Thermodynamic properties of calcium alkali phosphates Ca(Na,K)PO₄

Abstract

Calcium alkali phosphates Ca(Na,K)PO₄ are main constituents of bioceramics and thermochemically produced phosphorus fertilizers because of their bioavailability. Sparse thermodynamic data are available for the endmembers CaNaPO₄ and CaKPO₄. In this work, the missing data were determined for the low-temperature phase modifications of the endmembers CaNaPO₄ and CaKPO₄ and three intermediate Ca(Na,K)PO₄ compositions. Standard enthalpy of formation ranges from − 2018.3 ± 2.2 kJ mol⁻¹ to − 2030.5 ± 2.1 kJ mol⁻¹ and standard entropy from 137.2 ± 1.0 J mol⁻¹ K⁻¹ to 148.6 ± 1.0 J mol⁻¹ K⁻¹ from sodium endmember β-CaNaPO₄ to potassium endmember β′-CaKPO₄. Thermodynamic functions are calculated up to 1400 K for endmembers and the sodium-rich intermediate phase β-Ca(Na_0.93K_0.07)PO₄. Functions above 640 K are extrapolated because of the phase transition from low- to high-temperature phase. Impurities in the synthesized intermediate phases γ-Ca(Na_0.4K_0.6)PO₄ and γ-Ca(Na_0.35K_0.65)PO₄ and one additional phase transition around 500 K impeded the determination of high-temperature thermodynamic functions. In general, data for phase transition temperatures agree with the previously reported phase diagrams.

Introduction

Calcium alkali phosphates Ca(Na,K)PO₄ are of interest for fertilizer production and bioceramics used for bone tissue reconstructions. They possess higher bioavailability in comparison to the calcium phosphates such as whitlockite or apatite [1]. If used as bone substitution, they should be slowly decomposed and gradually replaced by a new fully functional bone [2, 3]. For the use as fertilizers, the nutrient elements should be available to plants. Therefore, calcium alkali phosphates are produced by thermochemical treatment of rock phosphate or sewage sludge ash at 800–1400 °C with alkali sulfates, carbonates or hydroxides [4,5,6].

The earliest description of CaNaPO₄ and CaKPO₄ dates back to the time of the development of the Rhenania process for thermochemical treatment of rock phosphate in the 1920s [6, 7] and the research continues until today [4, 8, 9]. Based on these procedures, a comparable thermochemical process was developed to treat phosphorus-rich sewage sludge ash for fertilizer production [5, 10, 11].

The phase diagram of the system CaNaPO₄-CaKPO₄ was investigated by Orlov et al. [12] and Znamierowska [13] (Fig. 1). The transformation temperatures from the low- to the high-temperature modification of calcium alkali phosphates are in the range of 823 K to 1033 K. The low-temperature modifications are divided by two miscibility gaps into three stability fields (β-, β′- and γ-phase) [13]. The sodium-rich β-phase includes the chemical composition ranging from CaNaPO₄ to Ca(Na_0.8K_0.2)PO₄ and the potassium-rich β′-phase that from CaKPO₄ to Ca(Na_0.05K_0.95)PO₄. The intermediate γ-phase, with a composition centered around Ca(Na_0.5K_0.5)PO₄ to Ca(Na_0.4K_0.6)PO₄, is a target phase for bioceramic applications [14, 15]. The endmember CaNaPO₄ was intensive characterized [12, 16,17,18], whereas CaKPO₄ was investigated less [12, 18, 19].

Figure 1

Phase diagram of CaNaPO₄–CaKPO₄ simplified after Znamierowska [13]. The grey circles represent investigated five samples at their preparation temperature 773 K. Pure CaKPO₄ has phase transitions at 877 K and 933 K

Incomplete thermodynamic data are available for the endmembers CaNaPO₄ and CaKPO₄ as well as for the intermediate phase Ca(Na_0.4K_0.6)PO₄. The transition enthalpy from the low- to the high-temperature modification (Δ_transH) of CaNaPO₄, Ca(Na_0.4K_0.6)PO₄ and CaKPO₄ is 8.2, 1.4 and 2.8 kJ mol⁻¹, respectively [12]. The database of FactSage, without citing data sources or specifying low- and high-temperature modifications, contains the standard entropy of 128.7 J mol⁻¹ K⁻¹ and enthalpy of formation of − 2033 kJ mol⁻¹ for CaNaPO₄. The enthalpy of formation for CaKPO₄ is calculated as − 2037.8 kJ mol⁻¹ [20]. To the best of our knowledge, there are no thermodynamic data for the potassium and sodium-containing intermediate calcium alkali phosphates. To close some of these gaps, this work presents heat capacity, standard entropy and standard enthalpy of formation for the endmembers β-CaNaPO₄ and β′-CaKPO₄ as well as for three intermediate phases.

Experimental methods

Synthesis

Low-temperature modifications of each calcium alkali phosphate Ca(Na,K)PO₄ were synthesized using protocols that avoid the use of excessive heat in order to remain well below the transformation temperature to the high-temperature modification. Other syntheses, e.g. β-CaNaPO₄ by Morris et al. [21] and Ben Amara et al. [16], Ca₂NaK(PO₄)₂ by Schneider et al. [14] and β-CaKPO₄ by Franck et al. [22], include steps above the transformation temperatures. The syntheses of β-CaNaPO₄ and the sodium-rich intermediate phase β-Ca(Na_0.93K_0.07)PO₄ completely below the transformation temperature are only possible if all chemicals can be initially dissolved in water and thoroughly mixed. Thus, stoichiometric mixtures of Ca(NO₃)₂·2H₂O, H₃PO₄, NaNO₃ and KNO₃ (all p.a. grade) were completely dissolved in double-distilled water. The solution was treated at first at 383 K to evaporate the water and subsequently the powder at 573 K for 3 h in a muffle furnace to remove NO_x. The dry material was ground in a vibrating tungsten-carbide cup mill (Pulverisette, Siebtechnik). The sintering process at 773 K was interrupted by grinding and re-grinding after 12, 24, 48, 96 and 192 h. However, β′-CaKPO₄ and the intermediate phases γ-Ca(Na_0.4K_0.6)PO₄ and γ-Ca(Na_0.35K_0.65)PO₄ could not be synthesized in a pure form using this protocol. Therefore, the syntheses of intermediate phases and β′-CaKPO₄ were modified and combined with the protocol of Schneider et al. [14]. The chemicals CaHPO₄·2H₂O, Na₂CO₃ and K₂CO₃ (all p.a. grade) were thoroughly mixed and heated to 1173 K for 1 h. Afterwards they were ground, sintered at 773 K with interruption of re-grinding after 12, 24, 48, 96 and 192 h, as described for the β-phases. Additionally, α-CaKPO₄ was prepared with the same chemicals as β′-CaKPO₄ but heated to 1273 K for 6 h. Afterwards, this sample was quenched in water.

Sample characterization

Chemical analysis

The chemical composition was analysed by inductively coupled plasma mass spectrometry (ICP-MS; Thermo iCAP Q, Dreieich, Germany) after digestion with aqua regia (chemicals p.a. grade) of 0.05–0.1 g sample in a microwave. Blanks, KCl (p.a. grade), NaCl (p.a. grade) and a reference material D826-1 for Ca and P were used to evaluate the analytical accuracy and precision. All digestions and ICP-MS measurements were carried out in triplicates.

Powder X-ray diffraction (PXRD)

PXRD measurements of the samples were performed in transmission geometry over a 2θ range from 5° to 70°, with a step size of 0.0009° (D8 Discover, Bruker AXS) and with Cu Kα₁ (λ = 1.54056 Å) radiation. The diffraction patterns were recorded with a Lynxeye detector. The PXRD were evaluated using the Match! Software (version 3.6). Full-profile refinements were done for the samples β-CaNaPO₄ and β-Ca(Na_0.93K_0.07)PO₄ (TOPAS version 6). The content of impurities was estimated additionally by the area method. This method is more closely described in the supplementary information.

Calorimetry

Acid-solution calorimetry

For the solution calorimetric experiments at T = 298.15 K, we used a commercial IMC-4400 isothermal microcalorimeter (Calorimetry Sciences Corporation), modified for the purposes of acid-solution calorimetry [23]. The liquid bath of the calorimeter was held at a constant temperature of 298.15 K with fluctuations smaller than 0.0005 K. The calorimetric solvent was 25 g of 5 N HCl contained in a polyetheretherketone (PEEK) cup with a total volume of 60 mL. The cup was then closed with a PEEK screwable lid and inserted into the calorimeter well. The calorimeter stabilized after ≈ 8 h. During the stabilization and the experiment, the solvent was stirred by a SiO₂ glass stirrer by a motor positioned about 40 cm from the active zone of the instrument. The samples were pressed into a pellet and weighed on a micro-balance with a precision of 0.002 mg (as stated by the manufacturer). The pellets were then dropped through a SiO₂ glass tube into the solvent and the heat produced or consumed during the dissolution was measured. The heat flow between the reaction cup and the constant temperature reservoir was then integrated to calculate the caloric effect. A typical experiment lasted 70–80 min and the end of the experiment was judged from the return of the baseline to the pre-experiment position. The calorimeter was calibrated by dissolving ≈ 20 mg pellets of KCl in 25 g of deionized water. Prior to each calibration measurement, the potassium chloride was heated overnight in the furnace to 800 K to remove the adsorbed water. The expected heat effect for the calibration runs was calculated from Parker [24].

Relaxation calorimetry (PPMS)

Heat capacity (C_p) was measured by relaxation calorimetry using a commercial Physical Properties Measurement System (PPMS, from Quantum Design, San Diego, California). With due care, accuracy can be within 1% for 5 K to 300 K, and 5% for 0.7 K to 5 K [25]. Powdered samples were wrapped in a thin Al foil and compressed to produce a ≈ 0.5 mm thick pellet which was then placed onto the sample platform of the calorimeter for measurement. Heat capacity was measured in the PPMS in the temperature interval 2 K to 300 K.

Differential scanning calorimetry (DSC and STA)

The heat capacities between 259 and 640 K for sample β-CaNaPO₄ and 282 K and 464 K for other samples were measured by differential scanning calorimetry (DSC) using a Perkin Elmer Diamond DSC. Details of the method are described in Benisek et al. [26].

C_p measurements at 310–1370 K were done with a Netzsch STA 449 C Jupiter thermoanalytical system (STA). The sample was loaded into platinum crucibles covered by platinum lids. Accuracy was regularly tested by the determination of C_p of sapphire. Accuracy of transformation enthalpy (Δ_transH) measurements was tested by measurements on quartz, SiO₂. The measured Δ_transH of 0.543 ± 0.007 kJ mol⁻¹ (in duplicate) compares roughly to the 0.625 kJ mol⁻¹ from Hemingway [27], the deviation being 13%.

Results and discussion

Chemical and phase characterization

The chemical compositions of the synthesis products were determined by ICP-MS and compared to the nominal elemental mass fractions calculated from the ratios of the starting chemicals (Table 1). The accuracies for calcium and phosphorus were between 97 and 103 wt% compared to the expected values and for sodium and potassium slightly below 100 wt% for all samples. Chemical analysis by ICP-MS was confirmed by X-ray fluorescence spectroscopy (data not shown).

Table 1 Chemical composition of calcium alkali phosphates Ca(Na,K)PO₄, measured in triplicates. Chemical composition is normalized to the molar proportions to 1 P atom per formula unit in the row below the mass fraction

All samples were crystalline, with sharp PXRD peaks. The models for the sample β-CaNaPO₄ (space group Pn2₁a; a = 20.3806(3) Å; b = 5.40518(9) Å; c = 9.1562(2) Å; V = 1008.6(1) Å³) and the sample β-Ca(Na_0.93K_0.07)PO₄ (Pn2₁a; a = 20.4633(9) Å; b = 5.4124(2) Å; c = 9.1771(4) Å; v = 1016.4(1) Å³) were refined by using ICSD entry 35629 (supplementary Fig. S1 and S2). The indexing of the diffraction pattern of β-Ca(Na_0.93K_0.07)PO₄ documents expansion of the unit cell in comparison to β-CaNaPO₄, because of incorporation of the larger K atoms in its structure. Furthermore, there are additional CaK₂P₂O₇ reflexes in the PXRD data for the sample β-Ca(Na_0.93K_0.07)PO₄ (Fig. S2). The amount of CaK₂P₂O₇ (COD entry 9007955) was quantified by Rietveld refinement as 3 wt%. Application of the semiquantitative area method yielded 4 wt% (see supplementary information). Combined with the results of chemical analysis, the assumed composition of this sample is 96 wt% β-Ca(Na_0.93K_0.07)PO₄ and 4 wt% CaK₂P₂O₇.

The PXRD peaks of the two samples of γ-Ca(Na_0.4K_0.6)PO₄ and γ-Ca(Na_0.35K_0.65)PO₄ are slightly shifted to lower 2θ angles compared to the data entry Ca₂NaK(PO₄)₂ (ICDD PDF entry 00-051-0579) (Fig. S3). Furthermore, similar refinement is achieved with lattice parameters for a monoclinic system postulated for Ca(Na_0.4K_0.6)PO₄ by Orlov et al. [12]. Both samples contain impurities but the quantification of these impurities by Rietveld refinement was not possible, because a structural model for Ca₂NaK(PO₄)₂ [14] or Ca(Na_0.4K_0.6)PO₄ [12] is not available. Therefore, the content of impurities was estimated by the area method. Accordingly, γ-Ca(Na_0.4K_0.6)PO₄ contains 5 wt% whitlockite (ICSD entry 91525), 1 wt% CaK₂P₂O₇, and 0.5 wt% hydroxylapatite (Ca₅(PO₄)₃OH) (COD entry 9011095). γ-Ca(Na_0.35K_0.65)PO₄ contains 2 wt% hydroxylapatite as an impurity.

There are very few structural data for the low-temperature phase β′-CaKPO₄. Our data could be refined with lattice parameters a = 5.492(4) Å; b = 9.603(2) Å; c = 16.245(5) Å; β = 68.9(1); V = 799.6(4) Å³ from Orlov et al. [12]. Evdokimov et al. [3] described this phase as “a monoclinic distortion of the hexagonal structure of α-CaKPO₄” and, therefore, the reflexes of β′-CaKPO₄ coincide largely with those of α-CaKPO₄ (Fig. S4). The sample β′-CaKPO₄ contains 2 wt% CaK₂P₂O₇, 3 wt% whitlockite and 3 wt% hydroxylapatite as impurities, estimated with the area method.

Acid-solution calorimetry

The enthalpies of formation of calcium alkali phosphates Ca(Na,K)PO₄ were determined by acid-solution calorimetry. All samples and reference compounds dissolved rapidly, and the results were reproducible, with uncertainties typical for this type of measurements. The measured bulk enthalpies in 5 N HCl are weighed sums of the enthalpies of dissolution (Δ_dissH) of the Ca(Na,K)PO₄ phase and the impurities. To evaluate and correct for the latter, we also measured Δ_dissH for K-whitlockite [Ca₁₀K(PO₄)₇] as 336.7 ± 0.3 kJ mol⁻¹ and Δ_dissH for hydroxylapatite [Ca₅(PO₄)₃OH] as − 124.3 kJ mol⁻¹. The Δ_dissH for CaK₂P₂O₇ was not measured but calculated from enthalpy of formation of CaK₂P₂O₇ of − 3286.9 kJ mol⁻¹ from Zdukos et al. [20]. The calculated value is − 69.4 kJ mol⁻¹. Δ_dissH of the impurities were used to correct the bulk Δ_dissH values in order to achieve the Δ_dissH values for pure calcium alkali phosphates.

The corrected Δ_dissH in 5 N HCl decreases monotonically with increasing potassium content in the Ca(Na_1−nK_n)PO₄ solid solution from − 52.23 ± 0.30 kJ mol⁻¹ for sample β-CaNaPO₄ to − 62.69 ± 0.14 kJ mol⁻¹ for the sample β′-CaKPO₄ (Table 3). The corrected Δ_dissH are used to calculate deviations of the intermediate phases Ca(Na_1−nK_n)PO₄ from the enthalpy of theoretical mixtures of the endmembers (Fig. 2). This calculation is comparable to the determination of enthalpy of mixing [28].

Figure 2

Deviation (ΔH) of experimental Δ_dissH of synthesized Ca(Na_1−nK_n)PO₄ from calculated Δ_dissH for theoretical mixtures of β-CaNaPO₄ and β'-CaKPO₄

The enthalpy of formation from elements in their standard state has to be calculated via a thermochemical cycle (Table 2) with respect to the known enthalpies of carefully chosen substances called the reference compounds. Special attention should be given to these reference compounds. Their formation enthalpies must be known with high accuracy and precision.

Table 2 Thermodynamic cycle for calcium-alkali-phosphates used for acid solution calorimetry

Table 3 Thermodynamic properties of five calcium alkali phosphates derived from calorimetric measurements based on corrected values of Δ_dissH and S°. Data in italics in the first row are from the modelling software FactSage [35]. All enthalpy and Gibbs free energy values in kJ mol⁻¹, all entropies in J mol⁻¹ K⁻¹

The thermochemical cycles were constructed with KH₂PO₄, NaH₂PO₄·2H₂O, CaO and H₂O as the reference compounds (Table 2). The Δ_dissH of CaO was reported by Majzlan [23]. The dissolution enthalpy of KH₂PO₄ in 5 N HCl at T = 298 K at low final molalities was measured as 24.61 ± 0.46 kJ mol⁻¹ [29]. In our study, the Δ_dissH of 25.08 ± 0.28 kJ mol⁻¹ was measured for higher final molalities. The dissolution enthalpy of NaH₂PO₄·2H₂O in 5 N HCl at T = 298 K was measured in our study as 29.83 ± 0.32 kJ mol⁻¹; no data are available in the literature for comparison.

There are only few data for the formation enthalpies of the phosphates KH₂PO₄ and NaH₂PO₄·2H₂O. The formation enthalpy of KH₂PO₄ can be found in Rud’ko et al. ([30]; − 1561.9 kJ mol⁻¹), Wagman et al. ([31]; − 1586.3 kJ mol⁻¹), and Luff and Reed ([32]; − 1573.6 kJ mol⁻¹). Majzlan [23] re-determined the formation enthalpy of KH₂PO₄ as − 1573.6 kJ mol⁻¹ and adopted the value from Luff and Reed [32], as done in this paper. For NaH₂PO₄·2H₂O, only the data in Wagman [31] are available and are adopted in this work.

The calculated enthalpies of formation from elements in their standard state (Δ_fH°) decrease with increasing potassium content from − 2018.3 ± 2.2 kJ mol⁻¹ (β-CaNaPO₄) to − 2030.5 ± 2.1 kJ mol⁻¹ (β′-CaKPO₄) (Table 3).

Standard entropy and Gibbs free energy of formation

Standard entropies were calculated by heat capacity data of PPMS. These data of the Ca(Na,K)PO₄ samples in the temperature range 2 to 303 K were fitted with polynomials:

$$ \mathop \sum \limits_{N = 0}^{11} A_{N } T^{N} $$

Experimental data in the temperature range 2–15 K could not be fitted with a simple (N = 3) or extended Debye (N = 3, 5, 7) model. Therefore, the low-temperature data were fitted and extrapolated to 0 K with polynomials with N = 1–7. The physical meaning of all these terms is not clear but the achieved fits are satisfactory. The temperature range 15–300 K was subdivided in three sections (15–60 K, 60–160 K and 160–300 K) and fitted with orthogonal polynomials. All coefficients (A_N) are listed in the supplementary information. The relative deviation between experimental data and fits [28] are shown in Fig. 3 for β-CaNaPO₄ and in the supplementary information for the other phases. The integration according to these orthogonal polynomials for heat capacity was used to calculate standard entropies S°:

$$ S^\circ = \mathop \int \limits_{0}^{15} \frac{{F_{1} \left( T \right)}}{T}{\text{d}}T + \mathop \int \limits_{15}^{60} \frac{{F_{2} \left( T \right)}}{T}{\text{d}}T + \mathop \smallint \limits_{60}^{160} \frac{{F_{3} \left( T \right)}}{T}{\text{d}}T + \mathop \smallint \limits_{160}^{300} \frac{{F_{4} \left( T \right)}}{T}{\text{d}}T $$

Figure 3

Deviation of experimental C_p and fitted C_p of PPMS (empty circles) against temperature for β-CaNaPO₄. The inset shows the temperature range 2–10 K. Experimental C_p of DSC (filled circles) plotted against fitted C_p of PPMS

The calculated standard entropies are, just as the enthalpies, weighed averages of the entropy of the Ca(Na_1−nK_n)PO₄ phases and the impurities. The S° value for hydroxyapatite of 390.4 J mol⁻¹ K⁻¹ was taken from Robie and Hemingway [33]. The S° values for the other impurities were estimated with the Neumann-Kopp rule, stating that the entropy of a phase is the sum of entropies of its suitably chosen components. In this case, the components are simple oxides and their entropies were taken from Robie and Hemingway [33]. The estimated entropies of K-whitlockite and CaK₂P₂O₇ are 828.5 J mol⁻¹ K⁻¹ and 246.6 J mol⁻¹ K⁻¹, respectively. The S° values for the Ca(Na_1−nK_n)PO₄ phases, corrected for the impurities, increase from 137.2 ± 1.0 J mol⁻¹ K⁻¹ for β-CaNaPO₄ to 148.6 ± 1.0 J mol⁻¹ K⁻¹ for β′-CaKPO₄ (Table 3). Their deviations from entropies of theoretical mixtures of the endmembers are displayed in Fig. 4.

Figure 4

Deviation (ΔS) of experimental S° of synthesized Ca(Na_1−nK_n)PO₄ from calculated S° for theoretical mixtures of β-CaNaPO₄ and β'-CaKPO₄

The calculated Gibbs free energies of formation decrease with increasing potassium content from − 1896.5 ± 2.4 kJ mol⁻¹ for β-CaNaPO₄ to − 1908.1 ± 2.4 kJ mol⁻¹ for β′-CaKPO₄. The Δ_fG° for β-CaNaPO₄ is 12 kJ mol⁻¹ higher than that in the database of the software FactSage (Table 3).

Superambient heat capacity measurements

Heat capacity (C_p) measurements using the DSC method were done between 259 and 640 K for sample ß-CaNaPO₄ and between 282 and 464 K for the other samples. The agreement between PPMS (2–303 K) and DSC data is very good. The differences at 295 K were < 0.3% for all samples. Additionally, semiquantitative C_p measurements (named as STA) between 310 and 1370 K were done with a Netzsch STA thermoanalytical system. The latter data were used for the characterization of the phase transitions (between 850 and 1000 K for the studied phases) and only used up to 640 K for fitting C_p’s for ß-Ca(Na_0.93K_0.07)PO₄ and ß′-CaKPO₄. Phase transition temperatures depend on chemical composition and show a hysteresis during heating and cooling cycles (Fig. 5).

Figure 5

Phase transition region for (a) ß-CaNaPO₄, (b) β-Ca(Na_0.93K_0.07)PO₄, (c) γ-Ca(Na_0.4K_0.6)PO₄, (d) γ-Ca(Na_0.35K_0.65)PO₄ and (e) ß′-CaKPO₄. The dotted line is for first heating, the solid curve for second heating and dashed line for cooling. The shaded area is the area used for calculation of the transition enthalpy [kJ mol⁻¹] which value is displayed. T onset [K] stands below graphs and T peak [K] above graphs

Phase transition

The phase transitions for ß-CaNaPO₄ and ß-Ca(Na_0.93K_0.07)PO₄ take place in two steps [13]. She reported that the low-temperature phase β-CaNaPO₄ and high-temperature α-CaNaPO₄ coexist between 883 K and 933 K. In contrast, Orlov et al. [12] postulated only one transition at T around 950 K.

Experimental C_p data for β-CaNaPO₄ show anomalies at 894 K and 945 K (Fig. 5a). The temperature difference between the two anomalies (50 K) and T onset of 926 K for the second peak correlates well with the data of Znamierowska [13] (Fig. 1). The temperature onset of 894 K for ß-Ca(Na_0.93K_0.07)PO₄ is 30 K below the onset temperature for β-CaNaPO₄. The data for ß-Ca(Na_0.93K_0.07)PO₄ show an additional small peak at T = 1071 K which can be explained neither by the existing phase diagrams nor a pyrophosphate impurity (Fig. 5b).

γ-Ca(Na_0.4K_0.6)PO₄ and γ-Ca(Na_0.35K_0.65)PO₄ have each a single C_p anomaly, in agreement with the phase diagram. The temperature onset for sample γ-Ca(Na_0.35K_0.65)PO₄ is 20 K below that for γ-Ca(Na_0.4K_0.6)PO₄ (Fig. 5c, d), although the phase diagram suggests that the difference should be only 10 K [13].

The potassium endmember β′-CaKPO₄ transforms also in two steps with peak temperatures of 898 K and 928 K and onset temperature of 910 K for the second peak (Fig. 5e). Comparable data are reported by Znamierowska [13] (Fig. 1). Additionally, a broad C_p anomaly is observed in the temperature interval 950–1010 K.

Enthalpy and entropy of phase transition

Data for enthalpy and entropy of phase transition are listed in Table 4. The enthalpy of phase transition of the intermediate phases γ-Ca(Na_0.4K_0.6)PO₄ and γ-Ca(Na_0.35K_0.65)PO₄ is smaller compared to other phases, because both low- and high- temperature phases of the intermediate samples are structurally related [14]. The more complex change from orthorhombic (β-CaNaPO_4, β-Ca(Na_0.93K_0.07)PO₄) and monoclinic (β′-CaKPO₄) to hexagonal symmetry results in higher enthalpy values. The transition enthalpies of the intermediate phases are comparable to the transition enthalpy of Ca(Na_0.4K_0.6)PO₄ reported by Orlov et al. [12] but lower enthalpies are reported for β-CaNaPO₄ and β′-CaKPO₄. Calculated transition enthalpies (Δ_trsH) and entropies (Δ_trsS) in Table 4 are considered in the thermodynamic functions for β-CaNaPO₄ at the onset temperature (Table 6).

Table 4 Thermodynamic properties of the phase transformations. T^trs is onset temperature of phase transition

Determination of superambient heat capacity

For the high-temperature regions, the DSC data were combined with STA data and estimate for the highest temperature (1400 K). In the region of 640–1400 K, phase transitions of the studied phosphates precluded accurate and precise C_p determination. The function used by Berman and Brown [34] was selected to allow for extrapolation to T = 1400 K.

$$ C_{P} { }\left( T \right) = { }A_{0} + { }A_{1} { } \times { }T^{ - 0.5} + { }A_{2} { } \times { }T^{ - 2} + { }A_{3} { } \times { }T^{ - 3} $$

The procedure chosen here agrees with the way the data in the FactSage database were derived, where the data were adjusted to reproduce the main features of the known phase diagrams. We acknowledge that there should be separate C_p functions for the low- and high-temperature phases but the data available do not allow for fitting in such a detail. Instead, just as practiced in FactSage, a single function is proposed. Given that these structures are closely related, and the transitions are only displacive in their nature, such treatment may be deemed acceptable. For calculating one single function, we force the extrapolation to a fixed point at T = 1400 K and C_p = 182.5 J mol⁻¹ K⁻¹, a value higher than the Dulong–Petit limit of 174.6 J mol⁻¹ K⁻¹, in agreement with the FactSage approach, and STA data for β-CaNaPO₄ and β′-CaKPO₄. Coefficients of all fits are summarized in Table 5.

Table 5 Heat capacities for the studied phases, fitted to polynomials C_P (T) = A₀ + A₁ · T^−0.5 + A₂ · T⁻² + A₃ · T⁻³ [34]. These polynomial-coefficients were determined between 260 and 640 K (DSC data) for sample β-CaNaPO₄; between 280 and 640 K (DSC/STA data) for samples β-Ca(Na_0.93K_0.07)PO₄ and β′-CaKPO₄; and a fixed C_p = 182.5 J·mol⁻¹ K⁻¹ at T = 1400 K

Heat capacities were fitted well in the temperature ranges from 260 to 640 K for sample ß-CaNaPO₄ and from 280 to 640 K for β-Ca(Na_0.93K_0.07)PO₄ and ß′-CaKPO₄. These were extrapolated to 1400 K for these three samples (Fig. 6a, b, d, Table 5). In the measured range for ß-CaNaPO₄, our data are consistently lower than those listed in FactSage (Fig. 6a). An explanation of the difference is difficult because it is not clear if the FactSage data are based on measurements or estimates. Furthermore, the values for the thermodynamic functions, calculated from the fit polynomials, are reported for β-CaNaPO₄ in Table 6.

Figure 6

Data (series DSC and STA) and fits of heat capacity for (a) ß-CaNaPO₄, (b) β-Ca(Na_0.93K_0.07)PO₄, (c) γ-Ca(Na_0.4K_0.6)PO₄ and γ-Ca(Na_0.35K_0.65)PO₄ und (d) ß′-CaKPO₄. The curve “FactSageData” in (a) was calculated from entries in the database of the software FactSage. C_p anomaly between 480 and 580 K in c) is the β′-to-γ phase transition (Fig. 1). Light grey circles of STA represent values with low accuracy above 1100 K

Table 6 Thermodynamic functions for ß-CaNaPO₄ (M_r = 158.04 g mol⁻¹). Data for phase transition at 926 K are listed in Table 5

Measurements for the intermediate compositions γ-Ca(Na_0.4K_0.6)PO₄ and γ-Ca(Na_0.35K_0.65)PO₄ compositions show a similar C_p increase between 380 and 460 K. Fitting was impeded due to the β′ to γ-phase transition between 480 to 580 K (Fig. 6c), also displayed in phase diagram (Fig. 1).

The heat capacity for β′-CaKPO₄ was fitted between 280 and 640 K. The extrapolated C_p values also fixed at 182.5 J mol⁻¹ K⁻¹_, differ strongly from the semiquantitative C_p measurements between 1000 and 1400 K (Fig. 6d).

Conclusion

Thermodynamic properties of technically important calcium alkali phosphates Ca(Na,K)PO₄ have been determined by different calorimetric measurements. Based on these data, standard enthalpies of formation and standard entropies were calculated for the low-temperature phases β-CaNaPO₄, β-Ca(Na_0.93K_0.07)PO₄, γ-Ca(Na_0.4K_0.6)PO₄, γ-Ca(Na_0.35K_0.65)PO₄ and β'-CaKPO₄. Besides, the sodium endmember all samples contain apatite, whitlockite, or pyrophosphate as impurities. All thermodynamic data were corrected for these impurities. The deviation is < 1% for standard entropy and < 0.1% for standard enthalpy of formation in case experimental values are compared with corrected ones.

The phase transformations from low- to high-temperature phases took place in the temperature range 890 K to 950 K which agrees with the phase diagram reported by Znamierowska [13]. Additionally observed heat capacity anomalies could not be matched to phase transition reactions of calcium alkali phosphates or any impurities. Further investigations are required to interpret these anomalies.

Heat capacity measurements at superambient temperatures were challenging because of phase transitions and low accuracy of the used high-temperature calorimetric method. Thus, heat capacities between 640 and 1400 K were estimated by extrapolation for samples β-CaNaPO₄, β-Ca(Na_0.93K_0.07)PO₄ and β'-CaKPO₄.

In summary, the determined thermodynamic properties of Ca(Na,K)PO₄ are valuable data for thermodynamic calculations that can improve manufacturing bioceramics and the production of phosphorus fertilizers containing these calcium alkali phosphates.