A Thermodynamic Perspective of the Composition Dependence of Bulk Modulus in Terms of Electron Density and Molar Volume

The variation of bulk modulus with composition of alloy phases is a core issue in any thermodynamic theory of alloy formation. Though it emerges from fundamental theory that elastic property, especially the bulk modulus is crucially dependent on interstitial electron density ( ρ b ) distribution and its variation with respect to alloy composition, a coherent thermodynamic treatment connecting bulk modulus with electron density together with its composition dependence, is still lacking. The present study addresses this issue for solid solution alloys. A phenomenological analysis of the composition dependence of bulk modulus ( B T ) of single phase alloys has been presented in terms of bonding charge density ( ρ b ) and its corresponding change with atomic volume ( V ). This link is developed using the fundamental interrelationship existing between bulk modulus, electron density, and molar volume. The change in bonding charge density (Δ ρ b ) with composition ( x ) has been modeled using an exponential scaling relation with respect to the corresponding change in atomic volume (Δ V ). This scaling relation is based on the concept of the universal binding energy relation, which in turn results in a simple exponential variation of bulk modulus with composition-induced change of atomic volume. It is also shown that a common functional representation namely, B T ( V ) ≈ B o exp{ C × (Δ V )}, can be obtained for the temperature, pressure, and composition dependence of bulk modulus in terms of corresponding changes in volume (Δ V ). The constant C takes context-dependent meaning and values. The applicability of this exponential relation towards representing the effect of composition on bulk modulus has been satisfactorily demonstrated for many substitutional alloy systems.