Thermal conductivity of nanocomposites with high volume fractions of particles
Introduction
Over the past few decades, significant research efforts have been devoted to the thermal properties of particulate composites, due to their many technological applications ranging from mechanical structures to electronics [1], [2], [3]. Among the wide variety of composites, those based on the dispersion of particles into a continuous matrix have attracted special interest for their ability to improve the thermal performance of the matrix by increasing its usually low thermal conductivity. The effects of microstructural features of these composites play a determinant role on their macroscopic properties; however the understanding of these effects is not well established to date, especially at high volume fractions of particles and sub-micron particle sizes.
The embedding of micro/nano-sized particles with high thermal conductivity in a polymeric matrix is one of the most common and successful methods used to enhance the thermal conductivity of the matrix. Balandin research group [4], [5], [6] has shown that, among a wide variety of materials for the filler particles, the most promising ones to maximize the enhancement of the thermal conductivity of composites are based on carbon materials of high thermal conductivity, as the carbon nanotubes, nanoplatelets, graphene oxide nanoparticles and graphene flakes. For instance, enhancements above 100% have been reported for a small concentration of 1% of carbon nanotubes or graphene loading [4], which have a good coupling to the matrix materials and a geometry that favors the heat conduction through them.
Most analyses on the thermal conductivity of these composites have been performed based on the effective medium approximation (EMA) models, which are based on the Fourier law of heat conduction. For a summary of them the books by Milton [1] and Torquato [3] are recommended. Two other approaches to address the modeling of the thermal conductivity of composites and closely related problems have also been proposed. The first one is based on the phonon-hopping model developed by Braginsky [7] and applied successfully to describe the thermal conductivity of quantum dot superlattices [8] and doped nanocrystalline diamond films [9]. The second method is based on the modification of the Callaway–Klemens approach for materials with fillers’ grains, as was done by Balandin research group [10], [11].
Most of the EMA models have been successful in predicting the thermal conductivity of composites when the volume fraction of particles is small and their sizes are of the order of micrometers or larger. Even though these macro/micro-composites have the required enhancement of the thermal conductivity of the basic matrix (the greater the particle volume fraction is, the greater the enhancement is); they undergo some major disadvantages due to the large sizes of the particles. This is mainly because high particle concentrations are required to obtain appreciable improvements in the thermal conductivities of these composites, which increases their costs and have severely limited their use in practical applications.
One of the most interesting of these models was proposed by Nan et al. [12] who, based on the multiple scattering theory [13], derived a formula for the effective thermal conductivity of composites with arbitrarily oriented spheroidal particles, where the effects of the particle size, shape, orientation, volume fraction and interfacial thermal resistance are considered. However, Nan et al. model has two main drawbacks: (1) it cannot be applied for composites with nano-sized particles, because the model is based on Fourier law, which is not applicable when the particles size is of the order of or smaller than the mean free path of the energy carriers, as it is the case of the nanoparticles. (2) It is just valid in the dilute limit, such that the interactions among the particles are not considered.
The first of these drawbacks has been addressed by Minnich and Chen [2] for spherical and cylindrical particles and more recently by Ordonez-Miranda et al. [14], for spheroidal metallic and non-metallic particles. In both of these works, the bulk mean free path of the energy carriers has been modified, to take into account their multiple interfacial scattering with the particles, and a modified EMA model has been proposed, which could be suitable for macro-, micro- and nano-composites [2], [14]. Even though these works overcome one of the drawbacks of Nan et al. model, its applicability is suitable just in the dilute limit of particles, which is the second drawback of Nan et al. approach. Therefore, a model involving not only the nanoscale effects but also high volume fraction of particles is desirable.
In this work, by combining the well-known Bruggeman integration principle [15], [16] and Ordonez-Miranda et al. model [14], formulas for predicting the effective thermal conductivity of particulate composites with spherical and cylindrical particles are derived. These results could be suitable for situations involving high volume fractions of nanoparticles, where the interfacial scattering of the energy carriers and the thermal interaction among neighboring particles needs to be considered [2], [14], [17].
Section snippets
Theoretical model and solutions
In general, the effective thermal conductivity k of a composite with a small volume fraction f of particles (f ≪ 1) can be written as [3]:where km and kp are the thermal conductivities of the matrix and particles, respectively; and P stands for other properties like the particles size, shape and orientation, and the interfacial thermal resistance. Most of the EMA models derived under the framework of the Fourier law can be written in the form of Eq. (1) [1],
Discussion of the results
In this section, the comparison of the proposed model with Nan et al. results [12], for the thermal conductivity of composites made up of copper particles embedded in a silicon matrix is carried out. It is important to mention that even though the heat conduction through metal–semiconductor interfaces is currently not well-understood, Nan et al. [12], Wong and Bollampally [22], and Wang and Yi [23] have shown that the predictions of the EMA models are in good agreement with the experimental
Conclusions
Formulas for predicting the thermal conductivity of composites with high volume fractions of spherical and cylindrical nanoparticles have been derived and analyzed. In the dilute limit, the obtained results coincide with those previously reported, while for high volume fractions of particles, they exhibit a remarkable difference, which increases when the radius of the particles scales down to the order of the mean free path of the energy carriers and decreases when the interface thermal
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