INTRODUCTION

Phase diagrams of inorganic salt systems are of interest due to their wide application. Variations in the cationic and anionic compositions of salts offer a means to manufacture materials intended for solar batteries, phosphors, electrolytes, and for other functional purposes [1–7]. Eutectic salt mixtures are used as heat-storing materials [8]. Reciprocal salt pairs, of which ion-exchange reactions are typical, are valuable for progress in the theory of physicochemical analysis [9–12].

The three-component reciprocal system comprised of fluorides and chlorides of calcium and barium is a complex subject matter of study; its boundary binary systems have differing types of phase diagrams. Figure 1 images the development of the T–x–y diagram of the Ca,Ba||F,Cl three-component reciprocal system, with the T–x diagrams of the boundary binaries adjoining to its lateral sides. In the CaF2–CaCl2 system, the incongruently melting compound CaFCl, a peritectic, and a eutectic are formed [13, 14] (Fig. 1). The CaCl2–BaCl2 system has the same type of phase diagram with the incongruently melting compound CaBaCl4 [15]. The BaF2–BaCl2 system features the congruently melting compound BaFCl and two eutectics [13], and the CaF2–BaF2 system features a limited solid solution series with a minimum and peritectic equilibrium [17, 18] (Fig. 1). Thus, the boundary binary systems are represented by three different types of phase diagrams, which makes the Ca,Ba||F,Cl subject matter an interesting model of phase equilibria and chemical interactions between components.

Fig. 1.
figure 1

Development of the T–x–y diagram of the Ca,Ba||F,Cl ternary reciprocal system and T–x diagrams of the boundary binaries. *Predicted value.

Full size image

THEORETICAL ANALYSIS

Figure 2 shows the liquidus projection on the composition square of the Ca,Ba||F,Cl three-component reciprocal system. All data on the constituent two-component systems and on invariant three-component compositions were analyzed while working with reference literature [15–17]. The binary compounds on the sides add secants and partition the ternary system into triangles. The partition into stable simplexes was carried out using graph theory [19]. The adjacency matrix for the vertices of the Ca,Ba||F,Cl system was thus obtained (Table S1).

Fig. 2.
figure 2

Liquidus projection on the composition square of the Ca,Ba||F,Cl ternary reciprocal system. * Predicted value.

Full size image

A logical expression was derived from the adjacency matrix, whose solution identified a set of unconnected graphs. By writing out the missing vertices, we obtained a set of stable simplexes:

$$\begin{gathered} {{{\text{X}}}_{{\text{1}}}}{{{\text{X}}}_{{\text{3}}}}{{{\text{X}}}_{{\text{7}}}}{\kern 1pt} :{\text{ }}~~~{\text{Ca}}{{{\text{F}}}_{{\text{2}}}}{-} {\text{Ba}}{{{\text{F}}}_{2}}{-} {\text{Ba}}{{{\text{F}}}_{{\text{2}}}}{\kern 1pt} \cdot {\kern 1pt} {\text{BaC}}{{{\text{l}}}_{2}}; \hfill \\ {{{\text{X}}}_{{\text{1}}}}{{{\text{X}}}_{{\text{4}}}}{{{\text{X}}}_{7}}{\kern 1pt} :{\text{ }}~~~{\text{Ca}}{{{\text{F}}}_{2}}{-} {\text{BaC}}{{{\text{l}}}_{2}}{-} {\text{Ba}}{{{\text{F}}}_{2}}{\kern 1pt} \cdot {\kern 1pt} {\text{BaC}}{{{\text{l}}}_{2}}; \hfill \\ {{{\text{X}}}_{{\text{1}}}}{{{\text{X}}}_{{\text{4}}}}{{{\text{X}}}_{5}}{\kern 1pt} :~~~~~{\text{Ca}}{{{\text{F}}}_{2}}{-} {\text{BaC}}{{{\text{l}}}_{{\text{2}}}}{-} {\text{Ca}}{{{\text{F}}}_{{\text{2}}}}{\kern 1pt} \cdot {\kern 1pt} {\text{CaC}}{{{\text{l}}}_{{\text{2}}}}; \hfill \\ {{{\text{X}}}_{{\text{4}}}}{{{\text{X}}}_{{\text{5}}}}{{{\text{X}}}_{6}}{\kern 1pt} :~~~~{\text{BaC}}{{{\text{l}}}_{2}}{-} {\text{Ca}}{{{\text{F}}}_{{\text{2}}}}{\kern 1pt} \cdot {\kern 1pt} {\text{CaC}}{{{\text{l}}}_{{\text{2}}}}{-} {\text{CaC}}{{{\text{l}}}_{2}}{\kern 1pt} \cdot {\kern 1pt} {\text{BaC}}{{{\text{l}}}_{2}}; \hfill \\ {{{\text{X}}}_{{\text{2}}}}{{{\text{X}}}_{{\text{5}}}}{{{\text{X}}}_{6}}{\kern 1pt} :~~~~{\text{CaC}}{{{\text{l}}}_{2}}{-} {\text{Ca}}{{{\text{F}}}_{2}}{\kern 1pt} \cdot {\kern 1pt} {\text{CaC}}{{{\text{l}}}_{{\text{2}}}}{-} {\text{CaC}}{{{\text{l}}}_{{\text{2}}}}{\kern 1pt} \cdot {\kern 1pt} {\text{BaC}}{{{\text{l}}}_{{\text{2}}}}. \hfill \\ \end{gathered} $$

Figure S1 images the phase tree of the Ca,Ba||F,Cl system comprised of five stable triangles interconnected by four stable secants. The phase tree is the basis for predicting phases that would crystallize; however, it is necessary to take into account the nature of interaction between the components, for example, the formation of congruent or incongruent compounds and areas of limited interstitial solid solutions.

In addition to the compounds formed on the binary sides (congruently melting BaFCl and incongruently melting CaBaCl4 and CaFCl), the Ca,Ba||F,Cl reciprocal system features 12 ion-exchange reactions at conversion points (denoted on Fig. S2). The reactions and the standard enthalpies and Gibbs free energies calculated for them can be found in the supplementary materials.

The reaction products are connected by solid lines on Fig. S2, and the starting products are connected by dashed lines. Therefore, the reaction products are the vertices of the stable elements of the phase tree (Fig. S1) and, accordingly, will crystallize from the melt.

3D-MODELING OF THE PHASE ASSEMBLAGE OF THE SYSTEM

A 3D model of the phase assemblage in the Ca,Ba||F,Cl ternary reciprocal system was designed based on the experimentally determined melting temperatures of individual compounds, the compositions and melting temperatures at binary and ternary invariant points in the KOMPAS 3D v21 program [20, 21] in the (composition)–(phase transition temperature) space (Fig. 3). The base of the model is the composition square (in equiv. percent); the temperature is plotted along the vertical axis in the range of 500–1500°С [22].

Fig. 3.
figure 3

3D model of the phase assemblage of the Ca,Ba||F,Cl system. *Predicted value.

Full size image

EXPERIMENTAL

Experimental characterization of samples was carried out on an MOM Q-1500 derivatograph in the controlled heating mode from 25 to 1100°C at 20 K/min in platinum crucibles for a derivatograph with thermocouples attached to the bottom. The sample size was 1 g. All compositions are in equivalent fractions expressed as percent [23–26]. The chemicals used were specialty grade CaF2, chemically pure grade BaF2, chemically pure grade BaCl2, and pure for analysis grade freshly calcined alumina used as the reference. The melting temperatures of the reagents corresponded to reference values [27]. X-ray powder diffraction (XRD) experiments were carried out on an Arl X’tra diffractometer using CuKα radiation with a nickel β-filter.

RESULTS AND DISCUSSION

The liquidus surface isotherms of the system in 50°C steps (Fig. 2) were obtained based on the 3D model by intersecting the horizontal plane with the liquidus surface. Calcium fluoride has the largest crystallization field (Fig. 2). Since minimum point min 1020 and a peritectic point p 1039 occur on the CaF2–BaF2 side, the liquidus surface in the CaF2–BaF2–BaFCl simplex is predicted to feature wedging point R 1000*: L + (BaF2)ss ⇄ (CaF2)ss + (BaF2)\(_{{{\text{ss}}}}^{'}\), where (BaF2)ss and (BaF2)\(_{{{\text{ss}}}}^{'}\) are BaF2-base limited solid solution phases, (CaF2)ss is the CaF2-base limited solid solution (Fig. 4), T =1000 and 900°C. Hereafter, ss stands for the solid solution. The degree of curvature of the liquidus surface can be judged from the density of isotherm arrangement.

Fig. 4.
figure 4

1200, 1000, 900, and 700°С isothermal sections of the Ca,Ba||F,Cl system derived from the model.

Full size image

The sections of the phase assemblage model at 1200, 1000, 900, and 700°С (Fig. 4) and at 800, 750, 650, and 600°С (Fig. 5) were obtained as intersections of a set applicate horizontal plane with surfaces of the model. The liquid area becomes smaller, and two-phase and three-phase areas become larger, as temperature lowers. Polythermal sections of the BaF2–CaCl2 unstable diagonal and the CaBaCl4–CaFCl secant were derived from the model (Figs. 6 and 7) via identifying the lines of intersection of the vertical plane with the surfaces of the figure. The (BaF2)ss area is a BaF2-base limited solid solution. The section features ray \({{\bar {E}}_{1}}880\) radiating from the BaFCl corner, rays \({{\bar {E}}_{2}}776{\text{\;and\;}}{{\bar {P}}_{1}}~662\) from the CaF2 corner, and rays \({{\bar {E}}_{3}}~542{\text{\;and\;}}{{\bar {P}}_{2}}~542\) from the CaFCl corner directed to invariant points: ternary eutectics Е1 880, Е2 776, and Е3 542; and ternary peritectics Р1 662 and Р2 542.

Fig. 5.
figure 5

800, 750, 650, and 600°С isothermal sections of the Ca,Ba||F,Cl system derived from the model.

Full size image
Fig. 6.
figure 6

Polythermal section of the BaF2–CaCl2 unstable diagonal of the Ca,Ba||F,Cl system derived from the model.

Full size image
Fig. 7.
figure 7

Polythermal section of the CaBaCl4–CaFCl secant of the Ca,Ba||F,Cl system derived from the model.

Full size image

The phase-transformation temperatures upon crystallization were predicted for sample α (25% CaF2, 15% BaF2, 60% BaCl2) and sample β (10% CaF2, 30% BaF2, 60% BaCl2); the relevant figurative points appear in Fig. 2. Thermoanalytical (T, DTA, and curves TG) were also recorded for figurative points α and β (Figs. 6 and 7). Table 1 lists theoretical and experimentally measured values of these temperatures and phase transformation schemes. Relative deviations of the theoretical (predicted using the 3D model) and experimental temperatures were calculated using the relationship

Table 1.   Comparison of theoretically predicted and experimentally determined phase transformation parameters
Full size table
$${{\delta }_{i}} = \frac{{\Delta {{T}_{i}}}}{{T_{i}^{{{\text{exp}}}} + 273}} \times 100\%,$$

where \({{\delta }_{i}}\) is the relative deviation of the experimental and theoretical values of temperature for the ith peak; \(T_{i}^{{{\text{exp}}}}\) is the experimentally measured temperature in the ith peak; \(\Delta {{T}_{i}} = \left| {T_{i}^{{{\text{exp}}}} - T_{i}^{{{\text{theor}}}}} \right|\) is the absolute deviation of the experimental and theoretical \(T_{i}^{{{\text{theor}}}}\) values of temperature for the ith peak.

The δ values in Table 1 indicate a good convergence of results and the adequacy of prediction of phase-transformation temperatures in three-component reciprocal systems by means of 3D phase diagram models.

In order to confirm the occurrence of reaction at conversion point K (Fig. S2) and the stability of the salt pair, we performed thermal analysis of a 22.5% CaF2 + 77.5% BaCl2 sample, which was previously assigned to a quasi-binary eutectic [15]. Our results differed slightly from the previously published data [15]. The DTA curve featured two events (at 881 and 793°С); therefore, the composition did not correspond to a eutectic alloy (Figs. 8 and 9). The sample was then removed from the crucible and subjected to X-ray powder diffraction analysis, the results of which are shown in Fig. 10.

Fig. 8.
figure 8

Thermoanalytical heating and cooling curves for sample α (25% CaF2, 15% BaF2, 60% BaCl2).

Full size image
Fig. 9.
figure 9

Thermoanalytical heating and cooling curves for sample β (10% CaF2, 30% BaF2, and 60% BaCl2).

Full size image
Fig. 10.
figure 10

X-ray diffraction pattern of a 22.5% CaF2 + 77.5% BaCl2 sample (CaF2: PDF 00002-1302; BaCl2: PDF 01-072-0368).

Full size image

CONCLUSIONS

The thermoanalytical curves for a 22.5% CaF2 + 77.5% BaCl2 sample and XRD data (Fig. 8) imply that the CaF2–BaCl2 diagonal is a stable diagonal; that is, for the reaction at point K this salt pair is the product of the reaction CaCl2 + BaF2 ⇄ CaF2 + BaCl2. Therefore, the diagonal partition has been verified for the Ca,Ba||F,Cl ternary reciprocal system. The secant lines connected by vertices to the central diagonal are stable: CaF2–BaFCl, BaCl2–CaFCl (Fig. S2). The spatial model of the Ca,Ba||F,Cl phase diagram visually represents the phase assemblage as a set of spaces of phase states, i.e., phase areas. Each area describes certain phase equilibrium in the system and is separated from other areas by interfaces.