Cluster expansion of the Onsager matrix for dilute solid solutions: application to diffusion in Fe-(C, N, O) solid solutions

Obtaining analytical expressions for transport coefficients with interacting diffusing species is a long-standing challenge. So far, such expressions have been obtained in the infinitely dilute limit only, i.e. considering the interaction between two diffusing species at most, or in concentrated solid solutions where the local environment is averaged [1]. Recent progress now enables to compute transport coefficients in dilute systems from ab initio data for various diffusion mechanisms [2, 3] and crystal structures [4].

In this work, we propose an original framework to go beyond the infinitely dilute limit, and consider the effect of the interaction of at least three diffusing species on the mass transport properties of a solid solution. We show that transport coefficients for dilute solid solutions can be formally written as an expansion of cluster contributions, which we call cluster transport coefficients. These coefficients are intrinsic and equilibrium properties of each cluster and can be split into a well-defined scalar mobility and several matrix components that describe the cluster association and dissociation contributions to the overall kinetics of the system. We extend the self-consistent mean-field method to the computation of cluster transport coefficients for clusters larger than two components. Our methodology makes it possible to compute analytical transport coefficients beyond the infinitely dilute limit, and we demonstrate this on Fe-X solid solutions, where X=C, N and O.

[1] Nastar, Phys. Rev. B 90 144101 (2014)

[2] Schuler et al., Phys. Rev. B 93 224101 (2016)

[3] Messina et al., Phys. Rev. B 93 (2016) 184302.

[4] Trinkle, submitted to Phil. Mag. (arXiv:1608.01252)