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# Epoxy/graphite nanocomposites as dielectric resins with enhanced thermal conductivity

Renaud Metz, Lurayni Diaz-Chacon, Reinaldo Atencio, Philippe Dieudonne-George

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Renaud Metz, Lurayni Diaz-Chacon, Reinaldo Atencio, Philippe Dieudonne-George. Epoxy/graphite nanocomposites as dielectric resins with enhanced thermal conductivity. Electrical Engineering, 2022, 104 (6), pp.3969-3982. 10.1007/s00202-022-01595-4. hal-04723736

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Submitted on 8 Oct 2024

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Epoxy/Graphite Nano-Composites as Dielectric Resins with Enhanced Thermal Conductivity

Renaud METZ <fnref n="1" />,<fnref n="2" />,* , Lurayni DIAZ-CHACON <fnref n="3" />, Reinaldo ATENCIO<fnref n="4" />, Philippe DIEUDONNÉ-GEORGE<fnref n="1" />

Corresponding author: renaud.metz@gmail.com

ABSTRACT

Most electronic and electrical equipment are coated or encapsulated by epoxy resin due to its physical, chemical
and dielectric properties. However, this material has a major drawback: its low thermal conductivity (~ 0.2 Wm⁻<fnref n="1" />
K⁻<fnref n="1" />). In this context, we have developed and characterized new epoxy/carbon composites in order to improve
the thermal conductivity (κcomp) of this type of resin while maintaining its electrical insulating properties. We
have tested the potential of a range of industrial fillers: graphite nanoplatelets (GnP, exfoliated graphite) and
two reference specimens: Venezuelan petroleum coke and synthetic graphite. Large composite samples
(50x50x4 mm) were prepared by varying the graphite filler content, using an industrial high insulating
anhydride cured-DGEBA epoxy resin of high viscosity (~ 8.5-15 Pa.s) and heavy molecular weight (385gmol⁻<fnref n="1" />).
The best results are obtained from graphite nanoplatelets GnP M: the relative increase of thermal conductivity
reaches 100% (0.37 Wm⁻¹K⁻<fnref n="1" />) at load of 1.13 vol. % while maintaining the electrical resistivity above 10⁵ Ω.m.
These results are interpreted in terms of transport of acoustic phonons in a two-phase system composite and in
the paradigm of the effective medium approximation theory based on the Maxwell-Garnett-Shahil relation. We
show that the graphite-epoxy composite thermal conductivities obtained from different graphite powders can
be discussed as a function on the graphite volume fraction, the graphite filler thickness (Lc), their aspect ratio
and thermal boundary resistance (Rint). We propose to express Rint as a function of κcomp, κf (filler intrinsic
thermal conductivity), filler aspect ratio, and two constant values, one depending on the chemical bonding
strength at the interface between the graphite particles and the matrix, the other depending on the filler
dispersion homogeneity inside the matrix at the macroscopic scale.

<footnote n="1">¹ Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS- Univ Montpellier, Place Eugène Bataillon, Bâtiment 11, F-France.</footnote>

<footnote n="2">² Université Claude Bernard, Lyon1, 43 Avenue du 43 Boulevard du 11 Novembre 1918, 69100 Villeurbanne.</footnote>

<footnote n="3">³ Laboratorio de Materiales para Tecnologías Emergentes. Centro de Investigación en Tecnología de los Materiales. Instituto Venezolano de Investigaciones Científicas, Maracaibo, Venezuela.</footnote>

<footnote n="4">⁴ Departamento de Ciencias Químicas y Ambientales and °Centro de Investigación y Desarrollo de Nanotecnología (CIDNA), Escuela Superior Politécnica del Litoral, ESPOL, Campus Gustavo Galindo km 30.5 Vía Perimetral, Guayaquil, Ecuador</footnote>
INTRODUCTION

The progress in the miniaturization of electronic devices and the increase in the average life of electrical
devices have led to the study of materials for the coating of electrical and electronic components that allow an
efficient thermal dissipation in order to avoid the failures of the electronic devices from overheating [1-6]. Most
electronic or electrical coatings are made of epoxy resins due to their excellent mechanical properties, chemical
resistance and high insulating properties. However, these materials have very low thermal conductivities (0.19
Wm⁻¹K⁻¹) resulting in poor heat dissipation [7-9]. On the other hand, lightweight carbonaceous materials have
a very broad range of conductivity with a record for suspended graphene up to 5,000 Wm⁻¹K⁻¹ [10-19]. Thermal
transport in those materials present the feature to be dominated by phonons at operating temperatures of
electronic devices. The incorporation of low cost carbon materials [20] into polymer matrix during
polymerization process could be an emerging alternative for the improvement of the thermal conductivity of
epoxy resins without significantly affecting their electrical transport. Moreover, nanocomposites differ indeed
from conventional particulate composite materials due to the exceptionally high surface to volume ratio of the
filler phase. The area of the interface between the matrix and filler phase is typically an order of magnitude
greater than for conventional composite materials. As a result, a relatively small amount of nanoscale fillers
may have an observable effect on the macroscale properties of the composite. Several authors have carried out
the preparation of composites based on epoxy resins using graphite nanoplatelets as loading agents, achieving
significant improvements in thermal conductivity; however, preserving the dielectric properties of the resin
remains a challenge [21-24]. The thermal conductivity of a composite material depends on the intrinsic
conductivity of each component, their respective proportions and the thermal resistance (Rint) of the region or
interface located between the charge and the matrix. Rint is not well defined in the literature and depends a priori
on the dispersion of the fillers into the matrix, their crystallinity, their morphology (shape and aspect ratio), and
the strength of the chemical bonding between the fillers and the matrix. In this work, the chosen approach is to
disperse commercial graphite in an industrial epoxy resin without any additional chemicals and non-
sophisticated equipment. We have then developed and characterized new epoxy/carbon composites in order to
improve the thermal conductivity (kcomp) of this type of resin while maintaining its electrical insulating
properties. We have tested the potential of a range of industrial exfoliated graphite fillers: graphite nanoplatelets
(GnP) and two reference specimens: Venezuelan petroleum coke and synthetic graphite. Large composite
samples 50x50x4 mm were prepared by varying the graphite filler content up to 2.7 vol.%, using an industrial
high insulating anhydride cured-DGEBA epoxy resin of high viscosity (~8.5-15 Pa.s) and heavy molecular
weight 385 gmol⁻¹. This paper aims to understand the main parameters which might influence the thermal
conductivity by fitting experimental results obtained on our composites, with calculations using the Maxwell-
Garnett-Shahil relation [21] and to better understand why an anisotropic 2D (conformation) shape of the fillers
with a high aspect ratio helps to enhance the thermal conductivity in composites [25].
Materials

Graphite nanoplatelets (GnP) were purchased from XG Sciences, Inc.. GnP consists of small stacks (~10-150) of graphene obtained by acid intercalation and microwave heating [26]. The size and thickness of GnP are controllable during fabrication. The M and H grade (GnP M, GnP H) are flakes made of elementary thin particles having average thicknesses (L~c~) in the range of 6-8 nm and 15 nm respectively with an average equivalent particle length (L~a~) of 25 microns as XG Sciences specifications. The manufacturer reports oxygen content ≤ 1 wt.% and residual acid content ≤ 0.5 wt.%.

We also used synthetic graphite powder provided by Sigma Aldrich (96-99 wt.%) with average particle size (L~a~) ≤ 20 µm, retarded petroleum coke [27-28], with average particle size ≤ 70 µm, coming from the refinery of José Antonio Páez, Anzoátegui-Venezuela.

A Huntsman epoxy system is carried out as the thermosetting matrix polymer. The epoxy system includes three components: (i) The resin: diglycidyl ether of bisphenol A which is a viscous epoxy monomer (CY225), clear colorless/yellow liquid with a dynamic viscosity of 8.5-15 Pa.S, a density of 1,170-1,200 kgm⁻³ and an epoxy content of 5.10-5.30 equivalent.kg⁻¹ at 25°C. (ii) The hardener: Aradur HY925 is an aromatic cyclic anhydride acid (Tetrahydromethyl-1,3-isobenzofurandione), yellow liquid with a viscosity of 0.3-0.4 Pa.S and a density of 1,190-1,220 kgm⁻³ at room temperature. (iii) A catalyst, included in the resin, being a quaternary ammonium halide salt. The mixing ratio for these components is CY225:HY925 = 100:80 respectively.

Carbon fillers characterization

Morphological characterization of the fillers was done using a field emission scanning electron microscope (FESEM) from FEI Quanta 200 (Carl Zeiss, Germany). The specimens were prepared by both techniques: i) deposition of the raw powder on a conductive support and ii) deposition of a drop from the dispersion obtained after ultra-sonication during 30 minutes on a silicon wafer (SiO₂/Si) with 500 nm of thickness. Most of the images were observed under 1-10 kV acceleration voltage. A detailed analysis was carried out using a methodology known as Image Analysis by Particle Descriptors [29-30] taking into account particle length (L~a~) and thickness (L~c~).

The textural properties of carbon fillers were determined using the Brunauer–Emett–Teller (BET method) by N₂ adsorption measured at 77 K using a BET specific surface area analyzer by Micromeritics ASAP 2010 automatic physisorption analyzer with multi-gas option, all samples were outgassed at 110 °C under vacuum 24 h beforehand.

Powder X-ray analyzes were performed on an INEL XRD3000 diffractometer using a Cu radiation source, (K~α~, λ = 0.15418 nm), in transmission mode. The instrumental resolution β~i~ (rad) = 1.38 10⁻³ was determined from a long range ordered Y₂O₃ crystal powder. The samples were analyzed in glass capillaries, arranged in a rotating sample holder, allowing obtaining average values of the orientation of the crystallites during the experiment.
The measurements were corrected for the transmission values and empty capillary contributions. (110) and (002) X Ray reflections were fitted using a pseudo-Voight function in order to obtain information on the structural order of graphite fillers in the graphene plane direction, and in the perpendicular out of plane directions respectively. For exfoliate graphite (and synthetic graphite), the crystalline graphite fillers present a huge anisotropy with L_a (micron) >> L_c (nm). When the crystal size is less than ~ 150 nm, the Scherrer relation modified by Short and Walker for carbon materials [31] can be applied to evaluate the crystal size, otherwise it gives a correlation distance between defects. As L_c < ~150nm, the Scherrer relation is applied on the (002) reflection to evaluate an average graphite plate thickness L_c* (relation (i)) with L_c ~ L_c*. The Scherrer relation was also used from the (110) reflection to evaluate the correlation distance between defects L_a* (relation (ii)) in the graphene plane direction as L_a > ~150nm and L_a >> L_a*.

$$L_c^* = \frac{K_1 \lambda}{(\beta_{002} - \beta_i) \cos\theta_{002}} \quad (i)$$

$$L_a^* = \frac{K_2 \lambda}{(\beta_{110} - \beta_i) \cos\theta_{110}} \quad (ii)$$

where L_c* and L_a* were calculated from XRD results, as a function of the shape factors [31]: K_1 = 1 and K_2 = 1.84, β_{hkl} is the integral width (in radian) of the (hkl) reflection, β_i is the instrumental resolution and, finally 2θ the diffraction angle of the (hkl) reflection.

## Preparation and characterization of epoxy composites

The carbon fillers, previously heated at 120°C during 24 hours, were dispersed into the HY925, after that, the CY225 was added high speed stirring at 10,000 rpm, the resulting mixture was later degassed at 60 °C and then cured into an oven at 100 °C during 5 hours followed by post curing at 140 °C during 8 h. As reference system, a pristine epoxy resin was prepared. Nanocomposites with filler varying from 0.2 to 2.67 vol.% were processed. The densities used for the calculation of the loading vol.% are: synthetic graphite and GnP M, 2,300 kgm⁻³, petroleum coke, 2,100 kgm⁻³ and epoxy, 1,187 kgm⁻³ respectively. The dispersion of the graphitic particles was estimated by optical microscopy technique in transmission mode with contrast difference, composite films (~ 60 µm thick) were analyzed using a Leica polarization microscope model DM2500 P, using different objectives (2.5X, 20X, 10X, 32X) to observe the samples at different magnifications. Three images of each sample were taken for each magnification.

## Thermal and electrical conductivity measurements

The thermal transport performance of the fabricated composite was characterized by the transient plane source technique based on a disk sensor composed of a double spiral nickel wire for rapid thermal conductivity measurements [32-33]. In our study, a Hot Disk TPS 2500 unit and a Kapton insulated probe (radius 3.200 mm) were carried out to measure the thermal conductivity for the composites at room temperature (22°C). The sensor was sandwiched between both pieces of the same polished composite slab (thickness > 3.5 mm). The sample surfaces were smooth and parallel by means of 400 and 600 grit 3M emery paper. The contact between the flat
and thin TPS sensor and the polished sample was optimal, leading to a substantial reduction in contact thermal
resistance between both elements. A direct current was supplied to the sensor which has the role of heat source
and temperature monitor at the same time. The hot disk method considers only data stemming from heat
propagation in the materials for its calculations. The determined values therefore represent the bulk properties
of the samples. For accuracy issue, the mean deviation parameter was considered correct when it was below 10-
³. For each composite, the heating power supplied by the sensor was fixed at 20 mW between 10 and 20 s. The
associated changes in temperature were recorded and from about seven thermal conductivity measurements
collected per sample a robust mean was calculated.

We used an electrical device that allowed the study of volumetric electrical resistivity of composites at high voltage. Each composite used for measurements was previously conditioned (free of organic and inorganic impurities), after which copper electrodes of 16 mm diameter were placed in the center of the sample. The tested particle composites are fairly large cylinders with a diameter of 25mm and a thickness of 3.4±0.2 mm. Electrical characterization is realized under an AC dielectric meter (Eaton [Sefelec] MGR 50, Dublin, Ireland) which delivers a voltage up to 5 kV with a maximum 5 mA current. (For security purposes the current was limited at 0.6 mA).

RESULTS AND DISCUSSION

Carbon fillers morphology

Analysis of images made of electron microscopy measurements (Fig. 1) allow to evaluate the distributions of
transverse lateral lengths L<sub>a</sub> of the different carbon fillers. Fig. 1a-c displays scanning electron microscopy
(SEM) images (imaging with secondary electrons) of graphite nanoplatelets GnP M, GnP H and synthetic
graphite respectively. Both synthetic graphite and nanoplatelets specimens appear like platelets with irregular
shape and with a wide size distribution (length L<sub>a</sub> and thickness L<sub>c</sub>). Graphite powders typically consist of
polydispersed aggregates of sub-micron elementary platelets which are stacked each other. The elementary
platelets exhibit polydispersity with varying lateral size and thickness. The zoom of Fig. 1a shows a relatively
large micronic bent platelet with transversal length (L<sub>a</sub>) up to 100 m but nanometric thickness (L<sub>c</sub>) less than
~100nm, as a result of the exfoliation process. The main difference of nanoplatelets *versus* graphite is that they
have on the one hand a crumpled paper look due to the presence of bent, folded and wrinkled particles and on
the other hand a population of about only around ~10% of rather thin platelets with thickness which may range
in the supplier specification, i.e. 6 and 15 nm for GnP M and H respectively.

*Retarded petroleum coke* (Fig. 1d) appears as aggregates which are made of elementary particles with more or less isotropic shape but with an apparent broad polydispersity of particle sizes in the micron scale.

We report in **Table 1** the specific surface area (SSA) measured on the raw materials.

It is worth to notice that the specific surface area of 135 and 65 m²/g given by the supplier for GnP M and H
are overestimated compared to our values (51 and 26 m²/g respectively), maybe due to an aggregation of the
nanoplatelets after being conditioned.
$L_a$ estimated from SEM images and the filler aspect ratio ($\frac{L_a}{L_c}$) are reported on Table 1. $\frac{L_a}{L_c}$ values range from 1444 for GnP M to ~1 for the petroleum cock, the latter being very badly crystallized.

Average $L_a$ values of graphite powders can then be obtained by SEM image analysis, but it is not the case for $L_c$ values which can hardly be estimated from SEM images. We propose then to combine SEM $L_a$ values and BET SSA results to assess $L_c$ values by using a simple geometric model of the nanoplatelets: square parallelepiped of edge $L_a$ and thickness $L_c$ and considering the platelets density of 2,200 kg m⁻³ (Fig. 2a).

The Thickness $L_c$ (iii) is then deduced by considering that the surface by volume unit (SSA) can be express as a function on $L_a$ and $L_c$ from the geometric model.

$$L_c = \frac{2 L_a}{(\rho L_a \text{ SSA}-4)} \qquad (iii)$$

Calculated $L_c$ (SEM-BET) values are then reported in Table 1.

Such calculated $L_c$ values can then give us a reliable order of magnitude of $L_c$. However, we have to consider that industrial fillers present an important $L_c$ polydispersity. SSA obtained are average values directly linked to this $L_c$ distribution, and we can then suppose that calculated $L_c$ values are then an average of this distribution. The size of the crystallites $L_c^*$ were also calculated from X-ray diffraction by analyzing the reflections (002) from relation (i). The fit of (002) reflections is shown on figure 3. For well crystalized natural graphite, the (002) reflection (Fig. 3a) can be fitted with a single pseudo-Voight contribution. In the case of synthetic graphite (Fig. 3b) and GnP M (Fig. 3c), the (002) reflection is fitted with two contributions. The broader one at lower 2theta values is attributed to the existence of a proportion of a less ordered turbostratic graphite in those samples. Calculated $L_c^*$ values are then obtained from the higher intensity and smaller width (002) peak.

$L_c^*$ values are reported and compared to $L_c$ values previously obtained (table 2).

From X ray analysis we should consider that $L_c^*$ values are not average values of the size distribution, but correspond to higher values of the distribution because the contribution of bigger crystals on the scattering intensity is much higher than the contribution of smaller crystals. As seen before, $L_c$ calculated values correspond to average values of the distribution and we have then $L_c^* - L_c$ which correspond to ~half of this size distribution. We can then evaluate the order of magnitude of the full width of the distribution of $L_c$ values: ~2 ($L_c^* - L_c$) of our graphite fillers indicated in table 2. We can then observe that the population of highly exfoliated fillers (low $L_c$ values) increases for exfoliates GnP H and GnP M samples with the higher proportion of highly exfoliated fillers in GnP M sample in accordance with SEM observations, but with a broad distribution of $L_c$ values in exfoliated samples.

### Epoxy composites characterization

Large specimens (100×70×4mm) were prepared from an industrial DGEBA resin (CY225/HY925), varying the volume fraction and the nature of the carbon fillers. (This resin whose hardener is an aromatic cyclic anhydride acid exhibits robust dielectric properties but its processing needs specific care due to its high viscosity ~ 8.5-15
Pa.s, which is a priori detrimental for sample degassing and for good fillers dispersion). The dispersion of the charges was studied by transmission optical microscopy performed on thin composite slices of ~60µm. Fig. 4 shows the filler dispersion in epoxy GnP M composite for different filler concentration. The observations were qualitatively the same for the other fillers. Fig. 4 shows that the carbonaceous fillers form large micronic aggregates, indicating the filler dispersion is not optimal as previously expected. In the case of GnP M, for load fractions greater than about 1.1 vol.%, the images suggest that a percolation threshold formed by graphitic aggregates is probably crossed. This microstructural change is important because this could mean that the composite is not anymore electrically insulating when such a percolation threshold is reached, above a filler content of about 1.6 vol.%. Such observations are in accordance with the following electrical conductivity measurements.

Such measurements carried out on composites, show that whatever the carbonaceous filler, the electrical ac
resistivity decreases with the amount of filler as shown on Fig. 5. In the case of retarded petroleum coke fillers,
the decrease is very limited. However, for crystalline graphitic fillers, the fall is dramatic after about 1.5 vol.%
for synthetic graphite and after a lower volume amount between 0.3 and 1 vol% for dispersed nanofillers. We
suggest as an explanation already reported [34], that in the case of fillers having a higher aspect ratio,
percolation threshold is achieved at a lower filler amount. The thermal conductivity properties of the
composites were then measured with hot disk technique based on the transient plane source (TPS). The best
thermal conductivity improvement was obtained with graphite nanoplatelets fillers (Fig. 6). The thermal
conductivity of the composite reaches 0.55 Wm⁻¹K⁻¹ for a maximum permissible load of 2.67 vol.% (limited by
the viscosity of the epoxy resin). The relative increase in thermal conductivity is 66% for 1.13 vol.%. This value
is clearly above what is expected from micronic filler and this value is fairly high, taking into account that the
DGEBA/anhydride industrial epoxy system has a high viscosity and heavy molecular weight. A previous
literature review has shown that among the studies dedicated to graphite-epoxy composites, there are two
families of epoxy system (DGEBA or DGEBF) [35]. Among both families, the best increase of thermal
conductivity is clearly obtained from the less viscous DGEBF resins as a result of a better dispersion of the
filler into the matrix. The diffusion of some monomers and polymers into the pores and galleries of
agglomerates help exfoliation and dispersion during the curing process [35]. The strength of chemical
interactions at the interface epoxy-carbon filler also affects the loading capacity and a good dispersion especially
if there is a good chemical compatibility between both fillers and matrix. Moreover, such chemical compatibility
can favor strong chemical bonds between fillers and matrix, which permit to decrease the interfacial thermal
resistance (Rᵢₘ), another crucial parameter which play an important role on the composite thermal conductivity.
In Fig. 6, it appears also that no increase in thermal conductivity is observed for the epoxy-petroleum coke
composites, regardless of the charge rate. The structure of petroleum coke is very poorly crystallized and
consequently exhibits a fair low intrinsic thermal conductivity (of the order of that of the epoxy). The best result
is acquired from fillers of exfoliated GnP M graphite, whose aspect ratio is high on the order of 1,400.
Interestingly, Fig. 6 shows also a more rapid increase in the thermal conductivity for both GnP M, and GnP H
for filler rates greater than about 1.8 vol%. This increase is consistent with the appearance of another kind of
percolation threshold of the GnP aggregates in the composite as suggested by the observations of the optical
microscopy but which should be distinguished from the electrical conductivity threshold previously reported and which show that the resistivity decreases sharply for epoxy-GnP composites when the charge rate exceeds a lower limit of about 1.3 vol.%. Thus, the difference between thermal and electrical transport responses on the onset of the percolating network suggests to us that the effective channel for electric transport and thermal transport is not the same. It appears to us that the apparent percolation observed by electrical transport measurements (around 1.3 vol% of GnP M), is in accordance with the apparent tortuous percolation network of aggregates observed by optical transmission microscopy. We can suppose that a very tortuous percolated filler/aggregate network (at low filler loadings) can be sufficient to permit electrical conductivity (and loss of dielectric properties). In contrast, the thermal energy flow can be greatly favored by a non-tortuous percolating network (obtained for higher filler loadings around 1.8 vol%). Indeed, it can be observed on fig. 4 that the filler/aggregate network is less tortuous, when filler content is above 1.6 vol%. Our experimental results are then in accordance with such explanation, indicating then a first "electrical percolation threshold" around 1 vol% filler loading, and a second "thermal percolation threshold" around 1.8 vol% filler loading for GnP-epoxy composites.

## Theoretical Approach

The thermal conductivity of graphite crystals is anisotropic since it is greater in the direction of the graphene basal plane (along the transversal L<sub>a</sub> direction) due to collaborative vibrations of sp<sup>2</sup> covalent bonds (up to 2,000 Wm<sup>-1</sup>K<sup>-1</sup> for highly oriented graphite (HOPG)). In the latter, the atoms are positioned in a regular order according to a particularly rigid hexagonal structure. The transport of heat is mainly provided by the vibrations of atoms which can be described as the superimposition of normal modes of acoustic vibrations of low frequencies. In the out of plane direction (along the thickness L<sub>c</sub> direction), Van der Waals bonding gives an ineffective heat flow and the common value of the thermal conductivity is ~ 6 Wm<sup>-1</sup>K<sup>-1</sup>. Interestingly, for thickness below about 500 atomic layers, the out-of-plane thermal conductivity drops. A robust experimental approach by time resolved X-ray diffraction gives 0.7 Wm<sup>-1</sup>K<sup>-1</sup> for 35 nm thick graphite layer [36]. Several simulations and theory confirm disturbances in the phonon transport along the c-axis since a significant portion of phonons are not anymore in a diffusive regime. This results in a strong size dependence of κ with L<sub>c</sub> [37-38] meaning that experimental measurements of κ are limited by L<sub>c</sub>.

Within the framework of quantum mechanics, each proper or normal mode of vibration of a crystal lattice is associated with an elastic wave (k) whose energy is quantified: $(n + ½) \frac{h}{2π} ω$, n being an integer, h the Boltzmann constant and ω the phonon pulsation. The quantum of energy which separates two consecutive levels is called phonon. The mean free path of the phonons is limited by multiple interactions: geometric diffusion where the phonon collides with the edges of the crystal, defects in the distribution of isotopic masses <sup>12</sup>C/<sup>13</sup>C, crystalline defects (0, 1, 2 and 3D), poorly crystallized structures and finally crystal lattice anharmonicity, that is the diffusion between certain phonons sufficiently energetic (Umklapp processes). The latter intrinsic (i) mechanism is temperature dependent. It degrades the thermal current and limits the thermal conductivity at high temperature. The degree of crystal anharmonicity is characterized by the Gruneisen parameter γ, which enters
the expressions for the *Umklapp* scattering rates [39] and describes a change of internal energy (U) under pressure (P) at constant volume (V): $\gamma = V(\frac{\partial P}{\partial U})_V$. The phonon mean-free path (Λ) is related to the relaxation time ($\tau$) as $\Lambda = \tau \nu$ where $\nu$ stands for the phonon group velocity of a specific mode. In the relaxation-time approximation, various scattering mechanisms, which limit $\Lambda$, are additive that is, $\tau^{-1} = \sum \tau_j^{-1}$, where j enumerates the scattering processes previously mentioned. Thus, the thermal conductivity $\kappa_f = \kappa_i + \kappa_e$, $\kappa_i$ thermal conductivity limited by *Umklapp* scattering above room temperature and extrinsic when it is mostly limited by phonon-rough boundary or phonon-defect scattering, thus: $\tau^{-1} = \tau_{i,(T)}^{-1} + \sum \tau_{e,j}^{-1}$ and the thermal conductivity is expressed as: $\kappa_f = \sum \int C_v(\omega)\nu^2(\omega)\tau(\omega)d\omega$, $C_v$ being the heat capacity at constant volume [17].

In graphite, electron-phonon interactions are neglected insofar as phonons preferentially carry heat [17]. (This paradigm has been recently questioned as far as graphene; a single sheet of carbon is of concern since the electron contribution to heat conduction at room temperature might be about 10% [39] of the total $\kappa$). Nevertheless, by analogy with the kinetic theory of gases, and considering the phonon velocity as constant ($\nu = 10 \text{ kms}^{-1}$), the heat capacity ($C_v = 1.33 \ 10^6 \text{ Jm}^{-3}\text{K}^{-1}$) [16] and a mean maximum thermal conductivity value for highly oriented pyrolytic graphite (HOPG) of $2,000 \text{ Wm}^{-1}\text{K}^{-1}$ [17], the mean free path of the phonons $\Lambda = 450 \text{ nm}$, is extracted from the equation:

$$ \kappa_f = \frac{1}{3} C_v \nu \Lambda \qquad (iv) $$

which holds only for diffusive transport when crystallite sizes are far larger than the phonon mean free path. This latter mean free path is a rough mean of the absolute velocities, velocity ($\frac{d\omega}{dk}$) given for a set of phonon of $\omega$ mode. (Interestingly, with a more robust heat capacity of graphite (8.23 Jmol$^{-1}$K$^{-1}$) [40], a molecular weight 12.0107 g and a density of 2,300 kg m$^{-3}$, $C_v = 1.58 \ 10^6 \text{ Jm}^{-3}\text{K}^{-1}$ which gives a value of about $\Lambda_{HOPG} = 381 \text{ nm} \sim 15\%$ lower.)

In this experimental study, the powder fillers are a mixture of micronic aggregates with few isolated nanoplatelets. From XRD, the calculated average distance between defects ($L_a^*$) are lesser than 450 nm (Table 3). The mean free path of HOPG at room temperature is fairly high since it is more than 1/2 of the mean free path of phonons in graphene (L$_a$ direction) ~775 nm [17] and is comparable of the average distance between defects inside the platelets along L$_a$. The platelets result from harsh chemical and physical top-down approach to the synthesis and exhibit *a priori* far more defects than HOPG. To account for the size effects on heat conduction inside the nanoplatelets, the average phonon mean free path is rewritten according to the *formulae*:

$$ \Lambda = \left(\frac{1}{450} + \frac{1}{L_a^*}\right)^{-1} \qquad (v) $$

We report in Table 3 the estimated true mean free path and intrinsic thermal conductivity of the studied nanoplatelets using relation (iv). From Table 3, it appears that the resulting intrinsic thermal conductivity of the platelets (in L$_a$ direction) is lower than the one commonly reported in the literature: ~1,000 WK$^{-1}$m$^{-1}$ [41]. It is worth noticing that the thermal conductivity reaches about $2,200 \text{ Wm}^{-1}\text{K}^{-1}$ for a crystal with $L_a^* = 10 \text{ µm}$ (Figure 7).
To analyze our experimental data at the stage of the composite after curing, we used Maxwell-Garnett effective medium approximation (EMA) modified by Shahil et al. [21] to include the size of the fillers, aspect ratios (Nano-platelets are regarded as oblate spheroid $\frac{L_c}{L_c} \to 0$) and the thermal boundary resistance $R_{int}$. Assuming randomly oriented fillers, the thermal conductivity of the composites is expressed in function of the thermal conductivity of the filler and the matrix $\kappa_f$ and $\kappa_m$ respectively and the volume fraction of filler $f$:

$$ \kappa_{\text{composite}} = \frac{\kappa_f[3\kappa_m + 2f(\kappa_f - \kappa_m)]}{(3-f)\kappa_{nc} + f\kappa_m + \left(\frac{fR_{\text{int}}\kappa_m\kappa_{nc}}{L_c}\right)} \quad (\text{vi}) $$

The precise nature of $R_{int}$ is not clearly defined. It might arise from poor interface between the filler and the matrix resulting from the process carried out, but also unavoidable contact resistance between fillers or interfacial resistance between fillers and matrix. The units of the thermal boundary resistance are the units of thermal insulation (m²KW⁻¹) which is the reciprocal of heat transfer coefficient i.e. the quantity of heat that passes in unit time through a unit area of a plate of particular thickness when its opposite face differs in temperature by one Kelvin. Under uniform conditions, $R_{int}$ is expressed as the ratio of the temperature difference across a region and the heat flux density. The viscosity of the mixture might be detrimental for the filler dispersion, so as the achievement of specimens without any porosity after curing. However, beyond technical issues, interfacial resistance also named Kapitza resistance, comes from the impossibility of coupling perfectly the modes of the phonons between 2D density of states (nanoplatelets) and 3D amorphous organic bulk matrix owing to the large difference in the phonon density of states. The Maxwell-Garnet-Shahil relation shows that the thermal conductivity ($\kappa_{composite}$) depends inversely on $R_{int}$, but also relies on $L_c$ (i.e. the exfoliation), so that:

$$ \kappa_{\text{composite}} \propto \frac{L_c}{R_{\text{int}}} \quad (\text{vii}) $$

Consequently, the thermal conductivity increases with the exfoliation only if $R_{int}$ decreases more rapidly with $L_c : \frac{dR_{int}}{dL_c} > |dL_c|$. In other words, the interfacial surface between the fillers and the matrix or the number of atoms in direct contact with the matrix (which participate directly in the heat transfer), increases relatively more than the decrease of $L_c$ for a given mass of material: $|\Delta R_{int}| > |\Delta L_c|$.

In the case of an epoxy-GnP M composite, the experimentally values calculated for $\kappa_{composite}$ were fitted on Fig. 8 with the intrinsic thermal conductivity of the nanoplatelets of 428 Wm⁻¹K⁻¹ (Table 3) as a function of the volume fraction of the filler. The deduced value for the thermal boundary resistance is: $R_{int} = 2.2 \ 10^{-4} \text{ Km}^2\text{W}^{-1}$ (Fig. 8). The apparent intrinsic thermal conductivity of the platelets is therefore reduced by more than 55% compared to the commonly claimed thermal conductivity of isolated nanoplatelets.

For volume fractions between 0 and ~ 1.5%, the experimental values of $\kappa_{composite}$ are underestimated in relation with the calculated values. We attribute this difference to the presence of aggregates and the polydispersity of filler sizes, which may be relatively disadvantageous for increasing the thermal conductivity of the composite [25]. On the other hand, for volume fraction greater than ~ 1.5%, the experimental values are overstated compared to the calculated measurements. We assume that beyond 1.5 vol.%, the significant increase in thermal conductivity is induced by crossing the percolation threshold and beyond the working domain of the Maxwell-Garnet-Shahil relation.
We proceeded in the same way for the other carbon charges by fitting calculated $\kappa_{\text{composite}}$ to the experimental data. Thus, $R_{\text{int}}$ values have been deduced (Table 4). The calculation of $R_{\text{int}}$ does not depend on the volume fraction of charge in the composite. $R_{\text{int}}$ depends on the shape, surface, intrinsic thermal conductivity of the filler and/or filler aggregates and probably also depends on the chemical nature of the interphase region located between the charge and the matrix. $R_{\text{int}}$ is found to drop linearly with the exfoliation:

$$R_{\text{int}} = 3.1 \ 10^{-4} \ L_{\text{c}} - 9 \ 10^{-4} \qquad (\text{viii})$$

From (viii) and for a given mass of graphite, the decrease of the thermal boundary resistance with the exfoliation

$\frac{dR_{\text{int}}}{dL_{\text{c}}}$ needs to achieve $3 \ 10^{-4} \ \text{m}^2\text{KW}^{-1}$ per exfoliated graphite layer.

Another way to handle the (iv) equation is to extract the actual apparent $\kappa_f$ in our nanocomposite graphite-nanoplatelets- epoxy by fitting $R_{\text{int}}$ from the literature and varying $\kappa_f$ value. Several thermal boundary resistances are given in the literature, for instance $3.710^{-9} \ \text{m}^2\text{KW}^{-1}$ for graphene/epoxy [21], $> 10^{-6} \ \text{m}^2\text{KW}^{-1}$ for black carbon [22] and more recently $7.7 \ 10^{-8} \ \text{m}^2\text{KW}^{-1}$ for graphite/epoxy [42]. We used the former value taking equation (iv) in the validity domain, i.e. beyond thermal percolation (1.3 vol.) and found $k_f=24 \ \text{Wm}^{-1}\text{K}^{-1}$. This value is the apparent conductivity of the filler randomly oriented in the matrix. It is coherent if we take into account the fact that fillers oriented parallel to the heat flow have a maximum contribution ($428 \ \text{Wm}^{-1}\text{K}^{-1}$) although the one oriented perpendicular display almost no contribution due to their thickness ($\ll 0.7 \ \text{Wm}^{-1}\text{K}^{-1}$). This low apparent value is in line with Aliev et al. [43] who describe how bundling, tube-tube interconnections, defects or mis-alignments decrease the effective thermal conductivity of MWCNT from a theoretical $3000 \ \text{Wm}^{-1}\text{K}^{-1}$ to an actual value of $50 \ \text{Wm}^{-1}\text{K}^{-1}$. The value is also in agreement with the work of Veca et al. [44] who report a diffusivity of about $130 \ \text{mm}^2\text{s}^{-1}$ for thin films of carbon nanosheets prepared by filtering suspended nanosheets in dimethylformamide and drying in vacuum at 90°C. Details of the calculations are not reported but assuming a density of $2300 \ \text{kg m}^{-3}$ and a specific heat of $605 \ \text{J kg}^{-1} \text{K}^{-1}$, we estimate the thermal conductivity of $173 \ \text{Wm}^{-1}\text{K}^{-1}$ for an agglomeration of platelets in a sheet form (bulky-paper).

## CONCLUSIONS

We report on improving thermal conductivity while retaining high electrical resistivity of epoxy composites by incorporating carbon nanoplatelets. We have tested the potential of industrial fillers: graphite nanoplatelets (exfoliated graphite) and two reference fillers: Venezuelan retarded petroleum coke, and synthetic graphite. Large composite samples (50x50x4 mm) were prepared by varying the charge vol. %, in a matrix made of an engineering DGEBA/anhydride resin system of high viscosity ~ 8.5-15 Pa.s. and heavy molecular weight (385 $\text{gmol}^{-1}$). The best results were obtained from graphite nanoplatelets (GnP M industrial exfoliated graphite with high aspect ratio (~ 1,400)): the relative increase in thermal conductivity was 66% at 1.13 vol.%, while maintaining high electrical resistivity ($> 10^5 \ \Omega \cdot \text{m}$). Our experimental results, combined with theoretical thermal conductivity calculations, based on consideration of effective medium approximation theory, have led to a better understanding of the interfacial thermal resistance of graphitic charges. The improvement in transport at the interface is explained by the considerable increase in the surface area of contact between the particle assembly
and the polymer matrix. This surface ensures that a large number of atoms in direct contact with the matrix effectively participate in the transfer of heat. To increase the thermal conductivity during the exfoliation, it is needed to fulfill the condition: $\left|\frac{dR_{int}}{dL_c}\right| > |dL_c|$, i.e. the interfacial surface between the fillers and the matrix should increase relatively more than the decrease of $L_c$. A common strategy to get better phonon coupling is by establishing chemical bonds between the fillers and the matrix. However, this study suggests that starting with graphite nanoplatelets with a true high intrinsic thermal conductivity might improve substantially heat transfers at the interfaces. So far, industrial graphite fillers present too much large aggregates and bent, folded and wrinkled platelets which are associated to phonon scattering defects resulting to an apparent intrinsic thermal conductivity far lower than the expected one: ~ 1 k Wm⁻¹K⁻¹. Production of large graphite nanoplatelets with a controlled thickness remains a major challenge.

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ACKNOWLEDGMENTS

The authors would like to acknowledge Schneider Electric S.A, and are very grateful to David Bessières, Marina Hild from “Nanostructured materials for a sustainable development” Franco-Venezuelian PCP program, to Claude Castro-Gimenez from the French Embassy in Venezuela and the FONACIT (Venezuela) for their financial support and encouragements.