Modelling the growth and filling of creep-induced grain-boundary cavities in self-healing alloys

A set of numerical and analytical models is presented to predict the growth and contraction of grain-boundary creep cavities in binary self-healing alloys. In such alloys, the healing is realised by preferential precipitation of supersaturated solutes at the free surface of the cavity. The cavity grows due to the diffusional flux of vacancies towards the cavity, which is driven by the stress gradient along the grain boundary. Upon deposition of healing solute atoms on the cavity wall, effectively vacancies are removed from the cavity due to the inverse Kirkendall effect. The competition between the inward and outward vacancy fluxes results in a time-dependent filling ratio (i.e. the fraction of the vacancies removed from the original cavity) of the creep cavity. It is found that for stress levels lower than a critical stress σ_cr, the filling ratio can proceed to unity, i.e. to complete filling and annihilation of the pore. For applied stresses higher than σ_cr, complete filling is not achieved and the open volume of the creep cavity will continue to grow once a maximum filling ratio is reached at the critical time t_cr. The critical stress σ_cr, critical time t_cr, and time for complete filling t_h (if fully filling is achievable) are derived from the models for different combinations of parameters. The results from the analytical model and from previous nanotomography experiments are compared and are found to be in good agreement. Graphical abstract: [Figure not available: see fulltext.]