Investigation of the thermal conductivity enhancement mechanism of polymer composites with carbon-based fillers by scanning thermal microscopy

In order to elucidate the mechanism of enhancement of heat transfer in polymer composites, in this work, we investigated two types of polymer-carbon filler composite. This investigation was made using scanning thermal microscopy (SThM) using the Wollaston microprobe operated in active mode as a function of the carbon filler weight fraction within the polymer matrix. Samples consist of high-density polyethylene (HDPE) filled with 50 μm expanded graphite (EG) and polyvinylidene difluoride (PVDF) containing multiwall carbon nanotubes (MWCNTs). For HDPE/EG samples SThM images allow the detection of zones with a thermal conductance larger than that of the matrix for the highest studied filler’s concentration. These zones correspond to EG filler agglomerations within the polymer and explain the observed enhancement of the thermal conductivity k of the HDPE/EG composite. For PVDF/MWCNTs samples it is found from that k increases from 0.25 W.m -1 .K -1 for pristine PVDF to 0.37 W.m -1 .K -1 for PVDF nanocomposites filled with 8 wt.% MWCNTs. This k variation versus filler concentration is found in good

The insertion of fillers such as ceramics materials 4,5 , metallic materials [6][7][8][9] and carbonbased materials [10][11][12] is a popular strategy to enhance the thermal conductivity of polymer composites. Among all these fillers, carbon-based materials have aroused higher attention and interest of scholars due to their thermophysical properties exhibiting a thermal conductivity that can even reach values higher than several thousands of W.m -1 .K -1 for fillers such as carbon fibers and carbon nanotubes in their axial direction 13 . For such reasons, a large number of studies have been carried out to improve the polymer k using carbon-based nano-fillers [10][11][12][14][15][16][17] . For example, Chirtoc et al. 16 have introduced expanded graphite (EG) fillers into highdensity polyethylene (HDPE) and achieved increased thermal conductivity when the filler concentration increases in the range up to 0.06 graphite volume fraction. However, it was recognized from this study that the experimental results obtained using a modulated This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0099755 3 photothermal radiometry (PTR) method deviate from those predicted by the model of Nan et al. 18 for filler at higher concentrations 16 . A proposed reason was that the non-interacting particle assumption of the model is not valid as the filler concentration increases.
Furthermore, in addition to HDPE, several other types of polymers are widely employed as base matrix for nanocomposites, like polyvinylidenefluoride (PVDF), due to their excellent film formation, mechanical properties, high resistance towards chemicals, alongside a thermal, oxidative, and hydrolytic stability 19,20 . Specifically, Georgousis et al. 17 26,27 or have instead concerned ultra-localized calorimetry measurements 28,29 . The SThM capability to perform thermal conductance images and to detect local thermal conductivity inhomogeneities and distributions 25 have indeed been demonstrated.
It could provide more information to more deeply understand the heat transport enhancement mechanisms in polymer nanocomposites. Consequently, a methodology to estimate the effective thermal conductivity of polymer nanocomposites is required.
Hence, in order to verify the thermal conductivity results obtained by Chirtoc et al. 16 , and to obtain the complementary thermal conductivities for the samples of Georgousis et al. 17 , we will first give a comprehensive methodology developed to measure the effective thermal This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0099755 5 conductivity of polymer composites using SThM and will carry out additional measurements on the same sample types with this methodology. We will experimentally characterize heat transport in such polymer-carbon filler composites (HDPE/EG 16

Materials
High density polyethylene (HDPE) BP 5740 3 VA, supplied by British Petroleum UK, was used as matrix. Expanded graphite, particle sizes of 50 µm (EG50), Ecophit G, supplied by SGL Technologies GmbH. The PVDF (SOLEF1010, Solvay Solexis S.A., Belgium) was supplied in the form of pellets. The MWCNTs (Nanocyl7000, Belgium; purity of 90%, outer diameter of 9.5 nm and length 1.5 µm) were used as received.

HDPE/EG composites
HDPE was filled with 2 % and 10 % weight fraction (wt.%) of 50 μm EG in a Brabender Plasticorder PLE 331 apparatus as described in 16 . The thickness of samples after compression This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0099755 6 molding was around 0.3 mm. Optical observation suggests that large EG flakes were broken during preparation process and formed smaller fragments with observed size less than 10 μm 16 .

PVDF/MWCNTs composites
PVDF matrix with MWCNT fillers of an outer diameter of 9.5 nm and length 1. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

Experimental setup
The experiments were performed in ambient air environment using a NTEGRA-Aura AFM (NT-MDT, Russia) equipped with a SThM module allowing imaging the topography of the sample and the thermal conductance of a sample subsurface simultaneously. The SThM probe used in these works is the Wollaston wire probe 31 (Fig 1). This probe consists of a 5 µm diameter rhodium-plated (10 %) platinum filament (90 %) (Pt/Rh) core and a 35 µm silver shell.
The resistive element at the end of the tip is obtained by electrochemical etching of the silver shell, allowing stripping a 200 µm segment rhodium-plated platinum filament core previously bended into a V-shape. A mirror is glued on the cantilever to reflect the laser toward the AFM photodetector that allows measuring the force between the probe and the sample. We used this probe in active mode and dc regime, and assumed that, due to its relatively high electrical resistance compared to that of the silver shell, only the uncovered Pt/Rh wire generates heat by Joule effect within the probe. A linear relationship between the variation of electrical resistance and the mean probe temperature rise with respect to ambient temperature can be applied to the Pt/Rh resistive element:  respectively. One should note that throughout the manuscript the subscripts oc and ic denote the two steps of the measurements, out of contact and in contact, respectively.

Measurement modeling and calibration
The voltage difference − represent the variation of the thermal conductance of the probe between out of contact ( ) and in contact ( ) configurations: where − = 2 and − = 2 are the electrical power dissipated in the resistive element while the probe is out of contact and in contact with sample, respectively. The temperature at the apex of the Pt/Rh tip has been proved to be around 1.5 time of the mean temperature of the resistive wire 32 . However, it is very difficult to determine experimentally the tip apex part and its dimension. Furthermore, when not in vacuum, the thermal contact area This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0099755 9 is not just limited in the apex part due to heat conduction through the gas 32 . Therefore, we consider in eq. 2 that the whole Pt/Rh wire is isothermal and that the mean temperature ̅ was used to calculate the thermal conductances and .
As represented in Fig 2, which gives the thermal conductance network of the probe out of contact with a sample, can be written as: where corresponds to the heat dissipated to the environment by convection (radiation effect is small enough to be neglected 32 ) and corresponds to the heat conduction to the probe cantilever in Wollaston wire.
When the probe comes into contact with the surface of the sample, can be expressed as: changes to ′ due to the shielding effect of the sample from heat exchange between the tip and environment 22 . In addition, a new heat transfer channel towards the sample arises.
can be expressed as 32 : where is the thermal conductance corresponding to the heat transfer between the probe and the sample through solid-solid contacts, water meniscus and the gas, and is the thermal conductance of the sample. In the case of diffusive samples, can be expressed as 32 : This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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where is the effective radius of the thermal contact assumed discoidal (as exposed in Fig 2). Then the following equation can be obtained by combining eq. 2 to eq. 6, ∆ is a constant. With the assumption that and are invariable with [33][34][35][36] . We may plot − in a curve with the shape − = 1+ + , which can be used in the probe calibration for thermal conductivity measurements 32 .  . − . − ) and SiO 2 ( = . . − . − ). The equation in red is the expression of the fitting curve of the measurements. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0099755 11

Measurement method
The samples from the same series were cut into small pieces and glued on same sample holder to avoid changing the environment around the SThM probe when changing the sample.
Systematical topography analysis is important for properly interpreting the SThM measurements. On one hand, and strongly depend on mechanical contact between the probe and the sample surface, which is affected by the roughness 38  To study the thermal conductivity of a specific area, an average value of for a square containing at least 25 pixels was recorded. For the case of the samples with homogeneous distribution, five of such squares with low roughness were selected and taken into average. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. 12

Results and discussion
The effective thermal conductivity of the two studied material sets (see section 2.2) were measured applying the methodology previously described.  This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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13 volume, what is expected for a composite with low filler concentration such as 2 wt.%.
This result proves the reliable sensitivity of SThM in the studied thermal conductivity range. The deviation of the results obtained by SThM and PTR techniques verifies the inhomogeneity of dispersion of EG fillers at low concentration. In ref. 16 , it has been shown that the experimental result from back-detection PTR method fit well with the effective thermal conductivity of composite model of Nan et al. 18 when filler charge is less than 6 vol.%. It is worth to note that this model is effective with the assumption regarding non-interacting fillers.
No agglomeration of EG observed from SThM is also favorable to the validation of this assumption.
On the contrary to 2 wt.% EG fraction sample, HDPE composite with 10 wt.% filler charge shows a different contrast pattern between thermal and topography images as circled in Fig 4c   (zones A and B). The dark area in thermal image indicates a higher heat transfer from the probe to the sample that corresponds to a higher local thermal conductance of the sample. Meanwhile, the same position on topography image has a shallower color associated with higher heights.
Furthermore, as explained above, the topography of the sample surface may induce some artifacts in the thermal images. This can be linked to the fact that the contact radius may reach more than 5 μm for Wollaston probe on polymeric materials 42 , which is comparable to the size of convex zones for the sample. The scanning of the SThM probe on a concave surface may induce more thermal exchange due to the increase of contact area through the air gap, which does not agree with our results. Therefore, the thermal signal contrast obtained can only be attributed to the variation of the thermal properties beneath the surface of sample. Besides, these This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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14 high thermal conductance regions are distributed as islands inside the sample. The agglomerations of EG fillers are supposed to be the reason behind this phenomenon. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. To further investigate and demonstrate our conjecture, the same scanning procedure was performed on another zone of the sample 90HDPE-10EG with 50μm×50μm area size. The mismatch of the topography and thermal contrast also appears clearly on these images as shown in Fig 6. The distribution curve of the signals is also represented in this figure. The distribution curve for the height of the sample presents only one peak, demonstrating that the surface is flat.

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Two peaks appear on the thermal signal distribution curve, indicating the existence of two levels of thermal conductance within the sample. These peaks cannot be correlated to the topography of the sample surface.
We can observe in the top right corner of Fig 6b a dark area where the sample thermal conductance is larger than elsewhere at the surface. This kind of relatively high thermal dissipation area with such large size only arises while the weight fraction of EG reaches 10 %.
The estimated diameter of this area, which is around 20 μm from thermal image, is larger than EG intercalation diameter 5-10 μm that is observed under optical microscope for HDPE98-EG2 sample 16 . The possible reason of the formation of large heat sink area may be related to the agglomeration due to Van der Waals interaction between graphene platelets 44 of the EG with a high specific surface area [45][46][47] .
All these results further prove the capability of SThM to detect the local thermal conductivity variation among different zones of the polymer composites, which is arisen by the fillers buried beneath the surface.
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.  This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
The dependence of crystallites size on filler concentration is proposed to explain the enhancement of thermal conductivity 53 . It is worth to note that in our measurement, the trend of the thermal conductivity results is in concurrence with the PVDF β and  phase relative percentage versus MWCNTs weight fraction. We can only assume their percentage in melting peaks. Used some mathematic, we can assume This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. The β phase PVDF is a kind of crystalline polar polymer with an all-trans conformation, which is suggested to be favorable for the heat transfer by increasing the mean free path of energy carriers 59 . The γ phase PVDF has significantly lower dipole moment than that of the β phase 60 . Consequently, it is reasonable to suppose that the formation of β and γ phase PVDF may facilitate the thermal transport considering that the electric and thermal current flows at macroscopic dimension share the same equation 49 . The β phase PVDF contribution thanks to higher polarity and crystallinity percentage will be more pronounced than the contribution of γ phase. In summary, the formation of MWCNTs heat conduction paths and a relatively higher crystallinity of PVDF matrix lead to the enhancement of the composite .  This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. For HDPE/EG samples SThM images allow the detection of zones with a thermal conductance larger than that of the matrix for the highest filler concentration studied. These zones correspond to EG filler agglomerates within the polymer and explain the observed enhancement of the thermal conductivity of the HDPE/EG composite. The results obtained by SThM are comparable but lower than those measured by PTR method, flash method and modeling. The reason may be that the inhomogeneous distribution of EG fillers can affect the representativeness of data collected by microscale probe. In this case, the account more heat transfer capability of polymer matrix than the real situation. This has been well demonstrated by the result of HDPE/EG composite with 2 wt.% fillers.

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For PVDF/MWCNTs samples it is found from that thermal conductivity increases from 0.25 W.m -1 .K -1 for pristine PVDF to 0.37 W.m -1 .K -1 for PVDF nanocomposites filled with 8 wt.% This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.