Characterizing the dielectric properties of carbon fiber at different processing stages

The polyacrylonitrile (PAN) fibers go through a series of chemical reactions in various processing temperatures/stages and finally turn into the so-called carbon fibers. Oxidization is the first stage, and it takes the largest proportion of the entire processing time for the tremendous change from a chain texture to a ladder texture. The pre-carbonization of carbon fibers is then achieved using a furnace with a higher processing temperature (typically at 700–900 °C). During the reaction processes, the color of the fibers changes from white (PAN) to light black (oxidation), and eventually to black (pre-carbonization). Characterizing the complex permittivity helps us determine the carbonization status of the fibers. This work employed the enhanced-field method (EFM) and the contour mapping method to determine the fibers’ dielectric properties for the first time. Results show that both the real and imaginary parts of permittivity increase as the processing temperature rises. The dielectric constants change from 2.82 (PAN) to 6.50 (pre-carbonization), and the loss tangents increase from 0.007 (lossless) to 0.089 (lossy). This study provides a simple and effective method for characterizing carbon fibers’ processing status and can be applied to the measurement of other fibrous materials.


Results
The dielectric properties of fibrous materials are challenging to measure because of the ultra-high aspect ratio (the length divided by the diameter) 28 . To overcome this problem, we cut the fibrous materials into pieces of 2-3 mm and cram them into a Teflon sample holder. The holder with fibrous materials inside is placed in a resonant cavity using EFM 25,26 . The EFM employs a single-port cavity. The reflection can be measured using a vector network analyzer (Anritsu, MS46122A). Figure 1 shows the measured reflections for three processing stages: PAN, oxidation, and pre-carbonization. For each part, five volumetric percentages from 20 to 60% are measured. Below the volumetric percentage of 20%, the fibers are not randomly scattered in Teflon holder, resulting in unwanted experimental errors. Above 60%, cramming the fibers into the holder becomes extremely difficult. That's why we focus our attention on the volumetric percentages of 20-60%. The materials are obtained from two companies: Formosa Plastic Corporation (shown in Fig. 1a-c) and Jing Gong Co., Ltd. (shown in Fig. 1d-f).
As shown in Fig. 1, the resonant frequencies are inversely proportional to the volumetric concentrations. The FWHMs (full width at half maximum) of all reflected signals become border and border, which indicates that the quality factors (Q-factors) are also inversely proportional to the volumetric concentrations. Note that although the network analyzer (MS46112A) is low-cost, its frequency resolution is 1 Hz with the stability of ± 1 ppm. The network analyzer allows us to determine the resonant frequencies and the quality factors accurately. The major challenge is to cram the fibrous materials into the Teflon holder. Figure 2 plots the resonant frequencies and the quality factors as functions of the volumetric concentrations for three processing stages: PAN, oxidation, and pre-carbonization, each with two suppliers. We can then extrapolate the resonant frequencies and the quality factors at 100% volumetric concentrations based on the observed linear trend. The effective medium method (extrapolated method) will be employed to determine the dielectric constants and loss tangents of carbon fibers with 100% volumetric percentage in different stages. The relations of resonant frequencies and volumetric percentages are plotted as Fig. 2a. The resonant frequencies are inversely proportional to volumetric percentages. But the relations between Q-factors and volumetric percentages are not simple. The relations of the reciprocals of Q-factors and volumetric percentages are plotted as Fig. 2b. The reciprocals of Q-factors are proportional to the volumetric percentages. Note that the reciprocal of a Q-factor is related to the loss tangents of the materials under test. Table 1 shows the resonant frequencies and Q-factors of carbon fibers at the full volume for the PAN, oxidation, and pre-carbonization stages using the effective medium method. The vector network analyzer (MS46112A) can determine the resonant frequencies and the quality factors for each sample, accurately. The resonant frequency of PAN is higher than those of oxidation and pre-oxidation. Also, the Q-factor of PAN is higher than those of oxidation and pre-oxidation. Despite the quantitative difference in values, the samples from both companies exhibit the same trend. With the measured resonant frequencies and the Q-factors, we can then extract the complex permittivity.
The effective medium method suggests that the linear extrapolation should help us to find the resonant frequencies and Q-factors at 100% volumetric percentage. The fitted curve will be y = mx + b , where y represents the resonant frequencies or the Q-factors, x is the volumetric percentage. m and b are the slope and the y www.nature.com/scientificreports/ -intercept. Due to the measuring uncertainty, there will be the mean squared error (MSE). The mean squared error will give rise to the errors of the regression parameters, i.e., m and b , which will then contribute to the worst errors of y when x = 1 . According to the propagation of error for a linear model, the extrapolated errors, �y = ±( √ �m 2 + �b 2 ) , are listed in Table 1. The deviations of the resonant frequencies are relatively small as expected, but the errors of the Q-factors are non-negligible which will influence the accuracy of the imaginary permittivity as discussed below.
The real and imaginary parts of the complex permittivity ( ε = ε ′ + iε ′′ ) are coupled through the Kramers-Kronig relations. The resonant frequency and the quality factor will be functions of the real part and the imaginary part of the permittivity, i.e., For lossless materials, ε ′ and ε ′′ can be determined independently 25,26 . The real part of the complex permittivity is related to the resonant frequency, it reads ε ′ = ε ′ (f r ) . But for the imaginary permittivity ε ′′ , it still depends on both f r and Q . Since f r helps us determine the real permittivity, we can express ε ′′ = ε ′′ (Q, ε ′ ) . However, for lossy materials, ε ′ and ε ′′ are closely coupled, the contour mapping technique is introduced to find the correct values 27 . The real permittivity is commonly normalized and expressed as the relative dielectric constant ( ε r = ε ′ ε 0 ). For the expression of the imaginary permittivity, it is normally denoted as the loss tangent ( tan δ = ε ′′ ε ′ ). Equations (1) and (2) can be rewritten as: The HFSS (High-Frequency Structure Simulator, ANSYS Inc.) provides reliable simulation results at microwave frequency. By changing the input parameters ε r and tan δ of the proposed cavity. A contour map is plotted in Fig. 3. Figure 3 verifies our understanding that for low-loss materials ( tan δ < 0.01 ), the resonant frequency depends merely on the real permittivity, The measured resonant frequency f r and quality factor Q in Table 1 can be used to find the corresponding ε r = ε ′ ε 0 and tan δ = ε ′′ ε ′ . For example, the resonant frequency and Q-factor of the PAN stage for Formosa Plastic are 2.411 GHz and 910. By mapping the contour lines of f r and Q , we obtain an intersection point at ε r = 3.19 and tan δ = 0.007 . The corresponding material's properties ( ε r , tan δ ) obtained from the measured data set ( f r , Q ) are marked in Fig. 3 and listed in Table 2. Table 2 shows that the relative dielectric constant ( ε r ) and the corresponding loss tangent ( tan δ ) increase as the processing temperature rises. For both companies (Formosa Plastic and Jing Gong), the PANs have the lowest dielectric constant and loss tangent, while the pre-carbonization fibers have the highest values. The loss (1) f r = f r (ε ′ , ε ′′ ), (3) f r = f r (ε r , tanδ),  www.nature.com/scientificreports/ tangent is generally related to the color of the material. A white or light yellow material is not a good microwave absorber with low tan δ , while a brown or black one is usually a good microwave absorber with high tan δ . Besides, the fibers shrink in size as the processing temperature rises, indicating that the densities of fibers increase. The increasing dielectric constant seems to be related to the increasing density. However, the increasing density is unable to explain the changes of the loss tangent. As the processing temperature increases, the fibers go through severe chemical reactions, not just physical densification. The complex permittivities of the fibers from the two companies have distinct values, but, interestingly, they have a similar trend. The better the carbonization, the higher the ε r and tan δ . In other words, characterizing the ε r and tan δ helps to evaluate the carbonization status of a PAN fiber. Note that the proposed approach is an auxiliary method, at least for now. It should be used together with other diagnostic techniques, such as FTIR, DSC, Raman spectrum, SEM, and XRD.

Discussions
Conductivity effect. The PAN is a dielectric material with extremely low electric conductivity, but the final product, i.e., the carbon fiber, is a good conductor under direct current (DC). This indicates that the conductivity of a carbonized fiber changes from extremely low (insulator) to high (conductor) during the processing stages 10 . It is essential to know the applicability of the proposed dielectric characterizing method.
The generalized dielectric constant contributes from the bound charges and the free charges, as shown below.
where Nf j is the number of bound electrons of the j-th kind per unit volume and Nf 0 is the number of free electrons per unit volume. ω is the driving frequency of the wave. ω j and γ j are the natural resonant frequency and the collision frequency of the j-th kind bound electrons, respectively. For free electrons, ω 0 = 0 and γ 0 is much greater than ω . For an ideal insulator like PAN, the contribution from the free electrons is zero because  Table 1, we can uniquely determine the relative dielectric constant and the loss tangent as marked in the figure and listed in Table 2. www.nature.com/scientificreports/ Nf 0 = 0 . The second term of Eq. (5) can be omitted. Therefore, the complex dielectric constant of an insulator is ε = ε ′ b + iε ′′ b , which considers the contribution of the bound charges only. When the number density of the free electrons is non-zero, i.e., Nf 0 = 0 , Eq. (5) can be rewritten as: where σ = e 2 Nf 0 mγ 0 with the condition γ 0 ≫ ω . Equation (6) tells that, The conductivity of the carbonized fiber ranges from 0.1 to 1000 Siemens/m 1,2,10 . The resonant frequency is around 2.45 GHz and the angular frequency ω = 1.54 × 10 10 rad/s . Even though at the case of the optimal conductivity, i.e., σ = 1000 S/m , we still have The conductivities at the early stages of the fabrication process, i.e., PAN, oxidation, and pre-carbonization, are low. It is fair and reasonable to claim that the contribution from the conductivity is negligible, which warrants the applicability of the proposed method. However, the authors admit that whether the proposed method is applicable to the carbonization/graphitization fibers is still under investigation, which is beyond the scope of this work.
It is worth noting that comparing with the dielectric constant measured in Ref. 29 , our measured dielectric constant is much smaller than theirs. The reason is that the fibers under study are anisotropic, and they measured the axial dielectric properties of the carbon fiber. Our measured fibers are much shorter and randomly orientated. In other words, our measurement provides an averaged dielectric properties of fibrous material and can be used to characterize the carbonization status.

Method
Enhanced-field method (EFM). The enhanced-electric field provides a stronger coupling with the dielectric properties of the specimens, resulting in a better resolution in characterization than that of the conventional perturbation method [25][26][27] . The specimens to be measured are PAN, oxidized, and pre-carbonized fibers. All of them are provided by two companies: Formosa Plastic Corporation and Jing Gong Co., Ltd. The fibers are cut into small pieces of 2-3 mm, randomly packed, and crammed into a Teflon holder, as shown in Fig. 4a. The outer diameter of the Teflon holder is ϕ 22.00 mm with a height of 9.00 mm. The inner diameter of the holder is ϕ 14.00 mm with a height of 7.00 mm. The capacity of the holder is 1.078 cm 3 . The Teflon holder is placed at the center of the cavity (Fig. 4b) and measured the frequency response with a vector network analyzer (VNA) in a range from 2.35 GHz to 2.6 GHz. The resonant frequency and quality factor can be derived from the measured frequency response (Fig. 4c).
The samples to be measured are fibrous with an extremely high aspect ratio. The traditional permittivity measurements require a sample with a fixed/specific shape. Here we solve the problem by cramming the fibers into the Teflon sample holder. The diameter of the fiber is around 5-10 μm. The free space wavelength of 2.45 GHz is 12 cm. Since the wavelength is several orders higher than the diameter of the fibers, the nonuniformity in the