Solar Energy Storage in Phase Change Materials: First-Principles

Article pubs.acs.org/JPCC

Solar Energy Storage in Phase Change Materials: First-Principles Thermodynamic Modeling of Magnesium Chloride Hydrates Philippe F. Weck*,† and Eunja Kim‡ †

Sandia National Laboratories, Albuquerque, New Mexico 87185, United States Department of Physics and Astronomy, University of Nevada Las Vegas, 4505 Maryland Parkway, Las Vegas, Nevada 89154, United States

‡

ABSTRACT: Thermal energy storage in salt hydrate phase change materials, such as magnesium chloride hydrates, represents an attractive option for solar energy applications. In this study, the structural, electronic, and thermodynamic properties of magnesium dichloride hexahydrate, MgCl2· 6H2O, and its dehydrated phases, MgCl2·nH2O (n = 4, 2, 1), were computed within the framework of density functional theory. Densities of states were predicted, and phonon analysis using density functional perturbation theory was performed at equilibrium volume to derive isochoric thermal properties (i.e., Helmholtz free energy, entropy, and isochoric heat capacity). Isobaric thermal properties (i.e., Gibbs free energy, isobaric heat capacity, and latent heat) were also calculated within the quasi-harmonic approximation. Overall good agreement is observed between the computed thermodynamic properties and the scarce experimental data available for these materials.

1. INTRODUCTION

carried out, an accurate assessment of the thermophysical properties of some of these materials is still missing. Among salt hydrate PCMs, magnesium dichloride hexahydrate, MgCl2·6H2O (also known as the mineral bischofite), is envisioned as a promising candidate for low-temperature energy storage applications.1,2,4−7 This PCM possesses a relatively high phase change enthalpy and heat capacity combined with a high thermal conductivity. Supercooling was observed in this material, with a phase change temperature Tpc ≃ 390 K (117 °C)4,7−11 and a crystallization temperature in the range Tcr = 353−370 K (90−97 °C).7,9 The phase change enthalpy of MgCl2·6H2O was estimated4,7,8,10,11 to be in range ΔpcH = 165−172 J g−1, and its thermal conductivity was measured to be 0.694 and 0.570 W m−1 K−1 at 363 K (90 °C) and 393 K (120 °C), respectively.4 MgCl2·6H2O thermally decomposes according to the successive reactions12

In the quest for renewable energy technologies to reduce CO2 emission produced from the combustion of fossil fuels, technologies developed for harvesting solar energy have emerged over the years as promising alternatives. However, the intermittent nature of solar energy remains a challenge and a major hindrance to its efficient and widespread utilization. Therefore, storage of excess solar energy appears crucial for bridging the gap between energy generation and consumption. The storage of thermal energy as latent heat in phase change materials (PCMs) represents an attractive option for low- and medium-temperature solar energy applications.1−3 The high energy storage density of PCMs compared to sensible heat energy storage systems is particularly beneficial for applications requiring lightweight and compact energy storage materials. Latent heat storage in PCMs is achieved by storing phase change heat at a nearly constant temperature. The availability of various phase change temperatures in the numerous PCMs developed for the 0−100 °C range is also particularly advantageous for different applications compared to sensible heat storage materials. PCMs are usually classified into three main categories: organics (e.g., paraffins, or fatty acids or esters, alcohols, glycols), inorganics (e.g., salt hydrates or metallic), and eutectics (e.g., mixtures of inorganic and/or organic). Comprehensive reviews of the characteristics, performance, and limitations of these groups of PCMs were reported in the literature.1−4 Although numerous studies of PCMs have been © 2014 American Chemical Society

MgCl2·6H 2O → MgCl2· 4H 2O + 2H 2O

(1)

MgCl2·4H 2O → MgCl2· 2H 2O + 2H 2O

(2)

MgCl2·2H 2O → MgCl2·H 2O + H 2O

(3)

where Tcr ≃ 107 and 153 °C for MgCl2·4H2O and MgCl2· 2H2O, respectively. MgCl2·H2O can further decompose into Received: November 21, 2013 Revised: January 31, 2014 Published: February 14, 2014 4618

dx.doi.org/10.1021/jp411461m | J. Phys. Chem. C 2014, 118, 4618−4625

The Journal of Physical Chemistry C

Article

Figure 1. Crystal structures of (a) MgCl2·6H2O (space group C2/m; Z = 2), (b) MgCl2·4H2O (space group Pbcn; Z = 4), (c) MgCl2·2H2O (space group C2/m; Z = 2), and (d) MgCl2·H2O (space group Pnma; Z = 4), optimized at the DFT/PBE level of theory. Color legend: Mg, orange; O, red; Cl, green; H, white.

Mg(OH)Cl·nH2O, HCl(g), and H2O at 193 °C, although this reaction is not considered of practical interest for storage applications due the detrimental effects of HCl gas. While the values of the phase change temperature and enthalpy of MgCl2·6H2O are in good agreement in the literature, some of the crucial thermodynamic data for this material show large discrepancies. For example, the isobaric molar heat capacity value reported by Yaws7,13 is 243.31 J mol−1 K−1 at 298.15 K and atmospheric pressure, significantly lower than the values of 315.02 and 337.03 J mol−1 K−1 measured at the same temperature by Kelley14 and Pilar et al.,7 respectively. Such large differences were attributed to the varying amount of crystallographic water in the MgCl2·6H2O samples characterized by calorimetry.7 In addition, thermodynamic data for MgCl2·nH2O (n = 4, 2, 1) are scarce and the experimental heat capacities reported for these compounds consist merely of linear extrapolations as functions of the temperature.15,16 In this study, the structural, electronic, and thermodynamic properties of MgCl2·nH2O (n = 6, 4, 2, 1) were computed within the framework of density functional theory (DFT). Densities of states were predicted and phonon analysis using density functional perturbation theory (DFPT) was performed at equilibrium volume in order to derive isochoric thermal properties (i.e., Helmholtz free energy, entropy, and the isochoric heat capacity). Isobaric thermal properties (i.e., Gibbs free energy and isobaric heat capacity) were also calculated within the quasi-harmonic approximation (QHA) and compared to experimental data. Details of our computational approach are given in the next section, followed by a discussion and analysis of our results. A summary of our findings and conclusions is given in the last section.

2. COMPUTATIONAL METHODS Structural Calculations. First-principles total energy calculations were performed using the spin-polarized density functional theory (DFT), as implemented in the Vienna ab initio simulation package17 (VASP). The exchange-correlation energy was calculated using the generalized gradient approximation18 (GGA) with the parametrization of Perdew, Burke, and Ernzerhof19 (PBE). The interaction between valence electrons and ionic cores was described by the projector augmented wave (PAW) method.20,21 The Cl(3s2,3p5), Mg(2p6,3s2), and O(2s2,2p4) electrons were treated explicitly as valence electrons in the Kohn−Sham (KS) equations, and the remaining core electrons together with the nuclei were represented by PAW pseudopotentials. The KS equation was solved using the blocked Davidson22 iterative matrix diagonalization scheme. The plane-wave cutoff energy for the electronic wave functions was set to 500 eV, ensuring the total energy of the system converged to within 1 meV/atom. All structures were optimized with periodic boundary conditions applied. Ionic relaxation was carried out using the quasi-Newton method, and the Hellmann−Feynman forces acting on atoms were calculated with a convergence tolerance set to 0.01 eV/Å. Structural optimizations and properties calculations were carried out using the Monkhorst−Pack special k-point scheme.23 The following k-point meshes were used for integrations in the Brillouin zone (BZ) of bulk systems: 3 × 3 × 3 for MgCl2·6H2O and MgCl2·4H2O, 3 × 3 × 5 for MgCl2· 2H2O, and 3 × 5 × 3 for MgCl2·H2O. The tetrahedron method with Blöchl corrections24 was used for BZ integrations. Ionic and cell relaxations of the bulk structures were performed simultaneously without symmetry constraints. Periodic unit cells were used in the calculations, with 42 atoms for MgCl2·6H2O (Z = 2), 60 atoms for MgCl2·4H2O (Z 4619



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= 4), 18 atoms for MgCl2·2H2O (Z = 2), and 24 atoms for MgCl2·H2O (Z = 4). Thermodynamic Calculations. The equilibrium structures of MgCl2·nH2O (n = 6, 4, 2, 1) optimized using the VASP software package were used to build a set of supercells. The forces exerted on atoms of the equilibrium structures were calculated using density functional perturbation theory25 (DFPT) with VASP at the GGA/PBE level of theory, and phonon frequencies were computed. Phonon analysis was performed at constant equilibrium volume to derive isochoric thermal properties (e.g., the Helmholtz free energy, entropy, and isochoric heat capacity). The Helmholtz free energy was calculated using the following formula: F=

1 2

∑ ℏω + kBT ∑ ln[1 − e−βℏω]

Table 1. Crystallographic Data of MgCl2·nH2O (n = 6, 4, 2, 1) Calculated at the GGA/PBE Level of Theory (Experimental Data Are Given in Parentheses) formula

MgCl2·6H2O

MgCl2·4H2O

MgCl2·2H2O

MgCl2·H2O

space group

C2/m

Pbcn

C2/m

Pnma

Z

2

4

2

4

9.91 (9.8607)a 7.16 (7.1071)a 6.12 (6.0737)a 90 (90)a 94.25 (93.758)a 90 (90)a 433.18 (424.736)a

7.38 (7.2557)b 8.42 (8.4285)b 11.00 (11.0412)b 90 (90)b 90 (90)b 90 (90)b 683.63 (675.226)b

7.90 (7.4279)c 8.50 (8.5736)c 3.70 (3.6506)c 90 (90)c 104.0 (98.580)c 90 (90)c 241.06 (229.886)c

8.98 (8.9171)c 3.69 (3.6342)c 11.58 (11.4775)c 90 (90)c 90 (90)c 90 (90)c 383.65 (371.954)c

a (Å) b (Å) c (Å) α (deg)

(4)

β (deg)

where ℏ is the reduced Planck constant, ℏω the energy of a single phonon with angular frequency ω, kB the Boltzmann constant, T the temperature of the system, and β = (kBT)−1. The entropy was computed using the expression S = −kB ∑ ln[1 − e−βℏω] −

1 T

∑

ℏω −1 e β ℏω

γ (deg) V (Å3) a

(5)

c

The isochoric molar heat capacity was calculated using the formula CV =

∑ kB(βℏω)2

e βℏω [e βℏω − 1]2

Mg−OH2 bond distances are in the range of 2.07 and 2.09 Å, and the H···Cl hydrogen bonds vary between 2.18 and 2.51 Å. The Mg atoms are positioned on 2a Wyckoff sites (2/m symmetry), the Cl atoms occupy 4i sites (m symmetry); the O and H atoms are on both 4i sites (m symmetry) and 8j sites (1 symmetry). The optimized structure slightly overestimates the lattice parameters of the structure characterized by Agron and Busing27 using neutron diffraction: a = 9.8607 Å, b = 7.1071 Å, c = 6.0737 Å; α = γ = 90°, β = 93.758° (V = 424.736 Å3; b/a = 0.7207, c/a = 0.6159). However, the computed b/a and c/a ratios are in excellent agreement with experiment. The computed equilibrium volume is 2.0% larger than the experimental estimate because of the fact that GGA calculations tend to overestimate the bond distances29 and that standard DFT cannot account accurately for long-range intermolecular forces.30 It can be inferred that some of the residual discrepancies between the computed and measured crystal structures might be reduced by using recently developed methods to correct DFT for dispersion.31 The Mg−OH2 bond distances determined experimentally are 2.057 and 2.062 Å, and the experimental H···Cl hydrogen bonds vary between 2.206 and 2.499 Å. MgCl2·4H2O belongs to the orthorhombic space group Pbcn (IT No. 60), with four formula units (Z = 4) per unit cell, and its computed lattice parameters are as follows: a = 7.38 Å, b = 8.42 Å, c = 11.00 Å; α = β = γ = 90° (V = 683.63 Å3; b/a = 1.14, c/a = 1.49). Each Mg2+ ion is coordinated by four water molecules in a square-planar arrangement, with two neighboring Cl− ions trans-positioned 2.59 Å from the Mg2+ ion with a mirror symmetry relative to this plane. The computed Mg− OH2 bond distances are in the range of 2.07−2.08 Å, and the H···Cl hydrogen bonds vary between 2.19 and 2.22 Å. The Mg atoms are positioned on 4c Wyckoff sites (0.2. symmetry), Cl atoms occupy 8d sites (1 symmetry), O atoms are on both 4c sites (0.2. symmetry) and 8d sites (1 symmetry), and all the H atoms occupy 8d sites (1 symmetry). The computed equilibrium volume is about 1.2% larger than the volume of the experimental structure characterized by Schmidt et al.28 using X-ray diffraction (XRD) with lattice parameters as

(6)

Further analysis from a set of phonon calculations in the vicinity of each computed equilibrium crystal structure was carried out to obtain thermal properties at constant pressure (e.g., the Gibbs free energy and isobaric heat capacity) within a quasi-harmonic approximation. The QHA mentioned here introduces a volume dependence of phonon frequencies as a part of the anharmonic effect. The Gibbs free energy is defined at a constant pressure by the transformation26 G(T , P) = min[U (V ) + F(T ; V ) + PV ] V

(7)

where minV[function of V] corresponds to a unique minimum of the expression between brackets with respect to the volume V; U is the total energy of the system, and P is the pressure applied. The isobaric heat capacity versus temperature was also derived as the second derivative of the Gibbs free energy with respect to T, i.e. CP = −T

∂ 2G ∂T 2

Agron and Busing, 1985.27 bSchmidt, Hennings, and Voigt, 2012.28 Sugimoto, Dinnebier, and Hanson, 2007.12

(8)

3. RESULTS AND DISCUSSION Structural Properties. The crystal unit cells of MgCl2· nH2O (n = 6, 4, 2, 1) optimized at the GGA/PBE level of theory are shown in Figure 1, and their computed and measured crystallographic parameters are summarized in Table 1. MgCl2·6H2O crystallizes in the monoclinic space group C2/ m (IT No. 12) with two formula units (Z = 2) per unit cell, and the computed lattice parameters are as follows: a = 9.91 Å, b = 7.16 Å, c = 6.12 Å; α = γ = 90°, β = 94.25° (V = 433.18 Å3; b/a = 0.72, c/a = 0.62). Each Mg2+ ion is octahedrally coordinated by six water molecules, with two neighboring Cl− ions per formula unit to compensate the charge of Mg2+. The computed 4620



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follows: a = 7.2557(3) Å, b = 8.4285(4) Å, c = 11.0412(6) Å; α =β = γ = 90° (V = 675.22(6) Å3; b/a = 1.1616, c/a = 1.5217). The computed b/a and c/a ratios are in good agreement with experiment. The Mg−OH2 bond distances determined experimentally are 2.052 and 2.058 Å, the Mg−Cl distance is 2.551 and the experimental H···Cl hydrogen bonds vary between 2.345 and 2.431 Å. MgCl2·2H2O adopts the monoclinic space group C2/m (IT No. 12), with two formula units (Z = 2) per unit cell and computed lattice parameters as follows: a = 7.90 Å, b = 8.50 Å, c = 3.70 Å; α = γ = 90°, β = 104.0° (V = 241.06 Å3; b/a = 1.07, c/a = 0.47). Each Mg2+ ion is coordinated by two water molecules along the b axis, with two neighboring Cl− ions bridging adjacent Mg2+ ions and forming polymeric chains propagating along the c axis, with Mg−Cl bond distances of 2.52 and 2.58 Å. The computed Mg−OH2 bond distance is 2.05 Å, and the H···Cl hydrogen bond distance between adjacent chains is 2.20 Å. The Mg atoms are positioned on 2a Wyckoff sites (2/m symmetry), and the Cl atoms occupy 4i sites (m symmetry); the O atoms are on 4g sites (2 symmetry), and the H atoms occupy 8j sites (1 symmetry). The calculated structure compares well with the structure characterized by Sugimoto et al.12 using XRD: a = 7.4279(4) Å, b = 8.5736(4) Å, c = 3.65065(16) Å; α = γ = 90°, β = 98.580(2)° (V = 229.886 Å3; b/a = 1.1542, c/a = 0.4915). The computed equilibrium volume is 4.6% larger than the experimental estimate. The Mg−OH2 bond distance determined experimentally is 2.008 Å, with Mg−Cl bond distances of 2.489 and 2.559 Å, and the experimental H−Cl hydrogen bonds vary between 2.447 and 3.072 Å. The orientations of the water molecules differ in observed and computed structures, thus explaining in part the larger predicted unit-cell volume compared with experiment. MgCl2·H2O crystallizes in the orthorhombic space group Pnma (IT No. 62), with four formula units (Z = 4) per unit cell, and its computed lattice parameters are as follows: a = 8.98 Å, b = 3.69 Å, c = 11.58 Å; α = β = γ = 90° (V = 383.65 Å3; b/a = 1.07, c/a = 0.47). Each Mg2+ ion is octahedrally coordinated by one water molecule at 2.02 Å and Cl− ions with Mg−Cl bond distances of 2.52, 2.53, and 2.58 Å; the structure consists of polymeric chains propagating along the b axis. The computed H···Cl hydrogen bond distance between adjacent chains is 2.29 Å. All the atoms in the structure are positioned on 4c Wyckoff sites (.m. symmetry). Close agreement is observed with the lattice parameters of the structure characterized by Sugimoto et al.12 using XRD: a = 8.9171(6) Å, b = 3.63421(18) Å, c = 11.4775(7) Å; α = β = γ = 90° (V = 371.95(4) Å3; b/a = 0.4075, c/a = 1.2871). The computed equilibrium volume is 3.1% larger than the experimental estimate. The Mg−OH2 bond distance determined experimentally is 1.918 Å, with Mg− Cl bond distances of 2.466, 2.551, and 2.560 Å, and the experimental H···Cl hydrogen bond between adjacent chains is measured to be 2.374 Å, i.e., appreciably longer than that in the relaxed structure. Electronic Properties. The total and partial densities of states (DOSs) of the equilibrium structures of MgCl2·nH2O (n = 6, 4, 2, 1) are displayed in Figure 2. The DOSs exhibit characteristic features of molecular solids, with very localized peaks, contrasting with the wide bands often seen in other solids with a larger degree of electron hybridization. In MgCl2·6H2O, the energy band gap separating the top of the valence band and the bottom of the conduction band is 5.10 eV, indicative of an insulator. The main contributions to

Figure 2. Total and partial densities of states (DOSs) of (a) MgCl2· 6H2O, (b) MgCl2·4H2O, (c) MgCl2·2H2O, and (d) MgCl2·H2O, per unit cell, computed at the DFT/PBE level of theory.

the total DOS near the Fermi level (i.e., EF = 0 eV) originate from the Cl(3p), O(2p), and H(1s) orbitals. The Cl(p) electrons contribute predominantly in the range −1.3 to +0.1 eV, the O(p) electrons in the ranges −1.8 to +0.1 eV, −4.2 to −3.1 eV, and −7.3 to −6.4 eV, and the H(s) electrons in the range −7.3 to −6.4 eV. The energy band gap in the vicinity of the Fermi level (EF = 0 eV) of MgCl2·4H2O is 5.17 eV, also indicative of an insulator. Similar to MgCl2·6H2O, the main contributions to the total DOS near EF originate from the Cl(3p), O(2p), and H(1s) orbitals. A strong hybridization between the Cl(3p) and O(2p) electrons is seen between −2.0 and +0.1 eV, while O(2p)− H(1s) hybridization occurs for lower-lying bands, below −3.2 eV and above −6.8 eV. Further widening of the band gap occurs with increasing dehydration level, as shown by the gap value of 5.39 eV for MgCl2·2H2O. As in the previous cases, the Cl(3p), O(2p), and H(1s) orbitals in this structure dominate the top of the valence band. The computed DOS is reminiscent of that of MgCl2· nH2O (n = 4, 6), with the Cl(3p) electrons featuring importantly between −2.3 and +0.2 eV, with some O(2p) hybridization in this range, while O(2p) are predominant in the narrower ranges −4.8 to −3.7 eV and −7.0 to −6.4. Strong O(p)−H(s) hybridization is predicted in the latter range. MgCl2·H2O is also an insulator, with a band gap of 5.76 eV. Similar to the compounds discussed above, the Cl(3p), O(2p), and H(1s) orbitals are predominant near the top of the valence band. The Cl(p) electrons contribute predominantly in the range −2.6 to +0.0 eV, with some degree of hybridization with 4621



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Figure 3. Thermal properties of (a) MgCl2·6H2O, (b) MgCl2·4H2O, (c) MgCl2·2H2O, and (d) MgCl2·H2O, per formula unit, calculated at constant equilibrium volume at the DFT/PBE level of theory. Experimental data for the entropy are from the FactSage FACTPS database.33

O(p) electrons, while the ranges −4.6 to −3.9 eV and −7.2 to −6.7 eV are characterized essentially by O(p) electrons, with some mixing with H(s) electrons in the latter range. Thermodynamic Properties. The thermal properties of MgCl2·nH2O (n = 6, 4, 2, 1), predicted using their phonon frequencies calculated with DFPT at the GGA/PBE level of theory, are reported in Figures 3 −5. Results for their thermal properties at constant equilibrium volume (i.e., the total energy

U, Helmholtz free energy F, entropy S, and isochoric heat capacity CV) and at constant, standard atmospheric pressure (i.e., the Gibbs free energy G and the isobaric heat capacity CP) are displayed. Limited thermal experimental data exist for these compounds. In Figure 3, the experimental values of the entropy from the FactSage FACTPS database33 are displayed for the sake of comparison; these reference data are based on the early calorimetric measurements of Kelley and Moore.14,32 The entropy predicted from first-principles is in very good agreement with experimental data for MgCl2·nH2O (n = 4, 2, 1): the computed entropy underestimates experimental data by ≃5% and ≃3% for MgCl2·4H2O and MgCl2·2H2O, respectively, and overestimates experiment by ≃2% for MgCl2·H2O. However, for MgCl2·6H2O, the predicted entropy is 14−17% lower than calorimetric data. As shown in Figure 3, the entropy increases steadily with temperature and decreases, for a fixed temperature, as the molar volume contracts with successive dehydration reactions from MgCl2·6H2O to MgCl2·H2O. This is consistent with the fact that a temperature increase (a molar volume decrease and the loss of water molecules) results in a larger (smaller) number of microstates in the system, W, which in turn increases (decreases) logarithmically the entropy according to Boltzmann’s entropy formula, S = kB log W, where kB is the Boltzmann constant. The monotonically decreasing Gibbs free energy curves and their derived isobaric molar heat capacities CP at atmospheric pressure for MgCl2·nH2O (n = 6, 4, 2, 1) are shown as functions of the temperature in Figure 4 and 5, respectively. For MgCl2·6H2O, large discrepancies are observed among the experimental estimates of CP(T) (cf. Figure 5a). For example, the isobaric molar heat capacity value reported by Yaws7,13 is 243.31 J mol−1 K−1 at 298.15 K and atmospheric pressure, significantly lower than the values of 315.02 and 337.03 J mol−1 K−1 measured at the same temperature by

Figure 4. Gibbs free energy of (a) MgCl2·6H2O, (b) MgCl2·4H2O, (c) MgCl2·2H2O, and (d) MgCl2·H2O, per formula unit, calculated at standard atmospheric pressure at the DFT/PBE level of theory. 4622



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Figure 5. Molar heat capacity of (a) MgCl2·6H2O, (b) MgCl2·4H2O, (c) MgCl2·2H2O, and (d) MgCl2·H2O, per formula unit, calculated at standard atmospheric pressure at the DFT/PBE level of theory.

Table 2. Coefficients of the Haas−Fisher Heat Capacity Polynomial CP(T) for MgCl2·nH2O (n = 6, 4, 2, 1)a

a

compound

a × 102 (T0)

b × 10−2 (T1)

c × 105 (T−2)

d × 103 (T−0.5)

e × 10−5 (T2)

SSDb

MgCl2·6H2O MgCl2·4H2O MgCl2·2H2O MgCl2·H2O

5.7643 −2.3567 2.0589 1.5498

−8.674 52.839 1.284 −0.858

4.521784 −72.422160 −5.574830 −3.357095

−5.066334 6.983317 −1.118995 −0.704519

5.97 −22.90 −0.21 0.71

0.004 0.006 0.124 0.001

The range of validity of the fits is 290−500 K. bSum of squared differences between calculated and fitted data.

Kelley14 and Pilar et al.,7 respectively. Such large differences were tentatively ascribed to the varying amount of crystallographic water in the MgCl2·6H2O samples characterized by calorimetry.7 The calorimetric data reported by Kelley and Moore32 between 54.1 and 295.8 K are also displayed in Figure 5. In the 298−385 K range, a simplified Maier−Kelley polynomial fit, CP = a + bT + cT−2 with c = 0, was used by Wagman et al.15 based on the measurements of Kelley,14 and Pilar et al.7 recently reported Maier−Kelley fits to their experimental data in the 298.15−387.15 K range with a declared uncertainty of ≃3%. As shown in Figure 5, the computed CP systematically underestimates the calorimetric measurements of Kelley and Moore and Pilar et al., while overestimating by ≃9% the value reported by Yaws at 298.15 K. Surprisingly, the agreement between computed and empirical CP values of Kelley improves as the temperature ramps up from 298 to 385 K, although one would expect a slight deterioration of this agreement due to the breakdown of the QHA. Large discrepancies are observed between the recent data of Pilar et al. and both the experimental data of Kelley14 and the calculations in the vicinity of 390 K, corresponding to the zone of phase transition between MgCl2·6H2O and MgCl2· 4H2O. The experimental isobaric molar heat capacity data for MgCl2·nH2O (n = 4, 2, 1) from Wagman et al.,15 with thermal extrapolation by Pabalan and Pitzer,16 are depicted along with the present predictions in Figures 5b to 5d. Similar to MgCl2· 6H2O discussed above, the experimental estimates of CP for those compounds consist merely of linear extrapolations as

functions of the temperature of data collected near room temperature.15,16 The ranges of validity claimed for those linear fits are 298−450 K for MgCl2·4H2O, 298−500 K for MgCl2· 2H2O, and 298−650 K for MgCl2·H2O. Within those temperature ranges, the heat capacity values computed in this study systematically underestimate experimental estimates by ≃6−7% for MgCl2·4H2O, ≃11−13% for MgCl2·2H2O, and ≃5−8% for MgCl2·H2O. Further experimental work is needed to improve upon those existing linear fits to the isobaric heat capacities. Toward the higher end of the their thermal stability range, the computed heat capacity values for these compounds are still well below their respective Dulong−Petit asymptotic value, i.e., CP = 3NR, where N is the number of atoms per formula unit and R = 8.314426 J K−1 mol−1 is the Regnault molar gas constant. The calculated Dulong-Petit limits are 523.8 J K−1 mol−1 for MgCl2·6H2O, 374.1 J K−1 mol−1 for MgCl2·4H2O, 224.5 J K−1 mol−1 for MgCl2·2H2O, and 149.7 J K−1 mol−1 for MgCl2·H2O. The thermal evolutions of the isobaric heat capacity calculated from first-principles for MgCl2·nH2O (n = 6, 4, 2, 1) were fitted over the 290−500 K range using a nonlinear least-squares regression to a Haas−Fisher type polynomial, i.e. C P(T ) = a + bT + cT −2 + dT −0.5 + eT 2

(9)

The resulting optimized coefficients are given in Table 2. The amount of molar latent heat stored when the material is heated, at constant atmospheric pressure, from an initial 4623



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ical properties of the MgCl2·nH2O (n = 4, 2, 1) dehydrated phases in order to derive consistent and reliable values of their phase change temperatures and enthalpies and latent heat storage capacities.

temperature T0 to its phase change temperature Tpc can be calculated as7 QP =

∫T

Tpc

0

C P(T )dT + ΔpcH

■

(10)

where ΔpcH is the molar phase change enthalpy at the phase change temperature Tpc and CP is the isobaric molar heat capacity of the solid phase. The recent experimental value of phase change enthalpy obtained by Pilar and co-workers7 using differential scanning calorimetry, i.e., ΔpcH = 34.6 ± 0.5 kJ mol−1 (170.1 ± 2.5 J g−1) at Tpc = 390.3 ± 0.2 K, was used in this study to calculate the amount of heat accumulated in MgCl2·6H2O from T0 = 298.15 K to Tpc. This value is in line with previous estimates4,7,8,10,11 in the range ΔpcH = 33.5−35.0 kJ mol−1 (165−172 J g−1). The Haas−Fisher polynomial fit to the computed CP(T) was also used in eq 10. The molar latent heat obtained from firstprinciples calculations is QP = 60.8 kJ mol−1, i.e., sligthly underestimating the experimental value of 68.2 kJ mol−1 (335 J g−1).7 A similar assessment of the amount of molar latent heat stored when MgCl2·nH2O (n = 4, 2, 1) crystals are heated to their respective phase change temperature can be made based on first-principles thermodynamic calculations. However, to the best of our knowledge, no consistent and reliable values of the phase change temperatures and enthalpies exist for these materials. Additional calorimetric studies and molecular dynamics simulations are needed to better characterize the thermochemical properties of the MgCl2·nH2O (n = 4, 2, 1) dehydrated phases.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: 505-844-8144. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

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REFERENCES

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4. CONCLUSION Using first-principles approaches, a comprehensive study of the structural, electronic, and thermodynamic properties of MgCl2· 6H2O and its dehydrated phases, MgCl2·nH2O (n = 4, 2, 1), was carried out. The calculated densities of states show that the band gap near the Fermi level increases steadily from 5.10 eV for MgCl2· 6H2O up to 5.76 eV for MgCl2·H2O. The computed entropy underestimates experimental data by ≃5% and ≃3% for MgCl2·4H2O and MgCl2·2H2O, respectively, and overestimates experiment by ≃2% for MgCl2·H2O. However, for MgCl2·6H2O, the predicted entropy is 14−17% lower than calorimetric data. Large discrepancies also exist among calorimetric data for the isobaric molar heat capacity of MgCl2·6H2O. For standard temperature and pressure conditions, the value predicted from calculations overestimates the value of 243.31 J mol−1 K−1 reported by Yaws and co-workers, while significantly underestimating the values of 315.02 and 337.03 J mol−1 K−1 measured by Kelley and Pilar et al., respectively. The heat capacities computed in this study also systematically underestimate experimental data by ≃6−7% for MgCl2·4H2O, ≃11−13% for MgCl2·2H2O, and ≃5−8% for MgCl2·H2O, although experimental data are scarce for these phases and further experimental work is needed to improve upon existing linear fits. The amount of heat accumulated in MgCl2·6H2O from 298.15 K to its phase change temperature of Tpc = 390.3 K was predicted from first-principles to be QP = 60.8 kJ mol−1, i.e., sligthly below the experimental value of 68.2 kJ mol−1. Additional calorimetric studies and molecular dynamics simulations are needed to better characterize the thermochem4624



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Solar Energy Storage in Phase Change Materials: First-Principles

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