Temperature dependence of thermal conductivity in hybrid nanodispersions

Abstract

We derive first a formula for the overall heat conductivity for dispersions of cuboid nanowires and recall similar formulas derived previously in Behrang et al., Appl. Phys. Lett., 2014, 104, 233111, 104, 063106 and Behrang et al., J. Appl. Phys., 2013, 114, 014305, by using the same method, for several other morphologies (dispersions of nano-size spheres, nano-size spheres of different sizes, nano-wires, rectangular cuboid nano-wires, and nano-spheres and nano-wires). The temperature dependence of the microscopic quantities (like the phonon mean free path) entering the formulas is known from the phonon kinetic theory. The formulas thus provide a setting for investigating the temperature dependence of the overall heat conductivity of dispersions. In general, we see that the interface that is present in hybrid nanodispersions influences significantly the thermal conductivity only in lower temperatures. This then means that the possible benefit of hybrid nanodispersions in fabricating electronic devices diminishes with increasing the temperature.

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1 Introduction

The heat and electric conductivities depend in heterogeneous materials on the nature of their homogeneous components, on the morphology, and on the nature of the interface among the components. Consequently, by varying these three elements a large family of materials with a wide spectrum of heat and electric conductivities can be created. Heterogeneous materials are therefore often used in the fabrication of electronic devices where such materials are needed.

In this paper we continue our investigation^1–3 of the heat conductivity in heterogeneous materials created by dispersing nano-size particles of various shapes and sizes in a homogeneous matrix. We follow the method pioneered by Minnich et al. in ref. 4. The Maxwell homogenization⁵ made in the first step is followed by the second step in which the Peierls kinetic theory of phonons⁶ is employed. The second step, needed due to the nano-size morphology, transforms the expressions for the effective overall heat conductivity obtained in the first step into expressions involving microscopic quantities (like for instance the phonon mean free paths, heat capacities, and the parameters characterizing interactions of phonons with the interface) entering the phonon kinetic theory. In this paper we concentrate in particular on one gain made in such a transformation. We can theoretically predict the temperature dependence of the overall heat conductivity. This is because the kinetic theory provides a setting for investigating the temperature dependencies of the microscopic quantities entering it. Having known these dependencies, and having an expression for the overall heat conductivity in terms of the microscopic quantities, we obtain the temperature dependence of the heat conductivity of heterogeneous material.

The paper is organized as follows. First, we apply the method used in ref. 1–3 to a new morphology, namely to dispersions of rectangular cuboid wires for which there is a large amount of microscopic simulation data with which our predictions can be compared. This first part gives us also an opportunity to recall general features of the setting that we have used for all the morphologies (namely for dispersions of nano-size spheres, nano-size spheres of different sizes, nano-wires, and nano-spheres & nano-wires). The temperature dependence of the overall heat conductivity is calculated and presented in the second part.

2 Dispersion of rectangular cuboid wires

The Maxwell homogenization in dispersions of nanowires of infinite length that are all oriented in one direction leads^7,8 to the following expression for the overall Fourier heat conductivityBy the symbol ϕ we denote the volume fraction of the dispersed phase, k is the thermal conductivity, and subscripts “m” and “p” stand for matrix and dispersed phases respectively. Depending on the direction of the dispersed particles against heat flow, the parameters γ and β take different values. γ → ∞ or β = k_i for rectangular cuboid wires aligned in the direction of the heat flow (denoted by ‖), and γ = 2 and β = k_m for rectangular cuboid wires oriented in the direction that is perpendicular to the heat flow (denoted by ⊥). In this paper we consider nano-rectangular cuboid wires of thickness a_p, width b_p, and length L with the orientation characterized by the angle θ with respect to the orientation of the heat flow.

In order to take into account the nano-size of the wires, we pass now to the phonon kinetic theory. The thermal conductivity of the homogeneous component arises in the kinetic theory in the form

where ω is the phonon frequency, T is the temperature, C denotes the volumetric specific heat capacity per unit frequency at the frequency ω, v the phonon group velocity, and Λ the phonon mean free path. By 0 ≤ s ≤ 1, we denote the probability of the specular scattering of phonons on the particle–matrix interface is presented. The phonon scattering is diffuse (“j = diff”) if s = 0 and specular (“j = spec”) if s = 1. The angle between the direction of wires and the direction of heat flux is denoted by θ. Note that θ = 0 if all the nanowires are oriented in the direction of the heat flow and θ = π/2 if all the nanowires are aligned in the direction that is perpendicular to the direction of the heat flow. We assume in addition that a_p − L face is exposed to heat flux when θ = π/2

We now continue to investigate further the mean free paths. We begin with the matrix phase. The Matthiessen rule is employed to express the influence of boundary scattering:

Λ⁻¹ = Λ_b⁻¹ + Λ_coll⁻¹ + Λ_TBR⁻¹ (3)

where Λ_b is the mean free path in the bulk, Λ_coll is the mean free path associated with the collision, and Λ_TBR the thermal boundary resistance mean free paths. We now adapt the method developed in ref. 1–3 to rectangular cuboid wires of length L and thickness a_p and width b_p and arrive atWe point out that 1/Ξ^(‖) = (1/L + ξ/b_p + ζ/a_p)cos θ and 1/Ξ^(⊥) = (1/b_p + ζ/a_p)sin θ are the characteristic lengths of nanowires in the ‖ and ⊥ directions, respectively. Note that and denote the probabilities of phonon-wire scattering when phonon scatters with faces a_p − L and b_p − L, respectively. The diffuse and specular behaviour of phonons at the matrix–particle interface is defined byBy the symbols t_mp and t_pm we denote probabilities of the transmission from matrix “m” to particles “p” and vice versa. Explicit expressions for these quantities in both specular and diffuse transmissions are given in ref. 1.

Next, we direct our attention to dispersed particles and phonon scattering inside them. For nanowire in ⊥ direction, we have

Note that and are the effective nanowire thickness and width, respectively.⁹

For the ‖ direction and when L ≫ a_p,b_p, the phonon kinetic theory provides (see ref. 10) the expression

where ψ and δ are the polar and azimuthal angles respectively. In order to take into account the influence of the length of nanowires, we modify expression (7) (following ref. 11) intonote that

When the angle between the orientation of wires and the heat flux is θ, then (6) and (8) combine to give

Λ(s,L,a_p,b_p,θ) = Λ^(∥)_p(L)cos² θ + Λ^(⊥)_psin² θ (9)

Expressions (4) and (9) help us to show the influence of boundaries and their properties created by incorporating nanoscale particle sizes into the homogeneous matrix. Substituting these expressions into (2), the thermal conductivities of the matrix and dispersed phases which are function of (s,L,a_p,b_p,θ) are determined. Thus, recalling (1) in its appropriate form, the overall heat conductivity coefficient of the nanodispersion of rectangular cuboid wires is written as

k(s,L,a_p,b_p,θ) = s[k^(‖,spec)(s,L,a_p,b_p,θ)cos² θ + k^(⊥,spec)(s,L,a_p,b_p,θ)sin² θ] + (1 − s)[k^(‖,diff)(s,L,a_p,b_p,θ)cos² θ + k^(⊥,diff)(s,L,a_p,b_p,θ)sin² θ] (10)

Next, we compare (qualitatively and quantitatively) our results with available results obtained by microscopic direct simulations. We make the comparison for Bi₂Te₃–Sb₂Te₃ and then SiGe which have been considered as candidates for nanostructure materials used in thermoelectric applications. The material parameters are presented in Table 1. Note that for the sake of consistency between the structure of our nanodispersion and the nanodispersions investigated previously in ref. 12 and 13, it is assumed that a_p = b_p.

Table 1 Material parameters used in calculations

Material	C [×10^6 J m^−3 K^−1]	v [m s^−1]	Λb [nm]	Density [kg m^−3]
Si (ref. 13)	1.001	2403	172	2330
Ge (ref. 13)	0.933	1308	133	5330
Bi2Te3 (ref. 12)	0.5	212	31	—
Sb2Te3 (ref. 12)	0.53	200	25.4	—

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Fig. 1a and b show the influence of the volume fraction ϕ and the size a_p of dispersed Sb₂Te₃ particles on the thermal conductivity of Bi₂Te₃–Sb₂Te₃ nanodispersion perpendicular to the heat flow direction. We note a good agreement with results obtained from the Boltzmann–Peierls transport equation for Bi₂Te₃–Sb₂Te₃ nanodispersion.¹²

Fig. 1 The effect of (a) volume fraction for a_p = 25.4 Å, and (b) particle size on the thermal conductivity of Bi₂Te₃–Sb₂Te₃ nanodispersion in ⊥ direction.

In Fig. 1a, we see that an increase in the volume fraction ϕ leads to a decrease of k^(⊥). This is due to the influence of the phonon–boundary scattering. For a fixed particle size, an increase in the volume fraction increases the chance of the phonon–boundary scattering which in turn causes the thermal conductivity to decrease.

Fig. 1b shows that k^(⊥) increases monotonically with increasing the size of dispersed Sb₂Te₃ particles. The reason behind this enhancement is the decrease of the probability of the phonon–boundary scattering with the increase of a_p leading to a weaker resistance against heat transfer. In other words, the role of the phonon–boundary interactions (i.e. boundary resistance) is suppressed at larger particle sizes and the phonon–phonon scattering (i.e. the internal resistance) dominates. Consequently, the thermal conductivities of the matrix and of the dispersed particles come close to their values in bulk where the boundary effects are absent and the macroscopic nature of the heat conduction prevails. Note that the boundary scattering is expressed in Λ_coll and Λ_TBR and the internal scattering in Λ_b. For large dispersed particles the obstacles to the motion of phonons caused by boundaries are smaller and consequently the thermal conductivity is higher.

We have also compared our results with the Monte Carlo simulations with SiGe nanodispersions.¹³ In Fig. 2a, Si wires are embedded in Ge matrix. The dispersion of Ge particles in Si matrix is presented in Fig. 2b. For both cases, our results are in good agreement with results of the Monte Carlo simulation in particular for smaller particle sizes. These results confirm again that the boundary scattering, caused by the presence of nanoscale particles, creates strong obstacles against the phonon mean free path which then results in lower thermal conductivity of the dispersion. We can indeed anticipate that a simultaneous decrease of the particle size and increase of the volume fraction of the dispersed phase will intensify impact of the phonon–boundary scattering. In other words, the probability of phonon–boundary scattering will be greater than the probability of phonon–phonon scattering at higher volume fractions and smaller particle sizes. This then leads to a significant reduction in the thermal conductivity.

Fig. 2 The sensitivity of the thermal conductivity of (a) Si nanowires embedded in Ge matrix for ⊥ direction, (b) Ge nanowires embedded in Si matrix for ⊥ direction, and (c) Si nanowires in Ge matrix for ‖ direction.

In Fig. 2c, we show the agreement between our results and results presented in ref. 13 for the thermal conductivity of nanodispersion of Si embedded rectangular cuboid wires in Ge matrix where Si wires are oriented in ‖ direction to the heat flow. Comparing to the thermal conductivity in ⊥ direction, higher thermal conductivity is expected in the ‖ direction due to less phonon–boundary scattering events.

As the final comment on Fig. 1a and 2, we can see a dramatic dependence of the thermal conductivity on the degree of homogeneity of the dispersion. This means that in these graphs the thermal conductivity curves show an exponential decay behavior with high sensitivity to an incorporation of dispersed phase at low volume fractions where a small amount of dispersed particles results in a significant reduction in the thermal conductivity of the dispersion. This sharp drop in the thermal conductivity can be interpreted by the concept of interparticle distance. In this region, a smaller change in the volume fraction of the dispersed phase at a certain particle size induces an intense reduction in the interparticle distance which then dramatically increases the chance of the phonon–boundary scattering which then leads to a sharp reduction in the thermal conductivity. For the central region of the graph we note that the thermal conductivity is almost independent of ϕ. In this region the size of the dispersed particles influences more the thermal conductivity rather than the volume fraction (the behavior investigated in ref. 14–17).

Fig. 3 depicts the influence of other parameters on the thermal conductivity. In order to take into account the role of the nanowire alignment, the orientation distribution function f(θ) presented in ref. 18 is considered:

where p and q are the shape parameters. We note that when p = q = 1 then the most probable orientation angle θ_mod = π/4; p = 1 and q > 1 or p > 1 and q = 1, θ_mod > π/4; when p = 1/2, θ_mod = 0; q = 1/2, θ_mod = π/2. The nanowires are distributed randomly if p = q = 1/2.

Fig. 3 Sensitivity of the thermal conductivity of the Si nanowires incorporated in Ge matrix to orientation of nanowires.

In Fig. 3, the dispersion of Si nanowires of thickness a_p, width b_p and length L in the Ge matrix is shown. It is observed that the thermal conductivity of dispersion decreases when the orientation of nanowires changes from 0 toward π/2. This reduction arises due to an increase in the probability of phonon–boundary scattering. In other words, once the orientation of the nanowires tends toward π/2, the effective cross section for scattering of phonons by boundaries increases. Consequently, phonons experience more boundary scatterings and as a result the thermal conductivity decreases. We also point out that the thermal conductivity of dispersion increases with increasing the probability of the specular scattering of phonons on the particle–matrix interface (i.e. with the increase of the specularity parameter “s”). One can see that the higher the specularity parameter, the lower the phonon confinement at the interface and consequently the larger the thermal conductivity. Finally, we note that phonons encounter more obstacles if the width of nanowire b_p decreases which then means that the thermal conductivity decreases.

3 Temperature dependence

Now we turn to the main objective of this paper. We use the expressions obtained in the previous section and similar expressions obtained in ref. 1–3 to investigate the thermal dependency of the effective heat thermal conductivity of hybrid heterogeneous materials.

We begin by specifying the temperature dependence of all the microscopic quantities entering the expression (1) and similar expressions obtained in ref. 1–3.

First, we assume (following ref. 19 and 20) that the phonon group velocity is independent of the temperature.

For the specific heat capacity, we assume^21,22

where ħ is the reduced Planck’s constant, k_B is the Boltzmann constant.

For the bulk mean free paths, we take

where B and θ are constant parameters determined by fitting experimental data.^20,21

The temperature dependence of the transmission coefficients (t^(diff)_mp and t^(diff)_pm) entering (5) is given in ref. 1

By substituting the above expressions into (2) we obtain^19,20

As for the phonon–boundary scattering, we limit ourselves hereafter only to the diffuse scattering.

As the case study, we take the dispersion of Si wires in Ge matrix. This type of heterogeneous materials is widely used in thermoelectric applications. First, we need to identify the constants entering the expression (13). For silicon and germanium, we use ref. 1 where the constants have been found by making the best fit with experimental results reported in ref. 23. For Si the values are: B = 5.753 × 10⁻²³ s² m⁻¹ K⁻¹ and θ = 199.2 K. For Ge the values are: B = 1.655 × 10⁻²² s² m⁻¹ K⁻¹ and θ = 78.92 K. Finally, we use the temperature dependence given in (4) and (9).

For the temperature dependence investigation we use the material parameters calculated using the Debye model. This then means that the Debye frequency cutoffs of Si and Ge are assumed to be 9.125 × 10¹³ s⁻¹ and 5.14 × 10¹³ s⁻¹ (ref. 24) and the phonon group velocities for Si and Ge 6400 m s⁻¹ and 3900 m s⁻¹,²⁵ respectively.

Fig. 4a and b, depict k^(⊥) as a function of the particle size (note that a_p = b_p) for ϕ = 0.05 and ϕ = 0.4 at four different temperatures. For lower temperatures, the thermal conductivity decreases monotonically with decreasing the particle size. This is due to the relative increasing of the boundary scattering area per unit volume. As the temperature increases, the sensitivity of the thermal conductivity to the particle size is weakened and we can observe a smooth trend at higher temperatures. We note that as the temperature increases, the impact of phonon–phonon scattering reflected by the Λ_b term is strengthened and at the same time the influence of the phonon–boundary scattering expressed in Λ_coll and Λ_TBR terms becomes less significant. In other words, reduction of the thermal conductivity by creating more obstacles against phonon transport is smaller at higher temperatures.

Fig. 4 Size dependent thermal conductivity of nanodispersions of Si nanowires embedded in Ge matrix for four different temperatures when (a) ϕ_p = 0.05 and (b) ϕ_p = 0.4, respectively.

In order to observe the dependence of k^(⊥) on the temperature for different particle sizes and volume fractions, Fig. 5a and b are presented. In both figures we can observe the same scenario in which a decrease in the thermal conductivity is a consequence of an increase in the temperature. Note that the influence of the particle size (phonon–boundary scattering) is not taken into account if a_p ≫ Λ_b. This means that the resistance against phonon transport caused by boundaries fades. This then implies that the mean free paths and consequently also the thermal conductivities of the matrix and dispersed particles tend toward their corresponding bulk values. At lower temperatures, one can see a strong dependence of the thermal conductivity on the particle size and the volume fraction, while, apart from particle size and volume fraction, almost the same thermal conductivities are observed at higher temperatures. To sum it up, the phonon–phonon scattering becomes more important and the phonon–boundary scattering less important as the temperature increases. We also see that the thermal conductivity of dispersion is almost independent of temperature for higher volume fractions and smaller particle sizes (see Fig. 5a for a_p = 10 nm and ϕ = 0.3). In other word, the influence of the boundary scattering dominates and the contribution of phonon–phonon scattering (i.e. internal scattering) becomes even smaller than that of boundary scattering when the nanodispersion is fabricated with higher volume fraction and smaller size dispersed particles. Generally speaking, the lower the temperature, the higher the contribution of phonon–boundary scattering (i.e. boundary resistance) while the higher the temperature, the more significant the phonon–phonon scattering (i.e. the internal resistance).

Fig. 5 Effect of temperature on the thermal conductivity of nanodispersions of Si nanowires within Ge matrix for three different volume fractions and particle sizes (a) a_p = b_p = 10 nm and (b) a_p = b_p = 60 nm.

Fig. 6a depicts the comparison between k^(‖) and k^(⊥). Since phonons experience less boundary scattering when they travel along the L direction, the higher thermal conductivity is expected in the ‖ direction. k^(‖) vs. ϕ is illustrated in Fig. 6b. When the volume fraction increases, a significant reduction of thermal conductivity is observed for lower temperatures, while the sensitivity of the thermal conductivity to the volume fraction becomes reduced as the temperature is elevated. Such behavior again confirms the importance of the boundary scattering at lower temperatures and the internal scattering at higher temperatures. In other words, when temperature is low (say 200 K) and volume fraction increases monotonically, the boundary scattering plays the key role and the thermal conductivity of nanodispersion (apart from either in ‖ or ⊥ directions) experiences a significant reduction due to an increase of the effective area for phonon–boundary scattering. Consequently, as the temperature increases (for example 1000 K) the scenario changes and the contribution of phonon–phonon scattering becomes important, thereby an increase in the volume fraction of the dispersed particles does not affect significantly the thermal conductivity. We see a weak sensitivity of the thermal conductivity of the nanodispersion to changes in the volume fraction (note an almost plateau curve for k^{‖ or ⊥} vs. ϕ).

Fig. 6 (a) Comparison between the thermal conductivity of nanodispersions in ‖ and ⊥ directions as a function of temperature for different particle sizes and ϕ_p = 0.25 and (b) thermal conductivity of nanodispersions as a function of the volume fraction for a_p = b_p = 25 nm and different temperatures. Nanodispersions of Si nanowires/Ge matrix has been illustrated.

We turn now our attention to our previous work (presented in ref. 3) where we have investigated hybrid dispersion of nanowires and nanospheres. We concentrate on the temperature dependence. The hybrid dispersion under investigation is composed of nanowires with radius of a^(w)_p, length of L^(w) and volume fraction of ϕ^(w)_p and nanospheres with radius of a^(sph)_p and volume fraction of ϕ^(sph)_p. In ref. 3, two different methods of homogenizations have been derived and performed on such hybrid dispersions. It was observed that the results of these different approaches are in excellent agreement. For the sake of simplicity, we take here the second method of homogenization presented in ref. 3. In the study of the influence of temperature, we assume that only diffuse scattering of phonons on the interfaces takes place. In Fig. 7, the thermal conductivity of the hybrid dispersion in the ⊥ direction is plotted against the volume fraction of spheres, ϕ^(sph)_p, for four different temperatures. Note that in this figure, a^(w)_p = 10 nm, a^(sph)_p = 100 nm, L^(w) ≫ a^(w)_p & a^(sph)_p and ϕ^(sph)_p + ϕ^(w)_p = ϕ^(t)_p, where ϕ^(t)_p is the total volume fraction of the dispersed components. In Fig. 7, ϕ^(t)_p = 0.4. For lower temperatures, we note that the thermal conductivity of the hybrid dispersion becomes smaller when the volume fraction of nanospheres decreases (ϕ^(sph)_p → 0). Provided a^(w)_p < a^(sph)_p, such behavior is a consequence of the confinement of the phonon transport by boundaries when ϕ^(w)_p → ϕ^(t)_p. When the temperature increases, another scenario emerges in which the sensitivity of the thermal conductivity to the composition of hybrid dispersion significantly decreases and almost a plateau line is observed (see Fig. 7 for 1000 K). This behavior is observed because the possibility of phonon scattering at high temperatures is mostly the internal (phonon–phonon) scattering and subsequently the role of the boundary (the phonon–boundary scattering) is smaller. Summing up, we note that fabrication of hybrid dispersions may not lead to a desired flexibility in influencing the thermal conductivity if the temperatures used in applications are high. We see that at higher temperatures, apart from the particle sizes, the thermal conductivity of the hybrid dispersion does not dramatically vary from those nanodispersions of single particle shape of nanowires (ϕ^(w)_p = ϕ^(t)_p) or nanospheres (ϕ^(sph)_p = ϕ^(t)_p). Fig. 8 and 9 also confirm such observation.

Fig. 7 Thermal conductivity of hybrid dispersion of Si nanospheres and nanowires in Ge matrix against the volume fraction of nanospheres for different temperatures when a^(w)_p = 10 nm, a^(sph)_p = 100 nm, L^(w) ≫ a^(w)_p & a^(sph)_p, θ = π/2 and ϕ^(t)_p = 0.4.

Fig. 8 Sensitivity of the thermal conductivity of hybrid dispersion of Si nanospheres and nanowires in Ge matrix to temperature for different particle sizes when L^(w) ≫ a^(w)_p & a^(sph)_p, θ = π/2, ϕ^(t)_p = 0.3 and ϕ^(sph)_p = 0.15.

Fig. 9 Thermal conductivity of hybrid dispersion of Si nanospheres and nanowires in Ge matrix as a function of the nanowire particle size for a^(sph)_p = 80 nm, L^(w) ≫ a^(w)_p & a^(sph)_p and temperatures of 200 K and 1000 K in ‖ and ⊥ directions when (a) ϕ^(t)_p = 0.2 and ϕ^(sph)_p = 0.15 and (b) ϕ^(t)_p = 0.6 and ϕ^(sph)_p = 0.2.

The dependence of the thermal conductivity of the hybrid dispersion in the ⊥ direction on the temperature is shown in Fig. 8. One can see a notable difference between the thermal conductivity of hybrid dispersions at low temperatures, while apart from the size of nanoparticles, all curves converge at high temperatures. For lower temperatures, the lower thermal conductivity is expected for a situation in which the influence of phonon–boundary scattering is more significant. Note that for a fixed volume fraction, the smaller the particle size, the stronger the phonon confinement by boundaries. In the opposite situation when a^(w)_p, a^(sph)_p ≫ Λ^(w)_b, Λ^(sph)_b (i.e. sign of macroscopic heat conduction where the boundary scattering is eliminated), the higher thermal conductivity of the hybrid dispersion is achieved at lower temperatures.

Since the boundary scattering does not influence the thermal conductivity at higher temperatures, the advantage of nanostructuration is diminished. Accordingly, no remarkable difference is found between the thermal conductivities of hybrid dispersions with different particle sizes.

As shown in Fig. 9a–b, the thermal conductivity of hybrid dispersions versus particle size of wires has been depicted for two temperatures of 200 K and 1000 K in ‖ and ⊥ directions when a^(sph)_p = 80 nm and L^(w) ≫ a^(w)_p & a^(sph)_p. In Fig. 9a, ϕ^(t)_p = 0.2 and ϕ^(sph)_p = 0.15, and ϕ^(t)_p = 0.6 and ϕ^(sph)_p = 0.2 are considered for Fig. 9b.

The above illustrations show that the thermal conductivity of the hybrid dispersion is insensitive to the composition of hybrid dispersion (i.e. ϕ^(t)_p, ϕ^(sph)_p & ϕ^(w)_p), particle size of nanospheres, particle size and orientation (either in ‖ or ⊥ directions) of nanowires at higher temperatures (see Fig. 9a and b for 1000 K). This means that the strength of phonon–boundary scattering created by incorporation of hybrid nanoparticles with different compositions, sizes, shapes and orientations within a homogeneous matrix reduces as the temperature is elevated. Therefore, for high temperature applications, we can not benefit from fabrication of hybrid dispersions.

On the other hand when the temperature is low, the influence of the phonon–boundary scattering is more significant than the influence of the phonon–phonon scattering and also the influence of changes in the composition, size, shape and orientation of hybrid nanoparticles.

4 Conclusion

In this paper we present two results. First, we adapt the method used in ref. 1–3 to nanodispersions of rectangular cuboid wires. The values of the coefficient of the overall heat conductivity calculated from the analytical formula derived in the paper agree well with results coming from direct simulations. We see that: (i) the smaller the particle size and the higher the volume fraction, the higher the phonon–boundary scattering and the lower the thermal conductivity. (ii) As the specularity of the interface increases, the confinement of phonons by boundaries decreases and the thermal conductivity increases. (iii) When more nanowires are aligned perpendicularly (θ → π/2) to the direction of heat flow then the chance of phonon–boundary scattering increases and the thermal conductivity is lowered.

Second, we systematically study the temperature dependence of the thermal conductivity of hybrid dispersions of particles of various shapes (spheres, circular and rectangular cuboid wires) and sizes. This investigation is based on the formulas (derived in this and the previous papers, ref. 1–3) in which the coefficient of the overall heat conductivity is expressed in terms of microscopic quantities (like the phonon mean free path) for which the temperature dependence is known from the phonon kinetic theory. As for the results, we see that at low temperatures the temperature dependence is sensitive to differences in dispersion morphologies. Such sensitivity disappears at higher temperatures.

Acknowledgements

This article was partially supported by Natural Sciences and Engineering Research Council of Canada. Authors thank Professor Lebon for valuable discussions and useful comments.

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