Vegard's law: a fundamental relation or an approximation?

Abstract

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.

Development of plurimetallic electrocatalysts prepared by decomposition of polymeric precursors for EtOH/O2 fuel cell

“…This indicates formation of an alloy which is in accordance with Vegard's Law. 16,33 However, as explained before, the introduction of Sn into Pt 64 Sn 15 Ru 13 Ni 8 /C does not elicit any shifts as compared to pure Pt.…”

“…This indicates formation of an alloy which is in accordance with Vegard's Law. 16,33 However, as explained before, the introduction of Sn into Pt 64 Sn 15 Ru 13 Ni 8 /C does not elicit any shifts as compared to pure Pt.The crystallite sizes, obtained by means of DebyeScherrer's equation, 27 are in the 7.40-22.50 nm range. The great difference observed among the different reflection planes attest to the high heterogeneity of the particles formed by the DPP method.…”

“…Generation of traces of CO 2 cannot be ruled out, as verified from the HPLC data in the range of the limit 33 Ni 22 /C nanoelectrocatalysts is higher than the quantity of acetaldehyde. These electrocatalysts contain a similar amount of Ru in their composition.…”

First-Principles Investigation of Structural, Electronic and Elastic Properties of HfX (X = Os, Ir and Pt) Compounds

“…where V M (HfX) is the volume (cm 3 /mol) of each HfX compound; Z B (HfX) is the bonding valence in (el/atom) rationalized from Vegard's law [44]:…”

Investigations on III-Nitrides nanostructures: application to Renewable Energies and Bio-Sensing

“…2.1 is quadratic). A complete study on this can be found in the work done by K.T Jacob, Shubra Raj and L. Rannesh [Jacob 2007]. Most interesting is the controversy on the bowing parameter of InGaN.…”