Probing disorder in pyrochlore oxides using in situ synchrotron diffraction from levitated solids–A thermodynamic perspective

Pyrochlore, an ordered derivative of the defect fluorite structure, shows complex disordering behavior as a function of composition, temperature, pressure, and radiation damage. We propose a thermodynamic model to calculate the disordering enthalpies for several RE2Zr2O7 (RE = Sm, Eu, Gd) pyrochlores from experimental site distribution data obtained by in situ high temperature synchrotron X-ray diffraction. Site occupancies show a gradual increase in disorder on both cation and anion sublattices with increasing temperature and even greater disorder is achieved close to the phase transition to defect fluorite. The enthalpy associated with cation disorder depends on the radius of the rare earth ion, while the enthalpy of oxygen disordering is relatively constant for different compositions. The experimental data support trends predicted by ab initio calculations, but the obtained enthalpies of disordering are less endothermic than the predicted values. Thermal expansion coefficients are in the range (8.6–10.8) × 10−6 K−1. These new experimental determinations of defect formation energies are important for understanding the stability of pyrochlore oxides and their disordering mechanisms, which are essential in the context of their potential applications in nuclear waste management and other technologies.

SCiENTifiC RePORts | (2018) 8:10658 | DOI: 10.1038/s41598-018-28877-x found a significant difference in Zr-La and Ti-La CA DFE, supporting the observation that titanate PYs may be unfavorable for radioactive waste immobilization as radiation damage causes amorphization rather than disordering 24 . All of the reported computational studies support experimental correlations that link observed PY -DF transformations with simple radius ratio (r A /r B ) rules 9 . However, the predicted energies (at least 2-3 eV) 20,22,24 appear to be too large for significant disordering to occur prior to melting. On the other hand, recent DFT studies on select PY oxides show that these energies are substantially smaller than those predicted with FFA methods 25 . It is clear that the computational studies need comparison with experimental determinations of DFE. Our goal is to develop an in situ experimental method for quantitative determination of site distributions from which DFE can be calculated by a thermodynamic model. In situ structural studies are essential because the high temperatures involved in equilibrium disordering make it questionable whether the disordering can be preserved on "quenching" the sample to ambient conditions. Indeed, numerous authors have studied functional properties of PY oxides with different degrees of disorder made by quenching from high temperatures, but they achieved very low percentages of disorder (5-22%) and did not reach the fully disordered state [26][27][28] .
The disordering in PY oxides may be viewed as an equilibrium reflecting the balance of a positive enthalpy of interchange on cation and on anion sublattices and a positive configurational entropy of disordering. Such gradual disordering in spinels has been described by a simple thermodynamic model, treating cation distribution as a chemical equilibrium at a given temperature 29 . The cation distribution as a function of temperature can be used to calculate the appropriate interchange enthalpies for both CA and AFP. In this study, a similar thermodynamic model is applied to the disordering in PY oxides, with independent reactions representing CA and AFP disorder. For the first time, we have adopted a high temperature in situ diffraction technique to induce equilibrium cation and anion disorder within the PY samples by thermal treatment close to the melting point using aerodynamic levitation and laser heating. The site occupancies (atomic sublattice disorder), lattice constants and positional parameters were refined at a number of temperatures by Rietveld structure analysis, and the obtained site occupancies were then subjected to thermodynamic analysis.

Results and Discussion
Composition and thermal expansion. Chemical analysis of the melt-quenched spheroids by microprobe showed stoichiometric composition within experimental error, see Table 1. The room temperature synchrotron X-ray diffraction pattern of all the melt-quenched compositions, showed intense and sharp patterns, and no impurity phases could be detected ( Figure S1 in SI). All the compositions showed DF-based diffraction peaks, (222), (400), (440) and (622). Also, superlattice reflections, (111), (311), (331) and (511)  For all the compositions tested, the unit cell size changes smoothly as a function of temperature till melting. No anomalous lattice constant behavior is detected at high temperature, confirming that the applied temperature corrections are reasonable. For comparison, the temperature range selected for linear regression analysis was fixed to 1123-2323 K. The variation of thermal expansion coefficients (TEC) vs. temperature is within reported uncertainty from the linear regression fit, so a temperature independent TEC is supported. Some general trends can be noticed from Table 2 and Fig. 1, TEC increases as the size of the lanthanide ion decreases, and the zirconate series has higher TEC than the hafnate series. TEC of studied compositions varies from (8.6 to 10.8) × 10 −6 K −1 , the lowest TEC being observed for La 2 Hf 2 O 7 . The highest TEC is seen for Sm 2 Zr 2 O 7 and Gd 2 Zr 2 O 7 = (10.8 ± 0.5) × 10 −6 K −1 and (10.6 ± 0.3) × 10 −6 K −1 . The TEC values within the studied temperature range, 1123-2373 K, measured by the levitation method, agree well with previously reported values of 8-11 × 10 −6 K −1 in the 298-1473 K temperature range 31 .  Figure S4 in SI). It is known that Nd 2 Zr 2 O 7 shows a PY to DF phase transition, with a reported transition temperature between 2493 and 2573 K 1,32 . In this study, the (311) reflection disappears at 2423 K, within the reported temperature range, confirming the existence of the phase transition. However, the presence of residual superlattice reflections till melting may be due to the presence of small PY domains within the DF matrix 33 . Also, inhomogeneity in specimen temperature could cause the observed residual superlattice reflections. For Sm 2 Hf 2 O 7 , the reflections of the PY structure intensify up to 2123 K, but further increase in temperature weakens the superlattice reflections, that then disappear completely, indicating transformation to DF just before melting at 2823 K ( Figure S5 in SI). For Sm 2 Zr 2 O 7 , the superlattice reflections disappear, indicating transformation to DF at 2323 K, and melting occurs at 2823 K ( Figure S6 in SI). For Eu 2 Zr 2 O 7 , previously we reported a reversible PY-DF phase transition at 2173 K during heating and 2073 K during cooling ( Figure S7 in SI) 5 Table S1 in the SI. Figure 2(a-d) shows the 2D synchrotron diffraction images tracking the structure change as a function of temperature and the corresponding Rietveld refinement plots. Since the sample is initially in the metastable DF phase (beads made by melt quench), during heating superlattice reflections (111), start to appear around 1473 K and become significant at 1673 K (Fig. 2a). Further increase in temperature to 1873 K shows minor superlattice reflections (Fig. 2b), while   Corresponds to thermal expansion coefficient at 298 K; ‡ Calculated as α = (a (2323K) − a (1123 K) )/a (1123K) /1200; number in parenthesis represents the decimal point variation. above this temperature no (111) reflection is visible (Fig. 2C), confirming the phase transition to DF. Similar behavior is also observed during cooling; from 2573 to 1773 K there is a (111) superlattice reflection indicative of PY structure (Fig. 2d). Thus, a stepwise first-order (with some hysteresis) reversible phase transformation from ordered (PY) to disordered (DF) is captured for Gd 2 Zr 2 O 7 . We conducted high-temperature differential thermal analysis (DTA) on Sm 2 Table 3). These observations confirm that aerodynamic levitation combined with in situ diffraction is a very reliable technique to study the phase equilibria of high-temperature ceramics. Figure 4(a,b) presents the temperature dependence of antisite cationic and oxygen Frenkel occupancies of Sm 2  From the current study, the percentages of cationic and anionic disorder as a function of temperature are given in Fig. 4(c). Gd 2 Zr 2 O 7 shows the greatest cationic and anionic disorder, 34.3 ± 1.9% and 16.8 ± 1.6%, respectively. It is well-known that heavier lanthanides do not form ordered zirconates with PY structure and Gd is considered the borderline lanthanide between PY and DF structures 9 , so its greater disordering is not surprising. Direct comparison of refinements is not fully appropriate since our study includes Rietveld refinement of both cation and anion disorder whereas the majority of published results refined only cation antisite disorder 26,36,37 . However, quenching PY samples such that they would retain different equilibrium degrees of disorder may be difficult or even impossible, particularly for lighter lanthanides which strongly prefer ordered structures. In contrast, aerodynamic levitation and laser heating in combination with in situ synchrotron diffraction is a versatile technique to obtain the actual degree of disorder under high temperature conditions. Previously, we have shown the systematic behavior of cation and anion disorder in Eu 2 Zr 2 O 7 during heating and cooling using this methodology 5 .
Change of 'x' (48f) as a function of temperature. Disordering as a function of temperature can also be observed by monitoring the changes in the 48f oxygen 'x' positional parameter. In a large number of PYs, the 'x' parameter lies well below 0.375 9 . In the present study, 48f oxygen 'x' parameters for Sm 2 (2) and 0.345 (2), respectively at room temperature. For Eu 2 Zr 2 O 7 and Gd 2 Zr 2 O 7 , since the initial melt-quench phases are DF, the given 'x' values are after quenching from the melt. As the 48f oxygen positional 'x' parameter changes smoothly with temperature, until reaching the PY-DF phase boundary, further increase in temperature abruptly raises the value toward 0.375 (see Fig. 4d). In Sm 2 Zr 2 O 7 , 'x' is 0.355(3) at 1123 K, showing slightly higher values as it has more disorder due to melt quench, and 'x' is lowered to 0.345(2) at 2273 K, indicating the change toward the ideal PY structure. Increasing the temperature to 2373 K raises 'x' to 0.352(2), and 'x' reaches a maximum of 0.361(5) at 2523 K close to the value for DF (0.375). In Gd 2 Zr 2 O 7 , during   7 5 , and the trend correlates well with the current study. The change in lattice constant, refined occupancies, 48f oxygen 'x' positional parameter, % disorder (both cation and anion), and agreement factors (R-factors namely Bragg R, R f -factor, and Chi2) during heating and cooling as a function of temperature are documented in Tables S2, S3 and S4 in SI.

Thermodynamic Model of Disordering
Any high temperature equilibrium phase transactions involving order-disorder reflects a balance between the enthalpy of disordering and the configurational entropy created by disorder on crystallographic sites on the available sublattices. The equilibrium disordering can occur gradually with temperature, representing a second order or more complex transition, or it can occur sharply at one temperature (sometimes with hysteresis), representing a first order transition. In real systems, a transition can show complex behavior, with gradual disordering over a temperature range culminating in a first order transition, or with some short-range order persisting in the high temperature nominally disordered phase. A first-order transformation is accompanied by an abrupt change in enthalpy and entropy. The Gibbs free energy difference between the two phases is and since ΔG = 0 at the equilibrium temperature, For cation antisite disorder: --   Eq. (3) represents the cation exchange equilibrium reaction involving one mole of ' A' and one of 'B. ' The equilibrium constant at a given temperature is written as: Eq. (4) represents the anion exchange equilibrium reaction involving six moles of oxide ions and one mole of vacant anionic sites per A 2 B 2 O 7 formula. The equilibrium constant at a given temperature is: The relation between equilibrium constant (lnK) and Gibbs free energy is, By substituting Eq. (6) & (7) in Eq. (1), the equilibrium constant can be written as follows The non-configurational entropy change associated with disordering is neglected, and we assumed the distribution of both cations and anions in a given sublattice is random. The configurational entropy (S conf ) for a PY phase involving four moles of cations (2 moles of A-site + 2 moles of B-site), six moles of oxygens (6 moles of 48f) and 1 mole of 8a vacant sites (whereas the 1 mole of 8b oxygens does not participate in Frenkel formation), is where 'x' is the fraction of 16d-site cations in the 16c-site and 'y' is the fraction of 48f oxygens in 8a vacant site. The total configuration entropy for both cation and anion distribution in a disordered PY (DPY) is given by S conf, DPY = S conf, cat + S Conf, ani . The configurational entropy of the fully disordered DF (i.e. A 1−x B x O 2−x/2 ▯ x/2 , x = 0.5 is fully disordered state) equivalent to PY stoichiometry including the disordering of 8b oxygens is constructed as follows: The configurational entropy for an ideal fully ordered PY phase is zero, and that of the completely disordered DF phase is 48.11 JK −1 mol −1 . The detailed configurational entropy calculations are provided in the supporting information.
The S conf for Sm 2 Zr 2 O 7 , Eu 2 Zr 2 O 7 , and Gd 2 Zr 2 O 7 based on site distribution at each temperature increment is given in the supporting information, Table S5. The entropy change for the phase transition from PY to DF is also calculated from the enthalpies obtained by DTA using Eq. (2). The calculated entropy changes during heating/cooling from DTA are 4.8 ± 0.9/−6.2 ± 0.9 and 3.5 ± 0.2/−4.0 ± 0.5 Jmol −1 K −1 for Sm 2 Zr 2 O 7 and Eu 2 Zr 2 O 7 , respectively (see , Table 3). Table 3 shows that the entropy change calculated from the enthalpies obtained by DTA using Eq. (2) is indeed similar to the ΔS conf from site distribution. For example, for Eu 2 Zr 2 O 7 , the entropy change for ordering during cooling is −4.0 ± 0.5 from DTA and −6.1 ± 0.4 Jmol −1 K −1 from the site distribution. Previously we reported the PY-DF phase transformation enthalpy for Eu 2 Zr 2 O 7 to be 37.8 ± 3.1 kJ·mol −1 by measuring the enthalpy of the solution in a molten oxide solvent of PY and a laser-melt-quenched DF phase of the same composition. However, the structural state of the samples was not characterized in detail, so the results cannot be compared directly to those from the in situ high temperature DTA and site distributions in the present work.
Shamblin et al. 14,15 and others 18,38 suggest substantial short-range order at the nanoscale, described as a weberite-type structure that is derivative of fluorite structure bearing a higher degree of order than DF, in radiation-damaged PY. However, at present, there is no evidence, pro or con, of weberite-like ordering in situ at high temperatures, in either the partially disordered PY or DF phases. In situ, high-temperature neutron studies would be desirable to investigate this further, as X-ray diffraction gives little information about the short-range order on the oxygen sublattice. Without further information about range order, we proceed below to apply a thermodynamic model that assumes random distributions of the ions on each sublattice constrained by the measured site occupancies. In the gradual disordering of PY with temperature, the favorable configurational entropy and the unfavorable energy (enthalpy) of disordering balance each other, resulting in greater disorder with increasing temperature. This balance can be described by an equilibrium constant analogous to that for spinel disordering 29 to calculate the disordering enthalpy from the site distribution. Using this approach, the cation and anion disorder follow simple interchange reactions.
The following assumptions are made based on Navrotsky and Kleppa's work on spinel disordering 32 :  12), (14) in (8) for anion Frenkel disorder and minimizing the free energy at a given temperature leads to the following relation for cation and anion interchange or disordering enthalpy: The molar enthalpies of interchange as a function of temperature using equations (15) and (16) are calculated for both CA and AFP disorder. Table 4 provides the calculated interchange enthalpy (enthalpy of disordering) values for RE 2 Zr 2 O 7 [RE = Sm, Eu, and Gd]; the disordering appears to reach equilibrium (similar site occupancies on heating and cooling and consistent calculated interchange enthalpies over the range of temperatures marked in bold and italic).The calculations of equilibrium constant also left out the highest temperature points in the region of the transition from PY to DF since that transition appears to have a first order component and represents changes distinct from the equilibrium disordering within the PY phase. The average interchange enthalpies for cation disorder (CA) are 114.9 ± 9.6, 109.1 ± 13.6 and 90.9 ± 5.4 kJ·mol −1 , and for anion Frenkel pair (AFP) disorder are 29.9 ± 3.7, 35.6 ± 3.5 and 35.9 ± 2.8 kJ.mol −1 for Sm, Eu, and Gd zirconates, respectively. The AFP interchange enthalpies show similar values for all three compositions and apparently do not depend on the size of the cation. The CA interchange enthalpy decrease with decreasing ' A' cation radius suggests greater disorder for smaller A-site cations at a given temperature. At any given temperature greater anion disorder exists than cation disorder.
Since numerous computational studies report somewhat different defect energies in various PY oxides by applying force field (FF) and density functional theory (DFT) calculations, we will not compare our experimental values individually, but discuss general trends. The energies obtained by FFA calculations for RE = Sm, Eu and Gd range between 360 and 400 kJ·mol −1 for cation disorder and 480 and 560 kJ.mol −1 for anion Frenkel disorder 20,21 . The FF modeling studies consider a single defect formation energies at 0 K, whereas in the current study the defect energy values are obtained on a structure that already contains a significant amount of disorder at room temperature (5-10%) and even more in the range of the measurements. So, the direct quantitative comparison may not be appropriate. The recent DFT calculations show lower values ranging between 160 and 200 kJ·mol −1 for cation disorder and negative values for anion Frenkel disorder, especially for compositions that lie near the PY-DF phase boundary 25 .
The current experimentally derived values range between 90 and 140 kJ·mol −1 for CA, and 30-36 kJ·mol −1 for AFP disorder, reasonably similar to the DFT calculations 25 . Both computational and experimental trends follow radius ratio r A /r Zr rules; i.e. the phase transition to DF shifts to lower temperatures as the radius of the A-site cation decreases and the lowest interchange enthalpy is associated with Gd 2 Zr 2 O 7 , which shows the greatest tendency to disorder.

Conclusions
We investigated the temperature-induced order-disorder phase transition in zirconate pyrochlores using a combination of in situ synchrotron X-ray diffraction and aerodynamic levitation with laser heating up to the melting points. The obtained diffraction pattern at each temperature was subjected to Rietveld analysis to extract the change in lattice constant, antisite cation/anion occupancies and phase transition temperatures. The lattice constant changed smoothly as a function of temperature, and the calculated thermal expansion coefficients fall in the range of 8.

Methods
Sample preparation and characterization. A series of stoichiometric rare earth pyrochlores (PY) with the general formula A 2 B 2 O 7 , where A = Rare earth (RE), B = Zr, Hf was synthesized by laser hearth melting. The starting materials used were high purity ZrO 2 (Aldrich; 99.99%), HfO 2 (Alfa Aesar; 99.9%), and RE 2 O 3 (Alfa Aesar, 99.9% or higher). Prior to synthesis, the RE 2 O 3 oxides were heated overnight at 1000 °C to remove any adsorbed moisture and carbon dioxide. Desired amounts of RE 2 O 3 [RE: La, Nd, Sm, Eu, Gd] and ZrO 2 or HfO 2 were finely ground and melted in a copper hearth into oblate spheroids 1.5-2.5 mm in diameter with a CO 2 laser using an experimental setup described in detail elsewhere 39,40 . The cooling profiles on crystallization were recorded using a spectropyrometer on samples aerodynamically levitated in air flow and analysed for thermal arrests. The composition and homogeneity of the samples were determined by wavelength dispersive electron probe microanalysis (Cameca SX100, Gennevilliers, France). The instrument was operated at an accelerating voltage of 15 kV, with a 20 nA beam current with a spot size of 1 µm. The calculated compositions are an average of several data points per sample. Thermal Analysis. Ultra-high-temperature differential thermal analysis (DTA) was carried out on Sm 2 Zr 2 O 7 , Eu 2 Zr 2 O 7 and Gd 2 Zr 2 O 7 using a Setaram Setsys 2400 (Caluire, France). Tungsten crucibles with lids were used in an argon flow of 20 mL/min. The laser melted samples were sealed in tungsten crucible using a custom semiautomatic welding chamber (Miller, Appleton, WI) to prevent carbon contamination. To account for thermocouple aging effects, temperature and sensitivity calibrations were performed before and after the sample measurements by melting Al 2 O 3 using a procedure described elsewhere 41 . High-temperature in situ diffraction on levitated samples. To gain insight into the essential structural features associated with temperature change, and to capture the gradual disordering in complex PY oxides, all the compositions were subjected to incremental heating using aerodynamic levitation and laser heating followed by in situ synchrotron diffraction at each temperature. The high-temperature in situ synchrotron X-ray diffraction experiments were performed at the Advanced Photon Source, Argonne National Laboratory, Illinois (USA) at beamlines 11-ID-C and 6-ID-D 42 . Two different wavelengths were used for the high-energy X-ray measurements, 0.139397 Å in beamline 6-ID-D, and 0.10798 Å in beamline 11-ID-C. A Perkin Elmer XRD 1621 high-sensitivity fast-readout large-area flat panel detector based on amorphous silicon was used to collect the scattered high-energy X-rays 43 . The sample-detector distance and tilt angle were set to the maximum distance (~1 m) to increase resolution. The sample temperature was measured by a single-band pyrometer (IR-CAS3CS; Chino, Tokyo, Japan) with emissivity set to 0.93. An unconditioned beam from a Synrad CO 2 laser was used for sample heating. The experimental arrangement, including the aerodynamic levitator, was described in detail by  Weber et al. [44][45][46][47] . The diffraction experiments at 11-ID-C were performed on samples 1.01.2 mm in diameter followed by experiments at 6-ID-D on 2.42.6 mm samples. The difference in bead dimension, necessitating slightly different acquisition and processing conditions, was the result of a lower power 100 W CO 2 laser used for the earlier experiments. The 2D X-ray images were collected by summing up 30 frames with 1.0 s exposures at beamline 11-ID-C, and 100 frames with 0.1 s exposures at beamline 6-ID-D. The obtained synchrotron diffraction rings were integrated using Fit2D 48,49 processing software. The sample-to-detector distance, beam position, and detector tilt was refined using the CeO 2 standard. In the case of the smaller diameter samples, the diffraction rings were masked for the nozzle background. The temperature of the diffracted volume was calculated from the surface temperature measured by a pyrometer, with correction determined from measured melting temperatures and the appearance of the amorphous halo in the X-ray diffraction pattern, as described previously 40,50 .
Rietveld structure refinement. The high-temperature in situ synchrotron diffraction data was grouped into two sets based on the quality of data obtained and beamline used. A complete Rietveld structure refinement was performed for high-temperature diffraction data on Sm 2 Zr 2 O 7 , Eu 2 Zr 2 O 7 , and Gd 2 Zr 2 O 7 using the FULLPROF program 51 . All the refinements were performed in the following manner. The background was selected manually, and a pseudo-Voigt function was used to model the peak shape. After successful refinement of the scale factor, lattice constant, peak shape, the antisite cation distribution (between 16d/16c crystallographic sites) and anion Frenkel (vacancy on a 48f site and oxygen on 8a site 21 ) was refined under the constraint of stoichiometric composition. Finally, the 48f 'x' positional and isothermal parameters were refined. For the high-temperature diffraction data from the La 2 Hf 2 O 7 , Nd 2 Zr 2 O 7 , Nd 2 Hf 2 O 7 and Sm 2 Hf 2 O 7 phases , pattern matching was performed using the Le Bail method 52 for unit cell refinement.