**3.4 High-temperature dielectric behaviors of multilayer C***f***/Si3N4 composites**

The evolution of *ε*′ and *ε*″ for multilayer C*f*/Si3N4 composites with temperature and frequency is illustrated in **Figure 10(a)** and **(b)**, respectively. It can be clearly seen that both *<sup>ε</sup>*′ and *<sup>ε</sup>*″ of C*f*/Si3N4 composites are enhanced with temperature. Similar phenomenon was also observed in other microwave-absorbing/shielding materials [24–26, 61–63]. Besides, as shown in **Figure 10(c)**, the loss tangent of C*f*/ Si3N4 composites remains higher than 0.6 over X-band and increases with increase in temperature. After comparison with low-loss *β*-SiC [64] as well as Si3N4, it is fairly clear that short carbon fibers are dominantly responsible for the frequency and temperature-dependent permittivity of C*f*/Si3N4 composites. This is attributed to the unique microstructure (named skin-core structure), which was confirmed from transmission electron microscope analysis and selected-area electron-diffraction patterns previously [65, 66]. As illustrated in **Figure 11**, the core region consists of graphitic basal planes with random orientation, whereas graphitic basal planes are parallel to the fiber axis in the skin region [66, 67]. Therefore, electron migration behaviors both within and between graphitic basal planes inside carbon fibers would lead to electric charge accumulation at interfaces between carbon fibers and Si3N4 matrix (usually referred to as space charge polarization) when exposed to electromagnetic field. Furthermore, this space charge polarization is supposed to be

#### **Figure 10.**

*Three-dimensional plots of dielectric properties of Cf/Si3N4 composites versus frequency and temperature: (a) real part, (b) imaginary part, (c) loss tangent, and (d) attenuation coefficient (reprinted with permission from Ref. [39]).*

**39**

**composites**

*Dielectric Responses in Multilayer Cf/Si3N4 as High-Temperature Microwave-Absorbing Materials*

enhanced with temperature rise since more and more electrons would be excited,

It should also be noted here that the lattice vibration also enhances with temperature rise, corresponding to the enhancement of scattering effect on migration of electrons. Fortunately, the energy of lattice vibration, which is described by phonons, is so small [68] that the scattering effect on electron migration could be

*Sketches of microstructure arrangement and electronic motion in carbon fiber (reprinted with permission from* 

Attenuation coefficient is a quantity that characterizes how easily a material can be penetrated by incident microwave. A large attenuation coefficient means that the beam is quickly "attenuated" (weakened) as it passes through the material. The intrinsic attenuation coefficient of multilayer C*f*/Si3N4 composites along

illustrated in **Figure 10(d)**. It can be clearly observed that attenuation coefficient of C*f*/Si3N4 composites enhances continuously with increase in frequency, and reaches a maximum value around 0.7 at 12.4 GHz, which is almost twice as much as that reported for C*f*/SiO2 composites [25]. This superior absorption performance suggests the multilayer C*f*/Si3N4 composites to be competitive high-temperature microwave-absorbing materials. Another noteworthy phenomenon in **Figure 10(d)** is that the absorption coefficient remains steady in the investigated temperature range regardless of the loss tangent. As discussed previously, the overall absorbing performance results from reflection on the surface and attenuation inside the materials. However, the increasing permittivity means more severe impedance mismatch between air and the absorbing material, corresponding to reduction of penetrated electromagnetic energy. It could be explained by the reduced part of energy, which compensates the enhanced loss tangent, and consequently leads to a relatively steady absorption coefficient of C*f*/Si3N4 composites with temperature rise.

2 -*S*<sup>21</sup> 2 ) is

with temperature and frequency derived from S parameters (i.e. *A* = 1-*S*<sup>11</sup>

**3.5 Modeling of temperature-dependent dielectric responses for C***f***/Si3N4**

As explicated in Section 3.2, the dielectric responses for multilayer C*f*/Si3N4 composites have been evaluated experimentally over X-band. Additionally, the corresponding theoretical relationship of complex permittivity versus frequency at room temperature has been successfully established, which could be expressed as:

which is in accordance with results in **Figure 10(a)** and **(b)**.

neglected reasonably.

**Figure 11.**

*Ref. [39]).*

*DOI: http://dx.doi.org/10.5772/intechopen.82389*

*Dielectric Responses in Multilayer Cf/Si3N4 as High-Temperature Microwave-Absorbing Materials DOI: http://dx.doi.org/10.5772/intechopen.82389*

**Figure 11.**

*Electromagnetic Materials and Devices*

The evolution of *ε*′

seen that both *<sup>ε</sup>*′

and *ε*″

and *<sup>ε</sup>*″

lattice *Eb* is distributed between 33.29 and 40.40 KJ/mol. The activation energy of electronic *Ea* is less than that of lattice *Eb*, which is mainly attributed to the binding force between the electrons and nucleus being lower than the covalent bonding force of lattice. Another important feature to be noticed is that the real permittivity of Si3N4 ceramics shows symmetrical features between the heating-up and cooling-down periods. The excellent thermo-stability of dielectric properties of Si3N4 ceramics has

established the foundation for high-temperature radar absorbing materials.

**3.4 High-temperature dielectric behaviors of multilayer C***f***/Si3N4 composites**

and frequency is illustrated in **Figure 10(a)** and **(b)**, respectively. It can be clearly

Similar phenomenon was also observed in other microwave-absorbing/shielding materials [24–26, 61–63]. Besides, as shown in **Figure 10(c)**, the loss tangent of C*f*/ Si3N4 composites remains higher than 0.6 over X-band and increases with increase in temperature. After comparison with low-loss *β*-SiC [64] as well as Si3N4, it is fairly clear that short carbon fibers are dominantly responsible for the frequency and temperature-dependent permittivity of C*f*/Si3N4 composites. This is attributed to the unique microstructure (named skin-core structure), which was confirmed from transmission electron microscope analysis and selected-area electron-diffraction patterns previously [65, 66]. As illustrated in **Figure 11**, the core region consists of graphitic basal planes with random orientation, whereas graphitic basal planes are parallel to the fiber axis in the skin region [66, 67]. Therefore, electron migration behaviors both within and between graphitic basal planes inside carbon fibers would lead to electric charge accumulation at interfaces between carbon fibers and Si3N4 matrix (usually referred to as space charge polarization) when exposed to electromagnetic field. Furthermore, this space charge polarization is supposed to be

*Three-dimensional plots of dielectric properties of Cf/Si3N4 composites versus frequency and temperature: (a) real part, (b) imaginary part, (c) loss tangent, and (d) attenuation coefficient (reprinted with permission* 

for multilayer C*f*/Si3N4 composites with temperature

of C*f*/Si3N4 composites are enhanced with temperature.

**38**

**Figure 10.**

*from Ref. [39]).*

*Sketches of microstructure arrangement and electronic motion in carbon fiber (reprinted with permission from Ref. [39]).*

enhanced with temperature rise since more and more electrons would be excited, which is in accordance with results in **Figure 10(a)** and **(b)**.

It should also be noted here that the lattice vibration also enhances with temperature rise, corresponding to the enhancement of scattering effect on migration of electrons. Fortunately, the energy of lattice vibration, which is described by phonons, is so small [68] that the scattering effect on electron migration could be neglected reasonably.

Attenuation coefficient is a quantity that characterizes how easily a material can be penetrated by incident microwave. A large attenuation coefficient means that the beam is quickly "attenuated" (weakened) as it passes through the material. The intrinsic attenuation coefficient of multilayer C*f*/Si3N4 composites along with temperature and frequency derived from S parameters (i.e. *A* = 1-*S*<sup>11</sup> 2 -*S*<sup>21</sup> 2 ) is illustrated in **Figure 10(d)**. It can be clearly observed that attenuation coefficient of C*f*/Si3N4 composites enhances continuously with increase in frequency, and reaches a maximum value around 0.7 at 12.4 GHz, which is almost twice as much as that reported for C*f*/SiO2 composites [25]. This superior absorption performance suggests the multilayer C*f*/Si3N4 composites to be competitive high-temperature microwave-absorbing materials. Another noteworthy phenomenon in **Figure 10(d)** is that the absorption coefficient remains steady in the investigated temperature range regardless of the loss tangent. As discussed previously, the overall absorbing performance results from reflection on the surface and attenuation inside the materials. However, the increasing permittivity means more severe impedance mismatch between air and the absorbing material, corresponding to reduction of penetrated electromagnetic energy. It could be explained by the reduced part of energy, which compensates the enhanced loss tangent, and consequently leads to a relatively steady absorption coefficient of C*f*/Si3N4 composites with temperature rise.

#### **3.5 Modeling of temperature-dependent dielectric responses for C***f***/Si3N4 composites**

As explicated in Section 3.2, the dielectric responses for multilayer C*f*/Si3N4 composites have been evaluated experimentally over X-band. Additionally, the corresponding theoretical relationship of complex permittivity versus frequency at room temperature has been successfully established, which could be expressed as:

$$\boldsymbol{\varepsilon}^{'}(o) = \frac{\boldsymbol{\varepsilon}\_1}{o^2} + \boldsymbol{\varepsilon}\_2 \tag{9}$$

$$
\varepsilon'(\alpha) = \frac{\alpha^2 \star c\_3}{c\_4 \cdot \alpha^3 \star c\_5 \cdot \alpha} \tag{10}
$$

where ω refers to the angular frequency (i.e., *<sup>ω</sup>* <sup>=</sup> <sup>2</sup>*<sup>f</sup>*), and *c*1, *c*2, *c*3, *c*4, and *c*5 are all pre-experimental parameters that are mainly associated with the surface density of short carbon fiber layers and thickness of Si3N4 layers. Herein, we first demonstrate this "room-temperature model" is still available at each evaluated temperature coverage up to 800°C or even higher with the help of nonlinear fitting technology. The best fitting curves of experimental data based on Eqs. (9) and (10) are depicted in **Figure 12**.

As expected, the measured results at each evaluated temperature agree quite well with the theoretical curve with coefficient of determination (R2 ) above 0.98. These observed results suggest that both *ε*′ and *ωε*″ of C*f*/Si3N4 composites are still inversely proportional to the frequency square *ω*<sup>2</sup> within the temperature range of 25–800°C. However, a universal model coupled with frequency as well as temperature is urgently needed. Actually, a great deal of effort has been made to model frequency dispersive behaviors of permittivity for dielectrics [45]. It is well established that the development of all dielectric relaxation models that came after classical Debye's could be explicated as:

$$
\varepsilon = \varepsilon\_{\text{oo}} + \frac{\varepsilon\_t - \varepsilon\_{\text{vo}}}{\mathbf{1} + j\alpha\sigma} \tag{11}
$$

**41**

as follows:

tivity in (*ε'*

, ε*''*

*Dielectric Responses in Multilayer Cf/Si3N4 as High-Temperature Microwave-Absorbing Materials*

heterogeneity of fibers' internal structure. This distribution property brings the

where *α* is a parameter which determines the width of the distribution of relaxation time. All parameters in Eq. (12) should be a function of temperature.

1 + (*j*)

<sup>2</sup> ] \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 1 <sup>+</sup> <sup>2</sup> () 1−*<sup>α</sup>* ⋅ sin\_\_\_ <sup>2</sup> <sup>+</sup> ()

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 2 1 <sup>+</sup> <sup>2</sup> ()

1−*<sup>α</sup>* ⋅ sin\_\_\_ <sup>2</sup> <sup>+</sup> ()

+ (*ε*″ + \_\_\_\_\_\_ *ε<sup>s</sup>* + *ε*<sup>∞</sup> <sup>2</sup> tan\_\_\_ <sup>2</sup> ) 2

should still be a circular arc with different radius. To put this into perspective, the experimental data are re-plotted in **Figure 13**, in which the best fitted circles based on Eq. (15) are marked as solid curves. As seen, high agreement between the proposed model with experimental data is observed. Besides, the center tends to

enhanced dielectric strength with temperature according to the Eq. (15). This may be attributed to the Enhanced electron concentration which participated in the

In addition, it is important to highlight some additional details. Firstly, the

be reduced to the classical Debye expression. From this point of view, the electronic polarization of short carbon fibers still follows the classic Debye relaxation process. Besides, the effects of temperature and electromagnetic interaction between neighboring short carbon fibers on relaxation time could be neglected reasonably.

Relaxation time also is one of key factors to analyze the dielectric behaviors for composites. After leaving out the effect of α, Eqs. (10) and (11) could be rewritten

<sup>τ</sup> <sup>⋅</sup> ε″ \_\_

It can be clearly seen from Eq. (16) that the real and imaginary parts of permit-

What is more, the slope of each line is exactly the inverse of relaxation time *τ* at a certain temperature evaluated. In this case, we can come to a conclusion that the relaxation time for multilayer C*f*/Si3N4 composites is weakly dependent on the temperature since the differences between fitted lines are quite modest. For clarity, a detailed plot of *τ* as a function of temperature is also illustrated as an inset in **Figure 14**. An increase from 216.1 to 250.2 ps has been derived when samples are

<sup>⁄</sup>ω) coordinate should be linear, which is also confirmed in **Figure 13**.

1−*<sup>α</sup>* ⋅ cos\_\_\_ 

> and *<sup>ε</sup>* ″

1−*<sup>α</sup>* ⋅ sin\_\_\_ 

1−*<sup>α</sup>* (12)

2(1−*α*) (13)

2(1−*α*) (14)

, which could be expressed as

) complex plane

) that Eq. (12) would

(15)

2 \_\_\_\_\_\_\_\_ 4cos<sup>2</sup> \_\_\_ 2

, *ε*″

<sup>=</sup> (*ε<sup>s</sup>* <sup>−</sup> *<sup>ε</sup>*∞)

axis with increase in temperature, suggesting

<sup>ω</sup> + ε<sup>∞</sup> (16)

Debye model (Eq. (11)) into more empirical and given by [25, 26, 70]:

Substitution of Euler's formula *ejx* <sup>=</sup> cos(*x*) <sup>+</sup> *<sup>j</sup>* sin(*x*) in Eq. (12) yields

*<sup>ε</sup>* <sup>=</sup> *<sup>ε</sup>*<sup>∞</sup> <sup>+</sup> \_\_\_\_\_\_\_ *<sup>ε</sup><sup>s</sup>* <sup>−</sup> *<sup>ε</sup>*<sup>∞</sup> *\_*

*<sup>ε</sup>*′ <sup>=</sup> *<sup>ε</sup>*<sup>∞</sup> <sup>+</sup> (*ε<sup>s</sup>* <sup>−</sup> *<sup>ε</sup>*∞) <sup>⋅</sup> [<sup>1</sup> <sup>+</sup> ()

*<sup>ε</sup>*″ <sup>=</sup> (*ε<sup>s</sup>* <sup>−</sup> *<sup>ε</sup>*∞) <sup>⋅</sup> ()

We finally obtain the relationship between *ε*′

*ε<sup>s</sup>* + *ε*<sup>∞</sup> <sup>2</sup> ) 2

Eq. (15) suggests that the locus of the permittivity in the (*<sup>ε</sup>*′

y-coordinates of fitted circular centers remain so small (~10<sup>−</sup><sup>3</sup>

(*ε*′ − \_\_\_\_\_\_

shift toward greater values of the ε'

polarization process occurs with temperature.

ε′ = \_\_1

*DOI: http://dx.doi.org/10.5772/intechopen.82389*

where *<sup>ε</sup><sup>s</sup>* and *ε*∞ are "static" and "infinite frequency" dielectric constants, respectively; *j* is the imaginary unit (i.e. equals to √ \_\_\_ −1); *ω* is the angular frequency; and *<sup>τ</sup>* is called the relaxation time. However, it is well known that the Debye model is originally developed for spherical polarizable molecules with a single relaxation time and without interaction between them [69]. Nevertheless, the short carbon fibers act as dipoles when exposed to altering electromagnetic fields oscillating in the microwave band, and electromagnetic field around each dipole is supposed to be coupled with that of neighboring dipoles. Furthermore, the relaxation time of chopped carbon fibers is supposed to be distributed over an interval rather than taking a constant value due to the variation of the fibers' length and the

#### **Figure 12.**

*(a) The real and (b) imaginary parts of permittivity for Cf/Si3N4 composites in X-band at the evaluated temperature (reprinted with permission from Ref. [39]).*

*Dielectric Responses in Multilayer Cf/Si3N4 as High-Temperature Microwave-Absorbing Materials DOI: http://dx.doi.org/10.5772/intechopen.82389*

heterogeneity of fibers' internal structure. This distribution property brings the Debye model (Eq. (11)) into more empirical and given by [25, 26, 70]:

$$\mathcal{E} = \mathcal{E}\_{oo} + \frac{\varepsilon\_t - \varepsilon\_m}{1 + \text{(jour)}^{1-\alpha}} \tag{12}$$

where *α* is a parameter which determines the width of the distribution of relaxation time. All parameters in Eq. (12) should be a function of temperature. Substitution of Euler's formula *ejx* <sup>=</sup> cos(*x*) <sup>+</sup> *<sup>j</sup>* sin(*x*) in Eq. (12) yields

SumIumou ou Euler's 10 ninuia  $\boldsymbol{\sigma} = \cos(\boldsymbol{\sigma}) \cdot \boldsymbol{\zeta} / \sin(\boldsymbol{\sigma})$  ni Eq. (12) ylemus

$$\boldsymbol{\varepsilon}' = \boldsymbol{\varepsilon}\_{\Leftrightarrow} + \frac{(\boldsymbol{\varepsilon}\_{l} - \boldsymbol{\varepsilon}\_{m}) \cdot \left[1 + (\boldsymbol{\sigma}\boldsymbol{\sigma})^{1-a} \cdot \sin\frac{a\boldsymbol{\pi}}{2}\right]}{1 + 2\left(\boldsymbol{\sigma}\boldsymbol{\tau}\right)^{1-a} \cdot \sin\frac{a\boldsymbol{\pi}}{2} + (\boldsymbol{\sigma}\boldsymbol{\sigma})^{2(1-a)}}\tag{13}$$

$$
\begin{aligned}
\mathbf{1} + \mathbf{2} \begin{pmatrix} \alpha \mathbf{r} \end{pmatrix} &\quad \cdot \sin \frac{\alpha \mathbf{r}}{2} + \begin{pmatrix} \alpha \mathbf{r} \end{pmatrix} \\\\
\mathbf{2} \\
\mathbf{3}
\end{aligned}
$$

$$
\mathbf{e}^{\prime} = \frac{(\varepsilon\_{i} - \varepsilon\_{\mathrm{eo}}) \cdot (\alpha \sigma)^{1-a} \cdot \cos \frac{\alpha \pi}{2}}{\mathbf{1} + 2 \left( \alpha \sigma \right)^{1-a} \cdot \sin \frac{\alpha \pi}{2} + \left( \alpha \sigma \right)^{2(1-a)}}
\tag{14}
$$

We finally obtain the relationship between *ε*′ and *<sup>ε</sup>* ″ , which could be expressed as

$$\left(\left.\varepsilon^{'} - \frac{\varepsilon\_{i} + \varepsilon\_{m}}{2}\right)^{2} + \left(\varepsilon^{'} + \frac{\varepsilon\_{i} + \varepsilon\_{m}}{2}\tan\frac{\underline{\alpha\pi}}{2}\right)^{2} = \frac{\left(\varepsilon\_{i} - \varepsilon\_{m}\right)^{2}}{4\cos^{2}\frac{\underline{\alpha\pi}}{2}}\tag{15}$$

Eq. (15) suggests that the locus of the permittivity in the (*<sup>ε</sup>*′ , *ε*″ ) complex plane should still be a circular arc with different radius. To put this into perspective, the experimental data are re-plotted in **Figure 13**, in which the best fitted circles based on Eq. (15) are marked as solid curves. As seen, high agreement between the proposed model with experimental data is observed. Besides, the center tends to shift toward greater values of the ε' axis with increase in temperature, suggesting enhanced dielectric strength with temperature according to the Eq. (15). This may be attributed to the Enhanced electron concentration which participated in the polarization process occurs with temperature.

In addition, it is important to highlight some additional details. Firstly, the y-coordinates of fitted circular centers remain so small (~10<sup>−</sup><sup>3</sup> ) that Eq. (12) would be reduced to the classical Debye expression. From this point of view, the electronic polarization of short carbon fibers still follows the classic Debye relaxation process. Besides, the effects of temperature and electromagnetic interaction between neighboring short carbon fibers on relaxation time could be neglected reasonably.

Relaxation time also is one of key factors to analyze the dielectric behaviors for composites. After leaving out the effect of α, Eqs. (10) and (11) could be rewritten as follows:

$$\mathbf{e}' = \frac{1}{7} \cdot \frac{\mathbf{e}'}{60} + \mathbf{e}\_{oo} \tag{16}$$

It can be clearly seen from Eq. (16) that the real and imaginary parts of permittivity in (*ε'* , ε*''* <sup>⁄</sup>ω) coordinate should be linear, which is also confirmed in **Figure 13**. What is more, the slope of each line is exactly the inverse of relaxation time *τ* at a certain temperature evaluated. In this case, we can come to a conclusion that the relaxation time for multilayer C*f*/Si3N4 composites is weakly dependent on the temperature since the differences between fitted lines are quite modest. For clarity, a detailed plot of *τ* as a function of temperature is also illustrated as an inset in **Figure 14**. An increase from 216.1 to 250.2 ps has been derived when samples are

*Electromagnetic Materials and Devices*

*ε*′

*ε*″

in **Figure 12**.

where *<sup>ε</sup><sup>s</sup>*

(*ω*) = \_\_ *c*1

(*ω*) <sup>=</sup> *<sup>ω</sup>*<sup>2</sup>

where ω refers to the angular frequency (i.e., *<sup>ω</sup>* <sup>=</sup> <sup>2</sup>*<sup>f</sup>*), and *c*1, *c*2, *c*3, *c*4, and *c*5 are all pre-experimental parameters that are mainly associated with the surface density of short carbon fiber layers and thickness of Si3N4 layers. Herein, we first demonstrate this "room-temperature model" is still available at each evaluated temperature coverage up to 800°C or even higher with the help of nonlinear fitting technology. The best fitting curves of experimental data based on Eqs. (9) and (10) are depicted

As expected, the measured results at each evaluated temperature agree quite

range of 25–800°C. However, a universal model coupled with frequency as well as temperature is urgently needed. Actually, a great deal of effort has been made to model frequency dispersive behaviors of permittivity for dielectrics [45]. It is well established that the development of all dielectric relaxation models that came after

*ε<sup>s</sup>* − *ε*<sup>∞</sup>

and *ε*∞ are "static" and "infinite frequency" dielectric constants, respec-

\_\_\_

*<sup>τ</sup>* is called the relaxation time. However, it is well known that the Debye model is originally developed for spherical polarizable molecules with a single relaxation time and without interaction between them [69]. Nevertheless, the short carbon fibers act as dipoles when exposed to altering electromagnetic fields oscillating in the microwave band, and electromagnetic field around each dipole is supposed to be coupled with that of neighboring dipoles. Furthermore, the relaxation time of chopped carbon fibers is supposed to be distributed over an interval rather than taking a constant value due to the variation of the fibers' length and the

*(a) The real and (b) imaginary parts of permittivity for Cf/Si3N4 composites in X-band at the evaluated* 

and *ωε*″

well with the theoretical curve with coefficient of determination (R2

0.98. These observed results suggest that both *ε*′

classical Debye's could be explicated as:

tively; *j* is the imaginary unit (i.e. equals to √

*temperature (reprinted with permission from Ref. [39]).*

are still inversely proportional to the frequency square *ω*<sup>2</sup>

*ε* = *ε*<sup>∞</sup> + \_\_\_\_\_\_

\_\_\_\_\_\_\_\_\_ + *c*<sup>3</sup>

*<sup>ω</sup>*<sup>2</sup> <sup>+</sup> *<sup>c</sup>*<sup>2</sup> (9)

*<sup>c</sup>*<sup>4</sup> <sup>⋅</sup> *<sup>ω</sup>*<sup>3</sup> <sup>+</sup> *<sup>c</sup>*<sup>5</sup> <sup>⋅</sup> *<sup>ω</sup>* (10)

) above

of C*f*/Si3N4 composites

<sup>1</sup> <sup>+</sup> *<sup>j</sup>* (11)

−1); *ω* is the angular frequency; and

within the temperature

**40**

**Figure 12.**

#### **Figure 13.**

*(a) Argand diagram of Cf/Si3N4 composites at different temperature, (b) detailed view for the region marked by black box in (a) (reprinted with permission from Ref. [39]).*

#### **Figure 14.**

*Plot of ε' as a function of* ε *''⁄* ω *with an inset of temperature-dependent relaxation time (reprinted with permission from Ref. [39]).*

heated from room temperature to 800°C. This gradually increasing trend may be ascribed to the enhancement of scattering effect between electrons for increase both the number and energy of electrons. Furthermore, the relaxation time *<sup>τ</sup>* (216.1–250.2 ps) is almost twice as much as a single cycle for time-harmonic electromagnetic wave in X-band (*<sup>t</sup>*<sup>0</sup> <sup>=</sup> <sup>1</sup> <sup>⁄</sup> *<sup>f</sup>* = 80.65–121.95 ps). As a result, electronic migration could not keep up with the pace of external alternating electronic field, which leads to continuous decrease of permittivity with frequency increase.

#### **4. Conclusion**

In this chapter, microwave dielectric properties of multilayer C*f*/Si3N4 composites fabricated via gelcasting and pressureless sintering were intensively studied in X-band. Firstly, a strong frequency dependence of the real and imaginary parts of permittivity at room temperature was observed at the X-band. Particularly, an

**43**

*Dielectric Responses in Multilayer Cf/Si3N4 as High-Temperature Microwave-Absorbing Materials*

equivalent RC circuit model concerning the frequency-dependent permittivity of multilayer C*f*/Si3N4 composites has been established. The predicted results reveal

quite well with the measured data. Secondly, high-temperature dielectric behaviors of Si3N4 ceramic show that both permittivity and loss tangent exhibit excellent

temperature range of 25–800°C. A revised dielectric relaxation model with Lorentz correction for as-prepared Si3N4 ceramics was established and validated by experimental data. The activation energy of electrons (15.46–17.49 KJ/mol) was demonstrated to be slightly smaller than that of lattice (33.29–40.40 KJ/mol). Finally, the microwave attenuation coefficient of multilayer C*f*/Si3N4 composites was inclined to be independent of temperature, and a maximum value of 0.7 could be achieved. The obvious positive temperature coefficient characteristic for permittivity is mainly attributed to the enhancement of electric polarization and relaxation of electron migration for graphitic basal planes of short carbon fibers. In addition to the fact that the room-temperature model concerning frequency-dependent permittivity is still available when extended into the full range of temperature coverage up to 800°C, an empirical equation with respect to temperature-dependent permittivity of multilayer C*f*/Si3N4 composites has been established. It is concluded that the measured complex permittivity of multilayer C*f*/Si3N4 composites is well distributed

plane. Furthermore, the relaxation time as a function of temperature also has been derived. Results suggest that the relaxation time for multilayer C*f*/Si3N4 composites increases from 216.1 to 250.2 ps when heated from room temperature to 800°C, and is almost twice as much as a single cycle for electromagnetic wave in X-band which leads to continuous decrease in permittivity with frequency increase. These findings point to important guidelines for analyzing high-temperature dielectric behaviors and revealing fundamental mechanisms for carbon fiber functionalized composites

The authors would like to acknowledge the generous funding from the National Key Research and Development Program of China (Grant No. 2017YFA0204600), the State Key Development Program for Basic Research of China (Grant No. 2011CB605804), and the National Natural Science Foundation

The authors declared that they have no conflicts of interest to this work.

thermo-stability with temperature coefficient lower than 10<sup>−</sup><sup>3</sup>

on circular arcs with centers actually kept around the real (<sup>ε</sup>

including but not limited to C*f*/Si3N4 composites.

**Acknowledgements**

**Conflict of interest**

of China (Grant No. 51802352).

) are inversely proportional to the frequency square, and agree

°C<sup>−</sup><sup>1</sup>

′

) axis in (<sup>ε</sup>

′ , ε ″

) complex

within the

*DOI: http://dx.doi.org/10.5772/intechopen.82389*

and (*<sup>ω</sup>* <sup>⋅</sup> *<sup>ε</sup>''*

that both *ε'*

*Dielectric Responses in Multilayer Cf/Si3N4 as High-Temperature Microwave-Absorbing Materials DOI: http://dx.doi.org/10.5772/intechopen.82389*

equivalent RC circuit model concerning the frequency-dependent permittivity of multilayer C*f*/Si3N4 composites has been established. The predicted results reveal that both *ε'* and (*<sup>ω</sup>* <sup>⋅</sup> *<sup>ε</sup>''* ) are inversely proportional to the frequency square, and agree quite well with the measured data. Secondly, high-temperature dielectric behaviors of Si3N4 ceramic show that both permittivity and loss tangent exhibit excellent thermo-stability with temperature coefficient lower than 10<sup>−</sup><sup>3</sup> °C<sup>−</sup><sup>1</sup> within the temperature range of 25–800°C. A revised dielectric relaxation model with Lorentz correction for as-prepared Si3N4 ceramics was established and validated by experimental data. The activation energy of electrons (15.46–17.49 KJ/mol) was demonstrated to be slightly smaller than that of lattice (33.29–40.40 KJ/mol). Finally, the microwave attenuation coefficient of multilayer C*f*/Si3N4 composites was inclined to be independent of temperature, and a maximum value of 0.7 could be achieved. The obvious positive temperature coefficient characteristic for permittivity is mainly attributed to the enhancement of electric polarization and relaxation of electron migration for graphitic basal planes of short carbon fibers. In addition to the fact that the room-temperature model concerning frequency-dependent permittivity is still available when extended into the full range of temperature coverage up to 800°C, an empirical equation with respect to temperature-dependent permittivity of multilayer C*f*/Si3N4 composites has been established. It is concluded that the measured complex permittivity of multilayer C*f*/Si3N4 composites is well distributed on circular arcs with centers actually kept around the real (<sup>ε</sup> ′ ) axis in (<sup>ε</sup> ′ , ε ″ ) complex plane. Furthermore, the relaxation time as a function of temperature also has been derived. Results suggest that the relaxation time for multilayer C*f*/Si3N4 composites increases from 216.1 to 250.2 ps when heated from room temperature to 800°C, and is almost twice as much as a single cycle for electromagnetic wave in X-band which leads to continuous decrease in permittivity with frequency increase. These findings point to important guidelines for analyzing high-temperature dielectric behaviors and revealing fundamental mechanisms for carbon fiber functionalized composites including but not limited to C*f*/Si3N4 composites.
