Mathematical Modeling of the Electrodeposition Process
Authors: Suggu Santosh Kumar , Mahapatro Anil 
Published: February 2020

Electroplating (1) or electro-deposition is a process which produces a dense, uniform, and adherent coating on a metal surface using electric current in an electrolytic cell. Usually, the metal to be coated is the cathode and the anode can be one of the two: sacrificial anode (dissolvable anode) or permanent anode (inert anode) (2). Electroplating is used for a various applications including corrosion protection. In this abstract we report the development of a mathematical model to predict the electro deposition of copper on cobalt chrome alloy in an electrochemical cell with copper and cobalt chrome alloy as the electrodes and copper sulfate as the electrolyte.
Mathematical Model: The following assumptions were made to develop the model: 1) Dilute solution theory applies. This states that there are negligible interactions between the solute species, 2) The current density distribution is uniform on the surface of the electrode 3) The current density takes the form of Butler-Volmer equation, which expresses the dependence of the current on the composition of the electrolyte solution adjacent to the electrode surface and on the relative activity of the solid state species and the exponential dependence of the current on the over potential, 4) The physical, transport and kinetic parameters are constant throughout the solution, 5) solution is isothermal and 6) no homogeneous chemical reactions occur in the electrolyte
The transport equation which is applied in the diffusion layer is based on the flux equation of the ionic species in the solution. Mass transport in the solution occurs because of migration in electric field, diffusion in concentration gradient and convection in a flow field. The general flux expression of a species i is given as follows.
Where Ni- molar flux, Zi - charge number, Ui - mobility, F – Faradays constant, Ci - initial concentration, Di – Diffusivity,
- potential in the electrolyte, V - velocity

The material balance equation for each ionic species at every point within the diffusion layer is as follows
Additional assumptions such as conservation of electro neutrality and a perfectly mixed electrolyte were assumed.
The models were solved using finite element method solver software COMSOL. The computation domain and the mesh were defined as seen in figure 1. Where mass transfer is assume to occur between the electrodes, through the electrolyte only.

Figure 2 shows the thickness change of the electrodeposited copper on cobalt chrome alloy versus time. From the simulated results we see that for the first 500 sec the coating hasn’t started, as the time goes by we see a steady coating of Cu on Co-Cr. At the end of 3000 sec we see that coating was at 0.020mm layer on the Co- Cr.

Figure 1: Defining computational domain (top) and mesh (bottom)

Figure 2: Copper coating thickness change on cathode as function of time
This demonstrates the preliminary applicability of the model in predicting electro deposition of copper on cobalt chrome substrate.

Reference

[1] A. Basile, A. I. Bhatt, A. P. O’Mullane, S. K. Bhargava, Electrochimica Acta 56, 2895, (2011)
[2] Z. Zheng, R. M. Stephens, R. D. Braatz, R. C. Alkire, L. R. Petzold, Journal of Computational Physics 227, 5184, (2008)