Two-Component Rare-Earth Fluoride Materials with Negative Thermal Expansion Based on a Phase Transition-Type Mechanism in 50 RF3-R’F3 (R = La-Lu) Systems

The formation of materials with negative thermal expansion (NTE) based on a phase transition-type mechanism (NTE-II) in 50 T–x (temperature–composition) RF3-R’F3 (R = La-Lu) systems out of 105 possible is predicted. The components of these systems are “mother” RF3 compounds (R = Pm, Sm, Eu, and Gd) with polymorphic transformations (PolTrs), which occur during heating between the main structural types of RF3: β-(β-YF3) → t-(mineral tysonite LaF3). The PolTr is characterized by a density anomaly: the formula volume (Vform) of the low-temperature modification (Vβ-) is higher than that of the high-temperature modification (Vt-) by a giant value (up to 4.7%). In RF3-R’F3 systems, isomorphic substitutions chemically modify RF3 by forming R1−xR’xF3 solid solutions (ss) based on both modifications. A two-phase composite (β-ss + t-ss) is a two-component NTE-II material with adjustable parameters. The prospects of using the material are estimated using the parameter of the average volume change (ΔV/Vav). The Vav 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 ΔV/Vav is achieved by changing T within a “window ΔT”. The available ΔT values are determined using phase diagrams. A chemical classification (ChCl) translates the search for NTE-II materials from 15 RF3 into an array of 105 RF3-R’F3 systems. Phase diagrams are divided into 10 types of systems (TypeSs), in four of which NTE-II materials are formed. The tables of the systems that comprise these TypeSs are presented. The position of Ttrans of the PolTr on the T scale for a short quasi-system (QS) “from PmF3 to TbF3” determines the interval of the ΔTtrans offset achievable in the RF3-R’F3 systems: from −148 to 1186 ± 10 °C. NTE-II fluoride materials exceed known NTE-II materials by almost three times in this parameter. Equilibrium in RF3-R’F3 systems is established quickly. The number of qualitatively different two-component fluoride materials with the giant NTE-II can be increased by more than ten times compared to RF3 with NTE-II.


Introduction
Trifluorides of rare-earth elements (REE), RF 3 , are polymorphic [1][2][3].In [4], the discovery of an effect called a density anomaly in lanthanide (Ln) trifluorides, LnF 3 , was reported.The anomaly occurs during the PolTr This is because the density of the high-temperature modification is higher than that of the low-temperature modification.This usually happens in the opposite direction.Such an anomaly has been described [4] for LnF 3 with Ln = Sm, Eu, and Gd.At that time, there was no interest in materials with NTE.The publication of [4] went unnoticed.In subsequent studies, there was no mention that any RF 3 other than ScF 3 [5][6][7] was an NTE material.
The polymorphism of PmF 3 was not known because all its isotopes are short-lived, and the synthesis of Pm is possible only in atomic reactors.Recently, a method of structural and chemical modeling in a "CeF 3 -GdF 3 " QS was developed [8].The composition Int.J. Mol.Sci.2023, 24, 14000 2 of 12 61 ( 58 Ce 0.5 Gd 0.5 )F 3 with an atomic number (Z) of promethium, Z = 61, was proposed and named "pseudo-PmF 3 ".It has the structural and thermal (PolTr) properties of 61 PmF 3 ."Pseudo-PmF 3 " polymorphism attached PmF 3 to dimorphic LnF 3 with Ln = Pm, Sm, Eu, and Gd, which are materials with NTE-II [9].
In this study, the concept [10] of the chemical modification of components of RF 3 -R'F 3 (R = La-Lu) systems was used to predict 50 RF 3 -R'F 3 systems (out of 105 possible) with two-phase composites having the giant NTE-II at the PolTr.
According to IUPAC, 17 REE were designated as R = Sc, Y, or La, and 14 Ln.When analyzing the periodicity in the properties of REE compounds, it is necessary to separate the d-element La from the 4f -elements Ln.If not necessary, the sum (La + Ln) is designated R.
The classification of materials with NTE presented in [10] cannot be considered complete.The authors of this article have written this statement.He identified materials with NTE-I (with NTE in a wide temperature (T) interval) and NTE-II (associated with the PolTr in a narrow "window ∆T").
To verify the necessity and/or sufficiency of the empirical features of NTE-II materials [10], it is necessary to verify their compliance with the fundamental rules for constructing equilibrium phase diagrams of chemical systems.The materials formed in the NTE-II systems must comply with thermodynamic rules.
The origin of the features of NTE-II materials is discussed using an example of the phase diagram of this system.Their manifestation in the phase diagram is in line with the rules for constructing phase diagrams.
Such a comparison showed that the common features of NTE-II materials according to [10] (the "window ∆T" and two-phase region) neither separately nor together can determine NTE-II materials.
The necessary and sufficient signs of NTE-II materials are polymorphism and density anomaly.The anomaly is that the formula volume (V form , the volume of one formula unit) of the low-temperature (β-) modification (V low ) is higher than that of the high-temperature (t-) modification (V high ).
Chemical modification by isomorphism of known RF 3 with NTE-II is possible only in a binary (and more complex) system.For trifluorides of R, such systems are RF 3 -R'F 3 .From 17 RF 3 , 136 systems are formed.From 15 fluorides (without fluorides of d-elements: ScF 3 has a structure different from RF 3 and YF 3 ), 105 systems are formed.They are considered in this work.
Four dimorphic fluorides form the structural subgroup (SSGr) B: PmF 3 , SmF 3 , EuF 3 , and GdF 3 .They have the PolTr with the giant NTE-II (∆V/V ~4.7% relative to the smaller value) [9].Such an anomaly is rare for inorganic compounds.The structural mechanism of the giant NTE-II for the β-→ t-PolTr (at heating) is unknown.
RF 3 -R'F 3 systems are the most suitable for the creation of fluoride NTE-II materials.RF 3 are ionic compounds with the simplest formula and high chemical stability.Their melting (T fus ) and PolTr (T trans ) temperatures provide diffusion processes that quickly achieve equilibrium.
A long homologous series of LaF 3 and 14 LnF 3 (Ln = Ce-Lu) consists of extremely chemically similar (ChProx) compounds with a minimum difference in the Z of cations in the series (∆Z = 1).
High-temperature chemical interactions of the components in the RF 3 -R'F 3 systems are described by the ChCl [16].ChCl translates the search for NTE-II materials from 15 RF 3 to a complete array of 105 RF 3 -R'F 3 systems.
ChCl reflects periodic changes in the structure and properties of the RF 3 homological series [16] and the systems formed by them.Unlike the "dimensionless" ChProx, the ChCl is based on the measurable parameter ∆Z = | N Z− M Z|.
The chemistry of the high-temperature interactions of components is visually reflected in the topological features (forms) of phase diagrams.The most common form of weak interaction in the RF 3 -R'F 3 systems is isomorphism.Depending on ∆Z, it varies from perfect to limited.
Stronger forms of interactions are two types of morphotropic transformations.Their topological features are invariant phase reactions: peritectic (MorphTrs-1) and eutectic (MorphTrs-2).The maximum chemical differences between the components and ∆Z (from 8 to 14) are accompanied by the presence of both MorphTr-1 and MorphTr-2 in one system or a complete rupture of the isomorphic miscibility [1].
A chemical design of the RF 3 -R'F 3 systems covers a complete array of 105 binary systems formed by REE fluorides (without ScF 3 and YF 3 ).It is based on studies of the chemistry of high-temperature interactions between components [1].
The structural and chemical modeling of "pseudo-61 PmF 3 " detected the low-temperature "hidden" PolTr in it, gave the values of the lattice parameters, and detected the giant NTE-II [9].
The short "from PmF 3 to TbF 3 " QS defines the limits of the "window ∆T" position on the T axis in the entire array of RF 3 -R'F 3 systems as very wide.
All these studies, especially in recent years, have prepared the formulation of the chemical design of two-component NTE-II materials with REE trifluorides with adjustable properties.
The full range of regulated parameters of fluoride NTE-II materials in the RF 3 -R'F 3 systems can be found from the analysis of the phase diagrams of the studied systems and the "from PmF 3 to TbF 3 " QS [8].The RF 3 -R'F 3 phase diagrams presented in this paper were obtained experimentally.The ratio of the components in them varies from 0 to 100 mol.%.
The aim of this study is to predict, using the principle of isovalent isomorphism, RF 3 -R'F 3 systems in which two-phase materials with NTE-II are formed.
Research objectives: (1) to identify the stages of the chemical design of RF 3 -R'F 3 systems for the search for fluoride NTE-II materials; (2) based on the ChCl of RF 3 -R'F 3 systems, give a forecast of 50 systems (out of 105) with two-phase composites having the giant NTE-II at the β-ss → t-ss PolTr (at heating); (3) using the concept of the "from LaF 3 to LuF 3 " QS [8], determine the range of T trans of components in the area in which the position of the "window ∆T" on the T axis in the "from LaF 3 to LuF 3 " QS is regulated; (4) evaluate the practical applicability of the (β-ss +t-ss) two-phase composite NTE-II materials from the kinetics of phase transformations in the subsolidus region of the studied RF 3 -R'F 3 systems in which they are formed.

Results and Discussion
Four Stages of the Chemical Design of Rare-Earth Fluoride Materials with Adjustable NTE-II Parameters This approach is called chemical design because it uses high-temperature chemical interactions in RF 3 -R'F 3 systems.
The chemical design of materials with NTE-II includes four stages.The first stage involves the choice of "mother" substances with the giant NTE-II.It is identical to the proposed paradigm [10].
The density anomaly at the PolTr is a necessary condition for NTE-II.By describing RF 3 as a single-component material with NTE-II, we are actually doing this for the first time in the field of materials science.
In the literature, the coefficients of a volumetric (α V ) or linear (α L ) thermal expansion CTE are called NTE-II parameters.For the PolTr in single-component RF 3 from the SSGr B, the CTE cannot be defined at ∆T = 0. Therefore, the CTE is not a parameter of NTE-II (see below and [10]).
The second stage involves the choice of isomorphism as a method of modifying the "mother" RF 3 .Isomorphism is the most common method for chemical modification of properties, including NTE-II.
The chemical design by chemical modification of simple RF 3 compounds with the giant NTE-II by means of isovalent isomorphism leads to binary RF 3 -R'F 3 systems as the simplest of the multicomponent systems.
Isovalent isomorphism is exceptionally effective in modifying the properties of REE trifluorides.Studies on the phase diagrams of 34 RF 3 -R'F 3 systems of chemically similar fluorides [1] have shown a wide development of isomorphism and provided digital data for the second stage of the chemical design.
The third stage of the chemical design is new in NTE-II search concepts.It is based on studies of the high-temperature chemistry of the RF 3 -R'F 3 systems.At this stage, the systems in which NTE-II materials are formed are predicted based on the chemical interactions of the components [1].
The possibility of using the chemical design is limited for most homologous series of REE compounds.The number of homologous series of REE compounds with the studied phase diagrams is usually small.It is often combined with an incomplete set of REE.There are no Pm compounds anywhere for the reason described above.The difference in the methods of studying compounds, sometimes separated by decades, reduces the comparability of data from different authors.
The long RF 3 homological series of 15 trifluorides (LaF 3 and 14 LnF 3 with Ln = Ce-Lu) is unique in the completeness of the study of individual trifluorides and the systems formed by them.It is also unique in terms of the high rate of equilibrium in the formation of NTE-II materials (see above).
Fluoride rare-earth NTE-II materials are formed in systems with one or two RF 3 components with the giant NTE-II [4,9].
Without ChCl, the analysis of such RF 3 -R'F 3 systems is impossible.ChCl, which was recently created [16], is the basis of the third stage of the chemical design of systems with multicomponent NTE-II materials.
The third stage is discussed in the following section.For the first time four types of systems (out of 10) in which NTE-II materials are formed were identified in ChCl.
The fourth stage of the chemical design for the study of materials with NTE-II is used for the first time.Quantitative data forming NTE-II were obtained from the analysis of the position of NTE-II materials on a phase diagram, their phase composition, and equilibrium phase reactions in a binary system.They are the values of the ∆V/V volume change of a two-phase composite in the interval T, determined by the "window ∆T".
The "window ∆T" and the ∆V/V = f (T) dependencies are calculated from the phase reaction curves.This stage is based on the thermodynamics of chemical systems, which has not been previously used for the analysis of NTE-II [10].
The proposed scheme for calculating the parameters of NTE-II materials can be applied to each of the studied RF 3 -R'F 3 systems with NTE-II.To achieve this, the liquidus and solidus curves of solid solutions (ss) and the changes in their lattice parameters (V form ) with composition must be determined in these systems.
The special role of structural changes during the PolTr in the formation of NTE-II materials separates them from normal materials possessing NTE-I, in which the growth of ∆V/V occurs with the growth of T.
To date, there is no data on the structural mechanism of NTE-II in RF 3 .Its detection should be the fifth stage in the chemical design of fluoride NTE-II materials.

Materials and Methods
The chemical design of materials with NTE-II among 105 RF 3 -R'F 3 systems is based on ChCl [16].

NTE-II and ChCl of the RF 3 -R'F 3 Systems
ChCl built on the inversely proportional dependency of the degree of ChProx of RF 3 on ∆Z (the difference in Z of cations) [16].ChCl is represented as a coordinate semi-square, Figure 1.

NTE-II and ChCl of the RF3-R'F3 Systems
ChCl built on the inversely proportional dependency of the degree of СhProx of RF3 on ΔZ (the difference in Z of cations) [16].ChCl is represented as a coordinate semi-square, Figure 1.The first level of the ChCl divides 15 RF3 by structural features (a type of structure and the presence of the PolTr) into four structural subgroups (SSGrs A-D) (Table 1).We highlighted the SSGr B in bold, emphasizing the presence of the PolTr with NTE-II.
Table 1.The SSGrs A-D of RF3 in the ChCl [1,16].The coordinate semi-square is a graphical representation of the number of combinations of 15 objects, two in each (C 15  2 ).It is formed by the intersections of 15 columns of the long homologous series from LaF 3 to LuF 3 (without ScF 3 and YF 3 ) and 14 horizontal rows from CeF 3 to LuF 3 .The upper right half of the square is discarded as a replicate.

RF3
There are 105 rectangles left.Each rectangle represents a separate RF 3 -R'F 3 system.The studied systems are indicated by the REE symbols.
The first level of the ChCl divides 15 RF 3 by structural features (a type of structure and the presence of the PolTr) into four structural subgroups (SSGrs A-D) (Table 1).We highlighted the SSGr B in bold, emphasizing the presence of the PolTr with NTE-II.

SSGrs
The GrSs and TypeSs include several systems.To compare the TypeSs and GrSs, two values of the ChProx in each are used: maximum |∆Z max | and minimum |∆Z min | (Table 2) [1,16].Only |∆Z max | is discussed here.The third level of the GrSs is partially shown in Figure 1 in color: for GrS-1 in yellow and dark yellow, for GrS-2 in blue and dark blue, and for GrS-3 in green (the colors are the same as in the ChCl in [16]).

TypeS-N (B-X)
Six TypeSs remained colorless in Figure 1.These TypeSs are divided into two groups.Four TypeSs belong to the first group: TypeS-1 (C-C'), TypeS-2 (A-A'), TypeS-4 (D-D') and TypeS-6 (C-D).The Z av of R from the SSG B cannot be realized in these SSGs.These systems do not contain materials with NTE-II.
The third stage of the chemical design allocates 50 RF 3 -R'F 3 systems (Figure 1) with RF 3 from the SSGr B (R = Pm, Sm, Eu, and Gd) as the sources of fluoride materials with adjustable NTE-II parameters.Only four TypeSs (TypeS-3, TypeS-5, TypeS-7, TypeS-9) (out of 10) contain RF 3 with the β-→ t-PTs (at heating) with NTE-II.
Table 2 lists the number of systems with NTE-II materials in each TypeS.

NTE-II Materials in TypeS-3 (B-B') from RF 3 of the SSGr B
The TypeS-3 (B-B') from the components of the SSGr B includes six systems, as shown in Table 2 and Figure 1.
According to the ChProx these components are heterogeneous.Three systems, PmF 3 -SmF 3 , SmF 3 -EuF 3 , and EuF 3 -GdF 3 , were distinguished from the others in the TypeS-3 by the maximum ChProx |∆Z| = 1 of the components.They are marked in dark yellow in Figure 1.
From full QS stands out the section of SSGr B. Let us call it short QS "from PmF 3 to GdF 3 (TbF 3 )".These systems are marked in dark yellow in Figure 1.This short QS is used to analyze the dependence of the position of RF 3 (R = Pm-Tb) T trans .
Three particular systems of the short QS are shown in Figure 2. NTE-II during the PolTr in the SmF 3 -EuF 3 system is indicated by an arrow.Similar PolTrs exist in two adjacent systems PmF 3 -SmF 3 and EuF 3 -GdF 3 .
From full QS stands out the section of SSGr B. Let us call it short QS "from PmF3 to GdF3(TbF3)".These systems are marked in dark yellow in Figure 1.This short QS is used to analyze the dependence of the position of RF3 (R = Pm-Tb) Ttrans.
Three particular systems of the short QS are shown in Figure 2. NTE-II during the PolTr in the SmF3-EuF3 system is indicated by an arrow.Similar PolTrs exist in two adjacent systems PmF3-SmF3 and EuF3-GdF3.They do not belong to the short QS, but are included in the list of systems from the TypeS-3 with NTE-II materials (Table 3).Continuous ss are formed between their components.The structural modifications of the components involved in the PolTrs cause NTE-II.These systems are marked in yellow in Figure 1.

No. B-B'
PmF3-GdF3 3 In Figure 2, the MorphTr in the GdF3-TbF3 system is added to the two PolTrs considered.This is the true MorphTr (according to V.M. Goldschmidt [11]).This is the MorphTr of the first type (MorphTr-1) between the β-ss and t-ss.The appearance of the MorphTr-1 in the GdF3-TbF3 system is caused, as in RF3, by a change in the structure type.The change proceeds in accordance with the mechanism of the MorphTr, an invariant (T = const) peritectic phase reaction with the melt (Liq) [1]: -Gd0.49Tb0.51F3 Liq (melt) + t-Gd0.57Tb0.43F3

NTE-II Temperature Control in the Short "from PmF3 to TbF3" QS
The position of the ΔV/Vform jump of ΔТ on the T axis plays a critical role for the practical use of NTE-II materials.Most applications require room temperature.In the short "from PmF3 to TbF3" QS ΔTtrans can be controlled by the RF3-R'F3 systems.
Dimorphic RF3 (R = Pm-Gd) have a uniquely large range of variation in Ttrans during the PolTrs.In RF3-R'F3 systems, there is a possibility of more "subtle" control of ΔTtrans by isomorphic substitutions.
The limit of Тtrans values for two-component materials with NTE-II is determined by the short QS of the full QS which contains RF3 of the SSGr B (R = Pm, Sm, Eu, and Gd).Curve a-b-c-d-e  They do not belong to the short QS, but are included in the list of systems from the TypeS-3 with NTE-II materials (Table 3).Continuous ss are formed between their components.The structural modifications of the components involved in the PolTrs cause NTE-II.These systems are marked in yellow in Figure 1.In Figure 2, the MorphTr in the GdF 3 -TbF 3 system is added to the two PolTrs considered.This is the true MorphTr (according to V.M. Goldschmidt [11]).This is the MorphTr of the first type (MorphTr-1) between the β-ss and t-ss.The appearance of the MorphTr-1 in the GdF 3 -TbF 3 system is caused, as in RF 3 , by a change in the structure type.The change proceeds in accordance with the mechanism of the MorphTr, an invariant (T = const) peritectic phase reaction with the melt (Liq) [1]: β-Gd 0.49 Tb 0.51 F 3 ↔ Liq (melt) + t-Gd 0.57 Tb 0.43 F 3

NTE-II Temperature Control in the Short "from PmF 3 to TbF 3 " QS
The position of the ∆V/V form jump of ∆T on the T axis plays a critical role for the practical use of NTE-II materials.Most applications require room temperature.In the short "from PmF 3 to TbF 3 " QS ∆T trans can be controlled by the RF 3 -R'F 3 systems.
Dimorphic RF 3 (R = Pm-Gd) have a uniquely large range of variation in T trans during the PolTrs.In RF 3 -R'F 3 systems, there is a possibility of more "subtle" control of ∆T trans by isomorphic substitutions.
The limit of T trans values for two-component materials with NTE-II is determined by the short QS of the full QS which contains RF 3 of the SSGr B (R = Pm, Sm, Eu, and Gd).Curve a-b-c-d-e, Figure 2, in the short "from PmF 3 (point a) to GdF 3 (Gd 0.57 Tb 0.43 F 3 , point e)" QS presents the dependence of T trans on Z. Materials with this ∆T trans and Z have NTE-II.
With the help of the chemical design of NTE-II materials in RF 3 -R'F 3 systems, T trans can be changed from −148 • C in PmF 3 [9] to 1186 ± 10 • C for the MorphTr-1 with t-Gd 0.57 Tb 0.43 F 3 [1].
According to [10], the "window ∆T" of known NTE-II materials does not exceed 600 • C. Fluoride NTE-II materials surpass it by almost three times.
Fluoride NTE-II materials have a problem of accessibility when they need to be used at standard (room) T. The continuation of the curve a-b in Figure 2 to the room T region corresponds to the PmF 3 -SmF 3 system with an inaccessible PmF 3 .
This problem is solved by the wide possibilities of carrying out structural and chemical modeling of the "lanthanide" for any average value of the atomic number Z av between 58 Ce and 71 Lu [9].The necessary short QS is selected for modeling.The model composition with Z av can be calculated for the required T trans .
Structural and chemical modeling of "lanthanides" with an intermediate Z av moves the NTE-II "window ∆T" of the adjustable interval on the T scale (see above).This provides a solution to the availability problem (PmF 3 ) and high cost of some REE, allowing the replacement of unavailable RF 3 .
The temperature range in which the chemical design can adjust T trans in RF 3 follows from the short "from PmF 3 to GdF 3 (Gd 0.57 Tb 0.43 F 3 )" QS (the curve a-e, Figure 2).The isomorphism continuously changes the T trans of the R 1-x R' x F 3 PolTrs from (−148 • C) in PmF 3 (and lower in the unexplored part of the NdF 3 -PmF 3 system) up to 1186 ± 10 • C for MorphTr-1 with t-Gd 0.57 Tb 0.43 F 3 .
In the other three systems, |∆Z max | increases to 2 and 3.This does not limit the formation of phases with NTE-II in the systems of the TypeS-3.

NTE-II Materials in the TypeS-5 (B-C)
There are 12 systems in the TypeS-5 (Table 4).The phase diagrams of the GdF 3 -TbF 3 and GdF 3 -DyF 3 systems are shown in Figures 2 and 3.For the TypeS-5 |∆Z max | = 6, which is twice that for the TypeS-3.

NTE-II Materials in the TypeS-7 (A-B)
The TypeS-7 contains 16 systems (Table 5) with |∆Z max | = 7 from RF 3 of different SSGrs.In the NdF 3 -PmF 3 system, the first component is the last in the SSGr A, and the second component is the first component in the subsequent SSGr B. As a result, the NdF 3 -PmF 3 system has |∆Z| = 1, which refers to the QS.This system is borderline, similar to the GdF 3 -TbF 3 system.The phase diagram of system has not yet been studied.
The high |∆Z max | makes β-ss limited, leaving continuous t-ss at high temperatures (Figure 4 and Table 2).
The TypeS-7 contains 16 systems (Table 5) with |ΔZmax| = 7 from RF3 o SSGrs.In the NdF3-PmF3 system, the first component is the last in the SSGr A, and component is the first component in the subsequent SSGr B. As a result, the N system has |ΔZ| = 1, which refers to the QS.This system is borderline, similar to TbF3 system.The phase diagram of this system has not yet been studied.
The high |ΔZmax| makes β-ss limited, leaving continuous t-ss at high tem (Figure 4 and Table 2).The TypeS-9 consists of 16 systems with one component from the SSGr B (Table 6).In this TypeS, |∆Z max | reaches 10.A large chemical difference in the Z values of the components is manifested in the difference in the structural types of the components.In the SSGr D, to which the second components of the TypeS-9 systems belong, the α-type appears at high temperatures.

NTE-II Materials in the TypeS-8 (A-С) and TypeS-10 (A-D)
The remaining TypeS-8 (A-C) and TypeS-10 (A-D) implement Zav, corres the Zs of R, which form fluorides from the SSGr B (from Pm to Gd).The first T |ΔZmax| = 10 (similar to TypeS-9).The second has the highest |ΔZmax| = 14 for The phase diagrams at high |ΔZmax| become very complicated.

NTE-II and Kinetics of Phase Formation in RF 3 -R'F 3 Systems
The features of NTE-II materials relate only to the equilibrium state.Structural transformations of the phases with NTE-II along the curves of their decay (formation) occur in the solid state.To achieve equilibrium, disinhibited changes in the phase composition are required.
The processes of ss ordering in the subsolidus and the formation of double compounds are usually inhibited.There are no such processes in the studied RF 3 -R'F 3 systems [1].
"Liquid-solid" transitions are disinhibited.In the 34 studied systems, the thermal effects of crystallization (liquidus and solidus) were well separated by thermal analysis [1].According to the kinetics of melt crystallization of the studied RF 3 -R'F 3 systems during thermal analysis and the thermal effects of phase transformations, two-phase composite NTE-II materials, β-ss +t-ss, formed during phase reactions are in equilibrium with a high probability.
In this paper, 50 (out of 105) RF 3 -R'F 3 systems in which these materials may form were selected using the principle of isovalent isomorphism.
Composite materials with NTE-II are also formed in MF 2 -RF 3 systems with M = Ca, Sr, Ba; R = Gd-Lu, Y, and NaF-RF 3 with R = Gd, Tb [1].The LaF 3 -type structure has a high isomorphic capacity (up to 35% mol.) in relation to REE ions.When part of the R 3+ cations is replaced with M 2+ (Na + ) cations (aliovalent isomorphism), berthollide phases with the t-type structure are formed [1,17,18].When the temperature decreases, their decay occurs over the temperature range (the t-→ β-PolTr with "window ∆T" > 0) with the formation of β-type phases [1].The thermodynamic aspects of the phase transitions in the NTE-II berthollide phases are not included in the objectives of this study.

Figure 1 .
Figure 1.The chemical design of 50 (out of 105) systems with the NTE-II materials (shown in color).The coordinate semi-square is a graphical representation of the number of combinations of 15 objects, two in each (C 15 2).It is formed by the intersections of 15 columns of the long homologous series from LaF3 to LuF3 (without ScF3 and YF3) and 14 horizontal rows from CeF3 to LuF3.The upper right half of the square is discarded as a replicate.There are 105 rectangles left.Each rectangle represents a separate RF3-R'F3 system.The studied systems are indicated by the REE symbols.The first level of the ChCl divides 15 RF3 by structural features (a type of structure and the presence of the PolTr) into four structural subgroups (SSGrs A-D) (Table1).We highlighted the SSGr B in bold, emphasizing the presence of the PolTr with NTE-II.

Figure 1 .
Figure 1.The chemical design of 50 (out of 105) systems with the NTE-II materials (shown in color).

Figure 2 .
Figure 2. The chemical design of Ttrans in the short QS "from PmF3 to GdF3(TbF3)".Curve a-b-c-d-e, in the short "from PmF3 (point a) to GdF3 (Gd0.57Tb0.43F3,point e)" QS presents the dependence of Ttrans on Z.The PmF3-EuF3, SmF3-GdF3, and PmF3-GdF3 systems of the TypeS-3 have |ΔZ| > 1.They do not belong to the short QS, but are included in the list of systems from the TypeS-3 with NTE-II materials (Table3).Continuous ss are formed between their components.The structural modifications of the components involved in the PolTrs cause NTE-II.These systems are marked in yellow in Figure1.

,
Figure 2, in the short "from PmF3 (point a) to GdF3 (Gd0.57Tb0.43F3,point e)" QS presents the dependence of Ttrans on Z. Materials with this ΔTtrans and Z have NTE-II.

Figure 2 .
Figure 2. The chemical design of T trans in the short QS "from PmF 3 to GdF 3 (TbF 3 )".Curve a-b-c-d-e, in the short "from PmF 3 (point a) to GdF 3 (Gd 0.57 Tb 0.43 F 3 , point e)" QS presents the dependence of T trans on Z.The PmF 3 -EuF 3 , SmF 3 -GdF 3 , and PmF 3 -GdF 3 systems of the TypeS-3 have |∆Z| > 1.They do not belong to the short QS, but are included in the list of systems from the TypeS-3 with NTE-II materials (Table3).Continuous ss are formed between their components.The structural modifications of the components involved in the PolTrs cause NTE-II.These systems are marked in yellow in Figure1.

Table 4 .*Figure 3 .
Figure 3.The GdF3-DyF3 system of the TypeS-5 (B-C) with the β-ss  t-ss PolTr wi 0 is marked with a red arrow.The two-phase (t-+ β-) region is highlighted in blue.

Figure 4 .
Figure 4.The scheme of the СeF3-GdF3 system of the TypeS-7 (A-B) with NTE-II materia phase (t-+ β-) region is highlighted in blue.

3. 4 . 3 .
NTE-II materials in the TypeS-9 (B-D)The TypeS-9 consists of 16 systems with one component from the SSGr B In this TypeS, |ΔZmax| reaches 10.A large chemical difference in the Z values o ponents is manifested in the difference in the structural types of the compone

Figure 5 .
Figure 5.The GdF3-ErF3 system from the TypeS-9 (B-D) with NTE-II materials.The two β-) region is highlighted in blue.

Figure 5 .
Figure 5.The GdF 3 -ErF 3 system from the TypeS-9 (B-D) with NTE-II materials.The two-phase (t-+ β-) region is highlighted in blue.The chemical design with the ChCl highlights 50 systems (out of 105) of the four TypeSs in which the presence of NTE-II materials is possible.One or both components of these systems are selected from dimorphic RF 3 from the SSG B(R = Pm, Sm, Eu, and Gd).They have the β-ss → t-ss PolTrs (at heating) associated with NTE-II.Only 11 of the 50 systems were studied.They are shown in bold in Tables 3-6.

3. 5 .
NTE-II Materials in the TypeS-8 (A-C) and TypeS-10 (A-D) The remaining TypeS-8 (A-C) and TypeS-10 (A-D) implement Z av , corresponding to the Zs of R, which form fluorides from the SSGrB (from Pm to Gd).The first TypeS-8 has |∆Z max | = 10 (similar to TypeS-9).The second has the highest |∆Z max | = 14 for all TypeS.The phase diagrams at high |∆Z max | become very complicated.

Table 2 .
Four TypeSs with components from the SSGr B with NTE-II materials.
* The studied systems are shown in bold.
* The studied systems are shown in bold.