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Thermal Characterization of Carbon Fiber-Reinforced Carbon Composites

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Abstract

Carbon fiber-reinforced carbon (C/C) composites consist in a carbon matrix holding carbon or graphite fibers together, whose physical properties are determined not only by those of their individual components, but also by the layer buildup and the material preparation and processing. The complex structure of C/C composites along with the fiber orientation provide an effective means for tailoring their mechanical, electrical, and thermal properties. In this work, we use the Laser Flash Technique to measure the thermal diffusivity and thermal conductivity of C/C composites made up of laminates of weaved bundles of carbon fibers, forming a regular and repeated orthogonal pattern, embedded in a graphite matrix. Our experimental data show that: i) the cross-plane thermal conductivity remains practically constant around (5.3 ± 0.4) W·m−1 K−1, within the temperature range from 370 K to 1700 K. ii) The thermal diffusivity and thermal conductivity along the cross-plane direction to the fibers axis is about five times smaller than the corresponding ones in the laminates plane. iii) The measured cross-plane thermal conductivity is well described by a theoretical model that considers both the conductive and radiative thermal contributions of the effective thermal conductivity.

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Acknowledgements

Authors acknowledge the financial support received by the “Fondo Sectorial Conacyt-Secretaría de Energía-Sustentabilidad Energética and the “Centro Mexicano de Innovación en Energía Solar (CeMIESol)” Grant no. 207450 within the Strategic Project No. 10 “Combustibles Solares y Procesos Industriales (COSOL-pi)”, as well as to the Instituto de Energías Renovables of the Universidad Nacional Autónoma de México, Universidad Autónoma Metropolitana (Unidad Iztapalapa), Universidad Autónoma de Chihuahua and to the Centro de Investigacion y de Estudios Avanzados del Instituto Politecnico Nacional (CINVESTAV), for their assistance with financial and strategic management. This work was also partially supported by Projects 192 “Fronteras de la ciencia” and 251882 “Investigación Científica Básica 2015”. Additionally, Juan Daniel Macias acknowledges the financial support received by the “Fondo Sectorial Conacyt-Secretaría de Energía-Sustentabilidad Energética” under the program: “Estancias Posdoctorales en Mexico”.

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Appendix

Appendix

1.1 Thermal conductivity modeling

According to diverse theoretical models reported in the literature, the effective thermal conductivity κeff(T) of C/C composites can be expressed in terms of the thermal conductivity of the fibers and matrix, fibers volume fraction [72], fibers mass fraction [30], graphitic planes direction [12], contact heat transfer rate among fibers [19], and the effective contact area between two plates [5]. Considering the thermal and geometrical properties of the individual components, here we model κeff(T) as the sum of its solid conductive ((κc(T)) and radiative ((κr(T)) contributions, as follows

$$ {\kappa}_{eff}(T)={\kappa}_c(T)+{\kappa}_r(T) $$
(4)

Concerning the conductive contribution, for a C/C composite made up of laminate bundles of fibers forming a regular orthogonal pattern and uniform delamination between fiber mats embedded in a solid matrix, as shown in Fig. 5, the in-plane (κ) and out-plane (κ) solid thermal conductivities of this bidirectional composite are given by:

$$ {\kappa}_{\parallel }={n}_v{\kappa}_{v\parallel}\left(\theta \right)+{n}_h{\kappa}_{h\parallel}\left(\theta \right) $$
(5a)
$$ {\kappa}_{\perp }={\kappa}_{1\perp } $$
(5b)

where κ1⊥ is the composite thermal conductivity perpendicular to unidirectional fibers (see Fig. 5), nv and nh are the number fractions of fibers in the vertical and horizontal directions (with nv + nh = 1) and κv(θ) and κh(θ) are the in-plane thermal conductivity of vertical and horizontal unidirectional fibers in a direction making an angle θ with the vertical axis calculated as follows [59]:

$$ {\kappa}_{v\parallel}\left(\theta \right)={\kappa}_{1\parallel }{\cos}^2\left(\theta \right)+{\kappa}_{1\perp }{\sin}^2\left(\theta \right) $$
(6a)
$$ {\kappa}_{h\parallel}\left(\theta \right)={\kappa}_{1\perp }{\cos}^2\left(\theta \right)+{\kappa}_{1\parallel }{\sin}^2\left(\theta \right) $$
(6b)

Where κ1∥ is the composite thermal conductivity parallel to unidirectional fibers, as shown in Fig. 5, considering that the numbers of fibers in vertical and horizontal directions in Fig. 5 are equal (nv = nh = 1/2), Eqs. (5a), (6a) and (6b) yields to:

$$ {\kappa}_{\parallel }=\frac{1}{2}\left({\kappa}_{1\parallel }+{\kappa}_{1\perp}\right) $$
(7)

Equation (7) explicitly shows that the in-plane thermal conductivity of the composite in Fig. 5 is independent of the direction in which is measured, as was first predicted by Pilling et al. [72].

Fig. 5
figure 5

Thermal conductivity components parallel and perpendicular to unidirectional fibers

The thermal conductivity κ1∥ of the unidirectional composite in Fig. 6 is simply given by the series model, as follows [73]:

$$ {\kappa}_{1\parallel }=f{\kappa}_{f\parallel }+\left(1-f-p\right){\kappa}_{m\parallel } $$
(8)

where f and p are the respective volume fractions of the fibers and pores, while κf and κm are the fiber and matrix thermal conductivities in the axial direction of the fibers, respectively.

Fig. 6
figure 6

Thermal conductivity components of a single fiber (Left) and considering a 3D C/C composite unit cell (Right)

On the other hand, the out-plane thermal conductivity κ1⊥ can be determined by means of the Bruggeman’s model [74] applied to cylindrical particles [75], as follows:

$$ \frac{\kappa }{\kappa_m}=a+\frac{1}{2}{\beta}^2+\beta \sqrt{a+\frac{\beta^2}{4}} $$
(9a)
$$ \beta =\left(1-a\right)\left(1-v\right) $$
(9b)
$$ \alpha ={\left(\frac{\kappa_m}{\kappa_p}+\frac{a_{\kappa }}{a_f}\right)}^{-1} $$
(9c)

where κm is the thermal conductivity of the continuous matrix and κp is the thermal conductivity of air inside the pores or the porous filler (fibers in this case), and they interact with an interface thermal resistance R, being aκ = m the Kapitza radius [76], af the average radius of the fillers of volume fraction v and κ is the overall composite thermal conductivity. Considering the fibers and matrix like an effective matrix with cylindrical pores represented by the fibers (see Fig. 7), if we take κm = κ1⊥ (p = 0), and κp = 0 (air thermal conductivity), R → ∞ and v = p in Eqs. (9a) to (9c), which render:

$$ {\kappa}_{1\perp }={\left(1-p\right)}^2{\kappa}_m $$
(10)

Equation (10) indicates that the presence of pores reduces the effective thermal conductivity of the matrix and fibers by a factor of (1 − p)2. The thermal conductivity κ1⊥(p = 0) can now be calculated by reusing Eqs. (9a) to (9c) with κm = κm, κp = κf and v = f/(1 − p) being the volume fraction of fibers in the solid matrix alone. After making these substitutions in Eqs. (9a) to (9c) and combining the result with Eq. (10), one obtains:

$$ {k}_{1\perp }={\left(1-p\right)}^2{k}_{m\perp}\left({a}_{\perp }+\frac{1}{2}{\beta}_{\perp}^2+{\beta}_{\perp}\sqrt{a_{\perp }+\frac{\beta_{\perp}^2}{4}}\right) $$
(11)

where β and a are defined by Eqs. (9b) and (9c), for κm, κf and v = f/(1 − p). Eq. (11) considers the effects of the radius a and volume fraction f of the fibers, as well as of the porosity p and the interface thermal resistance R. The thermal conductivities κ1∥ and κ1⊥ of a composite with unidirectional fibers thus determine the overall thermal conductivities κ and κ of the bidirectional composite shown in Fig. 5.

Fig. 7
figure 7

Matrix and fibers are considered like an effective matrix with cylindrical pores, represented by the fibers

1.2 In-plane and out-plane thermal conductivities of a single carbon fiber

To calculate the thermal conductivity as a function of the temperature of the C/C composites (Eq. 11), it is necessary knowing the data of the axial and transverse thermal conductivities of a single carbon fiber and the pure matrix (without carbon fibers). The effective transverse thermal conductivity of the pitch-based carbon fibers, was predicted by H.S Huang et al. [77]. They calculated the in-plane and out-plane thermal conductivity for straight graphite sheets. The value of the simulated equivalent transverse thermal conductivity was 50.8 W·m−1 K−1 for perfect fibers and the measured value for real fibers was 12 Wm−1 K−1. Therefore, we can take the value of 27 for the anisotropy factor (κ/κ), a value which is consistent with those reported in literature [3, 62]. We can also have assumed that this factor does not change significantly as function of temperature [30]. C. Pradere et al. [78] report the longitudinal thermal conductivity in the temperature range from 750 to 2000 K, of single carbon fibers made with the same raw materials and exposed to a heat treatment similar those C/C composites examined in this work. Out of plane thermal conductivity of the fibers in the temperature range from 750 to 2000 K can be adjusted to the expression [78]:

$$ {\kappa}_P(T)=5.5045+4.97x{10}^{-3}T-4.78x{10}^{-6}{T}^2+1.20x{10}^{-9}{T}^3 $$
(12)

1.3 Pure matrix thermal conductivity

There are different and varied methods to build C/C composites, usually these methods consist on impregnation of sheets of carbon fiber fabrics with precursors (i.e. phenol-formaldehyde resins or gas phase precursors such as propane) that give rise the matrix of the composite [79]. Once the sheets are impregnated, the composites are molded and cured, subsequently they are carbonized at 1273 K under vacuum. The processes of impregnation and carbonization are repeated several times to change the density and porosity of the composite. Under these conditions the thermal conductivity κC of the composite obeys the simple mixing rule for two phases system: κc = κfVf + κmVm, V is the volume fraction, and c, f, and m refer to composite, filler and matrix, respectively. Due to κfVf ⋙ κmVm, then κc ∼ κfVf. However, when the C/C composites are subjected to a high temperature treatment (HTT) at temperatures above 2273 K, the experimentally thermal conductivity value of the composite increases significantly with respect to that calculated from the simple mixing rule [12, 24]. In addition, it has been demonstrated that after several HTT cycles, the thermal conductivity value of the C/C composite is no longer affected. Due to the fact that the thermal conductivity of the single fibers does not change significantly during the HTT cycles of C/C composites, because the fibers were previously subjected to a similar HTT and their structural properties such as crystallinity, crystalline distribution, molecular orientation, carbon content, and the amount of defects, are well formed [6]. Then, an increase in the thermal conductivity of the C/C composite can be attributed to changes in the crystallographic parameters of carbon matrix (interlayer spacing d002, and stack height Lc) [31]. For all the above, C/C composites with same fiber fabric and matrix precursors but subjected to different HTT may have different thermal conductivities [80], and this behavior is most noticeable at high temperature. One way to determine and to distinguish the effect of components in the C/C composite (carbon fibers and matrix), is by obtaining the thermal conductivity of samples of pure matrix (without carbon fibers). A good approximation to thermal behavior of the pure matrix is to consider it as glass-like carbon (GC) material [31], then the effective thermal conductivity of the matrix (κme) can be expressed by: κme = ξκGC, where ξ is a constant that depends on volume fraction of solid matrix and porous, as well as the pore shape factor [81]. Effective thermal conductivity of the matrix out of plane can be expressed as [50]:

$$ {\kappa}_{me}(T)=-0.2199+0.0123\kern.02em T-1.08x{10}^{-5}{T}^5+4.46x{10}^{-9}{T}^3-7.12x{10}^{-13}{T}^4 $$
(13)

Solid thermal conductivity as a function of the temperature κc(T) of the C/C composites (Eq. 11) can be rewritten as:

$$ {\kappa}_c(T)={\left(1-p\right)}^2{\kappa}_{me}(T)\left[\alpha (T)+\frac{1}{2}\beta {(T)}^2+\beta (T)\sqrt{\alpha (T)+\frac{\beta {(T)}^2}{4}}\right] $$
(14)

1.4 Radiative thermal conductivity

Effective thermal conductivity in porous materials can be expressed in terms of the solid thermal conductivity and the radiative conductivity (thermal conduction by gas convection is usually ignored due to its low value with respect to the other heat transfer mechanisms). Radiative conductivity (κr(T)) is based on Rosseland diffusion approximation [82], and it can be expressed by [83]:

$$ {\kappa}_r(T)=\frac{16}{3}{\sigma}_{SB}\frac{n^2(T)}{\sigma_e(T)}{T}^3 $$
(15)

Where σe(T) is the Rosseland mean extinction coefficient [84] which is an average value of the spectral extinction coefficient (σe, ν(T)) weighted by the local spectral energy flux, σSB is the Stefan-Boltzmann’s constant (5.67 × 10−8 W·m−2 K−4), and n is the effective refractive index (n =1 in this work). The Rosseland mean extinction coefficient is defined as follows:

$$ \frac{1}{\sigma_e(T)}=\frac{\int_0^{\infty}\frac{1}{\sigma_{e,\nu }(T)}\ \frac{\partial I\left(T,\nu \right)}{\partial T}\ d\nu}{\int_0^{\infty}\frac{\kern0.75em \partial I\left(T,\nu \right)}{\partial T}\ d\nu} $$
(16a)
$$ \frac{\partial I\left(T,\nu \right)}{\partial T}=\frac{2{h}^2{\nu}^4}{\kappa_B{T}^2{c}^2}\frac{e^{\frac{h\nu}{\kappa_BT}}}{{\left({e}^{\frac{h\nu}{\kappa_BT}}-1\right)}^2} $$
(16b)

Where ∂I(T, ν)/∂T is the temperature derivative of Planck’s equation. For homogeneous media, and considering σe,ν (T) independent on thickness, according to Beer’s law:

$$ {\sigma}_{e,\nu }(T)d=-\ln \left[{\tau}_{n,\nu }(T)\right] $$
(17)

Being d the thickness of the sample and τn,λ (T) is the spectral transmittance percentage, which could be measured with a Fourier transform infrared (FTIR) spectrometer. Figure 8 shows the spectral transmittance percentage (τn,λ (T)) for two samples (600 and 800 ± 30 μm in thickness) obtained with a FTIR spectrometer, and the Rosseland mean extinction coefficient (σe(T)) calculated with Eq. (16a).

Fig. 8
figure 8

Spectral transmittance percentage (τn,λ (T)) obtained by FTIR spectrometry for two samples (600 and 800 ± 30 μm in thickness). The inset graph corresponds to Rosseland mean extinction coefficient (σe(T)) calculated as function of temperature (Eq. 16a)

Figure 9 shows the effective thermal conductivity (Eq. 4), the solid thermal conductivity (Eq. 14) and radiative conductivity (Eq. 15) as function of temperature. The effective thermal conductivity of the C/C composite for temperatures below 1000 K is dominated mainly by solid thermal conductivity (99.4%). However, for temperatures above 1000 K, the contribution of radiative conductivity increases appreciably until reaching the value of 4.5% of the effective conductivity of the C/C composite at 2000 K.

Fig. 9
figure 9

Effective thermal conductivity obtained from Eq. 4, the solid thermal conductivity based on Eq. 14, and radiative conductivity determined using Eq. 15 as function of temperature

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Macias, J.D., Bante-Guerra, J., Cervantes-Alvarez, F. et al. Thermal Characterization of Carbon Fiber-Reinforced Carbon Composites. Appl Compos Mater 26, 321–337 (2019). https://doi.org/10.1007/s10443-018-9694-0

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