A Multiscale Study of the Thermodynamics and Kinetics of Vacancy–grain-boundary Interactions

Abstract

Grain boundaries (GBs) influence many physical and mechanical properties of crystalline materials. This is primarily because of their interactions with other kinds of defects, such as point and line defects. Such interactions are known to depend on the structure and properties of individual GBs, i.e. the GB character. Although much research has been done in this regard, an understanding of the effect of GB character on such interactions is still incomplete.

The present study was undertaken to particularly understand the effect of GB character on vacancy--GB interactions in aluminium (Al). To achieve that, the study was done in five parts using a multiscale modelling approach which involved atomistic and continuum methods. First, using molecular statics (MS) simulations, Al bicrystals with symmetric [010]- and [101]-tilt GBs were constructed, and their stability was checked using molecular dynamics simulations. (A total of 32 boundaries were studied.) The GB structures were subsequently validated using the structural unit model (SUM). Second, vacancy formation and migration energies in the GBs were evaluated using MS simulations and the nudged elastic band method, respectively. These results were then used to obtain equilibrium vacancy concentrations and vacancy migration rates in the GBs. Third, using this data and the kinetic Monte Carlo (KMC) method, the anisotropic GB diffusivities were obtained by simulating random walks of a vacancy in the GBs with a focus on the role of the GB. Fourth, a multiple-site segregation equation was used to study the thermodynamics of vacancy segregation in cases where there is a supersaturation of vacancies, such as in-quenched or irradiated specimens. Segregation was studied as a function of GB character, grain size, temperature, and initial vacancy supersaturation. A phase-field method was then used to evaluate the kinetics of vacancy segregation at the GBs. Finally, quantities derived from the atomistic methods were used to evaluate differences in void nucleation and growth rates again in the presence of a vacancy supersaturation, and the role of GB character was explored.

The first part of the study reconfirms the result of earlier studies that a single parameter cannot be used to characterize GBs that crystallographically vary over a five-dimensional space. Furthermore, it was discovered that the structure of individual GBs could be modelled via a SUM. Some boundaries were identified as favoured GBs, and these could be described using a single repeating structural unit (SU). The structure of non-favoured GBs over the entire misorientation range about [101] tilt could be described by a linear combination of the SUs of favoured boundaries. Consistent mathematical descriptions of the equilibrium vacancy concentration and the overall vacancy jump rate in GBs are provided in the second part of the study. Using these descriptions, effective vacancy formation and migration energies were calculated for the GBs. In the third part of the study, KMC calculations were conducted to estimate GB diffusivities. The role of GB character on the activation energy for GB diffusion was also estimated.

The next two aspects of this study are related to a supersaturation of vacancies and effect of GB character on the segregation of excess vacancies and nucleation and growth rates of voids. In accordance with previous studies, it was observed that (compared to a more realistic multiple-site model) a single-site model overestimates the segregated vacancy concentration when the supersaturation was high and/or the grain size was large. Moreover, the difference between the models is pronounced at lower temperatures. Even though segregation of vacancies at GBs was observed to not drastically reduce the vacancy supersaturation in bulk, it substantially increased the probability of void nucleation at the GBs. The GB structure and, in particular, the presence of deep vacancy traps play a significant role in determining the void nucleation rate at GBs. Such deep traps were predominantly observed in low-angle GBs and GBs vicinal to the coherent twin boundary. As such, these were also observed to act as very good vacancy sinks and could better retard the nucleation of voids than bulk. Whereas, although the high-energy boundaries were observed to also act as good vacancy sinks, they were much more prone to void nucleation. On the other hand, the growth rate was found to be less sensitive to the GB character, with voids growing marginally faster along boundaries that have a higher diffusivity. Based on this, it was deduced that void-depleted zones that are experimentally observed in regions adjacent to the GBs are primarily due to the nucleation and growth of voids at GBs which consume the excess vacancies. Moreover, it was surmised that GBs with higher void nucleation rates are expected to have wider void-depleted zones.