Thermal conductivity of halide solid solutions: measurement and prediction.

Thermal conductivity of halide solid solutions: Measurement and prediction Aïmen E. Gheribi, Sándor Poncsák, Rémi St-Pierre, László I. Kiss, and Patrice Chartrand Citation: The Journal of Chemical Physics 141, 104508 (2014); doi: 10.1063/1.4893980 View online: http://dx.doi.org/10.1063/1.4893980 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Thermal diffusivity of alkali and silver halide crystals as a function of temperature J. Appl. Phys. 109, 033516 (2011); 10.1063/1.3544444 Thermal conductivity of molten alkali halides: Temperature and density dependence J. Chem. Phys. 130, 044505 (2009); 10.1063/1.3064588 Measurement of thermal diffusivity at high pressure using a transient heating technique Appl. Phys. Lett. 91, 181914 (2007); 10.1063/1.2799243 Thermal conductivity of molten alkali halides from equilibrium molecular dynamics simulations J. Chem. Phys. 120, 8676 (2004); 10.1063/1.1691735 Thermal conductivity in nickel solid solutions J. Appl. Phys. 81, 2263 (1997); 10.1063/1.364254

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THE JOURNAL OF CHEMICAL PHYSICS 141, 104508 (2014)

Thermal conductivity of halide solid solutions: Measurement and prediction Aïmen E. Gheribi,1 Sándor Poncsák,2 Rémi St-Pierre,2 László I. Kiss,2 and Patrice Chartrand1 1 Center for Research in Computational Thermochemistry, Chem. Eng., École Polytechnique, Montréal, Quebec H3C 3A7, Canada 2 Department of Applied Sciences, Université du Québec à Chicoutimi, Quebec G7H 2B1, Canada

(Received 19 May 2014; accepted 13 August 2014; published online 11 September 2014) The composition dependence of the lattice thermal conductivity in NaCl-KCl solid solutions has been measured as a function of composition and temperature. Samples with systematically varied compositions were prepared and the laser flash technique was used to determine the thermal diffusivity from 373 K to 823 K. A theoretical model, based on the Debye approximation of phonon density of state (which contains no adjustable parameters) was used to predict the thermal conductivity of both stoichiometric compounds and fully disordered solid solutions. The predictions obtained with the model agree very well with our measurement. A general method for predicting the thermal conductivity of different halide systems is discussed. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4893980] I. INTRODUCTION

The ability to predict thermal-transport properties as a function of temperature, stress and chemical composition of ionically bonded materials, such as halides, is often highly desirable in many industrial applications. Need is encountered in thermal barrier coating, glass manufacture, solar energy plant, nuclear fuel, or novel thermoelectric applications. Recently, interest in renewable energies has heightened due to environmental concerns and the increasing price of fossil fuel energy. Latent heat energy storage technology using a Phase Change Material (PCM) is one of the most important components of a solar energy installation. The choice of the material and the heat transfer mechanism are critical design considerations for optimum performance. Usually one wants to maximize the thermal conductivity of the material in order to provide minimum temperature gradients and thus facilitate the charge and discharge of heat. Salts1 are widely used as PCMs although they have low thermal conductivity. Salt mixtures exhibit lower thermal conductivities compared to the pure compounds. While a considerable number of theoretical and experimental work has been reported on thermal conductivity of pure salts, much less attention has been paid to solid solutions. Almost no experimental data of thermal conductivity or thermal diffusivity of salt solid solutions have been reported in the literature. In fact, there is only one set of experimental data of the thermal conductivity of solid salt solutions in the entire range of composition reported in the literature. This is for the KCl-KBr system, but only at very low temperature (maximum 15 K).2 Also, another data set has been reported, for the KCl-RbCl system, but for a single composition only (0.95KCl-0.05RbCl, up to 393 K3 ). The lack of experimental data is a significant barrier to quantitative understanding of the thermal transport properties of salt mixtures. Thus, it is clear that the theoretical prediction of thermal conductivity of the solid salt solutions is highly desirable from both industrial and fundamental research points of view. 0021-9606/2014/141(10)/104508/12/$30.00

The present work aims first to determine experimentally the thermal conductivity of a prototype solid salt solution as a function of composition and temperature. The prototype system chosen in this work is the NaCl-KCl pseudobinary. Given the apparent lack of experimental data, we also aim to propose a reliable theoretical method to predict the thermal conductivity of halide solid solutions by a model which requires no adjustable parameters linked to thermal transport properties. To achieve this, we have measured the thermal diffusivity of NaCl-KCl solid solutions from 373 K to 823 K over the complete composition range, in steps of 20 mol.%. The theoretical estimation method is based on the Debye approximation of phonon density of state. This is used to describe the phonon scattering process. For an electrical insulating material, the phonon mean free path is, overall, the key physical quantity to for analyses of the thermal transport properties and the underlying physics. The chemical effect upon the phonons mean free path will be calculated and analyzed in term of the fluctuation of the mass and the elastic strain field induced by the alloying effect. The rest of the paper is organized as follows. In Sec. II we give experimental details on sample preparation and measurement of the thermal diffusivity by the laser flash method. In Sec. III we present the formalism for prediction of the thermal conductivity of stoichiometric compound and solid solutions as a function of temperature. In Sec. IV we display and discuss our experimental results and theoretical predictions. The possibility for generalizing to all halide systems is also examined. Concluding remark and perspectives are given in Sec. V. II. EXPERIMENTAL DETAILS

The thermal conductivity (k) of the NaCl-KCl system was deduced from the thermal diffusivity (a), heat capacity, and density (ρ) data. Details of the experimental procedure are given below.

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A. Sample preparation

Analytical grade NaCl and KCl (both Anachemia) were milled and mixed in six different proportions: from 100% NaCl, to 100% KCl, in 20 mol.% steps. Disc shaped samples were prepared 3.2 cm diameter and 3 mm thickness. Such discs are necessary for thermal diffusivity measurements with the FlashLineTM 3000 Thermal Diffusivity System (Anter Corp). Compact (non-porous) round discs were prepared in a press (Stuers LaboPress 3), applying an auto-adjusted compression force of 50 kN at 453 K for 9 min. The discs were then gradually cooled to room temperature in six min, avoiding crack formation. The samples produced with this press had a diameter of 4.6 cm. This was reduced to the desired size of 3.2 cm by a sanding belt. To eliminate moisture and improve the homogeneity of the samples, they were heated to 7730 K in a stainless steel crucible, under argon atmosphere in a homemade furnace for 48 h. Both surfaces of the samples were painted with a high temperature resistant graphite-based dye in order to prevent light transmission and to increase energy absorption. They were then stored in sample holders containing desiccants in order to prevent exposer to water.

B. Thermal diffusivity measurement using the laser flash method

The laser flash method was used to determine thermal conductivity and diffusivity of small, relatively homogenous solid samples. The test consists of a short illumination (energy pulse) upon a disc-shaped sample on one side. The temperature at the opposite side was monitored. The FlashLineTM 3000 Thermal Diffusivity System uses a Xenon lamp to produce short pulses, while the rear surface temperature is detected and recorded with a liquid nitrogen cooled radiation thermometer. In this method there is direct contact with the sample, in order to minimize thermal perturbations. Fig. 1 shows a typical thermal history curve that can be measured on the rear face of a sample during the flash test. The decreasing part of the curve is due to the subsequent heat losses. The thermal diffusivity (a) can be determined from the thickness of the sample (L) and a characteristic time parameter of the temperature history curve (tc ), namely a = cons tan t.

L2 . tc

(1)

Generally, the time necessary to reach half of the maximum temperature rise on the rear face of the sample is used as characteristic time parameter. In this study, the Clark-Taylor method4 was used to evaluate the temperature history curves including the effect of heat losses. Usually, the thickness of the samples varied between 2 and 4 mm, depending on the estimated thermal diffusivity range of the material studied. The surfaces were painted with a well-absorbing layer, in order to maximise and smooth out the light absorption. The thermal conductivity of the can then be computed from the mass lattice heat capacity at constant pressure, Cpml , the density, ρ and a values of the sample using the relation k = aρCP m,l .

(2)

The NaCl-KCl equilibrium (incoherent) phase diagram is shown in Fig. 2, according to the thermodynamic optimization of Sangster and Pelton.5 It displays a minimum liquidus temperature at 927 K and a large miscibility gap governed by an incoherent phase separation below the consolute temperature of 777 ± 20 K. It was necessary to keep the samples at least 100 K below the minimum liquidus temperature, i.e., ∼830 K. This temperature is the safest maximum temperature which can be applied in the flash apparatus without damaging it by the formation of liquid. No measurement was carried out below 373 K, as earlier tests6 showed that The FlashLineTM 3000 Thermal Diffusivity System is less precise at low temperatures. III. THERMODYNAMICALLY CONSISTENT MODELLING OF THE THERMAL CONDUCTIVITY OF A PURE COMPOUND AND SOLID SOLUTION

In this section we detail briefly the theoretical models describing the thermal conductivity of pure compounds as a function of temperature as well as solid solutions as function of both temperature and composition. The models are already formulated in the literature. However, their parameterization method, using no experimental data related to transport properties, but only a few key experimental data points, is original. These key data points are easily measurable when samples are available, or can be estimated with a good accuracy using first principle calculations. Other studies found in the literature calculate the parameters of the model by fitting the temperature and the composition dependence thermal conductivity curve. A. Thermal conductivity of a compound as a function of temperature

FIG. 1. Principle of the flash method and typical measured temperature history on the rear surface of the sample.

The thermal conductivity in electrical insulating materials arises from the scattering of the phonons. At low temperatures it increases as T3 up to about 5% of Debye temperature (Θ D ). After this it decreases as T−1 or faster. Three-phonon scattering resulting from anharmonic Umklapp processes (U-processes) is the dominant scattering mechanism limiting the heat transfer at the “high temperatures,”7 typically greater than about Θ D /3. The most reliable theoretical approach to describe the thermal conductivity of insulating compounds in this region was proposed by Callaway7 and


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FIG. 2. The incoherent phase diagram of the NaCl-KCl system above 300 K. The two characteristic temperatures are the minimum liquidus temperature (927 K) and the incoherent consolute temperature (777 K).

later Holland.8 This approach is based on (i) the Boltzmann thermal equation,9 (ii) the Debye approximation of the phonon density of state, and (iii) a reliable expression single mode relaxation time (SMRT). For an isotropic material, and assuming the contributions of optical phonons to the thermal transport negligible, the lattice thermal conductivity can be formulated by solving the Boltzmann transport equation (BTE) using the Fourier law of conduction. In these conditions, the solution of BTE depends on temperature (T) and grain-size (d), and is expressed as follows:7, 8, 10 ω D τ (T , d, ω) 1 ω2 C V (T , ω) dω. (3) k (T , d) = 2 2π 0 υg (T , ω) Here ωD = kB θ D ¯−1 is the angular Debye frequency, υ g is the phonon group velocity in a quantum well averaged between the three acoustic branches, two transverse, and one longitudinal; kB is the Boltzmann constant, ¯ is the reduced Planck constant, τ (T, d, ω) is the combined phonon single mode scattering relaxation time, and C V is the heat capacity per atom at constant volume. The two different individual contributions to the combined phonon scattering relaxation time are: τ GB , that resulting from grain boundary scattering, and τ ph-ph , that resulting from phonon-phonon scattering. According to the Matthiensen rule (i.e., the additivity of the phonon scattering rate) τ is expressed as the sum −1 −1 (T , ω) + τph−ph (T , ω) . τ −1 (T , ω) = τGB

(4)

However, the phonon scattering induced by the grain boundaries can be neglected in this work. We point out that, strictly speaking, the description of the thermal conductivity of a polycrystalline material should also take into account the grain size distribution, the grain shape, and the grain orien-

tation. In the present situation the grain-size is large enough so that the microstructure effect upon the thermal conductivity can be neglected. According to first order perturbation theory, the U-processes SMRT can be expressed as11, 12 −1 (T , ω) = ω2 ψph−ph (T ) , τph−ph

(5)

where ψ ph−ph is a function which depends only on temperature. At high temperature the heat capacity and the phonon group velocity can be considered as independent of the phonon frequency. In these conditions, Slack11, 12 derived a reliable expression for the function . They showed that the integration of Eq. (4) can be written as a product of a “universal” constant, a characteristic constant which depends on material properties and a function describing the temperature dependence of the lattice thermal conductivity. The Slack expression can be written as11, 12 mθD∞ 0.59238 δ (T ) −θ ∞ / 3T e D . p γth∞ . · , (6) 2 ∞ 2 / 3 ¯ n γth T 3

k (T ) =

U niversal cons tan t

Material cons tan t

T emperature dependence term

where θD∞ and γth∞ are, respectively, the high temperature limit of the thermal Grüneisen parameter, and the Debye temperature m is the average of a single atom in the solid, n is the number of atoms per primitive unit cell (equal to 2 in the case of rock salts), p is a characteristic constant of the material defined by Julian13 as p = 2.593 × 10−3 · 2 (1 − 0.514/γth∞ + 0.228/γth∞ )−1 and expressed in unit of −2 −3 3 J s K . δ is the average volume per atom (ion) defined from the molar volume as δ 3 = Vm / (n0 NA ) (with n0 is the number of atom per molecule and NA the Avogadro number) the temperature dependence of δ is defined through the linear


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α (T )dT

thermal expansion, α l , as δ(T ) = δ0 e To l , where δ 0 is the value of δ at reference temperature T0 . For electrical insulating compounds, α l depends explicitly on θD∞ , the adiabatic bulk modulus, Bs , at 0 K and the cohesive energy, Ec.14–16 Consequently, the thermal conductivity of electrically insulating stoichiometric compounds can be, in principle, predicted from the key physical parameters θD∞ , γth∞ , Bs, Ec, and Vm (T0 ). B. Thermal conductivity of solid solutions as a function of temperature and composition

When forming solid salt solutions, the ions act as scattering centers for phonons which can cause a substantial decrease of the phonon mean free path and consequently a drastic lowering of the lattice thermal conductivity. This is due to differences in atomic mass, cationic size, and neighbor coupling forces between the solute and solvent cations. The chemical effect upon the SMRT was formulated originally by Klemens,10 based on physical considerations. Later, Abeles17 proposed a formalism which considers an effective radius difference term presumed to encompass the cationic size in the coupling force contributions. The effective radius difference term is, however, a phenomenological adjustable parameter.17–27 The aim of this work is to provide a methodology for the prediction of the thermal conductivity of solid solutions of halides to overcome the lack of data. Thus, one cannot consider any adjustable parameters in the description of the lattice thermal conductivity as a function of composition. The ratio of the thermal conductivity of a solid solution to that of the constitute materials was formulated by Abeles17 as arctan (u) . (7) k sol (T , x) = k id (T , x) · u Here ksol and kid are, respectively, the thermal conductivity of the solid solution and the thermal conductivity of the parent materials. This latter is due only to the Umklapp process (i.e., without fluctuation of mass, distance, and strain); u is the disorder scaling parameter defined as28

π θD∞ (x) m (x) id (8) u (T , x) = k (T , x) (x, T ), 2¯BS (T , x) where m is the average molecular weight and is the disorder scattering parameter, describing the influence of the solute effect upon the thermal conductivity. Following Slack29 and Abeles,17 the disorder scattering parameter is the sum of two independent contributions, namely a mass, M and a strain field fluctuation, S : = M + S .

(9)

The fluctuation mass term depends only on the average molecular weight of the solution, M, and the molecular weight of each constituent of the solution, mi . For a simple solid solution, like NaCl-KCl, the permutation of the species occurs only on one sublattice. The term is written as

2 m i −1 , xi (10) M (x, T ) = M i

where xi is the molar fraction of each constituent. The strainfield fluctuation is almost always approximated by the func tion ε xi [δi / δ − 1]2 as proposed by Abeles17 for the Si-Ge i

solid solution. The parameter ε is a “phenomenological” adjustable parameter. The original strain-field fluctuation term proposed by Klemens10 is related to physical properties of the solid solutions and is express as Bs,i (T ) −1 xi S (x, T ) = 2 BS (x, T ) i

−2Qanh γth∞

δi (T ) −1 δ (x, T )

2

,

(11)

where γth∞ and Qanh are, respectively, the average Gruneisen parameter of the solution and a parameter describing the difference of anhamonicity of each pair within the lattice. In the case of NaCl-KCl, it can be assumed that the pairs Na-Cl and K-Cl have the same anharmocity since the temperature dependence of the heat capacity at constant pressure and the thermal expansion are almost identical at high temperature. In this case Klemens10 has shown that Qanh ≈ 4.2. According to this formalism, the thermal conductivity of halides and solid solutions can be predicted from key physical properties, namely the Debye temperature, the Grüneisen parameter, the molar volume, and the adiabatic bulk modulus. The comparison between our experimental data and theoretical predictions is the aim of Sec. IV. IV. RESULTS AND DISCUSSION A. Coherent phase diagram

When materials, in particular alloys, are prepared under condition far from equilibrium, the microstructure is governed primarily by kinetic effects. In this case, coherent phases are first formed. The Gibbs energy difference between the coherent and incoherent phases is due to internal stress induced by coherency strains. The coherency strains arise when the lattice constants of the phases in coherent equilibrium are unequal. Accordingly, for the NaCl-KCl binary system, the large lattice mismatches of about 11% between NaCl and KCl (aNaCl = 5.64 Å and aKCl = 6.29 Å) lead without doubt to another metastable, coherent phase diagram. The coherent phase diagram is presented in Fig. 3 along with the set of compositiontemperature data points for which the thermal diffusivity has been measured. For a brief description of the procedure of calculation of the NaCl-KCl coherent phase diagram see the supplementary material.67 There are only two samples (373 K and compositions XKCl = 0.4 and XKCl = 0.6) which are inside the calculated coherent miscibility gap. The other samples are in the temperature-composition region where the solid solution is stable, for T > 777 K, or metastable, for T < 777 K. The experimental values of the thermal diffusivity of the NaCl-KCl system, including the pure compounds, are given in Table I together with the estimated relative errors. For each sample, five different laser flash experiments were performed; the thermal diffusivity and its error correspond, respectively, to the average value and the standard deviation.


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FIG. 3. The coherent phase diagram of the NaCl-KCl system above 300 K with the set of composition-temperature data points (solid grey square) for which the thermal diffusivity has been measured. The two characteristic temperatures which are: the minimum “azeotropic like” temperature (927 K) and the consolute point (453 K) of the coherent miscibility gap are shown in the figure. Outside the miscibility gap (dashed line) the solid solution (SS) is metastable and inside the miscibility gap the phase separation into two distinct solid solutions SS1 and SS2 occurs.

B. Comparison between the experimental and theoretical thermal conductivity of pure NaCl and KCl as a function of temperature

As mentioned previously, the thermal conductivity of the pure compounds can be predicted from reliable values of two key physical parameters, θD∞ and γth∞ . In prior work30 we proposed a method for determining the optimal values of θD∞ and γth∞ . This method is based on a simultaneous fitting of the temperature dependent thermal expansion, adiabatic bulk modulus, and heat capacity by a set of self-consistent physical models. In this work, we have used the same procedure to determine θD∞ and γth∞ for NaCl and KCl. Brief description of the fitting procedure and the comparison with experimental data is shown in the supplementary material.67 The values of

TABLE I. Average values of thermal diffusivity for each sample of the NaCl-KCl solid solution shown in Figure 1. The estimated error is deduced from standard deviation of the five different measurement performed for each sample. The thermal diffusivity is given in unit of 106 m2 s−1 . The value in bold character is the thermal diffusivity of the two sample inside the coherent miscibility gap. XKCl T (K) 373 473 573 673 773 823

0

0.2

0.4

0.6

0.8

1

Estimated error (%)

2.54 1.82 1.33 1.12 0.98 ...

2.43 1.74 1.25 0.96 0.81 ...

2.18 1.62 1.17 0.92 0.74 ...

2.39 1.72 1.23 0.97 0.78 ...

2.76 2.05 1.55 1.27 1.06 0.80

3.69 2.61 1.96 1.65 1.35 0.86

3 3 5 7 9 9

θD∞ and γth∞ calculated from the present fitting procedure are given in Table II, with available experimental data obtained from thermal properties. Our results are close to the average values reported in the literature. The thermal conductivities of NaCl and KCl were deduced from the thermal diffusivity and the assessed heat capacity and density are given in the supplementary material.67 The values are given in Table III, with the estimated error. The error consists of a ρ · Cp term, which is estimated to be 0.5% below 573 K and 1% above 573 K. Fig. 4 shows the predicted thermal conductivities of pure NaCl (Fig. 3(a)) and KCl (Fig. 3(b)) as a function of temperature, compared with the experimental data of this work and those reported in the literature. Also shown are molecular dynamic simulation (MD) results. Equation (17) thus shows excellent predictive capability for thermal conductivity as a function of temperature for both NaCl and KCl. Such exemplary accuracy is due mainly to the fact that the key parameters θD∞ and γth∞ were determinate by a self-consistent thermodynamic optimization procedure, which considers other thermal properties.

C. Comparison between the experimental and theoretical thermal conductivities of NaCl-KCl solid solutions as a function of composition and temperature, up to 823 K

The thermal conductivity is deduced from our data of the thermal diffusivity and the assessed value, of Cpml and ρ taken from Refs. 5, 60, and 61. The estimated error of the thermal conductivity is, as before, equal to the experimental error of thermal diffusivity plus the error arising from the term ρ Cpml .


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∞ obtained in this TABLE II. Comparison between the values of θD∞ and γth work (bold character) and experimental data reported in the literature with the associated error if available.

θD∞ (K)

This work NaCl

285

KCl

232

∞ γth

Literature values (with error if available)

This work

287a 280h,d,e,f,g 282 ± 1.2h 281j 279 ± 11l 287 ± 12n 304p 232a 231q 230d 231j 229 ± 0.8h 236 ± 18l 226 ± 18n

1.58b,e 1.58 ± 0.05h 1.64i 1.662 ± 0.025k 1.6±0.05m 1.656o 1.62r

1.65

1.41 1.39b 1.48 ± 0.05h 1.41s 1.45 ± 0.05m 1.49i

a

k

b

l

Reference 31. Reference 32. c References 33 and 49. d Reference 38. e Reference 35. f Reference 36. g Reference 37. h Reference 39. i Reference 41. j Reference 47.

Literature values (with error if available)

Reference 42. Reference 40. Reference 45. n Reference 46. o Reference 43. p Reference 44. q References 34 and 48. r Reference 49. s Reference 48. m

The error of the density reported by Walker is less than 0.2% and the error of the heat capacity reported by of Sangster and Pelton is about 1.5%. Hence, it is reasonable to fix the error on the term ρ · Cpml. at 2%. The values of the thermal conductivity of the NaCl-KCl system as a function of composition and temperature are given in Table III, together with the estimated error. As was explained above, theoretical prediction of the chemical effects upon the thermal conductivity of solid solutions required the thermal conductivity of the parent material (λid ) and three key physical properties (Table IV). These three quantities are lattice constant, the bulk modulus and the Debye temperature as a function of temperature and composition. According to Eq. (7), the thermal conductivity of the

FIG. 4. Experimental, molecular dynamics (MD) and theoretical thermal conductivity of pure NaCl (a) and KCl (b) versus temperature (see Refs. 50–59).

parent material is itself defined from these three key physical properties as shown in the following equation: k id (T , x) =

TABLE III. Thermal conductivity of NaCl and KCl deduced from the experimental thermal diffusivity (Table II). The estimated error is the sum of the error of thermal diffusivity and the term ρ · Cp. The thermal conductivity is given in units of W m−1 K−1 . T (K)

NaCl KCl Estimated error (%)

373

473

573

673

773

823

4.87 5.02 3.5

3.57 3.65 3.5

2.76 2.81 6

2.29 2.42 8

2.03 2.02 10

1.80 ... 10

0.59238 ∞ m (x) θD∞ (x)3 p γth ¯ n2 / 3 γth∞ (x)2 ·

δ (x, T ) −θD∞ (x) / 3T e . T

(12)

For halide solid solutions the composition dependences of θD∞ , BS , δ, and γ are well established in the literature. The composition dependence of the lattice parameter of most halide solid solutions obey the Vegard rule.62 The NaClKCl solid solutions show a maximum deviation of less than 0.4%.61, 63 It is thus reasonable to assume that δ (T , x) = (13) xi δi (T , x). i


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TABLE IV. Average values of thermal conductivity for each sample of NaCl-KCl solid solution shown in Figure 1. The estimated error is obtained from the error of the thermal conductivity and. Thermal conductivity y is given in units of W m−1 K−1 . Values in bold characters are the thermal diffusivities of the two samples inside the coherent miscibility gap. XKCl T (K) 373 473 573 673 773 823

1

0.8

0.6

0.4

0.2

0

Estimated error (%)

5.11 3.70 2.84 2.48 2.09

4.09 3.12 2.40 2.04 1.72

3.80 2.80 2.03 1.66 1.38

3.70 2.81 2.05 1.67 1.37

4.39 3.22 2.34 1.84 1.59 1.56

4.88 3.59 2.64 2.30 2.03 1.80

5 5 7 9 11 11

The excess heat capacity of mixing is due primarily to anharmonic effects, by assuming the additivity of the Debye harmonic heat capacity. The following relation is obtained64 ∞ (x)−3 . θD∞ (x)−3 = xi θD,i (14) i

This relation is known as the Kopp-Neumann rule. Generally, this rule is respected for most halide solid solutions. The adiabatic bulk modulus deviation from additivity of ionically bonded solid solutions was formulated by Flancher and Barsh.65 Their model shows good predictive ability for a large majority of halide systems.62 It has the advantage of containing only information of the pure compounds forming the solid solution. The Flancher and Barsh model is expressed as follows: BS (T , x) = xi BS,i (T , x) + xi xj i

×

i,j i=j

[(BS,i Vm,j )2 − (BS,j Vm,i )2 ](BS,i − BS,j ) BS,i BS,j Vm,i Vm,j

(T , x) . (15)

The key point of this work was to explore the possibility of predicting the chemical effect upon the lattice thermal conductivity of a halide solid solution from minimal data. That is, experimental and/or first principle information on a few key physical properties of pure compounds forming the solid solution. This is, in principle, possible by parameterizing Eqs. (8)–(12) by Eqs. (13)–(16). The accuracy of the proposed methodology is the purpose of the next discussion. The presentation of the results is as follows: for each temperature we represent two types of graphs: graph (a) with the predicted thermal conductivity compared to our experimental values and graph (b) showing the total and the various contributions of disorder scattering to the decrease of thermal conductivity. The degradation of each contribution to the thermal conductivity, Dki , is defined as k i (16) Dki (%) = 100 · 1 − lin , k

where k lin =

i

xi ki is the linear thermal conductivity, i = {0,

M, tot} and i = {0, M , tot }. The case i = 0 corresponds to the thermal conductivity of the parent material defined by Eq. (13) and i = M corresponds to the thermal conductivity of the solid solution (when only the mass fluctuation term is involved in the disorder scattering process). corresponds to the thermal conductivity of the solid including all the contribution to the disorder scattering process. The thermal conductivity degradation is thus defined as an excess term to the linear thermal conductivity, but, contrary to thermodynamic excess mixing properties (e.g., enthalpy or entropy of mixing) the ideal thermal conductivity is different from the linear thermal conductivity. The difference is due to a difference in Debye temperature of each constituent of the solid solution. The greater the difference between the Debye temperatures the greater the deviation from linear behaviour with composition of the ideal thermal conductivity. The thermal conductivity was not measured at the standard temperature of 300 K. In Fig. 5 we show the purely predicted thermal conductivity of the NaCl-KCl binary system at 300 K. Outside the miscibility gap is a single phase region of homogenous solid solution, and the thermal conductivity as a function of composition is predicted with the formalism elaborated in this work. Inside the coherent miscibility gap, there is a two-phase region. In this case a suitable mixture rule had to be applied for the two coexisting phases in order to predict the thermal conductivity. In this work, we considered the Effective Medium Theory model66 for estimating the thermal conductivity in the two phase region. Both the thermal conductivity and the thermal conductivity degradation curves at 373 K are shown in Fig. 6, with experimental data points. Of primary importance, the behaviour predicted at 300 K is experimentally observed at 373 K, i.e., it is a quasi-linear function of the thermal conductivity with composition inside the miscibility gap region. It has a quadratic shape in the metastable solid solution region. The predictive capability of the model is very satisfactory, seeing that the predictions for all studied compositions are within experimental error. Above the coherent consolute temperature (453 K) there is complete solid solution. The thermal conductivity and its degradation defined by Eq. (17) are quadratic functions of composition centred on the equimolar composition. Inside the miscibility gap the calculated thermal conductivity varies almost linearly with composition, but shows a slight positive deviation. This is due essentially to the fact that the thermal conductivity of the two terminal solid solutions forming the two phase mixture are very similar, respectively, 5.15 and 5.90 Wm−1 K−1 . These values result in the fact the thermal conductivity of the system is almost identical to the average between “parallel” and “series” model. It is highlighted that the thermal conductivity of the parent material is not a linear function of composition. This is understandable since the Debye temperature do not depend linearly on composition. The maximum degradation of the lattice thermal conductivity is about 30% around the equimolar composition. Half of this degradation is due to the mass difference between Na and K and the second half is due to the internal stresses which arise from the inclusion of


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FIG. 5. (a) Theoretical thermal conductivity of the NaCl-KCl system versus composition at 300 K. The extension of the thermal conductivity of the unstable solid solution in the miscibility gap region is shown as a dashed line. The thermal conductivity inside the miscibility gap is assumed to be linear, as discussed in the text. (b) The various contributions to the thermal conductivity degradation at 300 K versus composition XKCl for three given cases of the disorder scattering parameter ; = 0 (dashed-dotted line), = M (dashed line) and = M + s (solid line).

the guest ion into the host lattice. The abrupt variation of the thermal conductivity inside the miscibility gap is associated to a phase transition: from a single phase region to a two phase region. If the chemical mixing of the cations species is considered the shape of the thermal conductivity is represented by the dotted line in Figs. 5(a) and 6(a). However, the compositions of the two phases constituting the miscibility gap are fixed and thus the correct behaviour of the composition dependence of the thermal conductivity is properly represented by the full line inside the miscibility gap region in Figs. 5(a) and 6(a). In Figs. 7–11 we show predicted curves and our experimental data of the thermal conductivity (“a” parts) and the thermal conductivity degradation (‘b” parts) of the NaCl-KCl

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FIG. 6. (a) Theoretical thermal conductivity of the NaCl-KCl system versus composition at 373 K in comparison with our experimental data. The extension of the thermal conductivity of the unstable solid solution in the miscibility gap region is shown as a dashed line. The thermal conductivity inside the miscibility gap is assumed to be linear as discussed in the text. (b) The various contributions to the thermal conductivity degradation versus composition at 373 K, versus composition XKCl for three given cases of the disorder scattering parameter ; = 0 (dashed-dotted line), = M (dashed line), and = M + s (solid line) in comparison with the total degradation deduced from our experiments (filled square for measurement out of the miscibility gap region and open star for measurement inside the miscibility gap region).

solid solutions, respectively, at 473 K, 573 K, 673 K, and 773 K. For all temperatures and all compositions, the predicted thermal conductivities are within the error bars. The first conclusion that can be drawn from these results is that Eq. (17), parameterized with only key properties of pure compounds forming the solid solutions, appears to be suitable for the prediction of the thermal conductivity of halide solid solutions as a function of temperature. A pertinent analysis of the various degradation contributions to the thermal conductivity is the key to understanding the physical nature of the degradation of the thermal conductivity. The maximum deviation of total thermal conductivity degradation is observed at the equimolar composition. The


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FIG. 7. (a) Theoretical thermal conductivity of NaCl-KCl solid solutions versus composition at 473 K, compared with our experimental data. (b) The various contributions to the thermal conductivity degradation versus composition at 473 K, versus composition XKCl for three given cases of the disorder scattering parameter ; = 0 (dashed-dotted line), = M (dashed line), and = M + s (solid line) in comparison with the total degradation deduced from our experiments (filled square).

FIG. 8. (a) Theoretical thermal conductivity of NaCl-KCl solid solutions versus composition at 573 K in comparison with our experimental data points. (b) The various contributions to the thermal conductivity degradation versus composition at 573 K, versus composition XKCl for three given cases of the disorder scattering parameter ; = 0 (dashed-dotted line), = M (dashed line) and = M + s (solid line) in comparison with the total degradation deduced from our experiments (filled square).

amplitude of the total degradation increases with temperature; this is due only to strain field fluctuations since the ideal and the mass contributions to the total degradation decrease with temperature. The difference between the calculated and the experimental maximum degradation (at the equimolar composition) increases with temperature, expressed as Dk (Exp.) − Dk (Calc.) (%) σequimolar = Dk (Calc.)

composition dependence of the physical properties used to parameterize the model. However, this magnitude of error is still acceptable, since it is less than experimental error. At the equimolar composition, the ratio between each contribution to the thermal conductivity degradation and the total thermal conductivity degradation of the solid solution varies linearly with temperature and is expressed as

= 3.6 + 0.023 · (T − 300) . The predictive capacity of the model thus decreases with temperature. The difference between the predicted and experimental thermal conductivity degradation is ∼10% at 573 K and 15% at 773 K. This is the main limitation of the present model, probably due to the various approximations in the

Dk= 0 (Calc.) (%) = 18.2 − 0.016 · (T − 300) , Dk (Calc.) Dk= M (Calc.) (%) = 53.2 − 0.063 · (T − 300) , Dk (Calc.) Dλ=S (Calc.) (%) = 28.6 + 0.079 · (T − 300) . Dλ (Calc.)


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FIG. 9. (a) Theoretical thermal conductivity of NaCl-KCl solid solutions versus composition at 673 K in comparison with our experimental data. (b) The various contributions to the thermal conductivity degradation versus composition at 673 K, versus composition XKCl for three given cases of the disorder scattering parameter ; = 0 (dashed-dotted line), = M (dashed line), and = M + s (solid line) in comparison with the total degradation deduced from our experiments (filled square).

The temperature behaviors of the different contributions to thermal conductivity degradation are very different in sign and/or magnitude. At room temperature, the mass fluctuation term represent more than half of the total degradation of the thermal conductivity; about 30% and 20% of the degradation are due, respectively, to the strain field fluctuation and the thermal conductivity of the parent materials. These proportions are completely different at high temperature. Just below the minimum of the liquidus temperature (927 K), 80% of the thermal conductivity degradation originates from the strain field fluctuation and only 15% from the mass fluctuation term; about 5% is due to the thermal conductivity of the ideal parent material. The phonon mean free path of the solid solution can be estimated from the kinetic theory of gases. According to this

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FIG. 10. (a) Theoretical thermal conductivity of NaCl-KCl solid solutions versus composition at 773 K, in comparison with our experimental data. (b) The various contributions to the thermal conductivity degradation versus composition at 773 K, versus composition XKCl for three given cases of the disorder scattering parameter ; = 0 (dashed-dotted line), = M (dashed line), and = M + s (solid line) in comparison with the total degradation deduced from our experiments (filled square).

theory, the relationship between the lattice thermal conductivity and the mean free path of the phonons, lph. , can be written as k≈

1 C Cl , 3 V 0 ph.

(17)

where CV is the heat capacity of the solid per unit volume and C0 is the phonon speed. The phonon speed can be calculated from the adiabatic bulk modulus according to the relation C0 = Bs / ρ. In Figs. 11(a) and 11(b) we show, respectively, the phonon mean free path and the phonon mean free path degradation at 373 K, 573 K, and 573 K. The phonon mean free path degradation describes the deviation from the linearity of the phonon mean free path vs. composition. It is defined in the same way as the thermal conductivity


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this composition, the degradation decreases continuously with temperature 35% at 300 K to 65% at 773 K. Both temperature and the composition contribute to this degradation. The chemical effect upon the phonon mean free path, up to a certain composition (equimolar composition in the present case), is very weak. V. CONCLUSION AND PERSPECTIVE

FIG. 11. Estimated phonon mean free path (a) and phonon mean free path degradation (b) as a function of composition at 373 K, 573 K, and 573 K.

degradation, i.e., Dl

ph.

ACKNOWLEDGMENTS

⎤

⎡ (%) = 100 · ⎣1 − i

lph. xi lph.i

⎦.

In this work we present a model for predicting the chemical effect upon the lattice thermal conductivity of halide solid solutions. It is based on a localized continuum model, assuming a Debye approximation of the phonon density of state. A few key parameters, which are generally available from experiment or easily predicted ab initio, are required to parameterize the model. Therefore the model is purely predictive, since no experimental information on the thermal conductivity is required. Almost no information on the halide solid solutions is available in the literature. Valuable information on the physical properties of halide systems is increasingly required in many new technological applications, such as the thermal energy storage. Thus a model with high predictive capacity is very desirable for halide systems. In order to evaluate the predictive capability of the model, laser flash heating experiments have been performed to measure the thermal diffusivity of the prototype system NaCl-KCl. The model shows very good predictive ability for the thermal conductivity of pure compounds as a function of temperature and for solid solutions as a function of temperature and composition. The weakness of the model is shown by the fact that it slightly overestimates the degradation of the lattice thermal conductivity of solid solution at high temperature. In conclusion, we believe that the proposed model can predict the lattice thermal conductivity of solid halide solutions as a function of temperature and composition, at a high confidence level. In future work we will present a coupled, experimental-theoretical study of other halide systems to confirm the favourable predictive capability of the model.

(18)

The phonon mean free path is of nanoscale dimension and thus it is very small compared to the average grain size of the system (microscale dimension). This confirms the choice to neglect the grain size in the formalism. The phonon mean free path varies slightly in the range XKCl < 0.5 then increases quadratically up to pure KCl. In the range XKCl < 0.5, the phonon mean free path is almost constant at low temperature and starts to decrease slightly at high temperature. In the range XKCl > 0.5, the phonon mean free path increase quadratically up to pure KCl. The chemical effect upon the phonon mean free can be seen more clearly in the analysis of the degradation curve shown in Fig. 11(b). The degradation of the phonon mean free path shows as a regular quadratic function of composition, with a minimum at the equimolar composition. At

This research was supported by funding from the Natural Sciences and Engineering Research Council of Canada (NSERC) and Rio Tinto Alcan. We are grateful to Dr. James Sangster and Evguenia Sokolenko for their help during the writing of the manuscript. Particularly fruitful discussions with Dr. Sophia Ambre are extremely appreciated. 1 J. C. Gomez, “High-temperature phase change materials

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