Nanocrystalline (NC) materials are referred to those polycrystalline materials having a crystallite (grain) size usually inferior to 100 nm. Thermodynamic properties of NC materials are essentially different from conventional coarse-grained materials (with the same chemical composition). As the grain size is inferior to 100 nm, the role of grain boundary is very important in the characterization of thermodynamic functions and thermal properties of NC materials. Therefore, traditional thermodynamics being applied for coarse-grained materials is not applicable for NC materials. In order to deal with the thermodynamics of NC materials, a dilated crystal model is proposed and equation of state (EOS) and quasiharmonic Debye approximation (QDA) methods are used to calculate thermal properties of the grain boundaries. In this study, QDA and EOS methods are used to calculate Gi free energy in NC Fe. Since the Gi free energy for Fe, predicted by EOS and QDA methods, have an inaccurate (especially at temperatures higher than the ambient temperature), a term called as ?G Excess is proposed to modify the results. Thus, the modified QDA (MQDA) and modified EOS (MEOS) methods are introduced. Thereafter, the change in Gi free energy for ?-Fe to ?-Fe phase transformation (?G ??? ) via the grain size is calculated by MQDA and MEOS methods. The results obtained by the two methods are also compared and discussed. Then, the critical grain size, at which ?G ??? = 0, can be estimated at different temperatures and it is found that the allotropic transformation temperature would increase with increasing grain size. Because of same prediction by MQDA and MEOS method, the MEOS method (instead of EOS method) is used to calculate the total Gi free energy of each phase (?-Zr or ?-Zr and ?-Ti or ?-Ti) in NC Zr and Ti. Thereupon, the change in the total Gi free energy for ? to ? phase transformation (?G ??? ) via the grain size is calculated by this method. Similar to polymorphic transformation in other NC materials, the estimated transformation temperature in NC Zr and Ti (???) is reduced with decreasing grain size. Moreover, to predict accurate thermodynamics properties, MD simulation is used for studying structure of grain boundary in NC Fe and the changes of quantitative parameters of grain boundaries (such as excess volume and grain boundary thickness) versus grain size and temperature is survived. These results are demonstrated that the quantitative parameters of grain boundaries are not related to grain size and these parameters is related to temperature. Finally, with quantitative parameters obtained by this structure study, the changes of allotropic transformation temperature are studied. Comparison with experimental results is shown that these outcomes have a better coincidence.
