of the metal. It can be considered that a metal dissolves by means of two consecutive electron transfers followed by a transport step, according to the reaction scheme:

$$ {\text{Me}}{\left( {\text{0}} \right)}\xrightarrow{{r_{1} }}{\text{Me}}{\left( {\text{I}} \right)} + {\text{e}}^{ - } $$

(A1)

$$ {\text{Me}}{\left( {\text{I}} \right)}\xrightarrow{{r_{2} }}{\text{Me}}{\left( {{\text{II}}} \right)} + {\text{e}}^{ - } $$

(A2)

$$ {\text{Me}}{\left( {{\text{II}}} \right)}\xrightarrow{{r_{3} }}{\text{Me}}^{{{\text{2 + }}}} $$

(A3)

where r₁, r₂ and r₃ are the reaction rates for each elementary step. Me(I) and Me(II) are the reaction intermediates.

The charge balance at the electrode surface defines the faradaic current:

$$ \frac{{i_{F} }} {{AF}} = r_{1} + r_{2} $$

(A4)

where A and F are the electrode surface and the Faraday constant, respectively. The mass balance at the electrode surface, for each reaction intermediate, is described by the equations:

$$ \frac{{\operatorname{d} \theta _{1} }} {{\operatorname{d} t}} = r_{1} - r_{2} $$

(A5)

$$ \frac{{\operatorname{d} \theta _{2} }} {{\operatorname{d} t}} = r_{2} - r_{3} $$

(A6)

$$ \theta _{0} + \theta _{1} + \theta _{2} = \theta ^{0} $$

(A7)

where θ₀, θ₁ and θ₂ are the surface concentrations of Me(0), Me(I) and Me(II) species at the electrode surface and θ⁰ is the initial concentration of free sites at the electrode surface. r₁, r₂, r₃, θ₀, θ₁ and θ₂ will be a function of potential E and θ _i . If each elementary step obeys kinetics of first order, it can be expressed as:

$$ r_{1} = k_{1} \theta _{0} $$

(A8)

$$ r_{2} = k_{2} \theta _{1} $$

(A9)

$$ r_{3} = k_{3} \theta _{2} $$

(A10)

If a small sinusoidal perturbation of potential is applied around a defined steady state at a potential E₀, we can write the Taylor expansion for Eqs. (A4), (A5), (A6) and (A7) around this steady state in the Laplace plane. If the amplitude of the potential perturbation is small, only the terms of first order need be considered:

$$ \frac{{\Delta i_{F} }} {{AF}} = {\left( {\frac{{\partial r_{1} }} {{\partial E}} + \frac{{\partial r_{2} }} {{\partial E}}} \right)}\Delta E - \frac{{\partial r_{1} }} {{\partial \theta _{0} }}\Delta \theta _{2} + {\left( {\frac{{\partial r_{2} }} {{\partial \theta _{1} }} - \frac{{\partial r_{1} }} {{\partial \theta _{0} }}} \right)}\Delta \theta _{1} $$

(A11)

$$ {\left( {\frac{{\partial r_{1} }} {{\partial \theta _{0} }} + \frac{{\partial r_{2} }} {{\partial \theta _{1} }} + p} \right)}\Delta \theta _{1} = {\left( {\frac{{\partial r_{1} }} {{\partial E}} - \frac{{\partial r_{2} }} {{\partial E}}} \right)}\Delta E - \frac{{\partial r_{1} }} {{\partial \theta _{0} }}\Delta \theta _{2} $$

(A12)

$$ {\left( {\frac{{\partial r_{3} }} {{\partial \theta _{2} }} + p} \right)}\Delta \theta _{1} = \frac{{\partial r_{2} }} {{\partial E}}\Delta E + \frac{{\partial r_{2} }} {{\partial \theta _{1} }}\Delta \theta _{1} $$

(A13)

where p=jω is the Laplace variable. Dividing each term of Eqs. (A11), (A12) and (A13) by Δi _F an equation system for $Z_{F} = \Delta E/\Delta i_{F}$, $\Delta \theta _{1} /\Delta i_{F}$ and $\Delta \theta _{2} /\Delta i_{F}$ is defined, where Z _F is the faradaic impedance which can be determined by means of the Kramer’s rule:

$$ FAZ_{F} = \frac{{{\left| {\begin{array}{*{20}c} {1} & {{\frac{{\partial r_{2} }} {{\partial \theta _{1} }} - \frac{{\partial r_{1} }} {{\partial \theta _{0} }}}} & {{ - \frac{{\partial r_{1} }} {{\partial \theta _{0} }}}} \\ {0} & {{ - {\left( {\frac{{\partial r_{1} }} {{\partial \theta _{0} }} + \frac{{\partial r_{2} }} {{\partial \theta _{1} }} + p} \right)}}} & {{ - \frac{{\partial r_{1} }} {{\partial \theta _{0} }}}} \\ {0} & {{\frac{{\partial r_{2} }} {{\partial \theta _{1} }}}} & {{ - {\left( {\frac{{\partial r_{3} }} {{\partial \theta _{2} }} + p} \right)}}} \\ \end{array} } \right|}}} {{{\left| {\begin{array}{*{20}c} {{\frac{{\partial r_{1} }} {{\partial E}} + \frac{{\partial r_{2} }} {{\partial E}}}} & {{\frac{{\partial r_{2} }} {{\partial \theta _{1} }} - \frac{{\partial r_{1} }} {{\partial \theta _{0} }}}} & {{ - \frac{{\partial r_{1} }} {{\partial \theta _{0} }}}} \\ {{\frac{{\partial r_{1} }} {{\partial E}} - \frac{{\partial r_{2} }} {{\partial E}}}} & {{ - {\left( {\frac{{\partial r_{1} }} {{\partial \theta _{0} }} + \frac{{\partial r_{2} }} {{\partial \theta _{1} }} + p} \right)}}} & {{ - \frac{{\partial r_{1} }} {{\partial \theta _{0} }}}} \\ {{\frac{{\partial r_{2} }} {{\partial E}}}} & {{\frac{{\partial r_{2} }} {{\partial \theta _{1} }}}} & {{ - {\left( {\frac{{\partial r_{3} }} {{\partial \theta _{2} }} + p} \right)}}} \\ \end{array} } \right|}}} $$

(A14)

Appendix 2

When chloride ion is present in the acid medium the experimental impedance spectra are fitted to the equivalent circuit of Fig. 4. The faradaic impedance function for this equivalent circuit is:

$$ Z_{F} = \frac{{R_{2} {\left( {R + R_{1} } \right)} + {\left( {{\left( {L_{2} {\left( {R + R_{1} + R_{2} } \right)} + RR_{1} R_{2} C_{1} } \right)}j\omega - {\left( {R_{1} C_{1} L_{2} (R + R_{2} } \right)}} \right)}\omega ^{2} }} {{R_{2} + {\left( {R_{1} C_{1} R_{2} + L_{2} } \right)}j\omega - {\left( {R_{1} C_{1} L_{2} } \right)}}} $$

(A15)

If Eq. (A15) is compared with Eq. (11), the following equation system can be written:

$$ \frac{{\frac{{\partial r_{1} }} {{\partial \theta _{0} }}\frac{{\partial r_{2} }} {{\partial \theta _{1} }} + {\left( {\frac{{\partial r_{1} }} {{\partial \theta _{0} }} + \frac{{\partial r_{2} }} {{\partial \theta _{1} }}} \right)}\frac{{\partial r_{3} }} {{\partial \theta _{2} }}}} {{2\frac{{\partial r_{2} }} {{\partial E}}\frac{{\partial r_{1} }} {{\partial \theta _{0} }}\frac{{\partial r_{3} }} {{\partial \theta _{2} }} + 2\frac{{\partial r_{1} }} {{\partial E}}\frac{{\partial r_{2} }} {{\partial \theta _{1} }}\frac{{\partial r_{3} }} {{\partial \theta _{2} }}}} = F{\left( {R + R_{1} } \right)} $$

(A16)

$$ \frac{{\frac{{\partial r_{1} }} {{\partial \theta _{0} }} + \frac{{\partial r_{2} }} {{\partial \theta _{1} }} + \frac{{\partial r_{3} }} {{\partial \theta 2}}}} {{2\frac{{\partial r_{2} }} {{\partial E}}\frac{{\partial r_{1} }} {{\partial \theta _{0} }}\frac{{\partial r_{3} }} {{\partial \theta _{2} }} + 2\frac{{\partial r_{1} }} {{\partial E}}\frac{{\partial r_{2} }} {{\partial \theta _{1} }}\frac{{\partial r_{3} }} {{\partial \theta _{2} }}}} = F\frac{{{\left( {L_{2} {\left( {R + R_{1} + R_{2} } \right)} + RR_{1} R_{2} C_{1} } \right)}}} {{R_{2} }} $$

(A17)

$$ \frac{{\frac{{\partial r_{1} }} {{\partial \theta _{0} }}\frac{{\partial r_{2} }} {{\partial E}} + 2\frac{{\partial r_{1} }} {{\partial E}}\frac{{\partial r_{2} }} {{\partial \theta _{1} }} + \frac{{\partial r_{1} }} {{\partial E}}\frac{{\partial r_{3} }} {{\partial \theta _{2} }} + \frac{{\partial r_{2} }} {{\partial E}}\frac{{\partial r_{3} }} {{\partial \theta _{2} }}}} {{2\frac{{\partial r_{2} }} {{\partial E}}\frac{{\partial r_{1} }} {{\partial \theta _{0} }}\frac{{\partial r_{3} }} {{\partial \theta _{2} }} + 2\frac{{\partial r_{1} }} {{\partial E}}\frac{{\partial r_{2} }} {{\partial \theta _{1} }}\frac{{\partial r_{3} }} {{\partial \theta _{2} }}}} = \frac{{R_{2} R_{1} C_{1} + L_{2} }} {{R_{2} }} $$

(A18)

$$ \frac{{\frac{{\partial r_{1} }} {{\partial E}} + \frac{{\partial r_{2} }} {{\partial E}}}} {{2\frac{{\partial r_{2} }} {{\partial E}}\frac{{\partial r_{1} }} {{\partial \theta _{0} }}\frac{{\partial r_{3} }} {{\partial \theta _{2} }} + 2\frac{{\partial r_{1} }} {{\partial E}}\frac{{\partial r_{2} }} {{\partial \theta _{1} }}\frac{{\partial r_{3} }} {{\partial \theta _{2} }}}} = \frac{{R_{1} C_{1} L_{2} }} {{R_{2} }} $$

(A19)

$$ \frac{1} {{2\frac{{\partial r_{2} }} {{\partial E}}\frac{{\partial r_{1} }} {{\partial \theta _{0} }}\frac{{\partial r_{3} }} {{\partial \theta _{2} }} + 2\frac{{\partial r_{1} }} {{\partial E}}\frac{{\partial r_{2} }} {{\partial \theta _{1} }}\frac{{\partial r_{3} }} {{\partial \theta _{2} }}}} = F\frac{{R_{1} L_{2} C_{1} {\left( {R + R_{2} } \right)}}} {{R_{2} }} $$

(A20)

which can be simplified as in the previous case (two capacitive loops). If R and C₁ are very low and if it is considered that k₁≫k₂,k₃, the following equation system is obtained:

$$ \frac{{k_{2} + k_{3} }} {{2k_{1} k_{3} b_{2} \theta _{0} }} = FR_{1} $$

(A21)

$$ \frac{1} {{2k_{1} k_{3} b_{2} \theta _{0} }} = F\frac{{{\left( {R_{1} + R_{2} } \right)}L_{2} }} {{R_{2} }} $$

(A22)

$$ \frac{1} {{2k_{3} }} = \frac{{L_{2} }} {{R_{2} }} $$

(A23)

$$ \frac{{b_{1} + b_{2} }} {{2k_{1} k_{3} b_{2} }} = \frac{{R_{1} C_{1} L_{2} }} {{R_{2} }} $$

(A24)