Thermal transport in organic/inorganic composites

Abstract

Composite materials, which consist of organic and inorganic components, are widely used in various fields because of their excellent mechanical properties, resistance to corrosion, low-cost fabrication, etc. Thermal properties of organic/inorganic composites play a crucial role in some applications such as thermal interface materials for micro-electronic packaging, nano-porous materials for sensor development, thermal insulators for aerospace, and high-performance thermoelectric materials for power generation and refrigeration. In the past few years, many studies have been conducted to reveal the physical mechanism of thermal transport in organic/inorganic composite materials in order to stimulate their practical applications. In this paper, the theoretical and experimental progresses in this field are reviewed. Besides, main factors affecting the thermal conductivity of organic/inorganic composites are discussed, including the intrinsic properties of organic matrix and inorganic fillers, topological structure of composites, loading volume fraction, and the interfacial thermal resistance between fillers and organic matrix.

Keywords

thermal conductivity / organic/inorganic composites / effective medium theory / thermal percolation theory / interfacial thermal resistance

Introduction

Composite material consisting of two or more constituent materials is a good strategy to design materials with outstanding performance characterized by different constituents. Organic materials hold the advantage of lightweight, flexibility, corrosion stability, low cost and easy fabrication. Inorganic materials are introduced into organic matrices to utilize the above mentioned good qualities while enhancing particular properties such as strength, electrical conductivity, thermal conductivity, and magnetism. Organic/inorganic composite materials have been widely used in electronic packaging, thermoelectricity, solar cell, thermal barrier coatings, etc [¹–²].

Efficient thermal management has become a bottleneck for many industrial applications of organic/inorganic composites because the requirements of certain applications are quite rigorous [³–⁵]. For example, ultra-low thermal conductivity is favored in applications such as thermoelectric materials, with the main design concept of maintaining the intrinsic low thermal conductivity of organic materials [⁶–⁹]. On the contrary, it is more challenging to significantly increase the thermal conductivity of organic matrices to meet the demand of heat dissipation in many fields such as modern chip packaging, solar cell, and radiation-absorbent material. Organic/inorganic thermoelectric nanocomposites have recently been developed to combine the good electrical properties of inorganic constituents with the low thermal conductivity and flexibility afforded by organic materials [¹⁰–¹⁵]. Thermal interface resistance plays a major role in designing thermoelectric composite materials since large interfacial thermal resistance would sufficiently restrict the heat transport in composite [¹⁶–¹⁷].

In the field of electronic devices, thermal interface materials are indispensable in thermal management of micro-electronics. Thermal interface materials require flexibility, low cost, and high thermal conductivity. Both electrically conductive and electrically insulated materials are needed for electronic applications, as electrical insulators are applied to packaging and electrical conductor could provide electrical connectivity in computer [¹⁸]. In the past few years, many researchers have shown that the thermal conductivity of polymers could be greatly enhanced by the incorporation of high thermal conductivity inorganic components such as carbon nanotube (CNT) [¹⁹–²⁰], graphite [²¹–²²], silicon nitride [²³–²⁴], boron nitride nanotubes [²⁵], metal nanofibers [²⁶], and so on [²⁷–²⁹]. However, the improvement of thermal conductivity in these organic/inorganic composites is less than the expectation from the existing theories [³⁰,³¹], which may be attributed to the high interfacial thermal resistance between the inorganic fillers and organic matrix, together with the contact thermal resistance between the connected inorganic fillers [³²,³³]. In the case of CNT- or graphite-based organic composites, the geometry of connected CNT or rigid graphite nanoplatelets leads to a point contact with a small contact area of ~10⁻¹⁴ cm² [³⁴]. The weak interaction between the fillers would result in a large thermal contact resistance which suppresses the thermal transport along the connected fillers. Therefore, the interaction between the polymer matrix and inorganic fillers plays an important role to thermal transport in such materials [³⁵]. Besides, clusters of particles or wires can be formed in these composites. When particles or wires with high transport properties are randomly dispersed in the matrix with low thermal and electrical conductivities, the largest cluster can form a percolation network which connects the opposite boundary of the samples. Thus, conductive path of electrons and phonons can be established.

The physical mechanism of electrical conductivity in a composite consisting of conductive fillers embedded in an insulating polymer matrix has been well explained with the classical percolation theory [³⁶–³⁸]. The percolation theory predicts that when the volume fraction of fillers (\varphi) excesses a critical volume fraction (or percolation threshold) ({{\displaystyle \varphi }}_{\mathrm{c}}), the infinite conductive network would be formed and the electrical conductivities of the composites (σ) obey the universal scaling law [³⁹,⁴⁰]:

(1)\sigma ={{\displaystyle \sigma }}_{\mathrm{0}}{\left(\varphi -{{\displaystyle \varphi }}_{\mathrm{c}}\right)}^t,

where σ₀ is the pre-factor, which is determined by the intrinsic electrical conductivity of fillers, and the exponent factor t is dependent on the dimension of the system with t=1.6–2 for three dimensional systems. The critical volume fraction {{\displaystyle \varphi }}_{\mathrm{c}} is related to the aspect ratio of the inorganic fillers, because a larger aspect ratio will lead to the formation of the percolation network with a smaller volume fraction [⁴¹]. Although it has been confirmed by many experiments, the percolation theory remains largely a geometrical and empirical theory, in which many factors in real composite materials, such as the interface resistance, cannot be easily taken into consideration [⁴²]. Recently, it is reported that the thermal percolation behavior has been observed when the volume fraction of fillers reaches the electrical critical volume fraction [⁴³]. But up to now, the recognized thermal transport mechanism is that heat current could pass through both the inorganic fillers and the organic matrix. This is dissimilar to the case of electrical transport in which the contribution from the organic component is ignorable [⁴⁴,⁴⁵]. For this reason, there is no clear evidence of thermal percolation behavior, and it is still an issue of intense debate whether the enhancement in such composites should be interpreted in the concept of thermal percolation or the effective medium theory.

In this review, the latest development of thermal transport properties in the organic/inorganic composites will be summarized from the perspective of both experiment and theoretical studies. First, the observed enhancement of thermal conductivity in experiments are summarized by characterizing inorganic fillers into four types, namely, carbon-based fillers, metal fillers, ceramic fillers and hybrid fillers. Next, the thermal transport mechanism, which remains to be an open question now, is discussed. After that, some summaries and outlooks are given. And finally, the major thermal measurement techniques used in detecting the thermal conductivity of such composites are listed in the Electronic Supplementary Material.

Experimental researches in various composites

Carbon-based fillers

Carbon-based fillers with remarkable physical and mechanical properties have been considered to be the most suitable fillers used in organic/inorganic composites to obtain high thermal transport properties. Graphite, CNT and carbon black are traditional carbon-based fillers. Single graphene sheet shows an extreme high in-plane thermal conductivity of above 3000 W/(m·K) [⁴⁶,⁴⁷]. The thermal conductivity of graphite is as high as 100–400 W/(m·K) [⁴⁸]. Over the past few years, many studies have been conducted by adding graphite or graphene into the organic matrix, such as epoxy [⁴⁹,⁵⁰], polyphenylene sulfide [⁵¹], polystyrene [⁵²] and so on [⁵³], to improve their thermal conductivities.

Figure 1 shows the micro structure of graphene nanoplatelets (GnPs) with different volume fractions dispersed in the epoxy matrix. The sample was synthesized by Shtein et al. by using the planetary centrifugal mixer [⁵⁴,⁵⁵]. The high level of interface roughness shown in Fig. 1(a) indicates good homogenous fillers dispersion. As the loading volume fraction increases from 13% to 19%, the pathway between two connected GnPs increases as shown in Fig. 1(c) and (d), which would lead to the increase in the thermal conductivity of these composites. Figure 2 illustrates the thermal conductivity of different samples as a function of loading volume fraction and defect density. The defects of GnP provide additional phonon-defect scattering, thus cause a lower thermal conductivity of sample C₂ compared with sample H₁₅ at the same loading volume fraction. The optimal thermal conductivity was obtained to be 12.4 W/(m·K) when the filler volume fraction reached 25%, which corresponded to an enhancement of 6800% over the neat polymer (0.2 W/(m·K)). Sample with volume fraction larger than 25% is not studied. The reason should be attributed to the poor mechanical properties of the composites with a higher loading volume fraction during the sample preparation. The GnPs/epoxy composites prepared by using the ball milling technique were also investigated by Guo & Chen and much enhanced thermal conductivity was obtained [⁵⁶]. Eksik et al. presented the thermal conductivity of epoxy composites enhanced by using graphene coated poly-methyl-methacrylate balls, the core-shell form led to a more uniform dispersion and created a more contiguous phonon conduction pathway, resulting in a much higher thermal conductivity than just embedding GnPs into the epoxy resin [⁵⁷]. This approach may be efficient to improve the thermal transport properties in GnP-based organic composites at the low filler loading region.

In addition to the loading volume fraction [⁵⁸], the sheet size and alignment of graphene also play a significant role in determining the thermal conductivity of composites. For the phonon transport in organic/inorganic composites, the interfacial thermal resistance would arise from the different phonon spectra of different phases and the phonon-interface scattering [⁵⁹]. In general, more interfaces per unit volume in the system would lead to a lower thermal conductivity. Therefore, for a given loading fillers volume fraction, the thermal conductivity of the composites increases with the increasing filler sizes due to the smaller interfacial area between the fillers and matrix. Balandin et al. reported that the average size and alignment of GnP were significant physical parameters that affected the thermal transport in graphene laminate films, and linear relationship had been observed between the thermal conductivity and average GnP size [⁶⁰]. Recently, Kumar et al. fabricated the composite sample with highly self-aligned large-area reduced graphene oxide (rGO) dispersed in a poly (vinylidene fluoride-co-hexafluoropropylene) (PVDF-HFP) matrix by using the simple solution casting method followed by the low temperature chemical reduction process. Attributed to fewer defects in the graphene sp² structure, the large-area rGO exhibited a higher thermal conductivity than that of a small-area rGO [⁶¹], thus, providing a phonon transport path of lower thermal resistance in this type of composites [⁶²]. Moreover, Kim et al. investigated the thermal transport properties in polycarbonate composites filled with different sizes or thicknesses of GnPs via a melt-mixing method, and found that a larger lateral size and thickness of GnP increased the likelihood of reducing the phonon-scattering at the matrix-bonded interface and effectively improved the thermal conductivity [⁶³]. Similar increases of thermal conductivity in the organic composites with the incorporation of GnPs were also reported [⁶⁴,⁶⁵]. Li et al. reported that an anisotropic thermal expansion behavior was observed in an aligned multilayer graphene/epoxy composite. When temperature increases, the thermal expansion leads to an enhancement in the alignment and contact region between graphene and matrix, which could effectively reduce the interfacial thermal resistance and thus improve the thermal conductivity [⁶⁶]. They also found that the thermal conductivities both in parallel and perpendicular directions were obtained to be larger than those of dispersed multilayer graphene/epoxy composite.

CNTs, the cylinders with open or closed ends formed by wrapping one or more layers of graphene, are another mostly used carbon-based fillers due to the outstanding mechanical and thermal properties [⁶⁷]. However, the properties of these composites are limited by the quality of CNTs and the difficulties of dispersing and aligning for high loading fractions [⁶⁸]. Marconnet et al. prepared the sample with aligned multi-walled CNT (MWCNT) arrays infiltrated with epoxy by using chemical vapor deposition, which could increase the loading volume fraction up to 20% [⁶⁹]. The strongly nonlinear variation of axial thermal conductivity with volume fraction found in this composite was interpreted by the tube-matrix boundary resistance and tube-tube interaction. Lizundia et al. investigated the thermal properties of poly (L-lactide)/multi-walled CNT composite. They found that the thermal conductivity of composites was much lower than the value estimated from the intrinsic thermal conductivity of the nanotubes and their volume fraction, because of the large thermal resistance of 1.8±0.3 × 10⁻⁸ m²·K/W at nanotube-matrix interface [⁷⁰]. The reduction of thermal interface resistance and the improvement of nanotube dispersion within the polymer matrix are necessary for further improvement of the thermal conductivity. To reduce the interfacial thermal resistance, Cui et al. fabricated a uniform silica shell onto the surface of MWCNTs and then embedded them into the epoxy matrix. The incorporation of silica shell would reduce the module mismatch between the stiff CNTs and the soft epoxy matrix, which could effectively decrease the thermal interfacial resistance and thus yield a higher thermal conductivity [⁷¹]. This non-covalent interaction method does not induce defects into the structure of CNTs and enhances the interfacial area between the CNTs and the matrix. Therefore, it is helpful to improve the thermal conductivity of the composites [⁷²–⁷⁴]. Moreover, surface functionalization is demonstrated to be an effective method to increase the coupling between the CNT and the polymer matrix, but it also leads to the formation of defects and decreases the intrinsic thermal conductivity of CNTs [⁷⁵].

Bonnet et al. investigated the thermal properties of CNT/ polymethylmetacrylate composite thick films at room temperature [⁷⁶]. They found that the incorporation of 7% (vol) CNTs in the polymer matrix enhanced the thermal conductivity of the composite by 55% while the electrical conductivity is increased by several orders of magnitude. Figure 3 depicts the thermal conductivity contributed from the single-walled CNT(SWCNT) network, obtained by subtracting the contribution of the polymer matrix from the measured thermal conductivity, and electrical conductivity versus reduced volume fraction. The power law for both electrical and thermal conductivity is obtained with the critical volume fraction of 0.3% and power law exponent of 1.96 and 2.25 for thermal and electrical conductivity, respectively. They proposed that this result showed that the enhancement of thermal conductivity in the composite was quantified by the percolation of CNT network. However, the effect of interfacial thermal resistance between the CNT and matrix is not considered in this paper. After that, Kapadia et al. proposed that there was no percolation-like behavior of the thermal conductivity at or close to the predicted electrical percolation threshold, and all behaviors were attributed to the interfacial thermal resistance [⁷⁷]. The problem of exact physical mechanism of thermal transport in the composites requires further theoretical and experimental studies [⁶²,⁷⁶,⁷⁸].

Metal fillers

Metal fillers including powders, flakes and nanowires (NWs) of aluminum, copper, iron, bronze, silver, and nickel [⁷⁹,⁸⁰] also serve as good fillers as they hold the advantage of low interfacial thermal resistance, although their intrinsic thermal conductivity is not as high as that of carbon-based nanostructures. The enhancement of thermal conductivity attributes to the continuous conductive paths formed by metal fillers. Figure 4(a) is the photograph of the composite film with nickel NWs embedded in poly (vinylidene-fluoride) (PVDF) matrix [¹⁵]. As demonstrated in Fig. 4(b), the conductive path is formed by randomly dispersed nickel NWs with a weight fraction of 20%. As the loading fraction increases, more connections are formed as exhibited in Fig. 4(c)–(f), which is propitious for the thermal and electrical transport in such composites. However, metal fillers have the drawbacks of increasing the density, reducing the tensile strength, and reducing the breakdown strength of composites, thus limiting their applications when lightweight or flexibility are required, especially for composites requiring electrical insulation.

Gold and silver were frequently used because of their good conductivity and chemical stability. Balachander et al. produced an Au nanowire-filled polydimethylsioxane (PDMS) composite, whose thermal conductivity was ~5.0 W/(m·K) with a filler fraction of 3% (vol). They attributed the superior thermal properties to the tendency of the gold NWs to cold weld. Nano-rod fillers with a diameter>10 nm are unfavorable for cold welding and interconnectivity [⁸¹]. The 1-Tertradecanol/Ag NWs composite containing 11.8% (vol) of Ag NWs showed a high thermal conductivity of 1.46 W/(m·K) [⁸²]. Xu et al. introduced silver NW arrays into polycarbonate templates and the composite showed a thermal conductivity of 30.3 W/(m·K) with a filling fraction of 9% (vol) of Ag NWs [⁸³].

Copper was also used in different matrices to improve the conductivity. Zhu et al. compared two typical copper nanostructures, copper nanoparticles (NPs) with an aspect ratio of 1 and copper NWs with an aspect ratio of ~25. The thermal conductivity of dimethicone was improved from 0.15 W/(m·K) to 0.41 W/(m·K) and to 0.25 W/(m·K) by adding 10% (vol) of Cu NWs and Cu NPs, respectively. It coincided with the view that nanostructures with a high aspect ratio could easily form bridges between themselves and construct effective thermal conductive networks [⁸⁴]. Wang et al. produced Cu NWs with an aspect ratio of as high as 1×10² to 1×10³ and polyacrylate was chosen as the resin matrix. The thermal conductivity reached 2.46 W/(m·K) at a relatively low filler volume fraction of 0.9%(vol), enhanced by 1350% compared with the plain matrix. Moreover, they observed a sharp increase in thermal conductivity when loading fraction increased from 0.8% (vol) to 0.9% (vol) as displayed in Fig. 5. This finding indicated that the percolation threshold is around 0.9% (vol) and continuous Cu NWs network was formed [⁸⁵].

Nickel powder-epoxy resin composites were prepared by Nikkeshi et al. and the thermal conductivity of the composite increased by 7 times at 24.5% (vol) [⁸⁶]. Szostak and Andrzejewski investigated the influence of the size and morphology of fillers grain on the thermal diffusivity of composites by introducing Al, Cu, Fe, bronze power or flake into polyethylene. Adding 10% (wt) of bronze powder increased more than five times the thermal diffusivity and 20% (wt) of aluminum flake increased it by more than twice. More surprisingly, 10% (wt) of iron as a filler decreased the diffusivity [⁸⁷].

Ceramic fillers

Ceramics, such as alumina (Al₂O₃) [⁸⁸], aluminum nitride (AlN) [⁸⁹], boron nitride (BN) [⁹⁰,⁹¹], and silicon carbide (SiC) [⁹²], serve as good fillers due to their high thermal conductivity, high electrical resistivity, and superior thermal and chemical stability. Ishida and Rimdusit proposed that an enhanced thermal conductivity of 32.5 W/(m·K) could be achieved in the BN-filled polybenzoxazine system at its maximum filler loading of 78.5% (vol) [⁹³]. They investigated the effect of particle size on thermal conductivity using a steady-state measurement, and found that larger size fillers could induce a higher thermal conductivity due to forming fewer thermally resistant junctions of the polymer layers than the smaller particle size at the same loading fraction [⁹⁴,⁹⁵].

Li and Hsu investigated the thermal transport properties in the composites constructed by microsized and nanosized BN particles dispersed in the polyimide matrix, and found that the thermal conductivity was obtained to be 1.2 W/(m·K) for a mixture containing 30% (wt) of BN fillers [⁹⁶]. Huang et al. fabricated the thermally conductive epoxy nanocomposite samples by using polyhedral oligosilsesquioxane (POSS) functionalized BN nanotubes as fillers [⁹⁷]. POSS molecules could induce a local low dielectric constant area in the composites and modify the BN surface. The incorporation of modified BN nanotubes resulted in a thermal conductivity enhancement of 1360% in comparison with the pristine epoxy resin at a BN nanotube loading fraction of 30% (wt). As mentioned above, the alignment of fillers is an important parameter for determining the property of composites containing anisotropic fillers, such as BN platelets. As typical two-dimensional materials, BN platelets have a highly anisotropic thermal property with an in-plane thermal conductivity of about 600 W/(m·K) and a through-plane thermal conductivity of only 2–30 W/(m·K) [⁹⁸,⁹⁹]. Recently, the orientation effect on the thermal conductivity of the BN-filled polymer composites was investigated by Lin and coworkers [⁹⁸]. They modified the BN platelets with magnetic iron oxide nanoparticles and controlled the alignment of BN in the employed epoxy matrix by an external magnetic field [¹⁰⁰]. This epoxy composite with aligned BN platelets achieved an enhancement of thermal conductivity at a low filler loading of 20% (wt), which was 104% higher than that of unaligned counterpart.

Hybrid fillers

The strategy of hybrid fillers aims to further improve the thermal conductivity of composite by enhancing the connectivity to compose a more efficient percolation network, and by reducing the thermal contact resistance.

With hybrid fillers, it is possible to achieve a good thermal conductivity at lower loadings of filler, which leads to a lower density, better mechanical properties and lower cost. Goyal and Balandin found that the thermal conductivity of silver-epoxy composite could be improved by a factor of 6 when 5% (vol) of graphene were added [¹⁰¹]. Zhou et al. reported a 23.3 times increase of the thermal conductivity of epoxy when containing 5% (wt) of MWCNTs and 55% (wt) of micro-SiC [¹⁰²]. Lee et al. studied the synergistic effect between spherical and fibrous ceramic fillers, and found the synergistic effect of hybrid fillers could not be obtained above the maximum packing fraction. The role of structuring filler was reduced because of the already existing abundant thermal paths at the excessive filler loading level [¹⁰³].

Thermal contact resistance could be reduced by forming bridges between adjacent fillers or increasing the contact area between them. Fang et al. designed a nano-micro structure of two-dimensional micro-scale hexagonal boron nitride (h-BN) and 0-dimensional nano-scale α-alumina (α-Al₂O₃) hybrid fillers for epoxy composites. The composite showed a high thermal conductivity of 0.808 W/(m·K) (4.3 times that of epoxy) at 26.4% (vol) content of hybrid fillers with BN/Al₂O₃ at their optimal mass ratio of 4:1. Hyper-branched aromatic polymide (HBP) was grafted onto the surface of both BN and Al₂O₃ to achieve a desirable dispersion and a strong interface interaction. It was confirmed that Al₂O₃ particles contributed to better dispersion of BN and served as bridges between separated BN platelets to improve the interconnectivity within the heat conductive network [¹⁰⁴]. Wang et al. used silver nanoparticle-deposited BN nanosheets as fillers in epoxy and improved the thermal conductivity to 3.06 W/(m·K) at a loading of 25.1% (vol). The thermal contact resistance between the silver nanoparticle and the BN is relatively low after the decoration procedure, and the silver NPs could be sintered together during the curing process of epoxy, leading to the formation of the thermally conductive networks [¹⁰⁵]. Yu et al. observed a synergistic effect between the GnPs and SWCNTs in the enhancement of the thermal conductivity of epoxy composites. With a hybrid filler loading of 10% (wt), the thermal conductivity of the composite achieved 1.75 W/(m·K), surpassing the performance of individual GnP and SWCNT fillers. They attributed the enhancement of thermal conductivity mainly to the decrease of the thermal contact resistance as the contact area between GnP and SWCNT is larger than that of point contact between individual fillers [³⁴].

Table 1 gives a brief summary of the experimental progress in improving the thermal conductivity of organic/inorganic composites.

Thermal transport mechanism

The prediction and understanding of thermal transport properties in the composite has been a complex subject of research because the properties of composite materials depend on a lot of structural parameters and physical/chemical properties, such as the properties of organic matrix and inorganic reinforcement, particle shape and size, particle distribution, loading volume fraction, and particularly the interfacial thermal resistance between fillers and the organic matrix. It is known that the effective thermal conductivity of the composite comes from the contribution of both the additional fillers and the organic matrix. Over the past few decades, many significant theoretical efforts have been provided to the development of thermal transport in the composites. These efforts are summarize based on the effective medium theory and the thermal percolation theory in this section.

Effective medium theory

Any discussion of thermal transport in composites must begin with the effective medium theory (EMT). The commonly used methods to estimate the effective thermal conductivity in the mixtures or composites are Maxwell-Garnett (MG) EMT [¹⁰⁶] and Bruggeman EMT [¹⁰⁷] by ignoring the interface effect. The former is first proposed to calculate the electrical conductivity in the composites. Due to the identical dimensionless formulations, the heat transport problem can also be solved in the same way as the problem of electrical conductivity, dielectric constant, and magnetic permeability by just renaming the symbols in the formulas. The solution to this problem fits well with experimental data for thermal transport in a stationary random suppression of spherical particles, in which the volume fraction of fillers is limited to be small [¹⁰⁸]. In this case, the spherical fillers are considered to be isolated in the matrix with no interaction between them, and then effective thermal conductivity expected by MG EMT can be expressed as

(2){\displaystyle \frac{{{\displaystyle k}}_{\mathrm{eff}}}{{{\displaystyle k}}_{\mathrm{m}}}}={\displaystyle \frac{\left(\mathrm{1}-\varphi \right)\left({{\displaystyle k}}_{\mathrm{p}}+\mathrm{2}{{\displaystyle k}}_{\mathrm{m}}\right)+\mathrm{3}\varphi {{\displaystyle k}}_{\mathrm{p}}}{\left(\mathrm{1}-\varphi \right)\left({{\displaystyle k}}_{\mathrm{p}}+\mathrm{2}{{\displaystyle k}}_{\mathrm{m}}\right)+\mathrm{3}\varphi {{\displaystyle k}}_{\mathrm{m}}}},

where k_eff is the effective thermal conductivity of the composite, k_p the thermal conductivity of the fillers, and k_m the thermal conductivity of the employed matrix. The solution to Eq. (2) is calculated to the first order of \varphi by Maxwell. After that, a second order ({\varphi }^{\mathrm{2}}) formulation extended from Maxwell’s result is developed by Jeffrey [¹⁰⁹] and later modified by several authors [¹¹⁰–¹¹²]. All these calculations are still limited to small \varphi. The Bruggeman EMT based on the mean field approach has no limitation on the fraction of fillers. For a binary composite with spherical fillers randomly dispersed in organic matrix, Bruggeman EMT gives

(3)\varphi \left({\displaystyle \frac{{{\displaystyle {{\displaystyle k}}_{\mathrm{p}}-k}}_{\mathrm{eff}}}{{{\displaystyle k}}_{\mathrm{p}}+{{\displaystyle \mathrm{2}k}}_{\mathrm{eff}}}}\right)+\left(\mathrm{1}-\varphi \right)\left({\displaystyle \frac{{{\displaystyle {{\displaystyle k}}_{\mathrm{m}}-k}}_{\mathrm{eff}}}{{{\displaystyle k}}_{\mathrm{m}}+{{\displaystyle \mathrm{2}k}}_{\mathrm{eff}}}}\right)=\mathrm{0.}

For low volume fraction of fillers, the effective thermal conductivity expected by Bruggeman EMT is almost the same as that of MG EMT. All of these theories based on diffusive heat transport in the composites system only consider the particle shape and volume fraction as variables, which provide a good description for systems with micrometer or larger-size fillers, but fail to describe the thermal transport in nanofluids [¹¹³].

Some experiments on nanofluids made by CNT or metal NP suspended in organic solution indicated great enhancement in thermal conductivity and strong temperature dependence [¹¹⁴–¹¹⁶]. Keblinski et al. proposed four possible explanations such as Brownian motion of the fillers, molecular-level layering of the liquid at the liquid/filler interface, ballistic nature of heat transport in the nanoparticles, and the effects of filler clustering, for this anomalous increase [¹¹⁷]. These mechanisms are able to explain the thermal transport behavior partially. After that, a more comprehensive theoretical model consisting by stationary particle model and moving particle model was proposed by Kumar et al. to consider the efforts of particle radius, concentration, and the temperature of the medium on the thermal conductivity of composites [¹¹⁸]. The stationary particle model assumes that there are two parallel paths of heat transport through the system, one through the suspended fillers, and the other through the liquid. Assume that the liquid and particle fillers to be spheres of radii r_m and r_p, respectively, then the effective thermal conductivity is derived as

(4){{\displaystyle k}}_{\mathrm{eff}}={{\displaystyle k}}_{\mathrm{m}}\left[\mathrm{1}+{\displaystyle \frac{{{\displaystyle k}}_{\mathrm{p}}\varphi {{\displaystyle r}}_{\mathrm{m}}}{{{\displaystyle k}}_{\mathrm{m}}\left(\mathrm{1}-\varphi \right){{\displaystyle r}}_{\mathrm{p}}}}\right].

It can be seen that the effective thermal conductivity is proportional to the ratio k_p/k_m and volume fraction of fillers for , and inversely proportional to the radii of filler particles [¹¹⁹]. On the other hand, the moving particle model presents that the thermal conductivity of fillers denoted by k_p in Eq. (4) is proportional to its mean velocity (v_p). They proposed that the mean velocity of fillers could be taken as Brownian motion velocity at a given temperature, which can be estimated by the Stokes-Einstein formula. Therefore, the temperature dependence of k_eff is attributed to the mean velocity of filler particle with temperature.

Most studies mentioned above have focused on the idealized case of perfect interface contact. However, the incorporation of inorganic fillers in the organic-based composites would introduce the interfacial thermal resistance, known as Kapitza resistance (R_K) [¹²⁰], which plays a significant role in determining the thermal conductivity of the composites [⁶⁶,¹²¹]. The first two theoretical models taking the interfacial thermal resistance into consideration were proposed by Hasselman and Johnson [¹²²] and by Benvensite [¹²³], respectively, both starting from MG EMT. A more general EMT formulation was developed by Nan and coworkers to predict the effective thermal conductivity of arbitrary particulate composites with interfacial thermal resistance, based on the multiple-scattering theory [¹²⁴]. They considered the properties of the matrix and inorganic reinforcement, particle size and size distribution, volume fraction, interface resistance, and the effect of shape. For spherical fillers, the effective thermal conductivity as a function of interfacial thermal resistance and loading volume fraction can be expressed as

(5){{\displaystyle k}}_{\mathrm{eff}}={{\displaystyle k}}_{\mathrm{m}}{\displaystyle \frac{{{\displaystyle k}}_{\mathrm{p}}\left(\mathrm{1}+\mathrm{2}\alpha \right)+\mathrm{2}{{\displaystyle k}}_{\mathrm{m}}+\mathrm{2}\varphi \left[{{\displaystyle k}}_{\mathrm{p}}\left(\mathrm{1}-\alpha \right)-{{\displaystyle k}}_{\mathrm{m}}\right]}{{{\displaystyle k}}_{\mathrm{p}}\left(\mathrm{1}+\mathrm{2}\alpha \right)+\mathrm{2}{{\displaystyle k}}_{\mathrm{m}}-\varphi \left[{{\displaystyle k}}_{\mathrm{p}}\left(\mathrm{1}-\alpha \right)-{{\displaystyle k}}_{\mathrm{m}}\right]}},

where, the dimensionless parameter α is equal to R_Kk_m/a, where a is the radii of spherical fillers and R_Kk_m the so-called Kapitza radius. In the case of fiber fillers with length L and diameter d randomly embedded in the matrix, the effective thermal conductivity of the isotropic composites becomes [¹²⁵]

(6){{\displaystyle k}}_{\mathrm{eff}}={{\displaystyle k}}_{\mathrm{m}}{\displaystyle \frac{\mathrm{3}+\varphi \left({{\displaystyle \beta }}_x+{{\displaystyle \beta }}_z\right)}{\mathrm{3}-\varphi {{\displaystyle \beta }}_x}},

with

{{\displaystyle \beta }}_x={\displaystyle \frac{\mathrm{2}\left({{\displaystyle k}}_{\mathrm{p}}-{{\displaystyle k}}_{\mathrm{m}}\gamma \right)}{{{\displaystyle k}}_{\mathrm{p}}+{{\displaystyle k}}_{\mathrm{m}}\gamma }},\gamma =\mathrm{1}+{\displaystyle \frac{\mathrm{2}{{\displaystyle R}}_{\mathrm{K}}{{\displaystyle k}}_{\mathrm{p}}}{d}},

(7){{\displaystyle \beta }}_z={\displaystyle \frac{{{\displaystyle k}}_{\mathrm{p}}}{{{\displaystyle k}}_{\mathrm{m}}L+\mathrm{2}{{\displaystyle R}}_{\mathrm{K}}{{\displaystyle k}}_{\mathrm{p}}{{\displaystyle k}}_{\mathrm{m}}}}-\mathrm{1.}

Except for the effect of thermal interfacial thermal resistance between the fillers and matrix, the interaction between the filler particle is also a significant parameter for determining the thermal conductivity of composite [¹²⁶]. The crowding factor model was proposed by Ordonez-Miranda et al. based on the models suggested by Nan et al. [¹²⁵] and Ordonez-Miranda et al. [¹²⁷] considering the particle interactions by means of the crowding factor [¹²⁸]. This model is applied for predicting the thermal conductivity of composites made up of dielectric or metallic particles embedded in a dielectric matrix with the effects of both interfacial thermal resistance and the electron-phonon coupling included. Minnich and Chen proposed the modified effective medium approximation method which gave a closed-form expression for the thermal conductivity of the composites with nano-size spherical fillers included [¹²⁹]. Such method was based on the effective medium approximation method by considering the interfacial thermal resistance which was developed by Nan et al. [¹²⁴]. As the filler size reaches the order of mean free path, the incorporation of nano-inclusion would introduce an additive interface scattering which could modify the mean free path of phonon, as shown in Fig. 6 [¹³⁰], by applying Matthiessen’s rule and thus reduce the thermal conductivity of the matrix. The increased interface scattering could be characterized by the interface density, defined as the surface area of the nanoparticles per unit volume of composite, which is a primary factor in determining the effective thermal conductivity in the composites.

Thermal percolation theory

The existence of thermal percolation is still in debate. Because there are no thermal insulators, the thermal conductivity of most polymers is on the order of 0.1 W/(m·K) [¹³¹], thus the heat transport through the matrix cannot be ignored in the composite, which is dissimilar to the case of electrical conductivity. For the small volume fraction of fillers, the inclusions are discrete dispersed or form the finite clusters in the composites. As the volume fraction increases, more and more isolated fillers contact each other, and the infinite cluster will be formed when the volume fraction of fillers reach the electrical critical volume fraction predicted by classical percolation theory. How to consider the effect of continuous network and thermal interfacial thermal resistance between the fillers and the matrix is the key to understanding the thermal percolation mechanism in the composites.

It has been observed that the thermal conductivity of a composite material first increases very slowly and then increases very rapidly after a particular filler volume fraction. The thermal conductivity is nearly proportional to the additional amount of filler particles added [¹³²]. Wang et al. proposed the fractal model for predicting the effective thermal conductivity of nanofluid based on the fractal theory, which can well describe the disorder and stochastic process of clustering and polarization of nanoparticles within the mesoscale limit [¹³³,¹³⁴]. The fractal model is a modified multi-component Maxwell model by just substituting the effective thermal conductivity of the fillers clusters and the radius distribution function. Foygel et al. assumed that the thermal conductivity contributed from the oil could be eliminated by subtracting the thermal conductivity of oil from the measured results [³⁶]. The rest is attributed to the continue network formed by the additive CNTs, which can be described by the classical percolation theory. The critical volume fraction of CNT is obtained to be 0.03% due to the high aspect ratio and the percolation exponent factor fitted to be 1.24 [¹¹⁵]. In this case, the thermal resistance of the CNT or contact resistance between the connected CNTs is on the order of 10⁷‒10⁸ W/K, which is in agreement with the experimental data. In general, this assumption is able to be applicable only for an extremely small loading volume fraction, but it does not consider the influence of the interface between CNTs and the matrix in the total thermal conductivity of composites. The effects of thermal interface resistance on the thermal conductivity percolation in composites were examined by Devpura et al., who proposed that thermal interface resistance was most important below the percolation threshold and the increasing thermal interface resistance tended to increase the percolation threshold [¹³⁵].

Besides the theory based on the existing EMT and the classical percolation theory, there are also some researchers investigating the thermal transport in the composites by using the simulation method [¹³⁶–¹³⁸]. Kumar et al. applied the finite volume method to investigate the percolation effect on thermal diffusive transport in nanotube composites, and found that the role of percolation depends critically on the strength of tube-to-substrate and tube-to-tube contact resistances for tube-to-substrate conductivity ratios that are large enough [¹³⁹]. This model shows an agreement with the two-dimensional EMT for low tube densities, but departs significantly from it when tube-tube interaction becomes significant. The effective thermal conductivity varies linearly with the volume fraction of CNT in the low and high density regimes, and nonlinear behavior near critical volume fraction is found for a low thermal conductivity ratio k_m/k_p, low interfacial thermal resistance and high tube-tube conductance, which is indicative of percolation. It should be mentioned that the diffusive transport is valid when the length scale of simulation space is much larger than the phonon mean free path, or the filler density is higher than the critical volume fraction predicted by the classical percolation theory. Otherwise, the ballistic phonon transport should be taken into consideration [¹⁴⁰].

Yang et al. investigated the effect of phonon-interface scattering on the percolation of thermal conductivity in random NW composites by solving the phonon Boltzmann transport equation using Monte Carlo simulation [¹⁴¹]. They found that the forming of continuous percolating network did not cause the dramatic increase of the thermal conductivity in the composites due to the dominant role of phonon-interface scattering. As the volume fraction is below the critical value, the effective thermal conductivity changes slowly. In particular, when the size of NW is decreased to the order of mean free path of the matrix, the effective thermal conductivity in the composites decreases with increasing volume fraction of filler constituents initially, and starts to increase when further increasing \varphi. The reason for this is that the interface density plays a more important role in determining the thermal conductivity of the composites, which is able to effectively decrease the mean free path and then reduce the thermal conductivity of the employed matrix.

Currently, there is no reliable theory to predict the anomalous thermal conductivity of organic/inorganic composites, indicating the need for a new theory that properly accounts for the unique features in such composites. Many theoretical studies based on EMT and the thermal percolation theory have been conducted, but to dated, obtaining the thorough and comprehensive understanding of how heat is transferred in the composites poses the greatest challenge that must be overcome in order to realize the full potential of this new class of heat transfer materials.

Summary and outlook

The recent developments in the thermal transport properties of organic-based composites embedded with inorganic thermal conductors are reviewed from the perspective of both experiment and theoretical studies. It has been discovered that the thermal conductivity in such composites depends on a number of physical and chemical parameters of the inorganic reinforcement, such as particle shape and size, particle distribution, loading volume fraction, and the interfacial properties of connected-fillers interface and filler-matrix interface. It is worth pointing out that there are several important issues that may attract researchers’ attention in the future.

(1) The intrinsic thermal conductivity of the polymer matrix is a significant factor in determining the effective thermal conductivity of the composites. Increasing the intrinsic thermal conductivity of the polymer employed, such as by structure optimization, may be an effective method to achieve high thermal conductivity of composites.

(2) The thermal conductivity of polymer composites is determined by too many parameters of the fillers, such as loading volume fraction, size and shape, aspect ratio, dispersion and orientation, and so on. It is difficult to conclude that the thermal conductivity of a composite depends on a specific parameter without taking into consideration all other parameters. Thus, a big challenge for the future is to improve existing synthesis techniques or to develop new sample synthesis techniques that will make systematic study of a series of composites differ only in one parameter of the fillers they contain.

(3) High interfacial thermal resistance appears to be a critical factor in reducing the thermal conductivity of organic/inorganic composites. Surface treatment of fillers, such as surface modification or functionalization, can improve the adhesion between fillers and the polymer matrix and thus decrease the interfacial thermal resistance. However, the additional scattering of phonons by the defects induced by surface treatment will reduce the intrinsic thermal conductivity of fillers. Significantly more techniques should be developed to minimize the interfacial thermal resistance between the filler and the matrix, and further increase the effective thermal conductivity.

(4) New theories are needed. At present, most of the experimental results are explained by using EMT or its derivatives, limited in the low loading volume fraction. In this case, the interfacial thermal resistance between the fillers and matrix has a significant effect on determining the thermal conductivity of the composites. As the volume fraction of fillers increases, the continuous percolation network constructed by the fillers will be formed and may become another important parameter to further enhance the thermal conductivity of composites. How to effectively take into account the influence of percolated network and the interfacial thermal resistance between fillers and the matrix is the focus of current studies, which could provide guidance and support for further developing polymer-based composites with a high thermal conductivity for a wide variety of applications.

(5) In addition to thermal conductivity, other properties such as mechanical and dielectric properties of composites should be taken into consideration, which is also very significant for the application of organic/inorganic composites.

Significant progress has been made during the last few years in the research of thermal transport in organic/inorganic composite materials. However, the lack of theoretical understanding of the mechanisms limits the further development of this field. Further theoretical and experimental research investigations are needed to understand the heat transfer characteristics in such type of composites.

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