Article pubs.acs.org/JPCB
Thermal Conductivity of Molten Alkali Metal Fluorides (LiF, NaF, KF) and Their Mixtures Yoshiki Ishii,† Keisuke Sato,† Mathieu Salanne,‡,§ Paul A. Madden,∥ and Norikazu Ohtori*,⊥ †
Graduate School of Science and Technology, Niigata University, 8050 Ikarashi 2-no cho, Nishi-ku, Niigata 950-2181, Japan Sorbonne Universités, UPMC Univ Paris 06, UMR 8234, PHENIX, F-75005 Paris, France § CNRS, UMR 8234, PHENIX, F-75005 Paris, France ∥ Department of Materials, University of Oxford, Parks Road, Oxford 0X1 3PH, United Kingdom ⊥ Department of Chemistry, Faculty of Science, Niigata University, 8050 Ikarashi 2-no cho, Nishi-ku, Niigata 950-2181, Japan ‡
ABSTRACT: The thermal conductivities of molten alkali fluorides (LiF, NaF, and KF) and their mixtures (LiF−NaF, LiF−KF, and NaF−KF binaries and LiF−NaF−KF ternary) are predicted using molecular dynamics simulation with the Green−Kubo method. A polarizable ion model is used to describe the interionic interactions. All the systems except LiF−KF and LiF−NaF−KF mixtures follow a scaling law: it is proportionnal to mA−1/2(N/V)2/3, where mA is the arithmetic average of the ionic species masses in a given melt and N is the total number of ions included in the system volume V. In LiF−KF and LiF−NaF−KF mixtures a significant departure from the scaling law is observed. By examining separately the effects of the cation mass and size asymmetry in LiF−KF mixtures, we show that both of them account for half of the deviation. Finally, we observe that the temperature dependence of the thermal conductivity is very small in these molten fluorides.
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fluorides, whereas most of the applications employ mixtures of molten salts to lower the melting points. Recently, we have parametrized a series of interaction potentials, in the framework of the polarizable ion model (PIM),12,13 which can be used in molecular dynamics (MD) simulations of molten fluorides.3,4,14 The parameters were optimized by fitting the interionic forces acting on each ion and their induced dipoles to the same quantities calculated by density functional theory. There is no experimental information included in this procedure, and the resulting potentials were shown to provide predictions of high accuracy: The liquid structures as well as many thermodynamic (heat capacities) and transport (diffusion coefficients, electrical conductivities, viscosities) properties of many molten fluoride mixtures were determined and validated by comparing them with the corresponding experimental results when available.5,15,16 It was also confirmed that these potentials are accurately transferable to molten mixtures. In parallel, the expressions allowing for the evaluation of thermal conductivity using the polarizable ion model were derived.17 Again, very accurate predictions of experimentally measured values (where available) were shown for a series of simulations of molten alkali metal chlorides17 and for solid NaCl for various pressures.18 In another study, predictions of the thermal conductivity of solid MgO and MgSiO3 in the perovskite and postperovskite structure were performed under conditions
INTRODUCTION Molten fluorides are good candidates for being used as coolants and solvents in the next generation of nuclear fission reactors and as blankets in fusion reactors.1,2 Their main advantage is their durability under radiation. The precise knowledge of their thermal and transport properties at high temperatures is therefore useful for optimizing the composition of the melts depending on the target application.3−5 However, because molten fluorides are corrosive for silica glasses and many kinds of metals and ceramics that are often used in an experimental apparatus, some properties are extremely difficult to measure in the laboratory. This is particularly true for the thermal conductivity. This quantity is usually determined by the steadystate concentric cylinder method, and such measurements were made in molten alkali metal halides.6 Nevertheless, by using an alternative method based on forced Rayleigh scattering, Nagasaka and Nagashima showed that the former measurements are strongly affected by systematic error due to radiation and convection effects.7−9 However, their forced Rayleigh scattering method could not be applied to molten fluorides, due to the impossibility to have a stable container for performing the measurements (the same problem arises for LiBr and LiI). These authors have therefore tried to apply the principle of corresponding states to their experimental data obtained for a series of molten halides, to estimate the thermal conductivity of molten fluorides, LiBr, and LiI.10 Later, Galamba et al. applied the extended corresponding-states principle to them for the same purpose.11 These estimates might serve as rough guides for some uses, but they cannot be considered as precise predictions. In addition, they were only made in the case of pure alkali metal © 2014 American Chemical Society
Received: December 1, 2013 Revised: February 28, 2014 Published: March 17, 2014 3385
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that are relevant for the Earth’s deep mantle.19 The results were confirmed by an ab initio simulation study20 and by experiments.21 Therefore, our MD simulations can be expected to provide reliable data also for the thermal conductivity of molten fluorides, not only for pure salts but also for mixtures. In this paper, we report the calculated thermal conductivities for a series of molten alkali metal fluorides mixtures. We examine whether a scaling law that was recently suggested from calculations using simpler, nonpolarizable potentials still holds.22,23 The temperature dependence of the thermal conductivity was also studied.
Table 3. Calculation Conditions in the MD Simulations for Binary Mixtures with Varying Compositions (at Fixed Temperature) of Molten Fluorides and Results for the Thermal Conductivitiesa
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MODEL AND METHODS The thermal conductivity can be calculated by two routes, i.e., either via equilibrium simulations,24−28 using the Green−Kubo relations, or via nonequilibrium simulations.26,28−32 Here we have used the former method, which was well tested for molten salts.17,18 All the MD simulations were performed in the NVT ensemble, where N is the total number of ions included in the simulation cell, V is the volume of the cell which was set at the values fixed by the experimental density,33 and T is the temperature. All the calculation conditions are summarized in Tables 1−3. Note that there is a small disparity (of 2%) in the
system
xsalt1
d
NV−1
λ
σ
LiF−NaF (1300 K)
0.00 0.20 0.40 0.50 0.60 0.80 1.00 0.00 0.12 0.25 0.40 0.50 0.60 0.75 0.80 1.00 0.00 0.20 0.25 0.35 0.50 0.70 0.75 0.85 1.00
1928 1891 1854 1829 1811 1760 1720 1744 1736 1743 1788 1792 1797 1827 1837 1882 1806 1757 1764 1779 1764 1682 1694 1718 1675
55.31 58.71 62.75 64.86 67.45 72.76 79.89 36.15 37.23 38.83 41.67 43.14 44.70 47.82 48.95 54.00 37.44 40.93 42.45 45.80 50.55 56.88 60.05 67.39 77.80
1.138 1.079 1.111 1.165 1.286 1.430 1.790 0.552 0.573 0.613 0.683 0.707 0.776 0.871 0.915 1.106 0.548 0.535 0.589 0.568 0.695 0.947 0.937 1.280 1.756
0.019 0.039 0.078 0.081 0.106 0.113 0.193 0.024 0.031 0.044 0.060 0.078 0.066 0.043 0.037 0.020 0.033 0.114 0.137 0.196 0.302 0.384 0.349 0.278 0.159
NaF−KF (1300 K)
LiF−KF (1200 K)
Table 1. Calculation Conditions in the MD Simulations for Pure Molten Alkali Metal Fluorides and Results for the Thermal Conductivitiesa system
T
d
NV−1
λ
σ
LiF
1200 1300 1400 1300 1400 1500 1200 1300 1400
1770 1721 1672 1928 1865 1801 1865 1799 1734
82.18 79.90 77.63 55.31 53.49 51.66 38.66 37.31 35.95
1.812 1.790 1.715 1.138 1.072 1.014 0.612 0.571 0.537
0.223 0.193 0.161 0.019 0.018 0.020 0.033 0.027 0.031
NaF
KF
xsalt1 is the mole fraction of the first listed salt. The other symbols are explained in the caption of Table 1. a
density values provided in the database33 for pure NaF depending on whether the data from LiF−NaF or NaF−KF measurements are provided. This does not impact the thermal conductivity because we obtain values that differ by less than the standard error as can be seen in Table 3. The time step was set to 0.5 fs. The equations of motion were solved following the method by Martyna et al. where the Nose− Hoover chain thermostat was used for the temperature control.34 A large relaxation time of 10 ps was used for the latter, which ensures a weak coupling to the heat bath. Starting from crystal structure, after equilibration for 100 ps to reach the molten state, a series of 250 ps production runs were performed for each temperature, until a converged statistical average of the thermal conductivity could be obtained: 30 runs were performed for the pure salts (total simulation time of 7.5 ns) whereas 100 runs were necessary for the binary and ternary mixtures (25 ns). The pair additive form of the interionic potential includes charge−charge, charge−dipole, dipole−charge, dipole−dipole, repulsive, and two dispersion interaction terms, as follows: μi ·rijqj qiqj μi ·μj qi rij·μj ϕij(rij) = + f 4ij (rij) − f4ji (rij) + 3 3 rij rij rij rij 3
T is the temperature (in K); d is the density (in kg m−3); NV−1 is the number density (in nm−3); λ is the calculated thermal conductivity (in W m−1 K−1); σ is the standard error for λ (in W m−1 K−1). a
Table 2. Calculation Conditions in the MD Simulations for Binary and Ternary Mixtures with Varying Temperatures (at Fixed Composition) of Molten Alkali Metal Fluorides and Results for the Thermal Conductivitiesa system LiF−NaF (50:50 mol %) NaF−KF (40:60 mol %) LiF−KF (50:50 mol %)
LiF−NaF−KF (46:12:42 mol %) a
T
d
NV−1
λ
σ
1150 1300 1450 1050 1200 1350 1150 1200 1300 1450 950 1100 1250
1913 1829 1745 1938 1848 1758 1790 1764 1710 1630 1967 1867 1767
67.84 64.86 61.88 45.17 43.07 40.97 51.32 50.55 49.01 46.71 57.31 54.39 51.47
1.233 1.165 1.089 0.763 0.707 0.674 0.707 0.695 0.673 0.624 0.896 0.773 0.686
0.081 0.081 0.087 0.058 0.059 0.065 0.282 0.302 0.315 0.296 0.194 0.239 0.242
−
The symbols are explained in the caption of Table 1.
− 3386
3(rij·μi )(rij·μj ) rij 5 C8ij rij 8
f 8ij (rij)
+ Bij exp( −αijrij) −
C6ij rij 6
f 6ij (rij)
(1)
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where qi and μi are the charge and dipole moment of particle i, respectively, and f ijn is the damping function for short-range correction of interactions between charge and dipole and dispersion interactions:35 n
ij
f nij (rij) = 1 − cnije−bn × rij ∑
(bnij
1 VkB
∫0
where J = mod(I + 2,3) and K = mod(I + 2,3). Here again, I, J, and K equal 0 for the first cation, 1 for the second, and 2 for the third. The expression for energy current is
(2)
(α , β = e , z )
Lez 0A z 0z1 + Lez1A z1z 0 ⎞ 1 ⎛ L − ⎟ 2 ⎜ ee B ⎠ T ⎝
Aαβ =
Leα Lβα Leβ Lββ
B=
(4)
(6)
(7)
L z 0z 0 L z1z 0 L z 0z1 L z1z1
(8)
which involve jzI(t), defined as jz (t ) = I
∑ (qI − qF )v(i t ) i∈I
(9)
where I = 0 for the first cation and 1 for the second (qF = −1 is the formal charge of the anion). In the same way, the equation for ternary mixtures was derived in the present work as 2 ⎛ ∑I = 0 LezI A zIzJzK ⎞ 1 ⎜ ⎟ λ = 2 Lee − ⎟ B T ⎜⎝ ⎠
j≠i 2 ⎠ ⎝2 1 + ∑ ∑ (vi·rij)(Fijsr + Fijc,real ) + jec,recip 2 i j≠i
(12)
RESULTS AND DISCUSSION Figure 1 shows the running integrals of the autocorrelation functions defined in eq 5 for the pure alkali metal fluorides at 1300 K. Well-converged plateaus are clearly obtained for all of them, from which Lee, Lez, and Lzz are extracted to calculate the thermal conductivity. The values for λ are provided in Table 1 for the pure salts and in Tables 2 and 3 for the mixtures, together with the standard errors σ. These errors are larger for LiF-rich salts, which is due to the large values obtained for all the transport coefficients compared to the cases of NaF and KF, as can be seen in Figure 1. A similar behavior was observed for molten chlorides.17 A further examination of the data in Table 1 shows that the thermal conductivity of molten salts depends very weakly on the temperature. In fact, we have shown in our previous work that this dependence is simply due to the change of density of the melts with temperature.22 We note that although our values qualitatively agree with the measurements by Smirnov et al.,6 the latter observed a strong increase of the thermal conductivity with temperature. As discussed above, this increase is mainly due to radiation and convection effects that are not included in our calculations. In the following discussion, we will therefore ignore the temperature dependence, and only discuss density and composition dependences. Finally, we note that molten fluorides systematically have larger thermal conductivities than their chloride analogs, which makes them better candidates for applications in which heat transfer needs to be optimized. In our previous work, we have established a scaling law for the thermal conductivity.22 Indeed, we have shown on the basis of a series of simulations of alkali metal halides using simpler, non polarizable interaction potentials,22 that it is proportional to mA−1/2(N/V)2/3 for these melts, where mA is the arithmetic average of the ionic species masses in a given melt. We can see in Figure 2a that it remains valid when polarization interactions are included. As for the binary and ternary mixtures, there are some
where kB is the Boltzmann constant, je is the energy current, and jz is the charge current. The equation for binary mixtures is given in ref 23 as λ=
∑ 1 ϕij(rij)⎟⎟vi
■
(5)
1 ⟨j (t )j (0)⟩ 3 α β
+
where mi, vi, and ϕself i are respectively the mass, the velocity, and the self-energy of particle i, Fsrij is the short-range force that originates from the repulsive and the two dispersion interaction terms in eq 1, and Fc,real and jc,recip are the real-space force and the ij e reciprocal-space energy current, respectively. The two latter terms were evaluated using the Ewald summation technique, following the full description provided in ref 17. It is worth noting that the Green−Kubo method is advantageous because the system-size effects are reduced compared to nonequilibrium simulations.38,39
∞
Cαβ(t ) dt
∑ ⎜⎜ 1 mivi 2 + ϕiself i
(3)
A = Lez , B = Lzz
⎞
⎛
je =
and Cαβ(t ) =
L z 0z 2 L z1z 2 L z 2z 2 (11)
in which α is the Ewald parameter. The thermal conductivity, λ, of the pure molten salts was evaluated from the simulations using37
Lαβ =
L z 0z1 L z1z1 L z 2z1
× rij)
α 2 q π i
L A⎞ 1 ⎛ λ = 2 ⎜Lee − ez ⎟ ⎝ B ⎠ T
B=
Leγ Lβγ Lγγ
{Bij, αij, Cij6, Cij8, bijn, cijn} are a set of parameters determined from first-principles calculations in our previous works.4,14 The dipole moment induced on each ion is a function of the polarizability and electric field on the ion caused by charges and dipole moments of all the other ions. The instantaneous dipole moment is determined at every time step self-consistently using the conjugate gradient method by minimization of the total energy. The charge−charge, charge−dipole, and dipole−dipole contributions to the potential energy and forces on each ion are evaluated under the periodic boundary condition by using the Ewald summation technique.36 An additional self-energy term has to be included in the potential energy for every charged particle: ϕiself = −
Leβ Lββ Lγβ
Aαβγ =
k
k!
k=0
L z 0z 0 L z1z 0 L z 2z 0
Leα Lβα Lγα
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Figure 1. Transport coefficients of pure alkali metal fluorides at 1300K: (a) Lee, (b) Lez, and (c) Lzz. Figure 2. Density dependence of the thermal conductivity for (a) pure alkali metal fluorides and chlorides17 at various temperatures, (b) binary and ternary mixtures of alkali metal fluorides at various temperatures under constant composition, and (c) binary mixtures of alkali metal fluorides and chlorides23 at various compositions under constant temperature.
systems that show significant departures from the scaling equation, as shown in Figure 2b,c. On these plots, the solid line has the same slope as that in Figure 2a, which is fitted to the calculated thermal conductivities of pure alkali metal fluorides and chlorides. In particular, the thermal conductivity is much smaller than the scaling law prediction in the LiF−KF and LiF− NaF−KF systems. To show more quantitatively the extent of this departure from scaling laws, we show the composition dependence of the thermal conductivity of the three binary mixtures in Figure 3, as calculated in our MD simulations and as predicted from the scaling law and for an ideal mixing. The deviations for LiF−NaF and for NaF−KF are not significant as can be seen in panel d of Figure 3, whereas differences of as much as 30% are obtained in the LiF−KF case; this system also shows a strong departure from ideal mixing. It is the mixture for which there is the larger asymmetry between the cations, particularly in terms of mass and ionic size (which is reflected in the interaction potential). To understand further the roles of these two quantities, we have decided to perform a series of computer experiments based on the fact that the Na+ ion mass and ionic radius40 are intermediate between those of Li+ and K+ ions. The simulated systems consist of • pure NaF; • pure NaF where half of the Na+ ions have been assigned the interaction potential parameters of Li+ and the remaining half the ones of K+, all of them keeping the Na mass (virtual system 1);
• equimolar LiF−KF where all the Li+ and K+ ions have been assigned the interaction potential parameters of Na+ (virtual system 2); • equimolar LiF−KF. Four different temperatures were sampled. In each case the same density was used for the four simulations, which was ensured by setting the mass of Na+ ion to 23.02 g mol−1 (the arithmetic average of the masses of Li+ and K+) instead of 22.99 g mol−1. The corresponding results are summarized in Figure 4. We see that both the thermal conductivities calculated using only the mass of Na+ and that using only the NaF potential are located halfway between the NaF and LiF−KF systems. This shows that the deviation from the scaling law for LiF−KF mixtures results from additive effects of the asymmetry in mass and size between the two cations. The fact that deviations from the scaling law are observed in Figure 3 in the case of LiF−KF mixtures only and not for those of LiF−NaF and NaF−KF is rather intriguing. There are two possibilities: either (i) there also are some deviations, but they are too small to be detected due to the relatively large standard errors in our calculations, or (ii) there is a “threshold” in the asymmetry between the two cations for which deviations start to occur. To test this hypothesis, we have performed a second series of computer experiments, in which the virtual systems previously 3388
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defined are progressively “transmuted” into LiF−KF, either via its mass or via its interaction potential as depicted on the top and middle panels of Figure 5. Note that because the other
Figure 5. Variation of the thermal conductivity with (a) the mass of the cation or (b) the cation−anion repulsive potential parameter, BMF, during a “transmutation” from the virtual systems previously defined to equimolar LiF−KF.
parameters of the interaction potentials are very similar among Li+, Na+, and K+, in the second case we have only changed the cation−anion repulsion parameter BMF (see eq 1) for the sake of simplicity. It is immediately seen that the effects of asymmetry are progressive, even if more pronounced on the NaF side of the plot for the mass change and on the LiF−KF side of the plot for the interaction potential change. Overall, we can conclude that no evident threshold effect is observed, which means that asymmetry between cations should always lead to deviations from the scaling law proposed for the pure melts.
Figure 3. Composition dependence of the calculated thermal conductivity of binary mixtures (a) LiF−NaF, (b) NaF−KF, and (c) LiF−KF. xsalt1 indicates the molar fraction of the first listed salt. Dots correspond to the molecular dynamics values, the solid line represents the prediction based on the scaling law, and the dotted line represents ideal mixing behavior. (d) shows the deviation from the scaling law for the three mixtures.
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CONCLUSION The thermal properties of molten fluorides are of primary importance for evaluating their capabilities for being used as coolants in Generation IV nuclear reactors. Among these properties, the thermal conductivity is of particular interest, but no reliable experimental values are available in the literature. In this work we have determined the thermal conductivity for a series of molten alkali metal fluorides (LiF, NaF, and KF), including some binary and ternary mixtures. Our results show that molten fluorides have larger thermal conductivities than molten chlorides. In all cases, a very weak dependence toward temperature variations is observed. Regarding their density and composition dependence, in the pure salts the thermal conductivity follows a simple scaling law which was previously derived. In the case of mixtures, deviations from this scaling law are noticed for LiF−KF and LiF−NaF−KF mixtures. We have shown that this deviation is due to the asymmetry in mass and ionic size of the cations in these melts. This result is important because most of the applications of molten salts
Figure 4. Thermal conductivity as a function of density for pure NaF (open circles), pure NaF where the Na+ interaction potential parameters were set to either those of Li+ or K+ (open squares), an equimolar mixture of LiF−KF where the Li+ and K+ interaction potential parameters were set to those of Na+ (open reverse triangles), and an equimolar mixture of LiF−KF (closed triangles). 3389
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(15) Salanne, M.; Simon, C.; Groult, H.; Lantelme, F.; Goto, T.; Barhoun, A. Transport in Molten LiF-NaF-ZrF4 Mixtures: A Combined Computational and Experimental Approach. J. Fluorine Chem. 2009, 130, 61−66. (16) Pauvert, O.; Salanne, M.; Zanghi, D.; Simon, C.; Reguer, S.; Thiaudière, D.; Okamoto, Y.; Matsuura, H.; Bessada, C. Ion Specific Effects on the Structure of Molten AF-ZrF4 Systems (A+ = Li+, Na+, and K+). J. Phys. Chem. B 2011, 115, 9160−9167. (17) Ohtori, N.; Salanne, M.; Madden, P. A. Calculations of the Thermal Conductivities of Ionic Materials by Simulation with Polarizable Interaction Potentials. J. Chem. Phys. 2009, 130, 104507. (18) Salanne, M.; Marrocchelli, D.; Merlet, C.; Ohtori, N.; Madden, P. A. Thermal Conductivity of Ionic Systems from Equilibrium Molecular Dynamics. J. Phys.: Condens. Matter 2011, 23, 102101. (19) Haigis, V.; Salanne, M.; Jahn, S. Thermal Conductivity of MgO, MgSiO3 Perovskite and Post-perovskite in the Earth’s Deep Mantle. Earth Planet. Sci. Lett. 2012, 355−356, 102−108. (20) Dekura, H.; Tsuchiya, T.; Tsuchiya, J. Ab Initio Lattice Thermal Conductivity of MgSiO3 Perovskite as Found in Earth’s Lower Mantle. Phys. Rev. Lett. 2013, 110, 025904. (21) Dalton, D. A.; Hsieh, W.-P.; Hohensee, G. T.; Cahill, D. G.; Goncharov, A. F. Effect of Mass Disorder on the Lattice Thermal Conductivity of MgO Periclase under Pressure. Sci. Rep. 2013, 3, 2400. (22) Ohtori, N.; Oono, T.; Takase, K. Thermal Conductivity of Molten Alkali Halides: Temperature and Density Dependence. J. Chem. Phys. 2009, 130, 044505. (23) Takase, K.; Matsumoto, Y.; Sato, K.; Ohtori, N. Thermal Conductivity in Molten Alkali Halides: Composition Dependence in Mixtures of (Na-K)Cl. Mol. Sim. 2012, 38, 432−436. (24) Bernu, B. J.; Vieillefosse, P. Transport Coefficients of the Classical One-Component Plasma. Phys. Rev. A 1978, 18, 2345−2355. (25) Galamba, N.; Nieto de Castro, C. a.; Ely, J. F. Thermal Conductivity of Molten Alkali Halides from Equilibrium Molecular Dynamics Simulations. J. Chem. Phys. 2004, 120, 8676−8682. (26) Galamba, N.; Nieto de Castro, C. a.; Ely, J. F. Equilibrium and Nonequilibrium Molecular Dynamics Simulations of the Thermal Conductivity of Molten Alkali Halides. J. Chem. Phys. 2007, 126, 204511. (27) de Koker, N. Thermal Conductivity of MgO Periclase from Equilibrium First Principles Molecular Dynamics. Phys. Rev. Lett. 2009, 103, 125902. (28) Sirk, T. W.; Moore, S.; Brown, E. F. Characteristics of Thermal Conductivity in Classical Water Models. J. Chem. Phys. 2013, 138, 064505. (29) Pierleoni, C.; Ciccotti, G. Thermotransport Coefficients of a Classical Binary Ionic Mixture by Non-Equilibrium Molecular Dynamics. J. Phys. Cond. Matter 1990, 2, 1315−1324. (30) Muller-Plathe, F. A Simple Nonequilibrium Molecular Dynamics Method for Calculating the Thermal Conductivity. J. Chem. Phys. 1997, 106, 6082−6085. (31) Römer, F.; Lervik, A.; Bresme, F. Nonequilibrium Molecular Dynamics Simulations of the Thermal Conductivity of Water: A Systematic Investigation of the SPC/E and TIP4P/2005 Models. J. Chem. Phys. 2012, 137, 074503. (32) Armstrong, J. A.; Bresme, F. Water Polarization Induced by Thermal Gradients: The Extended Simple Point Charge Model (SPC/ E). J. Chem. Phys. 2013, 139, 014504. (33) Janz, G. J. NIST Molten Salts Database, 2nd ed.; NIST SRD 27; NIST: Boulder, CO, 1992. (34) Martyna, G. J.; Tobias, D.; Klein, M. L. Constant Pressure Molecular Dynamics Algorithms. J. Chem. Phys. 1994, 101, 4177−4189. (35) Tang, K. T.; Toennies, J. P. An Improved Simple Model for the van der Waals Potential Based on Universal Damping Functions for the Dispersion Coefficients. J. Chem. Phys. 1984, 80, 3726−3741. (36) Aguado, A.; Madden, P. A. Ewald Summation of Electrostatic Multipole Interactions up to the Quadrupolar Level. J. Chem. Phys. 2003, 119, 7471−7483. (37) Sindzingre, P.; Gillan, M. J. A Computer Simulation Study of Transport Coefficients in Alkali Halides. J. Phys. Condens. Matter 1990, 2, 7033−7042.
employ mixtures at eutectic compositions to reduce the operating temperatures. Because the scaling law cannot be applied, there is a strong need for building a reliable database based on molecular dynamics simulations. The situation will probably be even more dramatic when multivalent cations will be used, because a third source of asymmetry, the charge of the cation, will then be introduced.
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AUTHOR INFORMATION
Corresponding Author
*N. Ohtori: e-mail,
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was partially supported by Grant-in-Aid for Scientific Research (c) (Grant Nos. 21540382 and 24540397) from the MEXT-Japan. M.S. thanks the Japan Society for the Promotion of Science for a short-term invitation fellowship which enabled this collaboration.
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