Mixed ionic and electronic conductor oxides, in particular La1-xSrxCoyFe1-yO3-d (LSCF), have been widely used as the cathode materials in solid oxide fuel cells for high-temperature energy applications. The focus of this thesis is primarily on constructing the instability phase diagram of Sr segregations on LSCF surfaces at the experimentally relevant temperatures and oxygen partial pressures using the first-principles density functional theory (DFT). A generic first-principles free-energy functional is developed to obtain the nonstoichiometric oxygen vacancy concentrations for the bulk and surface phases. These results agree well with the corresponding thermo-gravimetry measurements, and furthermore suggest that the oxygen vacancies are energetically stabilized at surfaces for all temperatures and oxygen partial pressures, while such surface stabilization effects become stronger at higher temperatures and lower oxygen partial pressures. Based on these nonstoichiometric oxygen vacancy predictions, we construct the free-energy phase diagrams of the Sr-segregation reaction as a function of temperature, oxygen partial pressure, and CO2 partial pressure for both the bulk and surface LSCF phases. Our results suggest that Sr segregations strongly accumulate towards the LSCF surface phase where the oxygen vacancy nonstoichiometries are abundant. Our results also indicate that the Sr segregation reactions are significantly enhanced at high temperatures, low oxygen partial pressures, and high CO2 partial pressures. The computed reaction temperature ranges are consistent with the total reflection X-ray fluorescence (TXRF) measurements.