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 kox, 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 kt0  (’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 kred, taking into account that n = zO - zR.

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.