(a) TTTTtTTTTT (b) TTTTTTWT TTTTTT ▲BDF3400 ▲E130 ▣BDF3000 10 口E120 ●P.X-5 103 ● E105 △P.100-4 A E75 ◆ E55 102 0 E35 102 口 ●0 0 ●40 000000000000 10 101 ●40 AOO 100 Ao ◇● 100 阳 ◆P.120 310~1 10- VSC25 ■P.55 O IM7 11111L110-2 品mll102 100 10 102 103 100 101 102 103 Temperature(K) Temperature(K) Figure 3.3 (a)Comparison of the temperature variation of the thermal conductivity of pristine carbon fibers of various origins.Since scattering below room temperature is mainly on the crystallite boundaries,the phonon mean free path at low temperatures,i.e.below the max- imum of the thermal conductivity versus temperature curve is temperature insensitive and mainly determined by the crystallite size.The largest the crystallites the highest the thermal conductivity.Note that some VGCF and PDF of good crystalline perfection show a dielectric maximum below room temperature.For decreasing lattice perfection the max- imum is shifted to higher temperatures(Issi and Nysten,1998);(b)Temperature depend- ence of the thermal conductivity of the six samples of pitch-based carbon fibers heat treated at various temperatures,the same fibers with electrical resistivity is presented in Fig.3.1 (Nysten et al.,1991b). other carbon fibers (Bayot et al.,1989)and pyrocarbons(cfr.I,Figs 3.3 and 3.4).Such a behavior was explained in the frame of the weak localization theory (Bayot et al.,1989). As explained in I,weak localization generates an additional contribution to the low tem- perature electrical resistivity which adds to the classical Boltzmann resistivity.Indeed,in the weak disorder limit,which is also the condition for transport in the Boltzmann approxima- tion,i.e.when ke.11,where ke is the Fermi wave vector and I the mean free path of the charge carriers,a correction term,2,is added to the Boltzmann classical electrical con- ductivity,o2D Boltz.: c2D=o2DBol恤+δo2D (1) The additional term So2D accounts for localization and interaction effects which both predict a similar temperature variation (cfr.D). ©2003 Taylor&Francis
other carbon fibers (Bayot et al., 1989) and pyrocarbons (cfr. I, Figs 3.3 and 3.4). Such a behavior was explained in the frame of the weak localization theory (Bayot et al., 1989). As explained in I, weak localization generates an additional contribution to the low temperature electrical resistivity which adds to the classical Boltzmann resistivity. Indeed, in the weak disorder limit, which is also the condition for transport in the Boltzmann approximation, i.e. when kF.l 1, where kF is the Fermi wave vector and l the mean free path of the charge carriers, a correction term, 2D, is added to the Boltzmann classical electrical conductivity, 2D Boltz.: 2D 2D Boltz. 2D (1) The additional term 2D accounts for localization and interaction effects which both predict a similar temperature variation (cfr. I). BDF3400 BDF3000 P-X-5 P-100-4 P-120 VSC25 P-55 IM7 Temperature (K) Thermal conductivity (W m–1K–1) 100 101 102 103 Temperature (K) 100 101 102 103 E130 E120 E105 E75 E55 E35 103 102 101 100 10–1 10–2 103 102 101 100 10–1 10–2 (a) (b) Figure 3.3 (a) Comparison of the temperature variation of the thermal conductivity of pristine carbon fibers of various origins. Since scattering below room temperature is mainly on the crystallite boundaries, the phonon mean free path at low temperatures, i.e. below the maximum of the thermal conductivity versus temperature curve is temperature insensitive and mainly determined by the crystallite size. The largest the crystallites the highest the thermal conductivity. Note that some VGCF and PDF of good crystalline perfection show a dielectric maximum below room temperature. For decreasing lattice perfection the maximum is shifted to higher temperatures (Issi and Nysten, 1998); (b) Temperature dependence of the thermal conductivity of the six samples of pitch-based carbon fibers heat treated at various temperatures, the same fibers with electrical resistivity is presented in Fig. 3.1 (Nysten et al., 1991b). © 2003 Taylor & Francis
103 (wu) 102 0 00 8 10 口Lall O LaL A Thermal conductivity (lg) 100上1LL1 nteeei 0.335 0.338 0.341 0.344 0.347 doo2(nm) Figure 3.4 Dependence on the interlayer spacing doo2 of the in-plane coherence lengths and the phonon mean free path for boundary scattering,Ig (Nysten et al.,1991b). A magnetic field destroys this extra contribution(Bayot et al.,1989)and restores the clas- sical temperature variation predicted by the standard two band model (Klein,1964).This results in an apparent negative magnetoresistance. We have briefly discussed in I the positive and negative magnetoresistances in carbons and graphites and the interpretation of the latter in terms of weak localization effects.The positive magnetoresistance at low magnetic fields depends essentially on the carrier mobil- ities.The negative magnetoresistances,which was first observed in pregraphitic carbons by Mrozowski and Chaberski (1956)and later on in other forms of carbons,is a decrease in resistivity with increasing magnetic field.This effect was also observed in PAN-based fibers (Robson et al.,1972,1973),pitch-derived fibers (Bright and Singer,1979),and vapor-grown fibers (Endo et al.,1982)and was interpreted later on in the frame of the weak localization theory for two dimensional systems(Bayot et al.,1989). We present in Fig.3.2 the results obtained by Bright(1979)for the transverse magne- toresistance at 4.2 K for ex-mesophase pitch carbon fibers heat treated at different tempera- tures ranging from 1700C (sample D)to 3,000C(samples A,B,C,and F).It is worth noting that the four samples A,B,C,and F were all heat treated at the same temperature, but exhibited different residual resistivities (measured at 4.2K);3.8,5.1,7.0,and 6.6, 10-0cm respectively.Higher residual resistivities correspond to higher disorder.Samples G and E were heat treated at 2,500 and 2,000C respectively. It should be also noted that highly graphitized fibers,i.e.those heat treated at the highest temperatures,present large positive magnetoresistances,as expected from high mobility charge carriers.This explains why samples A and B which exhibit the lowest residual resis- tivities exhibit also large positive magnetoresistances,even at low magnetic fields.With increasing disorder,a negative magnetoresistance appears at low temperature,where the magnitude and the temperature range at which it shows up increase as the relative fraction of turbostratic planes increases in the material (Nysten et al.,1991a). The results obtained,which are presented in Fig.3.2,were later confirmed by Bayot et al. (1989)and Nysten et al.(1991a),who found the same qualitative behavior on different samples of pitch-derived carbon fibers. ©2003 Taylor&Francis
A magnetic field destroys this extra contribution (Bayot et al., 1989) and restores the classical temperature variation predicted by the standard two band model (Klein, 1964). This results in an apparent negative magnetoresistance. We have briefly discussed in I the positive and negative magnetoresistances in carbons and graphites and the interpretation of the latter in terms of weak localization effects. The positive magnetoresistance at low magnetic fields depends essentially on the carrier mobilities. The negative magnetoresistances, which was first observed in pregraphitic carbons by Mrozowski and Chaberski (1956) and later on in other forms of carbons, is a decrease in resistivity with increasing magnetic field. This effect was also observed in PAN-based fibers (Robson et al., 1972, 1973), pitch-derived fibers (Bright and Singer, 1979), and vapor-grown fibers (Endo et al., 1982) and was interpreted later on in the frame of the weak localization theory for two dimensional systems (Bayot et al., 1989). We present in Fig. 3.2 the results obtained by Bright (1979) for the transverse magnetoresistance at 4.2 K for ex-mesophase pitch carbon fibers heat treated at different temperatures ranging from 1700 C (sample D) to 3,000 C (samples A, B, C, and F). It is worth noting that the four samples A, B, C, and F were all heat treated at the same temperature, but exhibited different residual resistivities (measured at 4.2 K); 3.8, 5.1, 7.0, and 6.6, 104 cm respectively. Higher residual resistivities correspond to higher disorder. Samples G and E were heat treated at 2,500 and 2,000 C respectively. It should be also noted that highly graphitized fibers, i.e. those heat treated at the highest temperatures, present large positive magnetoresistances, as expected from high mobility charge carriers. This explains why samples A and B which exhibit the lowest residual resistivities exhibit also large positive magnetoresistances, even at low magnetic fields. With increasing disorder, a negative magnetoresistance appears at low temperature, where the magnitude and the temperature range at which it shows up increase as the relative fraction of turbostratic planes increases in the material (Nysten et al., 1991a). The results obtained, which are presented in Fig. 3.2, were later confirmed by Bayot et al. (1989) and Nysten et al. (1991a), who found the same qualitative behavior on different samples of pitch-derived carbon fibers. Figure 3.4 Dependence on the interlayer spacing d002 of the in-plane coherence lengths and the phonon mean free path for boundary scattering, lB (Nysten et al., 1991b). 103 102 101 100 0.335 0.338 0.341 0.344 0.347 Crystallite size (nm) d002 (nm) La// La⊥ Thermal conductivity (lB) © 2003 Taylor & Francis
4 Thermal conductivity 4.1 Electron and phonon conduction Around and below room temperature,heat conduction in solids is generated either by the charge carriers as is the case for pure metals or by the lattice waves,the phonons,which is the case for electrical insulators.In carbons and graphites,owing to the small densities of charge carriers,associated with a relatively large in-plane lattice thermal conductivity due to the strong covalent bonds,heat is almost exclusively carried by the phonons,except at very low temperatures,where both contributions may be observed.In that case,the total thermal conductivity is expressed: K=KE十KL (2) where KE is the electronic thermal conductivity due to the charge carriers and KL is the lattice thermal conductivity due to the phonons. We will show in Section 6.6 that,because of their large length to cross section ratio,it is possible to separate ke and KL in carbon fibers,when they contribute by comparable amounts as it is the case at low temperature for pristine fibers and at various temperatures for the intercalated material. In Fig.3.3a we present the temperature variation of the thermal conductivity of pristine carbon fibers of various origins and precursors.In Fig.3.3b we compare the temperature dependence of the thermal conductivity of the six samples of pitch-based carbon fibers heat treated at various temperatures(Nysten et al.,1991b).These are the same set of fibers which electrical resistivity is presented in Fig.3.1. 4.2 Lattice conduction It was shown that the lattice thermal conductivity of carbon fibers is directly related to the the in-plane coherence length (Nysten et al.,1991b;Issi and Nysten,1998).Thus thermal conductivity measurements allow to determine this parameter.It also enables to compare between shear moduli(C44)and provide information about point defects. In Fig.3.4,the dependence of the in-plane coherence lengths,La,and the phonon mean free paths for boundary scattering,IB,on the interlayer spacing doo2 is presented (Nysten et al.,1991b).One may see that the phonon mean free path for boundary scattering is almost equal to the in-plane coherence length as determined by x-ray diffraction,La.Thermal con- ductivity measurements may thus be used as a tool to determine this parameter,especially for high La values where x-rays are inadequate.One may also observe that the concentration of point defects such as impurities or vacancies,decreases with increasing graphitization. A naive way to understand how lattice conduction takes place in crystalline materials,is by considering the case of graphite in-plane,assuming that it is a two-dimensional(2D)sys- tem,which is not too far from the real situation around room temperature.The atoms in such a system may be represented by a 2D array of balls and springs and any vibration at one end of the system will be transmitted via the springs to the other end.Since the carbon atoms have small masses and the interatomic covalent forces are strong,one should expect a good transmission of the vibrational motion in such a system and thus a good lattice thermal con- ductivity.Any perturbation in the regular arrangement of the atoms,such as defects or atomic vibrations,will cause a perturbation in the heat flow,thus giving rise to scattering which decreases the thermal conductivity. ©2003 Taylor&Francis
4 Thermal conductivity 4.1 Electron and phonon conduction Around and below room temperature, heat conduction in solids is generated either by the charge carriers as is the case for pure metals or by the lattice waves, the phonons, which is the case for electrical insulators. In carbons and graphites, owing to the small densities of charge carriers, associated with a relatively large in-plane lattice thermal conductivity due to the strong covalent bonds, heat is almost exclusively carried by the phonons, except at very low temperatures, where both contributions may be observed. In that case, the total thermal conductivity is expressed: E L (2) where E is the electronic thermal conductivity due to the charge carriers and L is the lattice thermal conductivity due to the phonons. We will show in Section 6.6 that, because of their large length to cross section ratio, it is possible to separate E and L in carbon fibers, when they contribute by comparable amounts as it is the case at low temperature for pristine fibers and at various temperatures for the intercalated material. In Fig. 3.3a we present the temperature variation of the thermal conductivity of pristine carbon fibers of various origins and precursors. In Fig. 3.3b we compare the temperature dependence of the thermal conductivity of the six samples of pitch-based carbon fibers heat treated at various temperatures (Nysten et al., 1991b). These are the same set of fibers which electrical resistivity is presented in Fig. 3.1. 4.2 Lattice conduction It was shown that the lattice thermal conductivity of carbon fibers is directly related to the the in-plane coherence length (Nysten et al., 1991b; Issi and Nysten, 1998). Thus thermal conductivity measurements allow to determine this parameter. It also enables to compare between shear moduli (C44) and provide information about point defects. In Fig. 3.4, the dependence of the in-plane coherence lengths, La, and the phonon mean free paths for boundary scattering, lB, on the interlayer spacing d002 is presented (Nysten et al., 1991b). One may see that the phonon mean free path for boundary scattering is almost equal to the in-plane coherence length as determined by x-ray diffraction, La. Thermal conductivity measurements may thus be used as a tool to determine this parameter, especially for high La values where x-rays are inadequate. One may also observe that the concentration of point defects such as impurities or vacancies, decreases with increasing graphitization. A naive way to understand how lattice conduction takes place in crystalline materials, is by considering the case of graphite in-plane, assuming that it is a two-dimensional (2D) system, which is not too far from the real situation around room temperature. The atoms in such a system may be represented by a 2D array of balls and springs and any vibration at one end of the system will be transmitted via the springs to the other end. Since the carbon atoms have small masses and the interatomic covalent forces are strong, one should expect a good transmission of the vibrational motion in such a system and thus a good lattice thermal conductivity. Any perturbation in the regular arrangement of the atoms, such as defects or atomic vibrations, will cause a perturbation in the heat flow, thus giving rise to scattering which decreases the thermal conductivity. © 2003 Taylor & Francis
In order to discuss the lattice thermal conductivity results of isotropic materials,one generally uses the Debye relation: 3 (3) where C is the lattice specific heat per unit volume,v is an average phonon velocity,the velocity of sound,and I the mean free path which is directly related to the phonon relaxation time,T,through the relation I =v T.For a given solid,since the specific heat and the phonon velocities are the same for different samples,the sample thermal conductivity at a given temperature is directly proportional to the phonon mean free path. VGCF's heat treated at 3,000C,may present room temperature heat conductivities exceeding 1,000 WmK-(Fig.3.3a).The thermal conductivity of less ordered fibers may vary widely,about two orders of magnitude,according to their microstructure(Issi and Nysten,1998).At low temperature,the lattice thermal conductivity is mainly limited by phonon-boundary scattering and is directly related to the in-plane coherence length,La When scattering is mainly on the crystallite boundaries,the phonon mean free path should be temperature insensitive.Since the velocity of sound is almost temperature insen- sitive,the temperature dependence of the thermal conductivity should follow that of the spe- cific heat.Thus,the largest the crystallites the highest the thermal conductivity.Well above the maximum,phonon scattering is due to an intrinsic mechanism:phonon-phonon umk- lapp processes,and the thermal conductivity should thus be the same for different samples. Around the thermal conductivity maximum,scattering of phonons by point defects(small scale defects)is the dominating process.The position and the magnitude of the thermal con- ductivity maximum will thus depend on the competition between the various scattering processes(boundary,point defect,phonon,...).So,for different samples of the same mate- rial the position and magnitude of the maximum will depend on the point defects and La since phonon-phonon interactions are assumed to be the same.This explains why,by measuring the low temperature thermal conductivity,one may gather information about the in-plane coherence length La and point defects.This shows also that by adjusting the microstructure of carbon fibers,one may tailor their thermal conductivity to a desired value. Some VGCF and PDF of good crystalline perfection show a maximum below room temperature and,with decreasing lattice perfection the maximum is shifted to higher temperatures (Issi and Nysten,1998). Recently,the thermal conductivities of ribbon-shaped carbon fibers produced at Clemson University and graphitized at 2,400C and those of commercial round fibers graphitized at temperatures above 3,000C were measured and the data were compared.It was shown that, in spite of the difference in the heat treatment temperature,the two sets of fibers presented almost the same electrical and thermal conductivities.This clearly shows that,for a given HTT, spinning conditions have an important influence on the transport properties of pitch-based carbon fibers.By modifying these conditions,one may enhance these conductivities,which is important for practical applications since HTT is a costly process(cfr.Part 1,2 of this issue). Oddly enough,though the electrical and thermal conductivities of pristine carbon fibers are generated by different entities,charge carriers for the electrical conductivity and phonons for the thermal conductivity,a direct relation between the two parameters is observed at room temperature (Nysten et al.,1987).This is related to the fact that both transport properties depend dramatically on the structure of the fibers.They both increase ©2003 Taylor&Francis
In order to discuss the lattice thermal conductivity results of isotropic materials, one generally uses the Debye relation: (3) where C is the lattice specific heat per unit volume, v is an average phonon velocity, the velocity of sound, and l the mean free path which is directly related to the phonon relaxation time, , through the relation l v . For a given solid, since the specific heat and the phonon velocities are the same for different samples, the sample thermal conductivity at a given temperature is directly proportional to the phonon mean free path. VGCF’s heat treated at 3,000 C, may present room temperature heat conductivities exceeding 1,000Wm1 K1 (Fig. 3.3a). The thermal conductivity of less ordered fibers may vary widely, about two orders of magnitude, according to their microstructure (Issi and Nysten, 1998). At low temperature, the lattice thermal conductivity is mainly limited by phonon-boundary scattering and is directly related to the in-plane coherence length, La. When scattering is mainly on the crystallite boundaries, the phonon mean free path should be temperature insensitive. Since the velocity of sound is almost temperature insensitive, the temperature dependence of the thermal conductivity should follow that of the specific heat. Thus, the largest the crystallites the highest the thermal conductivity. Well above the maximum, phonon scattering is due to an intrinsic mechanism: phonon–phonon umklapp processes, and the thermal conductivity should thus be the same for different samples. Around the thermal conductivity maximum, scattering of phonons by point defects (small scale defects) is the dominating process. The position and the magnitude of the thermal conductivity maximum will thus depend on the competition between the various scattering processes (boundary, point defect, phonon, …). So, for different samples of the same material the position and magnitude of the maximum will depend on the point defects and La, since phonon–phonon interactions are assumed to be the same. This explains why, by measuring the low temperature thermal conductivity, one may gather information about the in-plane coherence length La and point defects. This shows also that by adjusting the microstructure of carbon fibers, one may tailor their thermal conductivity to a desired value. Some VGCF and PDF of good crystalline perfection show a maximum below room temperature and, with decreasing lattice perfection the maximum is shifted to higher temperatures (Issi and Nysten, 1998). Recently, the thermal conductivities of ribbon-shaped carbon fibers produced at Clemson University and graphitized at 2,400 C and those of commercial round fibers graphitized at temperatures above 3,000 C were measured and the data were compared. It was shown that, in spite of the difference in the heat treatment temperature, the two sets of fibers presented almost the same electrical and thermal conductivities. This clearly shows that, for a given HTT, spinning conditions have an important influence on the transport properties of pitch-based carbon fibers. By modifying these conditions, one may enhance these conductivities, which is important for practical applications since HTT is a costly process (cfr. Part 1, § 2 of this issue). Oddly enough, though the electrical and thermal conductivities of pristine carbon fibers are generated by different entities, charge carriers for the electrical conductivity and phonons for the thermal conductivity, a direct relation between the two parameters is observed at room temperature (Nysten et al., 1987). This is related to the fact that both transport properties depend dramatically on the structure of the fibers. They both increase g 1 3 C v 1 © 2003 Taylor & Francis