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—R) 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 (β−) and high-temperature LaF3 (t−). The change in a structural type is accompanied by a density anomaly: a volume of one formula unit (Vform) Vβ− >Vt−. The empirical signs of volumetric changes ΔV/V of NTE-II materials were considered. For the GdF3-TbF3 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 RF3-R′F3 systems modifies RF3 chemically by forming two-component t− and β− type R1−xR’xF3 solid solutions (ss). Between the two monovariant curves of ss decay, a two-phase area with ΔTtrans > 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 Vform corresponding to the selected T. According to the lever rule on a conode, Vform is calculated from the t−ss and β−ss compositions, which vary with T along two monovariant curves of ss decay. For the GdF3-TbF3 system, ΔV/V = f(T), ΔV/V = f(ΔT) and the “window ΔT” = f(x) dependencies were calculated.


Introduction
The search for new materials with negative thermal expansion (NTE) has been conducted via trial and error for many years.Recently, there has been a noticeable change in the paradigm for creating NTE materials.Chemical modification of known "mother" materials with a giant NTE was used [1].The method of chemically modifying "mother" substances was called "substitutions" [1].This implies isomorphism, which can be realized in a binary (and more complex) system.
The development and application of multicomponent fluoride materials with the participation of REE trifluorides-RF 3 -have been carried out since the mid-1970s [2][3][4].No work has been performed on the search for and study of NTE materials based on REE fluorides.
The 2nd type of NTE (NTE-II) includes materials with NTE based on a phase transitiontype mechanism.In such materials, a volume reduction when heating occurs during a polymorphic transformation (PolTr) [1].
RF 3 -R F 3 systems are the most suitable for studies of the common signs of NTE-II materials.They are ionic compounds with the simplest formula and high chemical stability.Their melting (T fus ) and PolTr (T trans ) temperatures ensure diffusion processes and the achievement of equilibrium.
REE trifluorides form a long homologous series of RF 3 (R = La − Lu) from extremely chemically similar compounds with a minimum difference in the atomic numbers (Z) of cations in the series (∆Z = 1).The lanthanide compression of cations in the LnF 3 (Ln = Ce − Lu) series leads to three types of structures and two morphotropic transformations (MorphTrans).
Among 17 RF 3 s, four dimorphic fluorides have been discovered: PmF 3 , SmF 3 , EuF 3 , and GdF 3 [2].SmF 3 , EuF 3 , and GdF 3 were described as having a density anomaly: a high-temperature form is denser than a low-temperature form [14]. Recently, PmF 3 was attached to them [15].The density anomaly is the main structural feature of PolTr in dimorphic RF 3 s.This makes them and RF 3 -R F 3 systems, which are formed by them, sources of NTE-II materials.
In this message, a new paradigm for the design of NTE-II materials using isovalent isomorphism [1] is considered based on phase diagrams.
In [16], the principle of isovalent isomorphism in RF 3 -R F 3 systems with one (or two) components with NTE-II was used to predict two-component two-phase compositions with NTE-II in 50 (out of 105) systems.Four stages for creating two-component NTE-II materials with adjustable operational parameters were considered.For systems with the studied phase diagrams, the temperature and composition ranges in which these parameters are regulated were estimated.
The connection of empirically found features of two-component NTE-II materials with the thermodynamic rules for the design of phase diagrams of chemical systems, Gibbs' phase rule (1879) and Kurnakov's continuity principle [17] (1936), is established.
It is shown that NTE-II follows from the fundamental laws of thermodynamics of chemical systems.In this study, the empirical signs of NTE-II [1] in a material based on its phase diagram are substantiated for the first time on the model GdF 3 -TbF 3 system [2].One of its components, GdF 3 , has a giant NTE-II.
A comparison of the rules for the phase diagram design with the thermophysical properties-coefficients of thermal expansion (CTE) of the materials formed in them-defines the universal signs of NTE-II.
The compliance of the formation of NTE-II materials with the fundamental rules of chemical thermodynamics provided a reasonable correction of their previously found empirical features.
A phase diagram of a system contains all the data for quantitative calculations of the values of the NTE-II parameters of the materials formed in this system.This report presents a scheme for calculating the parameters of an NTE-II material based on the experimental phase diagram of the selected GdF 3 -TbF 3 system.This scheme can be extended to any system with a studied phase diagram if it is equilibrium and if its components (one or both) are compounds with NTE-II.
The aims of this study are as follows: (1) a discussion of NTE-II empirical signs in oneand two-component fluoride materials on the basis of the fundamental rules for the chemical system design (the phase rule and the principle of continuity); (2) formulating the stages of the design of new fluoride NTE-II materials with adjustable parameters with isomorphic substitutions in "mother" RF 3 (R = Pm, Sm, Eu, and Gd) with PolTr (when heated); (3) consideration of the thermodynamic conditions (an "operating-temperature window ∆T" and a two-phase composite) for the NTE-II material formation in the model GdF 3 -TbF 3 system; (4) and a presentation of the method for calculating the NTE-II parameters controlled by isomorphism from the phase diagram of the GdF 3 -TbF 3 model system.
According to IUPAC, 17 REEs were designated R: Sc, Y, La and 14 Ln = Ce − Lu.When analyzing the periodicity of the properties of Ln compounds, it is necessary to separate La from the 4f elements Ln.If not necessary, the sum (La + Ln) was denoted by R. The inconvenience of the REE classification was noted by the IUPAC project (2015) regarding the La position.
Subscript designations of compositions in chemical formulas make it advisable to assign Z of R in the superscript position ( 57 Ce 0.5 64 Gd 0.5 F 3 ).Atomic weights with generally accepted superscript positions are not used in this work.

The Qualitative Description of the NTE-II Materials Signs
In the literature, one can find the opinion that a NTE-II material must (1) be two-phase, (2) have the "window ∆T", and (3) have the negative CTE that can be measured within the "window ∆T".
The thermal expansion law has the following form: The proportionality coefficient in the linear dependence (1) represents the volumetric CTE α V = dV/dT.It characterizes the rate of change, ∆V/V, as a function of T. NTE-II materials have negative α V within the "window ∆T", which can be up to 10 times greater than that of conventional materials [1].The total volume change ∆V/V was used in the literature to compare the densities of compounds with different structures.It is recommended [1] as an intrinsic index to indicate the potential of NTE-II.
Qualitative schemes of the ∆V/V change from T for an NTE-II material according to [1] are presented in Figure 1.The diagram in Figure 1a demonstrates the PolTr occurring "at a point" (T trans = const).At T trans , ∆V/V jumps at ∆T = 0 (the "window ∆T" = 0) [1].The scheme in Figure 1a from [1] corresponds to the phase rule for a single-component condensed system.The "window ∆T" is prohibited by the phase rule.
Int. J. Mol.Sci.2023, 24, x FOR PEER REVIEW 3 of 14 GdF3-TbF3 system; (4) and a presentation of the method for calculating the NTE-II parameters controlled by isomorphism from the phase diagram of the GdF3-TbF3 model system.According to IUPAC, 17 REEs were designated R: Sc, Y, La and 14 Ln = Ce − Lu.When analyzing the periodicity of the properties of Ln compounds, it is necessary to separate La from the 4f elements Ln.If not necessary, the sum (La + Ln) was denoted by R. The inconvenience of the REE classification was noted by the IUPAC project (2015) regarding the La position.
Subscript designations of compositions in chemical formulas make it advisable to assign Z of R in the superscript position ( 57 Ce0.5 64 Gd0.5F3).Atomic weights with generally accepted superscript positions are not used in this work.

The Qualitative Description of the NTE-II Materials Signs
In the literature, one can find the opinion that a NTE-II material must (1) be twophase, (2) have the "window ΔT", and (3) have the negative CTE that can be measured within the "window ΔT".
The thermal expansion law has the following form: ΔV/V = αV ΔT. ( The proportionality coefficient in the linear dependence (1) represents the volumetric CTE αV = dV/dT.It characterizes the rate of change, ΔV/V, as a function of T. NTE-II materials have negative αV within the "window ΔT", which can be up to 10 times greater than that of conventional materials [1].The total volume change ΔV/V was used in the literature to compare the densities of compounds with different structures.It is recommended [1] as an intrinsic index to indicate the potential of NTE-II.
Qualitative schemes of the ΔV/V change from T for an NTE-II material according to [1] are presented in Figure 1.The diagram in Figure 1a demonstrates the PolTr occurring "at a point" (Ttrans = const).At Ttrans, ΔV/V jumps at ΔT = 0 (the "window ΔT" = 0) [1].The scheme in Figure 1a from [1] corresponds to the phase rule for a single-component condensed system.The "window ΔT" is prohibited by the phase rule.During PolTr in a single-component system, CTE cannot be measured, because ΔT at the point of the PolTr is zero.The author [1] draws a fundamental conclusion from this: "The coefficients αV and αL are not intrinsic for NTE-II materials.The ΔV/V is the intrinsic index of the NTE potential".
The diagram in Figure 1a corresponds to simple "mother" substances, showing PolTr with the density anomaly of the modifications [1].
When the second component is added to a "mother" [1] simple substance, an isomorphic substitution of some of the cations occurs with the formation of ss.The term "substitutions" (Figure 4 in [1]) indicates isomorphism.The degree of freedom of a system increases by 1.The "window ΔT" > 0 appears, and the vertical line in the diagram (Figure During PolTr in a single-component system, CTE cannot be measured, because ∆T at the point of the PolTr is zero.The author [1] draws a fundamental conclusion from this: "The coefficients α V and α L are not intrinsic for NTE-II materials.The ∆V/V is the intrinsic index of the NTE potential".
The diagram in Figure 1a corresponds to simple "mother" substances, showing PolTr with the density anomaly of the modifications [1].
When the second component is added to a "mother" [1] simple substance, an isomorphic substitution of some of the cations occurs with the formation of ss.The term "substitutions" (Figure 4 in [1]) indicates isomorphism.The degree of freedom of a system increases by 1.The "window ∆T" > 0 appears, and the vertical line in the diagram (Figure 1a) acquires a slope (Figure 1b).In a binary system, PolTr occurs over a temperature range.If within the "window ∆T" a material has CTE < 0 that can be measured, such a material belongs to materials with NTE-II.
For a single-component dimorphic compound, ∆V/V, at the PolTr, is an intrinsic constant of a substance.The concept of the "total volume change" is applicable to it as a characteristic of PolTr.
The ∆V/V of a multicomponent material in a binary (and more complex) system generally varies with composition.Exceptions are the singular points of a phase diagram with invariant equilibria.
The "window ∆T" between two phases exists in any binary system with ss on the base of both structural modifications of components.This corresponds to the phase rule, not to the proof of the presence of NTE-II in a material.These are signs of a condensed T-x system with ss, based on its components.Many substances are polymorphic.Polymorphism is also not a sign of NTE-II.
In Figure 1, PolTr with the density anomaly is shown.The PolTr of a one-component material (Figure 1a) proceeds with the density anomaly and the "window ∆T" = 0.The PolTr of a two-component material (Figure 1b) also proceeds with the density anomaly but with the "window ∆T" > 0.
The qualitative signs of NTE-II, formulated based on empirical data, need to be tested on experimental systems with such materials.This is what the present study is dedicated to.
A necessary and sufficient condition for NTE-II is polymorphism combined with the density anomaly.Changing the structure with a decrease in ∆V/V at the PolTr (when heated) is of key importance for the emergence of NTE-II.

Signs of NTE-II in Ionic REE Fluorides
All RF 3 , except for volatile ScF 3 , are single-component condensed systems.Their PolTrs are invariant processes with T trans = const and the "window ∆T" = 0, which are described by the scheme in Figure 1a.
The proposed [1] chemical modification of "mother" simple RF 3 with the PolTr and NTE-II by isomorphism can be obtained only in a binary (and more complex) system, forming two-component NTE-II materials.Such materials are formed in RF 3 -R F 3 systems.
The objectives of this study include the construction of the qualitative and quantitative schemes of the PolTr in these systems using the example of the GdF 3 -TbF 3 system.The bank of phase diagrams of 34 studied RF 3 -R F 3 systems [2] serves as a scientific basis for this message.
Let us consider the signs of NTE-II materials in RF 3 -R F 3 systems based on empirical data from the literature.
Polymorphism is the first unconditional sign of any NTE-II material [1].The connection with PolTr defines the type of NTE material as the 2nd type [1].PolTr separates NTE-II from a conventional NTE (compression when heated over a wide T interval) and from normal materials (expansion when heated).
When investigating polymorphism, it is necessary to consider the peculiarities of this phenomenon.These are reflected in the discussion of polymorphism in IUPAC.It was said that its absence was most often caused by "a lack of financial resources for research".These are state parameter changes in the areas of high pressures and low temperatures, which incur large financial costs.Under such conditions, polymorphism can remain "hidden".
PmF 3 polymorphism is a vivid confirmation.Microquantities of PmF 3 were obtained in a nuclear reactor according to the Manhattan Project to determine its structural affiliation with the LaF 3 type [18].The cost of the reagent was not determined.However, its high cost has contributed to the lack of knowledge concerning PmF 3 .
PmF 3 polymorphism still remains "hidden" in terms of thermal analysis techniques.This occurs at low temperatures and is accompanied by a small thermal effect [15].Only the method of structural and chemical modeling allowed the evaluation of T trans in the model 61 (Ce 0.5 Gd 0.5 )F 3 ("pseudo 61 PmF 3 ") composition [15].
Polymorphism is a necessary but insufficient feature of NTE-II.Of the eight dimorphic RF 3 , only four (R = Pm, Sm, Eu, and Gd) possess a giant ∆V/V.
The second sign of NTE-II is the density anomaly at the PolTr.Under comparable conditions, the V form (the volume of one formula unit) of a high-temperature modification is lower than that of a low-temperature modification, V low (V high < V low , compression when heated).
Only β− → t− PolTr (when heated) of the three possible structural transitions yields a denser high-temperature t−modification.The volumes of dimorphic RF 3 are V β− > V t− (V t− and V β− are V form s of t− and β− types, respectively).In single-component SmF 3 , EuF 3 , and GdF 3 , density anomalies were noted for the first time [14].
Polymorphism and the density anomaly are determined by structural changes that occur during PolTr.Structural changes are key to understanding the NTE-II mechanism.
In [1], the density anomaly as a sign of NTE-II was not discussed.This is the main structural limitation of NTE-II.
The third sign of NTE-II is a two-phase composition [1].In a single-component RF 3 system, a two-phase area is prohibited by the phase rule.A two-phase area forms in binary systems between the two areas of homogeneity of the β-R 1−x R x F 3 -ss and t-R 1−x R x F 3 -ss based on RF 3 modifications.
The fourth sign of NTE-II is the "window ∆T", according to [1].The origin of this feature is not established [1].The "window ∆T" defines the second fundamental parameter of an NTE-II material.Its significance in the description and use of NTE-II materials plays a dual role.First, the "window ∆T" controls ∆V/V in a particular system, which determines the use of a material.Control of ∆V/V is considered in the example of the GdF 3 -TbF 3 system in this study.Secondly, the "window ∆T" "adjusts" a material to the temperature conditions of use.For one system, the value of the regulated ∆T is small.In RF 3 -R F 3 systems, the position of the "window ∆T" on the T scale can vary widely from low temperatures to melting.
The fifth sign of NTE-II is a measurable CTE.It is present only in two-component materials formed in RF 3 -R F 3 systems because their PolTrs occur over a temperature range ∆T.In a single-component RF 3 , CTE cannot be measured at PolTr (∆T = 0).
To realize NTE-II in dimorphic ionic fluorides RF 3 with R = Pm-Gd, their structural types must have features.The nature of the structural mechanism of NTE-II in RF 3 has not yet been clarified.

Two-Component NTE-II Materials with Adjustable Parameters in the GdF 3 -TbF 3 System
The GdF 3 -TbF 3 system was selected as a model [2] to describe the thermodynamic mechanism of the formation of NTE-II materials.Its phase diagram is shown in Figure 2. The black points represent thermal effects determined experimentally using the thermal analysis method.

Choosing a Model System
The first component of the GdF3-TbF3 model system, dimorphic GdF3, has a giant NTE-II (ΔV/V ~ 4.3%) [14,19] at the β-GdF3 → t-GdF3 PolTr (when heated).The choice of GdF3 from RF3 (R = Pm, Sm, Eu, and Gd) is due to its stability in terms of valency reduction.
The chemical modification of GdF3 due to isomorphism occurs when TbF3 is added.A new state parameter, composition x (mole % of TbF3), appears in the GdF3-TbF3 system.This increases the degree of freedom of the system by 1.
The PolTrs in two-component systems are a combination of peritectoid (one phase turns into two phases with an increase in T) and eutectoid (two phases turn into one phase

Choosing a Model System
The first component of the GdF 3 -TbF 3 model system, dimorphic GdF 3 , has a giant NTE-II (∆V/V ~4.3%) [14,19] at the β-GdF 3 → t-GdF 3 PolTr (when heated).The choice of GdF 3 from RF 3 (R = Pm, Sm, Eu, and Gd) is due to its stability in terms of valency reduction.
The chemical modification of GdF 3 due to isomorphism occurs when TbF 3 is added.A new state parameter, composition x (mole % of TbF 3 ), appears in the GdF 3 -TbF 3 system.This increases the degree of freedom of the system by 1.
The PolTrs in two-component systems are a combination of peritectoid (one phase turns into two phases with an increase in T) and eutectoid (two phases turn into one phase with an increase in T) equilibria.This is accompanied by the formation of a two-phase area [20].
According to the principle of continuity [17], two-component β-Gd 1−x Tb x F 3 and t-Gd 1−x Tb x F 3 phases form the two-phase area (β + t)-Gd 1−x Tb x F 3 with the ss, which have the β− and t− structural modifications of one-component GdF 3 .The properties of the β− and t− modifications change continuously when moving from the single-to two-phase area.
The GdF 3 -TbF 3 system has a convenient interval of thermal effects for the thermal analysis method, as set by the difference ∆(T fus -T trans ) of GdF 3 .This made it possible to study phase diagrams that contain a high frequency of the analyzed compositions (the black dots and red circles in Figure 2) when obtaining both liquidus and solidus curves.
Curves 3 and 4 of the Gd 1−x Tb x F 3 ss decay (formation) in Figure 2 come from point 1 (the PolTr of GdF 3 ).The Gd 1−x Tb x F 3 ss have the same types of structures (β− and t−) that lead to the giant NTE-II in GdF 3 .Anomalous volume relations remain between them: V β−ss > V t−ss (the compression occurs with an increase in T).
The homogeneity area of the two-phase composite (β + t)-Gd 1−x Tb x F 3 with NTE-II captures approximately 50 mol% of the TbF 3 composition axis.It is marked in Figure 2 in light green.
The ∆(T fus − T trans ) = 1228-1070 • C interval (the GdF 3 phase transformations) from the phase diagram [2] lies in the T region of disinhibited volumetric diffusion in a solid (Tamman's temperature is T tamm ~760 • C).This ensured an equilibrium necessary for the use of materials with NTE-II.

The Method of Calculating the Phase Composition of an NTE-II Material on the Example of the GdF 3 -TbF 3 Phase Diagram
The determination of the qualitative and quantitative compositions of a two-phase (β + t)-Gd 1−x Tb x F 3 composite is shown in the insert of The Two-Phase (β + t)-Gd 1−x Tb x F 3 Area with NTE-II At the temperature T β−ss , the figurative point intersects with curve 4. The decay of the β-Gd 1−x Tb x F 3 phase begins with the release of the t-Gd 1−x Tb x F 3 phase.The two-phase composite (β + t)-Gd 1−x Tb x F 3 is formed.The area of the two-phase composite is limited by curves 3 and 4.This is highlighted in light green in Figure 2.
The composition (x 1 ) of the t-phase at the beginning of the decay is determined by the intersection of the k 1 conode (the lower horizontal dotted line) with curve 3 (insert in Figure 2).
When moving the figurative point inside the two-phase area along the vertical red arrowed line from T β−ss to T t−ss (insert in Figure 2), the composition of t-Gd 1−x Tb x F 3 varies from x 1 to x o .The composition of β-Gd 1−x Tb x F 3 varies from x o to x 2 .
At T t−ss (the left edge of the k 2 conode), the figurative point intersects curve 3, and the composition of t-Gd 1−x Tb x F 3 becomes equal to the original one (x o ).PolTr, formally similar to the β-GdF 3 → t-GdF 3 PolTr, is completed.
However, because β-Gd 1−x Tb x F 3 and t-Gd 1−x Tb x F 3 are located on the different monovariant curves 3 and 4 and are separated by the interval ∆T, the PolTr proceeds over the temperature range.The "operating-temperature window ∆T" is formed [1].In the insert in Figure 2, "window ∆T" is indicated for composition x o by the red arrow.Its value is ∆T = T t−ss − T β−ss .
An example of calculating the average V form inside the "window ∆T" for the intermediate point of the PolTr at T (conode k 3 ) within the ∆T interval is shown in the insert in Figure 2. The mole fractions (m and n) of the t−ss and β−ss in the two-phase (β + t)-Gd 1−x Tb x F 3 area are shown in green and magenta, respectively.These values are calculated according to the lever rule.The concentrations of x t and x β of the t−ss and β−ss correspond to the extreme points of the k 3 conode.
The concentration x o of Gd 1−x Tb x F 3 at the beginning of the PolTr at V form of the t−ss (V t− ) and β−ss (V β− ) phases entering the two-phase composite (β + t)-Gd 1−x Tb x F 3 at T are calculated by ( 3) and ( 4): V form s V t− (x t ) and V β− (x β ) are calculated from the unit cell parameters of the t−ss and β−ss phases.The unit cell parameters of the t−ss and β−ss phases, in turn, are determined using Vegard's rule from the unit cell parameters of the GdF 3 and TbF 3 components [14,19].The unit cell parameter of TbF 3 was determined via the extrapolation of the RF 3 data [19].
Let us assume that the NTE-II composite material represents an equilibrium mechanical mixture of its constituent t−ss and β−ss phases.Then, the V form of the two-phase compositions (V t+β ) is calculated using (5) as the sum of the volumes (V t− + V β− ) of the t−ss and β−ss phases: V t+β = V t− + V β− (5) The average value of V form of a composite is variable.It is determined by T in the interval ∆T.
The Single-Phase t-Gd 1−x Tb x F 3 -ss Area without NTE At T t−ss , the figurative point intersects with curve 3. The decay of the β-Gd 1−x Tb x F 3 phase with the release of the t-Gd 1−x Tb x F 3 phase is completed.The figurative point falls into the t-Gd 1−x Tb x F 3 single-phase area.In this area, the material is single-phase, without NTE.
The experimental curves 3 (decay of the t−ss) and 4 (decay of the β−ss) [2] in the subsolidus region of the GdF 3 -TbF 3 system (black dots and red circles in Figure 2) were approximated using second-degree polynomials ( 6) and ( 7): T = 88.4 x 2 + 182.6 x + 1066.5 (7) Based on the described methodology, using Equations ( 6) and ( 7), it is possible to calculate the volume-temperature dependencies for any of the 34 RF 3 -R F systems studied experimentally [2].
Temperature is an active factor controlling ∆V/V, which is one of the parameters of NTE-II material prospects.The second factor is passive, that is, a fixed composition, x.In our case, this is the composition x o .If ∆V/V is approximately estimated via the scope of a material application, the composition can be changed in two ways.Small changes within the two-phase area are achievable in one system.For large changes, another system is searched using the RF 3 -R F 3 array [16].

NTE-II in the GdF 3 -TbF 3 System
The construction of a volume-temperature dependence is widely used in various fields of research to calculate the NTE for various purposes.∆V/V = f (T) dependence was used [1] to determine the CTE.We present the calculation of such dependencies for Gd 1−x Tb x F 3 at several x.Equation ( 8) is used to construct the curves: where V(t+β) is the two-phase composite volume at T′ and Vt− is the Vform of the t− type phase at the end point of the PolTr at Tt−ss (insert in Figure 2).The vapor pressure (P) for all RF3, except ScF3, is low.This allows us to neglect P and consider all RF3 discussed here to be condensed systems with invariant PolTrs with Ttrans = const.
For single-component GdF3, there is the invariant PolTr of the first kind at point 1 (Figure 2).At Ttrans = const, the two solid phases are in equilibrium β-GdF3 ↔ t-GdF3.The Equation ( 8) is used to construct the curves: where V (t+β) is the two-phase composite volume at T and V t− is the V form of the t− type phase at the end point of the PolTr at T t−ss (insert in Figure 2).∆V/V for (β + t)-Gd 1−x Tb x F 3 changes within ∆T trans = T t−ss -T β−ss (the "window ∆T") from 0 to 4.22%.The (β + t)-Gd 1−x Tb x F 3 composites have NTE-II over the "window ∆T".∆V/V is decreasing with increasing T, and the CTE is negative.
Beyond the "window ∆T", (β + t)-Gd 1−x Tb x F 3 refers to conventional materials without NTE and with the positive CTE.
The TbF 3 contents at the starting points of the β-Gd 1−x Tb x F 3 → t-Gd 1−x Tb x F 3 PolTr at T β−ss are indicated by x o = 0, 0.05, 0.1, 0.2, 0.29, 0.4, 0.51 numbers under the curves.The dependencies are constructed within the ∆T trans = T t−ss -T β−ss (the "window ∆T"), which is highlighted in light red in the diagram in Figure 3 for x = 0.29.
The PolTr in GdF 3 and Gd 0.49 Tb 0.51 F 3 and NTE-II The vapor pressure (P) for all RF 3 , except ScF 3 , is low.This allows us to neglect P and consider all RF 3 discussed here to be condensed systems with invariant PolTrs with T trans = const.
The ∆V/V = f (T) curve for GdF 3 (x = 0) in Figure 3 shows the absence (within the measurement error) of a slope between the β-GdF 3 and t-GdF 3 components.The shape of the curve for GdF 3 fully corresponds to the first scheme from [1] (Figure 1a) and to the phase rule.
The curve for the two-component Gd 0.49 Tb 0.51 F 3 ss (x = 0.51 in Figure 3) also has no slope (∆T trans = 0) because the PolTr of this composition represents an invariant process, a peritectic phase reaction.
The ∆V/V change within the two-phase area of the phase diagram (insert in Figure 2) consists of: (1) the changes in V β−ss and V t−ss with the changes in ss compositions along curves 3 and 4, (2) the differences in V β−ss and V t−ss of the two structural types (β−ss and t−ss) at T of the selected conode (k 3 in the insert, Figure 2), and (3) the quantitative ratio of the β−ss and t−ss phases (according to the lever rule) in the NTE-II composite on isothermal sections with T = const (the k 3 conode).This is the thermodynamic mechanism for controlling the parameters of materials via isomorphism in any "temperature-composition" (T − x) system.It follows from the analysis of the equilibrium phase diagram of a binary system and is not related to the sign of the ∆V/V change at the PolTr and NTE-II.
Owing to its fundamental thermodynamic nature, the mechanism of formation of two-phase composites is universal for compounds of any chemical class of substances.The volume reduction at the PolTr (when heated) makes the composite an NTE-II material.
The PolTr in the Two-Phase Composite Gd 0.71 Tb 0.29 F 3 (Gross Composition) According to the Phase Diagram and the NTE-II Parameters To describe an NTE-II material with the "window ∆T" formed in the GdF 3 -TbF 3 system, the gross composition of Gd 0.71 Tb 0.29 F 3 was selected.It is obtained by averaging the compositions of the three experimental points in the phase diagram (indicated by red circles in Figure 2).It corresponds to the dashed vertical I in Figure 2.
On the curve x = 0.29 in Figure 3, three V form values (red circles) obtained from the experimental points of the phase diagram (also red circles in Figure 2) are plotted.The concentration x = 0.29 is the average for these points.
The ∆V/V values obtained from the experimental data (red circles) of the phase diagram correlate well with the calculated ∆V/V = f (T) dependencies described by lines 3 and 4, between k 1 and k 2 .To characterize the CTE of (β + t)-Gd 1−x Tb x F 3 two-phase composites formed in the GdF 3 -TbF 3 system, the dependencies of the relative volume change ∆V/V on ∆T were constructed.The ∆V/V = f (∆T) dependencies for (β + t)-Gd 1−x Tb x F 3 (x = 0, 0.05, 0.1, 0.2., 0.29, 0.4, 0.51) are shown in Figure 4.
The ΔV/V values for the curves in Figure 4 were calculated using ( 2)-( 8).The temperature interval ΔT is calculated using ( 9) for every T' within the temperature interval of the β-Gd1−xTbxF3 → t-Gd1−xTbxF3 PolTr (when heated).ΔT = T' − Tβ−ss (9) The three ΔV/V values obtained from the experimental data with x = 0.29 (opened red circles) are shown in Figure 4.For x = 0 and 0.51 with invariant equilibrium, the dependencies in Figure 4 represent vertical lines (the "window ΔT" = 0).They coincide with each other and are marked in purple.The ∆V/V values for the curves in Figure 4 were calculated using ( 2)-( 8).The temperature interval ∆T is calculated using ( 9) for every T' within the temperature interval of the β-Gd 1−x Tb x F 3 → t-Gd 1−x Tb x F 3 PolTr (when heated).∆T = T' − T β−ss (9) The three ∆V/V values obtained from the experimental data with x = 0.29 (opened red circles) are shown in Figure 4.
For x = 0 and 0.51 with invariant equilibrium, the dependencies in Figure 4 represent vertical lines (the "window ∆T" = 0).They coincide with each other and are marked in purple.
For the two-phase composite (β + t)-Gd 1−x Tb x F 3 (0 < x < 0.51), the ∆V/V = f (∆T) dependencies are nonlinear and have different forms for different x values.The CTE of (β + t)-Gd 1−x Tb x F 3 varies within the "window ∆T".The dependence of ∆T trans (the "window ∆T") on the (β + t)-Gd 1−x Tb x F 3 composition x is shown in Figure 5.It is calculated for the area between curves 3 and 4 (Figure 2).The area of the β−ss → t−ss PolTr in Gd 1−x Tb x F 3 (0 < x < 0.51) is shown in light green in Figures 2 and 5.
The value of ΔТ (the "window ΔТ") in the GdF3-TbF3 system corresponds to the maximum (ΔТ = 22 °C) on the curve in Figure 5.This is much higher than the reproducibility of the T measurements in RF3-R′F3 systems (±3 °C).

Materials and Methods
Two-Component NTE-II Materials in RF3-R′F3 and MF2-RF3 Systems (M = Ca, Sr, Ba; The curve was approximated using a second-degree polynomial (10): ∆T trans = −325.3x 2 + 171.6 x − 0.184 (10) The approximated curve is shown in Figure 5 with a red line.It has a maximum of ∆T trans = 22 • C at x = 0.26.
The value of ∆T (the "window ∆T") in the GdF 3 -TbF 3 system corresponds to the maximum (∆T = 22 • C) on the curve in Figure 5.This is much higher than the reproducibility of the T measurements in RF 3 -R F 3 systems (±3 • C).

Materials and Methods
Two-Component NTE-II Materials in RF 3 -R F 3 and MF 2 -RF 3 Systems (M = Ca, Sr, Ba; R = La − Lu) The necessary and sufficient features (polymorphism with the density anomaly) for two-component composite NTE-II materials in the GdF 3 -TbF 3 system can be applied to any RF 3 -R F 3 system.
The two-phase (t−ss + β−ss) composites are NTE-II materials with adjustable parameters.The potential for using the material is estimated using the parameter of the average volume change ∆V/V av .The V av at a fixed gross composition of a system is determined by the β−ss and t−ss decay (synthesis) curves and the temperature T. The regulation of ∆V/V av is achieved by changing T within the "window ∆T".The available ∆T values are determined using phase diagrams.
Berthollide phases with the t− type structure formed in MF 2 -RF 3 systems with M = Ca, Sr, Ba; R = Gd -Lu, Y, and NaF-RF 3 with R = Gd, Tb [2] are also composite materials with NTE-II.The temperature interval "window ∆T" of the two-phase region (t−ss + β−ss) of berthollide phases is more than 300 K.
The "window ∆T" achievable in RF 3 -R F 3 systems is comparable with the "window ∆T" of conventional NTE materials.The "window ∆T" parameter plays a decisive role in controlling the performance characteristics of NTE-II materials when they are used as thermal expansion compensators in high tech.To date, no fluoride materials (single-or multicomponent) have been tested for use in this field.

Conclusions
The thermodynamic mechanism for the formation of two-component, two-phase NTE-II materials with controllable properties in a binary system is described.It follows from the equilibrium phase diagram.Owing to its fundamental nature, this mechanism is universal for compounds of any chemical class of substances.
A necessary and sufficient condition for the formation of a material with NTE-II is a combination of polymorphism and density anomaly.Under comparable thermal conditions, the V high of the high-temperature form is lower than that of the low-temperature form V low .
The "operating temperature window ∆T" determines the range of compositions and temperatures of the existence of the two-phase composite NTE-II material in a binary system.Its dimensions are given by the monovariant decay (formation) curves of ss, with the structures giving the density anomaly at the PolTr.
The volume change at the PolTr characterizes the NTE-II potential of a material in a particular system.
RF 3 (R = Pm, Sm, Eu, and Gd) represents a new class of fluoride single-component NTE-II materials.They possess the giant NTE of the 2nd type within the "window ∆T" = 0 temperature range.
Isomorphism in RF 3 -R F 3 systems chemically modifies RF 3 to form two-component R 1−x R x F 3 materials.R 1−x R x F 3 have the giant NTE-II if one or both of their components belong to the "mother" single-component dimorphic RF 3 with R = Pm, Sm, Eu, and Gd.

Figure 2 .
Figure 2. The GdF 3 -TbF 3 phase diagram: (1) the PolTr of GdF 3 with ∆T trans = 0, (2) the incongruent melting β-Gd 0.49 Tb 0.51 F 3 , (3) the t−ss decay curve, and (4) the β−ss decay curve.Insert: The scheme for the calculation of the qualitative and quantitative composition of the two-phase (β + t)-Gd 1−x Tb x F 3 composite.The experimental data are shown as closed black and open red (x ≈ 0.29) circles, "window ∆T" intervals as vertical red arrowed lines, concentration intervals as horizontal blue arrowed lines, and the two-phase (β + t)-Gd 1−x Tb x F 3 area in light green.

Figure 2 .
It is defined by the isothermal sections (conodes) k 1 , k 2 , and k 3 .The β-Gd 1−xo Tb xo F 3 → t-Gd 1−xo Tb xo F 3 PolTr (when heated) of the Gd 1−xo Tb xo F 3 ss with the composition x o begins at T β−ss (the k 1 conode) and ends at T t−ss (the k 2 conode).The Single-Phase β-Gd 1−x Tb x F 3 -ss Area without NTE The figurative point, which corresponds to the gross composition, x o , moves up (with the increase in T) along the vertical line, III, in the single-phase β−ss area to curve 4. In this area, a material is single-phase.It has a β− type structure and is a conventional material without NTE.