Thermodynamic assessment of the KF-ThF4, LiF-KF-ThF4 and NaF-KF-ThF4 systems

A thermodynamic assessment of the KF-ThF 4 binary system using the CALPHAD method is presented, where the liquid solution is described by the modiﬁed quasichemical formalism in the quadruplet approximation. The optimization of the phase diagram is based on experimental data reported in the lit- erature and newly measured X-ray diffraction and differential scanning calorimetry data, which have allowed to solve discrepancies between past assessments. The low temperature heat capacity of a -K 2 ThF 6 has also been measured using thermal relaxation calorimetry; from these data the heat capacity and standard entropy values have been derived at 298.15 K: C op ; m K 2 ThF 6 ; ð cr ; 298 : 15 K Þ ¼ 193 : 2 (cid:2) 3 : 9 ð Þ J (cid:3) K (cid:4) 1 (cid:3) mol (cid:4) 1 and S om K 2 ThF 6 ; cr ; 298 : 15 K ð Þ ¼ 256 : 9 (cid:2) 4 : 8 ð ) J (cid:3) K (cid:4) 1 (cid:3) mol (cid:4) 1 . Taking existing assessments of the relevant binaries, the new optimization is extrapolated to the ternary systems LiF- KF-ThF 4 and NaF-KF-ThF 4 using an asymmetric Kohler/Toop formalism. The standard enthalpy of forma- tion and standard entropy of KNaThF 6 are re-calculated from published e.m.f data, and included in the assessment of the ternary system. A calculated projection of the NaF-KF-ThF 4 system at 300 K and the optimized liquidus projections of both systems are compared to published phase equilibrium data at room temperature and along the LiF-LiThF 5 and NaF-KThF 5 pseudobinaries, with good agreement. (cid:1) 2020 The Authors. Published by Elsevier Ltd. ThisisanopenaccessarticleundertheCCBYlicense(http://


Introduction
The Generation IV International Forum, a group of fourteen member countries pursuing research and development for the next generation of nuclear reactors, has selected six nuclear energy systems [1,2]. Among these, the Molten Salt Reactor (MSR) is, in terms of safety and performance, one of the most promising nuclear reactor designs presently being studied. Its central characteristic is that the nuclear fuel is made of a molten fluoride (or chloride) salt instead of being a solid oxide or a metal. This liquid serves both as the fuel and coolant for the reactor. Two experimental MSRs have been built in the past: the Aircraft Reactor Experiment (ARE) [3] in 1954, and the Molten Salt Reactor Experiment (MSRE), which operated successfully between 1965-1969 [4]. A comprehensive knowledge of the physico-chemical properties of the salt is needed for the safety assessment and design of modern reactors, as the irradiated salt constitutes a complex and multi-component system.
The 7 LiF-NaF-KF-ThF 4 -UF 4 -AnF 3 (An= actinide) system has been proposed for the fuel of an actinide burner design [5], and still needs a full thermodynamic characterization. In particular, studies on many of the KF-containing systems are either absent in the literature or need to be revisited, namely KF-UF 3 (some intermediate compounds have been synthesized [6]), KF-UF 4 (phase diagram information exists [7], but a CALPHAD model is missing), KF-PuF 3 (there is no phase diagram information available), and KF-ThF 4 . Two sets of authors, Emelyanov and Evstyukhin [8] and Asker et al. [9] have studied the potassium fluoride-thorium fluoride binary system, with fair agreement in some regions of the system. However, they have reported conflicting interpretations in other regions, which need to be resolved. To this end, we present a reevaluation of the KF-ThF 4 binary system, using X-ray diffraction (XRD), Differential Scanning Calorimetry (DSC), and low temperature heat capacity measurements. A thermodynamic assessment using the CALPHAD method is moreover reported for the first time, where the Gibbs energy of the liquid solution is described using the quasi-chemical model in the quadruplet approximation. The assessment of the binary system is subsequently used to extrapolate to the ternary LiF-KF-ThF 4 and NaF-KF-ThF 4 systems.

Sample preparation for DSC measurements
The purity of the four constituent salts, LiF (ultra-dry), NaF, KF, (all from Alfa Aesar, 0.9999 0.00010.9999 AE 0.0001 1 mass fraction purity) and ThF 4 was confirmed using X-ray diffraction (XRD) and Differential Scanning Calorimetry (DSC). NaF and KF had to be dried further, for 4 h at 673 K in an open nickel boat under Ar flow, in order to reach the adequate purity for thermodynamic measurements. ThF 4 was synthesized in JRC-Karlsruhe as described in [10]. All salts were of white color, and were handled in either powder or pressed pellet form. The experimental compositions reported hereafter were prepared by mixing either powder or pellet fragments of the pure salts in the corresponding stoichiometric ratios. As fluoride salts are highly sensitive to water and oxygen, handling and preparation of samples took place inside the dry atmosphere of an argon-filled glove box, where H 2 O and O 2 content were kept below 1 ppm. (See Table 1)

Synthesis
The samples whose X-ray diffraction patterns are shown in this work were made by two methods. The first consisted in grinding powder mixtures and heating them inside a closed stainless steel crucible with a Ni liner in a tubular furnace under Ar flow. The second method consisted in heating powder mixtures in a DSC crucible above melting. The conditions are given below in Table 2.

Powder X-ray diffraction
X-ray powder diffraction (XRD) data were collected at room temperature (T = 293 AE 5 K 2 ) using a PANalytical X'Pert PRO Xray diffractometer and a Cu anode (0.4 mm x 12 mm line focus, 45 kV, 40 mA) by step scanning at a rate of 0.0104 Ás À1 in the range 10°< 2h<120°in a Bragg-Brentano configuration. The X-ray scattered intensities were measured with a real time multi strip (RTMS) detector (X'Celerator). Structural analysis was performed with the Rietveld and LeBail methods using the FullProf suite [11].

Differential Scanning Calorimetry
3D-heat flow DSC measurements were performed using a Setaram Multi-Detector HTC module of the 96 Line calorimeter under argon flow at a pressure of (0.10 AE 0.01 MPa 3 ). All samples were placed inside a nickel liner and encapsulated for the calorimetric measurements inside a stainless steel crucible closed with a screwed bolt as described in [12] to avoid vaporization at high temperatures. Two kinds of information were sought using the DSC technique: phase diagram equilibria points and mixing enthalpies. In all cases the measurement program began with one heating cycle reaching 1483 K and was maintained at that temperature for at least 300 s to ensure complete mixing and melting of the end-members. In general, this first cycle was followed by three successive heating cycles with a heating rate ranging between 4 to 10 KÁmin -1 , and 20-15-10-5 KÁmin -1 cooling rates. The procedure followed for the mixing enthalpy measurements is described in detail in Section 2.5.
Temperatures were monitored throughout the experiments by a series of interconnected S-type thermocouples. The temperature on the heating ramp was calibrated by measuring the melting points of standard high purity metals (In, Sn, Pb, Al, Ag, Au). The     [16] a The quoted uncertainty corresponds to the standard uncertainty. b Standard uncertainty not reported when not found in the reference. 1 The reported uncertainty corresponds to the standard uncertainty. 2 The reported uncertainty corresponds to the standard uncertainty. 3 The reported uncertainty corresponds to the standard uncertainty.
temperature on the cooling ramp was obtained by extrapolation to 0 KÁmin -1 cooling rate. The melting temperatures of pure compounds and transition temperatures of mixtures were derived on the heating ramp as the onset temperature using tangential analysis of the recorded heat flow, while the liquidus temperatures of mixtures were taken as the minimum of the last thermal event as recommended in [13]. The data measured on the cooling ramp were not retained for the phase diagram optimization due to the occurrence of supercooling effects. The uncertainty on the measured temperatures is estimated to be AE 5 K for the pure compounds and AE 10 K for mixtures 4 . The DSC measurements support the purity indicated by the suppliers and XRD data, as the heat flow signal for each of the four salts (LiF, NaF, KF, ThF 4 ) showed only one peak corresponding to the melting event, and no peaks that could be assigned to impurities. The measured onset temperatures are in good agreement with the literature: (1118 AE 5 K), (1268 AE 5 K), (1129 AE 5 K), and (1381 AE 5 K 5 ), respectively, vs. 1121.3 K (LiF, [14]), 1269.0 K (NaF, [14]), 1131.0 K (KF, [14]), and 1383.0 K (ThF 4 , [15]).

Enthalpy of mixing measurements
Enthalpies of mixing measurements were made in the same DSC calorimeter as the aforementioned equilibrium data, using a technique described in detail in [16]. The starting end-members KF and ThF 4 materials were pressed into pellets. The KF pellet (compound with the lowest melting point) was placed under the ThF 4 pellet, with a Ni liner separating them to avoid eutectic melting upon heating. Upon melting of KF, the Ni liner sank to the bottom and solid ThF 4 came into contact with molten KF, and melted too. The enthalpy of mixing is then calculated as the difference between the measured heat and the melting enthalpies of the end-members: The values used for the enthalpies of fusion of KF and ThF 4 were taken from the SGTE database [17] 4 The reported uncertainty corresponds to the standard uncertainty. 5 The reported uncertainties for LiF, NaF, KF, and ThF 4 correspond to the standard uncertainty. 6 The quoted uncertainty corresponds to the standard uncertainty.
run by using a silver standard in the reference crucible, as described in [16]. The sensitivity coefficient used to determine the mixing enthalpies was validated by testing on the endmembers. The enthalpy of fusion of ThF 4 was measured to be:  Table 3 along with those of KF and ThF 4 . These results give us good confidence in the chosen calibration factor. The errors reported in Table 7 are based on the propagation of the standard uncertainty of the sensitivity coefficient obtained from the calibration process.

Low temperature heat capacity
Low temperature heat capacity measurements were performed on m=7.29 mg 9 of K 2 ThF 6 in the temperature range T= (1.8-298.5) K using a PPMS (Physical Property Measurement System, Quantum Design) instrument with no applied magnetic field. A critical assessment of this thermal relaxation calorimetry method can be found in [18]. The contributions of the sample platform, wires, and grease were taken into account by a separate measurement of an addenda curve. From previous studies with standard materials and other compounds [19,20], the relative standard uncertainty was estimated at about 2 % above 270 K, 1% from 100 to 270 K, and reaching about 3 % at the lowest temperatures [18,19].

Thermodynamic modelling
Optimizations of the phase diagrams were carried out by the CALPHAD (CALculation of PHase Diagram) method [21] using the Factsage software [22]. To carry out such an optimization, the identity of the phases present in the system of interest must be known, as well as their respective Gibbs energy functions.

Pure compounds
The Gibbs energy function of a pure compound is given by: Þis the standard enthalpy of formation, S o m 298 ð Þis the standard absolute entropy, both evaluated at a reference temperature, in this case 298.15 K (throughout this work 298 will be understood to mean 298.15 K for simplicity), and C p;m is the isobaric heat capacity expressed as a polynomial: with more terms added if necessary. In this work, the Neumann-Kopp rule [23] applied to KF and ThF 4 was used to estimate the heat capacities of intermediate compounds in the absence of experimental data. The only exceptions were a-K 2 ThF 6 (cr) and b-K 2 ThF 6 (cr), for which a fit was made. The fit included the low temperature heat capacity points measured herein in the 250-300 K range as well as high temperature points given by the Neumann-Kopp rule in the 500-1500 K range. Moreover, the temperature-independent term of the heat capacity, a, was optimized such that (C p (K 2 -ThF 6 ,(cr),298.15 K) = 193.2 JÁK À1 Á mol À1 , the same value found by fitting of the low temperature heat capacity data of a-K 2 ThF 6 (cr) (see Section 5.3).
The thermodynamic data for all compounds in this study are listed in Table 4. The convention used throughout this paper is that the lower temperature phases are denoted with a; the opposite convention is used by [9,8] (respectively: Figs. A.1 and A.2). The data for both solid and liquid alkali fluorides (LiF, NaF, KF) and ThF 4 were taken from [14,24], respectively. All thermodynamic functions of the intermediate compounds in the LiF-ThF 4 and NaF-ThF 4 systems were derived by optimization in [24], while those for intermediate compounds in the KF-ThF 4 system were obtained in this work by optimization using phase equilibrium and mixing enthalpy data.
The transition temperature and enthalpy of transition of K 5  ; T tr = 825 AE 5) K were measured in this work by DSC using a similar procedure to the mixing enthalpies with a silver standard in the reference crucible. The experimentally determined values were implemented in the model without further optimization. No quaternary fluorides have been reported in the LiF-KF-ThF 4 system, while KNaThF 6 (which displays a phase transition) is the only quaternary fluoride reported in the NaF-KF-ThF 4 system [8], and its thermodynamic properties were studied by Mukherjee and Dash by means of DSC and solid electrolyte galvanic cell [25]. Based on the former technique, the authors derived the heat capacity in the temperature range 300-870 K, while with the latter, they obtained the Gibbs energy of formation in the temperature range 773-849 K from which they derived standard enthalpy of formation and standard entropy values at 298 K. The heat capacity reported by them results in an extremely stable phase, such that unreasonably high excess parameters would have to be applied to the liquid NaF-KF-ThF 4 solution in order to stabilize it even at high temperatures. For this reason, this work approximated the heat capacity function of KNaThF 6 using the Neumann-Kopp rule applied to NaF, KF, and ThF 4 . In addition, we have reassessed the enthalpy of formation and entropy at 298 K based on the experimental data of [25] and carefully selected auxiliary data (see Appendix for details). The final optimized standard enthalpy of formation and standard entropy yield a Gibbs energy which is only $1 % larger than the experimental value. 7 The quoted uncertainty corresponds to the standard uncertainty. 8 The quoted uncertainty corresponds to the standard uncertainty. 9 Standard uncertainties u are u(m) = 0.05 mg.

Solid solution
The total Gibbs energy function of the two-component solid solutions in the present system is given by: where X i are the molar fractions and (T) G o m;i (T) are the standard molar Gibbs energies of the pure end members. The excess Gibbs energy parameter is described using the polynomial formalism: where L i;j is a coefficient which may depend on temperature in the form of the general equation Solid solutions are formed in the NaF-KF and NaF-ThF 4 binary systems, with optimizations taken from the literature, shown in Eq. (7) [27] and (8) [28]:

Liquid solution
All excess Gibbs energy terms of liquid solutions presented here have been modelled using the modified quasi-chemical model proposed by Pelton et al. [29], in the quadruplet approximation. The quasi-chemical model is particularly well adapted to describe ionic liquids such as in the present system, as it allows to select the composition of maximum short-range ordering (SRO) by varying the ratio between the cation-cation coordination numbers Z A AB=FF and Z B AB=FF (the fluorine is in this case the only anion present). The quadruplet approximation assumes a quadruplet, composed of two anions and two cations, to be the basic unit in liquid solution, and the excess parameters to be optimized are those related to the following second-nearest neighbor (SNN) exchange reaction: where the fluoride anions are represented by F, and A and B denote the cations. Dg AB=F is the Gibbs energy change associated with the SNN exchange reaction, and has the following form: where Dg o AB=F and g ij AB=F are coefficients which may or may not be temperature-dependent, but which are independent of composition.
The dependence on composition is given by the v AB=F terms defined as: where X AA , X BB and X AB represent cation-cation pair mole fractions. The anion coordination number is finally fixed by conservation of charge in the quadruplet: where q i are the charges of the different ions, and Z F AB=FF is the anionanion coordination number, in this case fluorine-fluorine.
The cation-cation coordination numbers used in this work are listed in Table 5. These were chosen to represent the composition of maximum short-range ordering, where the Gibbs energy tends to have its minimum. In the case of KF-ThF 4 , the point of maximum SRO can be reasonably expected to lie near X(ThF 4 ) = 0.33, i.e. where the liquid solution seems to be especially stable as indicated by the low liquidus in the vicinity of that composition. Hence, the cation-cation coordination numbers were chosen to fix maximum SRO around X(Th 4 ) = 0.33. Similarly, the cation-cation coordination numbers in the LiF-ThF 4 and NaF-ThF 4 systems were chosen to fix maximum SRO around X(Th 4 ) = 0.25, because the liquidus those phase diagrams is lowest in the vicinity of that composition.
The optimized excess Gibbs energy parameters of the binary liquid solution in the KF-ThF 4 system are shown in Eq. (13). The parameters were optimized based on the enthalpy of mixing data and on the phase diagram equilibria points of the liquidus. The excess Gibbs energy parameters of the other binary liquid solutions needed to calculate the ternary systems are given in Appendix F.

Higher order systems
The ternary diagrams LiF-KF-ThF 4 and NaF-KF-ThF 4 have each been extrapolated from the constituting binary sub-systems using the asymmetric Toop formalism [30]. The salts belong to two groups of symmetry based on their tendency to remain as dissociated ionic liquids (LiF, NaF, KF) or to form molecular species in the melt (ThF 4 ). The optimized excess ternary parameters are: Dg KTh Na

Previous evaluations of the KF-ThF 4 system
Phase diagram studies of this system have been reported by Bergman and Dergunov [31], Asker et al. [9] (Fig. A.1), and Emelyanov and Evstyukhin [8] (Fig. A.2). Bergman and Dergunov report a phase diagram which is certainly too simple, with four eutectics and three congruent melting points corresponding to the compounds K 3 ThF 7 , KThF 5 , and KTh 3 F 13 ; these results markedly differ from those of the other two authors. Emelyanov and Evstyukhin [8] and Asker et al. [9] agree on the existence of K 5 ThF 9 , K 3 ThF 7 , KThF 5 , and K 2 ThF 9 , and the melting points they report for these compounds are similar. However, they report different ternary compounds for close compositions: KTh 3 F 13 (X(ThF 4 ) = 0.75, [9]) vs. KTh 6 F 25 (X(ThF 4 ) = 0.85, [8]); K 2 ThF 6 (X(ThF 4 ) = 0.33, [9]) vs. K 3 Th 2 F 11 (X(ThF 4 ) = 0.4 [8]). Unlike the former pair, for which similar melting points were reported, K 2 ThF 6 and K 3 Th 2 F 11 are reported to have dissimilar behavior: the former with a peritectic decomposition at 1023 K, and the latter with a congruent melting at 1160 K. In addition, these authors find different allotropic transformations: Asker et al. report two phases for K 5 ThF 9 and K 2 ThF 6 , while Emelyanov and Evstyukhi report two phases only for KTh 2 F 9 . Finally, Asker et al. suggest the existence of a solid solution close to Fig. 5. XRD pattern of a sample with composition X(ThF 4 ) = 0.418 after DSC measurements. Comparison between the observed (Y obs , in red) and calculated (Y calc , in black) Xray diffraction patterns. Y obs À Y calc , in blue, is the difference between the experimental and calculated intensities. The Bragg's reflection angular positions are marked in blue (K 2 ThF 6 ), and red (K 7 Th 6 F31). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 6. XRD pattern of a sample with composition at X(ThF4) = 0.857. Comparison between the observed (Y obs , in red) and calculated (Y calc , in black) X-ray diffraction patterns. Y obs À Y calc , in blue, is the difference between the experimental and calculated intensities. The Bragg's reflection angular positions are marked in blue (KTh 6 F25), and red (KTh 2 F 9 ). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) pure ThF 4 , yet Emelyanov and Evstyukhin do not. Our studies focused on exploring the differences found between the two sets of authors.

K 5 ThF 9
Attempts to synthesize K 5 ThF 9 did not yield pure K 5 ThF 9 , but a mixture with K 2 ThF 6 (diffractogram shown in Fig. 1). K 5 ThF 9 has orthorhombic symmetry space group Cmc2 1 [32]), with a diffractogram with many small reflections, while K 2 ThF 6 is hexagonal (space group P62 m [33]). The structure could be refined with the model proposed by Ryan and Penneman [32], with distorted anti-prism for K and Th, and pentagonal bipyramid for the polyhedra of K. The refined cell parameters and table of atomic positions are reported in Appendix C.
Our DSC data from a sample made with a stoichiometric mixture of KF and ThF 4 powders suggest a first transition taking place at (926 AE 10) K, which is higher than the transition temperature recorded by Asker et al. (908 K), but similar to the temperature that Evstyukhin and Emelyanov attribute to the first eutectic of the system (935 K). We assign the event to an allotropic transition as the thermogram at this composition shows three other thermal events which are coherent with the existence of a high temperature K 5 ThF 9 phase: eutectic, (968 AE 10 K), peritectic decomposition, (982 AE 10 K), and liquidus (1013 AE 10 K). With a measurement in the DSC using a silver reference method, the enthalpy of transition for K 5 ThF 9 was found to be D tr H o m ¼ 9:3 AE 0:8 ð ÞkJÁmol À1 .
5.1.2. K 3 ThF 7 K 3 ThF 7 is not stable at room temperature according to Asker et al. [9] (Fig. A.2). The main goal of the present investigation at this composition was to confirm its eutectoid decomposition into K 2 ThF 6 and K 5 ThF 9 around 840 K as reported by [9]. The XRD pattern indeed revealed a mixture of K 2 ThF 6 and K 5 ThF 9 , as illustrated in Figs. 2a and 2b. Although a satisfactory Rietveld refinement of the XRD pattern of the sample could not be obtained because of the poor crystallinity of the K 5 ThF 9 phase in the mixture, the main Bragg reflections of both phases could clearly be identified, such that we were able to confirm that K 3 ThF 7 is not stable at room temperature.

K 2 ThF 6
K 2 ThF 6 was successfully synthesized in pure form (hexagonal in space group P62 m [33], Fig. 3). A Rietveld refinement of the XRD data showed the sample was the low temperature hexagonal phase. In another attempt (Fig. 4), the synthesized sample was a mixture of the two phases reported in the literature, the high temperature phase being cubic and belonging to the space group Pm3m [33]. After a second annealing of this mixture at 873 K, below the transition temperature (918 K [9]), the diffractogram ( Fig. 4) no longer showed any Bragg reflections attributable to the cubic phase, indicating the complete transformation of the cubic phase to the hexagonal form (Fig. 3). The DSC measurement of pure a-K 2 ThF 6 shows an event at (952 AE 10) K which is higher than the temperature assigned by Asker    however. Rather, later studies identify K 7 Th 6 F 31 as the line compound in the neighborhood of X(ThF 4 ) = 0.5. This was first reported at Oak Ridge National Laboratory [34], where having studied other (AF:ThF 4 )=(7:6) compounds in alkali fluoride-thorium tetrafluoride systems, the authors predicted the existence of such a compound in the KF-ThF 4 system and confirmed its existence with thermal analysis of slowly cooled melts. Indeed, a melt of composition X (ThF 4 ) = 0.462 displayed a single event upon slow cooling and a single phase according to post-characterization by XRD. By comparing the XRD data of this phase with the spacings for Na 7 U 6 F 31 and K 7 U 6 F 31 , the authors were able to classify it as having rhombohedral symmetry. Moreover, the melting temperature, (1172 AE 2) K, was very close to that reported for KThF 5 : 1178 K [9] and 1166 K [8]. Brunton [35] and more recently Grzechnik et al. [36] were able to fully solve the crystal structure of K 7 Th 6 F 31 (R 3), both using single-crystal X-ray diffraction data. A sample of composition X(ThF 4 )= 0.418 (after having been subjected to a DSC measurement) was found in this work to be a mixture of K 2 ThF 6 and K 7 Th 6 F 31 according to the Rietveld refinement (Fig. 5) of the XRD data. Supporting this result is a sample of mole fraction X(ThF 4 )= 0.494 (between KF:ThF 4 = 7:6 and 1:1 compositions) measured in the DSC which did not show thermal events close to 950 or 1000 K (corresponding to the equilibria of K 2 ThF 6 ); these equilibria would be visible between K 2 ThF 6 and the putative compound KThF 5 if K 7 Th 6 F 31 were not a stable phase (see Fig. 11). Finally, due to the lack of convincing experimental evidence for the existence of KThF 5 , in particular the absence of reported Wyckoff positions, we discarded it from the phase diagram.

KTh 6 F 25
To discern whether the phase with the highest thorium content was KTh 3 F 13 or KTh 6 F 25 , we attempted a synthesis of both compounds (see Table 2). The XRD pattern of the sample with composition X(ThF 4 ) = 0.857 was found to be a mixture of KTh 2 F 9 (space group Pnma) and KTh 6 F 25 (space group P6 3 mmc) from a LeBail refinement (Fig. 6). Unreacted excess KF was possibly not detected by XRD. In the same manner, the sample with composition X(ThF 4 ) = 0.75 revealed reflections corresponding to KTh 2 F 9 and KTh 6 F 25 . Furthermore, KTh 3 F 13 is not mentioned in the literature outside the work of Asker et al., which brings further doubt on its existence. The phase was hence not retained in the present thermodynamic assessment.

Solid solution
Asker et al. [9] report the existence of a solid solution between KTh 3 F 13 and ThF 4 extending up to about 16 % KF in ThF 4 . No evidence for such solid solution was found in the present DSC measurements, however, for three compositions between KTh 6 F 25 and ThF 4 : X(ThF 4 )= 0.902, 0.942, 0.979. These measurements only showed two thermal events (Fig. 7) which we assign to the peritectic decomposition of KTh 6 F 25 and liquidus.    [16], NaF-ThF 4 (blue) [24], KF-ThF 4 (green, this work) and CsF-ThF 4 (black) [42]. The calculations were done at T = 1400 K considering KF(l) and ThF 4 (l) as initial states and the liquid solution as the final state.Circles: experimentally measured enthalpies of KF(l)-ThF 4 (s) mixing at 1131 K (see Table 7); squares: experimentally measured points by [16] at 1121 K (LiF(l)-ThF 4 (s), white) and 1383 K (LiF(l)-ThF 4 (l), black).The dashed green line corresponds to KF-ThF 4 at T = 1131 K. Initial states were KF(l) and (hypothetical) ThF 4 (l), the final state was the liquid solution (hypothetical for most of the composition range). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Table 7 Mixing enthalpy of the 1 À x ð Þ KF(l) + xThF 4 (l) system determined in this work at T = 1131 AE 10 K and (0.10 AE 0.01) d MPa. KF melts at the measurement temperature, while ThF 4 is solid, although the initial state is taken to be a hypothetical liquid. The final state was an undercooled liquid mixture. The scanning temperature range was from (303 AE 10) K to (1373 AE 10) K.
for the nine atoms in the formula unit) as the temperature approaches 298.15 K. The collected data exhibit an anomaly in the neighborhood of the 65-120 K range, evident in the plot of C p =T (Fig. 8b). Repeated operation of the instrument with diverse reference samples has shown that this is a systematic error attributable to the specific measuring puck upon which the sample is placed, and not an intrinsic feature of the measured material. This error is corrected by the fitted curve.
The thermodynamic functions of a-K 2 ThF 6 were derived at In the low temperature limit (T < 10.0 K), the phonon contribution can be adequately approximated by the harmonic lattice model [37], the form of which is given in Eq. 18: Three terms, with coefficients listed in Table 6, were used over the temperature range T= (1.9 to 10.0) K. The electronic contribution of the conduction electrons at the Fermi surface are represented with a linear term cT [41]. In this case, K 2 ThF 6 being a poor conductor, the electronic specific heat is nearly zero.
In the region 10.0 < T < 297.0 K the main contribution comes from the lattice term, modelled here with a combination of Debye Two Einstein functions were needed in order to achieve an adequate fit of the data, which was carried out excluding the problematic 65-120 K region. The fitted coefficients are listed in Table 6. The sum over n atoms is equal to 8.4, quite close to 9 as should be expected. The Debye and Einstein functions have the following forms: where the universal gas constant is denoted by R and is equal to The determination of the mixing enthalpy of the liquid solution is very useful in the assessment of a complex system such as KF-ThF 4 as it provides another dataset, besides the phase diagram points, to optimize the excess Gibbs energy terms of the liquid phase. Table 7 reports the values for the mixing enthalpies of the (K x Th 1Àx )F 4À3x liquid solution as determined in this work at the melting temperature of KF, i.e., (1131 AE 10) K. The range of investigated compositions is limited by the large difference in molar masses between the two salts: at high X(ThF 4 ) mole fraction, the mass of ThF 4 is much larger compared to the mass of the solvent KF, with lower melting temperature, such that complete mixing of liquid KF with ThF 4 , is difficult to obtain. However, complete mixing of the liquid solution compositions reported in Table 7 was ensured by the presence of one single event which corresponded to the combination of KF melting, ThF 4 melting, and mixing event, and the absence of other events such as the melting of residual ThF 4 , invariant equilibrium reactions, or the liquidus at the given composition. An example of a successful measurement is shown in Fig. A.5.
The experimental data obtained in this work and the curve predicted by our model are plotted for comparison (green) in Fig. 9a against the mixing enthalpies of the LiF-ThF 4 (red), NaF-ThF 4 (blue) and CsF-ThF 4 (black) systems as calculated respectively from the thermodynamic assessments of Capelli et al. [24], Beneš et al. [28] and Vozárová et al. [42]. The present experimental data and modelled mixing enthalpies are more negative than for the LiF-ThF 4 and NaF-ThF 4 systems, which is consistent with the increase in the alkali cation ionic radius. Depending on composition and temperature, liquid fluoride salts can form dissociated ions, molecular species, or even a polymeric network. The larger K þ ion offers a larger steric hindrance than Li + and Na + , isolating the coordination around Th 4þ in the (K,Th)F x liquid solution is also likely to be further stabilized compared to its analogue in (Li,Th)F x or (Na,Th)F x liquid solutions due to polarization effects: lighter alkali ions are more polarizing than K þ and thus are able to attract F À more strongly, leading to longer -Th 4þ À F À distances and weaker complexes. The most negative mixing enthalpy curve, corresponding to the largest cation Cs þ , fits this trend. Gibbs energies of mixing (Fig. 10a) also reflect this trend in stability, with the composition of maximum stability indicated by the minimum in the curve. This composition is related to the choice of cation-cation coordination numbers (Table 5), as is the maximum Þ= 0.005. b Standard uncertainties u are u(T)= 5 K for the pure end-members, u(T) = 10 K for mixtures. The pressure was (0.10 AE 0.01) MPa. 10 The quoted uncertainty corresponds to the standard uncertainty.
of the M-Th-F-F pair fractions (Fig. 10b), which is around X(ThF 4 ) = 0.25 for Li, Na systems and X(ThF 4 ) = 0.33 for K, Cs systems. The basicity of the systems can be qualitatively gauged from the shape and value of the M-Th-F-F maximum: a strongly basic system would lead to complete dissociation of the cation-cation pairs to result in perfect SNN ordering, i.e., a bond fraction of unity at the composition of maximum short-range ordering. In this case, the systems can be qualified as moderately to reasonably basic with a round shape for the M-Th-F-F fraction and maximum value between 0.55 and 0.79. Moreover the degree of basicity is closely linked with the stability trend just discussed: the maximum fraction of the Li-Th-F-F distributions is the lowest, those of K-Th-F-F and Cs-Th-F-F are the highest. It is interesting to note that not even CsF-ThF 4 is basic enough for Cs-Th-F-F to reach unity; it can be expected that FrF-ThF 4 would come closest.
The entropies of mixing are plotted in Fig. 9b. Capelli et al. [16] related the maximum for mixing entropy in LiF-ThF 4 at X(ThF4) % 0.25 to a higher content of free F À , observed in NMR studies by Bessada et al. [48]. The computed curves suggest that there would be less free F À , at least in the KF and CsF-based systems. Structural studies such as NMR or EXAFS on these systems at high ThF 4 compositions could help understand if this trend is correct.   Table 12). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

CALPHAD assessment of the KF-ThF 4 system
The KF-ThF 4 system (Fig. 11) was finally optimized based on our measured DSC equilibrium and mixing enthalpy data, all of which are presented in Table 8.There are 6 intermediate compounds:  Table 9. It can be seen that the liquidus temperatures measured in this work are sometimes slightly higher than those reported by the previous studies. This is most likely because the authors [9,8] used onset temperatures rather than minimum temperatures on the heat flow events for liquidus determination (see Fig. A.3 for an explanation of how these are determined in this work in a typical DSC measurement). In fact, the agreement with the previous studies [8,9] becomes much better if the onset temperatures of the liquidus equilibria as measured in this work (O, red) are selected (see Fig. 11).

LiF-KThF 5 pseudobinary section
Some equilibrium data were collected along the LiF-KThF 5 pseudobinary section in order to optimize the liquidus surface of the LiF-KF-ThF 4 ternary system. The section is relatively simple, as shown in Fig. 12. LiF is the first compound to crystallize below 25 mol $ 25 mol % X(KThF 5 ), beyond that the liquid solution is in equilibrium with K 7 Th 6 F 31 . Below the solidus the calculation shows that there are two ternary phase fields throughout the composition range: {LiF + Li 3 ThF 7 + K 7 Th 6 F 31 } and {Li 3 ThF 7 + KTh 2 F 9 + K 7 Th 6 F 31 }. Below $662 K the equilibrium is between LiF, KTh 2 F 9 , and K 7 Th 6 F 31 . The experimental points, however, suggest that the temperature range in which {LiF + Li 3 ThF 7 + K 7 Th 6 F 31 } and {Li 3 ThF 7 + KTh 2 F 9 + K 7 Th 6 F 31 } exist is much more narrow. The phase equilibria of the experimental points are given in Table 12.

LiF-KF-ThF 4 liquidus projection
The LiF-KF-ThF 4 system (Fig. 13) as calculated in this study is characterized by fifteen primary fields of crystallization and nineteen invariant points: nine quasi-peritectics, four saddle points, two peritectics, and four eutectics, the lowest of which is calculated at T = 755 K, very close to the LiF-KF eutectic: (X(ThF 4 ), X (LiF), X(KF)) = (0.023, 0.461, 0.516). DSC analysis of an experimental point in the neighborhood of this ternary eutectic (X(ThF 4 ), X (LiF), X(KF)) = (0.037, 0.463, 0.500) showed only one event at T = (753 AE 10 K), which can be assigned to the invariant reaction LiF(cr) + KF(cr) + K 5 ThF 9 (cr) = L. Hence, the point is a ternary eutectic. The agreement between the measured and calculated liquidus points is reported in Table 10. The solid phases in equilibrium with the liquid, compositions, and temperatures of all calculated invariant equilibria are listed in Table 10. The rest of the equilibria measured by DSC are listed in Table 11. There are no ternary stoichiometric compounds or solid solutions in the ternary system reported in the literature.

NaF-KThF 5 pseudobinary section
The NaF-KThF 5 pseudobinary section, studied by Emelyanov and Evstyukhin, is the only set of data reported in the literature related to the liquidus surface of the NaF-KF-ThF 4 ternary system. As measured by the authors, the pseudo-binary section shows only one intermediate compound: the quaternary fluoride KNaThF 6 , displaying a phase transition at 813 K [8], and incongruent melting at 938 K. In this work the a À b transition of KNaThF 6 In our calorimetric measurement we were not able to detect a third event which would correspond to the liquidus, and neither did [8] (Fig. 14). This might be due to a very close proximity of the incongruent melting event to the liquidus surface, making it hard to detect.  a Standard composition error is u jj ð (X(ThF 4 ), X(LiF), X(KF))jjÞ = 0.006. b Standard uncertainties u are u(T)=10 K. c Calculated as LiF + L 0 = L at the experimental composition, since the eutectic does not exactly match the experimental one. d Eutectic temperature calculated as 755 K in the vicinity of the experimental composition, see Table 11. Measurements done at (0.10 AE 0.01) MPa.     Table 16). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

NaF-KF-ThF 4 liquidus projection
The NaF-KF-ThF 4 (Fig. 16) system is an even more complex system, with one quaternary compound (displaying a phase transition), 19 primary crystallization fields and 31 invariant points (Table 14). Of these, nineteen are quasi-peritectic, five are saddle points, four are peritectic, and four are eutectic, with the lowest one occurring at 804 K and X(ThF 4 ) = (0.219, 0.234, 0.547). We have found the liquidus temperatures of eight samples (points shown in red) to closely match the calculated ones only in most instances (Table 15). All equilibria measured experimentally are given in Table 16.

Conclusions
A thermodynamic assessment for the KF-ThF 4 binary system using the CALPHAD method is reported for the first time in combination with XRD and calorimetric measurements. The main char-  a Standard composition error is u jj ð (X(ThF 4 ), X(NaF), X(KF))jjÞ = 0.006. b Standard uncertainties u are u(T)=10 K. The pressure was (0.10 AE 0.01) MPa.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Quasi-peritectic L 0 + KTh 2 F 9 + K 7 Th 6 F 31 = L + K 7 Th 6 F 31 1163 Liquidus L 0 + K 7 Th 6 F 31 = L a Standard composition error is u jj ð (X(ThF 4 ), X(NaF), X(KF))jjÞ = 0.006. b Standard uncertainties u are u(T) = 10 K. The pressure was (0.10 AE 0.01) MPa. capacity and solid electrolyte galvanic cell to derive the Gibbs energy of formation [25]. Herein we describe our recalculation of the standard enthalpy of formation and standard entropy based on the data published by the authors [25] and carefully selected auxiliary data.
The solid electrolyte galvanic cell used by Mukherjee  The reaction at the cathode is: and at the anode: The authors measured the electromotive force (e.m.f) as a function of temperature and found linear relationships with two different slopes, corresponding to the two phases of KNaThF 6 cr ð Þ [25]: one in the interval 773-848 K and one in the interval 848-973 K. In the lower temperature range of 773-848 K, the following expression was reported: E T ð Þ=V AE 0:0014 ¼ 0:2772 À 1:586 Á 10 À5 with the associated Gibbs energy of reaction given by:   Using carefully selected data reported in the literature (Table A.1, [14], [49], [50]), the Gibbs energy of formation of KNaThF 6 is derived as: The experimental heat capacity data points measured for the phase K 2 ThF 6 are listed in Tables A.3

Appendix E. Representative DSC curves
This section contains representative examples of DSC measurements carried out in this work and how the data were extracted. Fig. A.3 is a DSC heating ramp with two event corresponding to eutectic and liquidus temperatures. The intersection between the baseline and the line tangent to the first inflection point of a given event is the onset temperature, T onset , and the temperature assigned to all events except for the liquidus of mixtures. The intersection between the baseline and the line tangent to the second inflection point of a given event is the offset temperature, T offset . In this work, the minimum of the peak, T min , was chosen as the liquidus temperature except for congruent melting. Nevertheless, onset temperatures are also drawn in Fig. 11 as (O, red) for the liquidus events to illustrate this choice as a likely source of disagreement with previous authors. The standard uncertainty on the pressure is u(p) = 0.01 mPa.  Figs. A.3,A.4, or A.5, but a slightly more complex shape which arises from the endothermic contribution of the melting of KF and ThF 4 and the simultaneous exothermic mixing. Still, as the shading indicates, the total area of the mixing event is measured the same way as for all the other events: as the area constrained by the baseline and the curve which departs from it. Since the areas have opposite signs, they partially cancel each other out, and this is taken into account. The second thermal event corresponds to the melting of the silver reference, after the mixing event has concluded, such that they are both well resolved. The fusion of silver is endothermic, yet appears as exothermic because the silver sits in the reference crucible. The superimposed red DSC curve corresponds to a heating ramp of pure ThF 4 . Crucially, after the fusion of silver in the mixing enthalpy heating cycle there is no fusion event at the melting temperature of ThF 4 ; this is an indication of complete mixing. Data from mixing enthalpy measurements were only deduced for those runs which showed complete mixing. For every run, a sensitivity could be assigned based on the heat measured on the silver reference, as mentioned in Section 2.5. The total heat measured during the event at the melting temperature of KF, D meas H o T fus;KF À Á , is given by: D meas H o T fus;KF À Á ¼ S salt=Ag Á Q meas S Ag;i ðA:8Þ where Q meas is the heat measured per mole of sample during the mixing experiment, i.e. the gray area in Fig. A.5, S salt=Ag is the sensitivity coefficient determined during the calibration process, and S Ag;i is the sensitivity of silver during measurement i. Finally, the molar enthalpy of mixing can be derived based on Eq. 1.

Appendix F. Excess Gibbs energy parameters of the consitutent binary liquid solutions
The excess Gibbs energy parameters of the binary solutions needed to calculate the ternary systems were taken from existing assessments and are listed below for completeness.