Reference Correlations for the Density and Viscosity of Molten Alkali and Alkaline Earth Fluoride Salts Open Access

Molten fluoride salts have a wide range of applications as heat transfer and energy storage fluids, as well as in pyroprocessing and other metallurgical processes. Molten salts have been used¹ and are being considered^2–4 for use as coolants and as fuels in molten salt reactors because of many advantageous qualities such as high neutron moderation capability and low vapor pressures at high temperatures. Molten fluorides (namely LiF–BeF₂) are also being considered as blankets for fusion reactors, for which these fluorides would function as tritium producers and magnet radiation shields.⁵ Molten fluorides have been used as electrolytes in electrochemical manufacturing processes for many metals,^6–8 one notable example being aluminum production via the Hall–Heroult process.⁹ Finally, molten fluoride salts are being considered for pyrochemical separation of actinides from lanthanides for spent solid fuel forms.¹⁰ 

Because molten fluoride salts have many applications, critically assessed reference data of their thermophysical properties are necessary so that thermohydraulic, electrochemical, and pyrochemical processes can be accurately understood. George J. Janz conducted a significant effort to assess and tabulate molten salt thermophysical properties as functions of temperature; this effort supported the National Standard Reference Data System, which was established in 1963. Janz published several tabulations for fluorides and chlorides,^11–15 the last (and most exhaustive) of which was published in 1988.¹⁵ Since the conclusion of Janz’s work, updated reference correlations—including reference correlations for the thermal conductivity of inorganic molten salt compounds,¹⁶ the viscosity of inorganic molten salt compounds (excluding fluorides),¹⁷ and the viscosity of LiF–NaF–KF, LiF–BeF₂, and Li₂CO₃–Na₂CO₃–K₂CO₃¹⁸—have been determined in a number of works. In addition, several databases,^19–22 property reviews,^23–25 and comprehensive models^26–28 have been published to elucidate the thermophysical properties of key molten salt mixtures for energy applications. However, a gap exists in the literature published since the work of Janz (which concluded in 1988).¹⁵ A critical reassessment of density and viscosity has not been performed for molten fluoride compounds; yet, several publications have published new experimental data, and the understanding of property measurement methods has been advanced.²⁹ 

Knowledge of the thermophysical properties of pure compounds is crucial for estimating the thermophysical properties of several different molten salt mixtures that may be considered for the applications previously discussed. Simple mixing rules, such as additive molar volumes for density estimation and the Grunburg–Nissan³⁰ method for viscosity estimation, are based on ideal mixing assumptions and therefore rely on pure compound property data. Furthermore, pure compound property data form the basis for Redlich–Kister expansions.^31,32 Redlich–Kister expansions can be used to empirically address nonideal mixing, which tends to be significant for highly complexing mixtures. Pure compound density is also a necessary input for multiple thermal conductivity estimation models such as kinetic theory³³ and the Rao–Turnbull model.³⁴ Also, converting thermal diffusivity data to thermal conductivity data (or vice versa) requires accurate knowledge of density. Finally, a significant body of literature exists regarding molecular dynamics simulations that predict the thermophysical behavior of molten salt,³⁵ and those studies would benefit from updated, experimentally-based reference data that could be used for model validation.

Given the widespread need for and applicability of accurate density and viscosity values for molten fluoride compounds, the aim of this paper is to provide updated correlations that describe these properties as a function of temperature for all radiochemically stable alkali and alkaline earth fluorides. Estimated uncertainties are calculated based on two standard deviations (i.e., 95% confidence intervals) of global least-squares fits to primary reference data for each compound. Uncertainties have only been quantified for cases in which reference correlations are based on three or more primary datasets.

The selection of primary datasets is based on the following general rules: (1) high-quality datasets are always included as primary datasets, unless they are significantly inconsistent with multiple other high-quality datsets which are consistent; (2) any dataset that, based on quality alone, would otherwise be declared secondary may be considered primary if it is consistent with multiple primary datasets of high quality and there is a precedent for including the dataset as primary to ensure that at least three datasets are used to form a reference correlation; (3) datasets that are of lower quality but are consistent with multiple primary datasets of high quality are not considered primary datasets when there are already three or more primary reference correlations to form a reference correlation. The assessment of dataset quality is informed to a significant extent by quality rankings applied to fluoride compounds in a published technical reports.^36,37 These reports assess data quality on the basis of the application of the measurement method, calibrations, environmental controls, salt composition confirmation, uncertainty, and verifiability of the reported property value from measured data. The exact definitions of the varying levels of quality for each of these measurement aspects are conveniently organized in Table 1 of Rose.³⁷ Section 2 provides a detailed discussion of the selection of primary reference data for each individual compound.

A comparison of the reference correlations provided herein with those previously put forth by Janz¹⁵ is also provided. Notably, because of document unavailability or difficulty extracting quantitative data from literature, some references (which were generally published prior to 1970 in the USSR) are not considered in this analysis. Janz provides a fully exhaustive account of all density and viscosity measurements published prior to 1974.¹² 

In this section, the reasoning associated with the designation of primary and secondary references is provided, in association with the density and viscosity measurements for each fluoride compound considered herein. The logic generally follows the rules outlined in Sec. 1. The datasets associated with the references declared as primary are then used to quantify the reference correlations and the associated uncertainties in Sec. 3. As mentioned in Sec. 1, these designations rely to a significant extent on the determination of the quality of the underlying studies which supported the measurements; this is discussed in a reasonable amount of detail in this section, with additional information provided by Rose.^36,37

In this subsection, the reasoning associated with the designation of primary and secondary references for LiF, NaF, KF, RbF, and CsF density is provided. A summary of the references is provided in Table 1, in which the respective methods, sample purities, measured temperature ranges, and reported uncertainties are tabulated.

There are 15 accounts of molten LiF density measurements, as plotted in Fig. 1(a). Note that for this plot, and all subsequent plots which summarize duplicate datasets for a particular compound, the individual datapoints indicate measurements, whereas the trendlines represent fits to the measurement data; in some instances, datapoints have not been reported. There is strong consistency between density and the thermal expansion coefficient for all accounts, except for the measurements of Porter and Meaker,³⁸ Brown and Porter,³⁹ Belov et al.,⁴⁰ Matsushima et al.,⁴¹ Popescu and Constantin,⁴² and Danielyan and Belyaev;⁴³ therefore, these references are considered secondary. Mellors’s density data are also questionable because the density correlation extends lower than the known melting point of LiF (1121 K).⁴⁴ Of the remaining datasets that show good consistency, the data collected by Hill et al.,⁴⁵ Taniuchi and Kanai,⁴⁶ Katyshev et al.,⁴⁷ Paucirova et al.,⁴⁸ Hara and Ogino,⁴⁹ and Smirnov and Stepanov⁵⁰ are considered primary. Although the data collected by Yaffe and Van Artsdalen⁵¹ and Jaeger and Kahn⁵² are consistent with these primary datasets, these datasets are considered secondary for LiF specifically. Yaffe provided virtually no information regarding experimental details aside from the experimental method used, and Jaeger's report does not indicate whether measurements were conducted in a dry, temperature-controlled environment or whether salts were purified or their composition confirmed.

For the primary datasets considered, the references are summarized as follows. Hill et al.⁴⁵ used the Archimedean method to perform measurements of LiF, provided a detailed process for drying the salts before testing, performed impurity analysis before testing, and stabilized temperature to ±0.2 K. Hill does not provide calibration or uncertainty quantification details. Taniuchi and Kanai⁴⁶ used a two-sinker Archimedean method to perform measurements; this method implicitly accounts for surface tension effects. Taniuchi also used water and KNO₃ to perform calibrations, vacuum dried the salts before testing, and performed measurements under flowing argon. Taniuchi provided limited uncertainty quantification details and did not explicitly measure impurity content. Katyshev et al.⁴⁷ used the maximum bubble pressure (MBP) method to perform measurements, appropriately dried the reagents, detailed the purification of the argon gas used, and stabilized temperatures to ±2 K. The manometer and micrometer used were calibrated; however, the overall measurement system was not calibrated with a standard reference fluid. Paucirova et al.⁴⁸ used the Archimedean method to perform measurements and provided an experimental measurement accuracy. Paucirova provides a qualitative description of the purity of the salt, but no quantitative analysis was performed. Paucirova does not provide calibration details. Hara and Ogino⁴⁹ used a two-sinker Archimedean and MBP method to perform measurements, performed duplicate measurements, vacuum dried the salts, measured each of the samples for impurities, and provided a detailed discussion of sources of uncertainty. Smirnov used both the MBP and Archimedean methods to measure density, calibrated their systems with well-characterized compounds (KNO₃ and KCl), performed measurements under high-purity argon with temperature control within 1 K, and reported measurement precisions for both measurement systems.

There are 13 accounts of molten NaF density measurements, as plotted in Fig. 1(b). Porter and Meaker³⁸ measured data appear to be erroneously low (as is the case with LiF). Brown and Porter³⁹ and Popescu and Constantin⁴² measured more extreme thermal expansion coefficients not observed in the other measurement data. Because of these discrepancies, Porter, Brown, and Popescu are considered secondary references for NaF. For the remaining 10 measurements, Fig. 1(b) shows two clusters of measurement data: (1) the data measured by Edwards et al.,⁵³ Desyatnik et al.,⁵⁴ Artemov et al.,⁵⁵ and Rolin,⁵⁶ and (2) the data measured by Paucirova et al.,⁴⁸ Hara and Ogino,⁴⁹ Ogino et al.,⁵⁷ Fontana and Winand,⁵⁸ Jaeger and Kahn,⁵² and Smirnov and Stepanov⁵⁰ (although the values measured by Smirnov are somewhat low relative to this cluster).

Regarding the first cluster of data, Edwards performed measurements with the Archimedean method and performed preliminary heating of the samples before taking measurements (presumably for drying purposes). However, Edwards does not indicate how the measurement setup was calibrated, does not report on sample purity, and does not provide a description of environmental controls. Desyatnik performed measurements with the MBP method but did not provide information regarding instrument calibration, compositional analysis, and environmental controls. Artemov used the MBP method to measure the density of NaF and did report salt dehydration; Artemov’s method was identical to the method used by Katyshev et al.⁴⁷ However, other pertinent experimental details regarding calibration, compositional analysis, and environmental controls are not reported. Rolin appears to have used the Archimedean method to perform measurements, but virtually no other details are provided regarding how the measurements were performed. Because of the lack of experimental detail reported and the higher density values reported by this cluster of datasets, these datasets are also considered secondary.

Regarding the second cluster of data at lower density values, a summary of the experimental details provided by Hara and Ogino,⁴⁹ Jaeger and Kahn,⁵² Paucirova et al.,⁴⁸ and Smirnov and Stepanov⁵⁰ is provided in Sec. 2.1.1 (authors also measured LiF). Hara provides a particularly high level of detail, indicating that high-quality measurements were taken. Fontana and Winand⁵⁸ used the Archimedean method to perform measurements. Fontana took measures to account for liquid condensation on the bob, performed uncertainty quantification based on various sources of error, and performed measurements in a purified argon atmosphere; however, no account is provided of the purity of the samples measured. Ogino et al.⁵⁷ used the two-sinker Archimedean method to perform measurements, vacuum dried samples before mixing, provided quantitative purity details, and referenced previous reports that discuss uncertainty quantification in detail. Because of the good consistency between these six datasets in this cluster, and the particular quality with which measurements were performed by Ogino and Hara, the references associated with these data are considered primary (with the exception of Jaeger, for reasons discussed in Sec. 2.1.1).

There are 10 accounts of molten KF density measurements, as plotted in Fig. 1(c). As was the case for LiF and NaF, the data measured by Porter and Meaker³⁸ appear to be erroneously low for KF, and the data measured by Popescu and Constantin⁴² appear to be erroneously high. Additionally, Desyatnik’s 1979 measurements⁵⁹ appear to have an erroneously high thermal expansion coefficient, which may explain why he remeasured this same salt in 1981.⁵⁴ The remaining seven datasets (shown in a cluster in Fig. 2(c) show strong consistency, except for Mellors’s data, which extend below the melting point of KF.⁶⁰ Mellors’s data are thus considered a secondary reference for KF. Regarding the remaining six datasets, there is still reason to further exclude the data measured by Yaffe and Van Artsdalen⁵¹ and Desyatnik et al.⁵⁴ because of the limited experimental detail provided, as discussed in Sec. 2.1.1. There is also reason to exclude the measurement data of Darienko et al.⁶¹ because of (1) no information being provided on how the apparatus was calibrated, (2) a lack of salt composition analysis, and (3) lack of clarity on how well the temperature of the melt was maintained during the experiment. Consequently, the remaining three datasets, which are very consistent, are those of Taniuchi and Kanai,⁴⁶ Hara and Ogino,⁴⁹ and Smirnov and Stepanov;⁵⁰ these datasets are considered primary. An account of the reporting details provided in these references is provided in Sec. 2.1.1.

There are four accounts of molten RbF density measurements, as plotted in Fig. 2(d). Because RbF has significantly fewer independent measurements than LiF, NaF, and KF, the authors of this paper do not have a basis for defining consistency with other datasets to determine whether a particular measurement should be excluded from consideration as a primary reference. Rather, the decisions regarding the definition of references as primary or secondary was more predominantly based on the quality of the underlying studies generating the data, except in cases where cross-validation could be assessed with other measured compounds.

The measurement performed by Smirnov and Stepanov⁵⁰ is an obvious choice for a primary reference given its strong consistency with primary reference data for LiF, NaF, and KF, as well as the thoroughness of the measurement techniques applied (as discussed in Sec. 2.1.1). Jaeger and Kahn⁵² is a secondary reference for LiF, NaF, and KF because of a lack of compositional analysis and no details being provided on environmental controls. However, given the dearth of RbF data to select from, the strong consistency of Jaeger’s measurement for RbF density with Smirnov’s measurement for RbF density, as well as the strong consistency with primary references for other alkali fluorides, warrant the consideration of Jaeger as a primary reference. Katyshev et al.⁶² also has a reasonable case to be considered as a primary reference, even though a discrepancy in the measured thermal expansion coefficient of RbF exists between Katyshev’s measurements and both Jaeger’s and Smirnov’s measurements. The consideration of Katyshev as a primary reference is a result of (1) the careful salt drying procedure followed by Katyshev to ensure an anhydrous melt, (2) the noted experimental error in the measurement, and (3) the referenced⁶³ measurement process, which further describes how thermal expansion corrections were applied.

For RbF, the only reference that is considered secondary herein is Popescu and Constantin⁴² because Popescu’s measurements—in comparison with clusters of datasets that all show good consistency and that demonstrate great care being taken in measurements—appear systematically high for all alkali fluorides. Because the data measured by Jaeger would be considered secondary if there were more high-quality datasets to choose from for CsF, Jaeger is marked accordingly in Table 1. The mark (d) signifies that the limited experimental details provided do not convey that sufficient measures were in place to ensure experimental accuracy; this reference is chosen only because measurements of other salt compounds by Jaeger were validated based on this study.

There are five accounts of molten CsF density measurements, as plotted in Fig. 2(e). CsF is similar to RbF in that there are fewer independent measurements for CsF than for LiF, NaF, and KF; the authors of this paper do not have a basis for defining consistency with other datasets to determine whether a particular measurement should be excluded from consideration as a primary reference. Therefore, the same logic that was applied to select the datasets for RbF (Sec. 2.1.4) was applied for CsF. The measurements of Jaeger and Kahn,⁵² Katyshev et al.,⁶² and Smirnov and Stepanov⁵⁰ are considered primary, and the measurement of Popescu and Constantin⁴² is considered secondary. Additionally, Yaffe and Van Artsdalen⁵¹ is considered a secondary reference because essentially no information is provided regarding experimental details; however, Yaffe does show good consistency with higher-quality measurements for LiF and KF densities.

In this subsection, the reasoning associated with the designation of primary and secondary references for BeF₂, MgF₂, CaF₂, SrF₂, and BaF₂ density is provided. A summary of the references is provided in Table 2, in which the respective methods, sample purities, measured temperature ranges, and reported uncertainties are tabulated.

There are four accounts of BeF₂ density measurements, as plotted in Fig. 2(a). Because of toxic and carcinogenic effects associated with beryllium, laboratories must be specially equipped to handle beryllium; special handling requirements are likely one reason why the number of datasets for BeF₂ is limited. Furthermore, the incredibly high viscosity of BeF₂ also makes it a challenge to work with. There are no clearly overlapping datasets, but the variation between reported measured values at each temperature is ≤3%. Mackenzie’s measurement⁶⁵ was only performed for one temperature. This measurement is thus useful for comparison with other datasets but does not allow for the determination of the temperature trend associated with BeF₂ density; therefore, this measurement is considered a secondary reference.

Regarding the three remaining datasets, the authors of this paper employed sufficient environmental controls by using appropriate drying techniques and performing measurements under inert argon atmospheres. Cantor et al.⁶⁶ study reports compositional analysis from pretest impurity measurements of the salt studied, whereas Krylosov et al.⁶⁷ and Klimenkov et al.⁶⁸ studies report purification methods but no compositional analysis. Two of the datasets, Krylosov and Klimenkov, show good consistency in terms of temperature trend and extrapolated magnitude despite focusing on separate temperature ranges of 970–1170 and 1225–1280 K, respectively. These datasets are also the most recent of the available datasets. Both Krylosov and Klimenkov used the MBP method, whereas the two secondary references used the more traditional Archimedean method. Cantor reports results roughly 0.4%–1.0% lower than Krylosov but demonstrates a much smaller temperature dependence with a slope approximately half that of the Krylosov data. Cantor’s documentation suggests that the BeF₂ temperature dependence was not measured directly and was instead determined based on extrapolation of the volume expansivity of the BeF₂–LiF system as 100 mol. % BeF₂ was approached. Additionally, Cantor does caution that the effect of bubble accumulation on the bob could not be fully mitigated in experimentation. Thus, the data measured by Krylosov and Klimenkov are the primary references and are used to establish the reference correlation for BeF₂ density.

There are five accounts of MgF₂ density measurements, as plotted in Fig. 2(b). These datasets are consistent within 5% of the measured values across the full temperature range of 1525–2100 K. Each study employed the Archimedean method for density measurement, and Ogino and Takeda used a two-sinker Archimedean method to compensate for the effect of surface tension on the sinker wire. The spread of reported values is bounded above by data measured by Kulifeev et al.⁶⁹ and bounded below by data measured by Kirshenbaum et al.⁷⁰ and Takeda et al.⁷¹ Hara and Ogino⁴⁹ and Ogino et al.⁵⁷ considered a much smaller temperature range of 1675–1825 K, and the reported trends match almost exactly. Kulifeev’s measurements for all the alkaline fluorides measured (MgF₂, CaF₂ and BaF₂) tend to be outliers, as shown in Figs. 2(b)–2(e); therefore, Kulifeev is considered a secondary reference for all three salts. Although the quality of Kulifeev’s measurements tends to be comparable or lesser than that of other measurements of MgF₂,³⁷ Kulifeev did measure under inert argon and provide some details on purity. However, Kulifeev did not perform a full compositional analysis like Ogino et al.⁵⁷ and Hara and Ogino,⁴⁹ for example. Kulifeev reports <0.1% oxide and provides no other compositional details.

Regarding the remaining datasets, Kirshenbaum and Takeda additionally report (graphically) each collected data value, thereby enhancing, to some extent, the quality and reliability of the data. Hara and Ogino,⁴⁹ Ogino et al.,⁵⁷ and Takeda et al.⁷¹ each report measurement uncertainty ≤0.30%; particularly detailed uncertainty quantification was performed by Hara and Takeda. Notably, Kulifeev reports 3% uncertainty, and Kirshenbaum does not report uncertainty explicitly. Hara and Takeda include calibration of some or all components of the measurement devices, whereas Kirshenbaum and Ogino do not. Hara also reports duplicate trials of the density measurement. Each source reports salt purity of ≥98.0%. Hara, Kulifeev, and Ogino also quantify the composition of the impurities. Given the high quality of the datasets of Ogino, Hara, and Takeda, all are considered primary references. Given the reasonable level of quality of Kirshenbaum’s study and the strong consistency between Kirshenbaum’s measured values and the values from the primary references for MgF₂ (Ogino, Hara, and Takeda) as well as for other alkaline earth fluorides, Kirshenbaum is also considered a primary reference.

There are 10 accounts of CaF₂ density measurements, as plotted in Fig. 2(c). These data span roughly 1540–2300 K, and all values are consistent within 7% of the measured values at each temperature. Unlike the MgF₂ data, the data measured by Kulifeev et al.⁶⁹ constitute the lower bound for CaF₂, and the data measured by Winterhager et al.⁷² constitute the upper bound for CaF₂. Zhmoidin⁷³ employed the MBP method to determine density. Minato et al.⁷⁴ used a unique approach of electrostatic levitation of a molten salt drop and measured how sphere diameter changes with temperature. Each other study applied the Archimedean method, and Winterhager, Chen et al.,⁷⁵ and Takeda et al.⁷¹ used the dual-sinker approach to compensate for surface tension effects. The same logic that was applied to assess the datasets of Kirshenbaum, Hara, Ogino, Takeda, and Kulifeev in Sec. 2.2.2 was applied to these datasets. Kirshenbaum, Hara, Ogino, and Takeda are considered primary references for CaF₂, and Kulifeev is considered a secondary reference.

The remaining five datasets are assessed as having lower data quality. Minato calibrated some components of the experiment device, and the remaining references do not report any calibration. Salt composition was analyzed in detail by Winterhager, who reports salt purity as well as composition of the impurities. Chen and Minato report 99.99% salt purity, whereas Mitchell and Joshi⁷⁶ and Zhmoidin do not report salt purity. Chen, Mitchell, and Zhmoidin each provide bounds for temperature variation within the measurement device. Each source mentions inert atmospheres being used during testing and inert materials being used to contain the molten salt, except for Winterhager, who only details compatible materials. Zhmoidin discusses duplicate measurement trials. Winterhager and Zhmoidin provide experimental datapoints in addition to fits to experimental data, and the other sources present at least a figure that demonstrates the derived correlation.

Regarding correlation results, Chen’s measured values exhibit a significantly steeper temperature dependence than all other datasets, including high-quality references such as Takeda. Chen is therefore considered a secondary reference. There is one tight data cluster that comprises studies from Kirshenbaum, Zhmoidin, Takeda, Hara, and Ogino. However, close inspection of Zhmoidin’s data reveals a larger temperature dependence than the other four trends within this cluster. Zhmoidin provides the least amount of detail regarding sample purity of all datasets in the cluster. Minato’s data intersect the cluster but show significant deviation from linearity and reveal a large gap of unobserved temperatures at ∼1700–1780 K. Thus, the studies that are considered primary references are Hara, Ogino, Takeda, and Kirshenbaum.

There are four accounts of SrF₂ density measurements, as plotted in Fig. 2(d). Each of these datasets considers the range of ∼1725–1850 K, and Kirshenbaum et al.⁷⁰ dataset extends up to 2200 K. Although Hara and Ogino⁴⁹ reports the highest values and Kirshenbaum reports the lowest values, deviation is limited to 2.2% between these datasets. Each study employed the Archimedean method, and Takeda used the two-sinker Archimedean method. The same logic (regarding quality and dataset consistency) that was applied to MgF₂ and CaF₂ was applied to SrF₂, and the conclusion is that all four datasets for SrF₂ are considered primary. Hara, Ogino, and Takeda are high-quality references. Although Kirshenbaum is a lower-quality reference because of uncertainty not being reported, Kirshenbaum is consistent with the three high-quality references; therefore, all four of these datasets are considered primary references.

There are five accounts of BaF₂ density measurements, as plotted in Fig. 2(e). Figure 2(e) shows one tight data cluster that comprises the studies of Kirshenbaum et al.,⁷⁰ Takeda et al.,⁷¹ and Ogino et al.⁵⁷ Kulifeev et al.⁶⁹ is the upper bound of reported data and demonstrates a noticeably weaker temperature dependence than all other datasets. Bukhalova and Yagubian⁷⁷ reports a single measured value that is ∼7% lower than the provided correlations. Each study employed the Archimedean method, and Takeda used the two-sinker Archimedean method. The same logic that was applied to determine that Kirshenbaum, Takeda, and Ogino are primary references for MgF₂ and CaF₂ and that Kulifeev is a secondary reference for MgF₂ and CaF₂ (as discussed in Secs. 2.2.2 and 2.2.3) was applied for BaF₂. For BaF₂, these references maintain the same designations as those for MgF₂ and CaF₂. Bukhalova is considered secondary because only a single outlier datapoint is reported. Bukhalova also does not describe salt purity and only mentions that a platinum sinker was used; other environmental control details are not provided. Thus, Kirshenbaum, Takeda, and Ogino are considered the primary references for BaF₂.

In this subsection, the reasoning associated with the designation of primary and secondary references for LiF, NaF, KF, RbF, and CsF viscosity is provided. A summary of the references is provided in Table 3, in which the respective methods, sample purities, measured temperature ranges, and reported uncertainties are tabulated.

There are eight accounts of LiF viscosity measurements, as plotted in Fig. 3(a). Figure 3(a) shows one tight cluster of six datasets and two outliers. The cluster comprises the studies of Blanke et al.,⁷⁸ Abe et al.,⁷⁹ Ejima et al.,⁸⁰ Nguyen and Danek,⁸¹ Desyatnik et al.,⁵⁴ and Popescu and Constantin.⁸² Notably, the measurements of Blanke have a high standard deviation relative to the Arrhenius fit, causing individual measurement datapoints to fall outside the tight cluster. The three most recent of these datasets show very good consistency. Two other studies performed by Smirnov et al.⁸³ and Vetyukov and Sipriya⁸⁴ show similar results to one another and are ∼10% higher than the cluster.

All sources report measurement uncertainty ≤5%, except for Smirnov and Vetyukov, which do not report uncertainty. Abe and Popescu provide particularly detailed analyses of uncertainty propagation. Smirnov does not report on the methodology used, but all other studies used the oscillation technique. Only Abe and Ejima provide sample purities, which are 99.999 wt. % and 99.9 wt. % for these studies, respectively. Desyatnik describes the sample as “chemically pure,” Nguyen describes the sample as “prepared from ‘reagent quality’ salts,” and Popescu describes the sample as “Merck reagent high purity analytical grade,” but none of these studies provide a quantitative compositional analysis. Ejima describes an instrument calibration methodology that consisted of water and molten NaCl because of known viscosities, and Popescu used NaCl only. Regarding environmental controls, all studies except Vetyukov and Abe explicitly report a dry, inert atmosphere (helium or argon). However, Abe did indicate the viscometer was pumped down to 1 mPa, while the sample was tightly enclosed in the oscillating cup. Desyatnik, Smirnov, Blanke, and Popescu further provide the container material as platinum, molybdenum, or Inconel. Popescu describes a temperature stability of ±0.2 K, and no other references report on temperature stability. Abe’s measurements indicate particularly high precision with 3–4 separate measurements at each temperature; Abe’s measurements were performed with ascending and descending temperature. Ejima discusses “averaged values” provided but does not provide details of repeated measurements. Popescu, Nguyen, Blanke, and Desyatnik report precision based on the determined correlation. Abe, Ejima, and Vetyukov detail the numerical data that can be used to verify the temperature-dependent fit to the data.

The datasets provided by Ejima and Abe stand out as being of particularly high quality in terms of the underlying experimental details. These references show consistency with each other and other reasonably high-quality datasets; therefore, these references are obvious choices for primary references. Popescu’s study is nearly as high quality as Ejima’s, and Popescu’s data are very consistent with Ejima’s across all alkali fluoride measurements. Popescu is thus also considered a primary reference. Nguyen’s study is somewhat lacking from the standpoint of being able to verify the quality of the measurements, but Nguyen still provides a reasonable level of detail. The strong consistency of Nguyen with high-quality datasets for LiF and KF make Nguyen a primary dataset for LiF. All other references, either due to inconsistency or minimal reporting details for quality assessment, are considered secondary.

There are seven accounts of molten NaF viscosity measurements, as plotted in Fig. 3(b). Figure 3(b) shows one loose cluster and three outliers. The datasets of Smirnov et al.,⁸³ Abramov et al.,⁸⁵ and Desyatnik et al.⁵⁴ are higher than the datasets in the cluster. The tight cluster of datasets comprises the measurements of Ejima et al.,⁸⁰ Brockner et al.,⁸⁶ Kubikova et al.,⁸⁷ and Popescu and Constantin.⁸² Because the studies by Ejima and Popescu are identified as high-quality measurements with well-validated data in Sec. 2.3.1, Ejima and Popescu are considered primary references for NaF. The three outlier datasets of Abramov, Desyatnik, and Smirnov are assessed as follows to determine whether they should be considered as primary or secondary references. Abramov used an oscillation-based method to perform measurements, and virtually no account is provided of environmental controls or details regarding compositional purity—a stark contrast to the level of detail provided by Popescu and Brockner. The quality of Desyatnik’s and Smirnov’s datasets is assessed (relative to Ejima and Popescu) in Sec. 2.3.1. Desyatnik and Smirnov have been identified as lower quality than Ejima and Popescu. Therefore, all three outlier datasets (Abramov, Desyatnik, and Smirnov) are justifiably considered secondary.

Regarding the two other datasets in the tight cluster (Brockner and Kubikova), Brockner performed oscillation-based measurements, and Kubikova referenced the work of Silny and Danek,⁸⁸ implying that an oscillation-based technique was used. Kubikova and Brockner report highly precise results with standard deviations of 0.001 cP and 0.03%, respectively, but do not provide estimates of experimental uncertainty that account for systematic errors. Brockner does, however, provide a detailed discussion of potential sources of uncertainty and compares their data against other independent measurements. Whereas Kubikova does not describe a specific instrument calibration method, Brockner describes performing a preliminary control experiment with water and reports results within 0.1% of expected values. Regarding sample purity, only Kubikova provides a numerical value (99.9%), whereas Brockner describes a purification method of vacuum heating to melt, recrystallize, and select only the clear crystals. Environmental controls are described by Kubikova and Brockner, who both employed a purified inert gas atmosphere. These details indicate that Brockner performed a high-quality study with appropriate methods and provided a sufficient level of detail to characterize the uncertainty; conversely, Kubikova provided very little detail on the underlying measurement method, and data verifiability is limited. Therefore, Brockner is considered a primary reference for NaF, and Kubikova is considered a secondary reference for NaF.

There are seven accounts of molten KF viscosity measurements, as plotted in Fig. 3(c). Figure 3(c) shows one tight cluster and two outliers. The datasets of Smirnov et al.⁸³ and Desyatnik et al.⁵⁴ are high relative to the cluster. In Sec. 2.3.1, these datasets are identified as being of lower relative quality; they are thus considered secondary. The remaining datasets in the tight cluster are the measurements of Ejima et al.,⁸⁰ Kubikova et al.,⁸⁷ Popescu and Constantin,⁸² and Nguyen and Danek.⁸¹ For the same reasons discussed in Sec. 2.3.1 and Sec. 2.3.2, Ejima, Popescu, and Nguyen are considered the primary references in this cluster because of the higher quality of these studies.

The viscosities of molten RbF and CsF were measured by Ejima et al.⁸⁰ and Popescu and Constantin,⁸² as plotted in Figs. 3(d) and 3(e). These datasets show good consistency, agreeing within 2% across overlapping temperature ranges. As discussed, the studies of Ejima and Popescu are of high quality, so both references are considered primary for CsF and RbF. Smirnov et al.⁸³ also measured the viscosity of molten CsF. Relative to Ejima’s and Popescu’s results, Smirnov’s results are 25% higher at lower temperatures (≈1070 K) and 8% lower at higher temperatures (≈1180 K). Also, as discussed in Sec. 2.3.1, Smirnov’s study is of lower quality than those of Ejima and Popescu. Therefore, Smirnov is considered a secondary reference for CsF.

In this subsection, the reasoning associated with the designation of primary and secondary references for BeF₂, MgF₂, CaF₂, SrF₂, and BaF₂ viscosity is provided. A summary of the references is provided in Table 4, in which the respective methods, sample purities, measured temperature ranges, and reported uncertainties are tabulated.

There are three accounts of molten BeF₂ viscosity measurements, as plotted in Fig. 4(a). Moynihan and Cantor⁸⁹ used a rotating cylinder to perform measurements, Desyatnik et al.⁵⁴ used an oscillation damping technique, and Mackenzie⁶⁵ used a counterbalance technique. Moynihan and Mackenzie show reasonable agreement from ∼960 to 1100 K; at temperatures exceeding 1100 K, large discrepancies are observed. The data of Desyatnik exhibit a similar temperature trend to Moynihan but have significantly lower viscosity values. Identifying a primary reference is solely dependent upon an assessment of the studies’ respective levels of quality in the case of BeF₂ viscosity. A limited number of datasets are available, and large discrepancies exist between them, so no conclusions can be drawn about consistency with high-quality datasets.

A quality assessment of the documentation revealed that all three studies report uncertainty. Desyatnik states 2.0% uncertainty but does not provide an explanation regarding the quantification of this value. Moynihan offers a closer inspection of uncertainty (of 3%) from different sources in the experiment. These include <0.5% from the calibration constant, <0.6% from spindle depth measurement, and 2% for temperature uniformity, resulting in a combined uncertainty of ±3% overall. Mackenzie reports a higher uncertainty margin of 8%, associating this value primarily with load-velocity plots and expansion effects. Moynihan determined a calibration constant by measuring the viscosity of NBS standard oils P, OB, and N in a thermostated water bath. Desyatnik does not report on instrument calibration, and Mackenzie refers to previous works that describe calibration with NBS standard oil P for liquid GeO₂ measurements.⁹⁰ 

Regarding compositional analysis and sample purity, Desyatnik used anhydrous BeF₂ obtained from vacuum distillation of a commercially available sample. Similarly, Moynihan performed vacuum distillation at 800 °C in a nickel apparatus; distillation was followed by condensation on an air-cooled cold finger at 500 °C. This process produced a BeF₂ sample with >99.8% purity; major impurities are characterized as 0.13% oxygen, 0.02% iron, 0.01% sodium, and 0.003% magnesium. Mackenzie obtained BeF₂ with a quoted purity of >99%—the highest-concentration impurity being H₂O at 0.5% (in addition to aluminum, iron, magnesium, chromium, nickel, and manganese at significantly lower levels)—and further vacuum dried the sample under dry N₂. All studies report performing measurements in a dry, inert environment (either helium, N₂, or argon), and Moynihan reports the most accurate temperature control of ±0.2 °C. Moynihan and Mackenzie report individually measured viscosity values at each temperature, whereas Desyatnik only reports correlations as a function of temperature.

Because of the extent to which they report experimental details, Moynihan and Mackenzie stand out as having higher-quality studies than Desyatnik. Desyatnik is thus considered a secondary reference. Although the level of quality is similar for Moynihan’s and Mackenzie’s studies, two particular aspects of Moynihan’s study suggest that it should be considered a primary reference over Mackenzie. First, Moynihan cycled temperatures up and down multiple times, reporting no significant change in the temperature-dependent viscosity. Second, for molten salts (and for higher-viscosity fluids), the rotational method is more established than the counterbalance method as a viscosity measurement technique. Moynihan does suggest that Mackenzie’s measurements may have been high because of bubbles, but no evidence based solely on Mackenzie’s work supports this claim. Therefore, Moynihan is considered the only primary reference, and Desyatnik and Mackenzie are considered secondary references.

There are only two accounts of molten MgF₂ viscosity measurements, as plotted in Fig. 4(b). Takeda used an oscillating vessel viscometer to perform measurements,⁹¹ and Kulifeev et al.⁶⁹ used torsional vibration dampening to perform measurements (a similar method). A large discrepancy exists between these datasets, and no conclusions can be drawn about consistency with only two datasets. Therefore, the determination of a primary dataset is dependent on a quality assessment alone.

Regarding data uncertainty, Takeda provides an uncertainty evaluation based on different sources, including <0.2% from moment of inertia, <0.2% from crucible radius, and <3% variation from repeated measurements. Kulifeev reported an uncertainty of 10% but does not provide further detail regarding the uncertainty quantification. Neither study discusses instrument calibration. Takeda reports sample purity as 99.9%; experiments were conducted within a dry helium atmosphere that contained an oxygen-scavenging zirconium sponge. Kulifeev used an acetic acid wash and ammonium bifluoride treatment to prepare “analytical grade” salts, reducing oxide content below a reported 0.1 wt. %. Takeda reports temperature control within 0.5 K, and Kulifeev reports a temperature accuracy of 10 K.

Takeda generally provides more experimental details that suggest that the measurement was performed using sufficient environmental controls to achieve accuracy. Furthermore, Takeda’s density measurements of alkaline earth fluorides show better validation with independent studies in comparison to those performed by Kulifeev, suggesting that Kulifeev’s samples may have been impure. Therefore, Takeda is considered a primary reference for MgF₂ viscosity, and Kulifeev is considered a secondary reference.

There are seven accounts of molten CaF₂ viscosity measurements. Of these measurements, an extremely large spread exists in the measured viscosity values, as plotted in Fig. 4(c). The generally higher values are those measured by Schwerin,⁹² Zhmoydin,⁹³ Yanagase et al.,⁹⁴ and Davies and Wright.⁹⁵ Takeda et al.⁹¹ and Kapustin et al.⁹⁶ both measured values that are comparatively lower. Kulifeev et al.⁶⁹ performed measurements that yielded particularly low values.

For the loose cluster of generally high viscosity values, all measurements were performed using rotational methods (aside from Zhmoydin, who did not report a measurement method). Regarding the compositional details associated with this cluster, Schwerin provides the most detail regarding composition, indicating the SiO₂ and CaO content; notably, the CaO content was relatively high (1.2%). Yanagase indicates that the chromatographically measured fluorine content is within 0.2% of theory. Davies simply provides the quoted purity from the supplier, and Zhmoydin does not provide any compositional details. Neither Schwerin’s nor Zhmoydin’s reports indicate whether the measurements were performed under inert gas. Davies performed measurements under a N₂–H₂ mix, and Yanagase performed measurements under “dried argon and oxygen atmospheres.” Quantitative details regarding temperature control are not provided by the studies in this loose cluster, and only Davies provides an estimate regarding the experimental uncertainty. Davies is also the only study in this cluster that clearly reports that calibration was performed (with well-characterized oils). Schwerin implies that calibration was performed at some point (a calibration factor is used in the mathematics). Generally, this loose cluster of data has limited information regarding uncertainty quantification, and the experimental controls were not at the same quality level as those of other studies assessed and considered primary datasets in this paper.

Regarding the two middle datasets (Takeda and Kapustin), Takeda is evaluated in the previous section and is generally regarded as being a higher-quality study. Kapustin, like Takeda, used an oscillation-based measurement method and provided a total measurement uncertainty informed by uncertainty of individual aspects of the experiment. Kapustin calibrated the device with water, glycerol solutions, castor oil, and ultimately NaCl (as a reliability case) and performed measurements with “special purity” CaF₂. Kapustin does not provide details regarding temperature control and does not specify the atmospheric conditions; it can be deduced, however, that an inert environment was used based on the referenced technique by Gladkii et al.⁹⁷ Kapustin also mentions that graphite was used as the rotating or oscillating element in some previous CaF₂ measurements in literature. The use of graphite could be problematic because of salt–graphite interactions such as intrusion, as Kapustin describes. Notably, Schwerin used graphite as the rotating spindle material. Finally, an interesting note about the study by Kapustin is that “technological” grade CaF₂ that contained 5% oxides in addition to CaF₂ was measured for viscosity, in addition to the “special purity” sample. This resulted in a value of 2.4 cP across the temperature range of 1723–1873 K, agreeing reasonably well with Takeda.

The datasets of Takeda and Kapustin are reasonably consistent, and both studies are of higher quality than the other studies for CaF₂ viscosity. For these reasons, these two references are considered primary references, and the other references are considered secondary references.

There are two accounts of molten BaF₂ measurements, and there is one account of a measurement taken for molten SrF₂. Takeda measured the viscosity of molten SrF₂ and BaF₂,⁹¹ and Kulifeev measured the viscosity of molten BaF₂, as plotted in Fig. 4(d) and 4(e). For reasons discussed in Sec. 2.4.2, Takeda is considered a primary reference, and Kulifeev is considered a secondary reference.

Density reference correlations were calculated via least-squares regression for all alkali and alkaline earth fluorides. Because some of the studies provide actual measured values acquired at specific temperatures and some of the studies provide only linear relations, the density values sampled to inform the reference correlations had to be taken from the linear relations associated with each dataset for each study to achieve consistency. Twenty equally-spaced temperatures across each measured range for each study were sampled to prevent any weighting toward any particular primary dataset.

The 2σ values were able to be determined for all compounds except for BeF₂. For BeF₂, only two references are considered primary, and this number of references is insufficient for standard deviation calculations. Notably, for this particular method of quantifying uncertainty, common-cause errors could exist among the data associated with primary references for a given fluoride property assessment. These common-cause errors could systematically drive the reference correlation high or low relative to the true value. The quality assessments performed in this work are intended to reduce the likelihood that common-cause errors exist in the reference correlations (e.g., excluding density measurements performed using the Archimedean method that did not correct for surface tension effects as appropriate), but there is no method of guaranteeing that all primary references are entirely without common-cause errors. Regardless, because the uncertainty margins are 2σ values (as opposed to 1σ values), and because the uncertainty quantification method herein aligns with previously published methods for molten salt property reference correlations,^16–18 there is a high level of assurance that an appropriate level of conservatism was applied in evaluating uncertainty in this work. It is important to note that there is one major difference between these previous reference correlation studies,^16–18 and this work. Previous works used weights in the fitting process in inverse proportion to the square of the reported uncertainties. In this work, it was chosen to not use such weights, and instead to weigh all primary references equally, for two overarching reasons: (1) the underlying studies associated with the primary references identified herein are all of comparable levels of quality, and (2) the uncertainties of the primary references tend to be significantly more optimistic than the observed discrepancies in the data (as illustrated in Figs. 5–7), suggesting the reported uncertainties should not be used to inform uncertainty quantification, at least for the subset of thermophysical property data assessed in this paper.

Figure 5 shows the resultant discrepancies between the primary density datasets and the calculated reference correlations for the alkali fluorides. Figure 6 shows the resultant discrepancies between the primary density datasets and the calculated reference correlations for the alkaline earth fluorides. Comparisons with previously determined reference correlations by Janz¹⁵ are also shown. These comparisons include the margin of uncertainty that Janz estimated. Notably, Janz selected a single dataset from a single reference to represent a reference correlation for each compound, as opposed to averaging multiple datasets together. The references selected by Janz are cited in the captions for Figs. 5 and 6.

Regarding the alkali fluoride density discrepancies in Fig. 5, the previously determined correlations by Janz¹⁵ are consistent with those determined herein in terms of both values and consistency. In all cases for alkali fluorides, the margins of uncertainty determined by Janz overlap with those determined herein. Two minor but perhaps notable differences between Janz’s correlations and those determined herein are that the margin of uncertainty for NaF was reduced by approximately a factor of 2, and the margin of uncertainty for CsF was reduced by approximately a factor of 1.3.

Regarding the alkaline earth fluoride density discrepancies in Fig. 6, the previously determined correlations by Janz¹⁵ are reasonably consistent with those determined herein in terms of values, except that the margins of uncertainty are significantly lower herein because of the application of Eq. (4) to determine the margin of uncertainty. Therefore, this work provides increased confidence in the density correlations associated with MgF₂, CaF₂, SrF₂, and BaF₂. BeF₂ is the exception in that the correlation determined herein (which is based on the works of Krylosov et al.⁶⁷ and Klimenkov et al.⁶⁸) is outside the margin of uncertainty determined by Janz; a margin of uncertainty could not even be determined for this work because there were not at least three primary datasets for BeF₂ density. Additional studies will be necessary to measure the density of molten BeF₂ in order to quantify an accurate margin of uncertainty and to provide greater confidence in the findings of Krylosov et al.⁶⁷ and Klimenkov et al.,⁶⁸ which suggest that the coefficient of thermal expansion is not nearly as small as Cantor determined.⁶⁶ 

Viscosity reference correlations were calculated for all alkali fluorides via least-squares regression. Reference correlations were not able to be calculated for alkaline earth fluorides because of a lack of multiple consistent datasets determined to be primary datasets. Therefore, single datasets are suggested as reference correlations for the alkaline earth fluorides. Equation (4) was used to calculate—in the same manner as described for the density reference correlations in Sec. 3.1—the 2σ values for LiF, NaF, and KF. The 2σ values could not be calculated for RbF and CsF because only two datasets are considered primary references for each of these compounds. Table 6 provides the fit coefficients, temperature ranges, and 2σ values for all alkali and alkaline earth fluorides. Table 6 also provides melting points which have been used to ensure that the correlation temperature ranges do not fall below them.

Figure 7 shows the resultant discrepancies between the primary viscosity datasets and the calculated reference correlations for the alkali fluorides. For LiF, NaF, and KF, the uncertainty margins for the reference correlations determined herein overlap those determined by Janz;¹⁵ however, the uncertainties determined herein range from ∼2 to 4 factors larger than those determined by Janz. This signifies that the estimated uncertainties by Janz for these viscosity reference correlations may be too narrow, and the actual uncertainty margins in alkali fluoride viscosities are reasonably broader based on analysis of multiple primary datasets. Additional studies will be necessary to measure the viscosity of molten RbF and CsF in order to quantify an accurate margin of uncertainty for their associated reference correlations and to provide more confidence in the findings of Ejima et al.⁸⁰ and Popescu and Constantin.⁸² Furthermore, all alkaline earth fluorides require additional viscosity studies to define reference correlations (beyond selecting the reference of the highest quality) because none of the measurements for these compounds validate one another.

Plots of all reference correlations that were either calculated or determined in this work are summarized in Fig. 8, along with uncertainty margins where they are applicable. Molar volumes, which were calculated from the density correlations, are also summarized in Fig. 8.

A critical reassessment of the literature on the densities and viscosities of alkali and alkaline earth fluorides has enabled the calculation and determination of new reference correlations to describe these properties. This kind of assessment has not been performed for these compounds since Janz’s 1988 work for the National Standard Reference Data System. For most compounds, there were sufficient quantities of high-quality datasets to enable a least-squares fit and quantitative determination of an uncertainty margin to generate a reference correlation. In some cases (i.e., BeF₂ density, RbF viscosity, and CsF viscosity), only two primary datasets were identified, meaning that uncertainty quantification could not be performed. For alkaline earth viscosity correlations, only a single reference was put forth as a reference correlation, also resulting in an inability to quantify uncertainty. In comparison with previous reference correlations put forth by Janz, the uncertainty margins associated with the fits performed herein are comparable for alkali fluoride density, lower for alkaline earth fluoride density, and higher for alkali fluoride viscosity. The results of this study indicate that there is still a general need for additional high-quality density measurements for BeF₂ and additional high-quality viscosity measurements for RbF, CsF, and all alkaline earth fluorides.

This manuscript has been authored by UT-Battelle LLC under Contract No. DEAC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). This work is supported by the Nuclear Energy Advanced Modeling and Simulation and Molten Salt Reactor Program of DOE’s Office of Nuclear Energy. The authors would like to acknowledge N. Dianne Bull Ezell for providing guidance and strategic input that positively influenced this manuscript.

The authors have no conflicts to disclose.

The data that support the findings of this study are openly available in Molten Salt Thermal Properties Database–Thermophysical, at https://mstdb.ornl.gov/.