1 Introduction

Recently, the development of polymer composites with nanoscale modifiers has become an attractive new subject in materials science. In particular, most studies have been directed to carbon-composed structures such as fullerenes, carbon nanotubes (CNTs) [1, 2], and more recently graphene [3, 4]. Carbon nanotubes can be considered as large-aspect ratio particles, and therefore they can generate long-range interactions that could induce an increase in the thermal conductivity of the polymer [5]. This could be enhanced even more given the reported high thermal conductivity of individual carbon nanotubes due to the their graphite-like structure [6, 7]. However, interface effects between the nanotube and the matrix generally reduce significantly the thermal conductivity of the composite. This is one of the great challenges in materials science, to reduce as much as possible the mismatch between the polymer and the nanotube [8, 9]. Through theoretical predictions of thermal properties, high thermal conductivity values of CNTs have been calculated, and some of these high values have been found by experimentation. Enhancement of the thermal conductivity of polymer-based nanocomposites, by incorporation of CNTs, can be expected and has also been reported for CNT/polymer nanocomposites [10–12]. Parameters influencing the composite conductivities, e.g., the aspect ratios of the CNTs, shear rate, and nanostructure concentration, among others have been identified [13–15].

Thermal-diffusivity values of carbon nanotube composites have been reported in the literature. Xie et al. [16] studied a layer of chromium with multiwall nanotubes deposited on top of it by plasma-enhanced hot filament chemical vapor deposition on a glass substrate. They used a laser-flash (LF) technique to determine the thermal diffusivity, obtaining values of \(4 \text { cm}^{2}{\cdot } \text {s}^{-1}\) to \(4.5 \text { cm}^{2} {\cdot } \text {s}^{-1}\) for a temperature range of \(-50\,^{\circ }\text {C}\) to 200 \(^{\circ }\text {C}\). The data obtained by Mamunya et al. [14] reveal values in the range of \(1.2 \times 10^{-3} \text { cm}^{2}{\cdot }\text {s}^{-1}\) to \(1.33 \times 10^{-3} \text { cm}^{2}{\cdot } \text {s}^{-1}\) for poly(vinyl chloride)/CNT composites with concentrations of CNTs between 0 % and 7 % of volume using a periodic method [17, 18]. In this case, the amount of CNTs is too small to increase significantly the thermal conductivity, even if the CNT thermal conductivity is several hundred times larger than that of the polymeric matrix.

The purpose of this work is to analyze the effect of the diameter and concentration of CNTs on the enhancement of the thermal properties in polymeric fluid materials. Both polymer and filler participate in the heat transfer; therefore, the thermal-diffusivity values of such systems depend strongly on their concentration, physical properties, geometrical factors, and coupling between the polymer and filler. The thermal characterization of fluid samples is performed using the thermal-wave resonator cavity (TWRC), which has become, in the last few years, an attractive technique, in the thermal characterization of liquids and gasses, due to its simplicity, versatility, and accuracy [19, 20].

2 Materials and Experimental Methods

2.1 Preparation of Samples

Polyester resin, CAB-O-SIL, multiwalled carbon nanotubes of 50 nm to 80 nm diameter (MWCNT5080), and multiwalled carbon nanotubes with a diameter smaller than 8 nm (MWCNT8), both of \(0.5 \, {\upmu } \text {m}\) to 2 \({\upmu } \text {m}\) length (95 %, Nanostructured & Amorphous Materials Inc.), were used to obtain the studied samples. They were prepared at different MWCNT concentrations of 0.1 %, 0.2 %, 0.5 %, 1 %, 2 %, 3 %, and 4 % (volume fraction), for both kinds of MWCNTs. The MWCNTs were added to the polyester resin matrix and were sonicated for 3 min. It was necessary to add 0.5 % of CAB-O-SIL (volume fraction) to the matrix of polyester resin to increase the viscosity of the matrix and prevent the sedimentation of the nanotubes. The densities of the polyester resin, CAB-O-SIL, and MWCNT are \(1.17\,\text {g}{\cdot }\text {cm}^{-3},\, 2.2\,\text {g}{\cdot } \text {cm}^{-3}\), and \(1.9\,\text { g}{\cdot }\text {cm}^{-3}\), respectively. Micrographs of the MWCNTs were obtained using a field- emission scanning electron microscope (JEOL JSM-7600F).

2.2 Thermophysical Measurements

The thermal diffusivity of the samples was measured using the TWRC technique. Figure 1 shows schematically the experimental setup. It consists of a cavity of variable length in which the liquid sample is contained between a 500 \({\upmu } \text {m}\) thick circular silicon wafer (with a diameter of 7 mm sealed in a cylinder of steel) and the polyvinylidene fluoride (PVDF) sensor. An IR laser diode (Mitsubishi ML120G21 658 nm, 80 mW) was used as the excitation source; the laser was electronically modulated at a frequency \(f = 0.5 \text { Hz}\) by the internal sine oscillator of the lock-in amplifier (SR-830) and controlled by a diode laser driver (Melles-Griot-06DLD205). The laser light impinges on the outer surface of the silicon wafer, which acts as an optical-to-thermal power converter (thermal-wave generator) [21, 22]. As a consequence, the fluid is heated periodically at the modulation frequency of the incident laser beam. The temperature oscillations can be measured with the sensor (a pyroelectric PVDF film with a thickness of \(110\,{\upmu }\text {m}\)). The thickness of the cylindrical liquid container can be varied in steps with a resolution of 10 \({\upmu } \text {m}\). Before sending the signal to the lock-in amplifier, it was sent to a low-noise preamplifier (SR-560) to improve the signal-to-noise ratio. Cavity length scans were performed with a stepper motor coupled to a translation stage. All data acquisition and cavity length scans were automated with a home-made LabView program.

Fig. 1
figure 1

Experimental setup of the TWRC system used to measure the thermal diffusivity of the composite samples: (a) cylindrical thermal-wave emitter head containing (b) the silicon wafer, (c) sample container, (d) pyroelectric sensor, and (e) micrometer stage

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It has been shown that in the configuration used in this work, the phase \(\varphi \) of the signal measured by the pyroelectric sensor, for a sample of thermal diffusivity \({\upalpha }\) and thickness L, in the thermally thick regime \(\left( {L\gg \sqrt{\alpha /\pi f}/2} \right) \), follows the equation [22]:

$$\begin{aligned} \varphi =A-mL, \end{aligned}$$
(1)

where \(m=\sqrt{\pi f/\alpha },\, A\) is a constant independent of the thermal diffusivity, and f is the modulation frequency of the pumping laser. Hence, by performing a cavity length scan at a constant modulation frequency, the measured phase of the signal can be fitted using Eq. 1, and the sample thermal diffusivity is obtained from

$$\begin{aligned} \alpha =\frac{\pi f}{m^{2}}. \end{aligned}$$
(2)

3 Results

Figure 2 shows micrographs of the different MWCNTs used to prepare the samples. It can be observed that the nanotubes with a smaller diameter are more entangled than the nanotubes with larger diameters.

Fig. 2
figure 2

Micrographs of the MWCNTs used for the samples: (a, b) MWCNT diameter 50 nm to 80 nm and (c, d) MWCNT diameter \(<\)8 nm

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Fig. 3
figure 3

Typical results of the pyroelectric phase signal versus relative cavity length for four samples: MWCNT diameter of 50 nm to 80 nm (1 %

figure a
, 4 %
figure b
) and MWCNT diameter \(<\)8 nm (1 %
figure c
, 4 %
figure d
)

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Typical experimental results for the amplitude and phase of the TWRC signal for the analyzed samples are shown in Fig. 3. The effects induced by the charge of nanotubes in the matrix on the heat transport of the polymeric fluid can be seen by the differences in the slopes of the phase of the TWRC signal as a function of the length of the cavity. Fitting the experimental data using Eq. 1 provides the parameter m that is used in Eq. 2 to obtain the thermal diffusivity of the samples.

Fig. 4
figure 4

Thermal diffusivity obtained using the TWRC for composites of resin and concentrations of two types of carbon nanotubes. MWCNT diameter \(<\)8 nm (

figure e
) and MWCNT diameter of 50 nm to 80 nm (
figure f
)

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Thermal-diffusivity values of the samples are shown in Fig. 4. It is possible to observe that an increment in the concentration of carbon nanotubes leads to a rise in thermal diffusivity. For the lowest concentration (0.5 %), both types of samples show similar thermal-diffusivity values. However, for higher concentrations, composites with MWCNT5080 exhibit a higher thermal diffusivity as compared with MWCNT8 samples.

Fig. 5
figure 5

Normalized thermal conductivity for composites of resin with two kinds of MWCNT: MWCNT diameter \(<\)8 nm (

figure g
) and MWCNT diameter of 50 nm to 80 nm (
figure h
), as a function of the MWCNT concentration, with a theoretical fit to the Nan model (solid lines)

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The effective thermal conductivity was calculated using literature data and knowing that the thermal conductivity is given by \(k\, =\alpha \rho C_{p}\), where k is the thermal conductivity, \({\upalpha }\) is the thermal diffusivity, \({\uprho }\) is the density, and \(C_{p}\) is the specific heat capacity at constant pressure. In order to get the effective thermal conductivity, the following relations were used:

$$\begin{aligned} (\rho C)_{\mathrm{eff}}= & {} \left( {1-\phi } \right) \rho _{\mathrm{m}} C_{\mathrm{m}} +\phi \rho _{\mathrm{n}} C_{\mathrm{n}} \\ k_{\mathrm{eff}}= & {} \alpha _{\mathrm{eff}} (\rho C)_{\mathrm{eff}}, \end{aligned}$$

where \(\phi \) is the concentration of MWCNT and subindex m corresponds to the polyester matrix and n to the nanostructures. Polyester resin and MWCNT properties are, respectively: density- \(1.16 \text { g}{\cdot }\text {cm}^{-3}\) and \(1.9 \text { g}{\cdot } \text {cm}^{-3}\) and specific heat capacity-\(1.06 \text { J}{\cdot } \text {g}^{-1}{\cdot }\text {K}^{-1}\) and \(0.65 \text { J}{\cdot } \text {g}^{-1}{\cdot }\text {K}^{-1}\) [23, 24]. To interpret the experimental results, it is possible to use a theoretical model to correlate the experimental data for the effective thermal conductivity as a function of the volumetric MWCNT concentration known as Nan’s model [25]. The normalized effective thermal conductivity is defined as

$$\begin{aligned} \frac{k_{\mathrm{eff}} }{k_\mathrm{m}}=1+\frac{fL}{3d}\,\frac{k_\mathrm{c} /k_\mathrm{m }}{L/d+\left( {2a_K /d} \right) \left( {k_\mathrm{c} /k_\mathrm{m }} \right) }, \end{aligned}$$
(3)

where \(k_{\mathrm{c}}\) and \(k_{\mathrm{m}}\) are the thermal conductivities of the carbon nanotubes and matrix, respectively, d is the MWCNT diameter, L is its length, and \(a_{\mathrm{K}}\) is the Kapitza radius defined by \(a_{\mathrm{K}}=R_{\mathrm{K}} \, k_{\mathrm{m}}\), where \(R_K\) is the interfacial thermal resistance of the carbon nanotube. The normalized thermal conductivity and the fitted model are shown in Fig. 5. For concentrations lower than 1 %, no important differences for the thermal conductivity were observed when comparing samples with different kinds of MWCNTs. For concentrations above 1 %, the thermal conductivity for composites with MWCNT5080 is larger than the thermal conductivity for thinner nanotubes. The Nan’s model shows a good fit to the experimental results with interfacial thermal resistances of \(2.9 \times 10^{-7} \text { m}^{2} {\cdot }\text {K}{\cdot }\text {W}^{-1}\) for MWCNT5080 and \(3.9 \times 10^{-7} \text { m}^{2}{\cdot }\text {K}{\cdot }\text {W}^{-1}\) for MWCNT8, which are consistent with values reported in the literature for this type of materials [25].

Considering that the length of MWCNT5080 and MWCNT8 are the same, and due to the high variation of the diameter between these MWCNTs, an important change in the Kapitza radius and its related interfacial thermal resistance is presented.

Our results show that the interfacial thermal resistance and poor carbon nanotube dispersion due to smaller diameters of MWCNT8 than MWCNT5080 produce a substancial drop in the thermal enhancement.

4 Conclusion

Resins with different concentrations of MWCNTs of different diameters were obtained. It was found that CNTs with larger diameters exhibit better dispersion in the resin matrix. Using a thermal-wave resonator cavity, the thermal diffusivity of the samples was investigated. According to the results, an increment in the carbon nanotubes’ concentration leads to a rise in the thermal diffusivity in the composites of resin. Samples with MWCNTs of a larger diameter show a higher thermal diffusivity. These results confirm that the agglomeration of the thinner nanotubes reduces considerably the long-range interactions that could have favored heat transport. The theoretical model used to predict the behavior of the normalized effective thermal conductivity determined that the interfacial thermal resistance is higher in carbon nanotubes with a smaller diameter. Therefore, when designing composites with a charge of nanotubes, it is very important to reduce the effects of conglomeration as much as possible and use carbon nanotubes with a larger diameter to obtain a material with good thermal conductivity.