Analytical solution for transient grain-boundary filling by parallel diffusion

Derive an analytical solution for the transient filling of a grain boundary by hydrogen diffusion parallel to the boundary.

Background

The phase-field model is benchmarked against analytical solutions for hydrogen transport perpendicular to grain boundaries. The authors state that an analogous analytical treatment could not be identified for the transient process in which hydrogen fills a grain boundary through diffusion parallel to its plane. Developing such a solution would provide a corresponding validation benchmark for the model's parallel transport kinetics.

References

An analogous analytical solution for the transient filling of a GB by diffusion parallel to the boundary could not be identified; therefore, a corresponding benchmark was not included.

— A quantitative phase-field model for grain boundary trapping and diffusion of hydrogen  (2609.31367 - Feyen et al., 25 Sep 2026) in Appendix C: Grain boundary trapping with perpendicular flux