Thermal conductivity and thermal diffusivity of fullerene

Scientific Reports volume 12, Article number: 9603 (2022) Cite this article 866 Accesses 1 Citations 14 Altmetric Metrics details Owing to their outstanding characteristics, carbon based nanofluids

Scientific Reports volume 12, Article number: 9603 (2022) Cite this article

866 Accesses

1 Citations

14 Altmetric

Metrics details

Owing to their outstanding characteristics, carbon based nanofluids (CbNFs) have been applied to various advanced heat transfer and cooling technologies. It was claimed that these CbNFs can considerably improve the properties of the base working fluids. Among all the thermal characteristics, the thermal conductivity (λ) is regarded as the primary parameter to be considered for the application of nanofluids (NFs). In the present research study we measured for the first time both λ and thermal diffusivity (aT) of very stable fullerene (C60)-based NFs in liquid phase (1,2,3,4-tetrahydronaphthalene and 1,2-dicholorobenzene) by the transient multi-current hot wire technique at atmospheric pressure in a wide range of temperature (254–323 K). Similar to the base liquids (BLs), we observed a slight decrease in λ with an increase in temperature. Additionally, compared to the BLs λ was reduced upon the addition of C60. The results were compared with the predicted ones using different theoretical models. Not much variation in aT was observed between the C60 NFs and the corresponding BLs due partly to the small variation of λ with the addition of C60.

Carbon based family of nanomaterials (CbNFs) comprising of single- and multi-layer graphene, single- and multi-walled carbon nanotubes, graphite, graphene nanoplatelets, graphene quantum dots, graphene oxide, and so on have recently received major attention for the preparation of NFs (i.e. dispersion of CbNFs in base liquids, BLs) because of their various excellent and unique characteristics along with their superior thermal attributes as compared to conventional liquids1,2,3,4,5. The way the CbNFs bond to and through the molecules of the BLs govern the properties like rheology, thermal and electrical conductivity or light absorption/emission. These NFs have exhibited great potential in industrial applications such as solar thermal storage, heat pipes, and energy storage among other advanced heat transfer and cooling technologies; which is attributable to their greater thermal conductivity (λ) and convective heat transfer coefficients compared to the corresponding BLs2,5. In fact, λ improvement is the first expected advantage of using a nanofluid (NF) when applied as a thermal working fluid. When evaluated against metals or metal oxides (Au, Ag, Cu, Fe, CuO, Al2O3, ZnO, etc.), λ in carbon nanostructures is greater due to their high intrinsic λ, low density, strong C–C covalent bonds, and phonon scattering6. For instance, λ of carbon materials has a wide range that varies from 0.2 W/m K for diamond-like carbons to 6000 W/m K for single-walled carbon nano-tube (SWNT)7; which is superior to that of graphene (5300 W/m K for)8, double-walled carbon nano-tube (DWNT, 3986 W/m K) and multi-walled carbon nano-tube (MWNT, 3000 W/m K)5,7,9. Therefore, carbon materials can be applied either as heat insulators (e.g. diamond-like carbons) or heat superconductors (e.g. graphene).

It is worth noting that various experimental and theoretical studies have reported λ enhancement of NFs and the related affecting factors. The results indicated that λ of NFs are normally functions not only of the particle’s thermal conductivity, its concentration in a NF, size and shape, but also the environmental parameters such as the base fluid’s, pH value, surfactant, dispersing agent, and the standing time10. Various investigations have shown that higher λ enhancement of NFs could be obtained when lower λ base fluids were considered2. In addition, it has been reported that λ of NFs is atypically enhanced with a very low volume fraction of nano-additives11,12,13,14. For instance, in a first study by Choi et al.12, a 160% λ enhancement for 1.0 vol% multiwall carbon nano-tube (MWCNT) dispersed in synthetic poly (alpha-olefin) (PAO) oil was reported. Much lower λ enhancement (i.e. an order of magnitude smaller, as low as 7%) of carbon nano-tubes-based NFs were found in various other studies including functionalized MWCNT and different based fluids (water, oil, decene, ethylene glycol, glycerol, Refrigerant R113, etc.)14,15,16,17,18,19,20,21,22,23. In some studies, a decrease in time of λ was observed, especially by the first 10 days from the preparation of the NF, but the reduction rate also decreased with time9.

Contrastingly, some have reported that NFs containing graphene nanomaterials frequently exhibited greater λ compared with those including other nano-particles5. Yu et al.24 stated an increase of λ up to 86% for graphene and graphene oxide (GO) based ethylene glycol NFs (i.e. λ of the GO NF and graphene NF was 4.9 W/m K and 6.8 W/m K, respectively). Lower λ enhancements (below 27%) were claimed in other graphene nanosheets, graphite nano-particles and graphene quantum dots (GQD) based NFs prepared with water or ionic liquids3,25,26,27. It was indicated that, other than the presence of structural defects, inadequate stability, and restacking of graphene oxide, graphene nano-sheets are prone to be coiled, folded, and corrugated at the surface and edges affecting the heat transfer mechanisms and the subsequent measured λ.

Among the above cited CbNFs, much less attention has been given to nano-diamond particles and fullerene based NFs19,28,29,30. This may be attributed to the lower λ of these nanoparticles. For nano-diamond-water based NFs, the stated λ enhancements were up to 22.8%29,30. For fullerene C60 NFs, it was claimed 6% enhancement of λ for oil base liquid; but 3% reduction of λ for water base liquid, and a decrease of λ with increasing C60 volume fraction19,28.

In general, there exists a considerable scattering in the measured λ by different research groups for the same kind of NFs and even for the same volume fraction of nano-additive having the same size. Although an increase in the volume fraction of the nano-additive seems to always have a positive effect on λ of NFs, the enhancement rate is rather dissimilar in different research studies. Such disagreements are attributed to the different sample preparations and stabilization methods. The use of suitable surfactant(s) or dispersing agent(s) to prepare stable NFs, since the nano-additives can form aggregates because of strong van der Waals interactions, not only increases the viscosity but also reduces λ of NFs31,32,33,34. Moreover, even if λ of the nano-particle can directly affect λ of NFs (i.e. for a similar nano-particle content in a given NF, higher λ of the particle generally produces a greater enhancement in λ of the NF), this is not conclusive since λ of NFs is also affected by other aspects such as aggregation, Brownian motion, interfacial nano-layer, surface charge state, and thermal resistance of nanoparticles, among others10.

It must also be mentioned that some research studies have revealed an anomalous λ increment of some NFs with an increase of the nano-additive volume fraction within the base fluid12,35. This is still a controversial issue10,36. The measured λ was found to be significantly higher than the theoretical value predicted by the classical Maxwell model37 as adapted by Hamilton and Crosser38. This was explained by the nature of heat conduction in nano-particles and suspensions, the Brownian motion of nano-particles, the arranged structure at the solid/liquid interface, nano-particle clustering, etc.39,40,41. To date there is no consistent theory to predict the anomalous λ enhancement of NFs. Whether the differences are due to the previous reasons cited or to any other cause, it is still an open question worthy of further investigation both experimentally and theoretically42,43.

As stated earlier, the issue of stability in NFs is one of the causes for the observed contradictions in various published papers on λ measurements5. Achieving a homogenous dispersion with a long-term stability is essential in order to achieve greater thermal properties in the NFs and better heating performance for long periods of time. On the other hand, the bulk of studies in the literature have concentrated on studying λ enhancement in NFs using nano-additives with a much higher λ than the base fluids. The present research study is intended to investigate both the thermal conductivity, λ, and thermal diffusivity, aT, of very stable fullerene C60 based NFs prepared exclusive of any surfactant or dispersing agent. C60 has a spherical cage-like fused-ring structure that exhibits a different bond structure of carbon than the previously cited CbNFs and a slightly greater λ (0.4 W/m K)19,44 than the selected two base liquids, 1,2,3,4-tetrahydronaphthalene (C10H12) and 1,2-dicholorobenzene (C6H4Cl2). The C60 content in the NFs was up to 0.83 vol% in C10H12 and 1.64 vol% in C6H4Cl2. The measurements were carried out in a wide range of temperatures (254 – 323 K) at atmospheric pressure using the transient multi-current hot wire technique, recognized as the most reliable and precise method to measure λ of fluids as it is not adversely affected by convection45,46. The λ experimental results were compared to the theoretical predictions using different available theoretical models proposed to explain the atypical λ enhancement of NFs. The aT of the two base liquids used in the present study and C60 NFs has not yet been systematically studied.

One of the curious issues about the λ augmentation of NFs is whether the noted changes can be clarified by the existing effective-medium theoretical models. Various useful analytical models have been established to predict the actual λ of NFs; whether the nano-additive be spherical, cylindrical, or sheet based10,43. The most popular models for NFs like Maxwell, Bruggeman, and Timofeeva models, consider the nano-additive content and its λ together with that of the base fluid37,38,47,48. The known Brownian motion of micrometer or millimeter nano-additives in suspensions is commonly neglected due to their large size. Nonetheless, its effect was proved to have some impact on λ of NFs because of the involved micro-convection of the fluid surrounding the nano-additives. Some thermal conductivity computer models of NFs consider, for instance, the random Brownian motions through some factors influencing its intensity like temperature, particle size, and density10.

Various models have taken into consideration the influence of convective heat transfer caused by Brownian motion; such as, Sohrabi’s, Koo and Kleinstreuer's, Bhattacharya’s, and Xue's models49,50,51,52. Furthermore, in other models it was claimed to take into consideration the effect of a nano-particle’s clustering; such as, Prasher's, Wang's, and Evans' models53,54,55. Further model developments consider the effect of the interfacial nano-layer51,52,56,57,58,59.

It is apparent in all the previously proposed computer models that the particle volume fraction (ϕ), and λ of the nano-additive and base fluids should be considered. It is worth noting that the first commonly regarded computer model to predict λ of different types of suspensions/solutions is the Maxwell model37. This computer model was established for suspensions having low concentrations of homogeneously dispersed, hard spherical particles, and with no interactions between particles as follows10:

where ϕ is the volume fraction of the dispersed particles in the base liquid and the subscripts f and p refer to the base liquid and particle, respectively.

Wasp et al.10,60 suggested an analogous thermal conductivity model for NFs for a slightly higher concentration of nanoparticles, expressed as:

Based on homogenous spherical particles, Bruggeman61 proposed the following model:

To solve the above equation, Murshed et al.62 simplified the Bruggeman model by offering a direct solution given by:

where $\Delta ={\left(3\phi -1\right)}^{2}{\left(\frac{{\lambda }_{p}}{{\lambda }_{f}}\right)}^{2}+{\left(2-3\phi \right)}^{2}+2\left(2+9\phi -9{\phi }^{2}\right)\left(\frac{{\lambda }_{p}}{{\lambda }_{f}}\right)$

The thermal conductivity computer model proposed by Timofeeva et al.48 is also built on the effective medium theory and results in:

Xue et al.51 proposed a computer model to calculate λ of NFs based on carbon nano-tubes (CNT):

An apparent shortcoming is that the above computer models observably underestimate λ of those NFs prepared with nano-particles with a high λ. This is owed to the fact that the calculated theoretical λ increases with an increase in ${\lambda }_{p}/{\lambda }_{f}$ when the value is below 10 in the Maxwell, Wasp, and Bruggeman models. Conversely, an increase in the rate of the computed λ value slowly becomes moderate with an increase in the ${\lambda }_{p}/{\lambda }_{f}$ ; specifically, when the ${\lambda }_{p}/{\lambda }_{f}$ exceeds 20, the calculated λ of the NF continues nearly constant with an increase in ${\lambda }_{p}/{\lambda }_{f}$. Other methods, as the Timofeeva model, does not consider λ of the nano-particles; which alleviates that particular problem but causes others. Thus, those models do not seem appropriate for NFs created with nanoparticles with a high λ.

Xuan et al.63 established a computer model that considers the Brownian motion of the nano-additives and clusters together with Maxwell`s model:

where KB is Boltzmann constant, Rcl is the mean radius of gyration of the cluster, µf is the viscosity of the base liquid, ρp is the density of the nano-additive, and cp is its specific heat. Because the second term of Eq. (7) is not dimensionally homogeneous.

(i.e. must be in W/m K), this equation was revised by assigning the unit (m/s1/2) to the constant $(\frac{1}{2\sqrt{3\pi }})$ 64.

The obtained λ of the base fluids 1,2,3,4-tetrahydronaphthalene (C10H12) and 1,2-dicholorobenzene (ortho-dicholorobenzene, C6H4Cl2) are plotted in Fig. 1 as a function of temperature along with the published literature data for C10H1265. The presented standard deviations are related to 116–411 readings for each temperature. It can be seen that the reference data for C10H1265 is slightly higher than the measured values (< 2%) but within the margin of error confirming the adequate measurements of the hot wire. It is worth noting that the published λ values of C10H12 were given without inclusion of the corresponding associated errors and the followed measurement technique; while for C6H4Cl2 it is unfeasible to make any comparison due to the scarcity available data. For C6H4Cl2, λ decreases gradually with the increase of temperature being the slope −1.065 10–4 W/m K2. However, for C10H12, no clear decrease of λ with temperature was observed, provided that the slope of the reported data was very small −5.205 10–5 W/m K2. Within the full temperature range studied, λ of C10H12 is higher than that of C6H4Cl2 (9.8–15.7% for 273.9–313.4 K).

Measured λ of C60 NFs and base liquids C10H12 (a) and C6H4Cl2 (b) as a function of temperature (T). Microsoft Excel 2016 was used to generate these figures.

Similar to the base liquids (solvents), as shown in Fig. 2a, λ of the prepared NFs decreased with temperature, indicated by the slope being greater for C60 based C6H4Cl2 NF (−1.46 10–4 W/m K2) than for C60 based C10H12 (−8.45 10–5 W/m K2). Upon the addition of fullerene C60 in C6H4Cl2 (Fig. 2b), a slight reduction of λ was observed for the entire temperature range studied (−2.4% reduction for 1.64 vol% C60), while for C10H12 based NFs, as is depicted in Fig. 2b, a slow decrease in λ was observed with greater C60 concentration but only for temperatures above 303 K (2 , 3.6 and 3.8% reduction for 0.83 vol% C60 in C10H12 at 303.7, 313.2 and 323.2 K, respectively). This contradicts the previously reported λ enhancement of various types of carbon-based nano-additives (graphene, carbon nanotubes, nano-diamond particles, etc.)2,3,5,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,29,35,41; although λ of C60 (0.4 W/m K) is approximately 3.1% and 3.5% higher than that of the based liquids C10H12 and C6H4Cl2, respectively. In fact, λ of various types of carbon based nano-additives is extremely greater than that of C60 (e.g. 5300 W/m K for graphene8, 6000 W/m K for single walled carbon nano-tubes7, 3986 W/m K for double walled carbon nano-tubes7, and 3000 W/m K for multi-walled carbon nano-tubes5,7,9). In comparison to the base liquid, water, Hwang et al.28 also observed a lower λ value for water based C60 NFs, and λ decreased with a greater C60 volume fraction. For the same base liquid, water, MWCNT NFs exhibited greater λ values than that of water (e.g. For MWCNT NF, λ increased by approximately 7.0% at a volume fraction of 1.0%; whereas for C60 it decreased by approximately 3.0% for a volume fraction of 1.5%). In this case, λ of a C60 water based NF is less than that of water. For oil-based liquids, λ was improved for both C60 and MWCNT NFs (e.g. λ increased by 6.0% for 5 vol% of C60 and increased by 8.7% for 0.5 vol% of MWCNT). It was found that λ of MWCNT NF was considerably better than that of a C60 NF because λ of MWCNT is superior than that of C6028. For the same nano-additive loading, it is generally thought that the nano-additive having a higher λ induces a greater λ enhancement of the NFs. However, this conclusion is uncertain for specific types of NFs, since λ of the NFs is concurrently influenced by numerous other things such as aggregation, Brownian motion, interfacial nano-layer, surface charge state, and the thermal resistance of the nanoparticles10. More theoretical studies should be conducted in order to accurately reveal the governing mechanism(s) and justify the reduction in λ observed in our study.

Effect of temperature on λ of NFs prepared with 0.83 vol% in C10H12 and 1.64 vol% in C6H4Cl2 (a) and effect of C60 concentration (ϕ) on λ of NFs prepared with C10H12 for different temperatures (T) (b). Microsoft Excel 2016 was used to generate these figures.

As stated in the previous section thermal conductivity predicted models, the λ of the prepared NFs can be predicted using the mentioned theoretical models [Eqs. (1–7)]. The calculated λ values were plotted against the experimental ones in Fig. 3. In general, good agreements were observed for all models (Maxwell model [Eq. (1)], Wasp et al. model [Eq. (2), Bruggeman model [Eq. (3)], Timofeeva et al. model [Eq. (5)], Xue et al. model [Eq. (6)], Xuan et al. model [Eq. (7)] within 5% confidence except for the Timofeeva et al. model [Eq. (5)], which showed more than 5% deviation from the experimental values (i.e. maximum registered deviations were up to 7%). For small concentrations in volume fraction of C60 in C10H12 (< 0.6 vol.%), the calculated λ values from the Timofeeva et al. model agree well with the experimental one. However, for higher concentrations, this model overestimated λ. These conclusions are attributed to the fact that, in contrast to the other theoretical models, the Timofeeva et al. model does not consider λ of the nano-additive C60. Nearly the same deviations from the experimental data were obtained for the Maxwell, Bruggeman, and Xuan et al. models. The latter model (Xue) is based on the Maxwell model and the influence of Brownian motion of C60 was found to be negligible (4.2 10–4—6.7 10–4% for 1.64 vol% of C60 based C6H4Cl2 NF and 2.6 10–5 – 1.2 10–4% for 0.06 vol% of C60 based C10H12 NF, 2.6 10–4—1.2 10–3% for 0.6 vol% of C60 based C10H12 NF and 3.4 10–4 – 1.6 10–3% for 0.83 vol% of C60 based C10H12 NF). It should be mentioned that the contribution of Brownian motion of C60 increased with an increase in C60 content within the based fluid and with temperature, as can be expected from Eq. (7); although the obtained values were very low. The observed negligible influence of Brownian motion of C60 in both BLs, justifies the similarity of the registered temperature reliance of the λ of the NFs to the BLs. In fact, Brownian motion increases λ of NFs whereas the influence of aggregation is adverse for Brownian motion10. Some researchers reported a negative effect on λ when increasing the temperature. For SiO2 based water NFs, Masuda et al.66 observed a λ enhancement of approximately 10–11% at 31.85 °C, 9–10% at 46.85 °C, and 5–7% at 66.85 °C when the volume loading was increased from 1.1 to 2.3%. This indicates the different influences of temperature on λ of NFs through particle Brownian motion, dispersion stability, and particles clustering. In this study, because of the negligible Brownian motion of C60 and stability of the prepared NFs, the base liquids dictated the temperature dependency of the NFs (i.e. similar trends were observed for the base liquid and the corresponding NFs).

Predicted λ of C60 NFs based C10H12 (a) and in C6H4Cl2 (b) vs. the obtained experimental λ values. Microsoft Excel 2016 was used to generate these figures.

Taking into consideration the deviations of all data for both based liquids, among the considered model, Wasp et al. model60 is the best one (i.e. the obtained deviations from the measured values were −0.4 to 1.1% for 1.64 vol% of C60 in C6H4Cl2 NF; −0.6 −1.1% for 0.06 vol% of C60 in C10H12 NF; −1.5 to 2.2% for 0.6 vol% of C60 in C10H12 NF and 1.5—4.1% for 0.83 vol% of C60 in C10H12 NF).

Apropos that there is not any data for aT that has been reported for the base liquids used in this study, this was determined by the hot wire technique as explained in Measurements section, for both C10H12 and C6H4Cl2 together with the above mentioned C60 NFs. The results are plotted in Fig. 4 as a function of temperature, along with their associated inaccuracies.

Measured aT of C60 NFs together with the corresponding base liquids C10H12 (a) and in C6H4Cl2 (b) as a function of temperature (T). Microsoft Excel 2016 was used to generate these figures.

For C10H12, taking into consideration the standard errors, no clear variation of aT was detected with neither a change in temperature nor with the addition of C60. As indicated in our previous study67, the uncertainty associated to aT (up to 9.5%) acquired by the transient hot-wire method is greater than that of λ (up to 3%). However, a minor decrease with temperature was observed for C6H4Cl2 (i.e. 9.3% when the temperature was increased from 264.6 to 323.3 K). Similarly, as with C10H12, no variations were observed between aT values of C6H4Cl2 and the C60 related NFs for the entire temperature range studied. This result may be attributed partly to the small variation of the λ upon the addition of C60 and with temperature $({a}_{T}=\frac{\lambda }{{\rho c}_{P}})$.

For the first time, very stable fullerene (C60)-based NFs were prepared without any surfactant or dispersing agent using two base liquids C10H12 and C6H4Cl2; and both λ and aT were measured in liquid phase by the transient multi-current hot wire technique at atmospheric pressure in a wide range of temperature (254 – 323 K).

The obtained λ of C10H12 in this study is in very good accord with the values found from the only reported ones 27 years ago in65. However, for C6H4Cl2, it is impossible to produce any such comparison due to the lack of published data.

Similar to the base liquids, λ of both types of C60 NFs decreased slightly with an increase in temperature. However, in contrast to what it was expected (compared to the base liquids), λ of the C60 NFs was reduced upon the addition of C60 and decreased with an increase in its concentration. The obtained experimental λ values were compared to the values predicted using different theoretical thermal conductivity models, originally proposed to clarify the abnormal improvement of λ of NFs. In general, good accord was found, within 5% confidence, with the experimental data for the Maxwell [Eq. (1)], Wasp et al. [Eq. (2)], Bruggeman [Eq. (3)], Xue et al. [Eq. (6)], and Xuan et al. models [Eq. (7)]. More than 5% deviation, and up to 7%, was observed for the Timofeeva et al. model [Eq. (5)], mainly because of the overestimation of λ at high C60 concentrations in the NFs; ascribed to the fact that this model does not consider λ of the nano-particles. In general, the Brownian motion of C60 was found to be negligible in the studied NFs.

It was not possible to find any literature data for aT of C10H12 and C6H4Cl2 to be used in this study. For the tested base liquids, no clear variation of aT was detected between the C60 NF and the corresponding base liquid for the entire temperature range studied. This was partly due to the small variation of λ upon the addition of C60. For C10H12 no clear tendency was observed between aT and the temperature. However, a slight decrease (9.3%) was observed for C6H4Cl2 when the temperature was increased from 264.6 to 323.3 K.

There has been a significant amount of research on λ enhancement of different types of NFs, but the results differ even for the same NF. More importantly, the large enhancement in λ of NFs prepared with a small amount of nano-additive is regarded as anomalous and controversial. We measured for the first time both λ and aT of two types of stable fullerene-based NFs together with their base liquids, 1,2,3,4-tetrahydronaphthalene and 1,2-dicholorobenzene, by the transient multi-current hot wire technique at atmospheric pressure and different temperatures in the range 254 – 323 K.

We found that the obtained λ of 1,2,3,4-tetrahydronaphthalene is in very good agreement with the values reported in the literature more than 25 years ago confirming the adequacy of the developed hot-wire measurements. However, we did not find any literature data for λ of 1,2-dicholorobenzene. Similarly to the base liquids, we found that λ of both types of C60 NFs decreased slightly with the increase of the temperature. However, in contrast to what we expected, compared to the base liquids, λ of the NFs was reduced upon the addition of fullerene. We compared the obtained results with the predicted ones using different thermal conductivity theoretical models and good agreement was observed between them (up to 7% deviation). More theoretical studies should be conducted in order to accurately reveal the governing mechanism(s) and justify the reduction in λ observed in this research study, rather than the anomalous λ enhancement of NFs.

We didn’t find in the literature aT data of fullerene-based NFs or the base liquids 1,2,3,4-tetrahydronaphthalene and 1,2-dicholorobenzene. We didn’t detect much variation in aT between the C60 NFs and the corresponding base liquids tested at different temperatures due partly to the small variation of λ with the addition of C60.

The based liquids used to prepare the NFs, 1,2,3,4-tetrahydronaphthalene (C10H12) and 1,2-dicholorobenzene (ortho-dicholorobenzene, C6H4Cl2) having minimum purities of 99% were supplied by Sigma-Aldrich. The solvents were used without further purification or removal of any dissolved water or air. Fullerene C60 with approximately 0.7 nm in diameter and 99.5% purity was purchased from Sigma-Aldrich. To calibrate the λ measurements of the wire, dimethyl phthalate (C10H10O4, 99% purity, Sigma- Aldrich) with a known λ value was used following the ASTM D2717 standard test68.

The method commonly followed for NF preparation is known as the two-step method. With this method, the preparation procedures of nano-additive and NF are implemented independently. In this study, dry C60 was suspended into the base liquid, first stirred for 30 min at 150 rpm (Ika RCT basic) and then subjected to ultra-sonication for multiple 30-min intervals in order to remove any possible aggregations or colliding nanoparticles due to their high activity and interaction force. Then the dispersion was left overnight to assure that there was no precipitation of the nanoparticles. When the C60 nanoparticles were introduced to the clear base fluids it turned them a very dark purple/violet. The highest possible C60 loading in C6H4Cl2 was 2 wt% (1.64 v/v%) while in C10H12 it was found to be lower 1.35 wt% (0.83 v/v%). It is worth noting that no fullerene settlements were observed during the whole experimental time after NF preparation and even after all testing periods, so the fraction of contained C60 remained unchanged. No clear variation of λ with time was detected indicating very good dispersion stability of the prepared samples.

The transient hot-wire method is the most common method to quantify λ of different types of materials of both liquid and solid phases45,46,67. In this study, both λ and aT were measured using the transient hot wire experimental apparatus detailed in our previous studies67,69,70. The followed method is similar to the ones explained for the measurement of λ of several fluids67,69,70. Briefly, a platinum wire of 50 ± 0.001 μm diameter and 5.92 ± 0.07 cm length was used. Each end of the platinum wire was first soldered to tabs on a properly prepared, chemically resistant, flat frame, cut from a raw circuit board (substrate). Two leads (Teflon jacketed and isolated) were then soldered to each tab; the soldered tabs were subsequently covered by a thermal resistant epoxy. Before calibration a Teflon-based industrial coating was applied to the platinum wire having a thickness layer of less than 1 μm in order to prevent any electrical connection among the platinum wire and the sample.

The liquid sample is placed inside a cylindrical double-walled glass cell that has an internal diameter of 2.2 cm, an external diameter of 4.1 cm and a height of 13 cm. The temperature of the glass cell was controlled within ± 0.05 K by means of a working fluid, which flowed from a thermostatic bath (Lauda ECO RE630) through the jacket of the glass cell. The temperature of the samples were measured by a platinum resistance thermometer (F250 MKII Precision Thermometer, accuracy < ± 0.005ºC), which together with the hot wire, were inserted inside the glass cell containing the sample. The whole measurement cell was kept inside a climatic chamber (Mytron) set at the same temperature of the test temperature maintaining the humidity to around 40%.

To conduct the electrical measurements, we used a Keithley 2400 source-meter that allows operation as a current source and voltage meter simultaneously. The acquisition equipment was then connected to a desktop computer and a software code was developed to run the experimental test, register the measurement points, conduct a data fitting, and finally calculate aT and λ. The platinum thermometer was also coupled to the computer. In this study, the applied electrical currents (200, 250 and 300 mA) were first applied through two of the leads connected to the wire ends, while the voltage was measured concurrently by the other two leads.

To ensure the satisfactory measurements of the wire, first experimental runs using C10H10O4 at different temperatures were performed and the obtained λ values were compared to the reported ones following the ASTM D2717 standard test68 (See Fig. 5c,d). For each temperature of the sample, when steady-state has been established, the software was run under the established electrical currents, which were administered to the wire every 4 min. The recorded data consisted of time (t), the electric potential (V), the temperature (T), and the resistance of the wire (R0) at the beginning of the heating step. With each measurement, approximately 350 values of voltage (V) as a function of time were recorded and the measurements were performed for at least 6 and as many as 16 h depending on the set temperature. A typical heating run lasts for approximately 0.34–1.6 s (depending on the record rate). Figure 5a shows as an example the variation with time of the electric potential between the wire ends.

Typical discrete heating curve showing the voltage drop in the platinum wire, V, as a function of time for the C60 NF prepared by dispersing 0.6 v/v% in C10H12 at 293.15 K and 250 mA (The straight line represent the fitting to Eq. (8)) (a), variation of the wire resistance, R, with temperature, T (b), measured λ of C10H10O4 as a function of T following the ASTM D2717 standard test68 to calibrate the wire before (Initial data) and after (Final data) testing base liquids and NFs (c) and deviation from the tabulated values of the measured λ (d). Microsoft Excel 2016 was used to generate these figures.

Due to Joule heating, when an electrical current (I) is applied across the platinum wire, the temperature (T), resistance (R), and voltage difference (ΔV) of the wire increases depending on λ and aT of the sample surrounding the wire. For an infinite cylindrical straight wire, the change with time (t) of the voltage difference (ΔV) between two points separated by a distance (L) can be approximated for large time, $t > > \frac{{r_{0}^{2} \rho^{{}} c_{p} }}{4\lambda }$, by the following equation as reported elsewhere40,71,72.

where α is the temperature resistance coefficient of the Platinum wire, R0 is the electrical resistance of the wire at the beginning of the heating (t = 0), γ is Euler’s constant (γ = 0.5772), and L is the length of the wire. Equation (8) can be rewritten as:

where m is the slope of the {R ,T} curve at the initial temperature of the heating run (see Fig. 5b) and the parameter, β, (units of time) depends on aT of the sample and the wire radius (ro) as follows:

The resistance (R) is recorded for each measurement, temperature, sample, and electrical current. A mean value is calculated and the variation in resistance (R) with temperature (T) is plotted in Fig. 5b. The calculated slope (m) value is 0.0119 Ω/K.

λ of the sample is calculated from Eq. (9) as follows: the solid line in Fig. 5a depicts the fitting of the experimental points {Vi , ln(ti)} to Eq. (9) when the system reaches steady-state (i.e. t > > 412 ms; the first 150 points have not been utilized in determining the fitting). It must be pointed out that deviations in this study were not observed between the measured data and the straight lines in the long time asymptotic regime indicating that the measurements are free of natural convection. From the obtained intercept (B) and slope (S) of the straight lines, λ and aT, were determined using the following expressions:

where A is a constant of the wire that is determined based on its effective length. For the hot wire used in this study, A was found to be 0.015848 Ω/K.m. This was determined from the measured and reported λ values of dimethyl phthalate at different temperatures. The effective length of the wire (Leff) was calculated from the equation (Leff = m/(4πA)). The calculated value of 5.975 cm is very close to the measured length of the wire 5.92 ± 0.07 cm (1.0% deviation). This confirms that the measured aT and λ data in this investigation are absolute values. It is worth noting that the behavior of the electrical resistance of the platinum wire with temperature permits verification to the stability of the wire during the measurement run. Additionally, variances were not observed between the tests performed with the samples at the beginning, at the end, nor when switching between tested NFs.

All data generated or analysed during this study are included in this published article [and its supplementary information file].

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This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 753319 (Acronym: PCMNano). The authors thank Dr. José María Ortiz de Zárate for all needed support to develop the experimental part of this research study.

Department of Structure of Matter, Thermal Physics and Electronics, Faculty of Physics, University Complutense of Madrid, Avda. Complutense s/n, 28040, Madrid, Spain

Brian Reding & Mohamed Khayet

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The authors declare that both made substantial contributions to: Conception and design of the work and data analysis. Drafted the work and revised it critically for important intellectual content. Approved the version to be published. Agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved. BDR declare to be responsible of data acquisition.

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Reding, B., Khayet, M. Thermal conductivity and thermal diffusivity of fullerene-based nanofluids. Sci Rep 12, 9603 (2022). https://doi.org/10.1038/s41598-022-14204-y

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Received: 13 November 2021

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DOI: https://doi.org/10.1038/s41598-022-14204-y

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