Thermodynamic Origin of Negative Thermal Expansion Based on a Phase Transition-Type Mechanism in the GdF(3)-TbF(3) System

Multicomponent fluorides of rare earth elements (REEs—R) are phase transition-type negative thermal expansion (NTE-II) materials. NTE-II occurs in RF(3)-R′F(3) systems formed by “mother” single-component dimorphic RF(3) (R = Pm, Sm, Eu, and Gd) with a giant NTE-II. There are two structural types of RF(3) polymorphic modifications: low-temperature β-YF(3) (β−) and high-temperature LaF(3) (t−). The change in a structural type is accompanied by a density anomaly: a volume of one formula unit (V(form)) V(β)(−) >V(t)(−). The empirical signs of volumetric changes ΔV/V of NTE-II materials were considered. For the GdF(3)-TbF(3) model system, an “operating-temperature window ΔT” 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 RF(3)-R′F(3) systems modifies RF(3) chemically by forming two-component t− and β− type R(1−x)R’(x)F(3) solid solutions (ss). Between the two monovariant curves of ss decay, a two-phase area with ΔT(trans) > 0 (the “window ΔT”) forms. A two-phase composite (t−ss + β−ss) is an NTE-II material. Its constituent t−ss and β−ss phases have different V(form) corresponding to the selected T. According to the lever rule on a conode, V(form) is calculated from the t−ss and β−ss compositions, which vary with T along two monovariant curves of ss decay. For the GdF(3)-TbF(3) system, ΔV/V = f(T), ΔV/V = f(ΔT) and the “window ΔT” = f(x) dependencies were calculated.