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483
Mathematical Modeling of Inhibitor Transport in an Organic Coating

^{1}There have been numerous mathematical models of the electrochemistry in pits and crevices presented in literature. However, there has been a scarcity of models for the transient electrochemistry associated with an immersed organic coating. The authors have been developing mathematical models to address ionic transport in organic coatings.

The authors presented a one-dimensional model of such a system which employed the constraint of an open-circuit condition and the assumption that the corrosion potential and current density were functions of the local pH at the metal-coating interface.^{2} A two-dimensional model was also presented where the local polarization behavior, a function of pH, was coupled to the global polarization behavior by the constraint of a net zero current density.^{3}The strategy used in the mathematical models provides a framework for modeling the evolution of electrochemistry within the organic coating and electrolyte.

In this effort, the authors have investigated the electrochemistry of an organic coating with a delaminated region. An inhibitor is included in the model where its production is linked to a release rate in the coating as a function of local pH. The suppression of the corrosion activity at the metal-coating interface of the delaminated coating is based on a threshold local concentration being exceeded.

**Mathematical Model**

The conservation of a mass was expressed as Eq. 1 where the accumulation term is on the left-hand side and the net input of the flux and net rate of production of species by homogeneous reactions are on the right hand side, respectively. The parameter *c _{i}* was the local concentration,

*e*the porosity of the medium,

*t*time,

*N*the flux of a species, and

_{i}*R*the homogeneous production rate of the specie. In the electrolyte domain, the porosity was

_{i}set at unity. Six ionic species were considered; four being Na^{+}, Cl^{-}, OH^{-} and Al^{3+}. The inhibitor was considered to yield generic A^{+} and B^{-} ions upon dissolution into the coating. The B^{-}ion was considered as the species inhibiting the corrosion activity at the metal coating interface.

The model used the Nernst-Planck equation for the flux of a species as given in Eq. 2, where *u _{i}* is the mobility,

*D*the diffusion coefficient,

_{i}*z*the charge number, F the solution potential and

_{i}*F*Faraday's constant. The contribution of convection was considered negligible. The porosity of the coating was accounted for in the diffusion coefficient of the species,

*D*

_{i}^{*}, in the coating by using Eq. 3.

The experimental polarization behavior of 99.99% Al in de-aerated NaCl at different pH values was used to cast the polarization behavior of aluminum dissolution and hydrogen evolution as functions of pH.^{4} The current density at the metal coating interface *i _{Al}*for the aluminum dissolution reaction was calculated using a modified Tafel expression given in Eq. 4.

where β* _{Al}* is the Tafel slope, and E

_{Al}

^{0}, a

*reference potential and i*

_{Al}

^{0}, a reference current density, were functions of the local pH. The weighting parameter ω

*was used to account for the reduction in the current density at the interface due to the coating. The electrochemical potential*

_{Al}*V*was defined as

*V*=

*E*-Φ, where

*E*is the metal potential and Φ the solution potential. A similar expression was used for the hydrogen evolution reaction. The governing equations of conservation of species and electroneutrality were solved using a finite difference approach with the program developed in FORTRAN. Simulations are presented that show the evolution of the coating and electrolyte electrochemistry and metal potential in response to changes in the bulk pH, the coating porosity, and inhibitor production.

**Acknowledgements**

This work was supported by Boeing Research and Technology.

**References**

- H. Leidheiser, R. Granata,
*IBM J. Res. Develop.*, 32 (1988) 582. - Kerry N. Allahar, Michael Hurley, Erik Sapper, Darryl Butt, Research in Progress Symposium, NACE Corrosion/2012, Salt Lake City, Mar. 2012.
- Kerry N. Allahar, Michael Hurley, Erik Sapper, 223rd Electrochemical Society Meeting, Toronto, Canada, May 2013.
- W. Lee, S. Pyun,
*Electrochim. Acta.*, 44 (1999) 4041.