Estimating Free Energies of Formation of Titanate () and Zirconate () Pyrochlore Phases of Trivalent Lanthanides and Actinides
A linear free energy relationship was developed to predict the Gibbs free energies of formation (, in kJ/mol) of crystalline titanate (M2Ti2O7) and zirconate (M2Zr2O2) pyrochlore families of trivalent lanthanides and actinides (M3+) from the Shannon-Prewitt radius of M3+ in a given coordination state (, in nm) and the nonsolvation contribution to the Gibbs free energy of formation of the aqueous M3+ (). The linear free energy relationship for M2Ti2O7 is expressed as . The linear free energy relationship for M2Zr2O7 is expressed as . 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.
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 [1–4]. 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) . As a result, pyrochlore structure is the primary consideration as immobilization barriers for utilization of excess weapons-grade plutonium and other radioactive elements [6–8]. 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 . Rare earth (RE, also known an lanthanides) titanate pyrochlore (RE2Ti2O7, where RE = Lu to Sm, or Y) materials have potential use as solid electrolytes and mixed ionic/electronic conducting electrodes , catalysts , and ferroelectric/dielectric device components [10–13].
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 A2Ti2O7 and A2Zr2O7 such as La2Ti2O7 to Lu2Ti2O7, La2Zr2O7 to Lu2Zr2O7 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. Gd2Ti2O7 and CaZrTi2O7 doped with 3 wt% of 244Cm have been reported . The Gd2Ti2O7 phase and the more general RE titanate pyrochlore formulation (RE2Ti2O7) 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 . 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 (A2Ti2O7) and zircon bearing (A2Zr2O7) 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 CaZrTi2O7 and CaHfTi2O7 phases [18, 19] and the formation enthalpies of the zirconate  and titanate pyrochlores .
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.
2. The Rare Earth and Actinide Pyrochlore Structure
The natural mineral pyrochlore ((Ca, Na)2Nb2O6F) has a large number of both natural and synthetic analogs . The most extended group of synthetic pyrochlores to this date are oxides with the formula A2B2O7, where A and B are di- and pentavalent or tri- and tetravalent elements, respectively . RE2Ti2O7 pyrochlores have been widely studied [22, 23]. Among typical representatives of this group are gadolinium titanate Gd2Ti2O7 and calcium uranium titanate CaUTi2O7, the ceramics selected in the United States for immobilization of Pu and other radioactive elements. The actinide (An) titanate pyrochlores (e.g., actinium titanate pyrochlore, Ac2Ti2O7), are closely related in chemical and structural properties to the rare earth titanate pyrochlores (e.g., lanthanum titanate pyrochlore, La2Ti2O7).
In the A2B2O7 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 B4+ cation. In an pyrochlore formula, the choice of B4+ cation is thus limited to Ti4+ and Sn4+ and marginally Zr4+ or Ge4+. RE-zirconate pyrochlores (A2B2O7, where the B site cation is Zr4+) from La to Sm and RE-titanate pyrochlores (A2B2O7, where the B site cation is Ti4+) 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 . The fictitious phases of RE-pyrochlores can be expected to form across the entire trivalent RE and actinide series.
3. 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 [24–26], Sverjensky and Molling , and Sverjensky  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 , where M is the trivalent cation ( and X represents the remainder of the composition of solid, for instance, in M2(CO3)3, the trivalent cation M is La, Ce, Pu, and so forth, and X is ; in RE-zircone pyrochlore family, the trivalent M is La, Sm, Lu, and so forth and X is . The original Sverjensky-Molling linear free energy correlation was modified by the authors for trivalent cations as 
In (1) the coefficients and are characteristic of the particular crystal structure represented by , and is the Shannon-Prewitt radius of the cation in a given coordination state . is a coefficient related to the coordination number (CN) of the cation. In polymorphs, the structure family with smaller CN has higher value of than the family with higher CN . The parameter is the standard state Gibbs free energies of formation of the end member solids, and the parameter is the standard state Gibbs free energies of nonsolvation, based on a radius-based correction to the standard state Gibbs free energies of formation of the aqueous cation, . The , not the or the , of the cations directly contributes to containing the cation (). The and can be separated from as follows : Equation (1) was rearranged as
The coefficients and can be determined by regression if the Gibbs free energies of formation of three or more phases in one isostructural family are known.
4. 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  hydroxide  carbonate , 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 or the slope of (3) is only related to the stoichiometry of the solids. The slopes for all polymorphs of composition are the same within experimental error [27, 28]. Using the previously developed values of for trivalent oxide, hydroxide, carbonate, and sulfate phases of RE and actinides, we related the coefficient to the ratio between the charge of H, C, S, Ti, or Zr and their coordination numbers (CN) or the nearest neighbor in the oxyanions (OH- (hydroxide), (carbonate), (sulfate), , (titanate), and (zirconate)) (Figure 1). Based on this relationship we estimated values of 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 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 is about 0.001. The error of estimated Gibbs free energy of formation resulting from the error of is within 4 kJ/mol. On the other hand the coefficient is related to the structure or the nearest neighbor environment of the cation. The cation with higher CN will have lower value of . The values for trivalent hydroxide family (), carbonate family (), and sulfate family () are estimated to be equal to 791.70 kJ/mol nm , 197.24 kJ/mol nm , and 26.32 kJ/mol nm respectively. The value for pyrochlore family () should be lower than those for hydroxide and carbonate families and higher than that for calcite family. From the previously estimated values for the hydroxide (), carbonate (), sulfate (), and oxide () families of trivalent RE and actinides, we correlated the coefficient to the CN of the cation in the respective solid phase (Figure 2). The value obtained from this relationship for the perovskite family () is 2.8 kJ/mol nm and for the zirconate and titanate pyrochlore families (), 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 is within 4 kJ/mol.
According to Sverjensky and Molling , the coefficient 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 = 298.15 K and pressure = 1 atm.) formation enthalpies from oxides reported in the literature for RE titanate pyrochlores  and RE zirconate pyrochlores , we calculated the Gibbs free energies of formation as where is the standard state enthalpy of formation of the compound calculated using the thermochemical cycle shown in Table 2. The enthalpies of formation for perovskite structure were calculated from constituent oxides by the following equation : where is the tolerance factor for ABO3 perovskites . For an ideal perovskite structure (CN of the A site cation = 12) is equal to 1.0 . 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 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 () applicable to all trivalent RE and actinide perovskite and pyrochlore families based on the empirical parameter , in J/mol, defined as the difference between the measured entropies of formation of the oxides () and the measured entropies of formation of the aqueous cation ()) of RE and actinides as where is the charge of the cation ( for trivalent RE and actinides) and is the number of oxygen atoms combined with one atom of in the oxide (). in (6) refers to one oxygen atom and characterizes the oxygen affinity of the cation, . Experimental values of of RE and actinides were obtained from  and those of were obtained from . 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  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 crystalline solids [35–38]. The entropies of formation from constituent oxides () were estimated in this article from the empirical parameter by minimizing the difference between experimental entropies [39, 40] and the calculated entropies of formation from constituent oxides as The estimated and coefficients for titanate pyrochlores (Figure 3) are −58.47 and −0.2162 () and for zirconate pyrochlores (Figure 4) are 71.07 and −0.2162 (). The total error in the free energies of formation using 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 (M2(Zr2O3)3) and for the titanate (M2Ti2O7) and zirconate (M2Zr2O7) pyrochlore families are listed in Table 4.
5. 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 (TiO2) and zirconia (ZrO2), the effect of cations on the energies of the following pyrochlore formation reactions at room temperature were characterized:
The Gibbs free energies (across the reactions in (8) are all negative (Table 4). All pyrochlore phases are expected to be stable with respect to, and 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 calculate the free energies are shown in Figure 7. The calculated reaction energies show that large cations (e.g., La, Ce, and Pr) form 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 .
Several mixed oxides in the Ln2ScNbO7 series, with Ln = Pr, Eu, Gd, and Dy, were synthesized and found to crystallize in the cubic pyrochlore structure . Ce pyrochlore has been synthesized by sintering oxides of CeO2, CaTiO3, and TiO2 . 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.
The linear free energy relationship of Sverjensky and Molling was used to calculate the Gibbs free energies of formation of pyrochlore mineral phases (M2Ti2O7 and M2Zr2O7) from known thermodynamic properties of the corresponding aqueous trivalent cations of several lanthanides and actinides. The coefficients for the structural family of pyrochlore with the stoichiometry of M2Ti2O7 are estimated to be = 0.084, = −3640 kJ/mol, and = 82.30 kJ/mol nm and those for the M2Zr2O7 are estimated to be = 0.083, = −3920 kJ/mol, and = 83.18 kJ/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.
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