We have calculated the energy of three distinct grain configurations, namely completely connected, partially connected and unconnected configurations, evolving during a spheroidization of polycrystalline thin film by extending a geometrical model due to Miller et. al. to the case of spheroidization at both the surface and film-substrate interface. Stability diagram defining a stable region of each grain configuration has been established in terms of the ratio of grain size to film thickness vs. equilibrium wetting or dihedral angles at various interface energy conditions. The occurrence of spheroidization at the film-substrate interface significantly enlarges the stable region of unconnected grain configuration thereby greatly facilitating the occurrence of agglomeration. Complete separation of grain boundary is increasingly difficult with a reduction of equilibrium wetting angle. The condition for the occurrence of agglomeration differs depending on the equilibrium wetting or dihedral angles. The agglomeration occurs, at low equilibrium angles, via partially connected configuration containing stable holes centered at grain boundary vertices, whereas it occurs directly via completely connected configuration at large equilibrium angles except for the case having small surface and/or film-substrate interface energy. The initiation condition of agglomeration is defined by the equilibrium boundary condition between the partially connected and unconnected configurations for the former case, whereas it can, for the latter case, largely deviate from the equilibrium boundary condition between the completely connected and unconnected configurations because of the presence of a finite energy barrier to overcome to reach the unconnected grain configuration.
We have investigated detailed process of agglomeration of $TiSi_2$ thin film on (100) Si substrate as a model system for our geometric model for agglomeration based on the spheroidization at both the surface and film/substrate interface. Agglomeration occurs by a nucleation of holes at grain boundary vertices as a result of spheroidization at both interfaces and by their subsequent growth along grain boundaries in accordance with the prediction of our model. The critical condition of the ratio of the grain size to film thickness is predicted and confirmed to be in between 5 and 6 depending on the magnitude of free energy barrier.
We have investigated that residual stress effected on nucleation rate of holes at grain boundary vertices in thin film. Tensile stresses enhanced nucleation of holes, but compressive stresses suppressed it. Therefore , agglomeration is faster under a tensile stress than under a compressive stress.