A new approach to the estimation of electrocrystallization parameters

Abstract

To overcome the drawbacks in estimating electrocrystallization parameters using traditional methods, we propose a genetic algorithm using a novel crossover operator based on the non-convex linear combination of multiple parents to estimate the electrocrystallization parameters A (the nucleation rate constant), N₀ (the nucleation density) and D (the diffusion coefficient of Zn²⁺ ions) simultaneously in the general current-time expression of Scharifker and Mostany for nucleation and growth by fitting the whole current transients for zinc electrodeposition onto glassy carbon electrode immersed in the acetate solutions. By running the algorithm, we obtained for different step potentials, D values close to 2.10×10⁻⁶cm² s⁻¹, which are comparable to reported values. The values of A obtained for all step potentials are identical, 1.41×10⁹ s⁻¹, which indicates that zinc deposition onto glassy carbon electrode follows three-dimensional instantaneous nucleation and growth. In addition, from the values of N₀ obtained, one can observe that an increase in step potential leads to a higher N₀. These results show that our algorithm works stably and effectively in solving the problem of estimating the electrocrystallization parameters, and more importantly, it can be extended easily to a general algorithm to estimate multiple parameters in an arbitrary chemical model.

Introduction

In considering of the overlap of the diffusion fields of randomly growing nuclei, Scharifker and coworkers [1] proposed the following potentiostatic current–time expressions for instantaneous and progressive nucleation with diffusion controlled growth respectively

$$\text{I=}\text{zFD}{}^{\mathrm{1/2}}\text{c}\text{π}{}^{\mathrm{1/2}}\text{t}{}^{\mathrm{1/2}}\text{1−}\text{exp}\text{−N}{}_0\text{πkDt}\text{instantaneous}\text{nucleation}$$

$$\text{k≡}\text{8πcM/ρ}{}^{\mathrm{1/2}}$$

$$\text{I=}\text{zFD}{}^{\mathrm{1/2}}\text{c}\text{π}{}^{\mathrm{1/2}}\text{t}{}^{\mathrm{1/2}}\text{1−}\text{exp}\text{−Aπk′Dt}{}^2\text{/2}\text{progressive}\text{nucleation}$$

$$\text{k′≡}\text{4}\text{3}\text{8πcM/ρ}{}^{\mathrm{1/2}}$$

where z is the charge number of the reaction, D the diffusion coefficient, c the concentration of active cation, M the molar mass, ρ the density of deposit, and N₀ is the nucleation density (cm⁻²) while A is the nucleation rate constant (cm⁻² s⁻¹). The two expressions have been used widely for a great variety of systems [2], [3], [4], [5], [6], [7], including our previous publication describing the effects of organic additives on zinc deposition [5]. But they can deal only with two extreme cases in quantitative analysis, and the electrocrystallization of metals from many systems cannot be described by them (e.g. as Ag⁺ deposition onto a glassy carbon electrode from [Ag(NH₃)₂]⁺ solutions [8]). Moreover each equation can determine only one kinetic parameter, A or N₀. In view of these limitations, Scharifker and Mostany [9] deduced the following general current–time expression

$$\text{I=}\text{zFD}{}^{\mathrm{1/2}}\text{c}\text{π}{}^{\mathrm{1/2}}\text{t}{}^{\mathrm{1/2}}\text{1−}\text{exp}\text{−N}{}_0\text{πkD}\text{t−}\text{1−}\text{exp}\text{−At}\text{A}$$

in which one can estimate simultaneously the kinetic parameters A and N₀ with no necessity to classify the nucleation mechanisms. The most utilized approach [8], [9], [10], [11] for estimating these parameters in the general expression is to construct the following system of transcendental equations

$$\text{ln}\text{1−}\text{I}{}_{\mathrm{<mtext>m</mtext>}}\text{t}{}_{\mathrm{<mtext>m</mtext>}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{a}\text{+x−α}\text{1−}\text{exp}\text{−x}\text{α}\text{=0}\text{ln}\text{1+2x}\text{1−}\text{exp}\text{−x}\text{α}\text{−x+α}\text{1−}\text{exp}\text{−x}\text{α}\text{=0}$$

where a=zFD^1/2c/π^1/2, x=N₀πkDt_m, α=N₀πkD/A, using the single point, (I_m, t_m), on the experimental current transient where I_m is the current maximum and t_m is the corresponding time, and then solve it using the graphical method or the iterative method. Obviously the diffusion coefficient, D, must be known in advance. As for this approach, we considered that some drawbacks exist when it is applied to solve practical problems. First, during the process of constructing and solving the system of equations, only the current maximum point on the whole experimental current transient is used, which limits the accuracy of the electrocrystallization parameters. Secondly, prior to estimating N₀ and A, the values of D must be determined by additional experiments. Thirdly, the graphical method is too crude and can give rise to large errors, and the iterative method requires restricted conditions to be placed on the model, such as being continuous and derivable. Essentially, the parameters A, N₀ and D can be estimated simultaneously by fitting the whole current transient to the general expression.

Recently, there has been great interest in developing adaptive methods, namely genetic algorithms (GAs), for solving computational problems in many fields. The approach mimics the process of biological evolution and the mechanisms of natural selection and genetic variation. Suitable codings are used to represent possible solutions to a problem, and the search is guided by using some genetic operators and the principle of ‘survival of the fittest’. Due to the merits of self-adaptation, self-organization, self-learning, intrinsic parallelism and generality, GAs have been applied successfully in a wide range of economic, engineering and scientific computations [12]. The applications of GAs in chemistry are also very wide [13], [14].

To overcome the drawbacks in estimating electrocrystallization parameters using traditional methods, we propose a genetic algorithm to approach the task of parameter estimation in our experiments. To be specific, we use the genetic algorithm to estimate the values of D, N₀ and A in Eq. (3) by fitting the experimental current transients.

GAs can have various forms due to different representations, fitness evaluations and genetic operators which may vary with specific problems. Among all these components, genetic operators, including crossover and mutation, are considered as the most important parts in a genetic algorithm which can directly affect the effectiveness and the efficiency of the algorithm for solving the specific problem. In contrast to the genetic operators used by other researchers [15], [16], [17], we used a novel crossover operator based on the non-convex linear combination of multiple parents during the recombination of the population, which proved to work stably and effectively in solving the problem of electrocrystallization parameters estimation.