KF-BeF₂ system

The phase equilibrium information of the KF-BeF₂ system has been reported by three groups of investigators using thermal analysis. The experimental phase diagram of the KF-BeF₂ system was first established by Borzenkova et al. [44]. Four intermediate phases—K₃BeF₅, K₂BeF₄, KBeF₃, and KBe₂F₅—were detected in the KF-BeF₂ system, where only three compounds—K₃BeF₅, K₂BeF₄, and KBeF₃—were stable from room temperature to the melting temperature, whereas KBe₂F₅ was decomposed into KBeF₃ and BeF₂ at 278.0 °C. Afterwards, the KF-BeF₂ system was experimentally determined by Moore et al. [21] at ORNL. The four compounds were again discovered, where K₃BeF₅, K₂BeF₄, and KBe₂F₅ were melted congruently at 740.0 °C, 787.0 °C and 353.0 °C, whereas KBeF₃ was melted incongruently at 405.5 °C; these findings are substantially different from those of Borzenkova et al. [44]. Thus, four eutectic reactions ($\text{Liquid}\stackrel{\text{720}\text{.0}°\text{C}}{\to }{\text{KF+K}}_{\text{3}}{\text{BeF}}_{\text{5}}$, $\text{Liquid}\stackrel{\text{730}{\text{.0}}^{\text{o}}\text{C}}{\to }{\text{K}}_{\text{3}}{\text{BeF}}_{\text{5}}+{\text{K}}_{\text{2}}{\text{BeF}}_{\text{4}}$, $\text{Liquid}\stackrel{\text{330}{\text{.0}}^{\text{o}}\text{C}}{\to }{\text{KBeF}}_3+{\text{KBe}}_{\text{2}}{\text{F}}_{\text{5}}$, and $\text{Liquid}\stackrel{\text{323}{\text{.0}}^{\text{o}}\text{C}}{\to }{\text{KBe}}_{\text{2}}{\text{F}}_{\text{5}}{\text{+BeF}}_{\text{2}}$) and one peritectic reaction (${\text{Liquid+K}}_{\text{2}}{\text{BeF}}_{\text{4}}\stackrel{\text{390}{\text{.0}}^{\text{o}}\text{C}}{\to }{\text{KBeF}}_{\text{3}}$) occurred in the KF-BeF₂ system. Novoselova et al. [45] later measured the phase diagram of the KF-BeF₂ system, determining that K₃BeF₅ was incongruently melted at 740.0 °C with a peritectic composition of 0.25 mol BeF₂. Naturally, the KF-BeF₂ system reported by Novoselova et al. is characterized by three eutectic reactions ($\text{Liquid}\stackrel{\text{700}{\text{.0}}^{\text{o}}\text{C}}{\to }{\text{KF+K}}_{\text{3}}{\text{BeF}}_{\text{5}}$, $\text{Liquid}\stackrel{\text{327}{\text{.0}}^{\text{o}}\text{C}}{\to }{\text{KBeF}}_{\text{3}}{\text{+KBe}}_{\text{2}}{\text{F}}_{\text{5}}$, and $\text{Liquid}\stackrel{\text{346}{\text{.0}}^{\text{o}}\text{C}}{\to }{\text{KBe}}_{\text{2}}{\text{F}}_{\text{5}}{\text{+BeF}}_{\text{2}}$) and two peritectic reactions (${\text{Liquid+K}}_{\text{2}}{\text{BeF}}_{\text{4}}\stackrel{\text{740}{\text{.3}}^{\text{o}}\text{C}}{\to }{\text{K}}_{\text{3}}{\text{BeF}}_{\text{5}}$ and ${\text{Liquid+K}}_{\text{2}}{\text{BeF}}_{\text{4}}\stackrel{\text{405}{\text{.5}}^{\text{o}}\text{C}}{\to }{\text{KBeF}}_{\text{3}}$). To date, thermochemical data and thermodynamic parameters for the KF-BeF₂ system are still rare. Herein, the KF-BeF₂ system was thermodynamically optimized through the substitutional solution model in terms of the relatively accurate experimental data from Moore et al. [21].

RbF-BeF₂ system

A phase diagram of the RbF-BeF₂ system was constructed by two groups of investigators using thermal analysis. Grebenshchikov [46] first established the entire phase diagram of the RbF-BeF₂ system, in which four intermediate phases—Rb₂BeF₄, Rb₃Be₂F₇, RbBeF₃ and RbBe₂F₅—were detected. Three compounds—Rb₂BeF₄, RbBeF₃, and RbBe₂F₅—were stable and congruently melted separately at 792.0 °C, 464.0 °C, and 451.0 °C, whereas Rb₃Be₂F₇ was unstable and decomposed to Rb₂BeF₄ and RbBeF₃ when the temperature was higher than 423.5 °C. In addition, Rb₂BeF₄ and RbBe₂F₅ are not simple stoichiometric compounds and have a certain solid solubility. Eventually, four eutectic reactions ($\text{Liquid}\stackrel{\text{689}{\text{.0}}^{\text{o}}\text{C}}{\to }\text{RbF}+{\text{Rb}}_2{\text{BeF}}_4$, $\text{Liquid}\stackrel{\text{461}{\text{.0}}^{\text{o}}\text{C}}{\to }{\text{Rb}}_2{\text{BeF}}_4+{\text{RbBeF}}_3$, $\text{Liquid}\stackrel{\text{394}{\text{.0}}^{\text{o}}\text{C}}{\to }{\text{RbBeF}}_3+{\text{RbBe}}_{\text{2}}{\text{F}}_{\text{5}}$, and $\text{Liquid}\stackrel{\text{424}{\text{.0}}^{\text{o}}\text{C}}{\to }{\text{RbBe}}_{\text{2}}{\text{F}}_{\text{5}}+{\text{BeF}}_{\text{2}}$) were formed for the RbF-BeF₂ system. Subsequently, Moore et al. [21] reported the phase equilibrium information of the RbF-BeF₂ system and discovered four intermediate phases: Rb₃BeF₅, Rb₂BeF₄, RbBeF₃, and RbBe₂F₅. The presence of Rb₃BeF₅ versus Rb₃Be₂F₇ was the biggest difference between the works of Grebenshchikov and Moore et al. Moreover, the intermediate compounds Rb₃BeF₅, Rb₂BeF₄, and RbBe₂F₅ melted congruently, whereas RbBeF₃ melted incongruently. Consequently, four eutectic reactions ($\text{Liquid}\stackrel{\text{675}{\text{.0}}^{\text{o}}\text{C}}{\to }\text{RbF}+{\text{Rb}}_3{\text{BeF}}_5$, $\text{Liquid}\stackrel{\text{720}{\text{.0}}^{\text{o}}\text{C}}{\to }{\text{Rb}}_2{\text{BeF}}_4+{\text{Rb}}_{\text{3}}{\text{BeF}}_5$, $\text{Liquid}\stackrel{\text{383}{\text{.0}}^{\text{o}}\text{C}}{\to }{\text{RbBeF}}_3+{\text{RbBe}}_{\text{2}}{\text{F}}_5$, and $\text{Liquid}\stackrel{\text{397}{\text{.0}}^{\text{o}}\text{C}}{\to }{\text{RbBeF}}_3+{\text{BeF}}_2$) and one peritectic reaction (${\text{Liquid+Rb}}_2{\text{BeF}}_4\stackrel{\text{442}{\text{.0}}^{\text{o}}\text{C}}{\to }{\text{Rb}}_{\text{3}}{\text{BeF}}_5$) were observed in the RbF-BeF₂ system. Little thermochemical information about the RbF-BeF₂ system has been reported to date. The RbF-BeF₂ system was optimized in this study based on the substitutional solution model using the experimental data from Moore et al. [21], because their data were more accurate than those of Grebenshchikov [46].

CsF-BeF₂ system

Only Breusov et al. [47] reported the phase equilibrium information of the CsF-BeF₂ system using thermal analysis. The CsF-BeF₂ system contains four intermediate compounds—Cs₃BeF₅, Cs₂BeF₄, CsBeF₃ and CsBe₂F₅—where Cs₂BeF₄, CsBeF₃, and CsBe₂F₅ were discovered to melt congruently at 793.0 °C, 475.0 °C, and 480.0 °C, respectively, but Cs₃BeF₅ melted incongruently at 659.0 °C. Four eutectic reactions ($\text{Liquid}\stackrel{\text{598}{\text{.0}}^{\text{o}}\text{C}}{\to }\text{CsF}+{\text{Cs}}_3{\text{BeF}}_5$, $\text{Liquid}\stackrel{\text{449}{\text{.0}}^{\text{o}}\text{C}}{\to }{\text{Cs}}_2{\text{BeF}}_4+{\text{CsBeF}}_3$, $\text{Liquid}\stackrel{\text{393}{\text{.0}}^{\text{o}}\text{C}}{\to }{\text{CsBeF}}_3+{\text{CsBe}}_{\text{2}}{\text{F}}_5$, and $\text{Liquid}\stackrel{\text{367}{\text{.0}}^{\text{o}}\text{C}}{\to }{\text{CsBe}}_{\text{2}}{\text{F}}_5+{\text{BeF}}_2$) and one peritectic reaction (${\text{Liquid+Cs}}_2{\text{BeF}}_4\stackrel{\text{659}{\text{.0}}^{\text{o}}\text{C}}{\to }{\text{Cs}}_{\text{3}}{\text{BeF}}_5$) were the distinct features of the CsF-BeF₂ system. Thermochemical data for the CsF-BeF₂ system are still lacking. Similar to the systems mentioned above, a substitutional solution model was used to optimize this system based on experimental data from Breusov et al. [41].

KF-CsF system

The phase diagram of the KF-CsF system was solely obtained by Samuseva et al. [53] through thermal analysis. This system is eutectic with one eutectic point located at 625.0 °C and 0.570 mol CsF. In addition, the limited solid solubility of KF in CsF is 0.15 mol at 625.0 °C. To date, few thermochemical studies of the KF-CsF system have been reported. A substitutional solution model was used to optimize the KF-CsF system based on experimental data from Samuseva et al. [53].

RbF-CsF system

The phase diagram of the RbF-CsF system was investigated by Samuseva et al. [53] and Beneš et al. [54] using thermal analysis. The phase diagram reported by Samuseva et al. [53] shows a temperature minimum for the solidus without a corresponding minimum for the liquidus, which violates the phase rule. The liquidus of the RbF-CsF system was optimized by Sangster and Pelton [55] based on the liquidus experimental data of Samuseva et al. However, there was a significant deviation between the calculated phase diagram and liquidus experimental data obtained by Samuseva et al.

Subsequently, new measurements of the solidus and liquidus equilibria of the RbF-CsF system were obtained by Beneš et al. [54]. The phase diagram of the RbF-CsF system was first constructed using a quasi-chemical model based on the measured experimental data, where the calculated phase diagram agreed well with the experimental values. In this study, the RbF-CsF system was calculated using the substitutional solution model to maintain the consistency of the model. The experimental data from Beneš et al. were used to optimize the RbF-CsF system in the present work.

KF-RbF system

The KF-RbF system is isomorphous with a minimum point in the liquidus curve, which was systematically evaluated and optimized by Yin et al. [56]. The thermodynamic parameters obtained in that study were directly adopted in the present work.

Sub-ternary systems

Little relevant information has been reported to date for the ternary KF-RbF-BeF₂, KF-CsF-BeF₂, RbF-CsF-BeF₂, and KF-RbF-CsF systems. These ternary systems were investigated in this study based on the thermodynamic parameters of the binary systems using model extrapolation.

Thermodynamic optimization

The PanOptimizer module included in the PANDAT software, which is a C/C++ software package for optimizing thermodynamics and thermophysical properties, was used to optimize the thermodynamic parameters. The weighted least-squares technique was adopted during the optimization process, and a trial-and-error method was used to regulate the optimization weight factor until the optimized value agreed well with the experimental phase equilibrium data and the theoretically calculated value.

Thermodynamic model

Theoretical prediction of formation enthalpy

The formation enthalpies of the intermediate compounds ABeF₃, A₂BeF₄, A₃BeF₅, and ABe₂F₅ (A = K, Rb, and Cs) were predicted based on the Open Quantum Materials Database (OQMD), which is a high-throughput database includess many DFT energies of compounds. The formation enthalpy was used as the initial value to optimize the sub-binary systems of the KF-RbF-CsF-BeF₂ system. The relevant calculated equations of the formation enthalpy are as follows based on the DFT single-point energy of the corresponding compounds: ${\text{Δ}}_f{H}_{{\text{ABeF}}_3}=E_{\text{tot}}\left({\text{ABeF}}_{\text{3}}\right)-E_{\text{tot}}\left(\text{AF}\right)-E_{\text{tot}}\left({\text{BeF}}_{\text{2}}\right)$ (5) ${\text{Δ}}_f{H}_{{\text{A}}_{\text{2}}{\text{BeF}}_4}=E_{\text{tot}}\left({\text{A}}_{\text{2}}{\text{BeF}}_4\right)-2E_{\text{tot}}\left(\text{AF}\right)-E_{\text{tot}}\left({\text{BeF}}_{\text{2}}\right)$ (6) ${\text{Δ}}_f{H}_{{\text{A}}_{\text{3}}{\text{BeF}}_5}=E_{\text{tot}}\left({\text{A}}_{\text{3}}{\text{BeF}}_5\right)-3E_{\text{tot}}\left(\text{AF}\right)-E_{\text{tot}}\left({\text{BeF}}_{\text{2}}\right)$ (7) ${\text{Δ}}_f{H}_{{\text{ABe}}_{\text{2}}{\text{F}}_5}=E_{\text{tot}}\left({\text{ABe}}_{\text{2}}{\text{F}}_5\right)-E_{\text{tot}}\left(\text{AF}\right)-2E_{\text{tot}}\left({\text{BeF}}_{\text{2}}\right)$ (8) where $E_{\text{tot}}\left(\text{AF}\right)$, $E_{\text{tot}}\left({\text{BeF}}_{\text{2}}\right)$, $E_{\text{tot}}\left({\text{ABeF}}_{\text{3}}\right)$, $E_{\text{tot}}\left({\text{A}}_{\text{2}}{\text{BeF}}_4\right)$, $E_{\text{tot}}\left({\text{A}}_{\text{3}}{\text{BeF}}_5\right)$, and $E_{\text{tot}}\left({\text{ABe}}_{\text{2}}{\text{F}}_5\right)$ are the single points of AF, BeF₂, ABeF₃, A₂BeF₄, A₃BeF₅ and ABe₂F₅ (A = K, Rb, and Cs), respectively, as listed in Table 2.

AIMD

AIMD simulations were used to investigate the structure and properties of KF-RbF-CsF-BeF₂ systems with two different proportions (i.e., 3.50-28.92-21.78-45.80 mol% and 1.80-35.42-52.40-10.38 mol%), which are denoted as Mixtures 1 and 2, respectively, in the following sections. The initial simulation systems for the molten salts were constructed using a random insertion method, based on the densities predicted in the previous section. Mixtures 1 and 2 consisted of 140 atoms (2 K, 17 Rb, 12 Cs, 26 Be, and 83 F) and 118 atoms (1 K, 20 Rb, 29 Cs, 6 Be, and 62 F) atoms, respectively. Because the temperature has little effect on the structure of molten salts [58], only the 650 °C was investigated in this study.

AIMD based on DFT was performed using the Vienna Ab initio Simulation Package [59-62]. The exchange-correlation energy was described using the Perdew-Burke-Ernzerhof functional of the generalized gradient approximation [63, 64]. The electron–ion core interactions were described by the projector-augmented plane-wave [65] method, and an energy cutoff of 600 eV for the plane-wave expansion of the wave functions was used. The Г point was chosen to sample the Brillouin zone [66]. A time step of 1.0 fs and a simulation temperature of 650 °C were used for all AIMD simulations, and the convergence criterion for the total energy was set at 10^–5 eV.

The simulation process was as follows. First, 5 ps simulations for the two systems were conducted in an NPT ensemble with a Langevin thermostat to optimize the cell volumes using the method of Parrinello and Rahman [67, 68]. The corresponding equilibrium volumes were then obtained from the average of the last 3000 steps. Finally, 15 ps simulations for the two systems based on the equilibrium volumes were performed in an NVT ensemble with a Nosé thermostat [69, 70]. The systems reached equilibrium within 2 ps, and the structures and properties of the molten salt systems were obtained by analyzing the last 10,000 steps of the simulation.

The partial RDF is important for describing the atomic configuration in liquid systems and is defined as the probability of finding another atom at a distance $r$ from a center reference atom, as shown in Eq. (9) [71]: $g_{i,j}\left(r\right)=\frac{\text{d}{n}_j\left(r\right)}{4\pi r^2{\rho }_j\text{d}r}$ (9) where $\text{Δ}{n}_j$ is the number of $j$ atoms in the spherical shell between radii $r-0.5\text{Δ}r$ and $r+0.5\text{Δ}r$, $i$ stands for the center atoms, and ${\rho }_j$ is the atomic number density of element $j$ in the system. In a molten salt system, the first shell CNs of the central atom can be obtained by integrating the RDF within the cutoff radius ($R_{\text{cut}}$), as expressed in Eq. (10) [71]: $C{N}_{i,j}={\displaystyle {\int }_0^{R_{\text{cut}}}4\pi r^2{\rho }_j{g}_{r,j}\left(r\right)}\text{d}r$ (10) where $R_{\text{cut}}$ is the radius corresponding to the first valley of the RDF. The ADF describes the three-body correlation information and is defined as the average of the angles between the central atom and any two adjacent coordinated atoms in the system, as shown in Eq. (11) [72]: ${\theta }_{jik}=\langle {\cos }^{-1}\frac{r_{ij}^2+r_{ik}^2-r_{jk}^2}{2r_{ij}{r}_{ij}}\rangle$ (11) where $j$ and $k$ are the adjacent coordinated atoms within $R_{\text{cut}}$ of central atom $i$. The self-diffusion coefficients can be used to depict the transport characteristics of the molten salt and reflect the diffusivity of each component in the molten salt. They can be calculated using the mean square displacement (MSD) and Einstein formula, according to Eqs. (12) and (13) [72]: $MS{D}_A\left(t\right)=\frac{1}{N_A}{\displaystyle \sum _{i=1}^{N_A}{\left|r_i\left(t\right)-r_i\left(0\right)\right|}^2}$ (12) $D_A=\underset{t\to \infty }{\lim }\frac{1}{6t}MS{D}_A\left(t\right)$ (13) where $r_i\left(t\right)$ is the displacement of an A-type atom at time t and $N_A$ represents the total number of elements A.

Binary system

All phase diagrams were plotted using the PanPhaseDiagram module of the PANDAT software based on the optimized thermodynamic parameters listed in Table 3. The calculated phase diagrams will be discussed in the following section. The optimized formation enthalpies of the intermediate compounds from the sub-binary systems of the KF-RbF-CsF-BeF₂ system and the calculated values from the OQMD are shown in Table 4. The calculated formation enthalpy agrees with most values from the OQMD, although some deviation between them exists owing to the different temperatures used (i.e., the values in the OQMD were obtained at 0 K, and the present optimized data were acquired at 298.15 K). Table 5 compares the calculated values of the key invariant points with the experimental data of the sub-binary systems of the KF-RbF-CsF-BeF₂ system.

Figs. 1, 2, 3, 4 and 5 show the calculated phase diagrams of the AF-BeF₂ (A = K, Rb, and Cs) and A′F-CsF (A′ = K and Rb) systems with the corresponding experimental data [21, 47, 53, 54]. As clearly observed from Fig. 1, the calculated value of the KF-BeF₂ system shows a high level of consistency with the experimental data from Moore et al., especially for the invariable point. For instance, the congruent melting temperatures of K₃BeF₅, K₂BeF₄, and KBe₂F₅ at 736.7 K, 784.7 K, and 357.1 K, respectively, are in excellent agreement with the experimental values of 740.0 K, 787.0 K, and 358.0 K, respectively [21]. As shown in Fig. 2, good agreement exists between the calculated values of the RbF-BeF₂ system and the experimental data from Moore et al., except for some small deviation in the liquidus of the BeF₂ end member. Similarly, the calculated melting temperatures of the intermediate phases Rb₃BeF₅, Rb₂BeF₄, and Rb₂BeF₄ at 725.0 K, 800.0 K, and 464.0 K, respectively, are nearly equal to the experimental values of 725.0 K, 800.2 K, and 464.4 K, respectively [21]. As shown in Fig. 3, the calculated phase diagram of the CsF-BeF₂ system is in good agreement with the experimental data from Breusov et al. [47], although some slight deviation is present on the rich-BeF₂ side. In particular, the calculated values of the key invariant points (i.e., 598.0 °C and 0.140 mol BeF₂, 793.0 °C and 0.333 mol BeF₂, and 480.0 °C and 0.667 mol BeF₂) are almost the same as the experimental data (590.0 °C and 0.144 mol BeF₂, 793.0 °C and 0.333 mol BeF₂, and 480.0 °C and 0.667 mol BeF₂), as shown in Table 5. As notably demonstrated by Fig. 4, the present calculated phase diagram of the KF-CsF system is in satisfactory agreement with the experimental data from Samuseva et al. [53], where the present calculated eutectic point of 624.3 °C and 0.571 mol CsF is almost the same as the experimental point of 625.0 °C and 0.570 mol CsF. The optimized solid solution point of 624.3 °C and 0.950 mol CsF also agrees well with the experimental point of 625.0 °C and 0.850 mol CsF. Fig. 5 displays the optimized phase diagram of the RbF-CsF system, where the optimized value corresponds well with the experimental data from Beneš et al. [54]. The present optimized minimum temperature on the liquidus (670.8 °C) agrees with the experimental results from Beneš et al. (677.85 °C and 673.85 °C).

Fig. 1Full-screen View

Calculated phase diagram of the KF-BeF₂ system with experimental data from Moore et al

Fig. 2Full-screen View

Calculated phase diagram of the RbF-BeF₂ system with experimental data from Moore et al

Fig. 3Full-screen View

Calculated phase diagram of the CsF-BeF₂ system with experimental data from Breusov et al

Fig. 4Full-screen View

Calculated phase diagram of the KF-CsF system with experimental data from Samuseva et al

Fig. 5Full-screen View

Calculated phase diagram of the RbF-CsF system with experimental data from Beneš et al

Ternary system

The liquidus projections of the ternary BeF₂-KF-RbF, BeF₂-CsF-KF, BeF₂-CsF-RbF, and CsF-KF-RbF systems were calculated based on the thermodynamic parameters of the binary system using an extrapolation model. Figs. 6(a–d) show the liquidus projections of these ternary systems with the primary phases and temperatures formed during the solidification process. In Fig. 6(d), no ternary invariant point is observed for the CsF-KF-RbF system because of the inherent properties of its sub-binary system, that is, KF-RbF and RbF-CsF are isomorphous systems with a minimum in the liquids. The BeF₂-KF-RbF system shown in Fig. 6(a) is characterized by six eutectic and five transition reactions, and the BeF₂-CsF-KF system features seven eutectic reactions and three transition reactions, as shown in Fig. 6(b). The BeF₂-CsF-RbF system is also characterized by seven eutectic reactions, but has five transition reactions, as depicted in Fig. 6(c). The results show that complex reactions can occur in the BeF₂-KF-RbF, BeF₂-CsF-KF, and BeF₂-CsF-RbF systems during the solidification process. The detailed invariant reactions and temperatures of the BeF₂-KF-RbF, BeF₂-CsF-KF, BeF₂-CsF-RbF, and CsF-KF-RbF systems are listed in Table 6.

Fig. 6Full-screen View

Calculated liquidus projections of the ternary (a) BeF₂-KF-RbF, (b) BeF₂-CsF-KF, (c) BeF₂-CsF-RbF, and (d) CsF-KF-RbF systems

Quaternary system

The quaternary KF-RbF-CsF-BeF₂ system was further investigated in terms of the thermodynamic databases of its sub-binary systems using an extrapolation model. The component proportions of the quaternary KF-RbF-CsF-BeF₂ system were predicted based on its melting temperature. Table 7 shows the detailed melting point data with the corresponding component proportions; the melting temperature is not higher than 450.0 °C, which is a commonly acceptable temperature for the coolant in an MSR. As shown in Table 7, 10 kinds of KF-RbF-CsF-BeF₂ systems with different proportions were obtained through the thermodynamic calculations, where the content of BeF₂ varies from approximately 10 to 60 mol%. BeF₂-based molten salts commonly have much higher viscosities when the BeF₂ content of the molten salt is high (especially when the BeF₂ content is greater than 50 mol%). Therefore, the proportion of BeF₂ should be maintained within an acceptable range for practical applications. Thus, the quaternary KF-RbF-CsF-BeF₂ systems labeled No. (1)–(9) were chosen as relatively appropriate heat transfer media for MSRs based on the melting temperature (T_m.p. ≤ 450 °C) and viscosity (i.e., BeF₂ content ≤ 50 mol%). The local structures of the KF-RbF-CsF-BeF₂ molten salts with different portions will be further discussed based on first-principles calculations in the following section.

4.3.1

RDFs

In the AIMD calculations, the NPT ensemble simulations revealed that the equilibrium densities of Mixture 1 (KF-RbF-CsF-BeF₂, 3.50-28.92-21.78-45.80 mol%) and Mixture 2 (KF-RbF-CsF-BeF₂, 1.80-35.42-52.40-10.38 mol%) at 650 °C were 2.22 g/cm³ and 2.72 g/cm³, respectively. Fig. 7(a) presents the RDFs of the Be-F, K-F, Rb-F, and Cs-F ion pairs in the two molten salt systems. These systems exhibit some similar features. The RDFs for all cation–anion pairs have sharper and stronger first peaks, indicating an ordered arrangement of F^– around Be²⁺, K⁺, Rb⁺, and Cs⁺. As the distance increases, the fluctuations of the RDFs gradually decrease and tend towards 1, reflecting the short-range order and long-range disorder in the microstructure of the molten salt systems. The highest first peak is observed for the Be-F ion pair, with the peak valley being almost zero, indicating a strong interaction between Be²⁺ and F^–. This finding suggests that F^– has more difficulty escaping from the coordination shell of Be²⁺ compared to those of other cations. The relative interaction strength of the cations with F^– in the same system can be inferred from the heights of the first peaks for each ion pair as follows: Be²⁺ > K⁺ > Rb⁺ > Cs⁺. This correlates with the average charge of the cations, and a higher charge results in stronger interactions with F^–. Be²⁺ has the highest valence state and the smallest ionic radius among the cations, resulting in the strongest interaction. K⁺, Rb⁺, and Cs⁺ have the same valence state, and their interaction strength with F^– decreases with increasing ionic radius: Cs⁺ > Rb⁺ > K⁺.

Fig. 7Full-screen View

(a) RDFs of Be-F, K-F, Rb-F, and Cs-F pairs, and (b) their corresponding integral curves. The solid and dashed lines represent the cases of Mixtures 1 and 2, respectively

The first peak of the RDF of each ion pair in Mixture 1 is lower than that in Mixture 2, indicating a more ordered arrangement of F^– around the cations in Mixture 2. The first peak position of the RDF typically represents the average bond length of the ion pair (Table 8). The peak positions of Be-F, K-F, Rb-F, and Cs-F in Mixture 1 are 1.55, 2.63, 2.77, and 2.95, respectively, and 1.57, 2.51, 2.69, and 2.83 in Mixture 2. With decreasing Be²⁺ content in Mixture 2, the average bond length of the Be-F ion pairs remains relatively unchanged, which is attributed to the strong interaction between Be²⁺ and F^-. The Be–F bond length is almost the same as that in the LiF-BeF₂ system [48]. The peak positions for the other cation–anion pairs decrease significantly in Mixture 2 compared to those in Mixture 1. The bond length also reflects the strength of the ion pairs; shorter bond lengths indicate stronger interactions between the same ion pairs. Thus, the interactions between the cations and F^– in Mixture 2 are stronger than those in Mixture 1. Because the concentration of Be²⁺ is higher in Mixture 1, F^– preferentially binds with Be²⁺, thereby weakening its interactions with the other cations.

Table 8Full-screen View

First peak position and cutoff radius (R_cut) of the RDF (Å) and CNs of cations

	Mixture 1			Mixture 2
	First peak	Rcut	CNs	First peak	Rcut	CNs
	Be-F	1.55	2.45	3.72	1.57	2.43	3.93
K-F	2.63	3.69	6.14	2.51	3.61	4.23
Rb-F	2.77	3.97	6.73	2.69	3.93	4.89
Cs-F	2.95	4.11	7.03	2.83	4.09	5.01

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4.3.2

CNs

The CNs were obtained by integrating the RDFs of the ion pairs. Fig. 7(b) shows the integral values of the RDF for the Be-F, K-F, Rb-F, and Cs-F pairs in the two molten salt systems. In general, the values corresponding to R_cut in Fig. 7(b) are the CNs, and the calculated CNs for each cation are listed in Table 6. In Mixture 1, the CNs of Be²⁺, K⁺, Rb⁺, and Cs⁺ are 3.72, 6.14, 6.73, and 7.03, respectively, and in Mixture 2, they are 3.93, 4.23, 4.89, and 5.01, respectively. Except for Be²⁺, the CNs of the other cations in Mixture 1 are larger than those in Mixture 2, which is mainly owing to the higher concentration of F^– in Mixture 1. Additionally, for the K-F, Rb-F, and Cs-F ion pairs, the integral curves do not exhibit a clear plateau near R_cut, as is present for the Be-F pair. Therefore, these cations can localize F^– within a certain range, but there are no compact complex structures as Be²⁺. Furthermore, the proportions of CNs for all cations within the R_cut were calculated to study the coordination preference with F^–, as shown in Fig. 8. Be²⁺ predominantly forms 4-fold coordination structures in both systems, like in the LiF-BeF₂ system, where Be²⁺ is dominated by [BeF₄]^2– [48]. For the K-F, Rb-F, and Cs-F ion pairs, the coordinated structures in their first coordination shells range from 2-fold to 9-fold. The 6-fold and 7-fold coordination structures account for the largest two portions in Mixture 1, whereas the 4-fold and 5-fold structures account for the largest portions in Mixture 2.

Fig. 8Full-screen View

Proportions of CNs in (a) Mixture 1 and (b) Mixture 2

4.3.3

ADFs

To describe the three-body correlations of the ions in the molten salt systems further, the ADFs of F-Be-F, F-K-F, F-Rb-F, and F-Cs-F were calculated, as shown in Fig. 9. The angle of F-Be-F is mainly distributed in the range of 90°–130° both in Mixtures 1 and 2, with the maximum peak at approximately 109°. This corresponds to the slightly distorted tetrahedral [BeF₄]^2–, consistent with that in the LiF-BeF₂ system [1]. For the F-K-F, F-Rb-F, and F-Cs-F pairs, the angle distribution range is wide and different from that for the F-Be-F pair in both Mixtures 1 and 2, and various coordination structures ranging from 2-fold to 9-fold exist, as shown in Fig. 9. This is because K⁺, Rb⁺, and Cs⁺ do not form compact ion structures with F^–.

Fig. 9Full-screen View

ADFs of F-Be-F, F-K-F, F-Rb-F, and F-Cs-F in (a) Mixture 1 and (b) Mixture 2

4.3.4

Diffusion coefficient

The diffusion coefficients of the ions were calculated using Eqs. (12) and (13), as illustrated in Fig. 10. The diffusion coefficients of Be²⁺, K⁺, Rb⁺, Cs⁺, and F^– in Mixture 2 are $7.4\times {10}^{-5}$, $4.23\times {10}^{-5}$, $4.23\times {10}^{-5}$, $4.01\times {10}^{-5}$, and $6.67\times {10}^{-5}$ cm²/s, respectively. Owing to their small ionic radii and masses, Be²⁺ and F^– exhibit higher mobilities than the other ions. Conversely, characterized by its larger ionic radius and mass, Cs⁺ exhibits the lowest diffusion coefficient. As the proportion of BeF₂ increases in Mixture 1, the diffusion coefficients of Be²⁺, K⁺, Rb⁺, Cs⁺, and F^- decrease to $2.55\times {10}^{-5}$, $3.15\times {10}^{-5}$, $4.02\times {10}^{-5}$, $3.23\times {10}^{-5}$, and $3.87\times {10}^{-5}$ cm²/s, respectively. Reflecting on the structural characteristics of the two systems, the binding strength between the same ion pair in Mixture 2 is stronger than that in Mixture 1, but the interactions between Be²⁺ and F^– significantly influence the overall stability of the system. Because the concentration of Be²⁺ in Mixture 1 is considerably higher than that in Mixture 2, it is more likely to form larger clusters between Be²⁺ and F^–. This can obviously reduce the diffusion coefficients of all ions in Mixture 1, especially Be²⁺ and F^–.

Fig. 10Full-screen View

Diffusion coefficients of the ions in two molten salt systems at 650 °C