Abstract
A nonlinear version of Wagner’s classical model of internal oxidation is presented. This version accounts for the fact that precipitates may act as diffusion barriers by introducing a heuristic dependence of the diffusion coefficients of elements upon the local volume fraction of oxides formed. Remarkably, the now nonlinear problem still admits an analytic solution. This solution reveals a sudden and discontinuous transition from internal oxidation to formation of some external oxidized scale, when the fraction of oxide precipitates reaches some critical value which can be calculated explicitly in terms of the respective specific volumes of the matrix and the oxide.
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Notes
The slight dependence of the former volume upon the dissolved fraction of element A is neglected.
The fact that successive estimates of P and D O are constant in space and time is the key feature that allows for an analytic solution to be found to the nonlinear problem. Of course, it is a consequence of the (often wrong in practice!) assumption made by Wagner that the solubility product of the precipitate is vanishingly small.
It is worth noting incidentally that these qualitative predictions are supported by Brunac et al.’s [1] finite difference simulations of diffusion of O and Al and precipitation of Al2O3 in a Fe matrix, based on the model discussed.
This does not remain true for the critical nominal concentration of the oxidizable element, since the diffusion coefficients enter the relation connecting the two critical quantities.
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Leblond, JB. A Note on a Nonlinear Version of Wagner’s Classical Model of Internal Oxidation. Oxid Met 75, 93–101 (2011). https://doi.org/10.1007/s11085-010-9222-6
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DOI: https://doi.org/10.1007/s11085-010-9222-6