Analysis of Effect of Concentration Dependence of Exchange Current on Metal Electrodeposition into Template Nanopores

Abstract

The metal electrodeposition into the nanopores of template of porous anodic alumina type under the conditions of mixed kinetics of metal deposition is studied theoretically using analytical and numerical methods. Two main periods of the process are studied: the non-steady-state formation of diffusion layer in the template pores and much longer process of pore filling with the metal. The effect of nonlinearity of the concentration dependence of the exchange current density of metal electrodeposition on the variation of the current density with the time during the diffusion layer formation and pore filling with the metal is studied.

APPENDIX

Derivation of Equation for Metal Electrodeposition into Template Nanopores in the Quasi-Steady-State Approximation

Parameter c₀M/ρ controls the ratio between the characteristic diffusion time and the deposition time [24]. The ratio is small for the majority of aqueous solutions. Therefore, it can be always assumed that the deposition proceeds in the steady-state diffusion mode, except for a relatively short initial period of time of reaching the steady-state concentration profile.

Following the works [16, 23], we derive the equation in the quasi-steady-state approximation. Let us write the Fick’s laws for the outer diffusion layer and the diffusion layer inside a pore:

$$i = - \frac{{{{c}_{0}} - {{c}_{s}}}}{\delta }D{{z}_{ + }}F,$$

(A1)

$$i = - \varepsilon \frac{{{{c}_{s}} - {{c}_{p}}}}{L}D{{z}_{ + }}F,$$

(A2)

where c_s is the concentration at the template surface and c_p is the concentration at the pore bottom. The porosity ε in equation (A2) is introduced to match the current density inside the pores and in the outer diffusion layer.

Equations (A1) and (A2) enable us to express the concentration at the pore bottom c_p excluding the concentration at the template surface c_s. To do this, equation (A1) is multiplied by δ/(Dz₊F), equation (A2) is multiplied by L/ε (Dz₊F), and thus obtained equations are combined. As a result, we obtain:

$${{c}_{p}} = {{c}_{0}} + i\frac{{\varepsilon \delta + L}}{{\varepsilon D{{z}_{ + }}F}}.$$

(A3)

Assume that the kinetics of electrodeposition is described by the Tafel equation. Then, taking into account the concentration at the pore bottom and porosity ε, the equation for the current density of metal deposition can be written as follows:

$$i = - \varepsilon {{i}_{0}}{{\left( {\frac{{{{c}_{p}}}}{{{{c}_{0}}}}} \right)}^{\gamma }}{{e}^{{ - \frac{{{{\alpha }_{{\text{c}}}}F\eta }}{{RT}}}}}.$$

(A4)

Introducing the kinetic length δ_k = $\frac{{DF{{z}_{ + }}{{c}_{0}}}}{{{{i}_{0}}}}\exp \left( {\frac{{{{a}_{{\text{c}}}}F{{\eta }}}}{{RT}}} \right),$ equation (A3) can be written in the form similar to (A1) and (A2):

$$i = - \varepsilon \frac{{{{c}_{p}}}}{{{{\delta }_{{\text{k}}}}}}{{\left( {\frac{{{{c}_{p}}}}{{{{c}_{0}}}}} \right)}^{{\gamma \, - \,1}}}{{z}_{ + }}FD.$$

(A5)

Concentrations c_p and c_s can be excluded from (A1) using (A2) and (A3):

$$i = - \varepsilon \frac{{{{c}_{0}}D{{z}_{ + }}F}}{{L + \varepsilon {{\delta }_{{\text{D}}}} + {{\delta }_{{\text{k}}}}{{{\left( {1 + i\frac{{\varepsilon \delta + L}}{{{{c}_{0}}\varepsilon D{{z}_{ + }}F}}} \right)}}^{{1\, - \,\gamma }}}}}.$$

(A6)