Thermodynamic Origin of Negative Thermal Expansion Based on a Phase Transition-Type Mechanism in the GdF₃-TbF₃ System

Multicomponent fluorides of rare earth elements (REEs-) are phase transition-type negative thermal expansion (NTE-II) materials. NTE-II occurs in RF₃-R'F₃ systems formed by "mother" single-component dimorphic RF₃ (R = Pm, Sm, Eu, and Gd) with a giant NTE-II. There are two structural types of RF₃ polymorphic modifications: low-temperature β-YF₃ (-) and high-temperature LaF₃ (-). The change in a structural type is accompanied by a density anomaly: a volume of one formula unit (V) V_- >V_-. The empirical signs of volumetric changes ΔV/V of NTE-II materials were considered. For the GdF₃-TbF₃ model system, an "operating-temperature window Δ" and a two-phase composition of NTE-II materials follows from the thermodynamics of chemical systems: the phase rule and the principle of continuity. A necessary and sufficient sign of NTE-II is a combination of polymorphism and the density anomaly. Isomorphism in RF₃-R'F₃ systems modifies RF₃ chemically by forming two-component - and - type R_1-xR'_xF₃solid solutions (). Between the two monovariant curves of ss decay, a two-phase area with Δ_trans > 0 (the "window Δ") forms. A two-phase composite (-ss + -ss) is an NTE-II material. Its constituent -ss and -ss phases have different V_form corresponding to the selected . According to the lever rule on a conode, V_form is calculated from the -ss and -ss compositions, which vary with along two monovariant curves of ss decay. For the GdF₃-TbF₃ system, ΔV/V = (), ΔV/V = (Δ) and the "window Δ" = (x) dependencies were calculated.