Abstract
Nitrate molten salts are extensively used for sensible heat storage in Concentrated Solar Power (CSP) plants and thermal energy storage (TES) systems. They are the most promising materials for latent heat storage applications. By combining classical molecular dynamics and differential scanning calorimetry experiments, we present a systematic study of all thermostatic, high temperature properties of pure KNO_{3} and NaNO_{3} salts and their eutectic and ”solar salt” mixtures, technologically relevant. We first study, in solid and liquid regimes, their mass densities, enthalpies, thermal expansion coefficients and isothermal compressibilities. We then analyze the c_{P} and c_{V} specific heats of the pure salts and of the liquid phase of the mixtures. Our theoretical results allow to resolve a longstanding experimental uncertainty about the c_{P}(T) thermal behaviour of these systems. In particular, they revisit empirical laws on the c_{P}(T) behaviour, extensively used at industrial level in the design of TES components employing the ”solar salt” as main storage material. Our findings, numerically precise and internally consistent, can be used as a reference for the development of innovative nanomaterials based on nitrate molten salts, crucial in technologies as CSP, waste heat recovery, and advanced adiabatic compressed air energy storage.
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Introduction
Molten salts are the most used materials for thermal energy storage at high temperature. This is due to several physical properties that they exhibit, which are important in industrial applications related to energy. The first factor affecting the performance of a thermal energy storage (TES) system is the thermal stability of the materials used to store heat, i.e. the temperature interval where they are liquid. The thermal stability of nitrate molten salts (MNO_{3}, M = alkali metal), allows the heat to be stored between ≈520 K and ≈890 K, an extended range of very high temperatures. This interval is typical for pure alkali nitrates (LiNO_{3}, NaNO_{3}, KNO_{3}), while their mixtures show a lower freezing point (e.g. for the eutectic NaNO_{3}KNO_{3} mixture this is 495 K). This even larger thermal stability range fits the requirements of Concentrated Solar Power (CSP) plants which, as a consequence, use nitrate molten mixtures as a heat storage medium. By 2030, it is estimated a usage of ≈1.8 × 10^{9} tons of nitrate mixtures in CSP plants^{1}.
Despite this extensive use of molten salts in thermal energy management, many basic and challenging issues about these materials are still unresolved, both at the experimental and theoretical level. One crucial issue is the temperature dependence of c_{P}(T) in their liquid phase, especially concerning nitrate molten mixtures. Experimentally, different dependencies have been found, from increasing, to constant, to decreasing with T^{2,3,4,5,6,7}. For this reason, round robin tests have been launched within the scientific and technological communities^{8}.
A technologically relevant nitrate mixture is the so called “solar salt”, which has a weight fraction composition given by \({x}_{NaN{O}_{3}}=0.6\) and \({x}_{KN{O}_{3}}=1{x}_{NaN{O}_{3}}=0.4\). Industrially, for this mixture the most used empirical equation for the mass density ρ(T), at atmospheric pressure and within 530–890 K, is the function ρ(T) = α − β T, where α = 2.09 and β = 0.000636^{9}. The empirical relation used for the specific heat is the Tincreasing function c_{P}(T) = γ + δ T where γ = 1.443 and δ = 0.000172^{9} (the units of T, ρ(T) and c_{P}(T) are K, g cm^{−3} and Jg^{−1} K^{−1}, respectively). This last equation is the result of many experimental measurements and statistical analyses all based on the use of Differential Scanning Calorimetry (DSC)^{10,11}. However, an increasing c_{P}(T) behaviour is more typical of weakly interacting liquids^{12}, while nitrates belong to the class of strongly interacting ones.
In this rather complex experimental scenario, systematic theoretical results about thermostatic properties of molten salts and their mixtures are largely missing, while results for enthalpy H(T) and c_{P}(T) of pure salts have been reported in ref.^{13}.
As mentioned, empirical equations represent a key factor in establishing TES performances and have been crucial in the design of stateoftheart CSP plants built recently^{14}. Thus, revisiting them theoretically has a potentially high technological impact. Here, the validity of the mentioned c_{p} relation will be investigated in detail and critically reconsidered.
Clearly, improvements in the performances of storage systems, connected to the energy production and reduction of electricity costs (e.g., of CSP plants), heavily rely on the maximum optimization of the thermodynamic properties of well known and new saltbased mixtures^{15,16,17,18,19}.
In this work, we perform an extensive theoretical analysis, based on classical molecular dynamics (MD), of the temperature behavior of the thermostatic properties of NaNO_{3}, KNO_{3} and their mixtures, with emphasis on the thermal behaviour of the specific heats c_{P}(T) and c_{V}(T), technologically relevant especially for the mixtures. The results are first compared with our new DSC experiments and with previous measurements. Then, they are interpreted by establishing a link between the solidstate approach to collective vibrational modes in liquids^{20,21,22,23}, and the more standard gaslike approach^{24,25}.
Our analysis is devoted to (i) provide a theoretical characterization, both in the solid and liquid phases, of the thermostatic properties of nitrate molten salts; (ii) resolve the issues related to the c_{P}(T) thermal behavior in the liquid phase of relevant salt mixtures; (iii) validate the MD model^{13} accuracy, for further studies about, e.g., nanofluids where the suspending medium are the nitrate salts studied here. The first two points are relevant for all the applications in the CSP technology and more generally in the field of renewable energy; the third, to develop an accurate theoretical framework able to study the thermodynamic properties of saltbased fluids for thermal energy storage^{15,16,17,18,19}. Our calculations are supported by a selfconsistency check on the precision of our numerical methodology, only possible thanks to the fact that all relevant thermostatic properties were investigated.
Results
We will first investigate the thermostatic properties of pure potassium and sodium nitrates, in their solid and liquid regimes. Next, we will analyze the eutectic and “solar” mixtures with the ultimate goal to characterize the thermal behaviour of the specific heats in the liquid phases. All properties and methods to calculate them are described in the Methods Sections (MS).
Potassium Nitrate KNO_{3}
Melting temperature
In order to study via MD the KNO_{3} properties in a large temperature regime including various phases, the starting point is to locate the solidtoliquid transition temperature T_{M}, at P = 1 atm. We find \({T}_{M}^{MD}=592.5\pm 2.5\,{\rm{K}}\), which is very close to the experimental value, \({T}_{M}^{exp}=607\) K. This excellent agreement gives a first indication of the accuracy of the interatomic potentials used^{13}.
Moreover, it is also the result of a MD procedure to locate T_{M}, based on the temporal evolution of a twophase system, as described in MS 1.4. The melting point is also reported in ref.^{13}, where a different value (T_{M} = 513 K ± 17 K) was obtained via a thermodynamic integrationbased method.
By following our procedure, in Fig. 1 we show the evolution of the three Cartesian components of the mean squared displacement (msd) of K^{+} and \({{\rm{NO}}}_{3}^{}\) at T = 590 K and T = 595 K. At T = 590 K (\( < {T}_{M}^{MD}\)), the msd shows a typical solidlike constant behavior, while at T = 595 K (\( > {T}_{M}^{MD}\)), a typical liquidlike behavior.
Mass density and thermal expansion coefficient
Experimentally, solid KNO_{3} shows three polymorphic forms at P = 1 atm^{26}: a stable form at 299 K, denoted αKNO_{3}; a stable phase generated by heating at T = 403 K, βKNO_{3}; and a third, different metastable phase γKNO_{3} obtained by cooling down the system from high temperature, resulting from an alternative kinetic path.
As we aim to characterize the specific heats in various phases, we preliminary analyze the density ρ(T) and the enthalpy H(T) temperature behavior. The results are presented in Fig. 2. Focusing first on the solid phases between 273 K and ≈600 K, we find that by heating up the αKNO_{3} phase from T = 273 K, the calculated density shows a strongly nonlinear behavior in the range T = [273,400] K, Fig. 2(a). This indicates the formation of a new phase, corresponding to βKNO_{3}. By cooling down the latter from T = 450 K, the density is once again nonlinear, but the γKNO_{3} phase is obtained at room temperature.
In the liquid region, the density ρ(T) is, instead, a linear function with a MD slope that coincides with the experimental one^{27}. In this regime, our values confirm quantitatively the ones reported previously^{13}. The slope, normalized by ρ(T), gives the thermal expansion coefficient α(T), shown in Fig. 2(b). The difference between experiment and theory in α(T) reflects only the difference in the corresponding values of the density, which is within 5% in the entire Trange investigated (at T = 293 K, ρ = 2.023 g cm^{−3}, while ρ^{exp} = 2.101 g cm^{−3}; at T ≈ 600 K, ρ = 1.8 g cm^{−3}, while ρ^{ex} = 1.867 g cm^{−3}). A good agreement is also obtained for the volume change at the experimental melting point, T_{M} = 607 K. At \({T}_{M}^{MD}=592.5\,{\rm{K}}\), the MD value is \({\rm{\Delta }}{V}_{m}^{MD}/{V}_{S}^{MD}=4.03\)%, while the experimental change is \({\rm{\Delta }}{V}_{m}^{exp}/{V}_{S}^{exp}=3.3\)%^{27}. Again, these density results confirm the ability of the interatomic potentials^{13} in describing the experimental findings.
Enthalpy
We next calculate the enthalpy H(T) in the whole temperature interval. In Fig. 2(c,d), a nonlinear T behavior corresponding to the density one is found in the solid range. The two MD plots in Fig. 2(c) are obtained by heating up or cooling down the system, respectively (black and green symbols). The values at T = 273 K show that the stable phase is the αKNO_{3} (Fig. 2(d) inset), while the metastable γKNO_{3} phase should transform into the former with time^{28}. In the liquid phase we find the important result that our H(T) data are perfectly linear with the temperature. Finally, by extrapolating the solid and liquid H(T) to T_{M}, the computed value for the melting enthalpy is \({\rm{\Delta }}{H}_{M}^{MD}=130\,{\rm{J}}/{\rm{g}}\), which is to be compared with the experimental value of \({\rm{\Delta }}{H}_{M}^{\mathrm{Exp}}\approx 100\,{\rm{J}}/{\rm{g}}\)^{4}.
Specific Heats
By performing the derivative of the MD enthalpy plots (fitting polynomials for H(T) were used), we now show the obtained specific heats (Fig. 3), and compare them with our experimental results and existing experiments^{4}. The isobaric c_{P}(T) is presented in Fig. 3(a). Remarkably, due to the H(T) nonlinearity in the solid regime, the calculated c_{P} shows a transition peak in correspondence of the solidtosolid αKNO_{3} → βKNO_{3} transition (T ≈ 400 K)^{26} and, still in accordance with the experimental data, a slightly increasing behavior until the melting T_{M} is reached. Importantly, Fig. 3(b) shows that in the whole temperature range of liquid stability c_{P} has a constant value, \({c}_{P}^{MD}=1.518\) Jg^{−1} K^{−1}, in disagreement with previous theoretical results^{13}.
Figure 3(b) also shows the MD predicted c_{V}(T) and \({c}_{{V}_{0}}(T)\) behaviors calculated, respectively, at constant (V) and at fixed volume (V_{0}). The latter is chosen to be the volume immediately after the melting temperature^{29}. As it is clear from the internal energy of the liquid state (Eq. (5), Sec. 1.2), the \({c}_{{V}_{0}}(T)\) decrease is purely due to the smoothing and broadening of the peaks in the pair distribution functions g_{ij}(r), while in the c_{V}(T) decrease there is the additional contribution from the variation of V(T, P = 1 atm), via the density ρ(T).
In the gaslike approach to liquids^{24,25}, the g_{ij}(r) smoothing and broadening correspond to less defined coordination shells around given ions, from which they can more easily escape or jump at increased temperature. In the solidstate approach of refs^{20,21,22,23}, they correspond to a shorter time between two consecutive ion jumps, known as the Frenkel time τ_{F} (and to a larger frequency ω_{F}). This results in a temperature decrease of the number of transverse oscillating modes, which has been identified as the main cause of the decrease of \({c}_{{V}_{0}}(T)\)^{20,21,22,23}.
Since c_{P}(T) can be written as c_{P}(T) = c_{V}(T)(1 + γαT), where γ is the Grüneisen parameter and α is the coefficient of thermal expansion and since, typically, γ is almost Tindependent^{30}, it follows that c_{P}(T) is the product of a temperature decreasing term, c_{V}(T), and a temperature increasing term containing αT^{2,20}. As a result, c_{P}(T) must be less, or not at all sensitive to temperature changes, as we actually find. This behavior is also consistent with the experimental results found for molten alkali and alkali earth halides^{31} that, as KNO_{3}, belong to the class of low viscosity ionic liquids. Previous c_{P}(T) values^{13} differ from our results, since c_{P}(T) was determined by considering, on the top of the MD interaction model, internal degrees of freedom contributions to the kinetic ideal term that are Tdependent^{13}.
The overall agreement between experimental data and MD results in T = [400, 725] K is very good if compared to the accuracy found in the literature^{13}, the difference being <8%. However, we note that the c_{P} experimental data are in between the c_{P} and c_{V} theoretical results and show a slight tendency to decrease and oscillate with increasing T. This could be a consequence of experimental conditions closer to constant V than to constant P (sealed and small sample holder used in DSC experiments).
Isothermal compressibility
Our calculations reproduce correctly also the temperature dependence of the isothermal κ_{T}(T) in the liquid phase, Fig. 4. We evaluated κ_{T}(T) by using three different procedures, fully illustrated in Methods Section 1.3. The black plot stems from the relation κ_{T} = α^{2}T/[ρ(c_{P} − c_{V})]. The blue dot at T = 700 K is instead calculated via an alternative sequence of steps and the use of the equation κ_{T} = −V^{−1}(∂〈V〉/∂P)_{T}. Importantly, we can see that these data perfectly superpose. This represents a severe test to check our numerical precision and consistency, for all thermostatic quantities not directly obtainable as MD averages. We have a further confirmation of this consistency. After a suitable rescaling (MS, Eq. (9)) the blue dot value was reported in the inset of Fig. 4, together with the S_{NN}(k) BhatiaThornton structure factors^{25,32,33}, which we also calculated at T = 700 K. As it is well known^{25,32,33}, we have the relation \(\frac{1}{{\rho }_{N}{k}_{B}T}{\mathrm{lim}}_{k\to 0}{S}_{NN}(k)={\kappa }_{T}\). In the inset, we see that, indeed, our S_{NN}(k) curve tends to the right limit for small wave vectors k, approaching the rescaled κ_{T} that comes from a separate calculation. Also in this case the internal consistency is evident.
Sodium Nitrate NaNO_{3}
We present now our results for NaNO_{3}. To a large extent they exhibit similar features found in KNO_{3}.
Melting temperature
The NaNO_{3} melting temperature is calculated to be \({T}_{M}^{MD}=594\,{\rm{K}}\), which is higher than the experimental value \({T}_{M}^{Exp}=581\,{\rm{K}}\)^{26}, but still in a reasonable agreement. ref.^{13} reports 591 ± 18 K.
Mass density and thermal expansion coefficient
The mass density results are shown in Fig. 5(a). Also in this case, the experimental data in the solid range have a strong nonlinear behavior, as in KNO_{3}. In fact, at T = 433 K a secondorder phase transition begins from the low temperature stable phase, denoted IINaNO_{3}, to the high temperature phase, INaNO_{3}^{34}. The transition is complete at T = 544 K^{26}, and it is due to the activation of the rotational degrees of freedom of the nitrate groups^{35}. In this temperature range, the theoryexperiment discrepancy is found to be less than 4% at T = 293 K (ρ^{MD} = 2.181 g cm^{−3}, while ρ^{exp} = 2.257 g cm^{−3}), and the strong nonlinearity is also well reproduced. Also in this case our values confirm quantitatively the ones reported previously^{13}. In the liquid phase, both the ρ(T) linear behavior and the set of experimental data are perfectly reproduced. However, we need to consider that in the liquid range, the experimental data refer to systems where, intrinsically, both sodium nitrates and nitrites are present (see ref.^{2} and references therein), while our MD results refer to pure NaNO_{3}. Still, although this level of precision might be quite accidental, the interaction potentials used show to be, once more, highly accurate. This is also true for the liquid branch of the thermal expansion coefficient, shown in Fig. 5(b), where we see that the theoretical and experimental^{36} results essentially superpose.
Enthalpy
The enthalpy H(T) is reported in Fig. 5(c). As for the density, the enthalpy is nonlinear in the solid phase and linear in the liquid region. By extrapolating the solid and liquid enthalpies to the experimental \({T}_{M}^{exp}=581\) K, a value of \({\rm{\Delta }}{H}_{M}^{MD}=173\) J/g is obtained, in quantitative agreement with the experimental value, \({\rm{\Delta }}{H}_{M}^{Exp}\in 172187\) J/g^{2}.
Specific Heats
The c_{P} values are shown in Fig. 6. In the solid phase, c_{P} increases with T, and shows a peak in the same region where the secondorder phase transition is experimentally observed^{26,34}. This transition is also evident from the c_{P}(T) increase observed in our DSC experimental data, also reported in the same Figure. In the liquid phase, c_{P}(T), analogously to KNO_{3}, has a constant value for all the investigated temperatures (\({c}_{P}^{MD}=1.805\) Jg^{−1} K^{−1}), as shown in Fig. 6(a,b). Experimental c_{P}(T) data are either constant, as the MD results, or exhibit a decreasing behavior. The comparison between our experimental and theoretical results show a difference of ≈7%. Instead, our MD c_{V}(T) and \({c}_{{V}_{0}}(T)\) show a decreasing behavior.
Isothermal compressibility
Finally, as shown in Fig. 7, the Tbehavior of the experimental isothermal compressibility κ_{T}(T) in the liquid regime is also well reproduced by our MD modeling. A similar selfconsistency test as for KNO_{3}, based on the use of three calculation procedures was performed for NaNO_{3}. The test was successful, as shown by the coincidence of the black plot and the blue dot in Fig. 7 and by the correct behavior of the BhatiaThornton structure factors S_{NN}(k), tending to κ_{T} at the k → 0 limit. Hence, all the considerations on accuracy and precision made for the KNO_{3} isothermal compressibility apply here too.
Eutectic and “Solar” Mixtures
Molten salt mixtures are technologically extremely relevant, especially when considered in their liquid phase. The characterization of their thermostatic properties is important not only for a fundamental understanding of the physics of ionic liquids at high temperature, but also for energy applications in any system containing a heat storage component. In this respect, it is also important the development of viable simulation methods to determine technological relevant binary and ternary molten salt eutectics^{37}. In the following we will calculate, in the liquid phase only, the thermostatic properties of two relevant nitrate molten salt mixed systems: the eutectic and the “solar” mixture. Emphasis is given on the specific heats thermal behavior, where the available experiments still exhibit a high degree of controversy^{4,5,6,7}.
The eutectic NaNO_{3}KNO_{3} mixture has the chemical composition Na_{0.5}K_{0.5}NO_{3}. Due to the lower mass of Na, this corresponds to a 45.67% NaNO_{3}–54.33% KNO_{3} weight percentage composition. The “solar salt” mixture has a higher content of Na, with a chemical composition of Na_{0.641}K_{0.359}NO_{3} and a weight percentage of 60% NaNO_{3}–40% KNO_{3}.
Mass densities
The density plots for both systems are shown in Fig. 8. The theoryexperiment agreement is satisfactory. However, while for the pure salts the difference between experimental and modeling results is the same in the whole liquid phase, here the discrepancy increases as the temperature increases. This behavior could be related to the use of the Lorentz– Berthelot approximation^{38} to describe the crossed interaction between Na^{+} and K^{+} particles, which is the only approximation introduced in passing from the pure components to the mixtures.
Specific heats
The specific heats of the mixtures are shown in Fig. 9(a,b). Importantly, we found again that the calculated values of c_{P} are temperature independent. They are 1.673 Jg ^{−1} K^{−1} and 1.704 Jg ^{−1} K^{−1} for the eutectic and the “solar” mixture, respectively. The specific heat c_{V}(T), instead, is predicted to be a decreasing function of T. As the simulations allow to compute the specific heats by keeping either the volume or the pressure strictly constant, we speculate that all the observed systematic discrepancy between the c_{P}(T) experimental and MD results can be due to the fact that in DSC experiments aimed at measuring c_{P}(T), the control of the experimental conditions is rather challenging, especially in the case of ionic liquids. For instance, inside the sealed sample holders partially filled by a small amount of sample, a very high and variable pressure can be generated, with the sample changing its volume as the temperature is changed. Ionic samples present extra difficulties due to the nonwetting of the sample holder surfaces. This complexity is reflected in the variety of c_{P}(T) behaviors with temperature, as measured by DSC. As shown in Figs 3(b), 6(b) and 9(a,b), c_{P}(T) is found to decrease, to be constant and to increase. Our MD results clarify, once for all, that working at constant pressure a constant value of c_{P}(T) is produced, while at constant volume a decreasing c_{V}(T) is predicted.
As a conclusion for technological applications, we then propose to reconsider the use of empirical equations showing a c_{P}(T) temperaturedependent behavior for any molten KNa nitrate mixture (see, e.g., the increasing c_{p}(T) used industrially, Exp. 11^{9} in Fig. 9(b)). We estimate that such a behavior, bringing, e.g., to a 5% overestimate of the real c_{p}(T) value, corresponds to a loss of 0.75 h/day of electricity production in a CSP plant with a “solar” salt tank able to store energy for 15 h. On a yearly basis (and for a typical 50 MW plant), this traslates in the considerable loss of ≈12 GWh/year in the electricity production.
Due to the fact that the calculated c_{P} values of the pure and mixed salts are Tindependent, c_{P} can be plotted against the weight fraction x of KNO_{3} present in the mixture. This is done in Fig. 10, where we see that by interpolating the calculated values of the four investigated compositions (pure NaNO_{3}, eutectic, “solar salt” and pure KNO_{3}), c_{P} changes linearly. Although the experimental data in Fig. 10 refer to different temperatures, a linear behavior can be also identified. We then conclude that mixtures of KNO_{3} and NaNO_{3} behave as ideal mixtures. Thus, to determine the c_{P}(x) of any mixture, no extra measurements or simulations are needed, as it is sufficient to consider a linear combination of the pure salts values with their respective weight fractions, i.e. \({c}_{P}(x)=x{c}_{P}^{KN{O}_{3}}+\mathrm{(1}x){c}_{P}^{NaN{O}_{3}}\).
Isothermal compressibility
The MD values of the isothermal compressibility κ_{T}(T) are reported in Fig. 11. These data are the first prediction appearing in literature. By considering the accuracy of the MD results for the pure salts, highlighted in previous sections, they can be used as reference for further studies, both experimental and theoretical.
Discussion
In this work, by combining classical molecular dynamics (MD) simulations and differential scanning calorimetry (DSC) experiments, we investigated the thermostatic properties of nitrate molten salts, technologically relevant materials for thermal energy storage applications. We focussed, in particular, on the thermal behavior of the specific heats of KNO_{3}, NaNO_{3} and their eutectic and “solar” mixture, the latter known as “solar salt”. The motivation of our work is twofold: First, to the best of our knowledge, theoretical calculations about the thermostatic properties in the solid and liquid phase of these materials are largely missing. Second, there is a general lack of consensus about the experimentally measured specific heats as a function of temperature, especially for what concerns c_{P} in the liquid phases.
To address these issues, we first computed the mass density, the enthalpy and the thermal expansion coefficient of pure nitrate salts, as a function of temperature. Moreover, we calculated the melting transition temperature T_{M} and the enthalpy and the volume changes at T_{M}. We obtained an accurate theoretical description, reproducing quantitatively the available experimental data. This allowed to reproduce several nontrivial features of solidsolid and solidliquid phase transitions in the case of the pure salts.
Next, we characterized the specific heats in the pure salts, and, due to their technological relevance, in the mixtures. For all investigated liquid systems, we found a constant value of c_{P}(T), while c_{V}(T) is weakly decreasing with temperature.
Finally, we calculated the isothermal compressibility κ_{T}(T), encompassing other thermostatic properties, in excellent agreement with the experiment. This fact, together with a careful, selfconsistency check based on three independent procedures to calculate this quantity, fully validates our MD numerical scheme. Thus, we expect that our results for the eutectic and “solar” mixtures have the same degree of accuracy.
The constant value of c_{P}(T) in the liquid regime clarifies a complex experimental picture, especially in the case of the “solar” mixture, where more experimental data are available. In view of this, for this material we suggest to reconsider the empirical Tincreasing function c_{P}(T) = γ + δ T, used in the design of thermal energy storage components.
For the eutectic and “solar” salt our results also allow to say that these materials behave as ideal mixtures, i.e. the c_{P} of any mixture can be obtained from the c_{P} of the pure salts only.
We believe that our results are of general validity and not limited to the class of nitrate molten salts. They confirm many temperature trends observed in the thermostatic properties of strongly interacting liquids (e.g the condensed phases of alkali halides^{31}). They provide guidelines for researchers who perform experiments on the development of saltbased fluids for thermal energy storage. These include bulk nanomaterials and colloidal suspensions in ionic compounds.
Future studies will require theoretical reference data and validated models about basic materials that compose more complex nanomaterials, as the ones provided in this work.
Methods
Computational Methods
The used classical MD model is based on a version of the Fumi and Tosi pair interaction potential^{39,40}, i.e. the Buckingham potential, superimposed to a Coulomb potential. The interatomic parameters of the Buckingham potentials are taken from S. Jayaraman et al.^{13}. This parametrization has been chosen because: (i) it reproduces the liquid and crystal phase densities of the pure KNO_{3} and NaNO_{3} within 4% of the experimental data^{13}; (ii) it reproduces the MD partial pair distribution functions, g_{ij}(r), of the pure NaNO_{3} evaluated by A.K. Adya et al.^{41} with a BornMayerHuggins type of interaction potential. The latter reproduces the experimental structure factor extracted from the measured Xray diffraction intensity^{41}. No further refinements of the interaction parameters, and no other approximations beyond the Lorentz– Berthelot approximation^{38}, needed for the cross interaction parameters in the mixture case, are made. The model is sufficiently accurate for the purposes of the present study. Simulations have been performed with the LAMMPS code^{42,43}. Initial solid configurations are taken from ref.^{13}. The NaNO_{3} initial solid configuration at T = 293 K corresponds to the rhombohedral \(R\bar{{\rm{3}}}c\) group with Z = 6, a = 5.070 Å, c = 16.82 Å. The KNO_{3} initial solid configuration at T = 293 K corresponds to the orthorhombic Cmc2 group with Z = 16, a = 10.825 Å, b = 18.351 Å, c = 6.435 Å. The initial liquid configurations have been generated with the PACKMOL package^{44}. To study the pure salts, systems with 600 cations and 600 \({{\rm{NO}}}_{3}^{}\) ions, for a total of 3000 particles, were constructed. To study the mixtures, 600 \({{\rm{NO}}}_{3}^{}\) ions have been used while the cation numbers were 300 Na^{+} and 300 K^{+} for the equimolar mixture, and 384 Na^{+} and 216 K^{+} for the solar mixture. A timestep of 1 fs has been used. Systems were equilibrated using 10^{6} timesteps and run with further 10^{6} − 2 × 10^{6} timesteps. For the solid (liquid) systems, equilibrated configurations at the lowest (highest) studied temperature were used as input for the closest next higher (lower) temperature. Other relevant parameters used are: NoséHoover barostat time constant 0.5 ps; NoséHoover thermostat time constant 0.1 ps; Buckingham and Coulomb interaction cutoff distance r_{c} = 12 Å for systems with K^{+} ions, and r_{c} = 11 Å for other cases; longrange force calculation accuracy 10^{−4}. Production runs in NPT simulations were considered suitable for analysis only when the averaged equilibrated pressure was within 〈P〉 = 1 ± 1 atm. Within this range, the change in the averaged 〈H〉 is \(\le 1\times {10}^{2} \% \).
Thermodynamic Properties
Thermodynamically, the specific heats c_{V} and c_{P} are defined as:
where E is the internal energy, H the enthalpy and N the mole number, which is kept constant. These quantities are linked by the relation:
where α = V^{−1}(∂V/∂T)_{P} is the coefficient of thermal expansion, ρ = N/V is the density and κ_{T} = −V^{−1}(∂V/∂P)_{T} the isothermal compressibility. Eq. (3) can be also written in the form:
were the Grüneisen parameter, γ = α/(ρκ_{T}c_{V}), is introduced. Typically, γ ≈ 1 and it is almost Tindependent^{30}.
In the NVT ensemble, the internal energy E in Eq. (1) is expressed by the relation:
where x indicates the number fraction and g_{ij}(T, r) the ij pair distribution function. If in the considered Trange all the kinetic degrees of freedom are activated, E_{Kin} is a linear function of T giving a constant term contribution to c_{V}(T). Then the temperature contribution to c_{V}(T) only arises from the Tdependence of the density ρ and of the pair distribution functions g_{ij}(r) appearing in the E_{Pot} term. To separate the temperature effects of these two quantities, two sets of simulations have been conducted. In the first set, a liquid density ρ_{0}, chosen at a density close to the freezing point, is kept fixed for all the investigated T; in the second set, the ρ(T) values obtained from the NPT equilibration run are used.
Simulation Methods and SelfConsistent Check
To calculate the specific heats and other thermostatic properties in the liquid phase, the simulation is performed by a stepbystep cooling procedure. First, we worked in the NPT ensemble. Starting from a random distribution of particles^{44}, the system is equilibrated at the highest temperature, usually the experimental decomposition temperature where the salt molecules break. The temperature is then decreased, and the equilibrated configuration of the previous Tstep is used as a starting configuration for the next T. All equilibrated configurations are used to perform production runs and data analysis. When necessary, these NPT configurations are used as input for NVT ensemble simulations.
As for the specific heats, there are several ways to determine them from ensemble averages of MD trajectories^{38}. In this work, the direct evaluation via the relations
is preferred due to its stability and reliability. Their evaluation is functional to a scheme aimed at the determination of all thermostatic properties. Moreover, a final selfconsistency check is performed.
In detail:

1.
Fix the pressure, P = 1 atm, and the number of moles N.

2.
Perform NPT simulations for a set of temperatures T.

3.
Extract the functions: V(T), ρ(T), α(T) = V^{−1}(∂ 〈V〉/∂T)_{P}, H(T) and c_{P}(T) from Eq. (6).

4.
For each value V(T_{i}) perform four NVT simulations at temperatures T_{i} ± ΔT and T_{i} ± 2ΔT.

5.
Extract c_{V}(T_{i}) from Eq. (6), via a fourpoint differentiation relation of the total energy E(T)^{45}.

6.
Repeat the previous point for a set of temperatures and obtain the function c_{V}(T).

7.
Calculate the isothermal compressibility κ_{T}, via three different procedures. First, for each T_{i} perform a set of NPT simulations in the pressure interval P ∈ [1, 600] atm. Use the equation
$${\kappa }_{T}=\,{V}^{1}{(\partial \langle V\rangle /\partial P)}_{{T}_{i}},$$(7)to extract κ_{T}(T_{i}) from a linear differentiation.

8.
Repeat the previous point for a set of temperatures and obtain the function κ_{T}(T).

9.
Selfconsistency check:

(a)
Compare the κ_{T}(T) values from step 8 with a second set of values, from the equation
$${\kappa }_{T}={\alpha }^{2}T/[\rho ({c}_{P}{c}_{V})]$$(8)where α(T), ρ(T) and c_{P}(T) are obtained from step 3, and c_{V}(T) from step 6.

(b)
Evaluate the percentage difference of κ_{T} calculated in the two ways. This difference is a severe test for: (i) the achievement of equilibrated MD configurations (slope of the equilibrated energy functions 〈10^{−8} ÷ 10^{−9}); (ii) the numerical precision of the procedures to determine c_{P}, c_{V}, α, and κ_{T} itself.

(c)
κ_{T} can also be determined via a third procedure, from the total number density BhatiaThornton structure factor, S_{NN}(k)^{25,32,33}:

(a)
where
In these equations, ρ_{N} is the number density and \({\tilde{h}}_{\alpha \beta }(k)\) are Fourier transform of the cation and anion pair correlation functions h_{αβ}(r) = g_{αβ}(r) − 1. Here, the pair distribution functions g_{αβ}(r) are obtained from the NVT averages \({x}_{\alpha }{x}_{\beta }\rho {g}_{\alpha \beta }(r)={N}^{1}\langle {\sum }_{{i}_{\alpha }}{\sum }_{{j}_{\beta }}\delta (r+{r}_{{i}_{\alpha }}{r}_{{j}_{\beta }})\rangle \). This calculation of κ_{T} through microscopic structural quantities, is a further check of the overall numerical scheme precision.
Calculations in the solid phase follow the same lines sketched for the liquid phase. The only difference is in the sequence of the MD runs, as in this case, in general, the simulations at constant pressure were performed as a stepbystep heating procedure. As for KNO_{3}, we additionally adopted a stepbystep cooling procedure. In the initial configuration at the lowest temperature (T = 300 K), we considered the energetically stable crystallographic configuration at that temperature^{13,26}.
Method to Simulate the Melting Transition Temperature
To calculate the solidtoliquid transition temperature T_{M}, we adopted an approach based on the direct simulation of a twophases coexistence, with an explicit interface (Fig. 12). This approach has proven to be robust and reliable for systems of large size in particle number^{46,47,48}. Figure 12(a) shows a system with solid and liquid phases, previously (and separately) equilibrated at a temperature close to the expected T_{M}. By performing a simulation at \({T}_{L} > {T}_{M}\), the interface will move in order to suppress the solid phase (Fig. 12(b), upper panel), while at T_{S} < T_{M}, the interface will move in order to suppress the liquid phase (Fig. 12(b), lower panel). The evolution of the twophase system towards either the liquid or the solid phase is shown in Fig. 12(c). Here the mean squared displacement (msd) of the atoms is plotted as a function of the MD time steps. The solid will be characterized by a zero slope and the liquid by a slope related to the liquid selfdiffusion constant. The change in slope for increasing time steps will characterize the twotoone phase transition, in both cases. After having chosen a suitable temperature interval (which must include T_{M}), the procedure to obtain the transition temperature consists in starting from the configuration in Fig. 12(a) and performing a set of simulations by slowly decreasing (increasing) T_{L} (T_{S}) until the difference ΔT = T_{L} − T_{S} is small enough. In our simulations we chose ΔT ≈ 5 K, as smaller values make our results unstable and strongly dependent on the coupling rates with the thermostat and the barostat. This small value defines the accuracy of the calculated melting temperature, T_{M} = (T_{L} + T_{S})/2. This approach will also allow to calculate the volume and enthalpy changes at the transition, respectively ΔV_{M} = V_{L}(T_{M} + ΔT/2) − V_{S}(T_{M} − ΔT/2) and ΔH_{M} = H_{L}(T_{M} + ΔT/2) − H_{S}(T_{M} − ΔT/2).
Experimental
For the four investigated systems, we have performed a set of differential scanning calorimetry measurements (DSC). High purity sodium nitrate and potassium nitrate were provided by Sigma Aldrich and the salts were used without any further purification for the present study. In order to remove the final traces of moisture, the samples were heated at 293 K with a heating plate located inside a globe box, and then the samples were encapsulated using hermetic Tzero aluminium lids and pans at argon atmosphere. Pure salt, standard sapphire and a reference (only the Al crucible) were hermetically sealed at argon atmosphere. The sapphire and the samples weights were measured by using a microbalance to 4 decimals in milligram (Mettles Toledo. X6TU Model). The specific heat values of all the samples were measured by using modulated differential scanning calorimeter (MDSC) (TA instruments, Q2000) with specific heating program. The encapsulated samples were heated at 723 K and kept isothermal for 10 minutes (to stabilize the heat flux signal). Before that, Tzero heat flow was implemented at this temperature and the sample was equilibrated at 298 K and then ±1 K was executed every 120 seconds and stay again isothermal for 10 minutes. Finally, 2 K/min heat ramp was implemented till reach at 723 K.
Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
This work is dedicated to the memory of Prof. Sandro Massidda. Via our membership of the UK’s HPC Materials Chemistry Consortium, which is funded by EPSRC (EP/L000202), this work used the ARCHER UK National Supercomputing Service (http://www.archer.ac.uk). Nithiyanantham Udayashankar is acknowledged for experimental suggestions and useful discussions.
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B.DA. conceived and performed the theoretical calculations, which have been extensively discussed with A.F. M.K. and N.G. conducted the experiments. B.DA. and A.F. analyzed the results and wrote the manuscript.
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D’Aguanno, B., Karthik, M., Grace, A.N. et al. Thermostatic properties of nitrate molten salts and their solar and eutectic mixtures. Sci Rep 8, 10485 (2018). https://doi.org/10.1038/s41598018286411
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DOI: https://doi.org/10.1038/s41598018286411
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