Initial stages of misfit stress relaxation in composite nanostructures through generation of rectangular prismatic dislocation loops

Critical (necessary) conditions of misfit stress relaxation in bulk and hollow core-shell nanoparticles, bulk and hollow core-shell nanowires, and flat bi- and tri-nanolayers (composing of a substrate and one or two thin films, respectively) through generation of rectangular prismatic dislocation loops (PDLs) at the inner and outer interfaces of the composite nanostructures with subsequent extension of the PDLs into cores (substrates) or shells (films) are analyzed theoretically and discussed in detail. The critical conditions are considered for different shapes and places of generation of the PDLs. It is shown that the misfit parameter is the principal factor, which determines the energetic preference of PDL generation: the latter is energetically favorable when the misfit parameter reaches its critical value, which depends on the geometry parameters of a composite nanostructure and the PDL type. Regardless of the place of PDL generation, the most energetically favorable case is when the PDL is elongated along the interface. The generation of PDLs from the free surface is the most preferable for small thicknesses of the shells (films). Growth of the shell (film) leads to a change of places of PDL generation while maintains their preferential shape - elongated along the interface. Among the six composite nanostructures under consideration, the tri-nanolayers are the least stable to the PDL generation. The hollow nanoparticles and nanowires are more stable to the PDL generation than the bulk ones. The bi-nanolayers are the most stable to the PDL generation.