3 ELECTRONIC AND THERMAL PROPERTIES OF CARBON FIBERS J.-P Issi 1 Introduction Despite the great deal of attention that carbon nanotubes(CN),the"ultimate"carbon fibers, is attracting because of their fascinating scientific aspects,the physical properties of the macroscopic version of carbon fibers remain a topic of great interest.Transport properties are no exception,since the various structures of carbon fibers and their particular geometry, have lead to interesting observations,which could not be made on bulk carbons and graphites.To this should be added the practical aspects since carbon fibers find unique applications,particularly in the form of composites.Also,as is the case for the bulk mate- rial,some carbon fibers may be intercalated with various species leading to a significant modification of their physical properties. Because of their large length to cross section ratio,DC electrical resistivity measurements are relatively easy to perform on fibrous materials,in contrast to the case of bulk graphites (see Section 2).This allowed to separate the electronic and lattice contributions to the ther- mal conductivity using the Wiedemann-Franz law (Section 4).It also made possible high resolution electrical resistivity measurements leading to the discovery of quantum transport effects on intercalated (Piraux et al.,1985)and pristine fibers (Bayot et al.,1989).Also car- bon fiber-based composites exploit their unique mechanical and thermal properties for which they find applications as light mechanical systems and heat transfer devices. Transport properties have been reported for all varieties of carbon fibers including single wall(SWNT)and multiwall (MWNT)carbon nanotubes.Among macroscopic fibers,vapor grown carbon fibers(VGCF)are the most adequate for basic studies.They have the highest structural perfection when heat treated at high temperature.In that case,their transport prop- erties are much like those observed for highly oriented pyrolytic graphite(HOPG).They display the highest electrical and thermal conductivities with respect to other carbon fibers. Also,they may be intercalated with donor and acceptor species which results in higher elec- trical conductivities,as is the case for HOPG.Similarly,their thermal conductivity and thermoelectric power are notably modified by intercalation. The transport properties of carbon fibers are governed by the in-plane coherence length which mainly depends on the heat treatment temperature (cfr.e.g.Dresselhaus et al.,1988; Issi and Nysten,1998).Some commercial pitch-derived carbon fibers(PDF)heat treated at high temperature have low electrical resistivities and thermal conductivities close to 1,000 Wm-K-1,which largely exceeds that of copper.These high thermal conductivities associ- ated to the fact that they may be obtained in a continuous form make them the ideal candidates for thermal management applications(Allen and Issi,1985). ©2003 Taylor&Francis
3 ELECTRONIC AND THERMAL PROPERTIES OF CARBON FIBERS J.-P. Issi 1 Introduction Despite the great deal of attention that carbon nanotubes (CN), the “ultimate” carbon fibers, is attracting because of their fascinating scientific aspects, the physical properties of the macroscopic version of carbon fibers remain a topic of great interest. Transport properties are no exception, since the various structures of carbon fibers and their particular geometry, have lead to interesting observations, which could not be made on bulk carbons and graphites. To this should be added the practical aspects since carbon fibers find unique applications, particularly in the form of composites. Also, as is the case for the bulk material, some carbon fibers may be intercalated with various species leading to a significant modification of their physical properties. Because of their large length to cross section ratio, DC electrical resistivity measurements are relatively easy to perform on fibrous materials, in contrast to the case of bulk graphites (see Section 2). This allowed to separate the electronic and lattice contributions to the thermal conductivity using the Wiedemann–Franz law (Section 4). It also made possible high resolution electrical resistivity measurements leading to the discovery of quantum transport effects on intercalated (Piraux et al., 1985) and pristine fibers (Bayot et al., 1989). Also carbon fiber-based composites exploit their unique mechanical and thermal properties for which they find applications as light mechanical systems and heat transfer devices. Transport properties have been reported for all varieties of carbon fibers including single wall (SWNT) and multiwall (MWNT) carbon nanotubes. Among macroscopic fibers, vapor grown carbon fibers (VGCF) are the most adequate for basic studies. They have the highest structural perfection when heat treated at high temperature. In that case, their transport properties are much like those observed for highly oriented pyrolytic graphite (HOPG). They display the highest electrical and thermal conductivities with respect to other carbon fibers. Also, they may be intercalated with donor and acceptor species which results in higher electrical conductivities, as is the case for HOPG. Similarly, their thermal conductivity and thermoelectric power are notably modified by intercalation. The transport properties of carbon fibers are governed by the in-plane coherence length which mainly depends on the heat treatment temperature (cfr. e.g. Dresselhaus et al., 1988; Issi and Nysten, 1998). Some commercial pitch-derived carbon fibers (PDF) heat treated at high temperature have low electrical resistivities and thermal conductivities close to 1,000 Wm1 K1 , which largely exceeds that of copper. These high thermal conductivities associated to the fact that they may be obtained in a continuous form make them the ideal candidates for thermal management applications (Allen and Issi, 1985). © 2003 Taylor & Francis
PAN-based fibers(PAN)are also continuous fibers,but they are generally disordered and their conductivity levels are rather low.One exception is the Celanese GY70,which exhibits a room temperature thermal conductivity of almost 200 WmK-when it is heat treated at very high temperature (Nysten et al.,1987). Some carbon nanotubes may have electrical conductivities comparable to VGCF heat treated at high temperature(Issi and Charlier,1999).The thermal conductivity has not been measured on a single CN,but one may expect for such materials very high values associ- ated to their unique mechanical properties. VGCF's heat treated at high temperature are semimetallic with very few charge carriers as compared to metals.Also,like HOPG,they have generally more than one type of charge carriers,which complicates the analysis of electronic transport properties data.Owing to the small density of charge carriers,associated with a relatively large lattice in-plane thermal conductivity,heat is almost exclusively carried by the lattice vibrations above the liquid helium temperature range. Because of the favorable length to cross section ratios (Chieu et al.,1982;Issi,1992), four-probe electrical DC measurements may be readily performed on intercalated fibers contrary to the case of the bulk material.High resolution electrical resistivity measurements have thus been performed on graphite fibers allowing the investigation of weak localization effects and the separation of the ideal resistivity from the residual resistivity in spite of their very low residual resistivity ratio (RRR)-the ratio of the resistance at 300K to that at 4.2K (Piraux et al.,1986a,b). A few comprehensive reviews have been recently published on the transport properties of pristine and intercalated carbon fibers(Dresselhaus et al.,1988;Issi,1992;Issi and Nysten, 1998).We shall refer to them when necessary for more detailed information. In chapter 3,volume I of this series,to which we shall refer hereafter as I,we have discussed the basic aspects of the transport properties of carbons and graphites(Issi,2000). We have pointed out in I that there were no qualitative differences between the basic trans- port phenomena in bulk and fibrous materials.So,we will mainly concentrate here on the specific aspects related to fibers which were not discussed in detail in I.These are essen- tially due to their geometry and their very small cross sections.We will also discuss in some detail the thermal conductivity of fibers,since it was not presented in I.We will refer,when necessary,to the basic concepts developed in I.Thus,in this chapter emphasis will be placed on: the thermal conductivity of pristine fibers; the effect of intercalation on the transport properties; the thermal and,to a lesser extent,the electrical conductivity of composites; the experimental difficulties associated to measurements on fibrous materials. The chapter is organized as follows.First,we shall point out some experimental aspects specific to fibrous materials,including CNs(Section 2).Then we shall discuss the electri- cal conductivity (Section 3)the thermal conductivity (Section 4)and the thermoelectric power(Section 5)of various types of pristine carbon fibers.The effect of intercalation on these transport properties will be discussed in Section 6.Then,after briefly showing how transport measurements may be used to characterize carbon fibers (Section 7),we will consider the situation for carbon fiber composites (Section 8). ©2003 Taylor&Francis
PAN-based fibers (PAN) are also continuous fibers, but they are generally disordered and their conductivity levels are rather low. One exception is the Celanese GY70, which exhibits a room temperature thermal conductivity of almost 200Wm1 K1 when it is heat treated at very high temperature (Nysten et al., 1987). Some carbon nanotubes may have electrical conductivities comparable to VGCF heat treated at high temperature (Issi and Charlier, 1999). The thermal conductivity has not been measured on a single CN, but one may expect for such materials very high values associated to their unique mechanical properties. VGCF’s heat treated at high temperature are semimetallic with very few charge carriers as compared to metals. Also, like HOPG, they have generally more than one type of charge carriers, which complicates the analysis of electronic transport properties data. Owing to the small density of charge carriers, associated with a relatively large lattice in-plane thermal conductivity, heat is almost exclusively carried by the lattice vibrations above the liquid helium temperature range. Because of the favorable length to cross section ratios (Chieu et al., 1982; Issi, 1992), four-probe electrical DC measurements may be readily performed on intercalated fibers contrary to the case of the bulk material. High resolution electrical resistivity measurements have thus been performed on graphite fibers allowing the investigation of weak localization effects and the separation of the ideal resistivity from the residual resistivity in spite of their very low residual resistivity ratio (RRR) – the ratio of the resistance at 300K to that at 4.2K (Piraux et al., 1986a,b). A few comprehensive reviews have been recently published on the transport properties of pristine and intercalated carbon fibers (Dresselhaus et al., 1988; Issi, 1992; Issi and Nysten, 1998). We shall refer to them when necessary for more detailed information. In chapter 3, volume 1 of this series, to which we shall refer hereafter as I, we have discussed the basic aspects of the transport properties of carbons and graphites (Issi, 2000). We have pointed out in I that there were no qualitative differences between the basic transport phenomena in bulk and fibrous materials. So, we will mainly concentrate here on the specific aspects related to fibers which were not discussed in detail in I. These are essentially due to their geometry and their very small cross sections. We will also discuss in some detail the thermal conductivity of fibers, since it was not presented in I. We will refer, when necessary, to the basic concepts developed in I. Thus, in this chapter emphasis will be placed on: ● the thermal conductivity of pristine fibers; ● the effect of intercalation on the transport properties; ● the thermal and, to a lesser extent, the electrical conductivity of composites; ● the experimental difficulties associated to measurements on fibrous materials. The chapter is organized as follows. First, we shall point out some experimental aspects specific to fibrous materials, including CNs (Section 2). Then we shall discuss the electrical conductivity (Section 3) the thermal conductivity (Section 4) and the thermoelectric power (Section 5) of various types of pristine carbon fibers. The effect of intercalation on these transport properties will be discussed in Section 6. Then, after briefly showing how transport measurements may be used to characterize carbon fibers (Section 7), we will consider the situation for carbon fiber composites (Section 8). © 2003 Taylor & Francis
2 Experimental challenges There are some specific problems associated to the measurement of the transport properties on fibrous materials which are not encountered in bulk materials.It is obvious that this should be the case for measurements on individual nanotubes where samples are of submi- cronic sizes and are quite difficult to handle.For other reasons(cfr.below),it also applies for the measurement of the thermal conductivity on carbon fibers. For fibers with diameters around 10 um,electrical resistivity,magnetoresistance,and thermoelectric power measurements do not generally present serious problems.The prob- lems encountered in the case of electrical resistivity measurements are due to the nature of the samples.Indeed,the diameters of the fibers are not the same along a given filament and from one filament to another of the same batch.Also,the cross sections of the samples are not always cylindrical.This makes it difficult to calculate the conductivity (or resistivity) from the measured resistance.So,the determination of the fiber cross section introduces large uncertainties in the estimation of the absolute values of the resistivities or conductivi- ties.Since electrical parameters such as electrical current and voltage,may be measured with great accuracy,the data obtained are more accurate with regard to temperature varia- tion than with regard to absolute magnitudes.Fortunately,for the interpretation of the exper- imental results it is more important to know the temperature variation than the absolute values.These problems are not met in magnetoresistance and thermoelectric power meas- urements since the knowledge of the samples cross sections is not needed to calculate these transport coefficients. More generally speaking,it is a rather easy task to measure the electrical resistivity, except for extreme cases of very low or very high values.However,the measurement of sam- ples of submicronic sizes requires a miniaturization of the experimental system,which may in some instances attain a high degree of sophistication.This is particularly true for the case of single CNs,where one has to deal with samples of a few nm diameter and about a um length(Issi and Charlier,1999).One has first to detect the sample,then apply to it electri- cal contacts,which means,in a four-probe measurement,four metallic conductors,two for the injected current and two for measuring the resulting voltage.This requires the use of nanolithographic techniques(Langer et al.,1994).Besides,one has to characterize the CN sample which electrical resistivity is measured in order to determine its diameter and helicity,which leads to the knowledge of the electronic structure. Thermal conductivity is a very delicate measurement to perform on a single fiber and pro- hibitively difficult on a single CN.This explains why little attention had been paid to the thermal conductivity of carbon fibers until the beginning of the 1980s and while still no data are available on single CNs.More generally,thermal conductivity measurements are time consuming and very delicate to perform.This is particularily true for samples of small cross- sections,as it is the case for carbon fibers (Piraux et al.,1987).Indeed,since the fibers are usually of small diameters(~10>m)it is difficult to make sure that the heat losses in the measuring system do not by far exceed the thermal conductance of the samples measured. The thermal conductance is defined as the heat flow through the sample per unit tempera- ture difference.One may measure a bundle of fibers to increase the thermal conductance with respect to heat losses,but it is not always possible to realize samples in the form of bun- dles.Moreover,in order to be able to get an insight into the mechanisms of the thermal con- ductivity of these fibers,it is necessary to measure the temperature variation of this property over a wide temperature range on a single well characterized fiber,as was done for VGCFs (Piraux et al.,1984). ©2003 Taylor&Francis
2 Experimental challenges There are some specific problems associated to the measurement of the transport properties on fibrous materials which are not encountered in bulk materials. It is obvious that this should be the case for measurements on individual nanotubes where samples are of submicronic sizes and are quite difficult to handle. For other reasons (cfr. below), it also applies for the measurement of the thermal conductivity on carbon fibers. For fibers with diameters around 10m, electrical resistivity, magnetoresistance, and thermoelectric power measurements do not generally present serious problems. The problems encountered in the case of electrical resistivity measurements are due to the nature of the samples. Indeed, the diameters of the fibers are not the same along a given filament and from one filament to another of the same batch. Also, the cross sections of the samples are not always cylindrical. This makes it difficult to calculate the conductivity (or resistivity) from the measured resistance. So, the determination of the fiber cross section introduces large uncertainties in the estimation of the absolute values of the resistivities or conductivities. Since electrical parameters such as electrical current and voltage, may be measured with great accuracy, the data obtained are more accurate with regard to temperature variation than with regard to absolute magnitudes. Fortunately, for the interpretation of the experimental results it is more important to know the temperature variation than the absolute values. These problems are not met in magnetoresistance and thermoelectric power measurements since the knowledge of the samples cross sections is not needed to calculate these transport coefficients. More generally speaking, it is a rather easy task to measure the electrical resistivity, except for extreme cases of very low or very high values. However, the measurement of samples of submicronic sizes requires a miniaturization of the experimental system, which may in some instances attain a high degree of sophistication. This is particularly true for the case of single CNs, where one has to deal with samples of a few nm diameter and about a m length (Issi and Charlier, 1999). One has first to detect the sample, then apply to it electrical contacts, which means, in a four-probe measurement, four metallic conductors, two for the injected current and two for measuring the resulting voltage. This requires the use of nanolithographic techniques (Langer et al., 1994). Besides, one has to characterize the CN sample which electrical resistivity is measured in order to determine its diameter and helicity, which leads to the knowledge of the electronic structure. Thermal conductivity is a very delicate measurement to perform on a single fiber and prohibitively difficult on a single CN. This explains why little attention had been paid to the thermal conductivity of carbon fibers until the beginning of the 1980s and while still no data are available on single CNs. More generally, thermal conductivity measurements are time consuming and very delicate to perform. This is particularily true for samples of small crosssections, as it is the case for carbon fibers (Piraux et al., 1987). Indeed, since the fibers are usually of small diameters (~105m) it is difficult to make sure that the heat losses in the measuring system do not by far exceed the thermal conductance of the samples measured. The thermal conductance is defined as the heat flow through the sample per unit temperature difference. One may measure a bundle of fibers to increase the thermal conductance with respect to heat losses, but it is not always possible to realize samples in the form of bundles. Moreover, in order to be able to get an insight into the mechanisms of the thermal conductivity of these fibers, it is necessary to measure the temperature variation of this property over a wide temperature range on a single well characterized fiber, as was done for VGCFs (Piraux et al., 1984). © 2003 Taylor & Francis
In order to measure the temperature variation of the thermal conductivity of a single VGCF or a small bundle of continuous fibers one should use a sample holder specially designed to reduce significantly heat losses.This sample holder,which was designed for measuring samples with very small thermal conductances,is described in detail elsewhere (Piraux et al.,1987).It is based on the principle of a thermal potentiometer,adapted to measure the thermal conductivity of brittle samples of very small cross-sections and low thermal conductances (10-6-10-2WK-)over a wide temperature range. In some cases the fibrous geometry presents some advantages with respect to bulk car- bons and graphites.In lamellar structures like that of graphites the electrical conductivity is highly anisotropic.This anisotropy is even higher in their acceptor intercalation compounds where it may exceed 10.In that case measuring the in-plane DC electrical resistivity on bulk samples presents a real problem,since one needs a sample of extremely large length to cross section ratio to insure that the electrical current lines be parallel.Fibers have very high length to cross section ratios and thus electrical resistivity measurements could easily be realized on them. 3 Electrical resistivity We have introduced in I what we believe are the most important features pertaining to the electrical resistivity of carbons and graphites.We have insisted on the effect of the semi- metallic band structure on the electronic behavior in general,and,more particularly,on elec- tron scattering mechanisms in these materials.This applies also for VGCFs heat treated at high temperatures and to some CNs.We have also discussed in some length the effect of weak localization.Incidentally,the particular geometry and the defect structure of carbon fibers have allowed the observation of these quantum transport effects for the first time in carbons and graphites (Piraux et al.,1985). We have seen that the Boltzmann electrical conductivity for a given group of charge carri- ers is proportional to the charge carrier density and mobility.Concerning the scattering mech- anisms,the main contributions to the electrical resistivity of metals,p,consists of an intrinsic temperature-sensitive ideal term,pi,which is mainly due to electron-phonon interactions and an extrinsic temperature independent residual term,p,due to static lattice defects. As is the case for any solid,the temperature dependence of the electrical resistivity of var- ious classes of carbon-based materials is very sensitive to their lattice perfection.The higher the structural perfection,the lower the resistivity.Samples of high structural perfection exhibit room temperature resistivities below 107cm,while partially carbonized samples exhibit resistivities higher than 10-20 cm which generally increase with decreasing tem- perature.An intermediate behavior between these two extremes is represented by curves which depend less on the heat treatment temperature (HTT)and does not show significant temperature variations(cfr.I). The first comprehensive measurements of the temperature dependence of the electrical resistivity of pitch-based carbon fibers were performed by Bright and Singer(1979).They investigated radial as well as random samples heat treated at various temperatures ranging from 1,000 to 3,000C.They have shown that the magnitude of the resistivity,as well as its temperature variation,depends on the heat treatment temperature.The same kind of obser- vations were made for VGCFs. In I we have presented the temperature dependence of the electrical resistivity of many carbon materials,including fibers.As an additional illustrative example we show in Fig.3.1 ©2003 Taylor&Francis
In order to measure the temperature variation of the thermal conductivity of a single VGCF or a small bundle of continuous fibers one should use a sample holder specially designed to reduce significantly heat losses. This sample holder, which was designed for measuring samples with very small thermal conductances, is described in detail elsewhere (Piraux et al., 1987). It is based on the principle of a thermal potentiometer, adapted to measure the thermal conductivity of brittle samples of very small cross-sections and low thermal conductances (106 –102WK1 ) over a wide temperature range. In some cases the fibrous geometry presents some advantages with respect to bulk carbons and graphites. In lamellar structures like that of graphites the electrical conductivity is highly anisotropic. This anisotropy is even higher in their acceptor intercalation compounds where it may exceed 106 . In that case measuring the in-plane DC electrical resistivity on bulk samples presents a real problem, since one needs a sample of extremely large length to cross section ratio to insure that the electrical current lines be parallel. Fibers have very high length to cross section ratios and thus electrical resistivity measurements could easily be realized on them. 3 Electrical resistivity We have introduced in I what we believe are the most important features pertaining to the electrical resistivity of carbons and graphites. We have insisted on the effect of the semimetallic band structure on the electronic behavior in general, and, more particularly, on electron scattering mechanisms in these materials. This applies also for VGCFs heat treated at high temperatures and to some CNs. We have also discussed in some length the effect of weak localization. Incidentally, the particular geometry and the defect structure of carbon fibers have allowed the observation of these quantum transport effects for the first time in carbons and graphites (Piraux et al., 1985). We have seen that the Boltzmann electrical conductivity for a given group of charge carriers is proportional to the charge carrier density and mobility. Concerning the scattering mechanisms, the main contributions to the electrical resistivity of metals, , consists of an intrinsic temperature-sensitive ideal term, i , which is mainly due to electron–phonon interactions and an extrinsic temperature independent residual term, r, due to static lattice defects. As is the case for any solid, the temperature dependence of the electrical resistivity of various classes of carbon-based materials is very sensitive to their lattice perfection. The higher the structural perfection, the lower the resistivity. Samples of high structural perfection exhibit room temperature resistivities below 104 cm, while partially carbonized samples exhibit resistivities higher than 102 cm which generally increase with decreasing temperature. An intermediate behavior between these two extremes is represented by curves which depend less on the heat treatment temperature (HTT) and does not show significant temperature variations (cfr. I). The first comprehensive measurements of the temperature dependence of the electrical resistivity of pitch-based carbon fibers were performed by Bright and Singer (1979). They investigated radial as well as random samples heat treated at various temperatures ranging from 1,000 to 3,000 C. They have shown that the magnitude of the resistivity, as well as its temperature variation, depends on the heat treatment temperature. The same kind of observations were made for VGCFs. In I we have presented the temperature dependence of the electrical resistivity of many carbon materials, including fibers. As an additional illustrative example we show in Fig. 3.1 © 2003 Taylor & Francis
14 TTTTTTTTTTTTTTTTTTTTTT门 ▲E130 0E120 5a00000000000 00 ■■■■■■■■■国里 0 ●E105 10 A E75 8 ■E55 A0E35 6 中m0000000口0口0口口 ◇ ◇ d 4 2 0 50 100 150 200 250 300 Temperature(K) Figure 3.I Temperature dependence,from 2 to 300K,of the zero-field electrical resistivity of six samples of pitch-based carbon fibers heat treated at various temperatures (Nysten etal.,1991a). 6 T=4.2K 5 B 0 D -1 F 3 G 0 0.5 1.5 B(Tesla) Figure 3.2 Transverse magnetoresistance for ex-mesophase pitch carbon fibers heat treated at different temperatures ranging from 1,700(sample D)to 3,000C(samples ABCF). Samples A,B,C,and F,which were heat treated at the same temperature,exhibit different residual resistivities(measured at 4.2 K):3.8,5.1,7.0,and 6.6 X 10cm respectively.Samples G and E were heat treated at 2,500 and 2,000C,respectively (from Bright.1979). how this temperature dependence may vary with the heat treatment temperature,i.e.with crystalline perfection.In this figure the temperature dependence of the electrical resistivity of six samples of pitch-based carbon fibers heat treated at various temperatures are com- pared (Nysten et al.,1991a).Except for the E35 fibers,the resistivies of all the samples investigated decrease with increasing temperature.This dependence was also observed on ©2003 Taylor&Francis
–3 –2 –1 0 1 2 3 4 5 6 0 0.5 1 1.5 ∆/0 (%) B (Tesla) 0 T = 4.2 K A B C D E F G how this temperature dependence may vary with the heat treatment temperature, i.e. with crystalline perfection. In this figure the temperature dependence of the electrical resistivity of six samples of pitch-based carbon fibers heat treated at various temperatures are compared (Nysten et al., 1991a). Except for the E35 fibers, the resistivies of all the samples investigated decrease with increasing temperature. This dependence was also observed on 0 2 4 6 8 10 12 14 0 50 100 150 200 250 300 Temperature (K) E130 E120 E105 E75 E55 E35 Electrical resistivity ( µ Ω m) Figure 3.1 Temperature dependence, from 2 to 300 K, of the zero-field electrical resistivity of six samples of pitch-based carbon fibers heat treated at various temperatures (Nysten et al., 1991a). Figure 3.2 Transverse magnetoresistance for ex-mesophase pitch carbon fibers heat treated at different temperatures ranging from 1,700 (sample D) to 3,000 C (samples ABCF). Samples A, B, C, and F, which were heat treated at the same temperature, exhibit different residual resistivities (measured at 4.2 K): 3.8, 5.1, 7.0, and 6.6 104 cm respectively. Samples G and E were heat treated at 2,500 and 2,000 C, respectively (from Bright, 1979). © 2003 Taylor & Francis
(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
with the in-plane coherence length.As a practical result of the direct relation between these transport coefficients for fibers with the same precursor,once the electrical resistivity is measured one can determine the thermal conductivity. 5 Thermoelectric power We have introduced in I the two mechanisms responsible for the thermoelectric power, a diffusion and a phonon drag mechanism and have given an expression for the diffusion thermoelectric power.From this expression,it was found that the diffusion thermoelectric power for a degenerate electron gas varies as the inverse of the Fermi energy,or carrier den- sity.This explains why semimetals like graphites exhibit higher partial diffusion thermo- electric powers than metals or graphite intercalation compounds (GICs).We have also presented in I the temperature variation of the thermoelectric power of a graphite single crystal. We present in Fig.3.5 the temperature dependence of the thermoelectric power of six samples of pitch-based carbon fibers heat treated at various temperatures.The samples investigated are the same whose electrical resistivities are presented in Fig.3.1 and thermal conductivities in Fig.3.3b.In Fig.3.6 the earlier results of Endo and co-workers (1977)on the temperature dependence of the thermoelectric power of vapor grown(benzene-derived) carbon fibers are shown.In this figure VGCFs heat treated at two different temperatures are compared to the as-grown material. It may be seen from all these curves that,as is the case for the bulk material,the thermo- electric power of carbon fibers is very sensitive to lattice perfection.For as grown fibers or fibers heat treated at low temperatures,the thermoelectric power is low and does not vary significantly with temperature;the room temperature thermoelectric power may even be 25111111 ±-E130 20 0-E120 ◆—E105 ★-E75 —E55 0-E35 8 0 0 50 100 150 200 250 300 Temperature(K) Figure 3.5 Temperature dependence of the thermoelectric power of six samples of pitch-based carbon fibers heat treated at various temperatures,the same fibers with electrical resistivity is presented in Fig.3.1 and thermal conductivity in Fig.3.3b (Issi and Nysten,1998). ©2003 Taylor&Francis
with the in-plane coherence length. As a practical result of the direct relation between these transport coefficients for fibers with the same precursor, once the electrical resistivity is measured one can determine the thermal conductivity. 5 Thermoelectric power We have introduced in I the two mechanisms responsible for the thermoelectric power, a diffusion and a phonon drag mechanism and have given an expression for the diffusion thermoelectric power. From this expression, it was found that the diffusion thermoelectric power for a degenerate electron gas varies as the inverse of the Fermi energy, or carrier density. This explains why semimetals like graphites exhibit higher partial diffusion thermoelectric powers than metals or graphite intercalation compounds (GICs). We have also presented in I the temperature variation of the thermoelectric power of a graphite single crystal. We present in Fig. 3.5 the temperature dependence of the thermoelectric power of six samples of pitch-based carbon fibers heat treated at various temperatures. The samples investigated are the same whose electrical resistivities are presented in Fig. 3.1 and thermal conductivities in Fig. 3.3b. In Fig. 3.6 the earlier results of Endo and co-workers (1977) on the temperature dependence of the thermoelectric power of vapor grown (benzene-derived) carbon fibers are shown. In this figure VGCFs heat treated at two different temperatures are compared to the as-grown material. It may be seen from all these curves that, as is the case for the bulk material, the thermoelectric power of carbon fibers is very sensitive to lattice perfection. For as grown fibers or fibers heat treated at low temperatures, the thermoelectric power is low and does not vary significantly with temperature; the room temperature thermoelectric power may even be –5 0 5 10 15 20 25 0 50 100 150 200 250 300 Temperature (K) E130 E120 E105 E75 E55 E35 TEP ( µ V K–1) Figure 3.5 Temperature dependence of the thermoelectric power of six samples of pitch-based carbon fibers heat treated at various temperatures, the same fibers with electrical resistivity is presented in Fig. 3.1 and thermal conductivity in Fig. 3.3b (Issi and Nysten, 1998). © 2003 Taylor & Francis