Published values of ΔoxG, ΔoxH a ΔoxS for some mixed oxides in the system CaO-SrO-Bi2O3-Nb2O5-Ta2O5
Mixed oxides in the system CaO–SrO–Bi2O3–Nb2O5–Ta2O5 possess many extraordinary electric, magnetic and optical properties for which they are used in fabrication of various electronic components. For example Sr2(Nb,Ta)2O7 and (Sr,Ca)Bi2(Nb,Ta)2O9 are used for ferroelectric memory devices, CaNb2O6, Sr5(Nb1–xTax)4O15 and Bi(Nb,Ta)O4 for microwave dielectric resonators and Ca2Nb2O7 as non-linear optical materials and hosts for rare-earth ions in solid-state lasers. Ternary strontium bismuth oxides SrBi2O4, Sr2Bi2O5, and Sr6Bi2O9 are of considerable interest due to a visible light driven fotocatalytic activity.
To assess the thermodynamic stability and reactivity of these oxides under various conditions during their preparation, processing and operation, a complete set of consistent thermodynamic data, including heat capacity, entropy and enthalpy of formation, is necessary. Some of these data are available in literature. Akishige et al.  have been measured the heat capacities of Sr2Nb2O7 and Sr2Ta2O7 single crystals in the temperature range 2-600 K. The results have been only plotted and the values of S°m(298) have not been calculated. A commensurate transformation of Sr2Nb2O7 at TINC = 495 K has been observed accompanied by changes in enthalpy and entropy of ΔH = 291 J mol-1 and ΔS = 0.587 J K-1 mol-1. The heat capacity of Sr2Nb2O7 has been also measured by Shabbir at al.  in the temperature range 375-575 K. They have observed a phase transition at TINC = 487 ± 2 K connected with ΔH = 147 ± 14 J mol-1 and ΔS = 0.71 ± 0.10 J K-1 mol-1. The heat capacities of polycrystalline and monocrystalline SrBi2Ta2O9 and Sr0,85Bi2,1Ta2O9 have been measured by Onodera at al. [3–5] at 80-800 K. Morimoto at al.  have reported the results of the heat capacity measurements of SrBi2(NbxTa1-x)2O9 (x = 0, 1/3, 2/3 a 1). The temperature dependences of heat capacities show lambda-transitions with maxima at the Currie temperature TC = 570 ± 1 K, 585 ± 2 K, 625 ± 3 K a 690 ± 2 K for x = 0, 1/3, 2/3 and 1, respectively. Using EMF (electromotive force) measurements, Raghavan has obtained the values of the Gibbs energy of formation from binary oxides, ΔoxG, for some niobates [7,8] and tantalates [9,10] of calcium. His results are summarized in Table 1. The same technique has been employed by Dneprova et al.  for ΔoxG measurement for CaNb2O6 and Ca2Nb2O7. Their results presented in Table 1 are not significantly different from the results of Raghavan. Using the CALPHAD approach , Yang et al.  have assessed thermodynamic data for various mixed oxides in the SrO–Nb2O5 system. The same approach has been used by Hallstedt et al. for the assessment of thermodynamic data of mixed oxides in the systems CaO–Bi2O3  and SrO–Bi2O3 . Besides equilibrium data, values of the enthalpy of formation  of mixed oxide have been considered. Later on, these systems have been studied by EMF method by Jacob and Jayadevan [17,18] and temperature dependences of ΔoxG for various mixed oxides have been derived. These data have been included into the thermodynamic re-assessment of the CaO-SrO-Bi2O3 system .
(J K-1 mol–1)
|CaNb2O6||–75.82 – 0.03345T||1245-1300||–75.82||33.45|||
|CaTa4O11||–36.982 – 0.029T||1250-1300||–36.98||29.0|||
|CaNb2O6||–175.73 + 0.02259T||1100-1276||–175.73||–22.59|||
|Ca2Nb2O7||–212.54 – 0.02218T||1100-1350||–212.54||22.18|||
|Sr2Nb10O27||–1125.69 + 0.35069T||298-5000||–1125.69||–350.69|||
|SrNb2O6||–325.04 + 0.05865T||298-5000||–325.04||–58.65|||
|Sr2Nb2O7||–367.43 + 0.03993T||298-5000||–367.43||–39.93|||
|Sr5Nb4O14||–746.72 + 0.05101T||298-5000||–746.72||–51.01|||
|Ca5Bi14O26||–125.90 – 0.055T||298-1300||–125.9||55.0|||
|CaBi2O4||–27.60 – 0.003T||298-1300||–27.6||3.0|||
|Ca4Bi6O13||–97.60 – 0.008T||298-1300||–97.6||8.0|||
|Ca2Bi2O5||–42.20 – 0.003T||298-1300||–42.2||3.0|||
|SrBi2O4||–63.86 – 0.0018T||298-1300||–63.86||1.8|||
|Sr2Bi2O5||–118.75 + 0.024T||298-1300||–118.75||–24.0|||
|Sr3Bi2O6||–109.60 + 0.0024T||298-1300||–109.60||–2.4|||
This review brings a summary of our results [20–30] focused on calorimetric determination of heat capacity, entropy end enthalpy of mixed oxides in the system CaO–SrO–Bi2O3–Nb2O5–Ta2O5. Temperature dependences of molar heat capacity in a broad temperature range were evaluated from the experimental heat capacity and relative enthalpy data. Molar entropies at T = 298.15 K were calculated from low temperature heat capacity measurements. Furthermore, the results of calorimetric measurements of the enthalpies of drop-solution in a sodium oxide-molybdenum oxide melt for several stoichiometric mixed oxides in the above mentioned system are reported from which the values of enthalpy of formation from constituent binary oxides were derived. Finally, some empirical estimation and correlation methods (the Neumann-Kopp’s rule, entropy-volume correlation and electronegativity-differences method) for evaluation of thermodynamic data of mixed oxides are tested and assessed.
Nineteen mixed oxides in the system CaO–SrO–Bi2O3–Nb2O5–Ta2O5 with stoichiometry CaBi2O4, Ca4Bi6O13, Ca2Bi2O5, SrBi2O4, Sr2Bi2O5, CaNb2O6, Ca2Nb2O7, SrNb2O6, Sr2Nb2O7, Sr2Nb10O27, Sr5Nb4O15, BiNbO4, BiNb5O14, BiTaO4, Bi4Ta2O11, Bi7Ta3O18, Bi3TaO7, SrBi2Nb2O9, and SrBi2Ta2O9 were prepared, characterized and examined. The samples were prepared by conventional solid state reactions from high purity precursors (CaCO3, SrCO3 Bi2O3, Nb2O5 and Ta2O5). A three step procedure was used consisting of an initial calcination run of mixed powder precursors and subsequent double firing of prereacted mixtures pressed into pellets. The phase composition of the prepared samples was checked by X-ray powder diffraction (XRD). XRD data were collected at room temperature with an X’Pert PRO (PANalytical, the Netherlands) θ-θ powder diffractometer with parafocusing Bragg-Brentano geometry using CuKα radiation (λ = 1.5418 nm). Data were scanned over the angular range 5–60° (2θ) with an increment of 0.02° (2θ) and a counting time of 0.3 s step–1. Data evaluation was performed by means of the HighScore Plus software.
The PPMS equipment 14 T-type (Quantum Design, USA) was used for the heat capacity measurements in the low temperature region [31-35]. The measurements were performed by the relaxation method  with fully automatic procedure under high vacuum (pressure ~10–2 Pa) to avoid heat loss through the exchange gas. The samples were compressed powder pellets. The densities of the samples were about 65 % of the theoretical ones.
The samples were mounted to the calorimeter platform with cryogenic grease Apiezon N (supplied by Quantum Design). The procedure was as follows: First, a blank sample holder with the Apiezon only was measured in the temperature range approx. 2–280 K to obtain background data, then the sample plate was attached to the calorimeter platform and the measurement was repeated in the same temperature range with the same temperature steps. The sample heat capacity was then obtained as a difference between the two data sets. This procedure was applied, because the heat capacity of Apiezon is not negligible in comparison with the sample heat capacity (~8 % at room temperature) and exhibits a peak-shaped transition below room temperature . The manufacturer claims the precision of this measurement better then 2 % ; the control measurement of the copper sample (99.999 % purity) confirmed this precision in the temperature range 50–250 K. However, the precision of the measurement strongly depends on the thermal coupling between the sample and the calorimeter platform. Due to unavoidable porosity of the sample plate this coupling is rapidly getting worse as the temperature raises above 270 K and Apiezon diffuses into the porous sample. Consequently, the uncertainty of the obtained data tends to be larger.
A Micro DSC III calorimeter (Setaram, France) was used for the heat capacity determination in the temperature range of 253–352 K. First, the samples were preheated in a continuous mode from room temperature up to 352 K (heating rate 0.5 K min–1). Then the heat capacity was measured in the incremental temperature scanning mode consisting of a number of 5–10 K steps (heating rate 0.2 K min–1) followed by isothermal delays of 9000 s. Two subsequent step-by-step heating were recorded for each sample. Synthetic sapphire, NIST Standard reference material No. 720, was used as the reference. The uncertainty of heat capacity measurements is estimated to be better than ±1 %.
Enthalpy increment determinations were carried out by drop method using high-temperature calorimeter, Multi HTC 96 (Setaram, France). All measurements were performed in air by alternating dropping of the reference material (small pieces of synthetic sapphire, NIST Standard reference material No. 720) and of the sample (pressed pellets 5 mm in diameter) being initially held at room temperature, through a lock into the working cell of the preheated calorimeter. Endothermic effects are detected and the relevant peak area is proportional to the heat content of the dropped specimen. The delays between two subsequent drops were 25–30 min. To check the accuracy of measurement, the enthalpy increments of platinum in the temperature range 770–1370 K were measured first and compared with published reference values . The standard deviation of 22 runs was 0.47 kJ mol–1, the average relative error was 2.0 %. Estimated overall accuracy of the drop measurements is ±3 %.
The heats of drop-solution were determined using a Multi HTC 96 high-temperature calorimeter (Setaram, France). A sodium oxide-molybdenum oxide melt of the stoichiometry 3Na2O + 4MoO3 was used as the solvent. The ratio of solute/solvent varied from 1/250 up to 1/500. The measurements were performed at temperatures of 973 and 1073 K in argon or air atmosphere. The method consists in alternating dropping of the reference material (small spherules of pure platinum) and of the sample (small pieces of pressed tablets 10–40 mg), being initially held near room temperature (T0), through a lock into the working cell (a platinum crucible with the solvent) of the preheated calorimeter at temperature T. Two or three samples were examined during one experimental run. The delays between two subsequent drops were 30–60 min. The total heat effect (ΔdsH) includes the heat of solution (ΔsolH), the heat content of the sample (ΔTH), and, for the carbonates, the heat of decomposition (ΔdecompH) to form solid CaO or SrO and gaseous CO2. Using appropriate thermochemical cycles, the values of the enthalpy of formation of mixed oxides from the binary oxides and from the elements at 298 K were evaluated. The temperature dependence of the heat capacity of platinum  was used for the calculation of the sensitivity of the calorimeters.
2.1. Characterization of prepared samples
The XRD analysis revealed that the prepared samples were without any observable diffraction lines from unreacted precursors or other phases. The lattice parameters of the oxides were evaluated by Rietveld refinement  and are summarized in Table 2 together with the values of theoretical density calculated from the lattice parameters.
2.2. Evaluation of temperature dependence of heat capacity at low temperatures
The fit of the low-temperature heat capacity data (LT fit) consists of two steps. Assuming the validity of the phenomenological formula Cpm = βT 3 + γelT, at T → 0 where β is proportional to the inverse cube root of the Debye temperature ΘD and γelT is the Sommerfeld term, we plotted the Cpm/T vs. T 2 dependence for T < 8 K to estimate the ΘD and γel values. Since all compounds under study are semiconductors with a sufficiently large band gap, the non-zero γel values are supposed to be either due to some metallic impurities or to a series of Schottky-like transitions resulting from structure defects. Nevertheless, they are negligible in most cases (typically < 0.5 mJ K–2 mol–1) and can be ignored in further analysis. As an example, the results of heat capacity measurements on CaNb2O6 and LT fit for T < 10 K is shown in Fig. 1.
|Oxide||a (nm)||b (nm)||c (nm)||α (°)||β (°)||γ (°)||d (g cm–3)||Ref.|
In the second step of the LT fit, both sets of the Cpm data (relaxation time + DSC) were considered. Analysis of the phonon heat capacity was performed as an additive combination of Debye and Einstein models. Both models include corrections for anharmonicity, which is responsible for a small, but not negligible, additive term at higher temperatures and which accounts for the difference between isobaric and isochoric heat capacity. According to literature , the term 1/(1 – αT ) is considered as a correction factor.
The acoustic part of the phonon heat capacity is described using the Debye model
where R is the gas constant, ΘD is the Debye characteristic temperature, αD is the coefficient of anharmonicity of acoustic branches and xD = ΘD/T. Here the three acoustic branches are taken as one triply degenerate branch. Similarly, the individual optical branches are described by the Einstein model
where αEi and xEi = ΘEi/T have analogous meanings as in the previous case. Several optical branches are again grouped into one degenerate multiple branch with the same Einstein characteristic temperature and anharmonicity coefficient. The phonon heat capacity then reads
All the estimated values were further treated by a simplex routine and a full non-linear fit was performed on all adjustable parameters.
The values of relative enthalpies at 298.15 K, Hm(298.15) – Hm(0), were evaluated from the low-temperature Cpm data (LT fit) by numerical integration of the Cpm(T) dependences from zero to 298.15 K. Standard deviations (2σ) were calculated using the error propagation law. The values of standard molar entropies at 298.15 K, Sm(298.15), were derived from the low-temperature Cpm data (LT fit) by numerical integration of the Cpm(T)/T dependences from zero to 298.15 K. A numerical integration was used with the boundary conditions Sm = 0 and Cpm = 0 at T = 0 K. Standard deviations (2σ) were calculated using the error propagation law. All calculated values are summarized in Table 3.
(J K–1 mol–1)
(J K–1 mol–1)
(J K-1 mol–1)
|CaBi2O4||151.3||26470 ± 158||188.5 ± 3.3||1.9|||
|Ca4Bi6O13||504.1||85079 ± 507||574.1 ± 8.8||–23.8|||
|Ca2Bi2O5||197.4||33735 ± 201||231.3 ± 2.9||6.6|||
|SrBi2O4||155.6||29601 ± 169||206.1 ± 1.1||4.0|||
|Sr2Bi2O5||201.9||38199 ± 219||261.2 ± 1.4||5.5|||
|CaNb2O6||171.8||28159 ± 170||167.3 ± 0.9||–8.1|||
|Ca2Nb2O7||218.1||35631 ± 215||212.4 ± 1.2||–1.1|||
|SrNb2O6||170.2||28722 ± 174||173.9 ± 0.9||–17.0|||
|Sr2Nb2O7||216.6||37977 ± 266||238.5 ± 1.3||–5.9|||
|Sr2Nb10O27||746.8||124150 ± 740||759.7 ± 4.1||–33.9|||
|Sr5Nb4O15||477.2||83340 ± 490||524.5 ± 2.8||–18.4|||
|BiNbO4||121.3||22120 ± 134||147.9 ± 0.8||5.0|||
|BiNb5O14||386.8||62639 ± 362||397.2 ± 2.1||–25.8|||
|BiTaO4||119.3||22021 ± 132||149.1 ± 0.8||3.3|||
|Bi4Ta2O11||363.2||66566 ± 384||449.6 ± 2.3||9.5|||
|Bi7Ta3O18||602.7||109760 ± 634||743.0 ± 3.8||8.6|||
|Bi3TaO7||235.2||44265 ± 254||304.3 ± 1.6||10.0|||
|SrBi2Nb2O9||286.4||49230 ± 292||327.2 ± 1.7||–12.2|||
|SrBi2Ta2O9||286.6||49060 ± 289||339.2 ± 1.8||–5.9|||
A comparison is given in Table 4 of the values of entropy of formation from binary oxides ΔoxS at 298 K calculated from our results and those from literature. The values of ΔoxS are calculated using the relation
where Sm(MO) and Sm(BO,i) stand for the molar entropies of a mixed oxide and a binary oxide i, respectively, and bi is a constitution coefficient representing the number of formula units of a binary oxide i per formula unit of the mixed oxide. The following values were used for calculation: Sm(CaO,298.15 K) = 38.1 J K–1 mol–1 , Sm(SrO,298.15 K) = 53.58 J K–1 mol–1 , Sm(Bi2O3,298.15 K) = 148.5 J K–1 mol–1 , Sm(Nb2O5, 298.15) = 137.30 J K–1 mol–1  Sm(Ta2O5, 298.15) = 143.09 J K–1 mol–1 . Furthermore, Sm(Sr2Nb2O7, 298.15) = 238.5 J K–1 mol–1 from this work can be directly compared with the value 232.37 J K–1 mol–1 obtained by numeric integration of the Cpm(T)/T dependences from zero to 298.15 K given in Ref. . It should be noted that the values of entropy assessed by thermodynamic optimization of phase equilibrium data are generally considered as less reliable as the values derived from low temperature heat capacity measurements. It is due to possible strong correlation between the enthalpy and entropy contributions to the Gibbs energy. So the obvious discrepancies between our values and data from assessments [12,19] could be explain in this way.
(J K-1 mol–1)
(J K-1 mol–1)
It should be noted that the thorough analysis of the Debye and Einstein contributions to the heat capacities reveals that the different vibrational modes contribute to the total values of ΔoxS to a different extent and partial compensation is possible in some cases.
2.3. Evaluation of heat capacity at temperatures above 298 K
For the assessment of temperature dependences of Cpm above room temperature, the heat capacity data from DSC and the enthalpy increment data from drop calorimetry were treated simultaneously by the linear least-squares method (HT fit). The temperature dependence of Cpm was considered in the form
thus the related temperature dependence of ΔHm(T) = Hm(T) – Hm(T0) is given by equation
The sum of squares which is minimized has the following form
where the first sum runs over the Cpm experimental points while the second sum runs over the ΔHm experimental points. Different weights wi (wj) were assigned to individual points calculated as wi = 1/δi (wj = 1/δj) where δi (δj) is the absolute deviation of the measurement estimated from overall accuracies of measurements (1 % for DSC and 3 % for drop calorimetry). Both types of experimental data thus gain comparable significance during the regression analysis. To smoothly connect the LT fit and HT fit data the values of Cpm(298.15) from LT fit were used as constraints and so Eq. (7) is modified
The numerical values of parameters A, B and C are now obtained by solving a set of equations deduced as derivatives of Fconstr with respect of these parameters and a multiplier λ which are equal to zero at the minimum of Fconstr. Assessed values of parameters A, B and C of Eq. (4) for mixed oxides are presented in Table 5.
As an example, the results of heat capacity measurements and relative enthalpy measurements on Bi7Ta3O18  are shown in Fig. 2. Empirical estimation according to the Neumann-Kopp’s rule (NKR) is also plotted for comparison.
The empirical Neumann-Kopp’s rule (NKR) is frequently used for estimation of unknown values of the heat capacity of mixed oxides [46–48]. According to NKR, heat capacity of a mixed oxide is calculated as a sum of heat capacities of the constituent binary ones
It was concluded [47,48] that NKR predicts the heat capacities of mixed oxides remarkably well around room temperature but the deviations (mostly positive) from NKR become substantial at higher temperatures. Mean relative error of the estimated values of Cpm(298.15 K) is 1.4 %. Calculated temperature dependences of ΔoxCpm = Cpm(MO) –
|Oxide||Cpm = A + B·T + C/T 2 (J K–1 mol–1)||Temperature range (K)||Ref.|
|A||103 B||10–6 C|
2.4. Evaluation of enthalpy of formation
The heats of drop-solution for the calcium and strontium carbonates and for the bismuth and niobium oxides were measured first. These data are necessary for the evaluation of the ΔoxH values for the mixed oxides, and furthermore, these data could be compared with the literature data [49–52]. For the AECO3 carbonates, the measured heat effect consists of three contributions:
The measurements were performed at 973 K. The values of ΔdsH(AECO3, 973 K) are given in Table 6 along with the values of ΔdsH(AEO, 973 K), which were derived based on the following thermochemical cycle (T0 ≈ 298 K):
The values ΔdecompH(CaCO3, 298 K) = 178.8 kJ mol–1, ΔdecompH(SrCO3, 298 K) = 233.9 kJ mol–1 and ΔTH(CO2, 298 → 973 K) = 32.0 kJ mol–1  were used for the calculations.
Next, the ΔdsH values of the binary oxides Bi2O3 and Nb2O5 were measured. Because the dissolution of Nb2O5 and of the mixed oxides at 973 K proceeds rather slowly, the higher temperature of 1073 K was used. The measured values ΔdsH are also given in Table 6.
The experimental values of ΔdsH for SrCO3 and CaCO3 are in quite good agreement with the literature data [49–51]. On the other hand, our results and the published  values of ΔdsH(Nb2O5) are quite different. It should be noted that a more endothermic value ΔdecompH(SrCO3, 298 K) = 249.4 kJ mol–1 is presented in the literature , which results in more exothermic value for ΔdsH(SrO) by 15.5 kJ mol–1.
ΔdsH for the mixed oxides was measured at 1073 K. The following thermochemical cycle was used for the calculation of ΔoxH for calcium and strontium niobates (T0 ≈ 298 K):
An analogous scheme was applied to calculate ΔoxH(BiNbO4). All of the experimental and calculated values are summarized in Table 7. The ΔoxH(298 K) values derived from high-temperature EMN measurements [7,8,11] for the CaO-Nb2O5 oxides and the assessed values from the phase diagram for the SrO-Nb2O5 oxides  are also presented in Table 7.
|Substance||T (K)||ΔdsH (kJ mol–1) a)||ΔdsH (kJ mol–1)|
|CaCO3||973||128.4 ± 10.1 (10)||119.70 ± 1.02 b)|
|CaO||973||–82.39||–90.70 ± 1.69 b)|
|SrCO3||973||131.4 ± 9.1 (7)||130.16 ± 1.66 d)|
134.48 ± 1.89 e)
|SrO||973||–134.47||–135.82 ± 2.48 d)|
–131.42 ± 1.89 e)
|Bi2O3||973||26.0 ± 2.9 (12)|
|Nb2O5||1073||141.8 ± 6.0 (11)||91.97 ± 0.78 h)|
|Substance||T (K)||ΔdsH (kJ mol–1) a)||ΔoxH(298 K)|
(kJ mol–1) b)
|ΔoxH (298 K)|
|CaNb2O6||1073||196.8 ± 20.7 (8)||–132.0 ± 23.8||–159.8 c)|
|Ca2Nb2O7||1073||195.7 ± 27.8 (8)||–208.0 ± 31.9||–147.3 c)|
|SrNb2O6||1073||180.50 ± 15.7 (4)||–167.9 ± 19.1||–325.0 f)|
|Sr2Nb2O7||1073||167.54 ± 34.7 (4)||–289.2 ± 37.5||–367.4 f)|
|BiNbO4||1073||132.61 ± 8.9 (7)||–41.9 ± 11.1|
Our values for the calcium niobates are in good agreement with Raghavan’s data [7,8], while the data from Dneprova et al.  are quite different. Moreover, a relation, ΔoxH(CaNb2O6) > ΔoxH(Ca2Nb2O7), that holds for the values from the work of Dneprova et al. is rather unexpected. The ΔoxH values for strontium niobates obtained based on the binary SrO-Nb2O5 phase diagram evaluation  are substantially more exothermic than our calorimetric data. These large differences in the ΔoxH values are not surprising in view of simultaneous differences in the ΔoxS values from the assessment  and those derived from low temperature dependences of the molar heat capacity of SrNb2O6 and Sr2Nb2O7 [24,26].
3. Empirical correlation S–V
A linear correlation between the standard molar entropy at 298.15 K and the formula unit volume Vf.u has been proposed by Jenkins and Glaser [54–56]. This approach was used in this work for mixed oxides in the CaO–SrO–Bi2O3–Nb2O5–Ta2O5 system. The linear relation is obvious (see Fig. 4) and the straight line almost naturally passes through the origin:
The average relative error in entropy is 8.2 %, the binary oxides CaO and Nb2O5 show the deviations around 20 %. It should be noted that, in this set of values, the simple analogy of NKR (Eq.(9)) provides a better prediction with an average relative error in entropy of 4.2 %.
Eq. (21) can be used for estimation of missing data. So, the estimated value Sm(Sr2Ta2O7) = 256.06 J K–1 mol–1 can be compared with the value 245.41 J K–1 mol–1 obtained by numeric integration of the Cpm(T)/T dependences from zero to 298.15 K given in Ref.  (relative deviation of –4.3 %). Simple calculation Sm(Sr2Ta2O5) = 2Sm(SrO) + Sm(Ta2O5) = 250.25 J K–1 mol–1 gives more reliable value (relative deviation 2.0 %).
4. Empirical estimation of enthalpy of formation
There are other mixed oxides in the system CaO–SrO–Bi2O3–Nb2O5–Ta2O5 for which the values of enthalpy of formation ΔfH or enthalpy of formation from binary oxides ΔoxH have not yet been determined. As a rough estimate, the values of ΔoxH calculated according to an empirical method proposed by the authors  can be used. In the case of Ca, Sr and Bi niobates the following relation holds for ΔoxH:
where XNb and XMe (Me = Ca, Sr or Bi) are Pauling’s electronegativities of the relevant elements, xNb and xMe are the molar fractions of the oxide-forming elements (xNb = nNb/(nNb + nMe) etc.), y is the number of oxygen atoms per one atom of oxide-forming elements and α and δ are the model parameters. Using Pauling’s electronegativities, XNb = 1.60, XCa = 1.00, XSr = 0.95, and XBi = 2.02, and the calorimetric values of ΔoxH obtained in this work, the values of α = 2.576 and δ = 1.50 were derived from the least-squares fit. The estimated ΔoxH values for calcium and strontium niobates are shown in Fig. 5. The values of ΔoxH that were calculated according to an empirical method proposed by Zhuang et al.  are displayed for comparison.
The above presented data derived from calorimetric measurements became the basis for thermodynamic database FS-FEROX  compatible with the FactSage software [59,60]. Missing data for other stoichiometric mixed oxides were estimated by the empirical methods described before: the Neumann-Kopp’s rule for heat capacities, the entropy-volume correlation for molar entropies and electronegativity-differences method for enthalpies of formation. At the same time, thermodynamic description of a multicomponent oxide melt was obtained analyzing relevant binary phase diagrams published in literature. The database and the FactSage software were subsequently used for various equilibrium calculations including binary T-x phase diagrams and ternary phase diagrams in subsolidus region. Thermodynamic modeling of SrBi2Ta2O9 and SrBi2Nb2O9 thin layers deposition from the gaseous phase were also performed to optimize the deposition conditions.
This work was supported by the Ministry of Education of the Czech Republic (research projects N° MSM6046137302 and N° MSM6046137307). Part of this work was also supported from the Grant Agency of the Czech Republic, grant No P108/10/1006. Low temperature experiments were performed in MLTL (http://mltl.eu/), which is supported within the program of Czech Research Infrastructures (project no. LM2011025).