Estimating Free Energies of Formation of Titanate (M2Ti2O7) and Zirconate (M2Zr2O7) Pyrochlore Phases of Trivalent Lanthanides and Actinides

A linear free energy relationship was developed to predict the Gibbs free energies of formation (ΔGf ,MvX, in kJ/mol) of crystalline titanate (M2Ti2O7) and zirconate (M2Zr2O2) pyrochlore families of trivalent lanthanides and actinides (M3+) from the ShannonPrewitt radius of M3+ in a given coordination state (rM3+ , in nm) and the nonsolvation contribution to the Gibbs free energy of formation of the aqueous M3+ (ΔG n,M3+ ). The linear free energy relationship for M2Ti2O7 is expressed as ΔGf ,MvX = 0.084rM3+ + 82.30ΔG n,M3+ − 3640. The linear free energy relationship for M2Zr2O7 is expressed as ΔGf ,MvX = 0.083rM3+ + 83.13ΔG0n,M3+ − 3920. Estimated free energies were within 0.73 percent of those calculated from the first principles for M2Ti2O7 and within 0.50 percent for M2Zr2O7. Entropies of formation were estimated from constituent oxides (J/mol), based on an empirical parameter defined as the difference between the measured entropies of formation of the oxides and the measured entropies of formation of the aqueous cation.


Introduction
Pyrochlore is a mineral that preferentially incorporates large amount of Pu, U (up to 30 wt%), and Th (up to 9 wt%) into its structure [1][2][3][4].Pyrochlores exist as large polyhedra with coordination numbers ranging from 7 to 8, which provides them with the ability to accommodate a wide range of radionuclide (e.g., Pu, U, Ba, Sr, etc.) as well as neutron poisons (e.g., Hf, Gd) [5].As a result, pyrochlore structure is the primary consideration as immobilization barriers for utilization of excess weapons-grade plutonium and other radioactive elements [6][7][8].Due to their high radiation tolerance, pyrochlores are largely used as combined inert matrix fuel forms and waste forms for the "burning" and final disposal of Pu and the minor actinides [8].Rare earth (RE, also known an lanthanides) titanate pyrochlore (RE 2 Ti 2 O 7 , where RE = Lu to Sm, or Y) materials have potential use as solid electrolytes and mixed ionic/electronic conducting electrodes [9], catalysts [5], and ferroelectric/dielectric device components [10][11][12][13].
In actual waste forms, due to the presence of several trivalent cations, the pure as well as solid solution phases of pyrochlores of RE with stoichiometry of A 2 Ti 2 O 7 and A 2 Zr 2 O 7 such as La 2 Ti 2 O 7 to Lu 2 Ti 2 O 7 , La 2 Zr 2 O 7 to Lu 2 Zr 2 O 7 as well as trivalent actinide bearing phases are expected to occur, and their thermodynamic properties are needed to assess the behavior of Synroc-based waste forms and to optimize Synroc fabrications.Gd 2 Ti 2 O 7 and CaZrTi 2 O 7 doped with 3 wt% of 244 Cm have been reported [14].The Gd 2 Ti 2 O 7 phase and the more general RE titanate pyrochlore formulation (RE 2 Ti 2 O 7 ) have been reported in both glass and glass-ceramic nuclear waste forms [15,16].
Actinides (3+, 4+, and 5+) are predicted to form the pyrochlore structure by substitutions on both the A and B sites [17].Only the largest of the actinides exist in nature and the others must be obtained synthetically, and such processes may yield only a few atoms of product.As the atomic number increases, the stability of the tripositive state of actinides increases and parallels with the RE that the known properties of the latter can be used to predict quite exactly the properties of the comparable actinides including their free energies of formation.Despite the broad interest in titanate (A 2 Ti 2 O 7 ) and zircon bearing (A 2 Zr 2 O 7 ) pyrochlores of RE and actinides, thermodynamic data for pure as well as fictitious pyrochlores and zirconolites of RE and actinides are limited except for the recent measurements of the Gibbs free energy of formation for CaZrTi 2 O 7 and CaHfTi 2 O 7 phases [18,19] and the formation enthalpies of the zirconate [20] and titanate pyrochlores [21].
In this study a linear free energy relationship was developed and used to estimate the free energies of formation of trivalent actinide and RE titanate and zirconate pyrochlore phases using the existing thermodynamic data.The free energy relationship is useful in estimating the thermodynamic properties of pure and fictitious phases required for the immobilization reaction construction of solid solution models for actual crystalline phases of pyrochlores.

The Rare Earth and Actinide
Pyrochlore Structure The natural mineral pyrochlore ((Ca, Na) 2 Nb 2 O 6 F) has a large number of both natural and synthetic analogs [15].The most extended group of synthetic pyrochlores to this date are oxides with the formula A 2 B 2 O 7 , where A and B are di-and pentavalent or tri-and tetravalent elements, respectively [15].RE 2 Ti 2 O 7 pyrochlores have been widely studied [22,23].In the A 2 B 2 O 7 pyrochlore-type structure the A site is usually occupied by large cations such as lanthanides (Ln), whereas smaller first-or second-row transition elements fit the B site better.The most stable pyrochlore structure is formed when the RE cation is combined with a diamagnetic B 4+ cation.In an A 3+ 2 B 4+ 2 O 7 pyrochlore formula, the choice of B 4+ cation is thus limited to Ti 4+ and Sn 4+ and marginally Zr 4+ or Ge 4+ .RE-zirconate pyrochlores (A 2 B 2 O 7 , where the B site cation is Zr 4+ ) from La to Sm and RE-titanate pyrochlores (A 2 B 2 O 7 , where the B site cation is Ti 4+ ) from Sm to Lu with the coordination of the RE cation of eight and the coordination of Zr and Ti of six have been identified [21].The fictitious phases of RE-pyrochlores can be expected to form across the entire trivalent RE and actinide series.

Theoretical Basis of the Free Energy Model
Directly analogous to the well-established Hammett linear free energy relationship for substituted aqueous organic species and reactions [24][25][26], Sverjensky and Molling [27], and Sverjensky [28] developed a linear free energy relationship to correlate the standard Gibbs free energies of formation of an isostructural family of crystalline phases to those of aqueous cations of a given charge.For the trivalent RE and actinide isostructural family, the chemical formula of solids may be represented as M v X, where M is the trivalent cation (M 3+ ) and X represents the remainder of the composition of solid, for instance, in M 2 (CO 3 ) 3 , the trivalent cation M is La, Ce, Pu, and so forth, and X is CO 3  2− ; in RE-zircone pyrochlore family, the trivalent M is La, Sm, Lu, and so forth and X is (Zr 2 O 7 ) 2− .The original Sverjensky-Molling linear free energy correlation was modified by the authors for trivalent cations as [29] In ( 1) the coefficients a MvX , b MvX and β MvX , are characteristic of the particular crystal structure represented by M v X, and r M 3+ is the Shannon-Prewitt radius of the M 3+ cation in a given coordination state [27].β MvX is a coefficient related to the coordination number (CN) of the cation.In polymorphs, the structure family with smaller CN has higher value of β MvX than the family with higher CN [27].The parameter ΔG 0 f ,MvX is the standard state Gibbs free energies of formation of the end member solids, and the parameter ΔG 0 n,M 3+ is the standard state Gibbs free energies of nonsolvation, based on a radius-based correction to the standard state Gibbs free energies of formation (ΔG 0 f ,M 3+ ) of the aqueous cation, M 3+ .The ΔG 0 n,M 3+ , not the ΔG 0 s,M 3+ or the ΔG 0 f ,M 3+ , of the cations directly contributes to ΔG 0 f ,MvX containing the cation (M 3+ ).The ΔG 0 n,M 3+ and ΔG 0 s,M 3+ can be separated from ΔG 0 f ,M 3+ as follows [27]: Equation ( 1) was rearranged as The coefficients a MvX , b MvX , and β MvX can be determined by regression if the Gibbs free energies of formation of three or more phases in one isostructural family are known.

Application of the Free Energy Model to Titanate and Zirconate Pyrochlore Phases
Following the procedure of Sverjensky and Molling we have developed linear free energy correlations for oxide [29] hydroxide [29] carbonate [30], and sulfate isostructural families of trivalent lanthanides and actinides (Table 1).The discrepancies between the calculated and measured data were found to be less than ±3.0% for all isostructural families (oxides, hydroxides, carbonates, and sulfates).Based on our results and results from other crystal families, the coefficient a MvX or the slope of ( 3) is only related to the stoichiometry of the solids.The slopes for all polymorphs of composition M v X are the same within experimental error [27,28] 2− (zirconate)) (Figure 1).Based on this relationship we estimated values of a MvX for the titanate and zirconate perovskite and pyrochlore families.High ratio of charge/CN indicates strong interaction between the trivalent cation and oxy-anions.
The values of a MvX calculated for the zircon and titanate pyrochlores from this relation are 0.083 and 0.084 and that for the perovskite structure is 0.2742 (Table 1).The estimated error of a MvX is about 0.001.The error of estimated Gibbs free energy of formation resulting from the error of a MvX is within 4 kJ/mol.On the other hand the coefficient β MvX is related to the structure or the nearest neighbor environment of the cation.The cation with higher CN will have lower value of β MvX [27]  hydroxide family (CN = 6), carbonate family (CN = 7.1), and sulfate family (CN = 9) are estimated to be equal to 791.70 kJ/mol nm [29], 197.24 kJ/mol nm [30], and 26.32 kJ/mol nm respectively.The β MvX value for pyrochlore family (CN = 8) should be lower than those for hydroxide and carbonate families and higher than that for calcite family.From the previously estimated β MvX values for the hydroxide (CN = 6), carbonate (CN = 7.1), sulfate (CN = 9), and oxide (CN = 7) families of trivalent RE and actinides, we correlated the coefficient β MvX to the CN of the cation in the respective solid phase (Figure 2).The β MvX value obtained from this relationship for the perovskite family (CN = 12) is 2.8 kJ/mol nm and for the zirconate and titanate pyrochlore families (CN = 8), respectively, 83.18 kJ/mol nm and 82.30 kJ/mol nm.The total error in the calculation of free energies of formation resulting from the estimated coefficient β MvX is within 4 kJ/mol.
According to Sverjensky and Molling [27], the coefficient b MvX reflects characteristics of the reaction type and conditions under which solid formation took place regardless of the valence of the cation or the stoichiometry of the solid.Using the experimentally measured values of standard state (temperature (T) = 298.15K and pressure = 1 atm.)formation enthalpies from oxides reported in the literature for RE titanate pyrochlores [21] and RE zirconate pyrochlores [20], we calculated the Gibbs free energies of formation as where ΔH 0 f ,MvX is the standard state enthalpy of formation of the M v X compound calculated using the thermochemical cycle shown in Table 2.The enthalpies of formation for perovskite structure (ΔH ABO3,OX ) were calculated from constituent oxides by the following equation [31]: where t is the tolerance factor for ABO 3 perovskites [31].
For an ideal perovskite structure (CN of the A site cation = 12) t is equal to 1.0 [31].The thermochemical cycles used to calculate the Gibbs free energies of formation from the experimental enthalpies of formation and entropies of formation of the RE and actinide zirconate and titanate pyrochlore and perovskite phases are shown in Table 3.
The entropies of formation (ΔS 0 f ,MvX ) are calculated and available only for few RE and actinide perovskite and pyrochlore phases.We developed a relationship to estimate the entropy of formation of RE and actinide perovskite and pyrochlore phases from constituent oxides (ΔS 0 f ,OX ) applicable to all trivalent RE and actinide perovskite and pyrochlore families based on the empirical parameter ΔS M z+ , in J/mol, defined as the difference between the measured entropies of formation of the oxides (ΔS 0 f ,MOn (c)) and the measured entropies of formation of the aqueous cation (ΔS 0 f ,M z+ (aq)) of RE and actinides as where z is the charge of the cation (z = 3 for trivalent RE and actinides) and x is the number of oxygen atoms combined with one atom of M in the oxide (x = z/2).ΔS M z+ in (6) refers to one oxygen atom and characterizes the oxygen affinity of the cation, M z+ .Experimental values of ΔS 0 f ,MOn of RE and actinides were obtained from [32] and those of ΔS 0 f ,M z+ (aq) were obtained from [33].The entropy of formation from constituent oxides is considered as the sum of the products of the molar fraction of an oxygen atom bound to the two cations ((i) RE or actinide cation and (ii) Zr or Ti cation) in the pyrochlore and perovskite structure.Vieillard [34] showed the dependence of a cation on the oxygen affinity by the difference of electronegativity between cation and oxygen.Previous authors have developed empirical relationship between Gibbs free energy of formation from constituent oxides and the oxygen affinity of cation for f ,OX ) in J/mol K and the empirical oxygen affinity parameter (ΔS M 3+ ) in J/mol K shown for the zirconate pyrochlore (M 2 Zr 2 O 7 ) family.The relationship is estimated as ΔS 0 f ,OX = 71.07− 0.2162ΔS M 3+ .Standard molar entropies of the oxides and those of aqueous cations are from [32,33], respectively.crystalline solids [35][36][37][38].The entropies of formation from constituent oxides (ΔS 0 f ,OX ) were estimated in this article from the empirical parameter (ΔS M z+ ) by minimizing the difference between experimental entropies [39,40] and the calculated entropies of formation from constituent oxides as The estimated A and B coefficients for titanate pyrochlores (Figure 3) are −58.47 and −0.2162 (R 2 = 1.0) and for zirconate pyrochlores (Figure 4) are 71.07 and −0.2162 (R 2 = 1.0).The total error in the free energies of formation using ΔS 0 f ,OX thus estimated is within 0.5 kJ/mol.The estimated Gibbs free energies of formation for the titanate perovskite and the zirconate perovskite are shown in Figures 5 and 6.The calculated (from ( 4)) standard state Gibbs free energies of formation using the experimentally measured enthalpy and estimated entropy values and the estimated standard Gibbs free energies of formation for the zirconate perovskites (M 2 (Zr 2 O 3 ) 3 ) and for the titanate (M 2 Ti 2 O 7 ) and zirconate (M 2 Zr 2 O 7 ) pyrochlore families are listed in Table 4.

Effect of Cations on the Formation of Solids
Using the estimated linear free energies of formation for the perovskite phases and the pyrochlore phases and the formation energies for rutile (TiO 2 ) and zirconia (ZrO 2 ), the effect of cations on the energies of the following pyrochlore formation reactions at room temperature were characterized: The Gibbs free energies (ΔG 0 r,MvX−OX ) across the reactions in ( 8) are all negative (Table 4).All pyrochlore phases Figure 4: A relationship for entropy of formation from oxides (ΔS 0 f ,OX ) in J/mol K and the empirical oxygen affinity parameter (ΔS M 3+ ) in J/mol K shown for the titanate pyrochlore (M 2 Ti 2 O 7 ) family.The relationship is estimated as ΔS 0 f ,OX = 58.47− 0.2162 ΔS M 3+ .Standard molar entropies of the oxides and that of aqueous cations are from [32,33], respectively. (kJ/mol) Figure 6: Linear free energy relationship of (1) for the isostructural family of M 2 (ZrO 3 ) 3 perovskite.ΔG r,OX (Ti-pyrochlore) Figure 7: A plot of the free energy of reaction for the formation of titanate pyrochlore from constituent oxides (ΔG r,OX ) by ( 8) in kJ/mol shown as a function of the radius ratio of the A site (RE and actinide) cation and the B site (Ti) cation (closed circles) with a nonlinear regression fit (solid line, R 2 = 0.80).The experimentally measured formation enthalpies of titanate pyro-chlores from constituent oxides (ΔH f ,OX ) of twelve rare earth cations (from [21]) are also shown as a function of the radius ratio (open squares) with a linear regression fit (dashed line, R 2 = 0.71) for comparison.
cycles for all solids are shown in Table 2. b Data are reported in Table 4. are expected to be stable with respect to M 2 O 3 , M 2 (TiO 3 ) 3 and TiO 2 even at room temperature.The zircon pyrochlores in (9) are less stable by 50.09 kJ/mol than the titanate pyrochlores at room temperature.These findings are consistent with the findings from previous studies.The reaction energies by (8) and the experimentally measured enthalpies of formation from constituent oxides used to cal-culate the free energies are shown in Figure 7.The calculated reaction energies show that large cations (e.g., La, Ce, and Pr) form Cationic radii are from [41].Values of ΔG 0 n of the cations were calculated using ΔG 0 f values obtained from [42,43] as described in [30].The calculated ΔG 0 f ,MvX of the perovskite and pyrochlore solid crystals are from (4) by the thermochemical cycles shown in Table 2.The estimated ΔG 0 f ,MvX values of the perovskite and pyrochlore solid crystals are from (1).Calculated ΔG 0 r,OX values of the solid titanate pyrochlore solid crystals are from (8) using the estimated ΔG 0 f ,MvX of the solids.All calculations are at 25 • C and 1 bar.
more stable pyrochlores than small cations (e.g., Lu, Tm, and Er) (Figure 7).The relationship between ionic radii of the cations and the formation energies is nonlinear as shown by previous studies [21].Several mixed oxides in the Ln 2 ScNbO 7 series, with Ln = Pr, Eu, Gd, and Dy, were synthesized and found to crystallize in the cubic pyrochlore structure [44].Ce pyrochlore has been synthesized by sintering oxides of CeO 2 , CaTiO 3 , and TiO 2 [45].These experimental observations are consistent with our prediction of the negative Gibbs free energy changes across reaction in (8).Although the reaction energy calculation is based on room temperature, the prediction is basically consistent with experimental observation at higher temperatures.

Conclusions
The linear free energy relationship of Sverjensky and Molling was used to calculate the Gibbs free energies of formation of pyrochlore mineral phases (M 2 Ti 2 O 7 and M 2 Zr 2 O 7 ) from known thermodynamic properties of the corresponding aqueous trivalent cations (M 3+ ) of several lanthanides and actinides.The coefficients for the structural family of pyrochlore with the stoichiometry of M 2 Ti 2 O 7 are estimated to be a MvX = 0.084, b MvX = −3640 kJ/mol, and β MvX = 82.30kJ/mol nm and those for the M 2 Zr 2 O 7 are estimated to be a MvX = 0.083, b MvX = −3920 kJ/mol, and β MvX = 83.18kJ/mol nm.Thermodynamic properties of fictive mineral phases can also be predicted from this method.These fictive phases cannot be synthesized in the laboratory or occur in the nature, but their thermodynamic properties are required for the immobilization reaction construction of solid solution models for actual crystalline phases.The estimation method is superior because the estimated Gibbs free energies of formation of zirconate and titanate pyrochlore phases are validated with experimentally measured enthalpy and entropy data.

20 U 1 Figure 3 :
Figure3: A relationship for entropy of formation from oxides (ΔS 0 f ,OX ) in J/mol K and the empirical oxygen affinity parameter (ΔS M 3+ ) in J/mol K shown for the zirconate pyrochlore (M 2 Zr 2 O 7 ) family.The relationship is estimated as ΔS 0 f ,OX = 71.07− 0.2162ΔS M 3+ .Standard molar entropies of the oxides and those of aqueous cations are from[32,33], respectively.

Table 1 :
Summary of regression analysis: Gibbs free energies of formation of bulk solids of trivalent lanthanides and actinides.Structure family a MvX b MvX , kJ/mol β MvX , kJ/mol nm R 2 MvX r M 3+ = a MvX ΔG 0 n,M 3+ + b MvX .All calculations are at 25 • C and 1 bar.

Table 2 :
Thermochemical cycles used to calculate formation enthalpies from elements for M 2 Ti 2 O 7 (ΔH 0 . Using the previously developed values of a MvX for trivalent oxide, hydroxide, carbonate, and sulfate phases of RE and actinides, we related the coefficient a MvX to the ratio between the charge of H, C, S, Ti, or . The β MvX values for trivalent MvX Y = 0.0517 − 0.0727X + 0.1797X 2 , R 2 = 0.9996 Figure 1: A relationship for coefficient a MvX and the ratio between charge and coordination number (CN) of the oxyanions oxide, hydroxide, sulfate, carbonate, perovskite and titanate and zirconate pyrochlore structural families.The charge/CN ratios for oxyanions in sulfate, carbonate, oxide, hydroxide, perovskite and titanate and zirconate pyrochlore families are 6/4, 4/3, 4/3, 1/1, 4/3 and 4/6, respectively.

Table 4 :
Ionic radii and thermodynamic data for aqueous cations and estimated standard Gibbs free energies of formation for perovskite and pyrochlore families of solids.