(Digital Presentation) A Mathematical Model for Localized Corrosion of Copper Under a Droplet
Authors: You Chen , Briggs Scott , Orazem Mark E. 
Published: July 2022

The Nuclear Waste Management Organization (NWMO) is responsible for the implementation of Adaptive Phased Management (APM), the federally-approved plan for the safe long-term management of Canada’s used nuclear fuel.1 Under APM, used nuclear fuel will ultimately be placed within a deep geological repository (DGR) in a suitable host rock formation. Part of evaluating the long-term performance and safety of the repository system is understanding the behavior of the copper coated container with respect to localized corrosion. Even though localized corrosion is not expected in the DGR design system, there is still a need to develop a mathematical model for localized corrosion of copper.
The presentation will describe the development of a 2-D axisymmetric time-dependent model for the localized corrosion of copper under a drop of electrolyte using the concept of an Evans droplet.2 The model accounts for two reversible electrochemical reactions on the metal surface (copper dissolution/plating and hydrogen (H2) evolution/reduction) and two irreversible electrochemical reactions (cupric ion (Cu2+) reduction and oxygen (O2) reduction). Dissolved cuprous ion (Cu+) reacts in a second chemical step to form the cuprous chloride (CuCl) salt film treated as a diffusion barrier, or it reacts to form the cuprous oxide (Cu2O) film that partially blocks the active area on the metal surface. The model also accounts for a series of homogeneous reactions involving copper chloride complex, copper hydroxide complex, copper carbonate complex, and superoxide species. The associated equilibrium constants were obtained from PHREEQC thermodynamic software.3 The model includes a total of 23 dependent spatial-temporal variables.
The mathematical model is developed using the finite-element method (COMSOL Multiphysics). The model includes coupled nonlinear conservation equations for ionic species, which include the contribution of diffusion, migration, local electroneutrality, and homogeneous reactions. The anodic and cathodic regions are not predefined but are rather determined by values of local concentration and potential from the simulation results. The surface coverage of film was calculated implicitly and expressed in terms of the thickness in units of monolayers. The model also accounted for the effect of porous CuCl film thickness on the surface oxygen concentration and potential applied on electrochemical reactions.
In this preliminary study, two extreme conditions were chosen such that the droplet boundary could either have an infinite supply of oxygen or it could have no external supply of oxygen for an initially air-saturated droplet. The results show a 3-year time-dependent radial distribution of anodic and cathodic current density, surface coverage of CuCl film and Cu2O film, and localized corrosion rates and depths. The results include a distribution of pH, potential, and concentrations of dissolved gaseous and ionic species through the entire droplet. The potential change as the system transitions from aerobic to anaerobic conditions matches the expected equilibrium potential difference.4 The model could be used to explore the influence of different parameters on the localized corrosion rate and depth, film thickness and growth, and each individual current profiles. The parameters may be adjusted to conform to the DGR environment.
References

NWMO, “Choosing a Way Forward. The Future Management of Canada’s Used Nuclear Fuel. Final Study,” Nuclear Waste Management Organization, Toronto, Ontario, 2005.

R. Evans, The Corrosion of Metals, E. Arnold & Company, London, 1926.

PHREEQC Version 3, United States Geological Survey, 202, https://www.usgs.gov/software/phreeqc-version-3

Cleveland, M. E. Orazem, and S. Moghaddam, “Response to “Comment on “Nanometer-scale corrosion of copper in de-aerated deionized water” [J. Electrochem. Soc., 161, C107 (2014)],” Journal of the Electrochemical Society, 163 (2016), Y5-Y11.

Acknowledgement
This work was supported by the Nuclear Waste Management Organization, Canada, under project 2000904.