**4. Discussion**

#### *4.1. Experiment vs. Calphad Predictions*

The Calphad approach is used widely in the high entropy alloys and ceramics field [74–76]. Senkov et al. demonstrated [74] that for high entropy alloys, Calphad computations of type and number of phases agreed well with experimental results only when more than half of binary systems were fully assessed; however, even when there are enough data on binary and ternary systems, Calphad predictions of transformation temperatures, phase compositions and fractions are challenging. Compared with metals, the databases for oxides are much more limited, especially for the temperature range addressed in this study. Nevertheless, it is instructive to compare our experimental results with Calphad predictions.

We used an open-access database created by Zinkevich [67]. It is based on his 2007 assessments of thermodynamic properties of pure rare earth sesquioxides from room temperature to the melting point. This database is often used as a starting point for the creation of new thermodynamic assessments for systems with rare earth oxides [77,78]. Without any modifications, the database allows modeling of phase equilibria using the ideal solution model (which assumes zero enthalpies of mixing and the

largest configurational entropy gains). A set of interaction parameters for the regular solution model (which accounts for mixing enthalpies) are included in the Zinkevich review [67], but they are based on a limited number of selected binary systems, not included in the database [70] and were not used in the present study. The phase fractions as a function of temperature from Calphad modeling are shown in Figure S14.

#### 4.1.1. C-B Transition

After calcination at 800 ◦C, C-type was a major phase in all compositions and was the only phase detected in the HE-Y sample (Table 2). Calphad computations correctly predicted the formation of the C-type single-phase solid solution in HE-Y and as a major phase in HE-Gd and HE-Nd. After annealing at 1100 ◦C, a single-phase C-type solid solution was retained in the HE-Y sample, B-type solid solution became a major phase in He-Gd sample He-Nd completely transformed to B-type structure. The results for HE-Nd and HE-Gd are consistent with Calphad predictions; however, in HE-Y the B-type phase was predicted to appear but was not observed experimentally.

Laser melting and splat quenching produced B-type solid solutions in all samples. Calphad calculations predicted B-type single-phase field for all compositions, with low-temperature boundary shifting from 900 to 1400 ◦C with decreasing average ionic radius of RE from HE-Nd to HE-Y. The heating and cooling of laser-melted samples in DTA and two months of annealing at 800 ◦C did not reverse the transformation. The B-C transformation below 1000 ◦C is known to be kinetically hindered in pure oxides as well [79]; thus, retaining the B phase in our experiments does not indicate slower di ffusion in multicomponent compositions.

#### 4.1.2. B-A-H-X Transitions

Calphad calculations correctly predicted B-A transformation in HE-Nd and HE-Gd samples. DTA experiments indicated that transformation proceeds over a 40–70 ◦C range; however, from Calphad computations, B-A transformation proceeds over a temperature range of several hundred degrees with a change in the fractions and compositions of phases at equilibrium. For HE-Y composition, B-type was predicted to transform directly to the H phase over a narrow biphase region. In contrast, experimental results indicate the B-A transformation in all samples. In agreemen<sup>t</sup> with the experiment, the formation of single-phase solid solutions in H and X structure types was predicted for all compositions. H-X transformation was predicted to proceed over an indistinguishably narrow temperature range, with e ffectively congruen<sup>t</sup> melting at circa ~2400 ◦C. The narrow biphasic regions on H-X transition and congruen<sup>t</sup> melting from Calphad modeling is consistent with single peaks in DTA.

## 4.1.3. Biphasic Fields

In a five-component system at constant pressure and temperature, the phase rule allows for up to six phases. In the experiments, we never observed more than two phases at any temperature. Calphad modeling showed mostly single-phase or biphasic fields and very narrow temperature ranges of three-phase co-existence (A, B, C and B, A, H). The absence of large biphasic fields for the B-A transformation from DTA measurements compared with Calphad computations might be attributed to not reaching equilibrium in scanning experiments below 2000 ◦C. However, the very narrow temperature ranges for H-X transformation and melting are predicted from Calphad and observed in DTA and are likely to be real. In experimental binary phase diagrams of rare earth oxides, the biphasic fields on melting and high temperature transformations are usually not resolved and often added as suggested dotted lines with expectation for them to occur [33,80,81]. Calphad modeling indicates that biphasic fields on A-H-X-L transitions in most binary rare earth oxide systems are often extremely small (~20 ◦C or less) [67].

It is tempting to assume that observed shrinkage of two-phase fields is the e ffect of the entropy. Indeed, in many phase diagrams, multiphase fields usually shrink with temperature as solid solution and melt ranges increase. However, that is not always the case. For example, the liquidus loop in the

Os-Re system at 3100 ◦C is as wide as in Cu-Ni system at 1300 ◦C and shows 20 at.% di fference in composition between solid and liquid phases. The narrow two-phase fields for high temperature phase transformations and melting are peculiar to intra-rare earth systems. It was demonstrated in rare earth metal binaries, which melt below 1600 ◦C and studied more extensively and with higher accuracy than oxide systems [82,83]. Apparent congruen<sup>t</sup> melting across intra-rare earth binaries was discussed in detail by Okatomo and Massalski [83]. Spedding et al. [84] concluded the study of Er-Y and Tb, Dy, Ho, Er binaries with the statement "the isothermal arrests observed for the melting and transformation temperatures show that there is no appreciable enrichment of one component over the other during these processes, which is what would be expected from the general experience encountered in trying to separate rare earths."

#### *4.2. Thermal Expansion and Volume Change on Mixing*

A peculiar feature of rare earth oxides is the negative volume change on temperature-induced C-B, B-A, and H-X transformations. In this work, volumes of C- and B-type solid solutions at room temperature were obtained from the analysis of quenched samples, and volumes of B-, A-, H-, and X-type solid solutions were obtained at temperatures of B-A and H-X transitions. This allowed calculation of volume changes on C-B transformation at room temperature, and for B-A and H-X transformation at transition temperatures. Axial thermal expansion of the monoclinic B-type phase was derived from unit cell parameters measured at B-A transition and on samples quenched to room temperature. In the sections below, we compare the behavior of high entropy compositions with pure oxides.

#### 4.2.1. Volume Change on C-B Transition

All rare earth sesquioxides can be obtained in their ground-state C-type structure at room temperature. The B-type structure is both a high pressure and a high temperature phase for sesquioxides from Sm to Ho, and high pressure phase for sesquioxides from Er to Lu, Y and Sc [85–88]. In Figure 5, volumes of C and B phases for samples studied in this work are plotted vs. average ionic radius together with volumes of pure sesquioxides. The volumes of B-type phase in HE samples show no deviation from the trend. For C-type solid solution, only HE-Y sample was obtained as single-phase (in HE-Nd and HE-Gd samples C-type phase coexisted with B-type phase). Nevertheless, the volumes of C-type phase in HE samples show good agreemen<sup>t</sup> with the trend.

**Figure 5.** Volumes of C and B phases (**a**) at room temperature and H and X phases (**b**) at transition temperature in pure oxides [30,67] compared with (La0.2Sm0.2Dy0.2Er0.2RE0.2)2O3 HE-RE compositions studied in this work, where RE–Nd, Gd, or Y, plotted vs. average ionic radius.

The volume contraction on C-B transition in HE samples (~8%, Table 2) is indistinguishable from those in constituent oxides, and the B-type phase is both a high temperature and high pressure phase for the studied compositions. This comparison gives no indication of strong deviations from ideal mixing in the solid solution in the B-type structure. For ideal solid solutions, Gibbs energy of mixing does not depend on pressure and there is no excess volume of mixing. For the equiatomic compositions studied in this work, the volume of an ideal solid solution is an average of corresponding volumes of constituent oxides, thus they would follow the trend of volume change vs. average ionic radius.

#### 4.2.2. Thermal Expansion of B-Type Solid Solutions

Taylor [89] reviewed thermal expansion data for pure rare earth sesquioxides and found no data on axial expansion for B-type phases. Ploetz et al. [90] studied the linear expansion of B-type Gd, Eu, and Sm sesquioxides by interferometry and reported the values (10.0–10.8) × 10−<sup>6</sup> /K for 30 to 850 ◦C range. Since the Ploetz et al. measurements were performed on polycrystalline samples, volumetric thermal expansion can be estimated as three times the linear expansion and corresponds to (3.0–3.2) × <sup>10</sup>−<sup>5</sup>/K. These values are in good agreemen<sup>t</sup> with average volume thermal expansion for multicomponent rare earth oxides measured in this work: (3.5 ± 0.2) × <sup>10</sup>−<sup>5</sup>/<sup>K</sup> from room temperature to the B-A transformation temperatures (1916–1975 ◦C).

#### 4.2.3. Volume Changes on B-A and H-X Transformations

The volume change on B-A transformation was refined for HE-Nd and HE-Gd samples as −2.5 ± 0.1% and for HE-Y sample as −3.1 ± 0.1% The volume change on H-X transformation was refined for He-Nd and HE-Y samples as−1.2±0.1 and−0.6±0.1%, respectively. Since A, H, and X phases were not quenchable in these compositions, our values were determined from diffraction experiments at the respective transition temperatures. The volume change on B-A and H-X transformation reported for pure oxides are about −2% and −0.5%, respectively [30,67]. Thus, within the resolution of the data, we do not observe any anomalies in volume change on B-A, and H-X transformation in the studied solid solutions compared with pure oxides.

#### 4.2.4. Volumes of H-Type Solid Solution vs. Pure Oxides

In Figure 5, the volumes of the H-type phase for HE-Nd and HE-Y are overlaid with values reported by Foex and Traverse [30] for pure oxides. Foex and Traverse's values correspond to the temperatures from 2120 ◦C for La2O3 to 2330 ◦C for Ho2O3 and Y2O3. The volume for the H phase for Sm2O3 and Gd2O3 refers to the temperatures 2200 and 2250 ◦C, respectively. Our data on the volume of H phase in HE-Nd and He-Gd from diffraction on levitated samples were assigned to temperatures 2202 ± 4 ◦C and 2254 ± 8 ◦C based on co-existence with X-type phase (Table 5) and DTA results. Thus, we can compare volumes of solid solutions vs. pure oxides at a similar temperature. The observed agreemen<sup>t</sup> in volume is within 2%. In contrast with the calculation of volume change on transition in coexisting phases, XRD measurement of the absolute values for the volumes at high temperature are affected by sample shift and temperature calibration. Thus, within the resolution of the data, we do not observe deviations from ideal volumetric mixing in the H-type solid solution for the studied compositions.

#### *4.3. Increase of Melting Temperature in HE-RE*

The measured melting temperature for HE-Nd is 2456 ± 12 ◦C. The error is assigned based on uncertainty in melting temperature of Y2O3 used for calibration, 2439 ± 12 ◦C [73]. If the solid solution were to melt congruently, one might estimate, to a very crude first approximation, its melting point as a weighted average of the endmember melting points [91], namely 2357 ± 16 ◦C. The uncertainties assigned to melting points of constituent oxides in HE-Nd (La,Sm,Dy,Er,Nd)2O3 composition are similar to Y2O3 (±15 vs. ±12 ◦C), however, Y2O3 melting temperature was established from independent measurements in ten laboratories [73], while values for Dy2O3 and Er2O3 are based on results of one group and regarded only as "probable" [91].

This observation of apparent increase of melting temperature compare with constituent oxides may be attributed to a combination of kinetic and thermodynamic factors. If the observed behavior does represent equilibrium, then the solidus–liquidus relations must be strongly perturbed by nonideal mixing behavior in the solid, liquid, or both.

To raise the melting temperature, nonideality must favor the solid over the liquid. This can occur by one or both of the following: negative deviations from ideality in the solid phase or positive deviations in the liquid phase. Negative deviations in the solid solution are generally associated with local ordering, hinting at a tendency toward compound formation, and resulting in negative heats of mixing. The volumetric behavior described above shows no evidence for ordering, but volume is not necessarily a good proxy for energetics. The other possibility is strongly positive deviations from ideal mixing in the molten oxides, leading, in the extreme, to liquid–liquid phase separation. We know very little about the structure and thermodynamics of molten binary or multicomponent rare earth oxide systems, except for a recent report from Nakanishi and Allanore [92] on non-ideal mixing in La2O3-Y2O3 melt, so this possible scenario cannot be assessed at present.

Increases in melting temperature in multicomponent systems compared with end members is not common in oxides, however, it is known to occur, most notably on the ZrO2-rich side of binaries with Dy-Yb and Y sesquioxides, where melting temperature increase over 100 ◦C compared with pure ZrO2 for the solid solution with 25 mol. % of Yb2O3 [93]. This almost certainly relates to the formation of favorable short-range order and defect clustering involving tetravalent and trivalent ions and oxygen vacancies. Such a short-range order probably is less important in the trivalent oxide systems studied here. In our system, it is probable that the melting temperature increase should be manifested at least in one of the related binary systems. No increase in melting temperatures was reported in studied binaries with La2O3 [33,94,95]. Sm, Dy, and Er are a common constituent in all HE compositions studied in this work. The study of liquidus in the binary systems, currently unknown, is needed to further identify the extent and source of the increase of melting temperatures observed in HE-RE compositions.
