Kirja

6. Electrochemical reaction kinetics

6.4. The effect of electric double layer

An electric double layer (EDL, see Chapter 5) has an effect on an electrode reaction through two factors. First, in order for an electron transfer to occur, the species involved in the reaction must reside so close to the surface of the electrode that an electron can be tunneled between the species and the electrode. In the previous chapter it was shown that at this close vicinity of the surface, the concentration of a charged species is different from its bulk concentration according to the Boltzmann distribution:


$\displaystyle c_i(x_2)=c_i^be^{-z_if\phi_2}$	(6.38)

If mass transfer needs to be taken into account, the bulk concentration is replaced by the surface concentration calculated from mass transfer equations.

Second, at the distance x₂ from the electrode the driving force of electron transfer is not $E-E^{0'}$ but $E-\phi_2-E^{0'}$ . Taking this into account, as well as Equation (6.38), the expression of k_ox, for example, becomes


$k_{ox}=k^0e^{\displaystyle -z_{\text{R}}f\phi_2}e^{\displaystyle\alpha nf(E-\phi_2-E^{0'})}=k^0e^{\displaystyle -(\alpha n+z_{\text{R}})f\phi_2}e^{\displaystyle\alpha nf(E-E^{0'})}$	(6.39)

The correction factor $e^{-(\alpha n+z_{\text{R}})f\phi_2}$ is known as the Frumkin correction. If the calculation of $\phi_2$ is feasible, the actual k⁰ is denoted as k_t⁰ (’t’ = true). The experimental value of k⁰, calculated from i₀ for example, is therefore


$k^0=k_t^0e^{\displaystyle -(\alpha n+z_{\text{R}})f\phi_2}$	(6.40)

An analogous result is achieved for k_red, taking into account that n = z_O - z_R.

Equation (6.38) can be derived from the equilibrium condition between the bulk solution and the electric double layer, $\tilde{\mu}_i(x_2)=\tilde{\mu}_i^b$ , that is strictly speaking only valid in the absence of electric current. The formally correct method would be to integrate the Nernst-Planck equation (3.34) from the bulk (x = $\delta$ ) to the distance x₂. The result of this procedure is known as the Levich correction:


$\displaystyle c_i(x_2)=e^{-z_if\phi_2}\left(c_i^b-\frac{i}{nFD_i} \int_{\delta}^{x_2}{e^{z_if\phi}dx}\right)$	(6.41)

Corrections to the EDL are usually not made because it would require the calculation of the potential profile at each electrode potential, which depends on the chosen microscopic model of the EDL. The calculation of the Frumkin correction is not common either, and experimental (yet apparent) values usually suffice perfectly well. Chapter 7 briefly looks at the effect of the EDL on electrochemical experiments in connection with respective techniques.