Effect of microstructure on the thermal conductivity of nanostructured Mg2(Si,Sn) thermoelectric alloys: An experimental and modeling approach

Acta Materialia 95 (2015) 102–110

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Effect of microstructure on the thermal conductivity of nanostructured Mg2(Si,Sn) thermoelectric alloys: An experimental and modeling approach Philippe Bellanger a, Stéphane Gorsse b,⇑, Guillaume Bernard-Granger c, Christelle Navone c, Abdelkrim Redjaimia d, Solange Vivès a a

Univ. Bordeaux, ICMCB, UPR 9048, 33600 Pessac, France Bordeaux INP, ICMCB, UPR 9048, 33600 Pessac, France c DRT/LITEN/DTNM/SERE/LTE, 38054 Grenoble, France d Institut Jean Lamour, UMR 7198, CNRS, France b

a r t i c l e

i n f o

Article history: Received 29 January 2015 Revised 6 May 2015 Accepted 6 May 2015

Keywords: Thermoelectrics Magnesium silicides Nanostructuration Thermal conductivity reduction Spark plasma sintering

a b s t r a c t In this work, we produce bulk nanostructured Mg2Si0.4Sn0.6 thermoelectric materials made of nanograins with sizes below 200 nm and containing a fine distribution of Sn-rich nanoparticles. These materials are obtained by the mechanical alloying followed by spark plasma sintering. The microstructure and transport properties, and their evolutions upon aging, are investigated. A model is developed to capture the different contributions to the phonon scattering processes arising from the nano/microstructural parameters. The calculations show quantitative agreement with the temperature and the temporal dependence of the lattice thermal conductivity of the nanostructured Mg2Si0.4Sn0.6 alloy. This work provides a general analytic approach for identifying the individual contributions of the microstructural parameters on the thermal conductivity which is a very important property controlling the performance of thermoelectric materials. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

1. Introduction Thermoelectric technology offers a mean of converting heat into electricity, or conversely, electricity into cooling. In this respect, applications of thermoelectricity such as waste heat recovery systems and air conditioning are regarded as capable of reducing the environmental impact of transports. Ideally, a thermoelectric material should be a thermal insulator and an electrical conductor with a large Seebeck coefficient. This is an antagonist combination of properties that is translated into a figure of merit, zT ¼ S2 rT=j, controlling the conversion efficiency [1]. In practice, the best candidates are medium band gap semiconductors for which the heat transport has been minimized without side effect on electron transport. The total thermal conductivity of semiconductors is the sum of contributions of different heat carriers: the electrons and the phonons. In order to decrease the total thermal conductivity while preserving the electronic conductivity, only the phononic transport ⇑ Corresponding author. http://dx.doi.org/10.1016/j.actamat.2015.05.010 1359-6454/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

should be obstructed. This can be achieved by multiplying the phonon scattering events in order to decrease the phonon mean free path. The scattering contributions arise from the interactions between the phonons themselves (normal and umklapp scattering), and the phonons and the microstructural features. These scattering modes include the interactions between phonons and solute atoms, grain boundaries, second phase particles and others defects such as dislocations. For some materials one scattering mode may dominate the total scattering probability leading to an important decrease of the lattice thermal conductivity. The idea behind nanostructuration and microstructure control of thermoelectric materials is to enhance the scattering contributions arising from phonon–grain boundary interaction in nanocrystalline compounds, from phonon-particles in nanoprecipitates containing alloys, and from phonon-interface in nanocomposites [2–6]. In the family of n-type thermoelectrics, those based on Mg2(Si,Sn) solid solutions are very attractive because they are compatible with industrialization in a large volume (low cost and

P. Bellanger et al. / Acta Materialia 95 (2015) 102–110

abundant constitutive elements) and the rules of Hygiene, Health and Environment (HSE). Two recent experimental works [7,8] report a very low thermal conductivity (<2 W/m K) for nano-sized particles containing Mg2Si0.4Sn0.6 alloys. Liu et al. [7] adopted a two-step solid reaction followed by spark plasma sintering, and determined the presence of Sn-rich precipitates with the size of several tens of nanometers distributed into the matrix phase, by transmission electron microscopy. Zhang et al. [8] prepared the nanostructured alloy by ball milling and hot pressing. The presence of nanoparticles was evidenced by transmission electron microscopy analysis, the composition was not investigated though. From these works, one can wonder about the nature and the effect of the nanoparticles on the thermal conductivity of Mg2(Si,Sn) alloys. We report in the present paper an effort to quantify the individual effect arising from the microstructure on the lattice thermal conductivity of nanostructured Mg2Si0.4Sn0.6 alloys. In an attempt to tune the properties through the control of the microstructure and to gain some design guidance, it is indeed necessary to quantify the different contributions to phonon scattering. Such quantitative understanding is also important to describe and predict the temporal evolution of the thermal conductivity that may occur during service use at high temperatures as a consequence of the microstructural evolution of the material [9]. Our approach involves two steps: (i) the preparation and the characterization of nanostructured Mg2Si0.4Sn0.6 alloys, and (ii) the development of a model for the lattice thermal conductivity that accounts the different contributions to the phonon scattering arising from the nano/microstructural parameters. In Section 2 we will present the experimental procedure used to produce the Mg2Si0.4Sn0.6 nanostructured alloy by ball milling and spark plasma sintering (SPS). We summarize the characteristics of nano/microstructure in Mg2Si0.4Sn0.6 alloy and its temporal evolution during aging at high temperatures in Section 3. A model for the thermal conductivity taking into account the various scattering processes that arise from the interactions between the phonons and the nano/microstructure is described in Section 4. The adjustable parameters entering in the expression of the thermal conductivity are optimized using experimental results of the heat capacity for the pure phases Mg2Si and Mg2Sn, and the composition dependency of the thermal conductivity of the Mg2(Si,Sn) solid solution. In Section 5, the model is compared with the temperature dependence of the thermal conductivity of the nanostructured Mg2Si0.4Sn0.6 alloys measured as a function of the aging conditions. The origin of the low thermal conductivity and its evolution during aging is discussed.

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temperature. Moreover, these Ta tubes were sealed in quartz tube to avoid their oxidation during aging treatments in tubular furnaces. The different materials were then cut and grounded for structure and microstructure characterizations. X-ray diffraction (XRD) was performed on the grounded pellets for phase identification (Philips PW 18-20 Panalytical X’pert). Metallographic examination was performed by scanning electron microscopy (SEM) (JEOL 6700 F) on mirror polished samples. Transmission electron microscopy (TEM) in bright and dark field modes (JEOL 2200 FS and ARM 200F) was performed on ion etched specimens. The equipment used was a Precision Ion Polishing System (PIPS) from Gatan. Samples were sieved, then pre-polished and fixed on a copper grid of diameter 3 mm and width 50 lm. Ionic polishing was carried out at 8 kV and an incident angle of 8°. High Angle Annular Dark Field (HAADF) was used to increase the contrast between chemical elements and local chemical composition variations were measured using STEM-EDX (Tecnai Osiris) on sample foils obtained from the focused ion beam (FIB) technique. The average grain size was estimated from the number of grains per unit area, NA, on bright field micrographs. The counting method employed is due to Jeffries [10]. Assuming pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi spherical grains, the average grain size is Dgrains ¼ 2 3=2pN A . In the case of nanoparticles, we can no longer considered the projected image of a thin foil as equivalent to section, since Dparticles < t (where t is the foil thickness). However, the estimate of Dparticles is straightforward by measuring the projected diameter of each particle on bright field micrographs, assuming spherical grains. The thermal conductivity was calculated according to the relationship j = aqCp, where a is the thermal diffusivity measured with the flash diffusivity method (LFA 457 Micro Flash), q the density and Cp the heat capacity measured with the differential scanning calorimetry technique (PerkinElmer DSC 8500). 3. Experimental results 3.1. Nano/microstructure of the Mg2Si0.4Sn0.6 alloy According to X-ray powder diffraction (XRPD), the ball milled powders consist of the solid solution phase Mg2Si0.4Sn0.6 with trace of pure Si (Fig. 1a) not visible on the bright field transmission electron micrograph of the ball milled powders (Fig. 2a).

2. Experimental procedure Mg2Si0.4Sn0.6 alloys were prepared by mechanical alloying and the powders thus obtained were sintered by SPS. Commercial high-purity shots of Si, and bars of Mg and Sn were weighed in stoichiometric amounts and sealed in ZrO2 jars under Ar atmosphere. The mechanical alloying was performed at 300 rpm during 60 h in a planetary ball mill (Retsch PM 100). The fine powders were sintered using SPS (Dr. Sinter lab 515S) at 890 K during 4 min and under 100 MPa. The density of the 10 mm diameter pellets was measured 3 times with the Archimedean method. These sintering conditions led to the reference state sample for our study and will be noted as the ‘‘as-SPS’’ material. Aging treatments were then performed on the as-SPS samples for 14 and 28 days at 500 °C because it is expected to be the maximum service temperature for the Mg2(Si,Sn) alloys. Each pellet was sealed in a Ta tube under Ar using the Tungsten Inert Gas technique to avoid any oxidation or Mg evaporation at high

Fig. 1. X-ray diffraction patterns on Mg2Si0.4Sn0.6 powders after mechanical alloying (a) and after SPS (b) processes. Both materials crystallize in the Fm-3m crystal structure.

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presence of nano grains with average size of 70 nm without noticeable coarsening effect occurring during the sintering process. 3.2. Evolution of the nano/microstructure during thermal treatments

Fig. 2. Transmission electron microscopy on Mg2Si0.4Sn0.6 powders obtained after mechanical alloying (a). (b) Scanning electron microscopy on fractured sintered pellets showing a small amount of porosities.

After SPS of the ball milled powders, the peaks of Si disappeared on the XRPD spectra of the pellet thus obtained (Fig. 1b). Scanning electron micrographs show the high density of the as-SPS pellets in which only few porosities are visible (Fig. 2b). A high relative density of 99% was confirmed by the Archimedean method. Transmission electron microscopy clearly evidences that as-SPS samples contain nanoscale particles (Fig. 3a–d) with an average size of 10 nm which is in agreement with published values [7,8]. Furthermore, the bright contrast of the nanoparticles on the high angle annular dark field scanning transmission electron micrograph (Fig. 4) indicates that they are heavier (Sn-rich) than the surrounded matrix. The nanoparticle has a spherical-like shape. Elemental composition mapping obtained by energy-dispersive X-ray spectroscopy (EDX) suggests these are nano-size Mg2Sn-rich second phase with a Sn/Si ratio around 70/30. Bright field transmission electron microscopy (Fig. 5a) indicates the

In order to rationalize the effect of the metallurgical parameters on the thermal conductivity of the Mg2Si0.4Sn0.6 alloys we need to generate various nano/microstructural states. This is the main purpose of the thermal treatments. In addition, this will allow us to investigate the stability of the nanostructure and the consequences of aging on the thermal conductivity. Fig. 5b and c shows the TEM images of the samples treated 14 and 28 days at 500 °C in comparison with the as-SPS sample (Fig. 5a). We can observe an important grain growth leading to an average grain size of 95 and 180 nm for the samples treated during 14 and 28 days, respectively. Surprisingly, there is no apparent evolution of the particle size. To summarize, the aging at 500 °C of the nanostructured Mg2Si0.4Sn0.6 alloy leads to a significant grain growth but seems to have a negligible effect on the nanoparticles (Table 1). 3.3. Thermal conductivity Fig. 6 shows the temperature dependence of the thermal conductivity as a function of the heat treatments applied to the alloy. The trends observed are consistent with previous reports. First, the thermal conductivity decreases with temperature due to the enhancement of the phonon–phonon interactions (Umklapp process). However, this effect is counterbalanced above 480 K (200 °C) by the bipolar contribution detailed in the next section. Another important observation is that thermal conductivities are shifted to higher values during aging. It is an important feature since this change in the material property degrades the

Fig. 3. High magnification TEM (a) showing nanoparticles embedded within grains and at grain boundaries. Particle average size and volume fraction are annotated below. Bright field TEM (b) on the as-SPS sample and its diffraction pattern (c) through the [0 1 1]a axe (a is the CFC matrix). Dark field TEM (d) associated by selecting the encircle diffraction spot presented on (c).

P. Bellanger et al. / Acta Materialia 95 (2015) 102–110

performance (zT) and could limit the in-service life of a thermoelectric generator (TEG). 4. Model development for the lattice thermal conductivity The present section describes a simple model that was developed to account the different contributions to the phonon scattering arising from the nano/microstructural parameters with the aim to capture the temperature and temporal dependence of the lattice thermal conductivity of the nanostructured Mg2Si0.4Sn0.6 alloy. In solids, the heat is carried by electrons and lattice vibrations. The electronic contribution dominates the thermal conductivity of metals while phonons are the main heat carriers in electrical insulators and both these mechanisms can significantly contribute in semi-metals and in some semi-conductors. In order to reproduce the thermal conductivity of Mg2(Si,Sn) alloys, one must take into account these two contributions:

jtotal ¼ jE þ jl where jE and jl represent the lattice and electronic parts, respectively. The electronic part, jE ¼ je þ jh þ jeh , results of the heat conduction by the electrons, je , the holes, jh , and the bipolar contribution, jeh [11]. This last mechanism is related to the endothermic creation of electrons-hole pairs at the hot side and their exothermic splitting at the cold side. It is added to the contributions of electrons and holes given by the Wiedemann–Franz law: je  le T  LT r, where le is the electron mean free path, L is the Lorentz factor and r the electrical conductivity. The electronic thermal conductivity was estimated from the measured electrical conductivity using the above equation, whereas we have used

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the values calculated from Ref. [12] for the bipolar thermal conductivity. We have now to consider the phenomenon of thermal conductivity due to phonons in order to quantitatively evaluate the thermal dependence of the lattice thermal conductivity of Mg2(Si,Sn) alloys according their composition and microstructural state. The treatment of the lattice thermal conductivity is based on the kinetic theory by considering phonon frequency dependent phonon mean free path:

jl ¼

1X 3 k

Z

C mKeff dx

ð1Þ

where C is the heat capacity, m is the group velocity, x is the frequency, Keff is the effective mean free path and k the polarization of the phonons. This expression, obtained from gas kinetic arguments, shows the importance of the heat capacity of the heat carriers, their velocity and their mean free path. The heat capacity is estimated using the Debye model that uses solid mechanics of a continuous medium made of mean mass connected by springs, and in which the vibration is treated as a superposition of waves of different frequencies. In this continuum approximation, the crystal behaves as a box containing a phonon gas. The anharmonicity is taken into account by considering that phonons behave like pseudo-particles undergoing collisions characterized by a relaxation time. Following the treatment of Callaway [13], Holland [14] and Wang [15] who assumed a Debye phonon spectrum in which the dispersion relation is linear but include a cut-off frequency, the density of states equals 3x2 =2p2 m3 and the phonon distribution is given by the Bose–Einstein function, the heat capacity and the lattice thermal conductivity take the following expressions:

Fig. 4. HAADF-STEM micrography at high magnification (a) of the as-SPS sample. In (b), (c) and (d) are presented the EDX cartographies in hyper-map mode of Si, Mg and Sn respectively.

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Fig. 5. TEM micrographs of observed grains for the as-SPS sample (a), material aged at 500 °C for 14 days (b) and 28 days (c).

Table 1 Average diameter of particles and grains for the as-SPS sample and aged samples at 500 °C for 14 and 28 days. as-SPS

500 °C 14 days

500 °C 28 days

Dgrains ðnmÞ

70

95

180

Dparticles ðnmÞ

10

10

10

C¼3

XZ

xck

0

k

 2 kB x2 hx expðhx=kB TÞ dx 2p2 m3k kB T ½expðhx=kB TÞ  12

ð2Þ

and

jl ¼

XZ k

0

xck

  x 2 Keff kB x2 h expðhx=kB TÞ dx 2p2 kB T ½expðhx=kB TÞ  12 m2k

ð3Þ

 is the Planck’s constant, T is where kB is the Boltzmann constant, h the absolute temperature, xck and mk are the maximum frequency and the sound speed for the acoustic phonon branch k (one longitudinal and two transverse), respectively. Microscopic investigations presented in Section 2 have revealed the presence of Mg2Sn-rich nanoparticles distributed in the nanocrystallized solid solution Mg2Si0.4Sn0.6. In order to take into account the scattering that arises from these different

microstructural features and evaluate their individual contribution, we have represented all phonon scattering processes by mean free paths which are a function of frequency and temperature. Assuming that the scattering processes are independent, the scattering probabilities are additive. In other words, the inverse of phonon mean free path is a sum of the inverse of (a) umklapp processes, (b) alloy disordering, (c) nanoparticles scattering and (d) grain boundary contributions according to Matthiessen’s rule: 1 1 1 1 K1 eff ¼ Kum þ Kss þ Kp þ Kgb

ð4Þ

This relationship, in which the influence of normal scattering is neglected, denotes that the combined effect of different scattering mechanisms leads to an effective mean free path smaller than the smaller of all of them. 4.1. Umklapp contribution For the umklapp scattering we use the most common form [16]: 2 A2 =T K1 =m um ¼ A1 T x e

ð5Þ

where A1 and A2 are two adjustable parameters. 4.2. Solid solution contribution For the solid solution contribution, the solute atom scattering was accounted by the effective medium approach using a Rayleigh-like expression as described by Klemens [17]:

K1 ss ¼

x4 d 3 Cs 4pm4

pffiffiffi where d is the radius of the solute atom in the host lattice ( 3=4 times the lattice parameter in the antifluorine structure), and Cs characterizes the scattering cross section of the solute atom. Abeles [18] derived an expression for Cs that takes into account the mass defect scattering and the strain scattering. In the case of a mixture of two kinds of atoms (in the present case, Si and Sn mix in the same sublattice), the scattering cross section is:

n

Cs ¼ xð1  xÞ ½DM=M2 þ ½Dd=d2 Fig. 6. Measured thermal conductivity versus temperature for the as-SPS sample (squares and purple dash line), and for thermal treatments at 500 °C, 14 days (circles and green dash line) and 28 days (triangles and red dash line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

o

M denotes the molar weight of the Mg2(Si,Sn) solid solution, DM is the mass difference and Dd the radius difference between the mixing atoms, respectively, and e is an adjustable parameter.

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4.3. Particle scattering

5. Comparison with the experimental results and discussion

In treatments of the scattering of phonons by particles, the mean free path Kp is calculated from the average distance traveled by the phonon between two scattering events:

We have 9 adjustable parameters xck , v k , A1 and A2 that describe the cutting frequencies, the velocities and the adjustable parameters for Umklapp contributions to phonon mean free path for each end-member Mg2Si and Mg2Sn, and e for the solid solution. We consider first the thermal dependence of the heat capacities of pure Mg2Sn and Mg2Si to assess the cut-off frequencies xck ðMg2 Sn;Mg2 SiÞ using Eq. (2) and the experimental data from Refs. [21,29]. The sound speeds mk ðMg2 Sn;Mg2 SiÞ were taken from the literature [15]. From the values in Table 2, the heat capacities as a function of temperature (Fig. 7) can be well described for both pure Mg2Sn and Mg2Si phases. Parameters A1 and A2 were optimized by fitting the experimental thermal conductivity of pure compounds Mg2Si and Mg2Sn from Refs. [21,29–36] using Eqs. (3) and (5). It can be seen in Fig. 8, that the calculated lattice thermal conductivities of the pure Mg2Sn and Mg2Si phase capture well the experimental measurements as a function of the temperature. The values for A1 and A2 are shown in Table 2. The remaining unknown parameter to evaluate the lattice thermal conductivity of the Mg2(Si,Sn) solid solution, from the description of the two end-members (Mg2Sn and Mg2Si), is related to the scattering cross section of the solute atoms which requires the knowledge of the evolution of the thermal conductivity as a function of the composition. The most complete set of experimental data is provided by the works of [37]. They have measured the room temperature thermal conductivity for solid solutions with different solute contents. One must stress that at room temperature where the thermal conductivity is measured, the lattice contribution largely dominates. From these data represented in Fig. 9, we can see how the thermal conductivity changes as a

Kp ¼ ðCeff NÞ1 where Ceff is the effective scattering cross section due to particles and N is the number of particles per unit volume (the density of scatterers). The contribution of scattering provided by polydispersed spherical nanoparticles has been investigated by Majumdar [19] who proposed a simple expression for the scattering cross section that takes into account both Rayleigh and geometrical regimes, at low and high phonon frequencies, respectively. We have used a more sophisticated model developed by Kim et al. [20] that captures the effects of differences in mass and bond stiffness on scattering: 1 1 C1 eff ¼ CRayleigh þ Cgeometrical

where Ceff is the harmonic average of the two extremes (short and long wavelength scattering regimes) according to Matthiessen’s rule, CRayleigh is the Rayleigh limit in which the scattering cross section is proportional to the fourth power of the particle radius and the phonon frequency, and Cgeometrical is the scattering cross section in the geometrical limit in which the scattering cross section reaches twice the projected area of the spherical particle and is independent of the phonon frequency:

CRayleigh ¼

 4 "    # 4p2 R2 xR DM 2 3 DK 2 þ M 4 K 9 m

Cgeometrical ¼ 2pR2 where R is the radius particle, and DM and DK denote the difference of density and force constant between the host matrix and the particle, respectively. The force constant, K, is taken proportional to Young’s modulus [21].

Table 2 Transverse and longitudinal velocities and cut-off frequencies as well as A1 and A2 adjustable parameters for Mg2Si and Mg2Sn compounds.

mTA =mLA

Ref.

(km/s)

4.4. Grain boundaries scattering

Mg2Si Mg2Sn

4.9/7.7 3.0/4.9

[15]

xcTA =xcLA (THz)

A1 (1018 Ks)

A2 (K)

47.1/69.63 24.8/49.0

9.5 9.9

70.4 51.6

It is well known that the thermal conductivity can be significantly affected by the grain size because it limits the phonon mean free path. Consequently, in treatments of the scattering due to grain boundaries, the phonon mean free path is a function of the grain diameter and the features of the grain boundaries (structure, thickness, composition changes due to segregation) which modify its thermal resistance [22–27]:

Kgb ¼ aDav g where a is an adjustable parameter that expresses the lower transmission of the grain boundary. According to Klemens [28] the grain boundary resistance arises from two contributions: the disorientation between grains which only provides a weak contribution to phonon scattering, and the disordering of the grain boundary, that is regarded as a thin layer in which the phonon velocity changes, which becomes independent of the frequency at high temperature as the grain size decreases. Since we lack information related to the characteristics of grain boundaries, and because we want to restrict the number of adjustable parameters, we have used the most widespread expression to calculate the effective phonon mean free path: Kgb ¼ Dav g , which tacitly assumes that phonons are randomly scattered at each grain boundary (a = 1).

Fig. 7. Heat capacity versus temperature of experimental data extracted from the literature for Mg2Si (circles a) [29] and (squares b) [21] and Mg2Sn (triangles c) [21]. Results from calculations (blue line for Mg2Sn and red line for Mg2Si) fit the experimental values after the adjustable parameter optimization process. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 8. Experimental lattice thermal conductivities of Mg2Si compounds (circle a) [34], (square b) [33] and (diamond c) [36] and Mg2Sn (triangle d) [31] as well as the calculated lattice thermal conductivities of Mg2Si (red) and Mg2Sn (blue) after parameters (A1 and A2) optimization, as a function of the temperature. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 9. Experimental (gray a) [37] and black b [38] circles and modeled (red line) lattice thermal conductivity at room temperature for Mg2Si1xSnx compounds (0 6 X(Sn) 6 1). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

function of the solute content. Using the weighted average parameters, PMg2 ðSi;SnÞ ¼ xSi PMg2 Si þ xSn P Mg2 Sn where P stands for xck , mk , A1 and A2, a value of 1040 for e allows the complete prediction of the room temperature of the lattice thermal conductivity (Fig. 9) as a function of the composition for the solid solution Mg2(Si,Sn). The value of this adjustable parameter is in agreement with a previous study where a combinatorial approach based on diffusion couple was used to measure the thermal conductivity as a function of the composition of the solid solution Mg2(Si,Sn) [39]. This analysis shows that we can reproduce the temperature dependency of the thermal conductivity of the pure phases Mg2Si and Mg2Sn, as well as the composition dependency for the solid solution Mg2(Si,Sn). We can now move toward the evaluation of the lattice thermal conductivity of the nanostructured Mg2Si0.4Sn0.6 materials and account the scattering contribution of the nanoparticles and the grain boundaries. The calculated lattice thermal conductivity at 380 K of a Mg2Si0.4Sn0.6 material nanostructured with Mg2Si0.3Sn0.7 nanoparticles is presented in Fig. 10 versus the particles radius and their volume fraction. The figure clearly evidences the beneficial effect of a high volume fraction of second phase nanoparticles on the lattice thermal conductivity decrease. The minimum observed in Fig. 10 for a particle radius of 15 nm depends on the particle composition and is the consequence of the occurrence of two scattering regimes. When R increases, the mean free path Kp decreases 1/R3 in the Rayleigh scattering regime, whereas it increases as R in the geometrical regime. Therefore, the lattice thermal conductivity decreases first with the particles radius then increases when the phonon mean free path becomes limited by the geometrical regime. In this study, the measured radius of the nanoparticles is around 5 nm which is not the optimum value as given by the calculations, but still leads to a decrease of the lattice thermal conductivity from 1.8 to 1.5 W m1 K1 at 380 K for a volume fraction of 10%. Fig. 11 illustrates that the present model satisfactorily captures the temperature and temporal evolution of the lattice thermal conductivity upon aging, by considering the microstructural parameters experimentally determined and summarized in Table 1. We can also extract the magnitudes of the individual contributions as illustrated in Fig. 11. Several authors have commented on the low thermal conductivity of alloys based on Mg2(Si,Sn) and claimed that it is reduced by the formation of nanoparticles [7,8]. However, as can be seen in Fig. 11, the low thermal conductivity of the nanostructured Mg2Si0.4Sn0.6 alloy can be attributed to the combine effect of the scattering processes arising from both the

Fig. 10. Calculated lattice thermal conductivity at 380 K of the Mg2Si0.4Sn0.6 compound nanostructured with Mg2Si0.3Sn0.7 nanoparticles as a function of the particle radius and their volume fraction. The dark lines represent the isolines for the lattice thermal conductivity.

P. Bellanger et al. / Acta Materialia 95 (2015) 102–110

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Fig. 11. Experimental lattice thermal conductivities (gray circles), obtained by subtracting the electronic contribution to the measured total thermal conductivity, versus temperature of as-SPS and aged samples at 500 °C for 14 and 28 days. Calculated lattice thermal conductivities taking into account all microstructural parameters are represented by the red line (The Lorenz number was calculated to be 2.23  108, 2.83  108 and 2.87  108 W X K2 for the materials as-SPS, aged at 500 °C during 2 and 4 weeks, respectively). Individual contributions to the lattice thermal conductivity are represented by the other lines with the following significations: solid solution only (black lines), solid solution with nanoparticles (green dash-dotted lines) and solid solution with grain boundaries (blue dash lines). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

grain boundaries and the distribution of second phase nanoparticles. We can also emphasize in this study that the grain boundaries scattering is the most important contribution to the decrease of the thermal conductivity. This can be explained by the small grain sizes ranging from 70 to 180 nm which effectively limits the phonon mean free path. Moreover, the phonon mean free path issued from nanoparticles is also a function of the difference of density between the nanoparticles and the matrix. Therefore, the introduction of nanoparticles such as Mg2Si or Mg2Sn would be more effective to scatter the phonons and to compete with the effects of the small grains but their effect is less effective here with the Mg2Si0.3Sn0.7 nanoparticle composition. However, upon thermal treatment, because of the grain growth, the thermal conductivity increases but the increment is limited by the remarkable stability of the nanoparticle size and density number (Fig. 11). 6. Conclusions In the present paper, nanostructured Mg2Si0.4Sn0.6 alloy was prepared by ball milling and spark plasma sintering. We have evidenced the presence of a high density of second phase nanoparticles embedded into the nanocrystallized matrix. We also found that the nanostructured Mg2Si0.4Sn0.6 compound possesses a very low lattice thermal conductivity that increases when the alloy is subject to thermal treatment as a response to a substantial grain growth. A simple model accounting the different contributions to the phonon scattering arising from the microstructural parameters allowed to reproduce the temperature and the temporal dependence of the lattice thermal conductivity of the nanostructured Mg2Si0.4Sn0.6 alloy. It has been used to rationalize the evolution of the thermal conductivity occurring during thermal aging. Several specific outcomes can be highlighted: 1. When the grain size is below 200 nm, the thermal conductivity is mainly limited by the grain boundaries.

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