Vegard's law has been used extensively in mineralogy, metallurgy and materials science for the past six decades. According to the law, unit cell parameters should vary linearly with composition for a continuous substitutional solid solution in which atoms or ions that substitute for each other are randomly distributed. Although the law was postulated on empirical evidence, several cases of both positive and negative deviations from this law have been documented. Its theoretical foundations have not been critically explored. Presented in this communication is an analysis of the law within the framework of solution thermodynamics. It is shown that the deviation from Vegard's law is expected even for thermodynamically ideal solutions when there is a significant difference in lattice parameters of the pure components. The law should be reclassified as an approximation valid for specific conditions. The approximation is valid for ideal solutions when the lattice parameters of the pure components differ by less than 5 %. For solid solutions with positive deviations from ideality, there will always be positive deviations from Vegard's law. For solid solutions with moderately negative deviations from ideality, positive deviation from linearity of lattice parameters caused by size mismatch can be compensated for by the attractive interaction between the components, resulting in compliance with Vegard's law.
Nine tie-lines between Fe-Ni alloys and FeTiO 3 -NiTiO 3 solid solutions were determined at 1273 K. Samples were equilibrated in evacuated quartz ampoules for periods up to 10 days. Compositions of the alloy and oxide phases at equilibrium were determined by energy-dispersive x-ray spectroscopy. X-ray powder diffraction was used to confirm the results. Attainment of equilibrium was verified by the conventional tie-line rotation technique and by thermodynamic analysis of the results. The tie-lines are skewed toward the FeTiO 3 corner. From the tie-line data and activities in the Fe-Ni alloy phase available in the literature, activities of FeTiO 3 and NiTiO 3 in the ilmenite solid solution were derived using the modified Gibbs-Duhem technique of Jacob and Jeffes [K.T. Jacob and J.H.E. Jeffes, An Improved Method for Calculating Activities from Distribution Equilibria, High Temp. High Press., 1972, 4, p 177-182]. The components of the oxide solid solution exhibit moderate positive deviations from Raoult's law. Within experimental error, excess Gibbs energy of mixing for the FeTiO 3 -NiTiO 3 solid solution at 1273 K is a symmetric function of composition and can be represented as:Full spectrum of tie-lines and oxygen potentials for the three-phase equilibrium involving Fe-Ni alloys, FeTiO 3 -NiTiO 3 solid solutions, and TiO 2 at 1273 K were computed using results obtained in this study and data available in the literature.
The Green's function technique is used in the scattering matrix formalism to compute the mean square displacement (MSD) of hydrogen interstitials and the reaction coordinate technique is used to calculate the hydrogen diffusion parameters in the intermetallic compound Fe,.,Ti,,, from normal pressure to 2.5 GPa pressure for low concentration of hydrogen. The MSD of the metal atoms surrounding the interstitial are found to be smaller than those of the host crystal in agreement with NbH,. The diffusion of hydrogen into the alloy is found to decrease with increase in pressure as experimentally observed in PdH,.
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