*3.3. Uncertainty*

During the data measurement, different heat conduction processes occur when the modulation frequency is changed. Then, the thermal properties can be determined by fitting the temperature variation against the frequencies. Wang et al. found that the uncertainty is related to the ratio of the thermal diffusion length *μi* to the layer thickness *Li* [11]. Based on their SiO2/Si sample, the numerical uncertainty for the phase shift method was around ±5%when *μi* and *Li* were in the same order, and it increased to ±15% when *μi*/*Li* was around 100. For the case in which *μi*/*Li* was less than 0.15, the thermal energy would not diffuse across the SiO2 layer. The phase shift method would not be suitable for this case, while the amplitude method showed a ±10% uncertainty. Xu et al. studied a similar SiO2/Si sample and achieved an experimental uncertainty of 5% based on the phase shift method and 10% with the amplitude method [33]. For thermal contact resistance measurement, the experimental sensitivity and uncertainty were limited by the uncertainty in the thermal conductivity discussed above. The sensible limit was reported to be 10−<sup>8</sup> m2K/W [11] for the phase shift method and 10−<sup>7</sup> m2K/W [33] for the amplitude method. Since the thermal contact resistance fell in the range of 10−9–10−<sup>7</sup> m2K/W, the amplitude method was not sensitive to the thermal contact resistance. It could thus accurately measure the thermal conductivity without knowledge of the interface.

#### **4. PT Measurement of Nanomaterials**

When the characteristic lengths of materials are reduced to the micro-/nanoscale, the thermal properties also significantly decrease due to the size effect. Based on this fact, investigation of thermal properties can be an efficient supplementary way to characterize the micro-/nanostructure in addition to the most commonly used micro-/nanoscale imaging.

#### *4.1. Nanostructure Analysis through Thermal Characterization*

Wang et al. [32,34] adopted the PT method and studied the axial thermal conductivity of multiwall carbon nanotubes (CNTs) prepared using plasma-enhanced chemical vapor deposition (PECVD). The CNT sample for PT measurement was composed of three layers. As in Figure 3a, the layers from top to bottom were a thin silicon wafer (14 μm thick), a layer of chromium (Cr, 70 nm thick), and the layer of vertically aligned CNTs. Between the Cr layer and CNTs, there was a thin nickel (Ni) film of a negligible thickness, which offered seeds for CNTs' growth. The Si wafer was transparent to the incident laser wavelength (1064 nm) and thermal radiation. Therefore, the incident laser heated the Cr film, the partial generated heat was conducted along the axial direction of the CNTs, and the radiation from the Cr surface was analyzed to obtain the axial thermal conductivity of the CNTs. The resultant thermal conductivity of 27.3 W/m·K was dramatically lower than the theoretical thermal conductivity of 1600–6600 W/m·K for single-wall CNTs, where phonons can

conduct heat in a perfect wall plane. Combined with the TEM result for the CNTs, it was found that the special structure of the Ni seeds led to the CNTs' walls being tilted with respect to the tube axis, as shown in Figure 3b. This unexpected structure raised a large number of boundaries along the axial direction and thus reduced the axial thermal conductivity of the CNTs. Here, though the PT method measured the CNTs' axial thermal conductivity as a bulk, the result greatly helped interpret the special growth mechanism of the CNTs.

**Figure 3.** Thermal and structural characterization of CNTs: (**a**) multilayered structure of CNTs; (**b**) schematics of CNTs' wall growth on a Ni particle with a special structure. Reprinted with permission from Ref. [32], Copyright (2022), AIP Publishing.

#### *4.2. Porosity Determination in Nanostructures*

For loosely assembled nanoparticles, porosity is an important parameter demonstrating the quality of the assembly, but it is hard to measure by mapping only the surface. Pores and cavities in the nanostructure generate additional defects and boundaries and then reduce the thermal properties of the assembled nanostructure. Based on this mechanism, Chen et al. [35] measured the effective thermal conductivity and volumetric heat capacity of a hydrogenated vanadium-doped magnesium (V-doped Mg) porous nanostructure using PT technique. Under the effect of cavities on the V-doped Mg composite (MgH2 was the main component), the effective volumetric heat capacity was apparently lower than that of the MgH2 bulk counterpart. The volumetric heat capacity ratio of nanostructure to bulk helped further reveal the porosity level *ϕ* of the nanostructure. The determined porosity level was validated through SEM observation. The porosity level ϕ was estimated to be 25–42% from SEM, and the ϕ calculated from the PT results was 9.0–39.4%, with an upper limit falling into the range of the SEM observation. It should be noted that the SEM observation scale (microscale) was much smaller than that probed with PT technique (~millimeter scale). Thus, the PT-determined porosity level is more applicable when the size of nanostructured assemblies reaches the macroscale. The intrinsic thermal conductivity of the solid part of the porous nanostructure was then determined to be ~3.5 W/m·K, while it had been ~1.9 W/m·K before excluding the effect of the cavities. PT technique provides a new and convenient way to characterize the porosity level of porous nanostructures, as well as intrinsic thermal conductivity.

#### *4.3. Nano-Crystalline Structure Evolution under Heating*

Heat treatment facilitates the transformation between amorphous and crystalline structures. It is hard to observe this kind of structural transformation with conventional imaging methods. Thermal properties can again be a good indicator showing the variation in the state of the crystalline structure because the amorphous and crystalline structures of the same material have differences in their thermal conductivities. Xu et al. [36] applied the PT method to study internal structure transformations of spider silk proteins under heat treatment based on thermal effusivity. Two spider silk protein films prepared from

two different types of spiders, *N. clavipes* and *L. Hesperus*, were studied, as shown in Figure 4. When elevating the heating temperature, the thermal effusivity of the protein films significantly increased because of the transformation from random coils (amorphous structure) to α-helices and antiparallel β-sheets (crystalline structure). Supplementary Raman studies of the films showed that the characteristic peak of protein started to shift when the heating temperature reached 60 ◦C. In the heating process in this low temperature range (lower than 60 ◦C), the increase in crystallinity was the main reason accounting for the increase in the thermal effusivity. As the temperature increased to more than 80 ◦C, the Raman characteristic peaks disappeared because the crystalline structures were destroyed due to H-bond breaking among molecular chains. Increases in both thermal conductivity (fewer boundaries) and volumetric heat capacity quickened the increasing rate of the thermal effusivity from 100 to 120 ◦C. In this work, the Raman spectra of the protein films were strongly affected by fluorescence induced by surface carbonization, especially in the high temperature range, while the thermal effusivity from the PT technique continuously responded well to the structure variation across the whole temperature range. Thus, PT technique could be a good candidate for nanostructure investigation when conventional methods are not applicable.

**Figure 4.** Thermal and structural characterization of spider silk protein film: (**a**) Raman spectra and (**b**) PT determined thermal effusivity of *N. clavipes* spider silk protein film; (**c**) Raman spectra and (**d**) PT determined thermal effusivity of *L. Hesperus* spider silk protein film. Reprinted with permission from Ref. [36], Copyright (2022), Elsevier.

#### *4.4. Considerations in the Measurement of Micro-/Nanomaterials*

The abovementioned PT technique is able to measure the cross-plane thermal conductivity, heat capacity (*ρc*p), and thermal contact resistance for a multilayered sample under the assumption of 1D heat conductance along the thickness direction. From Equations (5)–(8), it can be seen that the method determines the absolute value of thermal resistance for conductance (the sum of *L*/*κ* and *R*) and heat capacitance (*Lρc*p). For a certain layer *i* with a thin thickness *Li*, when its thermal resistance (*Li*/*<sup>κ</sup>i*) is much smaller than

the uncertainty Δ *R* of *R*, *R* will dominate the variation in the PT signal and the change in *Li*/*<sup>κ</sup>i* will not be sensed. The minimum thickness in the PT method should thus be larger than Δ *R*·*κi.* As alternatives, the TDTR and FDTR methods employ an ultrafast pulsed laser (femtosecond to nanosecond) to realize a nanometer-level thermal penetration depth and, thus, the thermal measurement of nanometer-thick coatings [14]. Raman-based thermal methods measure temperature according to the variation in the materials' characteristic peaks in the Raman spectrum. They are available for both suspended and supported films and need no metal coating on the top of samples [24,26,30,37,38].

Another concern about the PT method is the in-plane thermal conductivity. As described in the physical model, the heating laser spot size should be larger than the in-plane thermal diffusion length in each layer so that the 1D model is valid. However, if the sample has a high in-plane thermal conductivity (such as graphene, etc.), the 1D model is violated. A direct solution is to build a 3D heat conduction model and also consider the spatial distribution of the heating laser [39,40]. For suspended samples, the evaluation of the transient term in Equation (2) in the time domain and the steady-state temperature field mapping can be used achieve in-plane thermal conductivity measurement using the current experimental setup. Moreover, other methods, such as the TDTR [41], FDTR [14], and Raman-based thermal methods [25,42,43], can achieve both kinds of in-plane thermal conductivity measurement.

Furthermore, infrared thermal radiation is not material-specific. When measuring an individual nanostructure, such as a single nanoparticle, an IR detector may gather the thermal radiation from the nanoparticle and its surroundings/supporting materials. The determined thermal properties are thus averages one rather than the properties for a specific nanostructure. In contrast, Raman spectroscopy has a fingerprint feature and can respond to temperature changes and detect the temperature rise for individual micro- /nanostructures [24,38,44–47].
