Gibbs Free Energy and Size-Temperature Phase Diagram of Hafnium Nanoparticles

The Gibbs free energy of HCP, FCC, and BCC structures is calculated, and the size temperature phase diagram is obtained for hafnium nanoparticles. It is found that FCC, HCP, and BCC are small-size, low-temperature, and high-temperature stable phases, respectively. The temperature-induced structure transition is caused by the relative magnitude of lattice vibration (characterized by the Debye temperature) for different structures, while the size-induced structure transition originates from the different molar volumes, The observed HCP to BCC and HCP to FCC structure transitions are consistent with our model predictions. More importantly, we predict that there exists a new structure transition from FCC to BCC in the size and temperature ranges 3.6-14.6 nm and 1766-1801 K for spherical nanoparticles (alpha = 1) and 1586-1742 K and 4-14.8 nm for tetrahedral ones (alpha = 1.49), which has not been reported in the literature.