# Terahertz time-domain spectroscopy of boron nitride nanotube-reinforced PMMA composites

Min Zhai, Feilin Gou, Changhong Ke, Wenlong He, Cong Zhai, Alexandre

Locquet, D.S. Citrin

► To cite this version:

Min Zhai, Feilin Gou, Changhong Ke, Wenlong He, Cong Zhai, et al. Terahertz time-domain spectroscopy of boron nitride nanotube-reinforced PMMA composites. Polymer Testing, 2025, 147, pp.108812. 10.1016/j.polymertesting.2025.108812. hal-05088490

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# Terahertz time-domain spectroscopy of boron nitride nanotube-reinforced PMMA composites

Min Zhai<sup>a,*</sup>, Feilin Gou<sup>b</sup>, Changhong Ke<sup>b</sup>, Wenlong He<sup>a</sup>, Cong Zhai<sup>c</sup>, Alexandre Locquet<sup>d,e</sup>, D.S. Citrin<sup>d,e,*</sup>

## ARTICLE INFO

**Keywords:**

Terahertz time-domain spectroscopy
(THz-TDS)

Dielectric properties

BNNT

Effective medium theory (EMT)

## ABSTRACT

Polymethyl methacrylate (PMMA) reinforced with boron nitride nanotubes (BNNT) at weight fill factors $f_w$ of 0.2 and 0.5 wt % are characterized by terahertz time-domain spectroscopy (THz-TDS). It is known that the introduction of BNNTs in polymer composites can provide thermal conduction paths and significantly reduce interface thermal resistance. Despite the low values of $f_w$, the presence of the BNNTs has a significant effect on the optical constants, either due to the intrinsic contribution of the BNNTs or due to extrinsic effects such as induced strain in the PMMA matrix. Negligible THz birefringence observed in the polarization-resolved THz transmission measurements indicates a high degree of isotropy of the BNNT dispersed in the PMMA. Assuming an intrinsic effect and the validity of modified Maxwell-Garnett theory regardless of the inclusion shape, the frequency-dependent optical constants of BNNTs are extracted. The refractive index of BNNTs varies roughly linearly from $n(\nu = 500 \text{ GHz}) \approx 1.8$ to $n(\nu = 1.5 \text{ THz}) \approx 1.75$, while the absorption coefficient increases from $\alpha(\nu = 500 \text{ GHz}) \approx 10 \text{ cm}^{-1}$ to $\alpha(\nu = 1.5 \text{ THz}) \approx 45 \text{ cm}^{-1}$. The extracted THz dielectric properties of the BNNTs trend toward reported values for bulk h-BNs.

## 1. Introduction

Poly(methyl methacrylate) (PMMA) has been attracted tremendous attention to be promising polymeric materials for high-frequency microelectronic packaging due to the characteristics of lightweight, good flexibility, low dielectric constant, and easy molding and processing [1, 2]. The rapid progress of miniaturization, high integration and high power consumption of electronic components has posed huge challenges to the efficient and effective thermal management. The heat accumulation within related devices without effective heat dissipation path will greatly deteriorate the service life and reliability [3]. In order to address low thermal conductivity, poor mechanical properties, and low temperature resistance of polymeric materials, nanofillers, such as Carbon Nanotube(CNT) [4,5], are added into the PMMA matrix to enhance the mechanical properties as well as the power dissipation of polymer-based nanocomposites. Owing to high electrical conductivity of carbon-based materials, the practical applications of such thermal conductive composites in insulation situations are severely restricted.

Boron nitride (BN) is a synthetically produced crystalline compound with a structural similar to graphite, consisting of alternating substitutions of B and N for C atoms [6]. BN is a versatile inorganic compound appearing in a variety of allotropes, including cubic BN (c-BN) with a diamond-shaped crystalline arrangement and hexagonal BN (h-BN) with a layered graphite structure in which planar networks of hexagonal BN are regularly stacked [7]. Due to the confluence of several properties, such as low thermal expansion, high thermal conductivity, high electrical resistivity, and high mechanical strength [8], BN has attracted interest for applications, including in electronics, aerospace, and energy [9–11].

BN nanotubes (BNNT), as an analog of better-known carbon nanotubes (CNT) in morphology, can be envisaged as a rolled graphite-like sheet substituting C atoms with alternating B and N atoms [12]. BNNTs possess many properties similar to other forms of BN, such as high thermochemical stability and thermal conductivity and high electrical resistivity, that distinguish them from their carbon-based counterparts [13,14]. BNNTs have attracted interest as nanofillers in

<footnote><sup>a</sup> College of Electronics and Information Engineering, Shenzhen University, Shenzhen, 518060, Guangdong, China</footnote>

<footnote><sup>b</sup> Department of Mechanical Engineering, State University of New York at Binghamton, Binghamton, 13902, NY, USA</footnote>

<footnote><sup>c</sup> School of Transportation and Civil Engineering and Architecture, Foshan University, Foshan, 528000, Guangdong, China</footnote>

<footnote><sup>d</sup> School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, 30332, GA, USA</footnote>

<footnote><sup>e</sup> Georgia Tech-CNRS IRL2958, Georgia Tech-Europe, 2 Rue Marconi, Metz, 57070, Lorraine, France</footnote>

<footnote>* Corresponding authors.
E-mail addresses: zhai.min@szu.edu.cn (M. Zhai), david.citrin@ece.gatech.edu (D.S. Citrin).</footnote>
various polymeric systems to construct interconnected thermally conductive paths to achieve excellent thermal-mechanical and electrical insulating characteristics, compared to CNTs and graphene [15,16].

Terahertz (THz) time-domain spectroscopy (THz-TDS) is used to characterize the dielectric properties of electrically insulating materials and components, typically from ~100 GHz to a few THz [17–19]. THz electromagnetic waves may penetrate deeper into technologically important electrical insulators than visible and infrared light, providing structural information beneath the surface. The foundation of any study of structural properties is knowledge of the dielectric properties [20]. A number of material properties of BNNTs are tabulated in Refs. [14,21]; however, there is little data related to their far-infrared properties. In Ref. [22], four grades of hexagonal BN were investigated by THz-TDS. The refractive indices and absorption coefficients within the THz band are reported there. Ref. [23] presents a comparative study of the dielectric properties of the monolayer hexagonal BN (hBN) placed on various substrates by THz-TDS in the temperature range 80–280 K.

In this paper, the dielectric properties BNNT/PMMA with low weight fill factor $f_w$ 0.2 and 0.5 wt % are investigated at room temperature by THz-TDS. Anisotropy of the BNNTs dispersed in PMMA is studied using polarization-dependent THz transmission measurements. The presence of BNNTs in the nanocomposites has a significant effect on the optical constants in the THz band. This might be due to intrinsic or extrinsic effects (such as BNNT-induced strain in the PMMA, or strong electrostatic interactions across the BNNT/polymer interface as mentioned above). Still, assuming an intrinsic origin, considering the low content as well as the diameter of BNNTs (typically ~2 to 5 nm [24]), modified Maxwell-Garnett (MG) theory was used to extract the dielectric properties of the BNNTs from 200 GHz to 2 THz. The extracted optical constants of BNNTs trend in a fashion apparently consistent with published results on other forms of BN. The intrinsic or extrinsic origin of the significant modification of the THz dielectric properties due to the presence of BNNTs remains unknown.

## 2. Materials and experimental setup

The transmission THz experiments were carried out using a TPS Spectra 3000 (TeraView Ltd., Cambridge, UK), a pulsed, broadband commercial THz time-domain system, as described in Ref. [25]. All measurements were performed in an air-conditioned laboratory at 22 °C with humidity <48%. Before starting transmission measurements, dry $N_2$ was purged in the propagation path to eliminate undesired water-vapor effects (ringing after the main THz pulses and sharp absorption lines in the corresponding spectra [26]). Balancing the noise level in the spectrum as well as the spectral resolution, a Black-Harris 3-term apodization method was used to eliminate spectral artifacts irrelevant to sample absorption [27]. The spectra were recorded at room temperature, and 1800 scans were collected and averaged to achieve a good signal-to-noise ratio (SNR).

The employed BNNT LLC (Newport News, VA) were synthesized using high-temperature pressure (HTP) methods [28] and were received in the form of dry cotton-like puffballs with BN purity >99% based on the manufacturer data sheet. The BNNT/PMMA composite were manufactured by first dispersing BNNTs in a mixture of dimethylformamide (DMF) and acetone (1:1 weight ratio) using ultrasonication for 2 h. PMMA (50,000 g mol⁻¹ molecular weight, Sigma-Aldrich Co., Saint Louis, MS, USA) was added into the dispersion BNNT solution at a weight concentration of about 20%. The mixture was stirred at room temperature to achieve a uniform solution before being casted into thin slabs by molding. Finally, the composite slabs were dried in a vacuum oven at 60 °C for 96 h. The BNNT/PMMA composites slabs were fabricated using this method, and the corresponding weight fill factors $f_w$ of the samples are 0.2 and 0.5 wt %. Thicknesses of all specimens are identical, regardless of $f_w$. Attempts to increase $f_w$ may result in BNNT aggregation and high inhomogeneity.

<figure><img src="image_4.png" /><figcaption>Fig. 1. Photograph of the BNNT/PMMA 0.5 wt %, where the percentage refers to the weight fill factor $f_w$. The dimensions of the BNNT-based polymer nanocomposite sample are 1.5 cm (vertical) × 2.5 cm (horizontal) × 1 mm (thickness). The sample shown contains an air bubble toward the upper right. No data were taken near the air bubble.</figcaption></figure>

## 3. Results and discussion

Fig. 2(a) shows transmitted THz signals through PMMA, BNNT/PMMA 0.2 wt % and 0.5 wt %, where the percentage is $f_w$; the corresponding power spectra of transmitted signals are obtained after Fast Fourier Transform (FFT) and shown in Fig. 2(b). The reference transmitted THz signal was recorded before mounting specimens on the sample holder. There is only weak evidence of atmospheric water-vapor ringing or absorption lines in the reference and sample signals and spectra due to the purgation of dry $N_2$. The BNNT/PMMA composites show higher attenuation compared with the pure PMMA specimen; this will be quantified below.

Because the properties of the resulting BNNTs-based PMMA nanocomposites are largely dependent on the dispersion of the nanofiller within the polymer matrix, we also investigate the frozen-in molecular anisotropy of the BNNTs and any possible induced anisotropy in the PMMA. An anisotropic distribution of the BNNT orientation might be expected to give rise to an intrinsic birefringence effect, or possibly an extrinsic birefringence due to BNNT-induced stresses in the PMMA [22,29]. To check for any significant anisotropy, we performed polarization-resolved THz transmission measurements defining the polarization angle $\theta$ with respect to the long axis of the sample (see Fig. 1).

Fig. 3 displays the $\theta$ dependence of the transmitted signals as the polarization angle $\theta$ is varied for the BNNT/PMMA 0.5 wt % sample from $\theta = 0°$ (black curve) to 90° (cyan curve) in 30° steps. Negligible polarization dependence is found, which implies that any birefringence is very small in the THz range. Similar results (not shown) are also found for the BNNT/PMMA 0.2 wt % sample. It is worth pointing out that dielectric anisotropy may be related to various morphologies, spatial arrangements, and directional alignments of nanofillers in a matrix [30]. The polarization-independent measurements presented here indicate that the BNNT orientation distribution appears to be isotropic.

The refractive index $n(\nu)$ and attenuation coefficient $\alpha(\nu)$ of BNNT/PMMA composites as a function of frequency $\nu$ can be calculated from the Fourier transforms of the electric field of the transmitted THz signal through the sample and the reference signal in the absence of the sample, with amplitude and phase $E_{sample}(\nu)$, $\phi_{sample}(\nu)$ and $E_{ref}(\nu)$, $\phi_{ref}(\nu)$, respectively [20],

$$ \alpha(\nu) = -\frac{2}{d} \ln \left[ \frac{E_{sample}(\nu)}{T(\nu)E_{ref}(\nu)} \right], \quad (1) $$

$$ n(\nu) = 1 + \frac{c}{2\pi v d} [\phi_{sample}(\nu) - \phi_{ref}(\nu)] \quad (2) $$
<figure><img src="image_5.png" /><figcaption>Fig. 2. Comparison of (a) transmitted signal and (b) transmitted power spectra for air, PMMA, BNNT/PMMA for various $f_w$. The polarization angle was set to 0° with respect to the long axis of the sample studied.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 3. Azimuthal angle $\theta$-dependent the of transmitted signals for BNNT/PMMA 0.5%, measured by the polarization-sensitive THz-TDS transmission mode configuration. $\theta$ is the polarization angle with respect to the long axis of the sample (see Fig. 1) with $\theta = 0°$ (black curve), 30° (red curve), 60° (blue curve) and 90° (cyan curve), respectively. No significant birefringence is observed indicating the BNNTs orientation distribution and any possible induced stress in the PMMA are isotropic.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 4. Frequency-dependent (a) refractive index $n(v)$ and (b) attenuation coefficient $\alpha(v)$ for PMMA and BNNT/PMMA composite with various $f_w$.</figcaption></figure>

where $d$ is the sample thickness measured using caliper with 10 µm resolution, and $T(v)$ is the transmittance

$$T(v) = 1 - \frac{[n(v) - 1]^2}{[n(v) + 1]^2} \qquad (3)$$

The factor $T(v)$ in $\alpha(v)$ accounts for surface transmissions of THz waves.

Fig. 4(a) and (b) show $n(v)$ and $\alpha(v)$ for PMMA, BNNT/PMMA 0.2 wt % and wt 0.5%, respectively. The feature at ~1.66 THz in Fig. 4(b) is likely due to a residual water-vapor resonance despite the $N_2$ purge but low SNR in the high-frequency range of our experimental bandwidth might also play a rôle [31] [see Fig. 2(b)]. Fabry-Pérot oscillations in the raw data have been corrected using a fixed-point

iteration method described in Ref. [32]. Accounting for the thickness of the BNNT/PMMA composites, as well as the effective bandwidth of the experimental setup [33], we find that $n(v)$ for all specimens studied is approximately linear in $v \in [0.2$ THz, 1.6 THz]. The dispersion $dn/dv$ is ~-0.03 THz$^{-1}$ for pure PMMA, and -0.08 THz$^{-1}$ for both BNNT/PMMA 0.2 wt % and 0.5 wt %. The dispersion is slightly more severe in the BNNT-loaded nanocomposites, and shows very weak $f_w$ dependence. Compared to pure PMMA, The increase in the dispersion $dn/dv$ for BNNT/PMMA might attribute to the microstructural changes. The similar dispersion for both BNNT/PMMA 0.2 wt % and 0.5 wt % is originated from the same dimension of BNNTs in BNNT/PMMA composites [34]. We also observe that $\alpha(v)$ increases with $v$ with $\alpha(v) = 1$
<figure><img src="image_8.png" /><figcaption>**Fig. 5.** Frequency-dependent refractive index-absorption coefficient product $n(v)α(v)$ for PMMA and BNNT/PMMA composites with various $f_w$.</figcaption></figure>

<table><caption>**Table 1**

$K$ and $β$ of PMMA and BNNT/PMMA composites with various $f_w$ obtained from fits of $n(v)α(v)$ to Eq. (4).</caption><thead><tr><th>Material</th><th>K (cm<sup>-1</sup> s<sup>2</sup>)</th><th>β</th></tr></thead><tbody><tr><td>PMMA</td><td>22.54</td><td>1.167</td></tr><tr><td>BNNT/PMMA 0.2 wt %</td><td>29.24</td><td>1.081</td></tr><tr><td>BNNT/PMMA 0.5 wt %</td><td>33.5</td><td>1.044</td></tr></tbody></table>

THz) is 7.47 cm⁻¹ for PMMA, 16.41 cm⁻¹ for BNNT/PMMA 0.2%, and 18.51 cm⁻¹ for BNNT/PMMA 0.5%, respectively. $α(v)$ for the BNNT-filled PMMA composites is larger than for the host PMMA matrix across the entire frequency regime. Although $f_w$ in the samples studied is low, its influence on the optical constants is relatively significant. Whether this effect is intrinsic or extrinsic is to be determined and is explored below.

It has been observed that the $n(v)α(v)$ product follows the universal dielectric response or Jonscher's law in amorphous materials [20,35], such as PMMA. Namely,

$$n(v)α(v) = K(hv)^β \quad (4)$$

where $h$ is the Planck constant, $K$ is material-dependent [36], and the exponent $β$ is associated with the characteristics of the disorder, and is typically < 2. Fig. 5 plots $n(v)α(v)$ of PMMA and BNNT/PMMA composites with various $f_w$, as a function of $v$. The values of $K$ and $β$ are obtained after fitting to the product of measured $n(v)$ and $α(v)$, and are listed in Table 1. The relatively low value of $β$ for BNNT/PMMA composites, may indicate more disorder compared to the pure PMMA materials, and the degree of disorder and $K$ seem to be positively correlated with $f_w$.

For some purposes, it may be desirable to express the dielectric properties in other ways. The complex permittivity $ε$ ($ε = ε' + iε''$) of the BNNT-filled PMMA composites can be calculated from

$$ε' = n_c^2 - k^2, \quad (5)$$

$$ε'' = 2n_c k \quad (6)$$

where $k$ is the extinction coefficient, which corresponds to the imaginary part of complex refractive index $n_c = n + ik$ with

$$k = \frac{\alpha \lambda_0}{4\pi} \quad (7)$$

and $λ_0 = c/v$ is the wavelength in vacuum and $c$ is the in-vacuo speed of light.

Fig. 6(a) and (b) show the real and imaginary parts of the effective complex permittivity $ε(v)$ for BNNT/PMMA composites with various

$f_w$. Both $ε(v)'$ and $ε''(v)$ increase with $f_w$. Moreover, the dispersion in $ε'(v)$ for BNNT/PMMA composites is more pronounced than for PMMA (as it also is for refractive index), for a given value of $f_w$. $ε''(v)$ increases with $v$, reflecting similar behavior for $α(v)$.

As mentioned above, the origin of the $f_w$ dependence of the optical constants might have an intrinsic or extrinsic origin (or some combinations of the two). We tentatively assume that the optical constants reflect intrinsic properties of the BNNTs. Under this assumption, we seek to extract the BNNT optical constants from effective-medium theories (EMT). There are a range of such theories appropriate to various sorts of mixtures, including the Landau-Lifshitz-Looyenga model (applicable to mixtures with arbitrary shaped particles), Polder-van-Santen theory (considering the influence of the particle shape as well as orientations), as well as complex refractive-index models (applicable to mixture systems with low permittivity contrast) [37,38]. Ref. [39] discusses the application of various classical mixing formulae. Considering the extremely low BNNT concentrations (<1 wt %) and the isotropy of the BNNTs, the Maxwell-Garnett(MG) theory is adopted.

Although MG typically assumes the inclusions are spherical, some modifications can be made to remove any information about the inclusions shape as long as the effective medium is spatially uniform and macroscopically isotropic [40], and are given by [41]

$$ε_{eff} = ε_h \frac{1 + 2f \frac{ε_i - ε_h}{ε_i + 2ε_h}}{1 - f \frac{ε_i - ε_h}{ε_i + 2ε_h}} = ε_h \frac{ε_h + \frac{1+2f}{3}(ε_i - ε_h)}{ε_h + \frac{1-f}{3}(ε_i - ε_h)} \quad (8)$$

where $ε_{eff}$, $ε_h$, and $ε_i$ represent the complex permittivity of the effective medium obtained directly from the measurements, the host material obtained on the pure PMMA sample, and the inclusions, respectively. The optimized MG mixing formula only contains the permittivities of the host and the inclusions as well as the volume fraction of the latter.

The real and imaginary parts of the measured permittivity $ε$ of PMMA are 2.66 and 0.039 at 500 GHz, 2.62 and 0.058 at 1 THz, and 2.58 and 0.076 at 1.5 THz, respectively. $f_v$ is the volume fill factor of the inclusions, which can be obtained from the weight filling factor $f_w$ by

$$f_v = \frac{f_w / \rho_i}{f_w / \rho_i + (1 - f_w) / \rho_h} \quad (9)$$

where $ρ_i$ and $ρ_h$ correspond to the density of inclusions and host materials. We employ MG theory for isotropic composites, finding that modifications for anisotropy due to the particle geometry or distribution has negligible effect on the extracted parameters [41].

To apply the modified MG theory, the BNNTs volume fill factor $f_v$ should be calculated first using Eq. (9). Assuming that the densities of BNNTs and PMMA are 1.38 g/cm³ (literature values range from 1.3 to
<figure><img src="image_9.png" /><figcaption>Fig. 6. Frequency-dependent (a) real $\varepsilon'(\nu)$ and (b) imaginary $\varepsilon''(\nu)$ parts of the permittivity $\varepsilon(\nu)$ for PMMA and BNNT/PMMA composites with various $f_w$.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 7. BNNTs inclusions optical properties between 200 GHz to 1.6 THz: (a) $\varepsilon'$, (b) $\varepsilon''$, and (c) $\tan\delta$. The standard deviation of mean values is represented by the shaded gray area.</figcaption></figure>

<table><caption>**Table 2**
Comparison of dielectric properties of BNNTs and PMMA at selected frequencies $v$.</caption><thead><tr><th rowspan="2">Material</th><th rowspan="2">Dielectric properties</th><th colspan="3">Frequency v (THz)</th></tr><tr><th>0.5</th><th>1</th><th>1.5</th></tr></thead><tbody><tr><td rowspan="4">BNNTs</td><td>n</td><td>1.81</td><td>1.75</td><td>1.71</td></tr><tr><td>α (cm<sup>-1</sup>)</td><td>11.36</td><td>26.94</td><td>45.45</td></tr><tr><td>ε'</td><td>3.11</td><td>2.93</td><td>2.88</td></tr><tr><td>tan δ</td><td>0.05</td><td>0.06</td><td>0.07</td></tr><tr><td rowspan="4">PMMA</td><td>n</td><td>1.63</td><td>1.62</td><td>1.61</td></tr><tr><td>α (cm<sup>-1</sup>)</td><td>6.16</td><td>13.97</td><td>22.54</td></tr><tr><td>ε'</td><td>2.66</td><td>2.62</td><td>2.58</td></tr><tr><td>tan δ</td><td>0.01</td><td>0.02</td><td>0.03</td></tr><tr><td rowspan="2">h-BN [44]</td><td>n</td><td>2.01</td><td>2.01</td><td>2.02</td></tr><tr><td>α (cm<sup>-1</sup>)</td><td>~1</td><td>~3</td><td>~8</td></tr><tr><td rowspan="2">p-BN [45]</td><td>n</td><td>2.02</td><td>~2.1</td><td>2.12</td></tr><tr><td>α (cm<sup>-1</sup>)</td><td>~25</td><td>~40</td><td>~50</td></tr></tbody></table>

1.4 g/cm³ [42]) and 1.19 g/cm³ [43], the corresponding volume fill factor $f_v$ for $f_w$ is 0.18 vol % for the 0.2 wt % sample and 0.44 vol % for the 0.5 wt % sample. From the measured complex permittivity $\varepsilon_{eff}(\nu)$, as shown in Fig. 6, the optical constants of the BNNTs inclusions can be extracted.

Fig. 7(a) and (b) show $\varepsilon(\nu)'$ and $\varepsilon(\nu)''$ of the BNNT inclusions extracted from the transmission measurements made on BNNT/PMMA 0.2 wt % and 0.5 wt % composites. Frequently, the microwave community quote the optical constants in terms of $\varepsilon'(\nu)$ and the loss tangent

$$ \tan \delta(\nu), \\ \tan \delta(\nu) = \frac{\varepsilon'(\nu)}{\varepsilon''(\nu)}. \qquad (10) $$

For the convenience of those who prefer $n(\nu)$ and $\alpha(\nu)$, Fig. 8(a) and (b) gives the values for BNNT obtained from MG theory.

Table 2 summarizes the dielectric properties of BNNT and PMMA at $v = 500$ GHz, 1 THz, and 1.5 THz assuming the validity of MG theory for the problem at hand. While the optical constants of BNNT in the THz band have not been reported, $n(\nu)$ and $\alpha(\nu)$ of BNNT inclusions obtained with MG theory are within the range of values reported for bulk-h-BN and pyrolytic BN [p-BN, one form of boron nitride deposited through a chemical vapor deposition (CVD) process] [44,45], which are also shown in the table. The values of $n(\nu)$ for BNNT obtained from the modified MG theory are higher than those of PMMA, in broad agreement with $n(\nu)$ for bulk-h-BN. The dispersion in $n(\nu)$ for bulk-h-BN is quite weak (whether positive or negative is unclear), while the extracted BNNT dispersion is weakly negative. The extracted values of $\alpha(\nu)$ indicate high loss, though the comparison with Refs. [44,45] is equivocal.

Assuming that the BNNTs' contribution to the optical constants is indeed intrinsic, there are several candidate sources of differences with the reported values, starting with the fact that we are comparing BNNTs with bulk-h-BN. Beyond that, the low permittivity contrast between the BNNT and PMMA, as well as the low BNNT $f_w$ all contribute to possible errors in the extracted values. In addition, the sample thicknesses $d$ were obtained by caliper measurement, introducing its own error. The broad size distribution of BNNTs may also contribute. We also note that
<figure><img src="image_11.png" /><figcaption>Fig. 8. BNNTs inclusions optical properties between 200 GHz to 1.6 THz: (a) $n(\nu)$, and (b) $\alpha(\nu)$. The standard deviation of mean values is represented by the shaded gray area.</figcaption></figure>

MG theory assumes a homogeneous distribution of spherical inclusions, although accounting for a shape factor for elongated inclusions [41] does not lead to drastically different extracted values. Finally, we cannot discount the possibility of extrinsic effects (e.g., local stress, local modification of the matrix [46], trapping of air around the BNNTs), or some combination of intrinsic and extrinsic effects.

## 4. Conclusion

In this letter, BNNT-filled PMMA with low $f_w$ (0.2 wt % and 0.5 wt %) are characterized by THz time-domain spectroscopy. The dielectric properties, namely, refractive index $n(\nu)$, absorption coefficient $\alpha(\nu)$, complex permittivity $\epsilon(\nu)$, and loss tangent $\tan\delta(\nu)$, of the BNNT-filled PMMA nanocomposites are observed in transmission experiments. Despite the low BNNT fill factors, there is a significant modification of the dielectric properties due to the presence of the BNNTs. Negligible birefringence is observed indicating a high degree of isotropy in the BNNT orientational distribution within the PMMA matrix. Assuming that the BNNTs modify the optical constants through intrinsic effects, MG theory is used to extract the optical properties of the BNNT inclusions. Under these assumptions, we find the BNNT refractive index $n(\nu)$ varies weakly, but roughly linearly from ~1.8 at 500 GHz to ~1.75 at 1.5 THz. We note that if indeed significant electrostatic effects occur at the BNNT/polymer interface, EMT may not be valid. While our results are the first reported optical constants of BNNT in the THz band, reported values for bulk-h-BN, while sparse and exhibiting large scatter, reflect the trends in the values we extract for the BNNTs assuming the validity of MG theory for this purpose and the intrinsic effect of the BNNTs on the optical constants. The weight fill factors $f_w$ of the samples studied are quite low making it difficult to reliably extract the BNNT contribution; however, attempts at producing higher values of $f_w$ might contribute to aggregation of BNNTs in the PMMA matrix, thus reducing the quality of BNNT-PMMA composites.

### CRediT authorship contribution statement

**Min Zhai:** Writing – original draft, Visualization, Methodology, Funding acquisition. **Feilin Gou:** Writing – review & editing, Validation, Methodology. **Changhong Ke:** Writing – review & editing, Resources, Project administration. **Wenlong He:** Writing – review & editing, Funding acquisition. **Cong Zhai:** Writing – review & editing, Methodology. **Alexandre Locquet:** Writing – review & editing, Supervision, Project administration. **D.S. Citrin:** Writing – review & editing, Validation, Supervision, Funding acquisition.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

We gratefully acknowledge the support of Conseil Régional Grand Est and CPER SusChemProc. F.G. and C.K. acknowledge the support by the National Science Foundation (NSF), China under award CMMI 2009134. M.Z. acknowledge the support by Youth Innovation talent program of Guangdong Province, China (No. 2024KQNCX041) and Shenzhen Basic Research General Project (Natural Science Foundation), China (No. JCYJ20240813141427036). W.H. acknowledge the support by Shenzhen Science and Technology Program, China (No. KQTD20200820113046084) and National Key Research and Development Program of China (No. 2023YFF0719300).

### Data availability

Data will be made available on request.

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