Typical Cluster Sizes in Metal Electrodeposition

Abstract

Using the framework of the classical theory of nucleation, main cluster sizes occurring in the course of electrodeposition are determined. The problem is also considered from the standpoint of Cahn–Hillard theory. The focus is on the qualitative estimation of particle sizes in the course of nanonucleation.

Appendices

APPENDIX A

There are alternative approaches to the consideration of nucleation in the Gibbs model. So, in [8, 9] it has been found that in the course of a stochastic process of ion discharging on the electrode surface with the formation of adatoms, the steady-state rate J of cluster formation from g adatoms (g-cluster) can be expressed as:

$$J = {\text{const}} \times g!{{\left( {{{\theta }_{0}}} \right)}^{g}}\exp \left[ {\left( {g + a} \right){{zF\eta } \mathord{\left/ {\vphantom {{zF\eta } {RT}}} \right. \kern-0em} {RT}}} \right],$$

(A1)

where θ₀ is the equilibrium level of surface filling with adatoms, whereas the remaining notations are standard. The derivation of this formula is based on finding the average number of pairs, triples, and so on, that is, the clusters consisting of g = 2, 3, etc. particles for a given level of filling θ, that depends both on the equilibrium value of θ₀ and on the electrode potential. More precisely, it depends on overvoltage η and on the exchange current density J. J is defined as the total number of particles attached per unit time to all the n-clusters located on a unit surface.

After taking the logarithm, formula (A1) transforms to

$$\ln J = g\ln {{\theta }_{0}} + \ln \left( {g!} \right) + \left[ {{{\left( {g + \alpha } \right)zF\eta } \mathord{\left/ {\vphantom {{\left( {g + \alpha } \right)zF\eta } {RT}}} \right. \kern-0em} {RT}}} \right] + C,$$

(A2)

whence it follows that:

$$d\ln {J \mathord{\left/ {\vphantom {J {d\eta }}} \right. \kern-0em} {d\eta }} = \left( {g + \alpha } \right){{zF\eta } \mathord{\left/ {\vphantom {{zF\eta } {RT}}} \right. \kern-0em} {RT}},$$

(A3)

which completely coincides with the expression that has been derived based on absolutely different considerations within another concept, namely within the atomistic theory of nucleation [6].

It is interesting that the existence of a critical value of g, according to equations (A2)–(A3), is caused not by energy, as is the case of the classical interpretation, but by another reason. In order for this to become obvious, let us differentiate expression (A2) with respect to g under constant overvoltage. Such differentiation, if one uses the fact that d ln(g!)/dg ≈ ln g, the following relationship:

$${{d\ln J} \mathord{\left/ {\vphantom {{d\ln J} {dg}}} \right. \kern-0em} {dg}} = \ln {{{\theta }}_{0}} + \ln g + {{zF{\eta }} \mathord{\left/ {\vphantom {{zF{\eta }} {RT}}} \right. \kern-0em} {RT}},$$

(A4)

and since θ₀ ⪡ 1, that is, lnθ₀ ⪡ 0, then for some g this derivative passes through zero:

$$\ln g + {{zF{\eta }} \mathord{\left/ {\vphantom {{zF{\eta }} {RT}}} \right. \kern-0em} {RT}} + \ln {{{\theta }}_{0}} = 0,$$

or

$$g = \left( {{1 \mathord{\left/ {\vphantom {1 {{{{\theta }}_{0}}}}} \right. \kern-0em} {{{{\theta }}_{0}}}}} \right)\exp \left( {{{ - zF{\eta }} \mathord{\left/ {\vphantom {{ - zF{\eta }} {RT}}} \right. \kern-0em} {RT}}} \right).$$

(A5)

For a given g, the value of flux J passes through a minimum. Therefore, it is just the transition from the g*-cluster to the (g* + 1)-cluster that is determining, that is, it determines the rate of the entire further growth process (that in the steady-state mode is equal to the rate of this transition). This actually coincides with the definition of a critical nucleus: it is just the size from which a stable growth mode starts.

Comparison of (A5) and (9) shows that the equilibrium filling of the surface with adatoms, θ₀, is very small and lower than 10^–5, which corresponds to 10¹⁴ adatoms per square meter. Unfortunately, reliable experimental data concerning the level of equilibrium filling are absent in the literature.

APPENDIX B

As it has been already noted, an interesting fact consists in the fact that in the theory of nonlinear equations, a pendulum model with one degree of freedom (MM1) is presented as an example of an integrable nonlinear equation. It has also been established that for our purposes, both linearized and nonlinear MM1 lead to the same physical results (as it is known, fundamental ones) [51], and this result is very important for the size of the nucleation nucleus, too. In this regard, it seems that some mathematical analysis of obtaining a solution of a nonlinear equation belonging to MM1 could be of interest.

Let us consider Eq. (26). This equation can be formally transformed into the following equation (the validity of this procedure has been shown earlier):

$${{d\theta \left( \rho \right)} \mathord{\left/ {\vphantom {{d\theta \left( \rho \right)} {\sin \left\{ {\theta \left( \rho \right)} \right\}}}} \right. \kern-0em} {\sin \left\{ {\theta \left( \rho \right)} \right\}}} = {{ \pm d\rho } \mathord{\left/ {\vphantom {{ \pm d\rho } {{{r}_{{{\text{cc}}}}}}}} \right. \kern-0em} {{{r}_{{{\text{cc}}}}}}}.$$

(B1)

Next, if a new variable is introduced

$$t = \tan \left\{ {{{\theta \left( \rho \right)} \mathord{\left/ {\vphantom {{\theta \left( \rho \right)} 2}} \right. \kern-0em} 2}} \right\},$$

(B2)

then nonlinear equation (B1) by means of this substitution can be transformed into a linear one:

$${{dt} \mathord{\left/ {\vphantom {{dt} t}} \right. \kern-0em} t} = {{ \pm d\rho } \mathord{\left/ {\vphantom {{ \pm d\rho } {{{r}_{{{\text{cc}}}}}}}} \right. \kern-0em} {{{r}_{{{\text{cc}}}}}}},$$

(B3)

that can be readily integrated

$${\rho \mathord{\left/ {\vphantom {\rho {{{r}_{{{\text{cc}}}}}}}} \right. \kern-0em} {{{r}_{{{\text{cc}}}}}}} = \pm \ln t + {{{{\rho }_{0}}} \mathord{\left/ {\vphantom {{{{\rho }_{0}}} {{{r}_{{{\text{cc}}}}}}}} \right. \kern-0em} {{{r}_{{{\text{cc}}}}}}}$$

(B4)

(ρ₀/r_cc is chosen as the integration constant).

If one returns to the previous variable, one can obtain the following formula in the form of (29):

$$\theta \left( \rho \right) = 2\arctan \left\{ {\exp \left[ {{{ \pm \left( {\rho - {{\rho }_{0}}} \right)} \mathord{\left/ {\vphantom {{ \pm \left( {\rho - {{\rho }_{0}}} \right)} {{{r}_{{{\text{cc}}}}}}}} \right. \kern-0em} {{{r}_{{{\text{cc}}}}}}}} \right]} \right\}.$$

(B5)

The presented study, in our opinion, shows a mathematical relationship between linear and nonlinear models, which can be formulated as follows. If a coordinate transformation exists that transforms a non-linear equation into a linear one, then a nonlinear model is equivalent to a linear model.